University of Southern California Dissertations and Theses

Optimal clipped linear strategies for controllable damping

(USC Thesis Other)

Optimal clipped linear strategies for controllable damping

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Content OPTIMAL CLIPPED LINEAR STRATEGIES FOR
CONTROLLABLE DAMPING
by
Qian Fang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(Civil Engineering)
August 2021
Copyright 2021 Qian Fang
Dedication
To my parents, Jun Fang and Ying Qu.
To my parents-in-law, Shusheng Ma and Mingjing Fan.
To my husband, Tengyuan Ma, and our baby boys, Kevin and Cary.
ii
Acknowledgements
This dissertation proposes an alternate optimal clipped linear control strategy for controllable
damping. This effort would not have been possible without the support of my advisor, many
other professors and my family.
First and foremost, I would like to thank my respected advisor, Professor Erik A. Johnson,
for his brilliant experience, valuable expert advice and warm encouragement throughout the past
six years. I would also like to express my gratitude to Professor Steven F. Wojtkiewicz and his
group at Clarkson University, Professor Richard E. Christenson and his group at the University of
Connecticut, Professor Hideo Fujitani, Professor Yoichi Mukai and their groups at Kobe Univer-
sity, Dr. Eiji Sato from Hyogo Earthquake Engineering Research Center, Dr. Patrick T. Brewick of
United States Naval Research Laboratory, Dr. Subhayan De of the University of Colorado Boulder,
and finally my colleagues and friends at USC, for being so helpful and supportive during the past
couple years. I would also like to gratefully thank my Dissertation committee members: Profes-
sors Sami F. Masri, Roger G. Ghanem, Ketan D. Savla and Mihailo R. Jovanovic for their valuable
advice.
I would like to give special thanks to my dear parents, Jun Fang and Ying Qu, my parents-
in-law, Shusheng Ma and Mingjing Fan, and my husband, Tengyuan Ma, for their love, support,
care and encouragement, and my adorable baby boys, Kevin and Cary, for providing me so much
strength and sunshine!
The support of the National Science Foundation through awards 14-36018/14-36058 and 14-
46424 and the University of Southern California through the Viterbi Doctoral Fellowship are also
gratefully acknowledged.
iii
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables viii
List of Figures xvi
Abstract xxv
Chapter 1: Introduction 1
1.1 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter 2: Background 5
2.1 Structural control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Passive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Active control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Controllable semiactive control . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3.1 MR fluid damper . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Switched linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Semiactive control algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Comparative studies of semiactive control strategies using MR dampers . . 12
2.3.2 Clipped-optimal and clipped-LQR control . . . . . . . . . . . . . . . . . . 13
2.4 MPC and neural-net results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 3: Proposed Optimal Clipped Linear Strategies for SDOF Systems 18
3.1 Proposed optimal clipped linear strategies for a SDOF system . . . . . . . . . . . 18
3.1.1 LQR/Clipped-LQR control for a SDOF system . . . . . . . . . . . . . . . 20
3.1.2 Optimal passive linear viscous damping . . . . . . . . . . . . . . . . . . . 21
3.1.3 Proposed optimal clipped linear control (OCLC) . . . . . . . . . . . . . . 22
3.1.4 Linear active system without clipping . . . . . . . . . . . . . . . . . . . . 23
3.1.5 Optimization results with GWN excitation . . . . . . . . . . . . . . . . . . 25
3.1.6 Results with historical earthquake ground motions . . . . . . . . . . . . . 27
3.1.7 Evaluation of OCLC with different excitations . . . . . . . . . . . . . . . 30
iv
3.1.7.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.8 Evaluation for Kanai-Tajimi filtered excitations . . . . . . . . . . . . . . . 33
3.1.9 OCLC optimization over a suite of excitations . . . . . . . . . . . . . . . . 37
3.2 Parameter study of OCLC for SDOF systems . . . . . . . . . . . . . . . . . . . . 53
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4: Proposed Optimal Clipped Linear Strategies for 2DOF Systems 59
4.1 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 LQR/Clipped-LQR control for 2DOF systems . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Optimal passive linear viscous strategy . . . . . . . . . . . . . . . . . . . 63
4.3 Proposed optimal clipped linear control for 2DOF systems . . . . . . . . . . . . . 64
4.3.1 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Optimal active control for a 2DOF system . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter 5: Robustness of OCLC Strategies for SDOF and 2DOF Systems 74
5.1 Robustness analysis for the SDOF system . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Robustness analysis for the 2DOF system . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Chapter 6: Experimental Verification of Optimal Clipped Linear Strategies through
Real-time Hybrid Simulation 86
6.1 Real-time hybrid simulation (RTHS) . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Real-time hybrid simulation (RTHS) setup . . . . . . . . . . . . . . . . . . . . . . 88
6.2.1 Physical component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.2 Simulated structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 RTHS tests and numerical simulation for various systems . . . . . . . . . . . . . . 90
6.3.1 Cost metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.2 MR damper numerical model . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.3 Control law for commanded current . . . . . . . . . . . . . . . . . . . . . 93
6.3.4 RTHS and pure simulation of SDOF systems . . . . . . . . . . . . . . . . 96
6.3.4.1 SDOF RTHS and numerical simulation comparison: selected
time histories . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3.4.2 SDOF RTHS and numerical simulation comparison: response
statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.4.3 Evaluation of OCLC gain with respect to excitation changes for
SDOF systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.5 RTHS and pure simulation of 2DOF systems . . . . . . . . . . . . . . . . 113
6.3.5.1 Building models . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.5.2 Bridge deck model . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3.5.3 Base isolated structure . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
v
Chapter 7: E-Defense Shake Table Experiments and Real-time Hybrid Simulation Tests
of Controllable Damping Strategies 153
7.1 E-Defense shake table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 E-Defense experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2.1 Structural test specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2.2 MR damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.3.1 Cost metric J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3.2 Bang-bang control law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3.3 Current driver amplifier model . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.4 Semiactive controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.4 RTHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.5 Experimental and simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.5.1 E-Defense results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.5.2 RTHS results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.5.3 Numerical simulation revisited to better calibrate MR damper and friction
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.5.4 Results comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Chapter 8: Solutions of the Fokker-Planck-Kolmogorov Equation Associated with Ideal
Optimal Clipped Linear Control of a SDOF System Excited by Gaussian White Noise194
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.1.1 Fokker-Planck-Kolmogorov equation . . . . . . . . . . . . . . . . . . . . 195
8.1.2 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 Case study: solutions of stationary FPK equation for a linear oscillator . . . . . . . 200
8.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.2.2 Parametric study for accuracy and computation time . . . . . . . . . . . . 201
8.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.3 Solutions of stationary FPK equation for a SDOF system with a clipped linear control207
8.3.1 FPK equation for a SDOF system with CLC . . . . . . . . . . . . . . . . . 207
8.3.2 Finite difference scheme with variable mesh density . . . . . . . . . . . . 208
8.3.2.1 Computation of inverse V ondermonde matrix . . . . . . . . . . . 210
8.3.3 Calculation of statistics using a polynomial quadrature . . . . . . . . . . . 210
8.3.4 Approximation of Heaviside function . . . . . . . . . . . . . . . . . . . . 213
8.3.5 Parametric study for convergence . . . . . . . . . . . . . . . . . . . . . . 214
8.3.5.1 Effect of grid points on convergence . . . . . . . . . . . . . . . 214
8.3.5.2 Effects of discretization order n and time stepDt on convergence 217
8.3.6 Improved initial guess p
0
via matrix partition . . . . . . . . . . . . . . . . 223
8.3.7 Methods to solve the time marching Eq. (8.13) . . . . . . . . . . . . . . . 224
8.3.8 Exploit symmetry of p to refine the mesh without increasing computational
cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.3.9 FPK solutions of a passively-controlled system . . . . . . . . . . . . . . . 229
8.3.10 Finite difference solutions for the semiactive system . . . . . . . . . . . . 232
8.3.10.1 Finite difference solution with Heaviside function H[ ˙ q sgnu
d
] . . 237
vi
8.3.10.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8.4 Implication of OCLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.4.1 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Chapter 9: Summary, Conclusion and Future Directions 249
9.1 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Bibliography 254
vii
List of Tables
3.1 Optimization results for different control strategies of a SDOF structure excited by
GWN with t
f
= 10000 s. Note that theD columns denote percent change relative
to OCLC; positive numbers mean improvements in OCLC. . . . . . . . . . . . . . 27
3.2 Optimization results for different control strategies of a SDOF structure excited by
historical earthquakes. Note that theD columns denote percent change relative to
OCLC; positive numbers mean improvements in OCLC. . . . . . . . . . . . . . . 29
3.3 Peak responses and control force for various control strategies applied to a SDOF
structure excited by various excitations. Note that theD columns denote percent
change relative to corresponding OCLC value; positive numbers mean the OCLC
response is superior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Cross comparisons of cost metric J
a
and peak absolute acceleration for various
control strategies applied to a SDOF structure excited by various excitations. Note
that theD columns denote percent change relative to the OCLC designed for the
evaluation excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Cross comparisons of mean square and peak velocity for various control strate-
gies applied to a SDOF structure excited by various excitations. Note that theD
columns denote percent change relative to the OCLC designed for the evaluation
excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Cross comparisons of mean square and peak displacement for various control
strategies applied to a SDOF structure excited by various excitations. Note that
theD columns denote percent change relative to the OCLC designed for the evalu-
ation excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Approximate effective closed-loop natural frequency and damping ratio for each
control strategy, computed from mean square responses to a sampled GWN exci-
tation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.8 Optimal control gain results for Kanai-Tajimi filtered excitations with 4 rad=s
w
g
52 rad=s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Basic information of 20 earthquake records chosen from large-magnitude events
in the PEER NGA database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
viii
3.10 Optimization results for 20 earthquakes. . . . . . . . . . . . . . . . . . . . . . . . 47
3.11 OCLC optimization results for three cost functions F1a/b and F2 over all 20 earth-
quakes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.12 Cost change provided by an OCLC evaluated by one excitation relative to the
OCLC designed and evaluated for that excitation:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)
1, i= 1;:::;23,
for all twenty-five OCLC gains and twenty-three excitations. . . . . . . . . . . . . 50
3.13 Statistical analysis of cost change provided by an OCLC evaluated by one excita-
tion relative to the OCLC designed and evaluated for that excitation:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)

1, i= 1;:::;23, for each row in Table 3.12. . . . . . . . . . . . . . . . . . . . . . . 51
3.14 Statistical analysis of cost change provided by an OCLC evaluated by one excita-
tion relative to the OCLC designed and evaluated for that excitation:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)

1, i= 1;:::;23, for each column in Table 3.12. . . . . . . . . . . . . . . . . . . . . 52
3.15 Evaluation of OCLC gain (a

F1
;b

F1
) and (a

F2
;b

F2
) for six relative improvement
metrics R1–R6 subjected to twenty-three excitations. . . . . . . . . . . . . . . . . 54
3.16 Statistical analysis of OCLC gain (a

F1
;b

F1
) and (a

F2
;b

F2
) for six relative im-
provement metrics R1–R6 subjected to twenty-three excitations in Table 3.15. . . . 55
4.1 Passive linear viscous c
d
designed by GWN for different cost metrics with the
damping device in the first or second story of a 2DOF structure. . . . . . . . . . . 64
4.2 OCLC q q q

designed by GWN with t
f
= 1000 s for different cost metrics with the
damping device in the first or second story of a 2DOF structure. . . . . . . . . . . 66
4.3 Optimization results for different control strategies and cost metrics with the damp-
ing device in the first story of a 2DOF structure excited by GWN with t
f
= 1000 s.
Note that theD columns denote percent change relative to OCLC; positive numbers
mean improvements in OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Optimization results for different control strategies and cost metrics with the damp-
ing device in the second story of a 2DOF structure excited by GWN with t
f
=
1000 s. Note that theD columns denote percent change relative to OCLC; positive
numbers mean improvements in OCLC. . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Optimal active controlq q q

designed to reduce various cost metric responses to the
El Centro earthquake when the damping device is in the first story of a 2DOF
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ix
4.6 Comparisons of LQR and optimal active designed with 1940 El Centro for different
cost metrics with the damping device in the first story of a 2DOF structure excited
by various excitations. Note that the D columns denote percent change relative
to the corresponding LQR; positive numbers mean improvements in the optimal
active control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1 Responses of the SDOF system with perturbed stiffness using OCLC. . . . . . . . 76
5.2 Responses of the SDOF system with perturbed stiffness using CLQR. . . . . . . . 77
5.3 Responses of the SDOF system with perturbed damping using OCLC. . . . . . . . 77
5.4 Responses of the SDOF system with perturbed damping using CLQR. . . . . . . . 77
5.5 Responses of the 2DOF system with perturbed stiffness using OCLC. . . . . . . . 81
5.6 Responses of the 2DOF system with perturbed stiffness using CLQR. . . . . . . . 82
5.7 Responses of the 2DOF system with perturbed damping using OCLC. . . . . . . . 82
5.8 Responses of the 2DOF system with perturbed damping using CLQR. . . . . . . . 83
6.1 Sinusoidal excitation and electric current of preliminary tests to calibrate the MR
damper model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Optimal parameters for the MR damper model, where I is the current in Amperes
[68]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Characteristics of SDOF systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Comparison of numerical simulation and experiment results for SDOF systems
with differentw values subjected to GWN using bang-bang control,z = 0:025;D
is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . . . . . . 103
6.5 Comparison of numerical simulation and experiment results for SDOF systems
with differentw values subjected to El Centro with bang-bang control,z = 0:025;
D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . . . . . 103
6.6 Comparison of numerical simulation and experiment results for SDOF systems
with differentw values subjected to El Centro with bang-bang control, z = 0:05;
D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . . . . . 104
6.7 Comparison of numerical simulation and experiment results for SDOF systems
with differentw values subjected to Kobe with bang-bang control,z = 0:025;D is
the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . . . . . . . 104
6.8 Comparison of numerical simulation and experiment results for SDOF systems
with differentw values subjected to GWN with modified bang-bang control, z =
0:025;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . 105
x
6.9 Comparison of numerical simulation and experiment results for SDOF systems
with different w values subjected to El Centro with modified bang-bang control,
z = 0:025;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . 105
6.10 Comparison of numerical simulation and experiment results for SDOF systems
with different w values subjected to Kobe with modified bang-bang control, z =
0:025;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . 106
6.11 Evaluation of CLQR and GWN OCLC for SDOF systems subjected to El Centro
with bang-bang control, z = 0:025; D is the percent change of CLQR relative to
OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.12 Evaluation of CLQR and El Centro OCLC for SDOF systems subjected to GWN
with bang-bang control, z = 0:025; D is the percent change of CLQR relative to
OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.13 Evaluation of CLQR and GWN OCLC for SDOF systems subjected to El Centro
with modified bang-bang control, z = 0:025; D is the percent change of CLQR
relative to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.14 Evaluation of CLQR and El Centro OCLC for SDOF systems subjected to GWN
with modified bang-bang control, z = 0:025; D is the percent change of CLQR
relative to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.15 Characteristics of 2DOF systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.16 Mean square response statistics for 2DOF buildings with different fundamental
frequency w
1
for z
1
= 0:025 and r
a
= 10
13
kg
–2
; D is the percent change of
CLQR relative to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.17 Peak response statistics for 2DOF buildings with different fundamental frequency
w
1
forz
1
= 0:025 andr
a
= 10
13
kg
–2
;D is the percent change of CLQR relative
to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.18 Mean square response statistics for 2DOF buildings with different fundamental
frequency w
1
for z
1
= 0:025 and r
a
= 10
16
kg
–2
; D is the percent change of
CLQR relative to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.19 Peak response statistics for 2DOF buildings with different fundamental frequency
w
1
forz
1
= 0:025 andr
a
= 10
16
kg
–2
;D is the percent change of CLQR relative
to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.20 Numerical simulated mean square response statistics for 2DOF buildings with dif-
ferent fundamental frequencyw
1
values subjected to El Centro forz
1
= 0:025 and
r
a
= 10
16
kg
–2
and r= 15%;D is the percent change of CLQR relative to OCLC. 128
6.21 Numerical simulated peak response statistics for 2DOF buildings with different
fundamental frequencyw
1
values subjected to El Centro forz
1
= 0:025 andr
a
=
10
16
kg
–2
and r= 15%;D is the percent change of CLQR relative to OCLC. . . . 128
xi
6.22 Mean square response statistics for 2DOF buildings subjected to a non-design ex-
citation for z
1
= 0:025 and r
a
= 10
13
kg
–2
; D is the percent change of CLQR
relative to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.23 Peak response statistics for 2DOF buildings subjected to a non-design excitation
for z
1
= 0:025 and r
a
= 10
13
kg
–2
;D is the percent change of CLQR relative to
OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.24 Mean square response statistics for 2DOF buildings subjected to a non-design ex-
citation forz
1
= 0:025 andr
a
= 10
16
;D is the percent change of CLQR relative
to OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.25 Peak response statistics for 2DOF buildings subjected to a non-design excitation
for z
1
= 0:025 and r
a
= 10
16
kg
–2
;D is the percent change of CLQR relative to
OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.26 Mean square response statistics for the bridge deck model with control weight
r
a
= 10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 138
6.27 Peak response statistics for the bridge deck model with control weightr
a
= 10
13
kg
–2
;
D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . . . . . 138
6.28 Mean square response statistics for the bridge deck model with control weight
r
a
= 10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 139
6.29 Peak response statistics for the bridge deck model with control weightr
a
= 10
20
kg
–2
;
D is the percent change of CLQR relative to OCLC. . . . . . . . . . . . . . . . . . 139
6.30 Mean square bridge deck responses to a non-design excitation for control weight
r
a
= 10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 141
6.31 Peak bridge deck responses to a non-design excitation for control weight r
a
=
10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . 141
6.32 Mean square bridge deck responses to a non-design excitation for control weight
r
a
= 10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 141
6.33 Peak bridge deck responses to a non-design excitation for control weight r
a
=
10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . 141
6.34 Mean square response statistics for the base isolated structure with control weight
r
a
= 10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 149
6.35 Peak response statistics for the base isolated structure with control weight r
a
=
10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . 149
6.36 Mean square response statistics for the base isolated structure with control weight
r
a
= 10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 150
xii
6.37 Peak response statistics for the base isolated structure with control weight r
a
=
10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . . . 150
6.38 Mean square base-isolated structure responses to a non-design excitation for con-
trol weightr
a
= 10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . 152
6.39 Peak base-isolated structure responses to a non-design excitation for control weight
r
a
= 10
13
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 152
6.40 Mean square base-isolated structure responses to a non-design excitation for con-
trol weightr
a
= 10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . 152
6.41 Peak base-isolated structure responses to a non-design excitation for control weight
r
a
= 10
20
kg
–2
;D is the percent change of CLQR relative to OCLC. . . . . . . . . 152
7.1 Sinusoidal excitation and electric current of preliminary tests to calibrate the Bingham-
viscoplastic MR damper model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2 Different state-feedback controllers designed numerically and then tested at E-
Defense. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3 Statistics of the E-Defense structural response to 150% of the 1940 El Centro earth-
quake for different controllers USC1 to USC5.D denotes reductions (i.e., improve-
ments) relative to the baseline CLQR strategy designed using cost J
1
; bold values
denote the best performance in each column; gray text denotes incomparable quan-
tities: USC1/2/3 are designed for J
1
; USC4/5 are designed for J
2
. . . . . . . . . . . 173
7.4 Statistics of the E-Defense structural response to 40% of the 1995 Kobe earthquake
for different controllers USC1 to USC5. . . . . . . . . . . . . . . . . . . . . . . . 173
7.5 Statistics of the E-Defense structural response to 50% of the 1994 Northridge
earthquake for different controllers USC1 to USC5. . . . . . . . . . . . . . . . . . 173
7.6 RTHS results using modified friction and specimen model subjected to the El Cen-
tro earthquake for controllers USC1 and USC4. . . . . . . . . . . . . . . . . . . . 176
7.7 RTHS results using modified friction and specimen model subjected to the Kobe
earthquake for different controllers USC1 to USC5. . . . . . . . . . . . . . . . . . 176
7.8 RTHS results using modified friction and specimen model subjected to the Northridge
earthquake for controllers USC1 and USC4. . . . . . . . . . . . . . . . . . . . . . 176
7.9 Numerical simulation results using modified MR damper and friction force mod-
els when the structure is subjected to the El Centro earthquake and mitigated by
controllers USC1 to USC5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.10 Numerical simulation results using modified MR damper and friction force models
when the structure is subjected to the Kobe earthquake and mitigated by controllers
USC1 to USC5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
xiii
7.11 Numerical simulation results using modified MR damper and friction force mod-
els when the structure is subjected to the Northridge earthquake and mitigated by
controllers USC1 to USC5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.1 Comparisons of maximum absolute nodal error in the joint and two marginal den-
sity functions between normalized numerical and exact solutions and computation
time using 2nd, 4th, 6th, 8th and 10th order finite difference schemes, with M1
(LU decomposition) and Mb (non-conservative discretization), for various domain
boundary, grid density, time step and tolerance, for the stationary FPK equation
associated with a linear oscillator subjected to Gaussian white noise. . . . . . . . . 204
8.2 Comparisons of maximum absolute nodal error in the joint and two marginal den-
sity functions between normalized numerical and exact solutions and computation
time using different orders of finite difference schemes, withjqj
max
=j ˙ qj
max
= 8,
Dq=D ˙ q= 0:1,Dt= 0:02 and tol= 110
14
, and different methods M1 to M4 to
solve Eq. (8.13), for the stationary FPK equation associated with a linear oscillator
subjected to Gaussian white noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.3 Maximum absolute nodal error in the joint and two marginal density functions
between normalized numerical and exact solutions and computation time using
different orders of finite difference schemes, withjqj
max
= 8,j ˙ qj
max
= 8,Dq= 0:1,
D ˙ q= 0:1,Dt= 0:02 and tol= 110
14
, with M1 (LU), and Ma (conservative) and
Mb (non-conservative), for the stationary FPK equation associated with a linear
oscillator subjected to Gaussian white noise. . . . . . . . . . . . . . . . . . . . . . 206
8.4 Norms of absolute and relative errors between final and exact PDF values for the
four cases(Dt; n) in Figure 8.9 whene = 1:25 10
3
and t
f
= 300 s. . . . . . . . 222
8.5 Comparison of descriptive statistics for the OCLC PDF p(q;0) (probability mass
normalized) when ˙ q= 0 withe= 10
6
along with those of a standard multivariate
Gaussian distribution with the same covariance matrix. . . . . . . . . . . . . . . . 235
8.6 Comparison of descriptive statistics for the OCLC PDF p(0; ˙ q) (probability mass
normalized) when q= 0 withe= 10
6
along with those of a standard multivariate
Gaussian distribution with the same covariance matrix. . . . . . . . . . . . . . . . 235
8.7 Comparison of descriptive statistics for the OCLC marginal PDF of displacement
q (probability mass normalized) with e = 10
6
along with those of a standard
multivariate Gaussian distribution with the same covariance matrix. . . . . . . . . 236
8.8 Comparison of descriptive statistics for the OCLC marginal PDF of velocity ˙ q
(probability mass normalized) withe = 10
6
along with those of a standard mul-
tivariate Gaussian distribution with the same covariance matrix. . . . . . . . . . . . 236
8.9 Comparison of descriptive statistics for the OCLC PDF p(q;0) (probability mass
normalized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] with e = 10
6
and those of a stan-
dard multivariate Gaussian distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
xiv
8.10 Comparison of descriptive statistics for the OCLC PDF p(0; ˙ q) (probability mass
normalized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] with e = 10
6
and those of a stan-
dard multivariate Gaussian distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.11 Comparison of descriptive statistics for the marginal PDF of displacement q (prob-
ability mass normalized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] withe= 10
6
and those of
a standard multivariate Gaussian distribution with the same covariance matrix as
the
˜
H[ ˙ q sgnu
d
] solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8.12 Comparison of descriptive statistics for the marginal PDF of velocity ˙ q (probabil-
ity mass normalized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] with e = 10
6
and those of a
standard multivariate Gaussian distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8.13 SDOF OCLC optimization results found via Monte Carlo simulation excited by
GWN with t
f
= 10000 s (Section 3.1.5) and those found via FPK solution using
e = 1:25 10
8
, spatial discretization order n= 6, time step durationDt= 0:01 s,
a 401 425 mesh and t
f
= 20 s, and Eq. (8.45) solved by M4 (GMRES with pre-
conditioner) and p
0
computed assuming p
c
= 1. Note that theD column denotes
percent change relative to Monte Carlo Simulation result. . . . . . . . . . . . . . . 247
xv
List of Figures
2.1 Controllability of Kyobashi Seiwa Building [62]. . . . . . . . . . . . . . . . . . . 7
2.2 Paradigm for clipped optimal control law. . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Idealized model of passivity constraints for controllable dampers. . . . . . . . . . . 15
3.1 SDOF controlled building model [27]. . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 CLQR control force as a function of displacement and velocity for a GWN-driven
SDOF system with natural frequencyw = 2p rad=s and damping ratioz = 0:05. . . 21
3.3 Comparison of desired LQR force and clipped force for a GWN-driven SDOF
system with natural frequencyw = 2p rad=s and damping ratioz = 0:05. . . . . . 21
3.4 Cost metric J
a
as a function of passive linear viscous mass-normalized damper
coefficient c
d
for a SDOF system assuming an ideal GWN excitation. . . . . . . . 22
3.5 Dissipativity regions for the OCLC controllable damping for a SDOF system (drawn
assuming 1a and 1b have the same signs andw
2
(1a) 2zw(1b)). . . 24
3.6 Surface of cost metric J
a
as a function of stiffness factora and damping factorb
parameters for a linear active SDOF system without clipping for exact Lyapunov
solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.7 Surface of cost metric J
a
as a function of stiffness factor a and damping factor
b parameters for a linear active SDOF system without clipping for approximate
result, excited by GWN with t
f
= 10000 s, Dt = 0:02 s, computed using ode45
with relative tolerance 10
6
and absolute tolerance 10
8
. . . . . . . . . . . . . . . 25
3.8 Contours of SDOF cost metric J
a
as a function of a CLC’s stiffness factor a and
damping factorb parameters for GWN with t
f
= 100 s. . . . . . . . . . . . . . . . 26
3.9 OCLC control force as a function of displacement and velocity for the SDOF sys-
tem, designed for a GWN excitation with t
f
= 10000 s. . . . . . . . . . . . . . . . 26
3.10 First 10s of absolute acceleration response and control force of the SDOF structure
excited by a GWN realization and controlled by CLQR or El Centro-designed OCLC. 28
xvi
3.11 Contours of SDOF cost metric J
a
as a function of stiffness factor a and damping
factorb parameters for CLC for two historical earthquakes. . . . . . . . . . . . . . 29
3.12 First 10s of absolute acceleration response and control force of the SDOF structure
excited by the 1940 El Centro earthquake and controlled by CLQR or El Centro-
designed OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.13 First 10 s of absolute acceleration response and control force of the SDOF struc-
ture excited by the 1940 El Centro earthquake and controlled by CLQR or Kobe-
designed OCLC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.14 Cost metrics of OCLC designed for a Kanai-Tajimi excitation with a ground fre-
quencyw
g
and damping ratioz
g
= 0:32, compared with those of the GWN-designed
CLQR (LQR control gain K
LQR
=[w
2
2zw] for the SDOF system as listed in Sec-
tion 3.1.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.15 Contours of SDOF cost metric J
a
as a function of CLC stiffness factor a and
damping factor b parameters for 20 earthquakes (note: for excitations No. 6, 9,
14 and 17, the contour plot is in the(10)(10) range, but the minimum point
(a

;b

) found throughfminsearch is outside the(10)(10) range). . . . . . 42
3.16 Absolute acceleration response and control force comparisons of CLQR and OCLC
for a SDOF system excited by 20 earthquakes. . . . . . . . . . . . . . . . . . . . . 46
3.17 Optimal control gain parameters(a

;b

) as function of the structural parameters
w=(2p)2(0:5; 0:75; 1; 1:25; 1:5) andz2(0:025; 0:0375; 0:05; 0:075; 0:1) for
a SDOF system driven by GWN (t
f
= 10000 s, D= 0:02). . . . . . . . . . . . . . . 55
3.18 Optimal control gain parameters(a

;b

) as function of the structural parameters
w andz for a SDOF system driven by GWN (t
f
= 1000 s, D= 0:02). . . . . . . . . 57
4.1 2DOF controlled building model 1, damper in the first story [27]. . . . . . . . . . . 60
4.2 2DOF controlled building model 2, damper in the second story. . . . . . . . . . . . 60
4.3 Two “slices” of the CLQR control force as a function of the velocity states when
q= 0 and q=[0 1 cm]
T
, respectively, for minimizing the absolute acceleration
cost metric with the damping device in the first story of a 2DOF structure. . . . . . 63
4.4 Cost metrics as functions of the damper coefficient c
d
for the 2DOF system with a
damping device in the (a) first or (b) second story; circles denote the minimum of
each curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Slices of the CLC absolute acceleration cost metric J
a
(q q q) when the damper is in
the first story of a 2DOF structure for GWN with t
f
= 1000 s. . . . . . . . . . . . . 66
4.6 Two “slices” of the OCLC control force as functions of the velocity states when
q= 0 and q=[0 1 cm]
T
when minimizing the absolute acceleration cost metric
with the damping device in the first story of a 2DOF structure. . . . . . . . . . . . 67
xvii
5.1 Nominal SDOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 SDOF system with perturbed stiffness. . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 SDOF system with perturbed damping. . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Improvement in mean square and peak responses for the SDOF system with per-
turbed stiffness subjected to GWN. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Improvement in mean square and peak responses for the SDOF system with per-
turbed damping subjected to GWN. . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Nominal 2DOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.7 2DOF system with perturbed stiffness. . . . . . . . . . . . . . . . . . . . . . . . . 80
5.8 2DOF system with perturbed damping. . . . . . . . . . . . . . . . . . . . . . . . . 81
5.9 Improvement in mean square and peak responses for the 2DOF system (damper in
first story) with perturbed stiffness subjected to GWN. . . . . . . . . . . . . . . . 83
5.10 Improvement in mean square and peak responses for the 2DOF system (damper in
first story) with perturbed damping subjected to GWN. . . . . . . . . . . . . . . . 84
6.1 Schematic of real-time hybrid simulation for semiactive control devices. . . . . . . 87
6.2 RTHS physical experiment setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Hysteresis MR damper model [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 From top to bottom: subfigure one shows time histories of GWN commanded,
measured and the filtered currents; subfigure two indicates the power spectral den-
sities (PSD) of GWN commanded and measured currents; bottom two figures com-
pares frequency responses of the estimated transfer function from commanded cur-
rent to measured current, and approximated second-order filter. . . . . . . . . . . . 95
6.5 Paradigm for semiactive control law using a physical MR damper. . . . . . . . . . 95
6.6 Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to GWN using bang-bang control for OCLC and CLQR
strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.7 Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to El Centro using bang-bang control for OCLC and CLQR
strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.8 Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to GWN using modified bang-bang control for OCLC and
CLQR strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xviii
6.9 Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to El Centro using modified bang-bang control for OCLC
and CLQR strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.10 Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to GWN with CLQR (r
a
= 10
13
kg
–2
). . . . . . . . . . . . 117
6.11 Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to GWN with OCLC (r
a
= 10
13
kg
–2
). . . . . . . . . . . . 119
6.12 Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to El Centro with CLQR (r
a
= 10
13
kg
–2
). . . . . . . . . . 120
6.13 Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to El Centro with OCLC (r
a
= 10
13
kg
–2
). . . . . . . . . . 121
6.14 Numerical simulation of first floor absolute acceleration time history and the cor-
responding hysteresis componentaz in the control force for a 2DOF system (w
1
=
4p rad=s,z
1
= 0:025) subjected to El Centro with OCLC (r
a
= 10
13
kg
–2
). . . . . 122
6.15 Numerical simulation and RTHS bridge deck GWN response time histories with
CLQR with control weightr
a
= 10
13
kg
–2
. . . . . . . . . . . . . . . . . . . . . . 133
6.16 Numerical simulation and RTHS bridge deck GWN response time histories with
OCLC with control weightr
a
= 10
13
kg
–2
. . . . . . . . . . . . . . . . . . . . . . 134
6.17 Numerical simulation and RTHS bridge deck El Centro response time histories
with CLQR with control weightr
a
= 10
13
kg
–2
. . . . . . . . . . . . . . . . . . . 135
6.18 Numerical simulation and RTHS bridge deck El Centro response time histories
with OCLC with control weightr
a
= 10
13
kg
–2
. . . . . . . . . . . . . . . . . . . 136
6.19 Numerical simulation and RTHS base-isolated structure response time histories to
a GWN excitation with CLQR with control weightr
a
= 10
13
kg
–2
. . . . . . . . . 142
6.20 Numerical simulation and RTHS base-isolated structure response time histories to
a GWN excitation with OCLC with control weightr
a
= 10
13
kg
–2
. . . . . . . . . 144
6.21 Numerical simulation and RTHS base-isolated structure response time histories to
an El Centro excitation with CLQR with control weightr
a
= 10
13
kg
–2
. . . . . . 145
6.22 Numerical simulation and RTHS base-isolated structure response time histories to
an El Centro excitation with OCLC with control weightr
a
= 10
13
kg
–2
. . . . . . 146
6.23 Time history and PSD of control force measured and filtered in RTHS tests and
numerical simulation, when the base-isolated structure is subjected to a GWN ex-
citation with OCLC with control weightr
a
= 10
13
kg
–2
. . . . . . . . . . . . . . . 147
7.1 E-Defense Test specimen [90]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
xix
7.2 MR damper design schematic [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3 Bingham-viscoplastic model of MR damper. . . . . . . . . . . . . . . . . . . . . . 157
7.4 E-Defense commanded and measured currents, and low-pass-filtered current I
df
(the structure is subjected to a random excitation [nominal peak 100 cm=s
2
, fre-
quency range 0.1–30 Hz] using random 25 Hz square wave current control). . . . . 161
7.5 Commanded, measured and modeled MR damper current time histories over a
short duration using random 25 Hz square wave current control when the structure
is subjected to a random excitation (nominal peak 100cm=s
2
, frequency range 0.1–
30Hz): (a) E-Defense commanded and measured currents and the current predicted
by the filtered multilevel rate-limiter; (b) corresponding current ratese(t); (c) time
duration when the current increases with current rate larger than 240 A=s; (d) cor-
responding maximum current rate; (e) time duration when the current decreases
with current rate smaller than800 A=s; (f) corresponding minimum current rate. . 163
7.6 Graphical representation of saturator function sat
˙
I
max
˙
I
min
(e). . . . . . . . . . . . . . . 164
7.7 Current driver amplifier model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.8 E-Defense commanded and measured currents compared to two models of the cur-
rent: a simpler rate-limiter and the multi-level rate-limiter (the structure is sub-
jected to a random excitation (nominal peak 100 cm=s
2
, frequency range 0.1–
30 Hz) using random 25 Hz square wave current control). . . . . . . . . . . . . . . 165
7.9 E-Defense commanded and measured currents, current modeled with the multi-
level rate limiter, and modeled with a simple first-order low-pass filter f
p
=(s+ f
p
)
of the commanded current for f
p
= 50, 100, or 150s
–1
(the structure is subjected to
a random excitation (nominal peak 100 cm=s
2
, frequency range 0.1–30 Hz), using
random 25 Hz square wave current control). . . . . . . . . . . . . . . . . . . . . . 166
7.10 Approximated three-state linear system with a third state, the actual control force u. 167
7.11 RTHS setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.12 RTHS outline [47]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.13 (a) USC5 controller’s commanded and measured currents in the E-Defense experi-
ment when the structure is subjected to the Kobe excitation, and (b) the correspond-
ing currents in a pure SIMULINK numerical simulation using the actual E-Defense
measured displacement, velocity and control force to compute the commanded
current and the MR damper model to compute the “measured” current. The two
commanded currents — E-Defense and simulated — should be identical; as they
are not, it is suspected that some other controller (perhaps the “passive on” case)
was mistakenly used in this E-Defense test instead of the USC5 controller. . . . . . 171
7.14 Shake table and structure drift and acceleration using USC2 when the structure is
subjected to the Northridge excitation. . . . . . . . . . . . . . . . . . . . . . . . . 175
xx
7.15 Actual control force u measured in the RTHS test subjected to the Kobe excitation
and controlled by USC2, and the corresponding MR damper control force mod-
eled with sgn ˙ q and tanha ˙ q with a= 3000 s=m: (a) & (b) RTHS control force and
numerically simulated MR damper force using numerically simulated current and
velocity; (c) & (d) RTHS control force and numerically simulated MR damper
force using measured current and velocity from this RTHS test. . . . . . . . . . . . 177
7.16 Friction force F
fric
measured in the E-Defense test subjected to the Kobe excitation
and controlled by USC1, and the corresponding friction force modeled with sgn ˙ q
and tanhb ˙ q with b= 700 s=m: (a) & (b) E-Defense friction force and numerically
simulated friction force using numerically simulated velocity; (c) & (d) E-Defense
friction force and numerically simulated friction force using measured velocity
from this E-Defense test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.17 Structural responses measured or computed when the structure is controlled by
USC1 and subjected to the one of three scaled earthquake excitations. . . . . . . . 182
7.18 Structural responses measured or computed when the structure is controlled by
USC2, USC3 & USC5 and subjected to the Kobe earthquake. . . . . . . . . . . . . 183
7.19 Structural responses measured or computed when the structure is controlled by
USC4 and subjected to the one of three scaled earthquake excitations. . . . . . . . 184
7.20 Power spectral densities (PSD) of absolute acceleration measured from E-Defense,
calculated using E-Defense data, computed in the virtual structure in the RTHS
test and from numerical simulation, when the controller is USC1 (CLQR) and the
structure is subjected to the El Centro excitation. . . . . . . . . . . . . . . . . . . 186
7.21 Time history and PSD of the absolute accelerations measured at E-Defense, those
filtered from E-Defense, and those computed from RTHS using the USC1 (CLQR)
controller and the El Centro external excitation. . . . . . . . . . . . . . . . . . . . 187
7.22 Time history and PSD of the damper velocity and central difference approximated
velocity at E-Defense, and velocity computed from RTHS, and, for the PSD com-
parison only, the laser displacement PSD timesw
2
, using the USC1 (CLQR) con-
troller and passive on control, and the El Centro external excitation. . . . . . . . . 188
7.23 Actual control force u measured in the E-Defense test, measured in the RTHS test
and numerically simulated when the structure is controlled by USC1. . . . . . . . . 189
7.24 Actual control force u measured in the E-Defense test, measured in the RTHS test
and numerically simulated when the structure is controlled by USC2, USC3 &
USC5 and subjected to the Kobe earthquake. . . . . . . . . . . . . . . . . . . . . . 190
7.25 Actual control force u measured in the E-Defense test, measured in the RTHS test
and numerically simulated when the structure is controlled by USC4. . . . . . . . . 191
xxi
7.26 Time history and PSD of the control forces measured at E-Defense, those filtered
from E-Defense, and those measured from RTHS using the USC1 (CLQR) con-
troller and the Northridge earthquake excitation. . . . . . . . . . . . . . . . . . . . 192
8.1 A typical finite difference stencil [113]. . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 Finite difference discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.3 Pseudo time stepping approach for the solution of stationary FPK equation. . . . . 200
8.4 Polynomial fit stencil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.5 Approximation function
˜
H(z), z2[0:4;0:4], for differente values. . . . . . . . . 213
8.6 Peak trailing change during time stepkp(t) p(tDt)k
¥
, peak leading change
during time stepkDp(t)k
¥
and peak difference from final PDFkp(t) p(300 s)k
¥
for the first 1500 iteration steps (150 s) with different grids,e = 1:25 10
3
,Dt=
0:1 s, finite difference order n= 10; the solid black curve (147 195 mesh) has
grid points displayed in Figure 8.7. . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.7 One set of the finite difference mesh and normalized grid location whene= 1:25
10
3
, with a 147 195 mesh, and variableDq andD ˙ q; much finer grid points are
defined when ˙ q is close to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.8 Approximated Heaviside function
˜
H[u
d
˙ q] with different magnitudes of u
d
ranging
from 0 to max
q
ju
d
(q;0)j for the finite difference mesh in Figure 8.7, when e =
1:25 10
3
with a 147 195 mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.9 Peak leading change during time stepkDp(t)k
¥
and peak difference from final
PDFkp(t) p(300 s)k
¥
with different orders of finite difference method n and
time stepDt using a 147 195 mesh whene = 1:25 10
3
. . . . . . . . . . . . . 219
8.10 Surface plot of absolute errorjKp(t
f
)j as a function of q and ˙ q for the four cases
(Dt; n) in Figure 8.9 when e = 1:25 10
3
and t
f
= 300 s. Note that the scales
differ in the z-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.11 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 148:3 s) with p
0
provided by Monte Carlo simulation and solved
using the M1 (LU factorization) solution method, when e = 1:25 10
8
, Dt =
0:1 s, spatial discretization order n= 10, and a 135 425 mesh. . . . . . . . . . . . 226
8.12 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) with p
0
computed assuming p
c
= 1 and solved using the M1
(LU factorization) solution method, when e = 1:25 10
8
, Dt = 0:01 s, spatial
discretization order n= 4, and a 135 425 mesh. . . . . . . . . . . . . . . . . . . 226
xxii
8.13 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) with p
0
computed assuming p
c
= 1 and solved using the
M3 (GMRES) and M4 (GMRES with a preconditioner) solution methods, when
e = 1:25 10
8
, time step Dt = 0:01 s, spatial discretization order n= 4, and a
135 425 mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.14 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) with Eq. (8.45) solved by M1
0
(improved LU decomposition)
and p
0
computed assuming p
c
= 1, for different time step durationDt and spatial
discretization order n, whene = 1:25 10
8
using a 401 425 mesh. . . . . . . . 230
8.15 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) with Eq. (8.45) solved by M3 (GMRES) and p
0
computed
assuming p
c
= 1, for different time step durationDt and spatial discretization order
n, whene = 1:25 10
8
using a 401 425 mesh. . . . . . . . . . . . . . . . . . . 230
8.16 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) with Eq. (8.45) solved by M4 (GMRES with preconditioner)
and p
0
computed assuming p
c
= 1, for different time step durationDt and spatial
discretization order n, whene = 1:25 10
8
using a 401 425 mesh. . . . . . . . 230
8.17 Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) and error from the exact solutionkp(t) p
stat
k
¥
for the
passively-controlled system with similar level of damping as semiactive system,
with Eq. (8.45) solved by M4 (GMRES with preconditioner) and p
0
computed as-
suming p
c
= 1, when time stepDt = 0:01 s, spatial discretization order n= 6, and
a 401 425 mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.18 PDF when velocity ˙ q= 0 and displacement q= 0 (probability mass normalized) for
various e values and a corresponding standard multivariate Gaussian distribution
with the same covariance matrix as the FPK solution withe = 10
6
. . . . . . . . . 234
8.19 Marginal PDFs of displacement q and velocity ˙ q (probability mass normalized) for
various e values and a corresponding standard multivariate Gaussian distribution
with the same covariance matrix as the FPK solution withe = 10
6
. . . . . . . . . 240
8.20 Surface and contour graphs of the OCLC PDF (probability mass normalized) when
e = 10
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.21 PDF when velocity ˙ q= 0 and displacement q= 0 (probability mass normalized)
when approximating H[u
d
˙ q] and H[ ˙ q sgnu
d
] (solved with e = 10
6
,Dt = 0:01 s,
t
f
= 10 s, n= 6, and p
0
computed assuming p
c
= 1 and the M4 [GMRES with a
preconditioner] solution method to solve Eq. (8.45)), and a corresponding standard
multivariate Gaussian distribution with the same covariance matrix as the FPK
solution using the
˜
H[ ˙ q sgnu
d
]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
xxiii
8.22 Marginal PDFs of displacement q and velocity ˙ q (probability mass normalized)
when approximating H[u
d
˙ q] and H[ ˙ q sgnu
d
] (solved with e = 10
6
,Dt = 0:01 s,
t
f
= 10 s, n= 6, and p
0
computed assuming p
c
= 1 and the M4 [GMRES with a
preconditioner] solution method to solve Eq. (8.45)), and a corresponding standard
multivariate Gaussian distribution with the same covariance matrix as the FPK
solution using the
˜
H[ ˙ q sgnu
d
]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.23 Surface and contour graphs of the OCLC PDF (probability mass normalized) using
˜
H[ ˙ q sgnu
d
] whene = 10
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.24 Contours of cost metric J
a
as a function of parametersa andb using FPK solutions
for the SDOF system subjected to GWN. . . . . . . . . . . . . . . . . . . . . . . . 246
xxiv
Abstract
In the past few decades, structural control research has focused mostly on passive and semiactive
vibration mitigation strategies. Among the latter, the clipped-optimal control paradigm [24] is one
of the most commonly used. With this control strategy, a primary controller is designed using an
optimal linear feedback control (e.g., LQR [linear quadratic regulator], H
2
, or H
¥
) assuming the
control device is a linear actuator that can exert any force. Then, a secondary controller is com-
manded to exert the force generated by the primary controller if it is dissipative and minimal force
if it is non-dissipative. However, if the optimization to design the optimal control law cannot con-
sider the inherent passivity constraints of controllable dampers, then the commanded forces may
often be non-dissipative for some structures and some performance objectives, causing frequent
clipping; the result effectively deactivates the controllable damper for most of the duration of the
response.
This study presents an alternate clipped linear control strategy. The clipped LQR (CLQR)
strategy is just one of a family of clipped linear strategies; herein, the optimal choice, considering
the dissipative nature of the controllable damper, from among this family is denoted the “Optimal
Clipped Linear Control” (OCLC). This OCLC strategy provides performance superior to all other
clipped linear strategies and can be far superior to CLQR when CLQR exhibits frequent clipping.
For convenience and for physical interpretation, the closed-loop OCLC system is parameterized
relative to a CLQR solution, and the parameters are chosen to minimize a response metric to a
particular excitation.
This proposed approach is first applied to a single degree-of-freedom (SDOF) structure model
subjected to a Gaussian white noise (GWN) excitation. To illustrate the optimality of OCLC, it is
xxv
compared to both CLQR and an optimal passive linear viscous damper. Then, the SDOF model,
with the OCLC and CLQR strategies, is excited by multiple historical earthquakes and Kanai-
Tajimi filtered excitations. OCLC designed for one specific excitation is also evaluated with other
excitations. A strategy to select the best control gain parameters for OCLC for a SDOF system is
proposed and validated. Moreover, the relationship between OCLC control gain parameters and
structural characteristics for SDOF systems is explored.
Next, the effectiveness of OCLC and CLQR strategies in reducing three different response met-
rics (absolute floor accelerations, inter-story drifts and ground-relative floor velocities) is studied
for a two-degree-of-freedom (2DOF) system with a control device either in the first or second story.
With a GWN ground acceleration, OCLC again reduces the cost metric and structural response bet-
ter than either the corresponding CLQR or an optimal passive linear viscous damper. Then, the
robustness of the proposed control strategy for both SDOF and 2DOF models is explored through
evaluating the controllable damper performance when the structure model differs from the nominal
ones used to design the OCLC strategy.
The proposed strategy is also tested for a physical magnetorheological (MR) damper. Real-
time hybrid simulation (RTHS) tests are conducted for a set of different structural systems and
OCLC shows variable levels of performance improvements over CLQR for different structures for
both numerical simulation and RTHS tests. The performance of OCLC designed for one specific
excitation is evaluated when subjected to other excitations through RTHS.
Next, shake table experiments are conducted at Japan’s NIED “E-Defense” laboratory, using
several controllable damping strategies designed to mitigate the responses of a full-sized base-
isolated structure specimen, with an MR fluid damper installed in the isolation layer. The experi-
mental results show that OCLC can provide performance superior to CLQR in minimizing absolute
acceleration while not increasing ground relative displacement significantly for the base-isolated
structure, and vice versa. E-Defense results are also compared with RTHS and pure simulation
results, and they match quite well.
xxvi
Finally, numerical solutions to the Fokker-Planck-Kolmogorov (FPK) equation associated with
ideal OCLC of a SDOF system excited by GWN are investigated using finite difference methods.
Some special considerations are presented: the Heaviside function is approximated by a hyperbolic
tangent function; variable grid spacing is applied for the finite difference discretization; a high or-
der polynomial quadrature method is proposed to calculate the statistics. The results indicated that
the mesh should be sufficiently fine for a convergent solution; a smaller time step and a lower order
finite difference scheme may increase the convergence rate for the semiactive system. Moreover,
a new choice of the initial guess for solving the time discretization equation is proposed and the
symmetry of solutions is utilized to reduce the computational effort. Further, the implications for
optimal clipped linear strategies using FPK solutions are indicated.
Keywords: controllable damping, optimization, dissipativity, Fokker-Planck-Kolmogorov (FPK)
equation, structural control
xxvii
Chapter 1
Introduction
Structural vibrations, especially due to strong wind and extreme earthquakes, are always a concern
for civil engineers and researchers. In the field of structural control, the goal is to develop devices
and strategies to mitigate dynamic responses (e.g., drifts and/or accelerations). Structural control
strategies can be divided into three categories: passive, active and semiactive [43]. Passive control
only dissipates energy but is not adaptive to different internal or external conditions; active control
is adaptive to changes of structural characteristics or external loadings, but may inject energy into
the system. On the other hand, semiactive systems have gained considerable attention because
of their superior performance compared with passive devices but with the adaptability of active
systems [44, 57, 58].
Over the past several decades, controllable semiactive devices have been widely studied with
applications ranging from industrial machines to buildings for mitigating vibrations due to external
disturbances. Some examples are magnetorheological (MR) fluid dampers [121], electrorheologi-
cal (ER) fluid dampers [18, 73], controllable tuned liquid dampers [98, 104], controllable friction
dampers [30, 31], and variable orifice dampers [36, 55, 115].
Extensive research has focused on developing control algorithms to design the commands for
a controllable damper; the clipped-optimal control [24] is the most commonly used paradigm.
This approach employs a primary linear controller designed assuming a linear actuator that can
exert any force, typically through an optimal linear feedback control such as a linear quadratic
regulator (LQR) [20, 122], optimal performance H
2
control [10], or robust H
¥
control [129]. Then,
1
a secondary controller “clips” the control force commanded by the primary controller to make the
device exert a force as close as possible to the desired control force; for an ideal controllable
damper that can exert any dissipative force, the secondary controller commands the force exactly
if it is dissipative, but commands zero force otherwise. However, since the optimization typically
used to design the primary control law does not consider the inherent passivity constraints of
controllable dampers (or other semiactive devices), commanded forces are often non-dissipative for
some structures and some performance objectives, causing frequent clipping; the result effectively
deactivates the controllable damper for most of the time history and negates its effectiveness.
This study proposes an alternate clipped linear control strategy. The clipped LQR (CLQR)
strategy is just one of a family of clipped linear strategies; herein, the optimal choice, considering
the dissipative nature of the controllable damper, from among this family is denoted the “Optimal
Clipped Linear Control” (OCLC). This OCLC strategy provides performance superior to all other
clipped linear strategies and can be far superior to CLQR when CLQR exhibits frequent clipping.
For convenience and for physical interpretation, the closed-loop OCLC system is parameterized
relative to a CLQR solution, and the parameters are chosen to minimize a response metric to a
particular excitation.
1.1 Overview of the Dissertation
Chapter 2 reviews the background for this study with an emphasis on semiactive controllable damp-
ing. It briefly introduces the concept and development of structural control. Then, switched linear
systems are introduced and the most popular semiactive control algorithm — clipped-optimal con-
trol — is reviewed. Model predictive control and neural network results are also reviewed to
indicate that a clipped linear control strategy exists and outperforms some nonlinear control strate-
gies.
Chapter 3 focuses on numerical simulations of single-degree-of-freedom (SDOF) systems, with
an OCLC strategy commanding an ideal controllable damper. First, the LQR and clipped LQR
2
strategies are discussed for a SDOF system. The disadvantages of CLQR are indicated and the
novel control strategy OCLC is proposed. Optimal strategies for several excitations are studied:
GWN, two historical earthquakes and a set of Kanai-Tajimi filtered excitations. For GWN, an
optimal passive viscous damper strategy (which is another in the same family of clipped linear
control gains) provides a comparison to further illustrate the superiority of OCLC. OCLC designed
for one specific excitation is also evaluated with other excitations. Next, the performance of OCLC
under a suite of ground motions selected from the PEER NGA database is evaluated and the final
optimal control gain for the specific SDOF system is determined through a comprehensive analysis.
Further, the relationship between optimal factors and structural characteristics (natural frequency
and damping ratio) is explored.
Chapter 4 extends the study of OCLC to two-degree-of-freedom (2DOF) systems subjected
to a GWN excitation. For each possible set of locations for the controllable damper, different
OCLC strategies are examined with three metrics — mean square structural absolute acceleration,
mean square structural ground-relative velocities, or structural inter-story drifts — each augmented
by the weighted mean square damper force so that the optimization is well-posed and has finite
solution. Three optimal control strategies are conducted — passive viscous damping, CLQR and
OCLC — through simulation and, for the passive viscous damping only, computed exactly via
a Lyapunov solution, and the optimization results are summarized. Moreover, an optimal active
control for the 2DOF system is evaluated and compared with LQR control.
Chapter 5 demonstrated the robustness of the OCLC for both SDOF and 2DOF systems to
imperfect knowledge of the structural parameters (and, therefore, of the structural dynamic char-
acteristics). OCLC is more efficient than CLQR for both SDOF and 2DOF systems even with
perturbed stiffness and damping factors. The improvements provided by the OCLC, relative to
CLQR, do not degrade much at all when the structure parameters are perturbed from that used to
design the OCLC strategy.
Chapter 6 further validates the effectiveness of OCLC using a physical MR damper. Numer-
ical simulation and real-time-hybrid-simulation (RTHS) tests are discussed for various systems
3
subjected to multiple excitations, using both CLQR and OCLC approaches. The performance of
different control strategies is evaluated and the RTHS results are compared to pure simulation re-
sults as well. Further, an OCLC designed for one particular excitation is evaluated when subjected
to a different excitation for different systems through both RTHS and pure simulation.
Chapter 7 presents shake table experiments of a full-sized base-isolated structure specimen,
with an MR fluid damper installed in the isolation layer, at Japan’s NIED “E-Defense” laboratory.
Different control strategies are designed numerically and applied to control the structural vibration.
The experimental results are also compared with corresponding RTHS and pure simulation results.
Chapter 8 investigates numerical solutions to the Fokker-Planck-Kolmogorov (FPK) equation
associated with an ideal OCLC of a SDOF system excited by GWN. A preliminary case study of an
uncontrolled linear oscillator system is conducted to provide implications of using finite difference
to solve the FPK equation of a nonlinear semiactive system. Next, different orders, grid spacing
and time step of finite difference discretization methods, as well as solution methods are discussed
to solve the FPK equation of the semiactive controlled system. A new choice of the initial guess
for solving the time discretization equation is proposed and the symmetry of solutions is utilized to
reduce the computational effort. The implications for optimal clipped linear strategies using FPK
solutions are indicated.
Chapter 9 summarizes the research work and concludes the Dissertation.
4
Chapter 2
Background
This chapter reviews the state-of-the-art of structural control, with particular emphasis on semi-
active control, including the basic concepts of key semiactive controllable damping devices and
strategies and switched linear systems, as well as recent research motivating this study.
2.1 Structural control
Structural control refers to reducing or mitigating structural dynamic responses due to external dis-
turbances, such as earthquakes and winds, and has been widely studied with applications ranging
from industrial machines to buildings. Structural control is important for designing new structures
as well as retrofitting subsistent structures. It is first applied in aerospace engineering, and then
extended to structural engineering. Since Yao [122] proposed an initial conceptual study by defin-
ing the concept of structural control as improving structural safety in 1972, structural control has
become a mature field but continue to attract the attention of structural engineers and researchers.
Structural control systems fall into three main categories discussed in the following paragraphs:
passive, active and semiactive [43, 59, 60].
5
2.1.1 Passive control
Passive control requires no external energy and dissipates energy in a fixed manner, typically via
friction or viscous fluid; since passive control only dissipates energy, it cannot destabilize the
system. However, passive control is unable to adjust the internal parameters to adapt to changes in
structural characteristics or loading conditions. For a passive control system, the structure usually
has one or more control devices designed to modify the stiffness or damping of the structure in an
appropriate manner, generating control forces that oppose the structure motion. Control forces are
developed as a function of the structural response at the location of the devices. Passive control may
depend on the initial design of the structure, on the frictional contact between structural elements or
on the use of impact dampers [43, 82]. Some examples include base isolation, buckling restrained
bracing, tuned mass dampers and tuned liquid dampers.
2.1.2 Active control
Active control is adaptive to different circumstances, but requires external energy to provide the
control force and may inject energy into the system; therefore, active control has the potential of
destabilizing the structure in the event of a power loss caused by catastrophic events. Some exam-
ples of active control are an active mass damper and an active brace. The first actively controlled
structure is the Kyobashi Seiwa Building in Tokyo in 1989, with an active mass damper (AMD)
control system inside the building. For the 11-story (33.1 m) building structure, two AMDs are
installed on the top of the building, along with a control computer, sensors and an observation
system, as shown in Figure 2.1 [62]. In addition to numerous implementation of active control
[95, 98], some approaches combine passive and active control, and are called hybrid control, such
as the hybrid mass damper (HMD) system in the Sendagaya INTES building built in Tokyo in 1991
[94].
6
Figure 2.1. Controllability of Kyobashi Seiwa Building [62].
2.1.3 Controllable semiactive control
Controllable dampers, also called smart or semiactive, are passive damping systems that have
some active component to adjust internal parameters or properties to maintain an optimal force
level. Some examples are magnetorheological (MR) fluid dampers [121], electrorheological (ER)
fluid dampers [18, 73], controllable tuned liquid dampers [98, 104], controllable friction dampers
[30, 31], and variable orifice dampers [36, 55, 115]. Semiactive controllable damping inherits not
only the controllable characteristics of active actuators but also the inherent energy dissipation
nature of passive dampers [53, 55, 84].
Semiactive control systems were proposed as early as the 1920s when patents were issued for
shock absorbers that utilized an elastically supported mass to activate hydraulic valving [104].
Semiactive systems were not used in structural engineering until 1983, when Hrovat et al. [44]
applied a semiactive tuned mass damper (TMD) to control wind induced vibrations in tall build-
ings. Since the 1990s, a variety of structures using semiactive controllable dampers have appeared,
such as the Kajima Technical Research Institute in Japan, built with active-variable-stiffness de-
vices in 1990 [97]. Further, extensive research on semiactive devices [72, 111] and design methods
7
[35, 84, 125] proceed at a rapid pace. Liu et al. [71] studied the performance of different semi-
active control strategies based on skyhook control and balance control for a base isolation system
and concluded that the semiactive system can always provide better isolation at higher frequen-
cies than a conventional passive system. Narasimhan et al. [78] developed a smart base-isolated
benchmark building, demonstrating its modeling and sample control designs, that can be used to
study a wide variety of semiactive control strategies for linear base-isolated buildings. Johnson
et al. [53] compared different dampers used in cables and demonstrated that semiactive dampers
can absorb more energy and, hence, exhibit better properties in reducing structural displacements
than other dampers. Liu et al. [72] proposed a new device using two controllable dampers and
two constant springs to solve the problem that controlling stiffness and damping variables are hard
to implement with conventional devices. Zapateiro et al. [125] designed semiactive controllers
based on backstepping and quantitative feedback theory techniques and tested them in a structure
with an MR damper through real-time hybrid testing, concluding that the controllers can efficiently
achieve vibration reduction. Gawthrop et al. [35] simulated a quarter car model using semiactive
damping combined with a hybrid control approach, and demonstrated that the control activity can
be reduced without significant reduction in the performance of decreasing the quarter car body
velocity relative to active and passive control. Weber [111] generated an optimal semiactive vibra-
tion absorber for harmonic excitations based on controllable semiactive dampers. As semiactive
control systems become more popular in practical application, researchers began to develop appro-
priate models for semiactive control devices, as well as improving their performance using optimal
algorithms and strategies.
2.1.3.1 MR fluid damper
Though there are many types of semiactive control devices, the MR fluid damper is one of the
most commonly studied. MR fluid dampers have a number of attractive characteristics for use in
vibration control applications. As indicated by the name, they are dampers that utilize MR fluids.
8
MR fluids, developed in the 1940s, are liquids that can transfer force because of their simply con-
trolled flow or shear properties and can be filled in vibration control devices [11]. A magnetic field
applied to the fluid causes the MR fluid particles to realign such that the yield strength of the fluid
is changed, thereby changing the fluid into a semisolid [48]. The damper then is able to produce
high levels of force while remaining controllability; this can be accomplished using only modest
power that can be readily provided by batteries. MR dampers have been shown to significantly
surpass the vibration control performance of comparable passive damping schemes, while requir-
ing only a fraction of the input power required by a fully active device. Several countries have
applied MR dampers in practical projects; the first full-scale implementation of MR dampers for
civil engineering applications was in the Nihon-Kagaku-Miraikan, the Tokyo National Museum of
Emerging Science and Innovation in 2001; the first full-scale implementation of MR dampers for
bridge structures was completed in the Dongting Lake Bridge in Hunan, China [97].
Meanwhile, MR dampers have been utilized in extensive theoretical and experimental studies.
Various types of MR damper models have been developed according to the dampers’ mechanical
characteristics. Spencer et al. [100] proposed a phenomenological modified Bouc-Wen model that
can efficiently mimic the behavior of a typical MR damper. Schurter and Roschke [91] established
an MR damper model using the Adaptive Neuro-Fuzzy Inference System (ANFIS), which was
demonstrated to satisfactorily represent the behavior of damper. Yang et al. [119] proposed a
novel MR damper model by incorporating a model of the current driver to power the device. Guo
et al. [37] utilized two parameters in an MR damper model to interpret the bi-viscous and hysteretic
behaviors of the damper. Kwok et al. [64] established a new hysteretic model for an MR damper
using particle swarm optimization method. Boada et al. [7] facilitated a recursive lazy learning
method based on neural networks to model the MR damper behavior. Case et al. [13] introduced a
multi-physics finite-element dynamic model for a small-scale MR damper.
9
2.2 Switched linear systems
Switched linear systems are dynamical systems that consist of a finite number of linear subsystems
and a logical rule that specifies the switching between these subsystems [69, 103]. Switched linear
systems have a long history, and rapid progress recently has generated many innovative achieve-
ments.
Hespanha et al. [41] employed the techniques of supervisory control to design a logic-based
switching control law for a nonholonomic system in the presence of parametric modeling uncer-
tainty. Bemporad and Morari [6] proposed a predictive control scheme to stabilize mixed logical
dynamical (MLD) systems, one kind of switched linear systems, on desired reference trajectories
with constraints, where the optimization was conducted through mixed integer quadratic program-
ming (MIQP). Asarin et al. [3] formulated a synthesis problem as finding switching controllers for
linear systems to avoid bad states when switching between different subsystems, and proposed a
novel algorithm to solve the problem by an iterative computation of reachable states, which utilized
a new approximation scheme for reachability analysis of linear systems. Zhao and Hill [126] set
up a framework of dissipativity theory for switched systems using multiple storage functions and
multiple supply rates, which described the exchange of “energy” between different subsystems.
Zhao et al. [127] developed a novel multiple discontinuous Lyapunov function approach to estab-
lish stability conditions for systems with a designed switching strategy, where fast switching and
slow switching were respectively applied to unstable and stable subsystems. Cheng et al. [16] ex-
plored a more general model with respect to a new switching rule by utilizing a non-homogeneous
sojourn probability approach.
Meanwhile, the analysis of dynamic behaviors of switched linear systems have attracted con-
siderable attention, such as stability, controllability, reachability and observability. For instance,
Hespanha [40] studied the stability of systems under arbitrary, slow and state-dependent switch-
ing; Ji et al. [49, 50] designed switching sequences for controllability and reachability realization
of switched linear systems; Egerstedt and Babaali [25] studied the observability and reachability
10
of switched linear systems with constant dynamics, i.e., with switching in only the measurement
or control matrices.
2.3 Semiactive control algorithms
The performance of the controlled system is highly dependent on the choice of a control algorithm.
There have been numerous studies of semiactive control algorithms applied to structures. Since
the mathematical description of a semiactive damper is highly nonlinear, the development of their
control strategies is far from straightforward [109]. Different control strategies for semiactive
devices have been developed to minimize structural responses [48].
Karnopp et al. [59] proposed a semiactive force generator which can respond to general feed-
back signals from a vibrating system in order to control the vibration but which does not require
the power supply of a servomechanism. The semiactive device was demonstrated to exhibit com-
parable vibration control performance to a fully active system when applied to a SDOF system.
Leitmann [66] proposed and summarized two control schemes: control based on minimizing
energy change and control based on Lyapunov stability theory, both of which allow for separate
and joint control of stiffness and damping coefficients of ER fluids for ER dampers. The former
scheme, minimizing the structure’s rate of energy change, i.e., the rate of work by forces on the
structure, was studied and validated through several numerical examples. The latter scheme com-
bined the task of minimizing the rate of change of the state norm and the proof of stability of the
controlled system, i.e., the derivative
˙
V(x) of a candidate Lyapunov function V(x) should be as
negative as possible to make the origin stable. It was concluded that control based on Lyapunov
stability theory is preferable and robust with respect to state measurement errors.
McClamroch and Gavin [74] proposed a decentralized bang-bang control law using ER dampers
by idealizing the nonlinearity characteristics of ER dampers and using Lyapunov-based control
synthesis. The objective was to minimize the rate at which the energy was transmitted to the struc-
ture. The Lyapunov function was selected to represent the total energy in the structure. Later,
11
Gavin [34] synthesized a decentralized bang-bang control law using Lyapunov’s direct method in
a vibration control system, which was designed to suppress earthquake-excited vibrations.
Iemura and Pradono [45, 46] studied the effectiveness of a variable damper employing pseudo-
negative stiffness control on cable-stayed bridges. It is known that the maximum dissipation of
energy occurs when the enclosed area of the force-displacement trajectory is maximized. A neg-
ative stiffness, which augments the amplitude of displacement, develops a force in the direction
of displacement, and can therefore, increase the total dissipated energy [8, 112]. The combination
of pseudo-negative stiffness and inherent stiffness of the deck-tower connections can therefore
increase the rigidity and damping ratio of hysteric force-deformation loop.
Choi et al. [17] presented a semiactive neural/fuzzy control strategy for reducing seismic re-
sponses using an MR damper, which has an inherent robustness, and easiness to treat the uncer-
tainties of input data from the ground motion and structural vibration sensors, and the ability to
handle the nonlinear behavior of the structure since an exact mathematical model of the structure
is no longer required. The advantage of this algorithm is that it can produce the required voltage
input for an MR damper such that the desired control force can be generated.
Johnson and Erkus [52] investigated the dissipativity and performance of semiactive systems
via linear matrix inequality (LMI) synthesis. A dissipativity index was proposed to modify a
standard linear quadratic regulator (LQR) using LMI theory and the corresponding dissipative
constraint was defined. A modified LMI-based LQR controller was obtained by attaching the
dissipativity constraint in its weak form. The numerical study indicated that the proposed method
can increase the dissipativity of a controller for both ideal and physical (MR) damper models, but
can only improve the semiactive performance significantly when an ideal damper is utilized.
2.3.1 Comparative studies of semiactive control strategies using MR dampers
Numerous comparative studies of various control strategies have been conducted using MR dampers
[23, 54, 121]. Agrawal et al. [1] addressed the comparisons of several semiactive and passive con-
trol systems for a benchmark cable-stayed bridge and proved the effectiveness of semiactive and
12
passive devices compared with the sample active controller. Xu et al. [117] applied various semi-
active control strategies and linear passive dampers to the phase I linear base-isolated benchmark
building, demonstrating that the semiactive friction controllers were the most effective. Liu et
al. [70] presented a comparison of four semiactive control strategies — namely energy minimiza-
tion, Lyapunov, fuzzy logic and variable structure system (VSS) fuzzy logic control — of a scaled
bridge using fail-safe MR dampers, and showed that the fuzzy logic and VSS fuzzy logic strategies
require far less input energy than other control cases. Cha et al. [14] compared the performance of
three semiactive controllers — the clipped optimal, the decentralized output feedback polynomial
and the simple passive controllers — to mitigate the response a three-degree-of-freedom frame
employing multiple MR dampers, through pure simulation and real-time hybrid tests as well.
2.3.2 Clipped-optimal and clipped-LQR control
Efficient and effective control algorithms can enhance the performance of semiactive controllable
devices. Among the semiactive control algorithms developed recently, the most often-employed
is the clipped-optimal control, proposed by Dyke et al. [24] in 1996. Consider an n degree-of-
freedom (DOF) system, which has the equation of motion given by:
M¨ q(t)+ C˙ q(t)+ Kq(t)=Mr ¨ q
g
(t)
¯
Bu(t) (2.1)
where q(t) is the vector of ground-relative structural displacements, r is the ground motion influ-
ence vector,
¯
B is the control force influence matrix, and ¨ q
g
(t) is the ground motion acceleration.
The state-space equation of motion is then:
˙ x= Ax+ B
g
¨ q
g
+ B
u
u (2.2)
where x=[q
T
˙ q
T
]
T
, A=
2
6
4
0 I
M
1
K M
1
C
3
7
5
, B
g
=
2
6
4
0
r
3
7
5
, B
u
=
2
6
4
0
M
1
¯
B
3
7
5
.
13
To determine the control force u, clipped-optimal control uses a primary controller to deter-
mine a desired control force and a secondary controller to command a device other than a perfect
linear actuator to achieve the desired control force, as shown in Figure 2.2. In this control strategy
Controller
Secondary
(Clipping)
Primary
(LQR)
desired
control
force
states or
measurements
clipped
control force
Figure 2.2. Paradigm for clipped optimal control law.
for controllable dampers, “clipping” is necessary because the resulting control forces of a pri-
mary controller are not always dissipative, and therefore not always achievable by the controllable
damper. The secondary controller may be a bang-bang type control for commanding a physical
device [24, 123] or, for an ideal controllable damper that can exactly exert any dissipative force, it
is a simple on-off switch that commands the force from the primary controller if it is dissipative
and commands zero force otherwise [53, 84]. The idealized model of passivity constraints for con-
trollable dampers is shown in Figure 2.3. Let the desired force vector u
d
=[u
d
1
u
d
2
::: u
d
n
u
]
T
for
the n
u
devices and v be a vector of corresponding velocities across the devices. Then the elements
of the actual force vector u=[u
1
u
2
::: u
n
u
]
T
for an ideal controllable damper are
u
i
(t)= u
d
i
(t)H[u
d
i
(t)v
i
(t)]=
8
>
<
>
:
u
d
i
(t); u
d
i
(t)v
i
(t)> 0
0; otherwise
(2.3)
where the sign conventions are chosen such that u
d
i
is dissipative when of the same sign as v
i
, and
H[] is the Heaviside unit step function.
Various approaches have been explored to encourage the dissipativity of the primary controller
to maximize the performance from a controllable damper. For instance, Johnson [51] incorporated
a weak form of the energy dissipation rate into an LQR control design optimization for a 9-story
14
Velocity
Control force
Dissipative
Dissipative Non-dissipative
Non-dissipative
Figure 2.3. Idealized model of passivity constraints for controllable dampers.
structure to encourage dissipative behavior. Aly and Christenson [2] presented an energy-based
probabilistic approach that can determine an equivalent semiactive system based on the proba-
bility of the dissipative control forces; further, Rezaee and Aly [86] extended the probabilistic
approach to multiple-degree-of-freedom (MDOF) systems by introducing a modal space control
and employing a Lyapunov function to enable the solution of highly nonlinear control systems with
smart dampers.
One commonly used clipped-optimal control strategy in structural control is the clipped LQR
(CLQR) control strategy, in which the primary controller is a linear state feedback gain designed for
an ideal linear actuator to minimize a quadratic metric of the system response to a GWN excitation.
For a linear, time-invariant, dynamical system described by state equation ˙ x= Ax+ Bu, where x
is the state vector and u is the control input (force), the LQR control gain minimizes an infinite-
horizon quadratic performance index of the form [128]:
Z
¥
0
[x
T
Qx+ 2x
T
Nu+ u
T
Ru]dt (2.4)
in which Q, N and R are weighting matrices. LQR can also be posed in a stochastic optimal control
formulation [5] where the state equation is ˙ x= Ax+Bu+Ew, where w is a stochastic disturbance,
and the cost metric is a mean square metric, i.e., the expected value of [x
T
Qx+ 2x
T
Nu+ u
T
Ru]
instead of its time integral. In both the deterministic and stochastic cases, the optimal linear state
feedback control force is u=Kx, where linear gain matrix K= R
1
(PB+ N)
T
, and P= P
T
15
satisfies the algebraic Riccati equation given by A
T
P+PA(PB+N)R
1
(PB+N)
T
+Q= 0. The
LQR and CLQR strategies are stochastic optimal strategies that implicitly assume the excitation is
an ideal Gaussian white noise; performance with other excitations may vary.
Such clipped control systems can be treated as switched linear systems. For the clipped-optimal
control, the switching law is given by Eq. (2.3), which ensures the stability of the overall system
since it only dissipates energy. However, as noted previously, any clipped-optimal strategy that
does not consider the inherent passivity constraints of controllable dampers, as shown in Fig-
ure 2.3, can result in commanded forces that are often non-dissipative for some structures and
some performance objectives, causing frequent clipping, leading the control strategy far from be-
ing optimal.
2.4 MPC and neural-net results
Elhaddad and Johnson [27] proposed a hybrid model predictive control (hMPC) strategy — a com-
bination of hybrid system models and model predictive control (MPC) — to compute controllable
damping device force commands that are optimized with the constraint that the damper can only
exert dissipative forces, resulting in significant performance improvements relative to a CLQR
strategy, though the optimal control force was found to be a very nonlinear function of structure
displacements and velocities. However, further study [26] indicated that a clipped linear strategy
that approximates the nonlinear hMPC may perform quite well; similarly, a study of neural net-
work control strategies [39] also found that a clipped linear strategy can be nearly as optimal as
a much more complicated neural net controller. In both of these cases, the approximate clipped
linear strategy was found to be somewhat different than the clipped LQR strategy, thus motivating
the question: what clipped linear gain is optimal when accounting for the clipping?
16
2.5 Summary
This chapter summarized the literature relevant to structural control, switched linear systems,
clipped-optimal control, CLQR control, MPC and neural-net results. In the subsequent chapters,
several control strategies will be studied for different systems, and it will be demonstrated that
the CLQR strategy is not optimal for some control objectives, and a new feedback design will be
proposed and shown to use the semiactive device much more efficiently.
17
Chapter 3
Proposed Optimal Clipped Linear Strategies for SDOF Systems
This chapter proposes an optimal clipped linear control (OCLC) strategy for SDOF systems. The
chapter is divided into three sections: the first presents the development of OCLC and demonstrates
its effectiveness for a specific SDOF system subjected to a variety of excitations; the second section
illustrates the relationship between the control gain parameters and structural parameters (stiffness
coefficient and damping ratio); the third section concludes the chapter.
3.1 Proposed optimal clipped linear strategies for a SDOF system
To give context for developing the optimal clipped linear feedback control strategy, consider first
a SDOF system, as shown in Figure 3.1.
m
Damper
Fixed Base
u
q
Ground Motion
k, c
Figure 3.1. SDOF controlled building model [27].
18
The ground-relative displacement q(t) of a SDOF structure, excited by base acceleration ¨ q
g
(t),
is given by equation of motion:
¨ q(t)+ 2zw ˙ q(t)+w
2
q(t)= ¨ q
g
(t) u(t) (3.1)
where an overdot (˙) denotes derivative with respect to time, w =
p
k=m is the natural frequency,
z = c=(2wm) is the damping ratio, and u(t) is a mass-normalized controllable damping force (with
orientation chosen such that positive u resists motion in the positive q direction). The system can
also be represented in a state-space form:
˙ x= Ax+ B
g
¨ q
g
+ B
u
u; A=
2
6
4
0 1
w
2
2zw
3
7
5
; B
g
= B
u
=
8
>
<
>
:
0
1
9
>
=
>
;
(3.2)
The explicit dependence of x, ¨ q
g
and u on time is omitted for notational clarity. The state vector is
x=[q ˙ q]
T
. The to-be-mitigated output of the system is the absolute acceleration:
¨ q
a
= ¨ q+ ¨ q
g
=2zw ˙ qw
2
q u= C
a
x+ D
a
u (3.3)
where C
a
=[w
2
2zw] and D
a
=1.
To facilitate comparisons with prior hMPC studies, this study uses the same structure model as
in Elhaddad and Johnson [26, 27]: m= 100 Mg, k= 3:948 MN=m and c= 62:833 kNs=m, which
results in natural frequency w = 2p rad=s and damping ratio z = 0:05. The initial conditions
are assumed quiescent, and the ground excitation ¨ q
g
is a stochastic Gaussian white noise (GWN)
process with intensity D= 0:02 m
2
=s
4
; i.e., E[ ¨ q
g
(t) ¨ q
g
(t+t)]= Dd(t), where d() is the Dirac
delta function. This scale of the excitation causes large but reasonable uncontrolled responses of
the model; note, however, that the formulation and result are independent of the excitation scale
factor as the clipped systems — both CLQR and OCLC — exhibit a homogeneity of order one
[87] and scale like a linear system but are not additive.
19
3.1.1 LQR/Clipped-LQR control for a SDOF system
CLQR control has proven to provide good performance in minimizing a cost metric such as struc-
tural displacements during strong earthquakes to mitigate damage to the structure; however, oc-
cupants comfort and safety of structure contents in the much more frequent moderate earthquakes
demand minimizing a serviceability cost metric such as absolute accelerations in the structure.
Actually, during an earthquake, non-structural components such as laboratory equipments and pip-
ings have the most significant damage and loss [105]. Minimizing displacement and acceleration
in different excitations are usually competing cost metrics and cannot be simultaneously achieved
with CLQR control [26].
Herein, the to-be-minimized serviceability cost metric J
a
is the mean square absolute structural
acceleration:
J
a
=E[ ¨ q
a2
]= lim
t
f
!¥
1
t
f
Z
t
f
0
¨ q
a2
(t)dt (3.4)
The cost metric J
a
can be written in the standard LQR form presented in Eq. (2.4), where u is a
scalar for a single device, with weighting matrices R= 1; N=[w
2
2zw]
T
and Q= NN
T
. Then,
the LQR solution (the desired force for CLQR) is:
u
LQR
(t)=K
LQR
x=k
LQR
q
q k
LQR
˙ q
˙ q=w
2
q 2zw ˙ q (3.5)
Therefore, for the perfect active system, the resulting closed-loop equation of motion is: ¨ q
LQR
=
¨ q
g
, resulting in the absolute acceleration ¨ q
aLQR
= ¨ q
LQR
+ ¨ q
g
= 0 and, subsequently, J
a
= 0 (how-
ever, the result is marginally stable, and slight error in w or z results in very large or unstable
response).
Next, CLQR is applied for the semiactive system, where u(t)= u
LQR
H[u
LQR
˙ q], and the closed-
loop equation of motion is piecewise linear:
8
>
<
>
:
¨ q= ¨ q
g
; u
LQR
˙ q> 0
¨ q+ 2zw ˙ q+w
2
q= ¨ q
g
; otherwise
(3.6)
20
(which is still marginally stable when the device is turned on; i.e., when u
LQR
˙ q> 0). The clipped
control law — the control force u as a function of displacement q and velocity ˙ q — is shown in
Figure 3.2; when u
LQR
˙ q 0, the control force is clipped to zero; otherwise, it remains the desired
LQR control force. Nevertheless, the desired LQR force is clipped quite often; therefore, the
device is turned off much of the time. As indicated in Figure 3.3, for one realization of the GWN
excitation, the device is inactive about half of the time, as frequent clipping occurs for this SDOF
system.
10
0
control force u [%mg]
20
0
40
velocity q
.
[cm/s]
10
5
0
displacement q [cm]
Figure 3.2. CLQR control force as a func-
tion of displacement and velocity for a GWN-
driven SDOF system with natural frequencyw =
2p rad=s and damping ratioz = 0:05.
desired u
LQR
(t)
control force u
0
0
2 4 6 8 10
time t [s]
off
on
off 48.1% of the time
clipped u(t)
Figure 3.3. Comparison of desired LQR force
and clipped force for a GWN-driven SDOF sys-
tem with natural frequency w = 2p rad=s and
damping ratioz = 0:05.
3.1.2 Optimal passive linear viscous damping
An optimal passive linear viscous damper is used to compare with the clipped strategies. The
viscous damping control force is u= c
d
˙ q, where c
d
is the mass-normalized damping coefficient.
From Eq. (3.1), the closed-loop passively-controlled state-space equation of motion is then
˙ x(t)= A
p
x(t)+ B
g
¨ q
g
(t); ¨ q
a
(t)= C
p
a
x(t); ˙ q(t)= C
p
v
x(t) (3.7)
where C
p
v
=[0 1], C
p
a
= C
a
+ c
d
D
a
C
p
v
and A
p
= A+ c
d
B
u
C
p
v
. The cost metric Eq. (3.4) can be
computed from simulation but, since this closed-loop system is linear, is also determined using
21
an exact Lyapunov solution: J
a
(c
d
jGWN)= C
p
a
XC
p
a
T
, where X= X
T
=E[xx
T
] is the solution to
the Lyapunov equation A
p
X+ XA
pT
+ B
g
DB
T
g
= 0. The passive system is optimized by choosing
damping coefficient c
d
to minimize the cost metric Eq. (3.4); for this system, the optimal value
of c
d
is 5.6549 s
–1
(the resulting effective damping ratio of the closed-loop system is 0.5), which
was found via MATLAB’sfminsearch [107] using the exact Lyapunov solution for J
a
(c
d
jGWN);
a graphical representation of the cost metric J
a
(c
d
jGWN) over a mass-normalized damping coeffi-
cient c
d
range is shown in Figure 3.4.
10
0
10
1
10
2
c
d
[s ]
0
0.2
0.4
0.6
0.8
1
1.2
cost J
a
(c
d
|GWN) [m
2
/s
4
]
optimal c
d
= 5.6549 s
Figure 3.4. Cost metric J
a
as a function of passive linear viscous mass-normalized damper coeffi-
cient c
d
for a SDOF system assuming an ideal GWN excitation.
3.1.3 Proposed optimal clipped linear control (OCLC)
The LQR control gain is just one of a family of linear control laws, designed to be optimal for the
non-clipped system with white noise excitation, but is no longer optimal for the clipped system.
The proposed optimal clipped linear control (OCLC) minimizes the cost metric J
a
in Eq. (3.4) for
the nonlinear clipped system by choosing a control gain K
OCLC
, which is likely different from the
gain K
LQR
that is chosen for the linear (non-clipped) system. The OCLC desired force could be
22
written, analogous to Eq. (3.5), as u
d
=K
OCLC
x=k
OCLC
q
q k
OCLC
˙ q
˙ q, but it is convenient to
parameterize relative to the LQR solution in Eq. (3.5) using nondimensional parametersa andb:
u
d
=w
2
(1a)q 2zw(1b) ˙ q (3.8)
Here, choosing a =b = 0 provides the LQR solution, and a =b = 1 is the structure without
controllable damping. Combining the equation of motion Eq. (3.1) and the generic linear feedback
force in Eq. (3.8), the system is now piecewise linear as depicted in Figure 3.5 and given by:
8
>
<
>
:
¨ q+ 2zwb ˙ q+w
2
aq= ¨ q
g
; u
d
˙ q> 0
¨ q+ 2zw ˙ q+ w
2
q= ¨ q
g
; otherwise
(3.9)
An optimization, then, must be performed numerically over stiffness factora and damping factor
b (each of which may be any real number) given the nonlinear (albeit piecewise linear) nature of
the system. Specifically, the OCLC is the one given by(a

;b

)= argmin
a;b
J
a
(a;bjf ¨ q
g
(t);0
t t
f
g). It should be noted that this approach does optimize to a particular design excitation.
However, this is, in fact, no different than other methods for finding any CLC: for example, CLQR
is designed with LQR, which implicitly assumes an ideal GWN excitation; even the active LQR
design, for an ideal GWN excitation, may be significantly suboptimal for any specific excitation,
including a finite-duration discrete-time GWN, as evaluated in further detail in Chapter 4 for a
2DOF example.
3.1.4 Linear active system without clipping
The semiactive system is highly nonlinear and no closed-form solution exists. On the other hand,
the linear active system has an exact Lyapunov solution; hence, the active performance can be
23
q
˙ q
u
d
= 0
u
d
< 0
u
d
> 0
¨ q+ 2zwb ˙ q+w
2
aq= ¨ q
g
u
d
is dissipative
¨ q+ 2zw ˙ q+w
2
q= ¨ q
g
u
d
is nondissipative
Figure 3.5. Dissipativity regions for the OCLC controllable damping for a SDOF system (drawn
assuming 1a and 1b have the same signs andw
2
(1a) 2zw(1b)).
evaluated by comparing the simulation results with the exact solutions. As stated previously, the
linear active system has the closed-loop equation of motion with u u
d
(no clipping):
¨ q+ 2zwb ˙ q+w
2
aq= ¨ q
g
(3.10)
Then, the mean square response, found by solving the Lyapunov equation, is:
E[ ¨ q
a2
]=a
2
w
4
E[q
2
]+ 4z
2
w
2
b
2
E[ ˙ q
2
]+ 2zw
3
abE[q ˙ q] (3.11)
whereE[q
2
]= D=(4zw
3
ab);E[ ˙ q
2
]= D=(4zwb) andE[q ˙ q]= 0 [93]. Then the cost metric J
a
can
be rewritten as:
J
a
E[ ¨ q
a2
]=
a
b
Dw
4z
+b(zwD) (3.12)
Figure 3.6 displays the surface of cost metric J
a
(a;b) computed via the exact Lyapunov solution,
and Figure 3.7 portrays the approximate results using SIMULINK [107]. When a and/or b are
negative, the responses blow up as the system is unstable; hence, only positive cases are considered
for this active system. The error relative to the exact Lyapunov solutions is calculated, finding
that, over most of the (a;b)> 0 region, the simulation exhibits relatively small error except for
the case when b = 0, which may be caused by several reasons: (a) infinitesimal time steps and
24
10
10
10
5 10
cost J
a
[m
2
/s
4
]
10
0
5
0
10
2
0
damping factor
stiffness factor
LQR Solution
Uncontrolled
Figure 3.6. Surface of cost metric J
a
as a func-
tion of stiffness factor a and damping factor b
parameters for a linear active SDOF system with-
out clipping for exact Lyapunov solution.
10
10
10
5 10
cost J
a
[m
2
/s
4
]
10
0
5
0
10
2
0
damping factor
stiffness factor
Figure 3.7. Surface of cost metric J
a
as a function
of stiffness factor a and damping factor b pa-
rameters for a linear active SDOF system without
clipping for approximate result, excited by GWN
with t
f
= 10000 s, Dt = 0:02 s, computed using
ode45 with relative tolerance 10
6
and absolute
tolerance 10
8
.
infinite time duration are unrealizable in numerical simulation, and the effect of both t
f
¥ and
1=dt¥ are amplified whenb = 0; (b)b = 0 is the singularity of the exact Lyapunov solution.
Nevertheless, the simulation results and the exact Lyapunov solutions are otherwise nearly identical
for the linear active system. The minimum occurs when a =b = 0, which is indeed the LQR
solution. The analysis demonstrates that the simulation matches very well with the exact solution
for the linear active system; thus, it is expected that this simulation approach will also provide
reasonably accurate results for the controllable passive system as well.
3.1.5 Optimization results with GWN excitation
To determine the OCLC, the system is simulated over a grid of CLC strategies with (a;b)2
(10)(10) over t2[0;t
f
] using a Gaussian pulse process excitation (i.e., discrete-time band-
limited white noise) with sampling time Dt = 0:02 s; assuming ergodicity, the mean square re-
sponses, computed using ode45 with relative tolerance 10
10
and absolute tolerance 10
12
, are
approximated by averaging over the duration of the simulation. The contour plot of cost metric
J
a
as a function of parameters a and b, for the SDOF structure using various CLCs, is depicted
25
in Figure 3.8. The final time t
f
= 100 s is selected because simulations with longer time durations
require greater computational effort, but the contour plot changes little with much longer t
f
, so this
shorter duration is sufficient for a fairly close evaluation of the cost metric contour. The cost met-
ric is less sensitive to changes inb thana; whena is negative, the cost metric begins to increase
rapidly.
To find the global optimal (a

, b

) that minimizes J
a
(a;b), a variety of optimization al-
gorithms could be adopted; this study uses MATLAB’s optimization toolbox function optimizer
fminsearch, which is a derivative-free downhill simplex method based on the Nelder-Mead sim-
plex algorithm [65]. The optimization results are summarized in Table 3.1, for three optimal control
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
-2
-1
0
1
2
3
4
5
Figure 3.8. Contours of SDOF cost metric J
a
as a
function of a CLC’s stiffness factora and damp-
ing factorb parameters for GWN with t
f
= 100s.
10
0
control force u [%mg]
20
0
40
velocity q
.
[cm/s]
10
5
0
displacement q [cm]
Figure 3.9. OCLC control force as a function of
displacement and velocity for the SDOF system,
designed for a GWN excitation with t
f
= 10000s.
strategies — passive viscous damping, CLQR and OCLC — through simulation with t
f
= 10000 s
(chosen larger than the previous 100 s because the optimization with fminsearch, using the min-
imum point found via the contour plot as the initial guess, requires far less computational effort
than the contour plot with the same time duration, and the optimization solutions are more ac-
curate with longer time durations), and, for the passive viscous damping only, computed exactly
via a Lyapunov solution. CLQR and optimal passive linear damping have cost metric values that
are 50% and 30% higher, respectively, than that of the OCLC with (a

;b

)=(0:3498;7:8759).
Passive simulation results are all slightly smaller than, but predictably very close to, the passive
26
Lyapunov solutions. Interestingly, mean square displacement and velocity using passive damping
are 20% and 55% smaller, respectively, with a lower control force level than OCLC, but at the
expense of higher mean square absolute acceleration, which is the metric to be minimized. Never-
theless, CLQR’s mean square controllable damping force is more than double that of the OCLC.
Further, CLQR’s mean square velocity is 77% larger and mean square displacement is 121% larger
than those provided by the OCLC. Finally, the CLQR-commanded controllable damper is inactive
48% of the time, whereas it is only inactive 27% of the time for the OCLC. Combined, these met-
rics all demonstrate that the OCLC uses the controllable damper much more efficiently than does
CLQR. Figure 3.9 depicts the OCLC control force u as a function of displacement q and velocity
˙ q. Figure 3.10 shows a time history comparison of the absolute acceleration and the control force
over the first 10 s of response. These graphs indicate smaller structural response and less frequent
clipping with OCLC relative to CLQR.
Table 3.1: Optimization results for different control strategies of a SDOF structure excited by
GWN with t
f
= 10000s. Note that theD columns denote percent change relative to OCLC; positive
numbers mean improvements in OCLC.
Control Design Response Computed (a

;b

) J
a
(D) E[u
2
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) Device off (D)
Strategy Excitation Via [m
2
/s
4
][%] [(%mg)
2
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [%] [%]
OCLC samp. GWN samp. GWN (0.3498,7.8759) 0.098 8.424 18.354 0.887 27.295
CLQR ideal GWN samp. GWN (0,0) 0.144 (47) 16.888 (100) 32.416 (77) 1.959(121) 47.703 (75)
Passive ideal GWN samp. GWN n/a 0.120 (22) 4.940 (–41) 14.866 (–19) 0.396(–55) 0
Passive ideal GWN ideal GWN n/a 0.126 (29) 5.289 (–37) 15.915 (–13) 0.403(–55) 0
3.1.6 Results with historical earthquake ground motions
To evaluate the performance of OCLC with different excitations, the SDOF system is also excited
by two historical earthquake ground motions — El Centro (north-south component of the 1940
Imperial Valley, CA, earthquake of magnitude 7.1, recorded at the Imperial Valley Irrigation Dis-
trict substation in El Centro, CA, over a 30s duration and sampled at 50Hz) and Kobe (north-south
component of the 1995 Hyogo-ken Nanbu Kobe earthquake of magnitude 7.2, recorded at the Kobe
Japanese Meteorological Agency JMA, Kobe, Japan, over a 32 s duration and sampled at 100 Hz).
The same cost metric Eq. (3.4) is used in this analysis as well. The contour plots of cost metric
27
0 2 4 6 8 10
Time [s]
0
1
Abs. accel. [m/s
2
]
CLQR
OCLC
(a) Absolute acceleration response
0 2 4 6 8 10
Time [s]
0
10
Control force [%mg]
CLQR
OCLC
(b) Control force
Figure 3.10. First 10 s of absolute acceleration response and control force of the SDOF structure
excited by a GWN realization and controlled by CLQR or El Centro-designed OCLC.
J
a
(a;bjEQ
i
), which is defined as
1
t
fi
R
t
fi
0
¨ q
a
i
2
(t;a;b)dt for historical earthquake i, for the system
using various CLCs subjected to 1940 El Centro and 1995 Kobe excitations, are shown in Fig-
ures 3.11a and 3.11b, respectively. The optimization results are summarized and compared with
CLQR in Table 3.2. Notably, when the external excitation is the 1995 Kobe earthquake, the CLQR
cost metric, mean square velocity and dispalcement are 81%, 206% and 187% larger, respec-
tively, using a 39% higher control force level than OCLC, and the CLQR-commanded controllable
damper is turned off 47% of the time, whereas it is only off 33% of the time for OCLC. Fig-
ures 3.12 and 3.13 show the resulting optimal trajectories of the absolute accelerations and control
forces, for CLQR and OCLC, during the first 10 s of the 1940 El Centro earthquake and the 1995
Kobe Earthquake, respectively. These results illustrate the efficacy of the proposed OCLC since
it provides seismic responses that are smaller than CLQR, and OCLC is clipped less frequently
than CLQR. Remarkably, the optimal a

is negative for the 1995 Kobe earthquake, resulting in
a negative stiffness in the equation of motion for the dissipative case, i.e., when the device is ac-
tive; while one may be concerned that this means the semiactive system is unstable because of
partly negative stiffness, the system is indeed stable and the controllable damper can never add
28
energy, only dissipate it. Nevertheless, there are a suite of near-optimal positive (a,b), with sub-
optimal performance but still superior to CLQR; e.g., when (a, b)= (0.05, 8.65), the cost metric
J
a
(0:05;8:65jKobe) is 0.073 m
2
=s
4
, which is still 42% smaller than that with CLQR.
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
-2
-1
0
1
2
3
4
5
(a) 1940 El Centro
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
-2
-1
0
1
2
3
4
5
(b) 1995 Kobe
Figure 3.11. Contours of SDOF cost metric J
a
as a function of stiffness factor a and damping
factorb parameters for CLC for two historical earthquakes.
Table 3.2: Optimization results for different control strategies of a SDOF structure excited by
historical earthquakes. Note that theD columns denote percent change relative to OCLC; positive
numbers mean improvements in OCLC.
Control Design Evaluation (a

;b

) J
a
(D) E[u
2
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) Device (D)
Strategy Excitation Excitation [m
2
/ s
4
] [%] [(%mg)
2
] [%] [cm
2
/ s
2
] [%] [cm
2
] [%] off [%] [%]
OCLC El Centro El Centro (0.0637,6.1087) 0.127 18.221 33.502 1.331 33.178
CLQR ideal GWN El Centro (0,0) 0.171 (34) 23.026 (26) 57.551 (72) 2.502 (88) 46.769 (41)
OCLC Kobe Kobe (–0.0996,8.0398) 0.069 14.101 14.160 0.698 32.927
CLQR ideal GWN Kobe (0,0) 0.126 (81) 19.568 (39) 43.323 (206) 2.003 (187) 47.391 (44)
To further illustrate that OCLC outperforms CLQR, the peak displacement, velocity, absolute
acceleration and control force for all three excitations are provided in Table 3.3. Notably, peak
displacement, velocity and absolute acceleration with CLQR are up to 52%, 68% and 52% larger,
respectively, with a 28% higher peak control force than those with OCLC.
29
0 2 4 6 8 10
Time [s]
0
1.5
Abs. accel. [m/s
2
]
CLQR
OCLC
(a) Absolute acceleration response
0 2 4 6 8 10
Time [s]
0
15
Control force [%mg]
CLQR
OCLC
(b) Control force
Figure 3.12. First 10 s of absolute acceleration response and control force of the SDOF structure
excited by the 1940 El Centro earthquake and controlled by CLQR or El Centro-designed OCLC.
Table 3.3: Peak responses and control force for various control strategies applied to a SDOF struc-
ture excited by various excitations. Note that the D columns denote percent change relative to
corresponding OCLC value; positive numbers mean the OCLC response is superior.
Control Design Evaluation q
max
(D) ˙ q
max
(D) ¨ q
a
max
(D) u
max
(D)
Strategy Excitation Excitation [cm] [%] [cm/s] [%] [cm/s
2
] [%] [%mg] [%]
OCLC samp. GWN samp. GWN 4.289 19.484 169.889 13.760
CLQR ideal GWN samp. GWN 5.783 (35) 28.215 (45) 230.364 (36) 23.178 (68)
OCLC El Centro El Centro 4.654 32.244 184.776 18.671
CLQR ideal GWN El Centro 5.567 (20) 41.005 (27) 221.948 (20) 22.331 (20)
OCLC Kobe Kobe 4.434 20.314 176.382 21.292
CLQR ideal GWN Kobe 6.765 (52) 34.169 (68) 268.397 (52) 27.193 (28)
3.1.7 Evaluation of OCLC with different excitations
In Section 3.1.5, the OCLC designed for a realization of GWN excitation was evaluated with
that same GWN realization. Similarly, in Section 3.1.6, the OCLC designed for the El Centro
and Kobe earthquakes were evaluated on these same design earthquakes. Since the OCLC differs
somewhat based on the design excitation, it is essential to cross validate the performance of an
OCLC designed for one excitation when subjected to a different excitation.
Tables 3.4, 3.5 and 3.6 provide the cross comparisons of the mean square and peak responses to
the three evaluation earthquakes when structural response is controlled by the four control strate-
gies, for absolute accelerations, velocities and displacements, respectively. OCLC designed for
30
0 2 4 6 8 10
Time [s]
0
1.5
Abs. accel. [m/s
2
]
CLQR
OCLC
(a) Absolute acceleration response
0 2 4 6 8 10
Time [s]
0
15
Control force [%mg]
CLQR
OCLC
(b) Control force
Figure 3.13. First 10 s of absolute acceleration response and control force of the SDOF structure
excited by the 1940 El Centro earthquake and controlled by CLQR or Kobe-designed OCLC.
one specific excitation always outperforms CLQR regardless of the evaluation excitation, provid-
ing important response reduction in the mean square absolute acceleration cost metric, as well
as peak absolute acceleration, mean square and peak displacement and velocity relative to those
with CLQR. When reducing cost metric J
a
, OCLC designed for the evaluation excitation always
performs best compared with the other two OCLCs, as is expected. For peak absolute accelera-
tion and mean square and peak displacement, OCLC designed with GWN and Kobe outperform
OCLC designed for the El Centro earthquake when the evaluation excitation is El Centro. Notably,
when the evaluation excitation is Kobe, the OCLC designed for the Kobe earthquake always pro-
vides largest response reduction in all structural responses relative to those of OCLC designed for
the other two excitations and CLQR. Specifically, CLQR has a mean square velocity over three
times that of OCLC designed with Kobe; actually, OCLC designed with Kobe provides reduction
of mean square velocity, regardless of the evaluation excitation, superior to the other two OCLCs
designed for GWN and El Centro excitations as well as superior to CLQR.
31
3.1.7.1 Observations
OCLC always provides performance superior to CLQR for both the stochastic Gaussian white
noise excitation and the historical earthquakes. Although the control was designed to minimize the
mean square absolute acceleration, the results show that OCLC can achieve important improve-
ments, compared to CLQR, in the mean square velocities and displacements as well, and OCLC
engages the device more frequently, all of which suggest that the CLQR strategy is not using the
damping device effectively. Moreover, the OCLC strategy results in peak responses and peak con-
trol force below those with CLQR, even though minimizing the mean square does not guarantee
minimizing the peak. Further, even though the control forces for CLQR are more aggressive,
CLQR still underperforms OCLC. Additionally, OCLC designed for one specific excitation can
still outperform CLQR in other excitations, further validating the superiority of OCLC.
The optimal gains differ for each ground motion as the design excitation influences responses
used in the optimization. Nevertheless, though the optimal factors a

and b

vary by design
excitation, all optimala

values are relatively close to zero, indicating that the closed-loop stiffness
should be close to that imposed by the CLQR solution; in contrast, the b

values are far away
from the origin, which means the controllable damper, when active, adds considerable damping
to the system. While the CLQR strategy, when active, attempts to eliminate both stiffness and
damping, the GWN-designed OCLC, when active, merely reduces the natural frequency by 41%
but increases the damping by a factor of 15 to about 66.6% of critical. While these clipped systems
are inherently nonlinear and, thus, have non-Gaussian response to a Gaussian excitation, one may
use the mean square responses to approximate the effective closed-loop stiffness and damping
ratios for the various CLC strategies. To do so, consider that a linear SDOF system with natural
frequency ¯ w and damping ratio
¯
z , driven by a (mass-normalized) GWN excitation with intensity
D, has displacement and velocity mean square responses given by [93] D=(4
¯
z ¯ w
3
) and D=(4
¯
z ¯ w),
respectively. By equating these expressions to a clipped system’s mean square displacement and
velocity responses to a sampled GWN excitation, one may solve for the effective frequency and
32
damping ratio. These results are provided in Table 3.7, which shows that the OCLC strategies
provide on the order of two-to-three times the effective damping levels of the CLQR.
3.1.8 Evaluation for Kanai-Tajimi filtered excitations
There are many methods to generate synthetic earthquake motions. A very common simple model
of earthquake-induced ground motion is the Kanai-Tajimi model proposed by Kanai [56] and
Tajimi [106], which is based on the investigation of the frequency content dominant at a local
site. They suggested that the ground acceleration below a structure can be approximated by the
absolute acceleration of a simple oscillator with a concentrated mass supported by a linear spring
and a dashpot and subjected to a bedrock acceleration white noise. The Kanai-Tajimi model can
be represented by the Laplace-domain filter:
F(s)=
2z
g
w
g
s+w
2
g
s
2
+ 2z
g
w
g
s+w
2
g
(3.13)
driven by zero-mean GWN withE[w(t)w(t+t)]= D
0
(t)d(t), where w
g
and z
g
are the natural
frequency and damping ratio of the “ground” oscillator, respectively, determined by the character-
istics of the local ground conditions, and D
0
(t) is either a time-varying or constant GWN intensity.
Then, the spectral density function of the ground motion is given by:
S
FF
(w)=
1+ 4z
2
g
(
w
w
g
)
2
[1(
w
w
g
)
2
]
2
+ 4z
2
g
(
w
w
g
)
2
S
0
(3.14)
where S
0
is determined by the strength of the seismic waves.
In practice, to use the Kanai-Tajimi model, the three parameters w
g
, z
g
and either S
0
or D
0
should be estimated from local representative earthquake records by means of statistical estimation
procedures. In this study, ranges and nominal values of w
g
and z
g
are determined according to
the histograms of Kanai-Tajimi frequency and damping ratio based on 140 western United States
strong motion records collected by Soong and Grigoriu [93]. Moreover, to test the relationship
33
Table 3.4: Cross comparisons of cost metric J
a
and peak absolute acceleration for various control
strategies applied to a SDOF structure excited by various excitations. Note that the D columns
denote percent change relative to the OCLC designed for the evaluation excitation.
Cost Metric J
a
Response to Peak Absolute Acceleration Response to
Evaluation Excitation Evaluation Excitation
Control Design GWN (D) El Centro (D) Kobe (D) GWN (D) El Centro (D) Kobe (D)
Strategy Excitation [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%]
CLQR ideal GWN 0.144 (47) 0.171 (34) 0.126 (82) 2.304 (36) 2.219 (20) 2.684 (52)
OCLC samp. GWN 0.098 0.143 (12) 0.091 (31) 1.699 1.631 (–12) 1.793 (2)
OCLC El Centro 0.106 (8) 0.127 0.078 (13) 1.829 (8) 1.848 1.884 (7)
OCLC Kobe 0.121 (24) 0.136 (7) 0.069 1.765 (4) 1.602 (–13) 1.764
Table 3.5: Cross comparisons of mean square and peak velocity for various control strategies
applied to a SDOF structure excited by various excitations. Note that theD columns denote percent
change relative to the OCLC designed for the evaluation excitation.
Mean Square Velocity Response to Peak Velocity Response to
Evaluation Excitation Evaluation Excitation
Control Design GWN (D) El Centro (D) Kobe (D) GWN (D) El Centro (D) Kobe (D)
Strategy Excitation [cm
2
/s
2
] [%] [cm
2
/s
2
] [%] [cm
2
/s
2
] [%] [cm/s] [%] [cm/s] [%] [cm/s] [%]
CLQR ideal GWN 32.416 (77) 57.551 (72) 43.323 (206) 28.215 (45) 41.005 (27) 34.169 (68)
OCLC samp. GWN 18.354 33.765 (1) 20.769 (47) 19.484 32.700 (1) 23.597 (16)
OCLC El Centro 18.598 (1) 33.502 21.083 (49) 21.667 (11) 32.244 22.589 (11)
OCLC Kobe 14.032 (–24) 24.988 (–25) 14.160 20.887 (7) 28.846 (–11) 20.314
Table 3.6: Cross comparisons of mean square and peak displacement for various control strategies
applied to a SDOF structure excited by various excitations. Note that theD columns denote percent
change relative to the OCLC designed for the evaluation excitation.
Mean Square Displacement Response to Peak Displacement Response to
Evaluation Excitation Evaluation Excitation
Control Design GWN (D) El Centro (D) Kobe (D) GWN (D) El Centro (D) Kobe (D)
Strategy Excitation [cm
2
] [%] [cm
2
] [%] [cm
2
] [%] [cm] [%] [cm] [%] [cm] [%]
CLQR ideal GWN 1.959 (121) 2.502 (88) 2.003 (187) 5.783 (35) 5.567 (20) 6.765 (52)
OCLC samp. GWN 0.887 1.063 (–20) 0.782 (12) 4.289 4.106 (–12) 4.514 (2)
OCLC El Centro 1.226 (38) 1.331 0.958 (37) 4.615 (8) 4.654 4.736 (7)
OCLC Kobe 1.266 (43) 1.217 (–9) 0.698 4.433 (3) 4.023 (–14) 4.444
Table 3.7: Approximate effective closed-loop natural frequency and damping ratio for each control
strategy, computed from mean square responses to a sampled GWN excitation.
Control Design Effective Closed-Loop
Strategy Excitation Frequency [rads/s] Damping Ratio
CLQR ideal GWN 4.07 37.9%
OCLC samp. GWN 4.55 60.0%
OCLC El Centro 3.89 69.0%
OCLC Kobe 3.33 107.0%
34
between the ground frequency and the OCLC gain, the OCLC gain is computed forw
g
= 4k rad=s,
k= 1;:::;13, with a fixedz
g
= 0:32, which is the mean value of the damping coefficients proposed
by Soong and Grigoriu [93]. To form a level basis for comparison, the intensity is selected for each
value ofw
g
such that the root mean square ground acceleration is constant [101]:
s
¨ x
g
= 0:12 g (3.15)
where g is the acceleration of gravity at the earth’s surface. Then D
0
is constant and given by:
D
0
=
4z
g
w
g
(4z
2
g
+ 1)
(0:12g)
2
(3.16)
To determine the OCLC for different Kanai-Tajimi filtered models, the system is simulated over
a grid of clipped linear strategies with (a;b)2(10)(10) over t2[0;t
f
= 50 s] (longer du-
rations may be needed for greater precision, but these are sufficiently long to clearly demonstrate
that OCLC performs better than CLQR for all of the Kanai-Tajimi simulations evaluated) using
a Kanai-Tajimi filtered Gaussian pulse process excitation (i.e., discrete-time band-limited white
noise) with sampling timeDt= 0:02 s; assuming ergodicity, the mean square responses, computed
using ode45 with relative tolerance 10
10
and absolute tolerance 10
12
, are approximated by av-
eraging over the duration of the simulation.
To find the global optimal (a

,b

) that minimizes J
a
(a;b), MATLAB’s fminsearch is used.
The optimal control gain parameters are summarized in Table 3.8 for a range ofw
g
, and constant
z
g
= 0:32. The optimal gains differ for each Kanai-Tajimi design excitation. Nevertheless, though
the optimal factors differ withw
g
, the optimala

values range in about[0:1;0:4] so the stiffness is
similar to the CLQR stiffness, but the b

values are far away from the origin, indicating that, for
this SDOF system, the controllable damper, when active, adds considerable damping to the system.
Moreover, the ground natural frequencyw
g
does not impact the optimal control gain significantly,
i.e., the frequency content of excitation does not affect the final optimization results severely.
The costs of 13 different OCLCs designed for each Kanai-Tajimi excitation and one CLQR
(LQR control gain K
LQR
=[w
2
2zw] for the SDOF system as listed in Section 3.1.1) evaluated
35
Table 3.8: Optimal control gain results for Kanai-Tajimi filtered excitations with 4 rad=sw
g

52 rad=s.
w
g
( rad=s) a

b

4 0.1409 13.2342
8 0.1006 10.1249
12 0.1864 10.3199
16 0.2003 5.2441
20 0.2047 5.1835
24 0.1882 5.4387
28 0.2295 6.9925
32 0.2004 5.6014
36 0.2077 5.3824
40 0.2082 5.8762
44 0.3176 5.9525
48 0.2015 5.0985
52 0.3722 5.2002
with different excitations, as functions of the ground natural frequencyw
g
are shown in Figure 3.14.
It can be concluded that: (a) OCLC always provides a consistent improvement over CLQR in
reducing the costs when the frequency contents of the excitation change; and (b) the excitation
frequency content does not have a dramatic influence on the performance improvements.
0 10 20 30 40 50 60
g
(rad/s)
0.6
1
10
60
cost J
a
[m
2
/s
4
]
OCLC
CLQR
Figure 3.14. Cost metrics of OCLC designed for a Kanai-Tajimi excitation with a ground frequency
w
g
and damping ratioz
g
= 0:32, compared with those of the GWN-designed CLQR (LQR control
gain K
LQR
=[w
2
2zw] for the SDOF system as listed in Section 3.1.1).
It is worth noting that the CLQR design could possibly be improved by designing for the four-
state 2DOF problem of ground plus structure; however, OCLC could also probably be improved
with a similar approach.
36
3.1.9 OCLC optimization over a suite of excitations
As indicated previously, the OCLC depends on the excitation, so it should be known a priori. To
evaluate the effects of this dependence, a suite of earthquake records is chosen to determine an
overall optimal control gain for this SDOF based upon a comprehensive analysis with multiple
excitations.
To ensure broad representation of different recorded earthquakes, 20 ground motions are cho-
sen from large-magnitude events around the world (United States, Canada, Japan, Iran, Turkey,
New Zealand, Italy, Taiwan and Russia) in the PEER NGA database. The distance d
r
to fault
rupture is used to separate the records into “far-field” (d
r
> 10 km) and “near-field” (d
r
10 km)
record sets. The near-field record set includes two subsets: (1) ground motions with strong pulses,
referred to as the “pulse” record subset, and (2) ground motions without such pulses, referred to
as the “no pulse” record subset. In this study, the same number of far-field and near-field records
are chosen, and pulse and no pulse records are divided evenly for near-field records. The selected
earthquake (EQ) records are listed in Table 3.9.
Table 3.9: Basic information of 20 earthquake records chosen from large-magnitude events in the
PEER NGA database.
No. Earthquake Year Station name Magnitude
Far-field
1 Northridge-01 1994 Beverly Hills-14145 Mulhol 6.69
2 Duzce Turkey 1999 Bolu 7.14
3 Hector Mine 1999 Hector 7.13
4 Imperial Valley-06 1979 El Centro Array #11 6.53
5 Kobe Japan 1995 Nishi-Akashi 6.9
6 Kocaeli Turkey 1999 Duzce 7.51
7 Landers 1992 Yermo Fire Station 7.28
8 Manjil Iran 1990 Abbar 7.37
9 Chi-Chi Taiwan 1999 CHY101 7.62
10 Friuli Italy-01 1976 Tolmezzo 6.5
Near-field
No pulse
11 Gazli USSR 1976 Karakyr 6.8
12 Nahanni Canada 1985 Site 1 6.76
13 Cape Mendocino 1992 Cape Mendocino 7.01
14 Chi-Chi Taiwan 1999 TCU054 7.62
15 Christchurch New Zealand 2011 Christchurch Botanical
Gardens
6.2
Pulse
16 Irpinia Italy-01 1980 Bagnoli Irpinio 6.9
17 Superstition Hills-02 1987 Parachute Test Site 6.54
18 Loma Prieta 1989 Saratoga-Aloha Ave 6.93
19 Chi-Chi Taiwan 1999 TCU065 7.62
20 Niigata Japan 2004 NIGH11 6.63
37
For each earthquake record, a family of CLCs is evaluated for its ability to mitigate the mean
square absolute acceleration cost of the SDOF system. The contour plots of costs as a function of
CLC control gain parametersa andb are shown in Figure 3.15 (note that, for excitations No. 6, 9,
14 and 17, the contour plot spans the(10)(10) range, but the minimum point(a

;b

) found
throughfminsearch is outside the(10)(10) range). As displayed in the figures, the contour
plots of costs for different excitations are similar to those presented in the previous subsections
for the El Centro and Kobe records. Comparisons of the time histories of the OCLC and CLQR
absolute acceleration responses and control forces are shown in Figure 3.16 for each earthquake.
OCLC always provides significant improvements in the absolute acceleration response reduction
with a smaller level of control force, as well as less frequent clipping compared to CLQR.
38
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(a) No. 1 (far-field, 1994 Northridge)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(b) No. 2 (far-field, 1999 Duzce)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(c) No. 3 (far-field, 1999 Hector)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(d) No. 4 (far-field, 1979 Imperial Valley)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(e) No. 5 (far-field, 1995 Kobe)
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(f) No. 6 (far-field, 1999 Kocaeli; minimum point out-
side(10)(10))
39
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(g) No. 7 (far-field, 1992 Landers)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(h) No. 8 (far-field, 1990 Manjil)
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(i) No. 9 (far-field, 1999 Chi-Chi; minimum point out-
side(10)(10))
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(j) No. 10 (far-field, 1976 Friuli)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(k) No. 11 (near-field, 1976 Gazli)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(l) No. 12 (near-field, 1985 Nahanni)
40
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(m) No. 13 (near-field, 1992 Cape)
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(n) No. 14 (near-field, 1999 Chi-Chi; minimum point
outside(10)(10))
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(o) No. 15 (near-field, 2011 Christchurch)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(p) No. 16 (near-field, 1980 Irpinia)
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(q) No. 17 (near-field, 1987 Superstition Hills; mini-
mum point outside(10)(10))
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(r) No. 18 (near-field, 1989 Loma)
41
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(s) No. 19 (near-field, 1999 Chi-Chi)
minimum of surface
0 5 10
stiffness factor
0
5
10
damping factor
0
1
2
3
4
5
6
(t) No. 20 (near-field, 2004 Niigata)
Figure 3.15. Contours of SDOF cost metric J
a
as a function of CLC stiffness factora and damping
factorb parameters for 20 earthquakes (note: for excitations No. 6, 9, 14 and 17, the contour plot
is in the (10)(10) range, but the minimum point (a

;b

) found through fminsearch is
outside the(10)(10) range).
42
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(a) No. 1 (far-field, 1994 Northridge)
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(b) No. 2 (far-field, 1999 Duzce)
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(c) No. 3 (far-field, 1999 Hector)
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(d) No. 4 (far-field, 1979 Imperial Valley)
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(e) No. 5 (far-field, 1995 Kobe)
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(f) No. 6 (far-field, 1999 Kocaeli)
43
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(g) No. 7 (far-field, 1992 Landers)
10 12 14 16 18 20
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
10 12 14 16 18 20
Time [s]
0
20
Control force [%mg]
(h) No. 8 (far-field, 1990 Manjil)
30 32 34 36 38 40
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
30 32 34 36 38 40
Time [s]
0
20
Control force [%mg]
(i) No. 9 (far-field, 1999 Chi-Chi)
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg]
(j) No. 10 (far-field, 1976 Friuli)
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg]
(k) No. 11 (near-field, 1976 Gazli)
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg]
(l) No. 12 (near-field, 1985 Nahanni)
44
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg]
(m) No. 13 (near-field, 1992 Cape)
30 32 34 36 38 40
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
30 32 34 36 38 40
Time [s]
0
20
Control force [%mg]
(n) No. 14 (near-field, 1999 Chi-Chi)
20 22 24 26 28 30
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
20 22 24 26 28 30
Time [s]
0
20
Control force [%mg]
(o) No. 15 (near-field, 2011 Christchurch)
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg]
(p) No. 16 (near-field, 1980 Irpinia)
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg] (q) No. 17 (near-field, 1987 Superstition Hills)
0 2 4 6 8 10
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
0 2 4 6 8 10
Time [s]
0
20
Control force [%mg]
(r) No. 18 (near-field, 1989 Loma)
45
30 32 34 36 38 40
0
2
4
Abs. accel. [m/s
2
]
CLQR
OCLC
30 32 34 36 38 40
Time [s]
0
20
40
Control force [%mg]
(s) No. 19 (near-field, 1999 Chi-Chi)
20 22 24 26 28 30
0
2
Abs. accel. [m/s
2
]
CLQR
OCLC
20 22 24 26 28 30
Time [s]
0
20
Control force [%mg]
(t) No. 20 (near-field, 2004 Niigata)
Figure 3.16. Absolute acceleration response and control force comparisons of CLQR and OCLC
for a SDOF system excited by 20 earthquakes.
46
The optimization results are summarized in Table 3.10, including the optimal control gain and
minimum cost; to facility comparisons with CLQR, the CLQR costs are listed as well. OCLC
always provides performance superior to CLQR for all earthquake records, with the largest im-
provement around 60%; even the smallest improvement is still about 20%. The optimal control
gain parametera

is near zero for all excitations, in the range[0:1768;0:1664], with some neg-
ative values; damping parameter b

ranges from 4.1375 to 11.1560. The results match those in
previous sections wherea

is close to zero whileb

is farther from origin.
Table 3.10: Optimization results for 20 earthquakes.
record set No. Year EQ OCLC(a

;b

) J
OCLC
a
J
CLQR
a
D*
[m
2
/s
4
] [m
2
/s
4
] [%]
Far-field
1 1994 Northridge (–0.1657,5.9587) 0.2496 0.4278 42
2 1999 Duzce (–0.1565,8.3000) 0.1277 0.1672 24
3 1999 Hector (–0.0411,9.9423) 0.0488 0.0837 42
4 1979 Imperial Valley (0.1599,8.9159) 0.0817 0.1225 33
5 1995 Kobe (0.1615,4.3361) 0.1033 0.1260 18
6 1999 Kocaeli (0.1593,10.3975) 0.1635 0.2786 41
7 1992 Landers (0.1584,9.4511) 0.0927 0.1765 47
8 1990 Manjil (0.1602,6.0432) 0.0780 0.0975 20
9 1999 Chi-Chi (0.1367,11.1560) 0.0999 0.1822 45
10 1976 Friuli (–0.1768,3.9847) 0.0165 0.0269 39
Near-field
No pulse
11 1976 Gazli (–0.0000,6.0006) 0.5590 0.8981 38
12 1985 Nahanni (–0.0401,4.1375) 0.1878 0.2798 33
13 1992 Cape (0.1543,7.2194) 0.2488 0.4407 44
14 1999 Chi-Chi (0.1654,10.2174) 0.0379 0.0563 33
15 2011 Christchurch (0.1664,10.0000) 0.2424 0.0686 53
Pulse
16 1980 Irpinia (0.0422,9.8280) 0.0266 0.0430 38
17 1987 Superstition Hills (–0.1511,10.4375) 0.8185 1.9983 59
18 1989 Loma (–0.0418,9.8783) 0.0900 0.1782 49
19 1999 Chi-Chi (0.0398,10.0000) 0.4712 0.8517 45
20 2004 Niigata (0.1605,7.0005) 0.0144 0.0204 29
*D is the percent change relative to CLQR; positive numbers mean improvements in OCLC.
The OCLC optimal control gain parameters have been calculated for each earthquake excita-
tion. However, a structure control typically must be coded with one specific control gain (a, b);
given the data in Table 3.10, as well as the GWN, El Centro and Kobe results in the previous
sections, what gain (a, b) would be near optimal for all of these records? For this selection,
three higher-level cost functions, as listed below, are defined to consider the performance for all
earthquake records; for each cost function, the CLC is optimized to obtain the optimal design
parameters:
F1a. Minimize the sum of the per-earthquake costs: min
a;b
S
20
i=1
J
a
(a;bjEQ
i
)
47
F1b. Maximize the sum of per-earthquake absolute cost improvement provided by a CLC relative
to CLQR: max
a;b
S
20
i=1
[J
a
(0;0jEQ
i
)
| {z }
CLQR
J
a
(a;bjEQ
i
)
| {z }
CLC
];
F2. Maximize the sum of per-earthquake relative cost improvement provided by a CLC relative
to CLQR: max
a;b
S
20
i=1
h
1
J
a
(a;bjEQ
i
)
J
a
(0;0jEQ
i
)
i
;
The resulting sets of optimal control gains are listed in Table 3.11. The optimal control gain
Table 3.11: OCLC optimization results for three cost functions F1a/b and F2 over all 20 earth-
quakes.
control gain parameters
a

b

F1a (min, sum of m.s. abs. accels.) 0.1341 9.3748
F1b (max, OCLC abs. improvement) 0.1341 9.3748
F2 (max, OCLC rel. improvement) 0.0792 7.7025
values are identical for F1a and F1b because both of them are designed to seek the absolute mini-
mum CLC cost; thus, there are two sets of candidates of optimal control gains, namely(a

F1
;b

F1
)
and (a

F2
;b

F2
), for this SDOF system. To demonstrate that the two sets of candidates would be
near optimal for the SDOF system regardless of excitations, the mean square absolute acceleration
cost metric is calculated using different OCLC gains when subjected to a variety of excitations.
Considering the twenty optimal control gains designed for each of the twenty earthquakes in Ta-
ble 3.9, and the three optimal control gains designed for GWN, El Centro and Kobe in the previous
Sections 3.1.5 to 3.1.7, i.e., (a

EX
i
;b

EX
i
), i= 1;:::;23 (note that the earthquake record No. 5 in
Table 3.9 and the Kobe in previous sections are different excitations as they were recorded by
different stations and have different magnitudes, time durations, etc.), as well as the two can-
didates(a

F1
;b

F1
) and(a

F2
;b

F2
), there are totally twenty-three excitations EX
i
, i= 1;:::;23 and
twenty-five OCLC gains(a

()
;b

()
), where() denotes either the design excitation EX
i
or the design
method F1 or F2. If the cost metric J
a
(a

()
;b

()
jEX
i
) computed using one OCLC gain (a

()
;b

()
)
when the system is subjected to an excitation EX
i
, differs significantly from the minimum cost
function J
a
(a

EX
i
;b

EX
i
jEX
i
) achieved using the OCLC gain (a

EX
i
;b

EX
i
) designed for and evalu-
ated by the same excitation EX
i
, then the minimum cost function is considered “very” sensitive to
48
OCLC gain changes and, thus, an OCLC that not designed to minimize the mean square absolute
acceleration cost for that excitation EX
i
, may not provide satisfactory performance in reducing
the absolute acceleration when the system is subjected to that excitation; otherwise, an OCLC
designed appropriately (e.g., the two candidates, (a

F1
;b

F1
) and (a

F2
;b

F2
), designed to minimize
some higher-level cost functions) would perform well in minimizing the absolute acceleration re-
gardless of excitations. The cost change using a particular OCLC gain, relative to the minimum
cost for that excitation, is given by:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)
1 (3.17)
Table 3.12 summarizes the relative cost change values computed using Eq. (3.17) for all twenty-
five OCLC gains and twenty-three excitations. The blue bolded numbers are the largest values
throughout that row, and the two purple italic ones are the negative values across the table. For
excitation No. 10, the minimum cost is most sensitive to the changes of OCLC gain parameters
because most large values are in that column. The very few negative values may be because the
OCLC gain(a

EX
i
;b

EX
i
) designed for that excitation EX
i
gives a local minimum instead of a global
minimum, and a different CLC gain corresponds to another local minimum which is more optimal
than the original one.
A statistical analysis of the mean, maximum, minimum and standard deviation values for each
row and each column of these relative cost change values in Table 3.12 is shown in Tables 3.13
and 3.14, respectively. The blue bolded entries are the largest values for each statistical metric.
As shown in Tables 3.12–3.14, for some excitations, e.g., No. 1, 10 and 17, the minimum cost
is sensitive to CLC gain changes; but, in all, the minimum cost is relatively insensitive to OCLC
gain changes, evidenced by no significant difference from the minimum cost, especially for the
two “overall” OCLC designs. Therefore, the final optimal control gain for this SDOF system can
be selected from the two candidates (a

F1
;b

F1
) and (a

F2
;b

F2
). To make the final decision, these
49
Table 3.12: Cost change provided by an OCLC evaluated by one excitation relative to the OCLC designed and evaluated for that
excitation:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)
1, i= 1;:::;23, for all twenty-five OCLC gains and twenty-three excitations.
Excitations
1 2 3 4 5
+
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
+
23
Control 1994 1999 1999 1979 1995
+
1999 1992 1990 1999 1976 1976 1985 1992 1999 2011 1980 1987 1989 1999 2004 1940 1995
+
Gain Northridge Duzce Hector Imperial
Valley
Kobe
+
Kocaeli Landers Manjil Chi-Chi Friuli Gazli Nahanni Cape Chi-Chi Christchurch Irpinia Superstition
Hills
Loma Chi-Chi Niigata El Centro Kobe
+
GWN
(a

EX 1
; b

EX 1
) 0.000 0.072 0.042 0.163 0.122 0.485 0.344 0.152 0.276 0.085 0.157 0.098 0.151 0.420 0.545 0.271 0.423 0.248 0.198 0.131 0.093 0.114 0.323
(a

EX 2
; b

EX 2
) 0.064 0.000 0.007 0.093 0.136 0.280 0.105 0.152 0.150 0.240 0.084 0.222 0.191 0.299 0.270 0.151 0.103 0.089 0.165 0.113 0.093 0.022 0.325
(a

EX 3
; b

EX 3
) 0.160 0.050 0.000 0.037 0.230 0.125 0.092 0.147 0.053 0.444 0.083 0.322 0.103 0.103 0.139 0.035 0.141 0.036 0.044 0.082 0.062 0.042 0.202
(a

EX 4
; b

EX 4
) 0.278 0.048 0.064 0.000 0.200 0.026 0.064 0.063 0.053 0.401 0.052 0.247 0.016 0.042 0.032 0.064 0.073 0.102 0.045 0.009 0.060 0.126 0.042
(a

EX 5
; b

EX 5
) 0.441 0.074 0.299 0.114 0.000 0.190 0.321 0.021 0.210 0.208 0.217 0.098 0.154 0.128 0.396 0.398 0.421 0.440 0.288 0.069 0.077 0.409 0.091
(a

EX 6
; b

EX 6
) 0.322 0.097 0.065 0.027 0.309 0.000 0.054 0.123 0.018 0.590 0.083 0.393 0.026 0.025 0.014 0.038 0.030 0.110 0.015 0.039 0.086 0.115 0.071
(a

EX 7
; b

EX 7
) 0.285 0.074 0.063 0.014 0.233 0.016 0.000 0.084 0.033 0.466 0.055 0.288 0.018 0.038 0.021 0.055 0.052 0.140 0.030 0.012 0.064 0.127 0.051
(a

EX 8
; b

EX 8
) 0.304 0.042 0.166 0.080 0.057 0.116 0.166 0.000 0.149 0.188 0.099 0.075 0.035 0.089 0.198 0.215 0.208 0.259 0.142 0.021 0.048 0.197 0.045
(a

EX 9
; b

EX 9
) 0.331 0.127 0.061 0.038 0.370 0.014 0.032 0.161 0.000 0.679 0.097 0.461 0.044 0.035 –0.001 0.023 0.043 0.096 0.008 0.037 0.094 0.118 0.101
(a

EX 10
; b

EX 10
) 0.097 0.174 0.116 0.227 0.207 0.756 0.523 0.243 0.398 0.000 0.292 0.062 0.158 0.569 0.654 0.365 0.953 0.418 0.430 0.248 0.078 0.223 0.365
(a

EX 11
; b

EX 11
) 0.099 0.042 0.082 0.081 0.070 0.234 0.186 0.063 0.172 0.117 0.000 0.051 0.057 0.149 0.293 0.103 0.579 0.170 0.124 0.061 0.017 0.116 0.130
(a

EX 12
; b

EX 12
) 0.129 0.113 0.183 0.164 0.051 0.462 0.339 0.124 0.354 0.111 0.062 0.000 0.124 0.226 0.517 0.211 0.661 0.271 0.254 0.145 0.041 0.157 0.198
(a

EX 13
; b

EX 13
) 0.254 0.012 0.112 0.019 0.088 0.074 0.102 0.028 0.094 0.242 0.052 0.123 0.000 0.058 0.082 0.134 0.143 0.198 0.087 0.006 0.032 0.144 0.035
(a

EX 14
; b

EX 14
) 0.321 0.092 0.071 0.024 0.293 0.015 0.027 0.116 0.019 0.566 0.077 0.374 0.024 0.000 0.011 0.045 0.038 0.108 0.013 0.014 0.082 0.127 0.065
(a

EX 15
; b

EX 15
) 0.316 0.080 0.065 0.018 0.283 0.011 0.036 0.106 0.034 0.536 0.074 0.348 0.017 0.022 0.000 0.052 0.018 0.110 0.019 0.028 0.080 0.137 0.059
(a

EX 16
; b

EX 16
) 0.199 0.066 0.027 0.031 0.236 0.053 0.090 0.107 0.042 0.459 0.056 0.311 0.061 0.053 0.050 0.000 0.195 0.064 0.023 0.026 0.064 0.062 0.125
(a

EX 17
; b

EX 17
) 0.134 0.062 0.023 0.089 0.252 0.246 0.158 0.178 0.170 0.494 0.118 0.370 0.198 0.243 0.139 0.112 0.000 0.027 0.118 0.104 0.074 0.060 0.357
(a

EX 18
; b

EX 18
) 0.159 0.051 0.013 0.032 0.213 0.137 0.079 0.146 0.044 0.436 0.070 0.314 0.105 0.126 0.108 0.035 0.092 0.000 0.069 0.056 0.064 0.034 0.202
(a

EX 19
; b

EX 19
) 0.203 0.074 0.023 0.027 0.245 0.067 0.093 0.113 0.032 0.482 0.050 0.326 0.066 0.063 0.059 0.009 0.166 0.041 0.000 0.035 0.062 0.063 0.130
(a

EX 20
; b

EX 20
) 0.264 0.027 0.129 0.030 0.077 0.070 0.109 0.023 0.104 0.230 0.061 0.109 0.015 0.047 0.132 0.151 0.192 0.192 0.095 0.000 0.036 0.163 0.035
(a

EX 21
; b

EX 21
) 0.178 0.045 0.103 0.087 0.045 0.173 0.151 0.046 0.154 0.141 0.036 0.053 0.050 0.117 0.200 0.104 0.304 0.188 0.117 0.028 0.000 0.131 0.083
(a

EX 22
; b

EX 22
) 0.066 0.019 0.001 0.070 0.108 0.256 0.168 0.100 0.135 0.218 0.056 0.189 0.111 0.225 0.222 0.128 0.248 0.076 0.214 0.066 0.069 0.000 0.238
(a

EX 23
; b

EX 23
) 0.520 0.068 0.234 0.018 0.192 0.069 0.186 0.039 0.061 0.416 0.181 0.279 –0.014 0.058 0.140 0.301 0.229 0.308 0.125 0.030 0.124 0.310 0.000
(a

F1
; b

F1
) 0.258 0.062 0.059 0.019 0.222 0.039 0.058 0.077 0.028 0.442 0.053 0.276 0.028 0.044 0.011 0.041 0.000 0.112 0.023 0.019 0.064 0.112 0.060
(a

F2
; b

F2
) 0.180 0.015 0.060 0.019 0.099 0.081 0.076 0.042 0.092 0.247 0.019 0.133 0.036 0.064 0.086 0.038 0.138 0.139 0.050 0.016 0.043 0.087 0.070
+
No. 5 is different than No. 22 as they were recorded by different stations and have different magnitudes, time durations, etc.
50
Table 3.13: Statistical analysis of cost change provided by an OCLC evaluated by one excitation
relative to the OCLC designed and evaluated for that excitation:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)
1, i= 1;:::;23,
for each row in Table 3.12.
Control gain Min Max Mean Standard
Deviation
(a

EX
1
; b

EX
1
) 0.000 0.545 0.214 0.149
(a

EX
2
; b

EX
2
) 0.000 0.325 0.146 0.092
(a

EX
3
; b

EX
3
) 0.000 0.444 0.119 0.103
(a

EX
4
; b

EX
4
) 0.000 0.401 0.092 0.098
(a

EX
5
; b

EX
5
) 0.000 0.441 0.220 0.146
(a

EX
6
; b

EX
6
) 0.000 0.590 0.115 0.148
(a

EX
7
; b

EX
7
) 0.000 0.466 0.097 0.116
(a

EX
8
; b

EX
8
) 0.000 0.304 0.126 0.082
(a

EX
9
; b

EX
9
) –0.001 0.679 0.129 0.171
(a

EX
10
; b

EX
10
) 0.000 0.953 0.329 0.238
(a

EX
11
; b

EX
11
) 0.000 0.579 0.130 0.120
(a

EX
12
; b

EX
12
) 0.000 0.661 0.213 0.162
(a

EX
13
; b

EX
13
) 0.000 0.254 0.092 0.071
(a

EX
14
; b

EX
14
) 0.000 0.566 0.110 0.143
(a

EX
15
; b

EX
15
) 0.000 0.536 0.106 0.136
(a

EX
16
; b

EX
16
) 0.000 0.459 0.104 0.109
(a

EX
17
; b

EX
17
) 0.000 0.494 0.162 0.121
(a

EX
18
; b

EX
18
) 0.000 0.436 0.112 0.102
(a

EX
19
; b

EX
19
) 0.000 0.482 0.106 0.114
(a

EX
20
; b

EX
20
) 0.000 0.264 0.100 0.073
(a

EX
21
; b

EX
21
) 0.000 0.304 0.110 0.071
(a

EX
22
; b

EX
22
) 0.000 0.256 0.130 0.083
(a

EX
23
; b

EX
23
) –0.014 0.520 0.168 0.140
(a

F1
; b

F1
) 0.000 0.442 0.092 0.108
(a

F2
; b

F2
) 0.015 0.247 0.080 0.057
two candidates are evaluated for each excitation separately to obtain the following six relative
improvement metrics:
R1. Per-excitation relative cost improvement provided by an OCLC relative to CLQR: 1
J
a
(a

()
;b

()
jEX
i
)
J
a
(0;0jEX
i
)
;
R2. Per-excitation relative improvement of mean square displacement, defined as J
d
(a;bjEX
i
)=
1
t
fi
R
t
fi
0
q
2
i
(t;a;b)dt for an excitation EX
i
, provided by an OCLC relative to CLQR: 1
J
d
(a

()
;b

()
jEX
i
)
J
d
(0;0jEX
i
)
;
R3. Per-excitation relative improvement of mean square control force, defined as J
u
(a;bjEX
i
)=
1
t
fi
R
t
fi
0
u
2
i
(t;a;b)dt for an excitation EX
i
, provided by an OCLC relative to CLQR: 1
J
u
(a

()
;b

()
jEX
i
)
J
u
(0;0jEX
i
)
;
51
Table 3.14: Statistical analysis of cost change provided by an OCLC evaluated by one excitation relative to the OCLC designed and
evaluated for that excitation:
J
a
(a

()
;b

()
jEX
i
)
J
a
(a

EX
i
;b

EX
i
jEX
i
)
1, i= 1;:::;23, for each column in Table 3.12.
Excitations
1 2 3 4 5
+
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
+
23
Statistical 1994 1999 1999 1979 1995
+
1999 1992 1990 1999 1976 1976 1985 1992 1999 2011 1980 1987 1989 1999 2004 1940 1995
+
Metric Northridge Duzce Hector Imperial
Valley
Kobe
+
Kocaeli Landers Manjil Chi-Chi Friuli Gazli Nahanni Cape Chi-Chi Christchurch Irpinia Superstition
Hills
Loma Chi-Chi Niigata El Centro Kobe
+
GWN
Min 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 –0.014 0.000 –0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Max 0.520 0.174 0.299 0.227 0.370 0.756 0.523 0.243 0.398 0.679 0.292 0.461 0.198 0.569 0.654 0.398 0.953 0.440 0.430 0.248 0.124 0.409 0.365
Mean 0.222 0.063 0.083 0.061 0.174 0.160 0.142 0.098 0.115 0.337 0.087 0.221 0.071 0.130 0.173 0.123 0.218 0.158 0.108 0.056 0.064 0.128 0.136
Standard deviation 0.121 0.038 0.074 0.057 0.097 0.181 0.122 0.059 0.105 0.182 0.064 0.131 0.062 0.136 0.183 0.112 0.232 0.114 0.105 0.057 0.027 0.089 0.110
+
No. 5 is different than No. 22 as they were recorded by different stations and have different magnitudes, time durations, etc.
52
R4. Per-excitation relative improvement of maximum absolute acceleration, defined as J
max
a
(a;bjEX
i
)=
maxj ¨ q
a
i
(t;a;b)j, t2[0; t
fi
] for an excitation EX
i
, provided by an OCLC relative to CLQR:
1
J
max
a
(a

()
;b

()
jEX
i
)
J
max
a
(0;0jEX
i
)
;
R5. Per-excitation relative improvement of maximum displacement, defined as J
max
d
(a;bjEX
i
)=
maxjq
i
(t;a;b)j, t2[0; t
fi
] for an excitation EX
i
, provided by an OCLC relative to CLQR:
1
J
max
d
(a

()
;b

()
jEX
i
)
J
max
d
(0;0jEX
i
)
;
R6. Per-excitation relative improvement of maximum control force, defined as J
max
u
(a;bjEX
i
)=
maxju
i
(t;a;b)j, t2[0; t
fi
] for an excitation EX
i
, provided by an OCLC relative to CLQR:
1
J
max
u
(a

()
;b

()
jEX
i
)
J
max
u
(0;0jEX
i
)
.
The values of these metrics are shown in Table 3.15. A statistical analysis of the mean, max-
imum, minimum and standard deviation values is summarized in Table 3.16. Even though the
relative improvements provided by control gain (a

F1
;b

F1
) are larger than those with the control
gain(a

F2
;b

F2
) for most metrics and most excitations, OCLC(a

F1
;b

F1
) underperforms CLQR for
some metrics subjected to some excitations. For instance, (a

F1
;b

F1
) makes OCLC suboptimal
compared with CLQR for excitation No. 5 when the metric is R1 and for excitation No. 12 when
the metric is R4 (note that, for control force metrics R3 and R6, negative values do not mean
sub-optimality but just larger control force levels). Moreover, the negative values of control force
metrics R3 and R6 for (a

F1
;b

F1
) are of larger magnitude than those of control gain (a

F2
;b

F2
),
indicating that the control forces of the former are larger than those of CLQR and the latter OCLC
for some excitations. Therefore, the best choice for optimal control gain for this SDOF system is:
(a

F2
;b

F2
)=(0:0792;7:7025).
3.2 Parameter study of OCLC for SDOF systems
The OCLC gain has been evaluated in the previous sections for a specific SDOF system with
w = 2p and z = 0:05. This section explores the relationship of the control gain parameters a
53
Table 3.15: Evaluation of OCLC gain(a

F1
;b

F1
) and(a

F2
;b

F2
) for six relative improvement metrics R1–R6 subjected to twenty-three
excitations.
Excitations
Relative Control 1 2 3 4 5
+
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
+
23
Improvement Gain 1994 1999 1999 1979 1995
+
1999 1992 1990 1999 1976 1976 1985 1992 1999 2011 1980 1987 1989 1999 2004 1940 1995
+
Metric Northridge Duzce Hector Imperial
Valley
Kobe
+
Kocaeli Landers Manjil Chi-Chi Friuli Gazli Nahanni Cape Chi-Chi Christchurch Irpinia Superstition
Hills
Loma Chi-Chi Niigata El Centro Kobe
+
GWN
R1 J a
(a

F1
; b

F1
) 0.266 0.189 0.382 0.321 –0.002 0.390 0.444 0.139 0.436 0.115 0.345 0.143 0.420 0.297 0.520 0.357 0.591 0.438 0.434 0.278 0.207 0.386 0.279
(a

F2
; b

F2
) 0.312 0.225 0.382 0.320 0.099 0.365 0.435 0.167 0.401 0.235 0.366 0.239 0.415 0.284 0.484 0.359 0.534 0.425 0.419 0.280 0.222 0.400 0.272
R2 J d
(a

F1
; b

F1
) 0.707 0.597 0.709 0.618 0.628 0.601 0.673 0.556 0.621 0.687 0.666 0.687 0.742 0.536 0.696 0.657 0.669 0.695 0.572 0.622 0.617 0.688 0.513
(a

F2
; b

F2
) 0.641 0.497 0.640 0.539 0.529 0.514 0.603 0.458 0.548 0.617 0.611 0.616 0.710 0.464 0.615 0.572 0.579 0.621 0.507 0.545 0.534 0.608 0.438
R3 J u
(a

F1
; b

F1
) 0.385 0.154 0.490 0.388 0.080 0.506 0.559 0.099 0.536 –0.110 0.421 0.272 0.546 0.427 0.614 0.489 0.523 0.519 0.413 0.374 0.255 0.449 0.224
(a

F2
; b

F2
) 0.354 0.085 0.437 0.326 0.059 0.411 0.485 0.056 0.462 –0.058 0.390 0.268 0.552 0.348 0.530 0.402 0.431 0.452 0.337 0.323 0.213 0.374 0.166
R4 J
max
a
(a

F1
; b

F1
) 0.468 0.339 0.396 0.354 0.346 0.278 0.394 0.117 0.318 0.372 0.433 –0.053 0.168 0.313 0.409 0.185 0.394 0.336 0.170 0.386 0.224 0.397 0.287
(a

F2
; b

F2
) 0.411 0.266 0.335 0.288 0.379 0.211 0.326 0.028 0.311 0.387 0.371 0.109 0.127 0.296 0.323 0.124 0.326 0.264 0.176 0.405 0.241 0.347 0.246
R5 J
max
d
(a

F1
; b

F1
) 0.469 0.343 0.395 0.357 0.438 0.280 0.393 0.119 0.319 0.430 0.434 0.385 0.170 0.316 0.409 0.185 0.394 0.336 0.172 0.412 0.285 0.397 0.289
(a

F2
; b

F2
) 0.411 0.271 0.335 0.285 0.379 0.213 0.326 0.030 0.311 0.388 0.372 0.342 0.128 0.298 0.323 0.124 0.326 0.262 0.178 0.406 0.238 0.348 0.242
R6 J
max
u
(a

F1
; b

F1
) 0.457 0.204 0.279 0.272 0.129 0.117 0.360 –0.098 0.233 0.262 0.340 0.217 –0.230 0.253 0.306 0.184 0.440 0.238 0.142 0.314 0.129 0.350 0.275
(a

F2
; b

F2
) 0.418 0.168 0.247 0.230 0.178 0.112 0.309 –0.099 0.288 0.268 0.335 0.281 –0.185 0.246 0.254 0.120 0.364 0.188 0.160 0.325 0.124 0.315 0.237
+
No. 5 is different than No. 22 as they were recorded by different stations and have different magnitudes, time durations, etc.
54
Table 3.16: Statistical analysis of OCLC gain (a

F1
;b

F1
) and (a

F2
;b

F2
) for six relative improve-
ment metrics R1–R6 subjected to twenty-three excitations in Table 3.15.
Statistical (a

F1
;b

F1
) (a

F2
;b

F2
)
Metric Relative improvement metrics Relative improvement metrics
R1 J
a
R2 J
d
R3 J
u
R4 J
max
a
R5 J
max
d
R6 J
max
u
R1 J
a
R2 J
d
R3 J
u
R4 J
max
a
R5 J
max
d
R6 J
max
u
Min –0.002 0.513 –0.111 –0.053 0.119 –0.230 0.099 0.438 –0.058 0.028 0.030 –0.185
Max 0.591 0.742 0.614 0.468 0.469 0.457 0.534 0.710 0.552 0.411 0.412 0.418
Mean 0.321 0.642 0.374 0.306 0.336 0.225 0.332 0.566 0.322 0.274 0.284 0.212
Standard deviation 0.142 0.060 0.183 0.122 0.097 0.154 0.106 0.068 0.164 0.104 0.098 0.138
(a) Optimal stiffness parametera

(b) Optimal damping parameterb

Figure 3.17. Optimal control gain parameters (a

;b

) as function of the structural parameters
w=(2p)2(0:5; 0:75; 1; 1:25; 1:5) andz2(0:025; 0:0375; 0:05; 0:075; 0:1) for a SDOF system
driven by GWN (t
f
= 10000 s, D= 0:02).
55
and b with the structural characteristics (natural frequency w and damping ratio z ) of SDOF
systems subjected to GWN. To determine the OCLC gain for a specific SDOF system, the system is
simulated over a grid of clipped linear strategies with(a;b)2(10)(10) over t2[0;t
f
] using
a GWN with sampling timeDt= 0:02s and t
f
= 100s; the mean square responses, computed using
ode45 with relative tolerance 10
10
and absolute tolerance 10
12
, are computed by averaging over
the duration of the simulation. MATLAB’s fminsearch is then utilized to find the global optimal
(a

, b

) that minimizes J
a
(a;b), using the minimum point found in the grid search as the initial
guess, with t
f
= 10000s. The optimal control gain parameters(a

;b

) as function of the structural
parameters w=(2p)2(0:5; 0:75; 1; 1:25; 1:5), and z2(0:025; 0:0375; 0:05; 0:075; 0:1) for a
SDOF system driven by GWN are depicted in Figure 3.17. The results indicate that the optimal
stiffness parametera

is relatively insensitive to changes inw andz , whereas the optimal damping
parameterb

is approximately inversely proportional to damping ratioz .
To further demonstrate that the implications provided by Figure 3.17 hold for a broader range of
structural parametersw andz , the OCLC gain parametersa

andb

as functions of the structural
parametersw=(2p)= 0:5k, k= 1;:::;10 andz = 0:025k, k= 1;:::;8 are depicted in Figure 3.18
for a SDOF system subjected to a GWN with D= 0:02 and t
f
= 1000s (chosen because simulations
with a broader range of w and z require greater computational effort; this shorter time duration
is sufficient for a fairly close evaluation of the relationship between OCLC parameters and SDOF
structural parameters). The results are consistent with those in Figure 3.17: a

is insensitive to
changes inw andz ;b

is almost constant for a fixed damping ratioz and is inversely proportional
toz ;b

does not depend on natural frequencyw.
3.3 Summary
This chapter proposed an optimal clipped linear control (OCLC) approach for determining opti-
mal controllable damping strategies. This approach uses a quadratic performance metric similar to
LQR control but optimizes the metric for the nonlinear system with the controllable damper for a
56
0 1 2 3 4 5
Natural frequency: /(2 )
0
0.1
0.2
0.3
0.4
0.5
0.6
Optimal stiffiness parameter:
*
=0.025
=0.05
=0.075
=0.1
=0.125
=0.15
=0.175
=0.2
(a) Optimal stiffness parametera

0 1 2 3 4 5
Natural frequency: /(2 )
0
2
4
6
8
10
12
14
16
18
20
Optimal damping parameter:
*
=0.025
=0.05
=0.075
=0.1
=0.125
=0.15
=0.175
=0.2
(b) Optimal damping parameterb

Figure 3.18. Optimal control gain parameters(a

;b

) as function of the structural parametersw
andz for a SDOF system driven by GWN (t
f
= 1000 s, D= 0:02).
57
given excitation. The SDOF results showed that OCLC always provides performance superior to
both CLQR and a passive linear viscous damper for a stochastic GWN. OCLC also outperforms
CLQR for Kanai-Tajimi filtered excitations as well as a suite of historical earthquake excitations
for the SDOF system. Moreover, the final decision of optimal control gain was made via a compre-
hensive analysis. Further, the relationship between optimal control gain and structural parameters
(natural frequency and damping ratio) of SDOF systems was explored; the results indicated that
the optimal stiffness parameter a

is relatively insensitive to changes in w and z , whereas the
optimal damping parameterb

is approximately inversely proportional to damping ratioz .
In summary, the OCLC strategy has significant promise for use in semiactive control appli-
cations, particularly those with a need to improve a serviceability cost metric as compared to a
passive linear viscous damper and CLQR strategies.
58
Chapter 4
Proposed Optimal Clipped Linear Strategies for 2DOF Systems
This chapter explores OCLC for a 2DOF structure model with several configuration. The first
section introduces the numerical models. The second section presents the LQR/CLQR control for
the 2DOF systems. The third section discusses the proposed new control strategy OCLC for the
2DOF systems; the results are compared with CLQR and optimal passive linear viscous damper
strategies. Section four explores the optimal active linear feedback control for a 2DOF system and
compares the results with LQR control. The fifth section concludes this chapter.
4.1 Numerical models
To verify that OCLC’s superior performance for a SDOF system holds for MDOF systems, con-
sider a 2DOF structure, which may be a two-story structure or a 2DOF model of a more complex
structure. The effectiveness of several OCLC strategies is studied here for this structure with the
controllable damper in the first story and for the same structure but the damper located in the
second story, as shown in Figure 4.1 and Figure 4.2.
The ground-relative displacements q(t)=[q
1
(t) q
2
(t)]
T
of this 2DOF structure, excited by
base acceleration ¨ q
g
, are governed by the equation of motion
M¨ q(t)+ C˙ q(t)+ Kq(t)=Mr ¨ q
g
(t)
¯
Bu(t) (4.1)
59
m
1
m
2
Fixed Base
Damper
q
1
k
1
, c
1
q
2
k
2
, c
2
u
Ground Motion
Figure 4.1. 2DOF controlled building model 1,
damper in the first story [27].
m
1
m
2
Fixed Base
Damper
u
q
1
k
1
, c
1
q
2
k
2
, c
2
Ground Motion
Figure 4.2. 2DOF controlled building model 2,
damper in the second story.
where M=

[m
1
0]
T
[0 m
2
]
T

, C=

[(c
1
+ c
2
)c
2
]
T
[c
2
c
2
]
T

and K=

[(k
1
+ k
2
)k
2
]
T
[k
2
k
2
]
T

are the mass, damping and stiffness matrices of the structural system, respectively; r=[1 1]
T
is
the ground influence vector;
¯
B is the controllable damper force influence vector (where
¯
B=[1 0]
T
for a damper in the first story and
¯
B=[1 1]
T
in the second); and u(t) is the controllable damping
force (not mass normalized for this 2DOF system). The corresponding state-space system is given
by
˙ x= Ax+ B
g
¨ q
g
+ B
u
u; A=
2
6
4
0 I
M
1
K M
1
C
3
7
5
; B
g
=
8
>
<
>
:
0
r
9
>
=
>
;
; B
u
=
8
>
<
>
:
0
M
1
¯
B
9
>
=
>
;
(4.2)
where state vector x=[q
T
˙ q
T
]
T
.
An OCLC strategy is designed for each of three response metrics, minimizing a weighted sum
of the mean square damper force (to ensure that the controller will not command impractically
60
large force) and either the mean square structure inter-story drifts or ground-relative velocities or
absolute accelerations, respectively; i.e., to minimize the cost metrics
J
d
=
1
cm
2
E[q
2
1
+(q
2
q
1
)
2
+r
d
u
2
]
J
v
=
1
cm
2
/s
2
E[ ˙ q
2
1
+ ˙ q
2
2
+r
v
u
2
]
J
a
=
1
m
2
/s
4
E[ ¨ q
a
1
2
+ ¨ q
a
2
2
+r
a
u
2
]
(4.3)
wherer
()
is the weight on the mean square damper force, and must be chosen to trade off the con-
trol force level versus structural response. Note that here the cost metrics are unit-nondimensionalized
for convenience. The drifts, velocities and absolute accelerations are given by
¯ q=[q
1
(q
2
q
1
)]
T
= C
d
x+ D
d
u
˙ q= C
v
x+ D
v
u
¨ q
a
= ¨ q+ r ¨ q
g
=M
1
Kq M
1
C˙ q M
1
¯
Bu= C
a
x+ D
a
u
(4.4)
where output matrices C
d
=

[11]
T
[0 1]
T
[0 0]
T
[0 0]
T

, C
v
=[0 I], C
a
=

M
1
K M
1
C

,
and force feedthrough matrices D
d
= D
v
= 0, D
a
=M
1
¯
B. To compare with the results in El-
haddad [26], the numerical parameters of the uncontrolled 2DOF model are m
1
= m
2
= 100 Mg,
k
1
= k
2
= 15:79173 MN=m, c
1
= 112:5 kNs=m, and c
2
= 0 — which results in natural frequen-
ciesw
1
7:769rad=s andw
2
20:326rad=s (corresponding to 1.236 and 3.236Hz), and damping
ratioz
1
=z
2
0:02. The initial conditions are assumed quiescent. The ground excitation ¨ q
g
is a
stochastic Gaussian white noise process with intensity D= 0:04 (chosen so that the structure has
large but reasonable response without control).
A preliminary parametric study of cost metric J
()
and mean square control force as a function
of corresponding mean square damper force weightr
a
for active LQR control indicated thatr
a
=
10
16
kg
–2
in J
a
can lead to root-mean-square (RMS) active damper force sufficiently large to
induce better comparison of OCLC and CLQR strategies. The preliminary study also found that
the superiority of OCLC over CLQR in reducing drift J
d
and velocity J
v
metrics are not as dramatic
61
as that for reducing absolute acceleration metric J
a
, but also determined that there is a wide range
of control weightsr
d
andr
v
that produce very similar (and moderate) comparison results between
OCLC and CLQR for these two cost metrics J
d
and J
v
. Since this study is intended to demonstrate
different levels of superiority of OCLC over CLQR, but not explore the exact trade off between
control effort and response reduction (which depend largely on specific application), an extensive
control weight parameter tuning is omitted, and the valuesr
d
= 10
16
kg
–2
s
4
andr
v
= 10
16
kg
–2
s
2
are used herein.
4.2 LQR/Clipped-LQR control for 2DOF systems
Similar to the analysis of the SDOF system, LQR and clipped-LQR control strategies are first
determined for the 2DOF system. These are designed for ideal GWN with cost weighting matrices
Q = C
T
()
C
()
, N = C
T
()
D
()
and R = r
()
+ D
T
()
D
()
, where the subscript () is “d”, “v” or “a”
denoting drift, velocity or absolute acceleration. Then, the LQR gain is K
LQR
= R
1
(PB+ N)
T
and the desired LQR force is:
u
LQR
(t)=K
LQR
x=k
LQR
q
1
q
1
k
LQR
q
2
q
2
k
LQR
˙ q
1
˙ q
1
k
LQR
˙ q
2
˙ q
2
(4.5)
which is a function of the four states. For CLQR, the clipping law is given by:
u
CLQR
(t)= u
LQR
(t)H[u
LQR
(t)D ˙ q
i
(t)]=
8
>
<
>
:
u
LQR
(t); u
LQR
(t)D ˙ q
i
(t)> 0
0; otherwise
(4.6)
where i is 1 (damper in first floor) or 2 (second floor), andD ˙ q
i
is the i
th
interstory velocity; i.e.,
D ˙ q
1
˙ q
1
and D ˙ q
2
˙ q
2
˙ q
1
. Eq. (4.6) can also be written as a function of the states: u
CLQR

u
CLQR
(x). Figures 4.3a and 4.3b show the CLQR control force surfaces when q= 0 and when
q=[0 1 cm]
T
, respectively, as functions of ˙ q
1
and ˙ q
2
(both matching prior results [26]) for
minimizing the absolute acceleration cost metric when the damper is in the first story.
62
10
10
0
control force u [%m
1
g]
20
0
velocity q
.
2
[cm/s]
0
velocity q
.
1
[cm/s]
(a) u
CLQR
([0 0 ˙ q
1
˙ q
2
]
T
)
10
10
0
control force u [%m
1
g]
20
0
velocity q
.
2
[cm/s]
0
velocity q
.
1
[cm/s]
(b) u
CLQR
([0 1 cm ˙ q
1
˙ q
2
]
T
)
Figure 4.3. Two “slices” of the CLQR control force as a function of the velocity states when q= 0
and q=[0 1 cm]
T
, respectively, for minimizing the absolute acceleration cost metric with the
damping device in the first story of a 2DOF structure.
4.2.1 Optimal passive linear viscous strategy
To compare against the clipped strategies, a passive linear viscous damper force, u= c
d
D ˙ q
i
with
damping coefficient c
d
, is also evaluated. From Eq. (4.1), the closed-loop passively-controlled
state-space equation of motion is then
˙ x(t)= A
p
x(t)+ B
g
¨ q
g
(t) (4.7)
where A
p
= A+ c
d
B
u
C
p
v
, C
p
v
=[0 0 1 0] for i= 1 and [0 0 1 1] for i= 2, and C
p
()
=
C
()
+ c
d
D
()
C
p
v
. For the GWN excitation, the cost metric is determined through simulation and
using an exact Lyapunov solution: J
()
= tr[C
p
()
XC
p
()
T
]+r
()
c
2
d
C
p
v
XC
p
v
T
, where X= X
T
=E[xx
T
]
is the solution to the Lyapunov equation A
p
X+ XA
pT
+ B
g
DB
T
g
= 0, and tr[] is the trace of a
matrix. Figures 4.4a and 4.4b present the normalized cost metric J
()
=J
uncontrolled
()
over a damping
coefficient c
d
range for a damper in the first or second story, respectively, with minimum cost
values noted with circles. The optimal values of c
d
for all cost metrics are shown in Table 4.1.
63
10
2
10
4
10
6
10
8
10
10
damper coefficient c
d
[N s/m]
0
1
2
3
4
5
6
J
( )
/J
( )
uncontrolled
drift
vel.
abs. accel.
(a) Damper in the first story
10
2
10
4
10
6
10
8
10
10
damper coefficient c
d
[N s/m]
0
1
2
3
4
5
6
J
( )
/J
( )
uncontrolled
drift
vel.
abs. accel.
(b) Damper in the second story
Figure 4.4. Cost metrics as functions of the damper coefficient c
d
for the 2DOF system with a
damping device in the (a) first or (b) second story; circles denote the minimum of each curve.
Table 4.1: Passive linear viscous c
d
designed by GWN for different cost metrics with the damping
device in the first or second story of a 2DOF structure.
Control Design Cost Damper in the Damper in the
Strategy Excitation Metric first story second story
Passive ideal GWN Drift c
d
= 3:1623 MN s=m c
d
= 1:1482 MN s=m
Passive ideal GWN Veloc. c
d
= 3:2359 MN s=m c
d
= 1:7378 MN s=m
Passive ideal GWN Abs. accel. c
d
= 1:4125 MN s=m c
d
= 0:8128 MN s=m
4.3 Proposed optimal clipped linear control for 2DOF systems
Again, it is convenient to parameterize K
OCLC
relative to the LQR solution in Eq. (4.5) using four
nondimensional parametersq q q =[q
1
q
2
q
3
q
4
]
T
as
u
d
=K
OCLC
x=(1q
1
)k
LQR
q
1
q
1
(1q
2
)k
LQR
q
2
q
2
(1q
3
)k
LQR
˙ q
1
˙ q
1
(1q
4
)k
LQR
˙ q
2
˙ q
2
(4.8)
so that q
1
= q
2
= q
3
= q
4
= 0 provides the LQR solution, and q
1
= q
2
= q
3
= q
4
= 1 is the
uncontrolled structure. Combining Eq. (4.1) and Eq. (4.8), the system is now piecewise linear with
the equations of motion
M¨ q(t)+
¯
C˙ q(t)+
¯
Kq(t)=Mr ¨ q
g
(t); u
d
˙ q
i
> 0
M¨ q(t)+ C˙ q(t)+ Kq(t)=Mr ¨ q
g
(t); otherwise
(4.9)
64
where
¯
C= C
¯
B
h
(1q
3
)k
LQR
˙ q
1
(1q
4
)k
LQR
˙ q
2
i
, and
¯
K= K
¯
B
h
(1q
1
)k
LQR
q
1
(1q
2
)k
LQR
q
2
i
.
The resulting optimization is now over a four-dimensional parameter space, which makes im-
possible direct graphical illustration of the cost metric, but “slices” in lower dimensions can be
utilized to demonstrate that there is a global minimum. The approach for optimizing the OCLC
is the same regardless of the design excitation, but, to explain the approach, a GWN realization is
used here.
Since there is no closed-form solution for the stochastic responses of this nonlinear system,
numerical simulations over t2[0;t
f
], using a Gaussian white noise excitation approximated by a
corresponding Gaussian pulse process with sampling time Dt = 0:025 s, is used to compute the
responses and approximate their mean squares. MATLAB’s fminsearch function is then used to
choose the parameter vectorq q q that minimizes the mean square cost metric Eq. (4.3) using various
initial guesses to find the global minimum. To ensure that the optimization solution is indeed
at a global minimum, slices of the cost metric J
a
(q q q) for the damper in the first story are shown
in Figure 4.5 (for t
f
= 1000 s, chosen because these graphs require many simulations and, while
this duration is not as accurate as longer durations, preliminary studies showed that this shorter
duration was sufficient for a fairly close evaluation of the cost metric surfaces); similar graphs
could be shown for J
d
(q q q) and J
v
(q q q). All slices are locally convex, each with a minimum at nearly
the same location as found via fminsearch with t
f
= 10000 s: q q q

[0:64 1:10 0:98 0:43]
T
.
Figure 4.6 depicts the resulting control force slices when the displacement vector q= 0 and when
q=[0 1 cm]
T
, respectively, as functions of ˙ q
1
and ˙ q
2
for minimizing the absolute acceleration
cost metric when the damper is in the first story. Comparing Figures 4.3 and 4.6, the region in
which the OCLC force is clipped is 71% smaller than that for the CLQR force for both the q= 0
and q=[0 1 cm]
T
cases.
4.3.1 Optimization results
The OCLC, CLQR and optimal passive linear viscous strategies are determined for each of the
three cost metrics for a GWN excitation. The optimizations here use simulations with t
f
= 1000 s
65
minimum of surface
0 5 10
3
0
5
10
4
0
5
10
15
minimum of surface
0 5 10
2
0
5
10
4
0
5
10
15
minimum of surface
0 5 10
2
0
5
10
3
0
5
10
15
minimum of surface
0 5 10
1
0
5
10
4
0
5
10
15
minimum of surface
0 5 10
1
0
5
10
3
0
5
10
15
minimum of surface
0 5 10
1
0
5
10
2
0
5
10
15
Figure 4.5. Slices of the CLC absolute acceleration cost metric J
a
(q q q) when the damper is in the
first story of a 2DOF structure for GWN with t
f
= 1000 s.
Table 4.2: OCLCq q q

designed by GWN with t
f
= 1000s for different cost metrics with the damping
device in the first or second story of a 2DOF structure.
Control Design Cost Damper in the Damper in the
Strategy Excitation Metric first story second story
OCLC samp. GWN Drift q q q

=[0:66 0:99 1:16 0:16]
T
q q q

=[1:44 0:08 1:08 0:04]
T
OCLC samp. GWN Veloc. q q q

=[ 0:08 0:13 0:22 0:13]
T
q q q

=[ 0:92 1:21 0:92 0:90]
T
OCLC samp. GWN Abs. accel. q q q

=[ 0:63 1:08 0:94 0:43]
T
q q q

=[ 0:54 6:60 0:85 0:40]
T
and, for the passive viscous damping only, computed exactly via a Lyapunov solution. Table 4.2
lists the values of OCLC optimal parameter vector q q q

for different cost metrics for the 2DOF
system when the damper is in the first or second story.
The response metrics for a first-story (i= 1) damper are reported in Table 4.3 for the three
performance metrics; Table 4.4 shows the corresponding results for a second story (i= 2) damper.
Bolded values are the response metrics that are intended to be minimized in each strategy; the red
italic values are the maximum improvements in reducing different metrics that OCLC can achieve
compared to CLQR and the passive viscous strategies throughout that column of the table.
66
10
10
0
control force u [%m
1
g]
20
0
velocity q
.
2
[cm/s]
0
velocity q
.
1
[cm/s]
(a) u
OCLC
([0 0 ˙ q
1
˙ q
2
]
T
)
10
10
0
control force u [%m
1
g]
20
0
velocity q
.
2
[cm/s]
0
velocity q
.
1
[cm/s]
(b) u
OCLC
([0 1 cm ˙ q
1
˙ q
2
]
T
)
Figure 4.6. Two “slices” of the OCLC control force as functions of the velocity states when q= 0
and q=[0 1 cm]
T
when minimizing the absolute acceleration cost metric with the damping
device in the first story of a 2DOF structure.
When the damper is in the first story, OCLC has the smallest value for all three cost metrics.
Specifically, OCLC provides moderate improvements to the drift and velocity metrics, but sig-
nificant performance improvements for the absolute acceleration metric, even though CLQR uses
a mean square force level that is 20–24% larger than that of OCLC. For drift reduction, OCLC
performs comparably to the passive linear viscous damper for reducing the first-story drift, but
OCLC is able to simultaneously significantly reduce the second-story drift as well, though does so
with larger first-floor accelerations (due to the larger damping force switching on and off). Linear
passive damping has drift and velocity metrics 32%–74% larger than those of OCLC and 33%–
42% for absolute acceleration metric, though the mean square damping forces are smaller (note,
however, that the passive damper, even with larger damping coefficient, cannot match the OCLC
performance). Notably, when minimizing absolute accelerations, CLQR’s mean square accelera-
tions and drifts are two to two-and-a-half times larger than those with OCLC; further, CLQR is
even worse than the passive strategy.
When the damper is in the second story, the OCLC cost minimization performance is also the
best among all strategies. When reducing drifts, OCLC performs comparable to the passive linear
viscous damper, for reducing both first-story and second-story drifts, though does so with larger
first-floor and second-floor accelerations; CLQR’s cost metric and first-floor mean square absolute
acceleration are 32% larger than and four times that of OCLC, respectively, with the mean square
67
damping force four times that of OCLC. Further, when the cost metric is to minimize mean square
velocity, CLQR uses a mean square damping force that is 58% larger than that of OCLC though has
a cost metric just slightly worse, and the absolute accelerations are about 60% larger than those of
OCLC. When minimizing absolute accelerations, CLQR has a cost value about four times that of
OCLC, and the CLQR drifts, velocities and accelerations, are two to three-and-a-half times larger
than OCLC; the passive control strategy, with damping force levels below that of OCLC, does not
provide performance even close to that of OCLC but is clearly superior to the CLQR strategy.
4.3.2 Observations
In all, OCLC outperforms CLQR and an optimal passive linear viscous damping for this 2DOF
system. The damper is recommended to be installed in the first story, evidenced by much smaller
cost metric values and control force levels, especially when the minimization objective is an abso-
lute acceleration metric. Further, OCLC designed to reduce the absolute acceleration may be the
best for this system because this OCLC provides very good reductions in all three metrics: drifts,
velocities and absolute accelerations, while using moderate control force levels. Some additional
observations are as follows:
1. A preliminary study found that a linear active control designed for minimizing the absolute
acceleration cost metric exhibits better performance than OCLC, though with a much higher
force level compared to OCLC.
2. When the damper is in the first story and minimizing J
a
, CLQR clips about half of the time; in
contrast, OCLC is optimized for the nonlinear system with consideration of the controllable
damper and is clipped a quarter of the time.
3. The linear passive control performs well by increasing the modal damping of the system, but
still underperforms the OCLC in minimizing cost functions.
4. OCLC can decrease drifts, velocities and absolute accelerations simultaneously when mini-
mizing J
a
for the 2DOF structure with a damper in the first or second story;
68
Table 4.3: Optimization results for different control strategies and cost metrics with the damping device in the first story of a 2DOF
structure excited by GWN with t
f
= 1000 s. Note that theD columns denote percent change relative to OCLC; positive numbers mean
improvements in OCLC.
Control Design Eval. Cost J
()
(D) E[u
2
] (D) E[q
2
1
] (D) E[(q
2
q
1
)
2
] (D) E[ ˙ q
2
1
] (D) E[ ˙ q
2
2
] (D) E[ ¨ q
a
1
2
] (D) E[ ¨ q
a
2
2
] (D)
Strategy GWN GWN Metric [%] [(%m
1
g)
2
] [%] [cm
2
] [%] [cm
2
] [%] [cm
2
/s
2
] [%] [cm
2
/s
2
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%]
Unctrld. – samp. – – – – – 5.868 – 2.347 – 367.636 – 926.478 – 2.879 – 5.853 –
Unctrld. – ideal – – – – – 5.629 – 2.261 – 355.556 – 890.314 – 2.830 – 5.637 –
OCLC samp. samp. Drift 0.401 248.6 0.126 0.251 10.002 48.022 2.163 0.627
Passive ideal samp. Drift 0.529 (32) 105.5 (–58) 0.187 (48) 0.332 (32) 10.351 (3) 68.767 (43) 0.527 (–76) 0.829 (32)
Passive ideal ideal Drift 0.546 (36) 125.7 (–49) 0.195 (55) 0.338 (35) 12.327 (23) 71.914 (50) 0.746 (–66) 0.844 (35)
CLQR ideal samp. Drift 0.409 (2) 242.1 (–3) 0.114 (–10) 0.271 (8) 11.819 (18) 48.387 (1) 2.218 (3) 0.676 (8)
OCLC samp. samp. Veloc. 48.27 1277.0 0.346 0.201 11.208 36.941 11.654 0.502
Passive ideal samp. Veloc. 79.11 (64) 108.0 (–92) 0.182 (–47) 0.337 (68) 10.072 (–10) 69.030 (87) 0.532 (–95) 0.840 (67)
Passive ideal ideal Veloc. 84.23 (74) 129.0 (–90) 0.191 (–45) 0.343 (71) 12.033 (7) 72.183 (95) 0.760 (–93) 0.855 (70)
CLQR ideal samp. Veloc. 48.50 (0.4) 1580.2 (24) 0.362 (5) 0.214 (6) 10.509 (–6) 37.840 (2) 14.453 (24) 0.533 (6)
OCLC samp. samp. Abs. Accel. 0.866 85.8 0.708 0.190 26.841 68.510 0.392 0.474
Passive ideal samp. Abs. Accel. 1.152 (33) 50.0 (–42) 0.400 (–44) 0.283 (49) 23.624 (–16) 81.298 (–12) 0.448 (14) 0.705 (49)
Passive ideal ideal Abs. Accel. 1.231 (42) 55.0 (–36) 0.411 (–42) 0.288 (52) 25.990 (–3) 84.466 (23) 0.513 (31) 0.718 (51)
CLQR ideal samp. Abs. Accel. 1.774 (105) 103.3 (20) 1.713 (142) 0.370 (95) 78.515 (193) 161.925 (136) 0.850 (117) 0.923 (95)
Table 4.4: Optimization results for different control strategies and cost metrics with the damping device in the second story of a 2DOF
structure excited by GWN with t
f
= 1000 s. Note that theD columns denote percent change relative to OCLC; positive numbers mean
improvements in OCLC.
Control Design Eval. Cost J
()
(D) E[u
2
] (D) E[q
2
1
] (D) E[(q
2
q
1
)
2
] (D) E[ ˙ q
2
1
] (D) E[ ˙ q
2
2
] (D) E[ ¨ q
a
1
2
] (D) E[ ¨ q
a
2
2
] (D)
Strategy GWN GWN Metric [%] [(%m
1
g)
2
] [%] [cm
2
] [%] [cm
2
] [%] [cm
2
/s
2
] [%] [cm
2
/s
2
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%]
Unctrld. – samp. – – – – – 5.868 – 2.347 – 367.636 – 926.478 – 2.879 – 5.853 –
Unctrld. – ideal – – – – – 5.629 – 2.261 – 355.556 – 890.314 – 2.830 – 5.637 –
OCLC samp. samp. Drift 1.228 218.2 0.855 0.352 74.692 76.927 2.094 1.789
Passive ideal samp. Drift 1.265 (3) 98.8 (–55) 0.789 (–8) 0.467 (32) 76.005 (2) 71.752 (–7) 1.145 (–45) 0.615 (–66)
Passive ideal ideal Drift 1.288 (5) 102.5 (–53) 0.804 (–6) 0.474 (35) 79.558 (7) 74.439 (–3) 1.198 (–43) 0.656 (–63)
CLQR ideal samp. Drift 1.621 (32) 827.3 (279) 1.070 (25) 0.471 (34) 121.078 (62) 51.346 (–33) 8.146 (289) 6.930 (287)
OCLC samp. samp. Veloc. 129.5 2665.8 0.679 0.830 68.490 60.740 24.847 24.300
Passive ideal samp. Veloc. 136.7 (6) 148.0 (–94) 0.784 (15) 0.558 (–33) 90.235 (32) 46.459 (–24) 1.532 (–94) 0.536 (–98)
Passive ideal ideal Veloc. 143.4 (11) 156.0 (–94) 0.802 (18) 0.568 (–32) 94.379 (38) 48.979 (–19) 1.630 (–93) 0.615 (–97)
CLQR ideal samp. Veloc. 137.2 (6) 4211.3 (58) 0.663 (–2) 0.910 (10) 57.408 (–16) 79.390 (31) 39.512 (59) 39.282 (62)
OCLC samp. samp. Abs. Accel. 1.255 103.0 0.823 0.581 60.981 103.515 0.595 0.660
Passive ideal samp. Abs. Accel. 1.705 (36) 70.0 (–32) 0.878 (7) 0.449 (–23) 73.112 (20) 100.114 (–3) 0.960 (61) 0.745 (13)
Passive ideal ideal Abs. Accel. 1.761 (40) 71.8 (–30) 0.890 (8) 0.454 (–22) 76.224 (25) 102.636 (–1) 0.992 (67) 0.769 (17)
CLQR ideal samp. Abs. Accel. 4.548 (263) 150.7 (46) 2.196 (167) 2.339 (303) 127.353 (109) 450.794 (335) 1.576 (165) 2.972 (350)
69
5. Smaller damper force weighting coefficientr
()
values may provide even further reductions
in structural responses, but they will not improve significantly when minimizing velocity and
drift cost metrics using OCLC because neither OCLC nor CLQR are clipped as frequently
as when minimizing absolute acceleration.
6. There is a practical limitation on the magnitude of the control force since the control design
depends on the details of structural assignments; thus, control weight r
()
values should be
decided in accordance with specific structural characteristics for practical applications.
7. The number of parameters to be optimized in the proposed OCLC strategy is the same as
the number of states, which means OCLC may be computationally expensive to design for
large-scale structural systems; hence, further research is required to facilitate application to
more complex structures.
4.4 Optimal active control for a 2DOF system
The proposed OCLC strategy optimizes a system given a particular external excitation, i.e., the
optimization result differs across different excitations for the same structure. An immediate reac-
tion may be concern that the external excitation is in fact unknown a priori and, thus, selecting
one typical optimization result for one design excitation may not be the “best”. However, it must
be emphasized that traditional LQR active control also optimizes responses to an implicit under-
lying excitation — it assumes the external excitation is ideal Gaussian white noise (GWN); this
motivates a question: is the LQR actually the optimal active control for a system subjected to an
excitation other than GWN?
To answer this question, consider using the same optimization approach proposed herein to
evaluate if there exists an active linear state feedback gain, other than the LQR gain, that outper-
forms the theoretical LQR. Consider again the 2DOF structure with the actuator in the first story
as used in Section 4.1. The optimal active control gain minimizes a cost metric J
()
by choosing a
70
control gain K
opt
, that may differ from the LQR control gain K
LQR
. Using the same parameteriza-
tionq q q as in Eq. (4.8),q
1
=q
2
=q
3
=q
4
= 0 provides the LQR solution, andq
1
=q
2
=q
3
=q
4
= 1
is the uncontrolled structure. MATLAB’sfminsearch is used to minimize each of the cost metrics
in Eq. (4.3), with a penalty function chosen to ensure that the real part of each eigenvalue of the
optimal closed-loop active system is negative, i.e., the linear feedback system is stable. An optimal
active control gain is designed for the 1940 El Centro earthquake for each cost metric. Table 4.5
lists the values of the optimal active control parameter vectorq q q

for different cost metrics for the
2DOF system when the damper is in the first story.
Table 4.5: Optimal active control q q q

designed to reduce various cost metric responses to the El
Centro earthquake when the damping device is in the first story of a 2DOF structure.
Control Design Cost Damper in the
Strategy Excitation Metric first story
Optimal active El Centro Drift q q q

=[69:9647:85 1:29 183:84]
T
Optimal active El Centro Veloc. q q q

=[ 0:75 0:77 0:21 2:68]
T
Optimal active El Centro Abs. Accel. q q q

=[ 0:29 0:80 12:21 11:76]
T
LQR ideal GWN Any q q q

=[ 0 0 0 0 ]
T
All six control strategies (for each of three cost metrics, both an LQR and an optimal active linear
gain) are then evaluated with both the 1940 El Centro and 1995 Kobe earthquakes. The results are
summarized and compared in Table 4.6.
The optimal active control designed to reduce absolute accelerations in the 1940 El Centro
outperforms LQR when the external excitation is either 1940 El Centro or 1995 Kobe, decreas-
ing the absolute acceleration cost metric by 85% and 98%, respectively, when subjected to the
El Centro and Kobe excitations, though with a larger control force leVeloc. Further, the optimal
active control designed to minimize inter-story drift and velocity cost metrics provides improve-
ments, though more modestly, over LQR. It is worth mentioning that the resulting optimal active
controller might be not practical because the control force might be too large to apply, e.g., when
reducing the drift cost metric, the control force of optimal active is almost twenty times that of
71
Table 4.6: Comparisons of LQR and optimal active designed with 1940 El Centro for different
cost metrics with the damping device in the first story of a 2DOF structure excited by various
excitations. Note that the D columns denote percent change relative to the corresponding LQR;
positive numbers mean improvements in the optimal active control.
Evaluation Excitation
1940 El Centro 1995 Kobe
Control Design Cost J
()
(D) J
()
(D)
Strategy Excitation Metric [%] [%]
LQR ideal GWN Drift 0.220 0.163
Optimal active El Centro Drift 0.145 (34) 0.115 (29)
LQR ideal GWN Veloc. 19.668 16.173
Optimal active El Centro Veloc. 17.058 (13) 13.816 (15)
LQR ideal GWN Abs. Accel. 0.002 0.001
Optimal active El Centro Abs. Accel. 2.89910
4
(85) 2.75110
5
(98)
LQR when evaluated by El Centro; however, if that is a concern, then the cost function weights
were improperly chosen from the start. The point of this analysis is to see whether LQR is optimal
or at least near-optimal for the chosen cost metric; clearly, LQR is not optimal, particularly for
reducing the absolute acceleration cost metric.
In conclusion, LQR is convenient to apply, as it can be designed analytically, but is not the
“best” active control strategy if the excitation is anything other than ideal GWN (even a finite-
duration discrete-time GWN realization). There does indeed exist an optimal active control gain
that outperforms LQR when accounting for the excitation. Thus, using a design excitation to
determine an OCLC, as proposed herein, is exactly what would be done to truly evaluate and
design an optimal active system.
4.5 Summary
In this chapter, OCLC applied to a 2DOF model, excited by a GWN excitation, was determined
by optimization over a four-dimensional parameter space. The number of control gain parameters
doubles the order of the system, leading to a four-dimensional optimization problem, which is
difficult to illustrate graphically. Therefore, the “slices” of the cost function multidimensional
surface were depicted and the global minimum was found. The optimization results were compared
to those of CLQR as well as those of an optimal passive viscous damping strategy. The results
72
demonstrated that OCLC performs very well when reducing the absolute acceleration response,
and still superior but more modestly reducing the displacement or velocity response metrics. The
location of the controllable damper also impacts the optimization results, since OCLC gains are
different, leading to different performance in reducing structural responses with the damper in
different stories. Finally, the evaluation of an optimal active control for the 2DOF system suggested
that LQR is not the “best” active control strategy if the excitation is anything other than ideal GWN,
thus, using a design excitation to determine an OCLC is similar to the procedure one might use to
design an optimal active control strategy.
73
Chapter 5
Robustness of OCLC Strategies for SDOF and 2DOF Systems
The models used in previous chapters are deterministic, but there are always uncertainties in the
real world [99]. A robustness analysis of these controllable damping strategies against uncertain-
ties in the structural system parameters is necessary to validate that the OCLC design maintains
robustness as well as optimality. In this chapter, the robustness to errors in both the SDOF and
2DOF structure models is analyzed by evaluating a performance improvement ratio for some re-
sponse metric p:
Improvement Ratio r=
p
CLQR
p
OCLC
p
OCLC
100% (5.1)
where positive r means OCLC is more effective than CLQR. Both CLQR and OCLC control
strategies are designed for the nominal structure, with parameters as given in the previous chapters,
but then applied to a structure model with a different stiffness and/or damping.
5.1 Robustness analysis for the SDOF system
A structure with nominal system parameters (m
0
, k
0
and c
0
), as shown in Figure 5.1, is used to
design the control system to minimize the mean square absolute acceleration Eq. (3.4). For the
SDOF system used herein, the same parameters used in Chapter 3 are considered to represent
the nominal system. Two different cases with perturbed parameters are evaluated herein: one
with structure stiffness perturbed by Dk, as shown in Figure 5.2 and the other with its damping
74
coefficient perturbed byDc, as shown in Figure 5.3. The magnitude of perturbation is in the range
of20% for stiffness coefficient perturbations and40% for the damping coefficient perturbations
(because damping is generally more uncertain than stiffness).
m
0
Damper
Fixed Base
u
q
Ground Motion
k
0
, c
0
Figure 5.1. Nominal SDOF system.
m
0
Damper
Fixed Base
u
q
Ground Motion
k
0
+Dk, c
0
Figure 5.2. SDOF system with perturbed stiffness.
The perturbed systems are used to simulate the responses using the control designed for the
nominal system, computing both mean square and peak responses to a Gaussian white noise exci-
tation with intensity D= 0:02 m
2
=s
4
through final time t
f
= 10000 s. Tables 5.1 and 5.2 tabulate
the response statistics for systems with perturbed stiffness controlled by the nominal OCLC and
CLQR, respectively. The corresponding improvement ratios are graphed as functions of the stiff-
ness perturbation level in Figures 5.4a and 5.4b for mean square and peak responses, respectively.
All improvement ratios are positive, indicating that OCLC is more efficient than CLQR even for
systems with perturbed stiffness; the improvement ratio is larger for mean square response im-
provement than for peak response improvement (which is expected since mean square is indeed
75
m
0
Damper
Fixed Base
u
q
Ground Motion
k
0
, c
0
+Dc
Figure 5.3. SDOF system with perturbed damping.
Table 5.1: Responses of the SDOF system with perturbed stiffness using OCLC.
Dk/k (%) –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
J
a
(E[ ¨ q
a2
]) [m
2
/s
4
] 0.088 0.091 0.093 0.096 0.098 0.101 0.103 0.105 0.108
E[ ˙ q
2
] [cm
2
/s
2
] 18.649 18.575 18.500 18.423 18.354 18.280 18.206 18.132 18.060
E[q
2
] [cm
2
] 1.144 1.067 1.000 0.940 0.887 0.839 0.795 0.755 0.719
E[u
2
] [(%mg)
2
] 7.684 7.873 8.062 8.243 8.423 8.597 8.772 8.944 9.116
¨ q
a
max
[cm/s
2
] 147.780 153.950 159.494 164.889 169.892 175.589 180.806 186.191 192.320
˙ q
max
[cm/s] 20.154 19.998 19.758 19.587 19.484 19.357 19.261 19.277 19.252
q
max
[cm] 4.619 4.532 4.438 4.367 4.289 4.218 4.152 4.093 4.035
u
max
[%mg] 12.68 12.88 13.18 13.50 13.76 14.02 14.31 14.60 14.90
a squared quantity). Notably, as indicated in Figure 5.4a, OCLC’s improvement in J
a
is insensi-
tive to the stiffness perturbation but mean square velocity, displacement and control force change
with stiffness perturbation, nevertheless, OCLC always exhibits excellent improvement relative to
CLQR. The improvement in peak response is relatively insensitive to the stiffness errors.
The response statistics for systems with perturbed damping are summarized in Tables 5.3 and
5.4. Figures 5.5a and 5.5b present the corresponding improvement ratios as functions of the damp-
ing perturbation. Even though using a wider range of perturbation compared to stiffness, OCLC
can still provide significant improvement in both mean square and peak responses; as expected,
the improvement ratio is larger for mean square response improvement than for peak response im-
provement. Again, the improvement in J
a
is relatively insensitive to the damping errors, though the
OCLC mean square and peak structural response performance gain over CLQR decreases modestly
when the actual damping is smaller than nominal. In summary, OCLC is relatively insensitive to
the changes of damping. Hence, OCLC satisfies one measure of robustness for this SDOF system.
76
Table 5.2: Responses of the SDOF system with perturbed stiffness using CLQR.
Dk/k (%) –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
J
a
(E[ ¨ q
a2
]) [m
2
/s
4
] 0.127 0.132 0.136 0.140 0.144 0.148 0.152 0.156 0.160
E[ ˙ q
2
] [cm
2
/s
2
] 35.732 34.805 33.962 33.146 32.414 31.691 31.004 30.390 29.815
E[q
2
] [cm
2
] 2.656 2.444 2.263 2.099 1.959 1.831 1.716 1.615 1.523
E[u
2
] [(%mg)
2
] 14.50 15.08 15.72 16.30 16.88 17.42 17.98 18.52 19.04
¨ q
a
max
[cm/s
2
] 196.663 201.250 212.723 222.443 230.372 237.199 243.427 247.100 250.914
˙ q
max
[cm/s] 30.126 28.480 28.447 28.330 28.214 28.210 28.134 27.598 27.300
q
max
[cm] 6.189 5.976 5.942 5.876 5.783 5.685 5.588 5.419 5.286
u
max
[%mg] 19.86 20.23 21.30 22.43 23.18 23.79 24.35 25.06 25.17
Table 5.3: Responses of the SDOF system with perturbed damping using OCLC.
Dc/c (%) –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
J
a
(E[ ¨ q
a2
]) [m
2
/s
4
] 0.106 0.102 0.100 0.098 0.098 0.099 0.100 0.101 0.103
E[ ˙ q
2
] [cm
2
/s
2
] 26.912 24.194 21.919 19.992 18.354 16.944 15.724 14.651 13.708
E[q
2
] [cm
2
] 1.223 1.118 1.029 0.952 0.887 0.830 0.781 0.735 0.694
E[u
2
] [(%mg)
2
] 8.123 8.137 8.194 8.289 8.423 8.584 8.778 8.966 9.178
¨ q
a
max
[cm/s
2
] 217.987 203.588 190.735 179.298 169.892 161.884 154.668 148.269 142.728
˙ q
max
[cm/s] 24.494 22.875 21.682 20.543 19.484 18.733 18.340 17.927 17.542
q
max
[cm] 5.514 5.146 4.819 4.530 4.289 4.081 3.895 3.730 3.588
u
max
[%mg] 15.70 15.07 14.54 14.10 13.76 13.48 13.24 13.39 13.55
Table 5.4: Responses of the SDOF system with perturbed damping using CLQR.
Dc/c (%) –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
J
a
(E[ ¨ q
a2
]) [m
2
/s
4
] 0.147 0.145 0.146 0.145 0.144 0.143 0.143 0.142 0.141
E[ ˙ q
2
] [cm
2
/s
2
] 33.576 33.231 32.990 32.696 32.414 32.117 31.836 31.544 31.249
E[q
2
] [cm
2
] 2.090 2.047 2.024 1.990 1.959 1.926 1.896 1.865 1.834
E[u
2
] [(%mg)
2
] 18.63 18.14 17.75 17.31 16.88 16.47 16.08 15.70 15.30
¨ q
a
max
[cm/s
2
] 235.169 233.675 232.890 231.476 230.372 229.545 227.729 225.795 224.533
˙ q
max
[cm/s] 29.118 28.751 28.573 28.389 28.214 28.085 27.923 27.660 27.620
q
max
[cm] 5.944 5.893 5.861 5.820 5.783 5.754 5.702 5.653 5.616
u
max
[%mg] 23.88 23.66 23.51 23.35 23.18 23.03 22.78 22.61 22.38
77
0 10 20
Δ k/k [%]
0
20
40
60
80
100
120
140
160
180
200
Improvement [%]
J
MS Vel.
MS Disp.
MS Control Force
(a) Improvement in mean square responses
0 10 20
Δ k/k [%]
0
20
40
60
80
100
120
Improvement [%]
Peak Abs. Accel.
Peak Vel.
Peak Disp.
Peak Control Force
(b) Improvement in peak responses
Figure 5.4. Improvement in mean square and peak responses for the SDOF system with perturbed
stiffness subjected to GWN.
0 20 40
Δ c/c [%]
0
20
40
60
80
100
120
140
160
180
200
Improvement [%]
J
MS Vel.
MS Disp.
MS Control Force
(a) Improvement in mean square responses
0 20 40
Δ c/c [%]
0
20
40
60
80
100
120
Improvement [%]
Peak Abs. Accel.
Peak Vel.
Peak Disp.
Peak Control Force
(b) Improvement in peak responses
Figure 5.5. Improvement in mean square and peak responses for the SDOF system with perturbed
damping subjected to GWN.
78
m
1
m
2
Fixed Base
Damper
q
1
k
1
, c
1
q
2
k
2
, c
2
u
Ground Motion
Figure 5.6. Nominal 2DOF system.
79
5.2 Robustness analysis for the 2DOF system
For brevity, perturbations to the 2DOF system are only presented here when the damper in the
first story, as shown in Figure 5.6, is designed to reduce the absolute acceleration cost metric J
a
in Eq. (4.3). Two different perturbed systems are examined: one with both k
1
and k
2
perturbed by
Dk (since k
1
= k
2
), as shown in Figure 5.7; one with c
1
perturbed byDc, as shown in Figure 5.8.
The magnitudes of perturbations are identical to those for the SDOF system. The excitation is a
Gaussian white noise process with intensity D= 0:04 m
2
=s
4
and t
f
= 1000 s.
m
1
m
2
Fixed Base
Damper
q
1
k
1
+Dk, c
1
q
2
k
2
+Dk, c
2
u
Ground Motion
Figure 5.7. 2DOF system with perturbed stiffness.
Tables 5.5 and 5.6 display the response statistics for systems with perturbed stiffness for OCLC
and CLQR, respectively. Figure 5.9 shows the improvement ratios for both mean square and peak
responses when the structural stiffness is perturbed. All responses exhibit positive improvement ra-
tios over the range of perturbations studied. It is worth noting that all mean square responses using
CLQR are roughly double (or more) those using OCLC, even though CLQR uses a mean square
control force that is about 20% larger even when the system is perturbed. Thus, OCLC maintains
significant mean square improvements even with significant perturbations of stiffness. Notably, the
improvement curves for peak responses show more variation compared to the flat pattern for mean
80
m
1
m
2
Fixed Base
Damper
q
1
k
1
, c
1
+Dc
q
2
k
2
, c
2
u
Ground Motion
Figure 5.8. 2DOF system with perturbed damping.
Table 5.5: Responses of the 2DOF system with perturbed stiffness using OCLC.
Dk/k (%) –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
J
a
[m
2
/s
4
] 0.777 0.801 0.823 0.845 0.866 0.887 0.907 0.926 0.945
E[ ¨ q
a
1
2
]) [m
2
/s
4
] 0.354 0.364 0.373 0.383 0.392 0.401 0.409 0.417 0.425
E[ ¨ q
a
2
2
]) [m
2
/s
4
] 0.423 0.437 0.449 0.462 0.474 0.486 0.497 0.509 0.520
E[q
2
1
] [cm
2
] 0.987 0.901 0.827 0.764 0.708 0.659 0.615 0.576 0.541
E[q
2
2
] [cm
2
] 1.935 1.767 1.621 1.496 1.386 1.289 1.203 1.125 1.056
E[u
2
] [(%m
1
g)
2
] 75.408 78.039 80.653 83.278 85.829 88.331 90.741 93.097 95.380
( ¨ q
a
1
)
max
[cm/s
2
] 3.174 3.136 3.586 3.883 4.129 3.654 3.334 3.356 3.384
( ¨ q
a
2
)
max
[cm/s
2
] 2.641 2.676 2.722 2.756 2.791 2.828 2.858 2.882 2.922
q
1 max
[cm] 0.038 0.037 0.036 0.035 0.034 0.033 0.032 0.031 0.029
q
2 max
[cm] 0.058 0.055 0.053 0.051 0.049 0.048 0.046 0.044 0.043
u
max
[%m
1
g] 38.698 39.447 41.454 43.075 44.069 44.564 44.573 44.603 44.856
square responses, especially for the absolute acceleration of the first story, showing that the stiff-
ness error has more influence on the peak response improvement; nevertheless, OCLC provides
comparable or much better peak response improvements for the perturbed system compared to the
nominal system.
Tables 5.7 and 5.8 list the response statistics for systems with perturbed damping. Figure 5.10
displays the corresponding improvement ratios as a function of the damping perturbation. Again,
both mean square and peak OCLC responses have positive improvement ratios over the larger
range of damping perturbations, indicating the superiority of OCLC compared with CLQR. Most
improvement ratios are relatively insensitive to the damping errors, except some modest changes
81
Table 5.6: Responses of the 2DOF system with perturbed stiffness using CLQR.
Dk/k (%) –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
J
a
[m
2
/s
4
] 1.557 1.622 1.677 1.732 1.774 1.799 1.840 1.893 1.959
E[ ¨ q
a
1
2
]) [m
2
/s
4
] 0.746 0.777 0.803 0.829 0.850 0.864 0.882 0.910 0.943
E[ ¨ q
a
2
2
]) [m
2
/s
4
] 0.811 0.845 0.874 0.902 0.923 0.936 0.957 0.983 1.016
E[q
2
1
] [cm
2
] 2.346 2.172 1.996 1.847 1.713 1.582 1.477 1.395 1.326
E[q
2
2
] [cm
2
] 4.653 4.300 3.956 3.665 3.393 3.126 2.914 2.745 2.608
E[u
2
] [(%m
1
g)
2
] 90.452 95.599 97.767 100.097 103.300 106.544 110.462 115.509 119.780
( ¨ q
a
1
)
max
[cm/s
2
] 5.149 5.213 5.121 5.082 5.051 4.990 5.103 5.558 5.414
( ¨ q
a
2
)
max
[cm/s
2
] 4.797 5.045 5.066 5.066 5.044 5.116 5.149 5.050 5.239
q
1 max
[cm] 0.060 0.058 0.055 0.051 0.047 0.044 0.043 0.041 0.042
q
2 max
[cm] 0.096 0.094 0.090 0.083 0.078 0.075 0.071 0.069 0.069
u
max
[%m
1
g] 76.024 79.668 78.472 79.817 79.389 76.447 81.542 76.279 83.444
Table 5.7: Responses of the 2DOF system with perturbed damping using OCLC.
Dc/c (%) –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
J
a
[m
2
/s
4
] 0.863 0.863 0.864 0.865 0.866 0.867 0.868 0.869 0.871
E[ ¨ q
a
1
2
]) [m
2
/s
4
] 0.390 0.390 0.391 0.391 0.392 0.392 0.393 0.393 0.394
E[ ¨ q
a
2
2
]) [m
2
/s
4
] 0.473 0.473 0.473 0.474 0.474 0.475 0.475 0.476 0.477
E[q
2
1
] [cm
2
] 0.699 0.702 0.704 0.706 0.708 0.710 0.712 0.715 0.717
E[q
2
2
] [cm
2
] 1.367 1.372 1.376 1.381 1.386 1.391 1.396 1.401 1.406
E[u
2
] [(%m
1
g)
2
] 90.428 89.271 88.120 86.979 85.829 84.677 83.533 82.398 81.277
( ¨ q
a
1
)
max
[cm/s
2
] 4.080 3.320 3.316 3.318 4.129 3.320 3.318 3.318 3.315
( ¨ q
a
2
)
max
[cm/s
2
] 2.726 2.741 2.758 2.775 2.791 2.807 2.824 2.839 2.855
q
1 max
[cm] 0.033 0.034 0.034 0.034 0.034 0.034 0.034 0.034 0.034
q
2 max
[cm] 0.049 0.049 0.049 0.049 0.049 0.050 0.050 0.050 0.050
u
max
[%m
1
g] 44.893 44.660 44.475 44.297 44.069 43.863 43.685 43.491 43.282
in peak absolute acceleration and control force in the first story (but both always very positive).
Therefore, OCLC is insensitive to the changes of damping and, in all, OCLC satisfies this test of
robustness to the changes of structural parameters for this 2DOF model.
5.3 Summary
This chapter conducted a robustness analysis of OCLC for both SDOF and 2DOF systems, which
may be summarized with the following:
1. OCLC is more efficient than CLQR for both SDOF and 2DOF systems even with perturbed
stiffness and damping factors;
82
Table 5.8: Responses of the 2DOF system with perturbed damping using CLQR.
Dc/c (%) –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
J
a
[m
2
/s
4
] 1.830 1.815 1.803 1.789 1.774 1.759 1.744 1.730 1.715
E[ ¨ q
a
1
2
]) [m
2
/s
4
] 0.892 0.881 0.871 0.861 0.850 0.840 0.830 0.820 0.810
E[ ¨ q
a
2
2
]) [m
2
/s
4
] 0.939 0.935 0.931 0.928 0.923 0.919 0.914 0.910 0.906
E[q
2
1
] [cm
2
] 1.802 1.779 1.758 1.736 1.713 1.691 1.668 1.647 1.625
E[q
2
2
] [cm
2
] 3.529 3.493 3.462 3.429 3.393 3.358 3.324 3.289 3.255
E[u
2
] [(%m
1
g)
2
] 114.771 111.774 108.919 106.081 103.300 100.611 98.075 95.592 93.125
( ¨ q
a
1
)
max
[cm/s
2
] 5.293 5.233 5.170 5.299 5.051 5.248 4.967 4.948 4.911
( ¨ q
a
2
)
max
[cm/s
2
] 5.163 5.125 5.104 5.072 5.044 5.022 5.000 4.979 4.956
q
1 max
[cm] 0.049 0.049 0.048 0.048 0.047 0.047 0.047 0.046 0.046
q
2 max
[cm] 0.080 0.079 0.079 0.079 0.078 0.078 0.077 0.077 0.076
u
max
[%m
1
g] 83.737 75.418 81.643 80.534 79.389 78.495 69.822 82.736 81.430
0 10 20
k/k [%]
0
20
40
60
80
100
120
140
160
Improvement [%]
J
a
MS Abs. Accel. 1
MS Abs. Accel. 2
MS Disp. 1
MS Disp. 2
MS Control Force
(a) Improvement in mean square responses
0 10 20
k/k [%]
0
20
40
60
80
100
120
140
160
Improvement [%]
Peak Abs. Accel. 1
Peak Abs. Accel. 2
Peak Disp. 1
Peak Disp. 2
Peak Control Force
(b) Improvement in peak responses
Figure 5.9. Improvement in mean square and peak responses for the 2DOF system (damper in first
story) with perturbed stiffness subjected to GWN.
83
0 20 40
c/c [%]
0
20
40
60
80
100
120
140
160
Improvement [%]
J
a
MS Abs. Accel. 1
MS Abs. Accel. 2
MS Disp. 1
MS Disp. 2
MS Control Force
(a) Improvement in mean square responses
0 20 40
c/c [%]
0
20
40
60
80
100
120
140
160
Improvement [%]
Peak Abs. Accel. 1
Peak Abs. Accel. 2
Peak Disp. 1
Peak Disp. 2
Peak Control Force
(b) Improvement in peak responses
Figure 5.10. Improvement in mean square and peak responses for the 2DOF system (damper in
first story) with perturbed damping subjected to GWN.
84
2. The improvements provided by the OCLC, relative to CLQR, do not degrade much at all
when the structure parameters are perturbed from that used to design the OCLC strategy.
Therefore, OCLC satisfies this test of robustness to the changes of structural parameters for both
models.
85
Chapter 6
Experimental Verification of Optimal Clipped Linear Strategies
through Real-time Hybrid Simulation
The previous chapters introduced the novel OCLC strategy and applied it in simulation to different
structural systems with an ideal damper model. This chapter explores the effectiveness of OCLC
for different structural systems using a physical magnetorheological (MR) damper through both
numerical simulation and real-time hybrid simulation (RTHS) tests. This chapter is divided into
four sections. Section one introduces the basic concepts of RTHS; section two explains the labo-
ratory components of a series of RTHS experiments performed in May 2019 at the University of
Connecticut (UConn); in section three, numerical simulation and RTHS tests are discussed for var-
ious structural systems subjected to multiple excitations using both CLQR and OCLC approaches;
the performance of OCLC designed for one excitation is also evaluated when the structure is sub-
jected to different excitations; section four summarizes this chapter.
6.1 Real-time hybrid simulation (RTHS)
Experimental testing is important to further illustrate the superiority of OCLC control approaches
for physical controllable dampers. Considering cost, scale, technology, convenience and sustain-
ability, Hakuno et al. [38] proposed a seismic experimental method in which a substructure is
86
excited by an actuator and the vibration response of the rest of the structure is calculated on a com-
puter. This experimental method has been developed and improved via various research and now is
called real-time hybrid simulation (RTHS), during which physical experiments of large-scale struc-
tural components assemblies are subjected to real-time loading while the remaining parts (typically
most) of the structure are simulated in the computer.
Relative to purely physical experiments, RTHS is preferred as it allows for repeatable tests in
an economic and efficient manner. In RTHS, the system is divided into simulated and physical
components. For studies of semiactive control, the physical component is usually a semiactive
controllable device, e.g., an MR fluid damper, and the simulated component is typically the excited
structural model, as shown in Figure 6.1.
Physical component
(Actual MR dampers)
Simulated component
(Remaining structural model)
Actual system
Real-Time
Hybrid Simulation
(RTHS)
Figure 6.1. Schematic of real-time hybrid simulation for semiactive control devices.
RTHS is capable of isolating the most critical component from a semiactive controlled struc-
ture, conducting physical tests on the separated components in real time, fully capturing any rate
dependencies simultaneously. This relatively new testing method is motivated by the recent ad-
vances in computing power, digital signal processing, and hydraulic control [19]. Inspired by the
recent developments of hybrid simulation techniques, RTHS has been implemented by a variety
of projects and experiments, and have been addressed in numerous research studies. For instance,
87
Reinborn et al. [85] conducted RTHS tests to obtain global dynamic response of structural sys-
tems by testing full-scale substructures using shaking tables while simultaneously applying the
boundary of the specimens actively controlled dynamic forces generated by a digitally compen-
sated controller. Christenson et al. [19] executed RTHS experiments for three large-scale MR
fluid dampers with a three-story steel frame structure model, addressed stability issues related
with RTHS and proved the capability to conduct real-time hybrid testing within the Network for
Earthquake Engineering Simulation. To address the performance of RTHS, Carrion et al. [12] pro-
posed a method that implemented a model-based feedforward compensator for the actuator. Shao
et al. [92] also conducted RTHS tests of a three-story structure with loading imposed by shake
tables and auxiliary actuators in real time, proving the versatile implementation of inertial forces
and a force-based substructure. Zapateiro et al. [125] proposed new semiactive control strategies
through the combination of back-stepping and quantitative feedback theory techniques and demon-
strated the effectiveness by applying the strategies to a large-scale three-story building with an MR
damper via RTHS. Cha et al. [15] validated the performance of four semiactive control algorithms
for a large-scale realistic moment-resisting frame using a large-scale 200 kN MR damper via pure
simulations and RTHS. Ashasi-Sorkhabi et al. [4] investigated the dynamic behavior of a struc-
ture with a tuned liquid damper (TLD) via RTHS on a shake table. Miah et al. [75] validated, via
RTHS, a novel semiactive control approach that utilized the unscented Kalman filter for the control
of structures with uncertainties.
In this chapter, the novel structural control strategy OCLC is validated with a variety of struc-
tural models for a physical MR damper via both numerical simulation and RTHS experiments.
6.2 Real-time hybrid simulation (RTHS) setup
Real-time hybrid simulation (RTHS) tests were conducted in May 2019 at the University of Con-
necticut (UConn) in conjunction with Professor Richard Christenson and graduate student Connor
88
Ligeikis. The experimental setup involving the MR damper, an actuator and a force sensor is
shown in Figure 6.2.
Figure 6.2. RTHS physical experiment setup.
6.2.1 Physical component
The physical component of the hybrid simulation is a small-scale Lord Corporation RD-8041-1
MR fluid damper. The damper is 248 mm (9.76 in) in length and has a stroke of 73.9 mm (2.91 in).
Its maximum operating temperature is 71

C (160

F). The maximum control force that can be ex-
erted by the damper is F
max
= 1:3345kN. The damper is controlled with a current-driven command
signal, i.e., a current command is passed as a voltage value to the input of a current driver that gen-
erates a current that is the same as (or very close to) the command. Preliminary study suggested
that the maximum current input that is safe for the MR damper is I
max
= 1 A. The measured coil
resistance of the MR damper is approximately 4:4W.
6.2.2 Simulated structure
The simulated structure is a seismically excited SDOF or 2DOF structure excluding the MR
damper. The model is simulated using SIMULINK [107].
During a RTHS test, first, the structure SIMULINK model is uploaded to a Speedgoat perfor-
mance real-time target machine, which is used to achieve the dynamical responses of the simulated
substructure in real time at a sampling frequency of 1024 Hz using the fourth-order Runge-Kutta
89
solver. The Speedgoat machine provides displacement commands to a Parker Hannifin Corpora-
tion analog controller (Model 23-7030), which uses the displacement feedback measured with a
Micropulse LVDT to control a servo-hydraulic actuator. The actuation system, which is comprised
of a Quincy-Ortman cylinder, is attached to a MOOG servo-valve and has a stroke of 190 mm
(7.5in), maximum speed of 760mm=s (30in=s), maximum force of 9kN (2kips), and a bandwidth
up to 40 Hz. The MR damper is mounted to the actuator at one end and a rigid steel bracket at
the other. The restoring damper force is measured by a PCB Piezotronics, Inc., ICP force sensor
(Model 208C04). A PCB signal conditioner (Model 483C28) provides a 4 mA constant current
to the force sensor. The measured restoring force is fed back into the SIMULINK model on the
Speedgoat machine to complete the RTHS feedback loop.
An Advanced Motion Controls 12A8 Pulse Width Modulation (PWM) servo drive amplifier
with 24-volt power supply (Advanced Motion Controls Model PS2 300W) is used in current
control mode to regulate the damper coil current that is directly measured by a Pico TA018 60A
AC/DC current probe. Furthermore, overlarge temperature changes due to heating caused by en-
ergy dissipation might change the MR fluid viscosity and the associated control force exerted by
the damper; thus, a fan is used to provide cooling and to prevent the damper from overheating dur-
ing RTHS tests; a handheld infrared temperature gun is employed to measure temperatures at the
center of the damper cylinder casing to ensure that the temperature of MR damper will not exceed
its maximum operation temperature. All physical data is collected by a Data Physics SignalCalc
Mobilizer dynamic signal analyzer.
6.3 RTHS tests and numerical simulation for various systems
To evaluate the effectiveness of OCLC, both CLQR and OCLC control strategies are developed and
tested for a variety of structural system models, specifically a series of SDOF and 2DOF structure
models with different structural properties: SDOF and 2DOF building models with various dy-
namic properties, as well as a 2DOF model of an elevated highway bridge deck and a 2DOF model
90
of a base-isolated shear building structure. The performance metrics of different control strategies
are contrasted with each other, and RTHS test results are compared to numerical simulation results
as well.
6.3.1 Cost metric
A serviceability cost metric is defined to minimize a weighted sum of the mean square damper
force (to ensure that the controller will not command impractically large force) and the mean
square structure absolute accelerations
J
a
=E[S
n
i=1
¨ q
a
i
2
+r
a
u
2
] (6.1)
where n is the system order and r
a
is weight on the mean square damper force (r
a
is chosen to
be zero for the SDOF systems, as that optimization is well-posed without the control force, but
nonzero for the 2DOF systems).
6.3.2 MR damper numerical model
The MR damper model used in numerical simulation is a hysteresis model [64], as shown in
Figure 6.3. This relatively simple model utilizes the hyperbolic tangent function to capture the
hysteretic force-velocity behavior of the MR damper. The hysteresis model can be represented by
u= c
MR
˙ x+ k
MR
x+az+ u
0
z= tanhb ˙ x+d sgnx
(6.2)
where x and ˙ x are the damper piston displacement and velocity, respectively, k
MR
and c
MR
refer
to the damper stiffness and viscous damping coefficients, respectively, z is the hysteretic variable
given by the hyperbolic tangent function, a is the scale factor of the height of the hysteresis, b
is the damper velocity scale factor that defines the slope of the hysteretic loop, the product of d
91
and the sign of the displacement determines the width of the hysteresis, and u
0
is the damper force
offset.
displacement
damping force
hysteresis
spring
dashpot
Figure 6.3. Hysteresis MR damper model [64].
The behavior of an MR damper might change over time due to temperature variations, fatigue
effects and changes in the fluid viscosity; thus, the model parameters need to be recalibrated [68].
A series of sinusoidal displacement tests were conducted by Mr. Ligeikis at different amplitudes
and frequencies: the actuator was driven by sinusoidal displacement with a fixed amplitude and
frequency; the current applied to the MR damper was held at a constant level in each case. The
loading conditions and currents are listed in Table 6.1. A constrained optimization was performed
Table 6.1: Sinusoidal excitation and electric current of preliminary tests to calibrate the MR damper
model.
Amplitude [in] Frequency [Hz] Electric current [A]
0:1 5
0, 0:25,:::, 1
0:25 1
0:25 3
0:5 1
0:5 2
using MATLAB’s fmincon to identify the parameters (k
MR
, c
MR
, a, b, d, u
0
) that minimized the
root-mean-square (RMS) error between the measured and simulated damper forces for each of the
test cases. Polynomials were fit to the identified parameters as functions of current using a least-
square approach. The optimal values of all parameters for the updated model are summarized in
Table 6.2, where I is the current measured in the MR damper. Note that both k
MR
and u
0
were
92
not curve fitted; rather, they were set to be zero based on engineering judgement (though there is a
static force offset u
0
for this damper, the piezoelectric force sensor used in the experimental setup
is not possible to measure static forces; thus, u
0
is set to be zero. This judgement is not an issue
in dynamic simulations where offset forces can be eliminated by subtracting mean response). It is
worth noting that the MR numerical damper model cannot completely capture the real damper’s
behavior for certain sinusoidal displacement tests, e.g., when the amplitude is 0.25in and frequency
is 1 Hz, the force-displacement graph of numerical model has a smaller area than that of real
MR damper; further, a model validation study of a series of random displacement and constant
current tests [68] indicated that the RMS error between the measured and simulated damper forces
increases as the applied current increases.
Table 6.2: Optimal parameters for the MR damper model, where I is the current in Amperes [68].
Parameter Units Value
c
MR
lbs s=in 3:8538+ 26:4962I 12:2027I
2
k
MR
lbs=in 0
a lbs s=in 21:1479+ 310:9319I 80:6280I
2
b s=in 2:9432 1:3363I
d – 0:4988+ 0:9001I 0:5338I
2
u
0
lbs 0
6.3.3 Control law for commanded current
The secondary controller commands the MR damper to generate the desired optimal control force
u
d
; hence, the current signal I must be determined appropriately; two algorithms are used herein.
The first algorithm is the bang-bang control proposed by Dyke et al. [24], choosing the command
signal
I= I
max
Hf(u
d
u
m
)u
m
g (6.3)
where I
max
is the maximum current that can be (safely) commanded to the current driver, u
m
is the
MR damper measured control force (for pure numerical simulation, u
m
u, where u is the actual
control force generated by the MR damper), and H() is the Heaviside unit step function.
93
The other secondary control algorithm is the modified bang-bang control law developed by
Yoshida and Dyke [123], which can mitigate the high acceleration effect caused by large changes
in forces when the dominant frequency of the system under control is low. In the modified control
law, the commanded current can be any value between 0 and I
max
, given by
I= I
d
Hf(u
d
u
m
)u
m
g; I
d
=
8
>
<
>
:
mju
d
j; ju
d
j F
max
I
max
; ju
d
j> F
max
(6.4)
where F
max
is the maximum control force that the MR damper can generate and m = I
max
=F
max
is
the coefficient relating current to the force.
In addition, preliminary tests with the MR damper indicated that the coil of MR damper in-
troduces its own dynamics into the system. These dynamics are found to be well approximated
by a second-order time lag in the response of the device to changes in the command input and
are modeled with a second-order low-pass filter from commanded current I to measured current
I
a
. The filter model, estimated by the author using preliminary test data of a Gaussian white noise
(GWN) commanded current and the resulting measured current, is given by the unitless transfer
function
F(s)=
270:5874s+ 436260
s
2
+ 732:2022s+ 434320
(6.5)
Figure 6.4 compares the time histories, power spectral densities (PSD) and frequency responses of
commanded, measured and filtered currents.
The block diagram of the full control algorithm and its components to command the current is
shown in Figure 6.5.
94
22.91 22.915 22.92 22.925 22.93 22.935 22.94 22.945 22.95
time [s]
0
1
2
3
current [A]
commanded
measured
filtered
10 10
0
10
1
10
2
Frequency [Hz]
10
10
0
current PSD [A
2
/Hz]
commanded
measured
10 10
0
10
1
10
2
Frequency [Hz]
10
10
0
10
1
magnitude [1]
TF
filter
10 10
0
10
1
10
2
Frequency [Hz]
0
100
phase [deg]
TF
filter
Figure 6.4. From top to bottom: subfigure one shows time histories of GWN commanded, mea-
sured and the filtered currents; subfigure two indicates the power spectral densities (PSD) of GWN
commanded and measured currents; bottom two figures compares frequency responses of the esti-
mated transfer function from commanded current to measured current, and approximated second-
order filter.
Eq. (6.3) or
Eq. (6.4)
Controller
Primary control filter F MR Damper
desired
control
force u
d
I I
a
states or
measurements
commanded
control force u
commanded control force u
Figure 6.5. Paradigm for semiactive control law using a physical MR damper.
95
6.3.4 RTHS and pure simulation of SDOF systems
For a SDOF system, the ground-relative displacement q(t), excited by base acceleration ¨ q
g
, is
given by equation of motion
¨ q(t)+ 2zw ˙ q(t)+w
2
q(t)= ¨ q
g
(t) u(t) (6.6)
where an overdot (˙) denotes derivative with respect to time, w is the natural frequency, z is the
damping ratio, ¨ q
g
(t) is ground acceleration, and u(t) is a mass-normalized controllable damping
force. A family of SDOF systems are generated by varying w and z for a fixed mass ratio r=
F
max
=mg, where F
max
= 1:3345 kN is the maximum control force the MR damper can generate
[68] and m is the structure mass. The characteristics of the SDOF systems are shown in Table 6.3.
Table 6.3: Characteristics of SDOF systems.
Parameters Parameter values
m 1133:6 kg (r= 12%)
w 2p; 4p; 6p; or 8p rad=s
z 0:025 or 0:05
The controlled systems are subjected to three scaled ground excitations, namely, Gaussian
white noise (GWN, duration 100 s, sampled at 1 kHz), El Centro (north-south component of the
1940 Imperial Valley, CA, earthquake of magnitude 7.1, recorded at the Imperial Valley Irrigation
District substation in El Centro, CA, over a 30 s duration and sampled at 50 Hz) and Kobe (north-
south component of the 1995 Hyogo-ken Nanbu Kobe earthquake of magnitude 7.2, recorded at
the Kobe Japanese Meteorological Agency JMA, Kobe, Japan, over a 32 s duration and sampled at
100 Hz). It should be noted that the earthquake data is resampled with a sample rate of 1024 Hz
assuming zero-order hold (i.e., piecewise constant) when performing the RTHS tests. To be con-
sistent with Chapter 3, r
a
= 0 is used in the cost metric Eq. (6.1). Further, to ensure that the
displacement of MR damper will not exceed its stroke limit, the GWN intensity is D= 0:005, and
the El Centro and Kobe earthquake excitations are both scaled by a factor of 0.2.
96
The CLQR and OCLC controllers are designed (with the second-order low pass filter) via
numerical simulation first to the SDOF systems. The numerical simulation studies show several
trends:
1. As the structural frequency w increases, the advantages of OCLC over CLQR improve for
GWN, El Centro and Kobe;
2. As the structural damping ratioz increases to 0.05, the performance improvement of OCLC
over CLQR does not vary significantly (as illustrated in Tables 6.5 and 6.6 for one example);
thus, due to limited time in the laboratory at UConn, only the case whenz = 0:025 is tested
in RTHS;
3. OCLC has similar performance improvements in reducing structural responses for the Kobe
earthquake excitation as it does for El Centro, e.g., as tabulated in Tables 6.5 and 6.7; there-
fore, again, due to limited time in the laboratory, the physical RTHS tests are only performed
for GWN and El Centro.
Moreover, to eliminate the possible offset and facilitate a fair comparison between RTHS and
numerical simulation results, for zero-mean GWN data, MATLAB’s commanddetrend is applied
to subtract off the mean value over the entire record time and the linear change for the absolute
acceleration and drift from both RTHS and numerical simulation; only the mean value over time is
subtracted off for El Centro data.
6.3.4.1 SDOF RTHS and numerical simulation comparison: selected time histories
The comparison of time history of absolute accelerations, drifts and control forces of SDOF sys-
tems with bang-bang and modified bang-bang secondary controllers using the CLQR and OCLC
control strategies, during the first 10 s of GWN and El Centro, are shown in Figures 6.6 to 6.9.
For brevity, only time history responses for the structure withw = 4p rad=s andz = 0:025 are de-
picted herein. CLQR has a larger magnitude of absolute acceleration and drift than those of OCLC;
e.g., the maximum drift when subjected to El Centro using bang-bang is about 5 mm and 7 mm for
97
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(a) Abs. accel. for OCLC
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(b) Abs. accel. for CLQR
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(c) Drift for OCLC
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(d) Drift for CLQR
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(e) Control force for OCLC
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(f) Control force for CLQR
Figure 6.6. Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to GWN using bang-bang control for OCLC and CLQR strategies.
98
OCLC and CLQR, respectively, during the first 10 s. The time histories of absolute accelerations,
drifts and control forces of RTHS match well with those of numerical simulation for both OCLC
and CLQR strategies using bang-bang and modified band-bang control, though with a slight time
shift that maybe due to a misalignment of when t = 0 s is between the experiment and numerical
simulation. The peak drifts of numerical simulation in each cycle are slightly larger on the order of
a half to one millimeter for a signal that has a peak of about 7 mm. The peak absolute acceleration
of numerical simulation in each cycle are overestimated, with a difference on the order of 0.1m=s
2
for a signal that has a peak of about 1.4 m=s
2
. When the system is excited by El Centro, again, the
numerical simulation and experimental results match well. For the drift using OCLC bang-bang
control, as shown in Figure 6.7c, experimental results deviate slightly from the numerical simula-
tion results at certain time steps, e.g., t= 2 s, though they are in the same phase; the control forces
also have relatively larger difference than other cases, as plotted in Figure 6.7e, which might be
due to slight imperfection in the MR damper numerical model used in the study.
6.3.4.2 SDOF RTHS and numerical simulation comparison: response statistics
The response statistics from the numerical simulation and RTHS experiment results for the SDOF
systems are summarized in Tables 6.4 to 6.10. An indicator function of commanded current,
denoted I
c
, is defined as:
I
c
(t)=
8
>
<
>
:
1; commanded current I(t) is on
0; otherwise
(6.7)
LetD=(p
CLQR
p
OCLC
)=p
OCLC
100% be the percent change of CLQR relative to OCLC, where
p is some metric: for response metrics, positiveD means OCLC is more effective than CLQR; for
the metricE[I
c
],D> 0 indicates OCLC clips less frequently than CLQR; for the mean square
control force metric, positiveD means OCLC uses less control effort than CLQR.
99
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(a) Abs. accel. for OCLC
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(b) Abs. accel. for CLQR
0 2 4 6 8 10
Time [s]
0
2
4
6
8
experiment
num. sim.
(c) Drift for OCLC
0 2 4 6 8 10
Time [s]
0
2
4
6
8
experiment
num. sim.
(d) Drift for CLQR
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(e) Control force for OCLC
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(f) Control force for CLQR
Figure 6.7. Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to El Centro using bang-bang control for OCLC and CLQR strategies.
100
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(a) Abs. accel. for OCLC
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(b) Abs. accel. for CLQR
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(c) Drift for OCLC
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(d) Drift for CLQR
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(e) Control force for OCLC
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(f) Control force for CLQR
Figure 6.8. Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,
z = 0:025) subjected to GWN using modified bang-bang control for OCLC and CLQR strategies.
101
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(a) Abs. accel. for OCLC
0 2 4 6 8 10
Time [s]
0
0.5
1
experiment
num. sim.
(b) Abs. accel. for CLQR
0 2 4 6 8 10
Time [s]
0
2
4
6
8
experiment
num. sim.
(c) Drift for OCLC
0 2 4 6 8 10
Time [s]
0
2
4
6
8
experiment
num. sim.
(d) Drift for CLQR
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(e) Control force for OCLC
0 2 4 6 8 10
Time [s]
0
5
10
experiment
num. sim.
(f) Control force for CLQR
Figure 6.9. Numerical simulation and RTHS time histories of a SDOF system (w = 4p rad=s,z =
0:025) subjected to El Centro using modified bang-bang control for OCLC and CLQR strategies.
102
Table 6.4: Comparison of numerical simulation and experiment results for SDOF systems with different w values subjected to GWN
using bang-bang control,z = 0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
][%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 5.2227 – 167.5744 – 9.9193 – 35.1441 – – – – –
OCLC 0.0323 0.0359 11.6092 9.5861 0.7018 0.8594 13.7256 11.8766 0.0911 0.5903 1.7222 2.9106
CLQR 0.0353 (9) 0.0358(–0) 17.5796 (51) 17.0419(78) 0.8021 (14) 0.7999 (–7) 16.1958 (18) 14.9523 (26) 0.0975 (–7) 0.4940 (16) 1.7202 (–0) 3.0379 (4)
4p
Uncontrol 5.6155 – 26.0019 – 10.6178 – 17.2216 – – – – –
OCLC 0.0664 0.0635 1.6814 1.5275 0.9643 1.0583 5.3677 5.0478 0.2243 0.6854 3.4284 5.0681
CLQR 0.0791(19) 0.0640 (1) 2.9471 (75) 2.4968 (63) 1.4002 (45) 1.3690 (29) 7.8218 (46) 7.2301 (43) 0.1424 (36) 0.5219 (24) 2.3300 (–32) 3.5927 (–29)
6p
Uncontrol 5.6713 – 5.5798 – 10.2016 – 8.7064 – – – – –
OCLC 0.0992 0.0900 0.4975 0.4258 1.1522 1.1556 2.9859 2.6883 0.2531 0.7210 3.8441 5.4374
CLQR 0.1211(22) 0.0961 (7) 0.9134 (84) 0.7584 (78) 1.3255 (15) 1.3961 (21) 3.4864 (17) 3.3652 (25) 0.1645 (35) 0.5432 (25) 2.4656 (–36) 3.7890 (–30)
8p
Uncontrol 5.9933 – 2.5660 – 10.7279 – 5.4742 – – – – –
OCLC 0.1372 0.1227 0.2108 0.1757 1.3191 1.3177 1.8973 1.7994 0.2650 0.7709 4.3150 5.6640
CLQR 0.1759(28) 0.1337 (9) 0.4273 (103) 0.3379 (92) 1.7607 (33) 1.5779 (20) 2.6578 (40) 2.3755 (32) 0.1814 (32) 0.5385 (30) 2.6429 (–39) 3.8568 (–32)
Table 6.5: Comparison of numerical simulation and experiment results for SDOF systems with differentw values subjected to El Centro
with bang-bang control,z = 0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 0.1064 – 53.5530 – 1.6907 – 32.0833 – – – – –
OCLC 0.0101 0.0100 2.3971 2.4999 0.3947 0.5267 6.4920 6.4820 0.0907 0.4104 0.6862 1.2841
CLQR 0.0114 (13) 0.0091(–9) 3.8252 (60) 2.9879 (20) 0.4293 (9) 0.3934 (–25) 8.8562 (36) 7.5431 (16) 0.0558 (38) 0.1732 (58) 0.5513 (–20) 0.7805 (–39)
4p
Uncontrol 0.2198 – 8.8359 – 2.2383 – 12.2293 – – – – –
OCLC 0.0234 0.0157 0.5903 0.3354 0.8463 0.8450 4.7919 4.7812 0.1323 0.5729 0.9504 1.9585
CLQR 0.0286 (22) 0.0194(24) 0.9678 (64) 0.7541(125) 1.2187 (44) 1.1480 (36) 7.1629 (49) 6.7114 (40) 0.0842 (36) 0.2683 (53) 0.8004 (–16) 1.1697 (–40)
6p
Uncontrol 0.1210 – 0.9813 – 1.6127 – 4.4266 – – – – –
OCLC 0.0150 0.0224 1.791110
6
0.0429 0.6957 1.2496 0.0083 1.3166 0.0750 0.6741 1.5473 1.1667
CLQR 0.0328(118) 0.0229 (2) 0.1879 ( 10
7
) 0.1706 (298) 0.8470 (22) 1.0608 (–15) 2.1736 (> 10
4
) 2.7682 (110) 0.0757 (–1) 0.3063 (55) 0.4098 (–74) 0.8590 (–26)
8p
Uncontrol 0.1784 – 0.4875 – 2.2702 – 3.5495 – – – – –
OCLC 0.0148 0.0225 7.510210
7
0.0205 0.6920 1.3275 0.0054 0.9080 0.0711 0.7049 1.5291 0.8706
CLQR 0.0395(166) 0.0230 (2) 0.0747 ( 10
7
) 0.0515 (152) 1.0638 (54) 0.8890 (–33) 1.5763 (> 10
4
) 1.3101 (44) 0.0619 (13) 0.3251 (54) 0.3320 (–78) 0.6102 (–30)
103
Table 6.6: Comparison of numerical simulation and experiment results for SDOF systems with differentw values subjected to El Centro
with bang-bang control,z = 0:05;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
][%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 0.0703 – 31.5436 – 1.4069 – 25.6231 – – – – –
OCLC 0.0101 – 2.2521 – 0.4092 – 6.2491 – 0.0855 – 0.6433 –
CLQR 0.0113 (12) – (–) 3.5367 (57) – (–) 0.4164 (2) – (–) 8.4292 (35) – (–) 0.0487 (43) – (–) 0.5087 (–21) – (–)
4p
Uncontrol 0.1259 – 5.0231 – 1.9400 – 10.3271 – – – – –
OCLC 0.0208 – 0.2357 – 1.0184 – 2.9871 – 0.0928 – 1.0381 –
CLQR 0.0268 (29) – (–) 0.8512 (261) – (–) 1.1649 (14) – (–) 6.7592 (126) – (–) 0.0743 (20) – (–) 0.6847 (–34) – (–)
6p
Uncontrol 0.0702 – 0.5971 – 1.4345 – 3.6918 – – – – –
OCLC 0.0149 – 9.669710
7
– 0.6960 – 0.0061 – 0.0750 – 1.5395 –
CLQR 0.0307(106) – (–) 0.1687 ( 10
7
) – (–) 0.8137 (17) – (–) 2.0630 (> 10
4
) – (–) 0.0644 (14) – (–) 0.3667 (–76) – (–)
8p
Uncontrol 0.0905 – 0.2602 – 1.8406 – 2.8610 – – – – –
OCLC 0.0149 – 3.435910
7
– 0.6940 – 0.0037 – 0.0700 – 1.5350 –
CLQR 0.0358(141) – (–) 0.0649 ( 10
7
) – (–) 1.0199 (47) – (–) 1.4946 (> 10
4
) – (–) 0.0502 (28) – (–) 0.2959 (–81) – (–)
Table 6.7: Comparison of numerical simulation and experiment results for SDOF systems with different w values subjected to Kobe
with bang-bang control,z = 0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
][%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 0.0406 – 24.1226 – 0.9212 – 20.9571 – – – – –
OCLC 0.0064 – 1.4857 – 0.3754 – 7.0978 – 0.0744 – 0.5273 –
CLQR 0.0074 (15) – (–) 2.8836 (94) – (–) 0.4628 (23) – (–) 9.7680 (38) – (–) 0.0215 (71) – (–) 0.3921 (–26) – (–)
4p
Uncontrol 0.0854 – 3.1303 – 1.0817 – 6.9279 – – – – –
OCLC 0.0116 – 0.2821 – 0.9474 – 3.3512 – 0.0607 – 0.4620 –
CLQR 0.0126 (9) – (–) 0.3506 (24) – (–) 0.5625 (–41) – (–) 3.0985 (–8) – (–) 0.0324 (47) – (–) 0.3563 (–23) – (–)
6p
Uncontrol 0.0867 – 0.7756 – 1.3762 – 4.5287 – – – – –
OCLC 0.0074 – 2.334310
6
– 0.6101 – 0.0127 – 0.0315 – 0.7589 –
CLQR 0.0175(137) – (–) 0.1048 ( 10
6
) – (–) 0.9386 (54) – (–) 2.4413 (> 10
4
) – (–) 0.0323 (–2) – (–) 0.2963 (–61) – (–)
8p
Uncontrol 0.0356 – 0.1081 – 1.0604 – 1.9611 – – – – –
OCLC 0.0074 – 1.051410
6
– 0.6179 – 0.0085 – 0.0317 – 0.7602 –
CLQR 0.0130 (75) – (–) 0.0215 ( 10
6
) – (–) 0.6737 (9) – (–) 0.9760 (> 10
4
) – (–) 0.0139 (56) – (–) 0.1830 (–76) – (–)
104
Table 6.8: Comparison of numerical simulation and experiment results for SDOF systems with different w values subjected to GWN
with modified bang-bang control,z = 0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
][%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 5.2227 – 167.5744 – 9.9193 – 35.1441 – – – – –
OCLC 0.0301 0.0305 12.3896 12.2801 0.6456 0.7068 13.8370 13.2603 0.3208 0.6377 1.8234 1.9640
CLQR 0.0337(12) 0.0326 (7) 18.9672 (53) 17.0678(39) 0.7386 (14) 0.7609 (8) 16.1361 (17) 14.9525 (13) 0.1984 (38) 0.4358 (32) 1.5253 (–16) 1.6992 (–13)
4p
Uncontrol 5.6155 – 26.0019 – 10.6178 – 17.2216 – – – – –
OCLC 0.0590 0.0573 1.5877 1.6976 0.8924 0.9517 5.0488 5.0615 0.4223 0.7464 3.0345 3.3228
CLQR 0.0751(27) 0.0677(18) 3.0708 (93) 2.9206 (72) 1.4227 (59) 1.4907 (57) 7.9222 (57) 7.8357 (55) 0.2530 (40) 0.5096 (32) 2.2387 (–26) 2.6429 (–20)
6p
Uncontrol 5.6713 – 5.5798 – 10.2016 – 8.7064 – – – – –
OCLC 0.0878 0.0861 0.4610 0.4849 1.0895 1.1740 2.8108 2.7397 0.4577 0.7918 3.6904 4.1410
CLQR 0.1141(30) 0.1026(19) 0.9268 (101) 0.8857 (83) 1.3262 (22) 1.4646 (25) 3.4859 (24) 3.4608 (26) 0.2718 (41) 0.5476 (31) 2.4393 (–34) 2.9602 (–29)
8p
Uncontrol 5.9933 – 2.5660 – 10.7279 – 5.4742 – – – – –
OCLC 0.1205 0.1203 0.1961 0.2019 1.2883 1.2916 1.8237 1.8321 0.4818 0.8244 4.2757 4.8882
CLQR 0.1650(37) 0.1449(20) 0.4249 (117) 0.3906 (93) 1.7740 (38) 1.7015 (32) 2.6779 (47) 2.5095 (37) 0.2766 (43) 0.5600 (32) 2.6560 (–38) 3.1628 (–35)
Table 6.9: Comparison of numerical simulation and experiment results for SDOF systems with differentw values subjected to El Centro
with modified bang-bang control,z = 0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 0.1064 – 53.5530 – 1.6907 – 32.0833 – – – – –
OCLC 0.0089 0.0088 2.1822 1.9816 0.3294 0.3304 6.3534 5.8114 0.1960 0.1993 0.6569 0.8420
CLQR 0.0108 (22) 0.0103 (17) 4.0712 (87) 3.7879 (91) 0.4480 (36) 0.4338 (31) 9.3208 (47) 8.7759 (51) 0.0962 (51) 0.1137 (43) 0.5339 (–19) 0.6999 (–17)
4p
Uncontrol 0.2198 – 8.8359 – 2.2383 – 12.2293 – – – – –
OCLC 0.0175 0.0151 0.2672 0.2410 0.6725 0.7386 2.9884 2.8848 0.2416 0.3662 0.9065 1.0965
CLQR 0.0259 (48) 0.0220 (46) 0.9593 (259) 0.9039 (275) 1.2364 (84) 1.2149 (64) 7.2785 (144) 7.1144 (147) 0.1486 (39) 0.1977 (46) 0.8241 (–9) 1.0760 (–2)
6p
Uncontrol 0.1210 – 0.9813 – 1.6127 – 4.4266 – – – – –
OCLC 0.0147 0.0287 3.276810
7
0.0136 0.6863 1.1128 0.0042 0.4332 0.5223 0.7571 1.5217 2.1166
CLQR 0.0293 (99) 0.0243(–15) 0.1834 ( 10
7
) 0.1820(1236) 0.8780 (28) 1.0668 (–4) 2.2583 (> 10
4
) 2.7659 (538) 0.1356 (74) 0.2351 (69) 0.4572 (–70) 0.7795 (–63)
8p
Uncontrol 0.1784 – 0.4875 – 2.2702 – 3.5495 – – – – –
OCLC 0.0147 0.0279 3.362810
7
0.0090 0.6864 1.1686 0.0042 0.4023 0.5210 0.7956 1.5177 1.6415
CLQR 0.0334(127) 0.0243(–13) 0.0656 ( 10
7
) 0.0552 (511) 1.0436 (52) 0.9469 (–19) 1.5449 (> 10
4
) 1.3866 (245) 0.1068 (80) 0.2593 (67) 0.3534 (–77) 0.5654 (–66)
105
Table 6.10: Comparison of numerical simulation and experiment results for SDOF systems with different w values subjected to Kobe
with modified bang-bang control,z = 0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
][%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [–] [%] [–] [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
Uncontrol 0.0406 – 24.1226 – 0.9212 – 20.9571 – – – – –
OCLC 0.0057 – 1.0512 – 0.3310 – 6.3405 – 0.1051 – 0.5375 –
CLQR 0.0072 (26) – (–) 3.1016 (195) – (–) 0.4715 (42) – (–) 9.9755 (57) – (–) 0.0483 (54) – (–) 0.3862 (–28) – (–)
4p
Uncontrol 0.0854 – 3.1303 – 1.0817 – 6.9279 – – – – –
OCLC 0.0091 – 0.0046 – 0.6284 – 0.4473 – 0.2034 – 0.7843 –
CLQR 0.0112 (23) – (–) 0.3368 (7296) – (–) 0.5369 (–15) – (–) 2.9510 (560) – (–) 0.0570 (72) – (–) 0.3559 (–55) – (–)
6p
Uncontrol 0.0867 – 0.7756 – 1.3762 – 4.5287 – – – – –
OCLC 0.0073 – 1.622510
7
– 0.6124 – 0.0036 – 0.2001 – 0.7539 –
CLQR 0.0155(114) – (–) 0.0994 ( 10
7
) – (–) 0.9490 (55) – (–) 2.4675 (> 10
4
) – (–) 0.0548 (73) – (–) 0.3232 (–57) – (–)
8p
Uncontrol 0.0356 – 0.1081 – 1.0604 – 1.9611 – – – – –
OCLC 0.0074 – 2.240210
6
– 0.6086 – 0.0133 – 0.1942 – 0.7556 –
CLQR 0.0121 (63) – (–) 0.0206 ( 10
6
) – (–) 0.6299 (4) – (–) 0.9231 (> 10
3
) – (–) 0.0278 (86) – (–) 0.1924 (–75) – (–)
106
Both CLQR and OCLC can decrease structural absolute acceleration and drift compared with
the uncontrolled case. OCLC provides various levels of performance improvements over CLQR
for different structural models for both numerical simulation and RTHS results, and is able to si-
multaneously reduce absolute acceleration and drift metrics significantly, even though the drift is
not included in the cost metric. Tables 6.4 to 6.10 report response statistics of various SDOF struc-
ture models, subjected to one of the three excitations, with OCLC (designed for the excitation),
CLQR and uncontrolled scenarios, in both numerical simulation and in RTHS tests.
For bang-bang control, when the external excitation is GWN, as shown in Table 6.4, numerical
simulation results illustrate that the performance improvements of OCLC over CLQR (i.e., decreas-
ing both the cost metric and mean square drift) increases as the natural frequencyw increases. No-
tably, CLQR has mean square absolute acceleration and drift metrics up to 28% and 103% larger,
respectively, than those of OCLC when w = 8p rad=s and about 45% larger for peak absolute
acceleration and drift when w = 4p rad=s. OCLC does not have quite so dramatic performance
improvements in the RTHS, but still provides performance superior to CLQR across all metrics.
Specifically, RTHS CLQR has mean square and peak drifts up to 92% and 43% larger, respectively,
than those of RTHS OCLC and up to 29% for peak absolute acceleration metric. Moreover, both
numerical simulation and RTHS indicates that CLQR clips more frequently than OCLC, evidenced
by CLQR’sE[I
c
] value being up to 36% smaller than that of OCLC, though the mean square control
force of OCLC is larger than or comparable to that of CLQR.
When the external excitation is El Centro, as listed in Table 6.5, the advantages of OCLC over
CLQR also improve asw increases for numerical simulation. Notably, whenw is larger, numeri-
cal simulation shows that CLQR has mean square and peak drifts more than several hundred times
those of OCLC and a cost metric that is up to two-and-a-half times larger than that of OCLC. Al-
though OCLC has much larger drifts in RTHS tests than in numerical simulation for systems with
largerw, OCLC can still provide significant RTHS performance improvements in decreasing drifts
while simultaneously reducing mean square absolute accelerations compared with CLQR. CLQR
clips much more frequently than OCLC, with the indicatorE[I
c
] up to 58% smaller than that of
107
OCLC. Table 6.6 shows the corresponding numerical simulation results when the structure damp-
ing ratio is increased from 0.025 to 0.05 (this configuration was not tested in RTHS); the OCLC
performance improvements are quite similar to the z = 0:025 case (Table 6.5), with significant
gains relative to CLQR across all metrics. When the external excitation is Kobe andz = 0:025, as
illustrated in Table 6.7, numerical simulation results are similar to those of El Centro (Table 6.5):
CLQR has a cost metric that is up to 137% larger than that of OCLC, and mean square and peak
drifts hundreds of times larger (RTHS tests were not performed for this case).
When modified bang-bang control is applied (Tables 6.8 to 6.10), again with damping ratio
z = 0:025, OCLC maintains its superiority over CLQR. When the excitation is GWN (Table 6.8),
RTHS and pure simulation results match well; CLQR has mean square absolute acceleration and
drift metrics up to 37% and 117% larger, respectively, than those of OCLC and up to 59% and 57%
for peak absolute acceleration and drift metrics, respectively. When the external excitation is El
Centro (Table 6.9), CLQR even has mean square and peak drifts more than several hundred times
larger than those of OCLC for numerical simulation; although the drifts in RTHS tests are very
different from those in numerical simulation, CLQR still has mean square and peak drifts up to
thirteen times that of OCLC, while slightly smaller mean square and peak absolute accelerations.
Further, OCLC uses the MR damper more effectively and efficiently, evidenced by less frequent
clipping (CLQR’sE[I
c
] value is up to 80% smaller than that of OCLC); OCLC, again, uses more
control effort than CLQR. Again, the numerical simulation results for Kobe (Table 6.10) are very
similar to those for El Centro (no RTHS tests were performed with the Kobe excitation).
A comparison of the numerical simulation and experiment results indicates that when the sys-
tems are subjected to GWN, the RTHS results match very well with the pure numerically simu-
lated results, especially for drift metrics; nevertheless, when the external excitation is the El Centro
earthquake, the numerical simulation and experimental results show non-negligible differences as
structural natural frequencyw increases, though OCLC remains clearly superior to CLQR. For the
higher-frequency w = 8p rad=s SDOF systems subjected to El Centro (Tables 6.5 and 6.9) using
OCLC control strategies, the experimental mean square drift is approximately 30,000 times the
108
numerically simulated drift, which may be due to an imperfect MR damper numerical model. In
addition, theE[I
c
] values are different for numerical simulation and RTHS, which may be because
that the commanded current is measured in RTHS and is not exactly zero even if it is close to zero
due to sensor noise; on the other hand, if the threshold of determining I
c
is changed, the results
will be much more closer. To verify this, a new indicator function I
0
c
is defined with a different
threshold (0.08 A other than 0 A):
I
0
c
(t)=
8
>
<
>
:
1; commanded current I(t)> 0:08 A
0; otherwise
(6.8)
E[I
0
c
] is computed for the case when natural frequency w = 2p, damping ratio z = 0:025 and
the SDOF structure is subjected to GWN using bang-bang secondary controller. E[I
0
c
]= 0:0911
for numerical simulation (as expected, the value is equal toE[I
c
]), and 0.1950 for RTHS; the new
indicator values are much more closer with each other than the values ofE[I
c
]= 0:0911 and 0.5903,
for numerical simulation and RTHS, as shown in Table 6.4.
In all, OCLC outperforms CLQR when applied to SDOF systems for both numerical simulation
and RTHS results.
6.3.4.3 Evaluation of OCLC gain with respect to excitation changes for SDOF systems
While CLQR implicitly assumes a GWN excitation, OCLC considers the external excitation in the
optimization process, resulting in different OCLC gains for different design excitations. Chapter 3
demonstrated the superior performance of an OCLC designed for one excitation when subjected
to other excitations via numerical simulation. In this section, OCLC performance is evaluated,
through both numerical simulation and RTHS, when the structure is subjected to excitations other
than the design excitation. Tables 6.11 to 6.14 show the SDOF mean square and peak absolute
acceleration and drift responses for numerical simulation and experimental results using OCLC
designed for one excitation but evaluated with the other excitation.
109
Table 6.11: Evaluation of CLQR and GWN OCLC for SDOF systems subjected to El Centro with bang-bang control, z = 0:025;D is
the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
][%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] – [%] – [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
OCLC 0.0118 0.0096 2.3742 1.7772 0.9555 0.6256 6.1668 5.6570 0.0899 0.1858 0.7019 0.8315
CLQR 0.0114(–3) 0.0091(–5) 3.8252(61) 2.9879(68) 0.4293 (–55) 0.3934 (–37) 8.8562 (44) 7.5431 (33) 0.0558 (38) 0.2199 (–18) 0.5513 (–21) 0.7805 (–6)
4p
OCLC 0.0232 0.0153 0.5503 0.3830 0.9888 0.7557 4.9684 4.2299 0.1145 0.3893 1.0083 1.3060
CLQR 0.0286(23) 0.0194(27) 0.9678(76) 0.7541(97) 1.2187 (23) 1.1480 (52) 7.1629 (44) 6.7114 (59) 0.0842 (26) 0.3234 (17) 0.8004 (–21) 1.1697 (–10)
6p
OCLC 0.0290 0.0193 0.1315 0.1091 1.0024 1.0018 2.0638 2.5999 0.0810 0.4269 0.7321 0.9998
CLQR 0.0328(13) 0.0229(18) 0.1879(43) 0.1706(56) 0.8470 (–16) 1.0608 (6) 2.1736 (5) 2.7682 (6) 0.0757 (7) 0.3651 (14) 0.4098 (–44) 0.8590 (–14)
8p
OCLC 0.0318 0.0208 0.0473 0.0369 1.0042 0.8687 1.2799 1.2563 0.0543 0.4236 0.5824 0.7015
CLQR 0.0395(24) 0.0230(11) 0.0747(58) 0.0515(40) 1.0638 (6) 0.8890 (2) 1.5763 (23) 1.3101 (4) 0.0619 (–14) 0.3894 (8) 0.3320 (–43) 0.6102 (–13)
Table 6.12: Evaluation of CLQR and El Centro OCLC for SDOF systems subjected to GWN with bang-bang control, z = 0:025;D is
the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] – [%] – [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
OCLC 0.0394 0.0477 15.3374 16.5766 0.7620 0.7368 16.0251 14.3648 0.2114 0.7315 3.1124 5.6710
CLQR 0.0353(–11) 0.0358(–25) 17.5796 (15) 17.0419 (3) 0.8021 (5) 0.7999 (9) 16.1958 (1) 14.9523 (4) 0.0975 (54) 0.6088 (17) 1.7202 (–45) 3.0379 (–46)
4p
OCLC 0.0724 0.0864 1.8734 1.7771 0.9829 0.9751 5.5877 4.7747 0.2980 0.7862 4.2493 9.0636
CLQR 0.0790 (9) 0.0640(–26) 2.9471 (57) 2.4968 (40) 1.4002 (42) 1.3690 (40) 7.8218 (40) 7.2301 (51) 0.1424 (52) 0.6361 (19) 2.3300 (–45) 3.5927 (–60)
6p
OCLC 0.4760 0.1521 0.0240 0.1644 1.3412 1.4878 1.0097 1.4719 0.8813 0.8343 44.0570 9.4664
CLQR 0.1210(–75) 0.0961(–37) 0.9134 (> 10
3
) 0.7584 (361) 1.3255 (–1) 1.3961 (–6) 3.4864 (245) 3.3652 (129) 0.1645 (81) 0.6580 (21) 2.4656 (–94) 3.7890 (–60)
8p
OCLC 0.5119 0.1885 0.0173 0.1757 1.4480 1.6354 0.7514 1.7994 0.8870 0.8667 44.6849 9.4256
CLQR 0.1759(–66) 0.1337(–29) 0.4273 (> 10
3
) 0.3379 (92) 1.7607 (22) 1.5779 (–4) 2.6578 (254) 2.3755 (32) 0.1814 (80) 0.6743 (22) 2.6429 (–94) 3.8568 (–59)
110
Table 6.13: Evaluation of CLQR and GWN OCLC for SDOF systems subjected to El Centro with modified bang-bang control, z =
0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
][%] [m
2
/s
4
][%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] – [%] – [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
OCLC 0.0094 0.0093 2.2100 2.1916 0.4037 0.4355 5.9964 5.8623 0.2066 0.1436 0.5958 0.7137
CLQR 0.0108(15) 0.0103(10) 4.0712 (84) 3.7879 (73) 0.4480 (11) 0.4338 (–0) 9.3208 (55) 8.7759 (50) 0.0897 (57) 0.1137 (21) 0.5339 (–10) 0.6999 (–2)
4p
OCLC 0.0189 0.0168 0.4804 0.4462 0.8258 0.8157 4.6918 4.6119 0.2234 0.2473 0.8308 1.0035
CLQR 0.0259(37) 0.0220(31) 0.9593(100) 0.9041(103) 1.2367 (50) 1.2090 (48) 7.2740 (55) 7.0998 (54) 0.1446 (35) 0.1977 (20) 0.8241 (–1) 1.0760 (7)
6p
OCLC 0.0245 0.0202 0.1257 0.1142 0.8729 0.9737 2.2830 2.4957 0.1895 0.2930 0.6000 0.8382
CLQR 0.0293(20) 0.0243(20) 0.1834 (46) 0.1820 (59) 0.8780 (1) 1.0668 (10) 2.2583 (–1) 2.7659 (11) 0.1336 (30) 0.2351 (20) 0.4572 (–24) 0.7795 (–7)
8p
OCLC 0.0276 0.0214 0.0443 0.0391 0.8272 0.8659 1.2249 1.2463 0.1433 0.3212 0.4569 0.6392
CLQR 0.0334(21) 0.0243(14) 0.0656 (48) 0.0552 (41) 1.0436 (26) 0.9469 (9) 1.5449 (26) 1.3866 (11) 0.1045 (27) 0.2593 (19) 0.3534 (–23) 0.5654 (–12)
Table 6.14: Evaluation of CLQR and El Centro OCLC for SDOF systems subjected to GWN with modified bang-bang control, z =
0:025;D is the percent change of CLQR relative to OCLC.
J
a
=E[ ¨ q
2
a
] E[q
2
] ¨ q
max
a
q
max
E[I
c
] E[u
2
]
w Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
[ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] – [%] – [%] [(%mg)
2
] [%] [(%mg)
2
] [%]
2p
OCLC 0.0329 0.0341 14.7330 16.4483 0.7981 0.7469 15.4470 14.0316 0.3439 0.6305 2.6552 3.5096
CLQR 0.0337 (2) 0.0326 (–5) 18.9020 (28) 17.0376 (4) 0.7391 (–7) 0.7614 (2) 16.1579 (5) 14.9674 (7) 0.1979 (42) 0.4358 (31) 1.5253 (–43) 1.6992 (–52)
4p
OCLC 0.0664 0.0652 0.8733 1.0563 0.9734 0.9839 3.6739 3.6949 0.5191 0.8586 4.4448 4.8111
CLQR 0.0750 (13) 0.0677 (4) 3.0686 (251) 2.9193 (176) 1.4195 (46) 1.4874 (51) 7.8938 (115) 7.8141 (111) 0.2524 (51) 0.5096 (41) 2.2387 (–50) 2.6429 (–45)
6p
OCLC 0.4785 0.2817 0.0199 0.0744 1.2961 1.5043 0.8980 1.2252 0.8912 0.9137 44.7384 21.7766
CLQR 0.1141(–76) 0.1026(–64) 0.9263 (> 10
3
) 0.8851 (1089) 1.3224 (2) 1.4609 (–3) 3.4726 (287) 3.4472 (181) 0.2713 (70) 0.5476 (40) 2.4393 (–95) 2.9602 (–86)
8p
OCLC 0.5112 0.3303 0.0170 0.2018 1.4707 1.8104 0.7468 1.8525 0.8914 0.9276 44.7195 21.3047
CLQR 0.1649(–68) 0.1448(–56) 0.4248 (> 10
3
) 0.3905 (94) 1.7710 (20) 1.6984 (–6) 2.6730 (258) 2.5096 (35) 0.2761 (69) 0.5600 (40) 2.6560 (–94) 3.1628 (–85)
111
When OCLC is designed for GWN but the structure is subjected to the El Centro earthquake,
and using bang-bang control, as shown in Table 6.11, OCLC still provides performance compara-
ble or superior to CLQR. CLQR has mean square and peak drift metrics up to 68% and 44% larger
than those of OCLC, though performs comparably to OCLC in minimizing the cost metric when
the structure natural frequency w = 2p rad=s. When w is larger than 2p rad=s, OCLC is able to
reduce drifts and absolute accelerations simultaneously. Specifically, CLQR has mean square abso-
lute acceleration and drift metrics up to 24% and 76% larger, respectively, than those of OCLC for
numerical simulation results, and even 27% and 97% larger for RTHS results. Although CLQR’s
numerically simulated peak absolute acceleration is smaller (e.g., when w = 6p rad=s), CLQR’s
experimental peak absolute acceleration is even larger compared to OCLC. When OCLC is de-
signed for El Centro but evaluated through GWN excitation using bang-bang control (Table 6.12),
CLQR has mean square and peak drift metrics up to multiple times that of OCLC, and the value
ofE[I
c
] is up to 81% smaller, though with cost metric, peak absolute acceleration and mean square
control force smaller than OCLC.
When OCLC is designed for GWN but evaluated with a structure subjected to the El Centro
earthquake using modified bang-bang secondary controller, as displayed in Table 6.13, OCLC
provides significant performance improvements for all SDOF systems. Furthermore, numerical
simulation responses closely match the RTHS results. CLQR has cost value and mean square
drift up to 37% and 103% larger, respectively, than those of OCLC and 50% and 55% for peak
absolute acceleration and drift metrics, with 35% smallerE[I
c
]. When OCLC is designed for El
Centro but evaluated with GWN, as shown in Table 6.14, for structures with natural frequencyw=
4p rad=s using a modified bang-bang secondary controller, CLQR has up to 13% and 51% larger
mean square and peak absolute accelerations, and even 251% and 115% larger mean square and
peak drifts compared with CLQR. When structure natural frequencyw increases, CLQR performs
better in decreasing the cost metric, but has mean square and peak drift metrics up to twelve and
three times those of OCLC in RTHS, and forty-five and four times those of OCLC in numerical
112
simulation. CLQR has more frequent clipping, asE[I
c
] is up to 70% smaller compared with OCLC;
CLQR also uses less control effort than OCLC.
Both pure simulation and RTHS results indicate that, for SDOF systems, the OCLC designed
for El Centro excitation for structures with larger w does not perform well in reducing the mean
square absolute acceleration when the structure is subjected to GWN, whereas the OCLC designed
for GWN performs quite well when the external excitation is El Centro. The likely reason for this
mixed performance is the more broad band nature of the GWN excitation.
6.3.5 RTHS and pure simulation of 2DOF systems
To verify that OCLC’s excellent performance for a SDOF system holds for MDOF systems, the
comparison study is subsequently performed for 2DOF systems with equation of motion given by:
M¨ q(t)+ C˙ q(t)+ Kq(t)=Mr ¨ q
g
(t)
¯
Bu(t) (6.9)
where M, K and C are the mass, stiffness and damping matrices of the structural system, respec-
tively:
M=
2
6
4
m
1
0
0 m
2
3
7
5
; K=
2
6
4
k
1
+ k
2
k
2
k
2
k
2
3
7
5
; C=
2
6
4
c
1
+ c
2
c
2
c
2
c
2
3
7
5
; (6.10)
r=[1 1]
T
is the ground influence vector;
¯
B is the control force vector indicating the location of
the controllable damper, and u(t) is the controllable damping force. For the 2DOF buildings and
2DOF base-isolated structures, the MR damper is in the first floor and base, respectively; hence
¯
B=[1 0]
T
; for the 2DOF bridge deck model, the MR damper is installed between the pier and the
deck and therefore
¯
B=[1 1]
T
. The 2DOF building models are varied by changing the masses
m
1
and m
2
and the dominant frequencyw
1
. The suite of 2DOF systems are listed in Table 6.15.
113
Table 6.15: Characteristics of 2DOF systems.
System type Parameters Parameter values
Building with various
characteristics
m
1
1133:6 kg(r= 12%) or 906:9 kg(r= 15%)
m
2
m
2
= m
1
k
1
2m
1
3
p
5
w
2
1
forw
1
= 2pn rad=s, n= 1;2;3
k
2
k
2
= k
1
C Rayleigh damping C= a
0
M+ a
1
K s.t.z
1
=z
2
= 0:025
Bridge
m
1
100 Mg
m
2
500 Mg
k
1
15:791 MN=m
k
2
7:685 MN=m
c
1
125:6 kN s=m
c
2
0 kN s=m
w
1
3:18 rad=s
w
2
15:5 rad=s
z
1
0.45%
z
2
4%
Base-isolated building
m
b
= m
1
6800 kg
m
s
= m
2
29485 kg
k
b
= k
1
232 kN=m
k
s
= k
2
11912 kN=m
c
b
= c
1
3:74 kN s=m
c
s
= c
2
23:71 kN s=m
w
1
2:51 rad=s
w
2
46:7 rad=s
z
1
2%
z
2
5%
114
For the 2DOF structure models, the control force weightr
a
in Eq. (6.1) is selected with differ-
ent values to evaluate the performance of different control strategies. To ensure that the physical
MR damper stroke limit is not exceeded, the 2DOF building excitations are the same as for the
SDOF system: GWN with intensity D= 0:005 and both El Centro and Kobe records scaled to
20% of original. For the 2DOF bridge deck and base-isolated structures, GWN with D= 5 10
5
and El Centro and Kobe records scaled to 5% of original are used. Further, only the bang-bang
secondary control law is used for these 2DOF systems (due to time limitation in the laboratory).
The performance of each controlled 2DOF system is evaluated with three excitations, GWN,
El Centro and Kobe, in numerical simulation and two excitations, GWN and El Centro, in RTHS
(the performance with Kobe is very similar to that with El Centro, as stated previously) and shown
in Tables 6.16 to 6.19. The suite of RTHS tests is also selected according to numerical simulation
results:
1. For the 2DOF buildings, when increasing the mass ratio, OCLC maintains its superiority
over CLQR (e.g., as seen from Tables 6.16 and 6.20); hence, only mass ratio r = 12% is
tested in RTHS experiments;
2. Preliminary numerical study of control weight r
a
values indicated that r
a
values smaller
than 10
13
kg
–2
can lead to RMS active damper force sufficiently large to induce better
comparison of OCLC and CLQR strategies [29]. Therefore, two control weight r
a
values,
r
a
= 10
16
kg
–2
and 10
13
kg
–2
, are chosen for 2DOF buildings, and r
a
= 10
20
kg
–2
and
10
13
kg
–2
are selected for 2DOF bridge deck and base-isolated structures.
Again, to eliminate the possible offset and facilitate a fair comparison between RTHS and numer-
ical simulation results, the mean value over the entire record time and the linear change for the
absolute acceleration and drift are subtracted off from both RTHS and numerical simulation for
zero-mean GWN data; only the mean value is subtracted off for El Centro data.
115
6.3.5.1 Building models
Selected time histories: Figures 6.10 to 6.13 show the first 10s, of the full 100s (GWN) or 30s (El
Centro) simulations, of the numerically simulated and experiment absolute acceleration, drift and
control force time histories of each story using CLQR and OCLC strategies for the 2DOF building
subjected to GWN and El Centro, whenw
1
= 4p rad=s,z
1
= 0:025, andr
a
= 10
13
kg
–2
. CLQR
has larger structural responses than OCLC; e.g., the second story maximum drift when subjected
to GWN is about 4.5 mm and 3 mm, respectively, for CLQR and OCLC during this 10 s time du-
ration (the corresponding peak statistics over the entire 100 s duration are shown in Table 6.16 and
discussed in subsequent paragraphs). The first story absolute accelerations oscillate more than,
and are larger than those of the second story. The RTHS test results have good agreement with
the numerical simulated results: the peak drifts and absolute accelerations of numerical simula-
tion in each cycle are slightly larger on the order of a half to one millimeter and 0.05 to 0.1 m=s
2
,
respectively. When the external excitation is GWN, the time history of first and second story ab-
solute accelerations and drifts of RTHS have good agreement with those of numerical simulation
for both CLQR and OCLC strategies, though the first floor accelerations exhibit high frequency
content in the RTHS that are not in the numerical simulation. The peak absolute acceleration dif-
ferences between numerical simulation and RTHS are around 3% for both CLQR and OCLC over
the entire time duration (100 s). When the system is subjected to the El Centro earthquake using
CLQR control, the RTHS time history also matches quite well with numerical simulation results,
though with some larger deviations from the numerical simulation responses during certain time
durations, e.g., 6 s t 8 s for first and second story drifts in Figures 6.12b and 6.12d. For OCLC
control, there are spikes in the time history of numerically simulated first floor absolute accelera-
tion and control force, as shown in Figures 6.13a and 6.13e; these acceleration spikes occur when
the control force changes abruptly, resulting in high local acceleration values [123]. This behavior
depends on the time lag in the generation of the control current modeled by Eq. (6.5), but is prop-
agated into the hysteresis parameters az in the MR damper representation equation Eq. (6.2), as
116
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
experiment
num. sim.
(a) 1st floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
experiment
num. sim.
(b) 1st story drift
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
2
experiment
num. sim.
(c) 2nd floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(d) 2nd story drift
0 2 4 6 8 10
Time [s]
0
5
10
15
experiment
num. sim.
(e) Control force
Figure 6.10. Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to GWN with CLQR (r
a
= 10
13
kg
–2
).
117
shown in Figure 6.14 for the numerical simulation results of first floor absolute acceleration and
the corresponding hysteresis componentaz in the control force.
Statistics of response to OCLC design excitation: Tables 6.16 to 6.19 display the mean square
and peak response comparison of numerical simulation and experimental results for 2DOF build-
ings with different fundamental frequency w
1
and control force weight values r
a
= 10
13
kg
–2
(Tables 6.16 and 6.17) andr
a
= 10
16
kg
–2
(Tables 6.18 and 6.19). OCLC is able to decrease ab-
solute accelerations and drifts simultaneously, and is always superior to CLQR, for both numerical
simulation and RTHS results. As fundamental structural frequency w
1
increases, the advantages
of OCLC over CLQR improve.
Tables 6.16 and 6.17 list the mean square and peak responses, whenr
a
= 10
13
kg
–2
. CLQR’s
cost metric and first and second floor mean square absolute accelerations for a GWN excitation are
up to 51%, 85% and 119% larger, respectively, than those of OCLC in numerical simulation, and
41%, 40% and 98% larger for the corresponding RTHS experiments; the mean square drifts are
even up to three times those of OCLC. CLQR also clips more frequently, with a 56% smallerE[I
c
],
though with a lower control force level than OCLC. OCLC’s superior performance over CLQR
improves when the excitation is El Centro. Specifically, CLQR has cost metric and first and second
floor mean square absolute accelerations up to 58%, 124% and 151% larger, respectively, than
those of OCLC in numerical simulation, and 56%, 176% and 106% larger for the corresponding
RTHS experiments. Notably, the first and second story mean square drifts are even up to nine and
two times those of OCLC, even though the drifts are not included in the cost function. OCLC also
clips less frequently than CLQR, indicating that OCLC utilizes the MR damper more effectively
and efficiently. The Kobe numerical simulation results are very similar to the El Centro results:
CLQR’s cost metric, first and second floor mean square absolute accelerations are up to 66%, 153%
and 220% larger, respectively, than those of OCLC, with mean square first-story drifts even forty
times that of OCLC. Further, the OCLC strategy results in peak responses below those with CLQR,
even though minimizing the mean square does not guarantee minimizing the peak. Remarkably,
118
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
experiment
num. sim.
(a) 1st floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
experiment
num. sim.
(b) 1st story drift
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
2
experiment
num. sim.
(c) 2nd floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(d) 2nd story drift
0 2 4 6 8 10
Time [s]
0
5
10
15
experiment
num. sim.
(e) Control force
Figure 6.11. Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to GWN with OCLC (r
a
= 10
13
kg
–2
).
119
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
experiment
num. sim.
(a) 1st floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
experiment
num. sim.
(b) 1st story drift
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
2
experiment
num. sim.
(c) 2nd floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(d) 2nd story drift
%
0 2 4 6 8 10
Time [s]
0
5
10
15
experiment
num. sim.
(e) Control force
Figure 6.12. Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to El Centro with CLQR (r
a
= 10
13
kg
–2
).
120
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
experiment
num. sim.
(a) 1st floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
experiment
num. sim.
(b) 1st story drift
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
2
experiment
num. sim.
(c) 2nd floor absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(d) 2nd story drift
0 2 4 6 8 10
Time [s]
0
5
10
15
experiment
num. sim.
(e) Control force
Figure 6.13. Numerical simulation and RTHS time histories of a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to El Centro with OCLC (r
a
= 10
13
kg
–2
).
121
0 2 4 6 8 10
Time [s]
0
0.5
0 2 4 6 8 10
Time [s]
0
5
10
Figure 6.14. Numerical simulation of first floor absolute acceleration time history and the cor-
responding hysteresis component az in the control force for a 2DOF system (w
1
= 4p rad=s,
z
1
= 0:025) subjected to El Centro with OCLC (r
a
= 10
13
kg
–2
).
122
when the external excitation is El Centro, the second floor peak absolute acceleration and first story
peak drift with CLQR are 106% and 206% larger, respectively, than those of OCLC.
OCLC also maintains its advantages over CLQR when the control weight r
a
= 10
16
kg
–2
(Tables 6.18 and 6.19). The test results also match the numerical simulation results very well.
The RTHS-tested performance improvements of OCLC over CLQR are similar to, or larger than,
those in the numerical simulation. When the external excitation is GWN, CLQR has up to 52%
larger cost metric, with even up to a 253% larger mean square first story drift, than those provided
by OCLC for building fundamental frequency w
1
= 4p rad=s. When the structure is subjected to
the El Centro earthquake, CLQR’s cost metric and the first and second floor mean square absolute
accelerations are up to 59%, 127% and 154% larger, respectively, than those of OCLC in numerical
simulation, and up to 58%, 184% and 114% larger for the corresponding RTHS experiments.
CLQR’s first story mean square drift is seven times that of OCLC in numerical simulation, and
nine times that of OCLC in RTHS. For Kobe excitation, the numerically simulated first story mean
square drift is more than forty times that of OCLC. Moreover, CLQR’s first story peak drift is three
times that of OCLC in RTHS tests when a building with fundamental frequencyw
1
= 4p rad=s is
subjected to the El Centro earthquake, and four times that of OCLC in numerical simulation when
a building with fundamental frequencyw
1
= 6p rad=s is excited by the Kobe earthquake.
Tables 6.20 to 6.21 provide the numerical simulated mean square and peak responses, respec-
tively, for 2DOF buildings with different fundamental frequencyw
1
values andz
1
= 0:025, mass
ratio r = 15% and control weight r
a
= 10
16
kg
–2
when the external excitation is the El Cen-
tro earthquake. As stated previously, OCLC maintains its advantages over CLQR in minimizing
structural absolute acceleration and drift responses when r= 15%, evidenced by performance im-
provements similar to those shown in Tables 6.18 to 6.19 when r = 12%. Remarkably, CLQR
has a cost metric, mean square absolute accelerations and drifts that are up to 54%, 128% and
129% larger, respectively, than those of OCLC and up to 100% and 108% larger for peak absolute
accelerations and drifts.
123
Table 6.16: Mean square response statistics for 2DOF buildings with different fundamental frequency w
1
for z
1
= 0:025 and r
a
=
10
13
kg
–2
;D is the percent change of CLQR relative to OCLC.
Design and J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Evaluation w 1 Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation [ rad=s] Strategy [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [–] [%] [–] [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN
2p
Uncontrol 0.5687 – 0.1920 – 0.3767 – 88.6414 – 35.2089 – – – – –
OCLC 0.1139 0.1179 0.0488 0.0589 0.0653 0.0589 12.2253 10.7964 6.0156 5.5957 0.2394 0.8158 6.8057 10.8247
CLQR 0.1780(36) 0.1410 (16) 0.0692 (42) 0.0648 (10) 0.1088 (67) 0.0762 (29) 27.5403 (125) 21.8495 (102) 10.0899 (68) 7.0334 (26) 0.1170 (51) 0.5746 (30) 2.4920 (–63) 5.8464 (–46)
4p
Uncontrol 1.3762 – 0.4484 – 0.9278 – 13.7182 – 5.4196 – – – – –
OCLC 0.2464 0.2117 0.1014 0.1058 0.1452 0.1058 1.5230 1.2104 0.8372 0.7076 0.3697 0.9654 10.2927 16.6041
CLQR 0.5044(51) 0.3572 (41) 0.1872 (85) 0.1479 (40) 0.3174 (119) 0.2094 (98) 5.1524 (238) 3.8296 (216) 1.8443 (120) 1.2124 (71) 0.1618 (56) 0.5745 (40) 3.5391 (–66) 7.3996 (–55)
6p
Uncontrol 1.5833 – 0.5494 – 1.0339 – 2.9525 – 1.1932 – – – – –
OCLC 0.3718 0.3115 0.1459 0.1558 0.2262 0.1558 0.4628 0.3621 0.2583 0.2238 0.3637 0.9616 11.0195 17.9493
CLQR 0.7005(47) 0.5291 (41) 0.2540 (74) 0.2105 (35) 0.4466 (97) 0.3186 (105) 1.3601 (194) 1.0610 (193) 0.5135 (99) 0.3653 (63) 0.1711 (53) 0.5896 (39) 3.0676 (–72) 6.3203 (–65)
El Centro
2p
Uncontrol 0.1776 – 0.0549 – 0.1226 – 27.9693 – 11.4614 – – – – –
OCLC 0.0370 0.0365 0.0117 0.0122 0.0253 0.0243 3.5192 2.3395 2.3225 2.2334 0.1476 0.5083 1.8346 3.3281
CLQR 0.0727(49) 0.0480 (24) 0.0245(109) 0.0195 (61) 0.0482 (91) 0.0285 (17) 10.8826 (209) 7.2076 (208) 4.4669 (92) 2.6214 (17) 0.1011 (31) 0.3214 (37) 1.0320 (–44) 2.0064 (–40)
4p
Uncontrol 0.4343 – 0.1297 – 0.3045 – 4.6504 – 1.7786 – – – – –
OCLC 0.0770 0.0589 0.0286 0.0193 0.0483 0.0396 0.2673 0.1666 0.2787 0.2284 0.1678 0.7809 4.2849 5.9575
CLQR 0.1854(58) 0.1347 (56) 0.0641(124) 0.0532(176) 0.1213 (151) 0.0815 (106) 1.8872 (606) 1.4773 (787) 0.7045 (153) 0.4716 (106) 0.1070 (36) 0.3614 (54) 1.1839 (–72) 2.6401 (–56)
6p
Uncontrol 0.2460 – 0.0759 – 0.1702 – 0.5185 – 0.1963 – – – – –
OCLC 0.0842 0.0647 0.0323 0.0219 0.0521 0.0428 0.0863 0.0629 0.0595 0.0487 0.1243 0.7978 2.7469 4.3742
CLQR 0.1392(39) 0.1074 (40) 0.0472 (46) 0.0417 (91) 0.0920 (77) 0.0657 (53) 0.2646 (207) 0.2252 (258) 0.1058 (78) 0.0752 (54) 0.0734 (41) 0.3783 (53) 0.5387 (–80) 1.5642 (–64)
Kobe
2p
Uncontrol 0.0847 – 0.0296 – 0.0551 – 12.7789 – 5.1512 – – – – –
OCLC 0.0200 – 0.0073 – 0.0127 – 1.2882 – 1.1665 – 0.0762 – 1.5248 –
CLQR 0.0334(40) – (–) 0.0123 (68) – (–) 0.0211 (66) – (–) 4.6623 (262) – (–) 1.9537 (67) – (–) 0.0762 (41) – (–) 0.4467 (–71) – (–)
4p
Uncontrol 0.1510 – 0.0424 – 0.1086 – 1.6370 – 0.6344 – – – – –
OCLC 0.0373 – 0.0127 – 0.0245 – 0.2089 – 0.1410 – 0.0798 – 1.5656 –
CLQR 0.0738(50) – (–) 0.0237 (86) – (–) 0.0502 (105) – (–) 0.7114 (241) – (–) 0.2914 (107) – (–) 0.0798 (39) – (–) 0.4608 (–71) – (–)
6p
Uncontrol 0.1916 – 0.0593 – 0.1323 – 0.4111 – 0.1526 – – – – –
OCLC 0.0318 – 0.0130 – 0.0188 – 0.0042 – 0.0217 – 0.0951 – 3.2665 –
CLQR 0.0932(66) – (–) 0.0328(153) – (–) 0.0603 (220) – (–) 0.1771 (4134) – (–) 0.0694 (220) – (–) 0.0951 (53) – (–) 0.4075 (–88) – (–)
124
Table 6.17: Peak response statistics for 2DOF buildings with different fundamental frequencyw
1
forz
1
= 0:025 andr
a
= 10
13
kg
–2
;D
is the percent change of CLQR relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Evaluation w
1
Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation [ rad=s] Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN
2p
Uncontrol 1.3891 – 1.9402 – 25.8916 – 18.6965 –
OCLC 0.9229 1.1712 0.8834 1.1712 13.7584 12.8654 8.5778 7.7977
CLQR 1.2492(35) 1.1584(–1) 1.2959 (47) 1.1170 (–5) 18.7749 (36) 15.6456 (22) 12.5637 (46) 10.8328 (39)
4p
Uncontrol 2.5086 – 3.2237 – 12.7199 – 7.7340 –
OCLC 1.4088 1.3700 1.4460 1.3700 5.1695 4.8753 3.4337 3.2327
CLQR 2.0277(44) 2.0998 (53) 2.5571 (77) 2.4780 (81) 9.2899 (80) 9.3801 (92) 6.1992 (81) 6.0113 (86)
6p
Uncontrol 3.3071 – 3.8322 – 6.1956 – 4.0948 –
OCLC 1.5617 1.5566 1.6855 1.5566 2.5659 2.3749 1.7974 1.5428
CLQR 2.1497(38) 2.1633(39) 2.7016 (60) 2.3401 (50) 4.2398 (65) 3.8056 (60) 2.9068 (62) 2.5263 (64)
El Centro
2p
Uncontrol 1.2249 – 1.6497 – 24.3513 – 15.8391 –
OCLC 0.9673 0.6038 0.6776 0.6802 8.0335 7.4079 6.4800 6.5150
CLQR 0.9643(–0) 0.8443(40) 1.0017 (48) 0.8418 (24) 14.5744 (81) 11.8200 (60) 9.7268 (50) 8.1430 (25)
4p
Uncontrol 1.4624 – 2.4693 – 8.5723 – 5.9521 –
OCLC 0.9903 0.9491 0.9584 0.9642 3.1093 2.1278 2.2984 2.3149
CLQR 1.5009(52) 1.5401 (62) 1.9715(106) 1.9786(105) 6.7851 (118) 6.5108 (206) 4.7665 (107) 4.7890 (107)
6p
Uncontrol 1.5299 – 1.8823 – 3.2745 – 2.0370 –
OCLC 1.0301 0.8505 1.3601 1.2330 1.5233 1.6663 1.4514 1.3075
CLQR 1.3764(34) 1.3819(62) 1.5003 (10) 1.5351 (25) 2.6604 (75) 2.5595 (54) 1.6239 (12) 1.6579 (27)
Kobe
2p
Uncontrol 0.8300 – 1.0236 – 15.3512 – 9.8565 –
OCLC 1.0561 – 0.6267 – 6.4448 – 5.9835 –
CLQR 0.9891(–6) – (–) 0.8343 (33) – (–) 12.3043 (91) – (–) 8.0810 (35) – (–)
4p
Uncontrol 0.9075 – 1.2383 – 5.1095 – 2.9775 –
OCLC 0.9880 – 0.8560 – 2.4515 – 2.0511 –
CLQR 1.0176 (3) – (–) 1.1048 (29) – (–) 3.9317 (60) – (–) 2.6832 (31) – (–)
6p
Uncontrol 1.4246 – 1.7306 – 3.3834 – 1.8548 –
OCLC 1.0076 – 0.8401 – 0.5225 – 0.8994 –
CLQR 1.1648(16) – (–) 1.4326 (71) – (–) 2.3600 (352) – (–) 1.5444 (72) – (–)
125
Table 6.18: Mean square response statistics for 2DOF buildings with different fundamental frequency w
1
for z
1
= 0:025 and r
a
=
10
16
kg
–2
;D is the percent change of CLQR relative to OCLC.
Design and J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Evaluation w 1 Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation [ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [–] [%] [–] [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN
2p
Uncontrol 0.5687 – 0.1920 – 0.3767 – 88.6414 – 35.2089 – – – – –
OCLC 0.1139 0.1172 0.0487 0.0586 0.0654 0.0586 12.3431 10.8733 6.0249 5.5923 0.2376 0.8258 6.7485 10.7693
CLQR 0.1809 (37) 0.1430(18) 0.0698 (44) 0.0650 (11) 0.1111 (70) 0.0779 (33) 28.0286 (127) 22.2785 (105) 10.3054 (71) 7.1989 (29) 0.1127 (53) 0.5630 (32) 2.3603 (–65) 5.6236 (–48)
4p
Uncontrol 1.3762 – 0.4484 – 0.9278 – 13.7182 – 5.4196 – – – – –
OCLC 0.2464 0.2126 0.1013 0.1063 0.1454 0.1063 1.4963 1.1975 0.8381 0.7142 0.3717 0.9648 10.3887 16.7352
CLQR 0.5165 (52) 0.3682(42) 0.1903 (88) 0.1507 (42) 0.3263 (124) 0.2174 (105) 5.2769 (253) 3.9386 (229) 1.8969 (126) 1.2596 (76) 0.1558 (58) 0.5593 (42) 3.3847 (–67) 7.1133 (–57)
6p
Uncontrol 1.5833 – 0.5494 – 1.0339 – 2.9525 – 1.1932 – – – – –
OCLC 0.3721 0.3089 0.1461 0.1545 0.2263 0.1545 0.4774 0.3715 0.2584 0.2228 0.3591 0.9601 10.5935 17.6816
CLQR 0.7119 (48) 0.5469(44) 0.2569 (76) 0.2152 (39) 0.4551 (101) 0.3317 (115) 1.3820 (189) 1.0908 (194) 0.5234 (103) 0.3806 (71) 0.1666 (54) 0.5685 (41) 2.9688 (–72) 5.8325 (–67)
El Centro
2p
Uncontrol 0.1776 – 0.0549 – 0.1226 – 27.9693 – 11.4614 – – – – –
OCLC 0.0371 0.0375 0.0116 0.0122 0.0254 0.0253 3.3096 2.1675 2.3369 2.3241 0.1568 0.5284 1.9267 3.5123
CLQR 0.0749 (51) 0.0501(25) 0.0249(114) 0.0205 (68) 0.0500 (97) 0.0296 (17) 11.2056 (239) 7.6364 (252) 4.6344 (98) 2.7265 (17) 0.1064 (32) 0.3200 (39) 1.0592 (–45) 2.0142 (–43)
4p
Uncontrol 0.4343 – 0.1297 – 0.3045 – 4.6504 – 1.7786 – – – – –
OCLC 0.0771 0.0586 0.0285 0.0192 0.0486 0.0394 0.2645 0.1654 0.2803 0.2273 0.1667 0.7820 4.3162 5.9339
CLQR 0.1881 (59) 0.1390(58) 0.0648(127) 0.0545(184) 0.1233 (154) 0.0845 (114) 1.9133 (623) 1.5126 (815) 0.7161 (155) 0.4889 (115) 0.1031 (38) 0.3571 (54) 1.1280 (–74) 2.4917 (–58)
6p
Uncontrol 0.2460 – 0.0759 – 0.1702 – 0.5185 – 0.1963 – – – – –
OCLC 0.0843 0.0651 0.0322 0.0219 0.0522 0.0432 0.0861 0.0631 0.0597 0.0492 0.1257 0.7962 2.7438 4.3790
CLQR 0.1406 (40) 0.1096(41) 0.0475 (48) 0.0424 (93) 0.0931 (78) 0.0672 (56) 0.2673 (210) 0.2291 (263) 0.1070 (79) 0.0770 (57) 0.0701 (44) 0.3758 (53) 0.5412 (–80) 1.4849 (–66)
Kobe
2p
Uncontrol 0.0847 – 0.0296 – 0.0551 – 12.7789 – 5.1512 – – – – –
OCLC 0.0200 – 0.0072 – 0.0127 – 1.2910 – 1.1733 – 0.0755 – 1.5237 –
CLQR 0.0335 (40) – (–) 0.0124 (71) – (–) 0.0212 (66) – (–) 4.6591 (261) – (–) 1.9634 (67) – (–) 0.0755 (38) – (–) 0.4438 (–71) – (–)
4p
Uncontrol 0.1510 – 0.0424 – 0.1086 – 1.6370 – 0.6344 – – – – –
OCLC 0.0373 – 0.0128 – 0.0246 – 0.2084 – 0.1416 – 0.0801 – 1.5863 –
CLQR 0.0748 (50) – (–) 0.0238 (87) – (–) 0.0509 (107) – (–) 0.7206 (246) – (–) 0.2959 (109) – (–) 0.0801 (41) – (–) 0.4594 (–71) – (–)
6p
Uncontrol 0.1916 – 0.0593 – 0.1323 – 0.4111 – 0.1526 – – – – –
OCLC 0.0319 – 0.0131 – 0.0188 – 0.0043 – 0.0216 – 0.0966 – 3.2482 –
CLQR 0.0934 (66) – (–) 0.0328(150) – (–) 0.0606 (223) – (–) 0.1778 (4022) – (–) 0.0697 (223) – (–) 0.0966 (55) – (–) 0.4185 (–87) – (–)
126
Table 6.19: Peak response statistics for 2DOF buildings with different fundamental frequencyw
1
forz
1
= 0:025 andr
a
= 10
16
kg
–2
;D
is the percent change of CLQR relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Excitation w
1
Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation [ rad=s] Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN
2p
Uncontrol 1.3891 – 1.9402 – 25.8916 – 18.6965 –
OCLC 1.0970 1.1858 0.8898 1.1858 13.7792 12.7847 8.6389 7.7809
CLQR 1.2564(15) 1.1678(–2) 1.3431 (51) 1.1425 (–4) 19.2205 (39) 15.9030 (24) 13.0351 (51) 11.0767 (42)
4p
Uncontrol 2.5086 – 3.2237 – 12.7199 – 7.7340 –
OCLC 1.3908 1.3312 1.4424 1.3312 5.1311 4.8349 3.4265 3.2259
CLQR 2.0103(45) 2.0622(55) 2.5764 (79) 2.5060 (88) 9.2983 (81) 9.3918 (94) 6.2425 (82) 6.0741 (88)
6p
Uncontrol 3.3071 – 3.8322 – 6.1956 – 4.0948 –
OCLC 1.6223 1.5856 1.7046 1.5856 2.6176 2.4424 1.8178 1.5548
CLQR 2.1946(35) 1.9615(24) 2.7456 (61) 2.4071 (52) 4.2609 (63) 3.8501 (58) 2.9546 (63) 2.6012 (67)
El Centro
2p
Uncontrol 1.2249 – 1.6497 – 24.3513 – 15.8391 –
OCLC 0.9678 0.6586 0.6988 0.7146 7.4591 6.6913 6.6886 6.8552
CLQR 0.9495(–2) 0.8776(33) 1.0117 (45) 0.8729 (22) 14.8321 (99) 11.8946 (78) 9.8241 (47) 8.4458 (23)
4p
Uncontrol 1.4624 – 2.4693 – 8.5723 – 5.9521 –
OCLC 1.0403 0.9280 0.9589 0.9527 3.0879 2.1336 2.3038 2.2869
CLQR 1.4769(42) 1.5399(66) 1.9707(106) 1.9974(110) 6.8007 (120) 6.5908 (209) 4.7636 (107) 4.8351 (111)
6p
Uncontrol 1.5299 – 1.8823 – 3.2745 – 2.0370 –
OCLC 1.0254 0.8756 1.3599 1.2405 1.5239 1.6716 1.4513 1.3144
CLQR 1.3845(35) 1.3837(58) 1.5054 (11) 1.5392 (24) 2.6792 (76) 2.5715 (54) 1.6293 (12) 1.6626 (26)
Kobe
2p
Uncontrol 0.8300 – 1.0236 – 15.3512 – 9.8565 –
OCLC 1.0556 – 0.6284 – 6.4566 – 5.9995 –
CLQR 1.0231(–3) – (–) 0.8322 (32) – (–) 12.3257 (91) – (–) 8.0597 (34) – (–)
4p
Uncontrol 0.9075 – 1.2383 – 5.1095 – 2.9775 –
OCLC 1.0128 – 0.8601 – 2.4471 – 2.0609 –
CLQR 1.0038(–1) – (–) 1.1090 (29) – (–) 3.9618 (62) – (–) 2.6927 (31) – (–)
6p
Uncontrol 1.4246 – 1.7306 – 3.3834 – 1.8548 –
OCLC 1.0149 – 0.8403 – 0.5272 – 0.8996 –
CLQR 1.1443(13) – (–) 1.4293 (70) – (–) 2.3666 (349) – (–) 1.5406 (71) – (–)
127
Table 6.20: Numerical simulated mean square response statistics for 2DOF buildings with different fundamental frequency w
1
values
subjected to El Centro forz
1
= 0:025 andr
a
= 10
16
kg
–2
and r= 15%;D is the percent change of CLQR relative to OCLC.
Design and J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q
2
q
1
)
2
] E[I c] E[u
2
]
Evaluation w
1
Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation [ rad=s] Strategy [m
2
/s
4
][%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [–] [%] [–] [%] [(%m
1
g)
2
] [%] [(%m
1
g)
2
] [%]
El Centro
2p
Uncontrol 0.1776 – 0.0549 – 0.1226 – 27.9693 – 11.4614 – – – – –
OCLC 0.0376 – 0.0119 – 0.0257 – 3.3044 – 2.3634 – 0.1153 – 1.9994 –
CLQR 0.0706(47) – (–) 0.0237 (99) – (–) 0.0470 (83) – (–) 9.7793 (196) – (–) 4.3499 (84) – (–) 0.1153 (15) – (–) 1.2909 (–35) – (–)
4p
Uncontrol 0.4343 – 0.1297 – 0.3045 – 4.6504 – 1.7786 – – – – –
OCLC 0.0764 – 0.0286 – 0.0480 – 0.2570 – 0.2769 – 0.1429 – 4.5367 –
CLQR 0.1673(54) – (–) 0.0581(103) – (–) 0.1092 (128) – (–) 1.6424 (539) – (–) 0.6338 (129) – (–) 0.1429 (36) – (–) 1.3685 (–70) – (–)
6p
Uncontrol 0.2460 – 0.0759 – 0.1702 – 0.5185 – 0.1963 – – – – –
OCLC 0.0839 – 0.0319 – 0.0519 – 0.0858 – 0.0593 – 0.1059 – 2.7856 –
CLQR 0.1323(37) – (–) 0.0457 (43) – (–) 0.0866 (67) – (–) 0.2387 (178) – (–) 0.0995 (68) – (–) 0.1059 (35) – (–) 0.7882 (–72) – (–)
Table 6.21: Numerical simulated peak response statistics for 2DOF buildings with different fundamental frequencyw
1
values subjected
to El Centro forz
1
= 0:025 andr
a
= 10
16
kg
–2
and r= 15%;D is the percent change of CLQR relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Evaluation w
1
Control Num. (D) Exper.(D) Num. (D) Exper.(D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation [ rad=s] Strategy [m/s
2
] [%] [m/s
2
][%] [m/s
2
] [%] [m/s
2
][%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
El Centro
2p
Uncontrol 1.2249 – 1.6497 – 24.3513 – 15.8391 –
OCLC 1.1777 – 0.6772 – 8.1347 – 6.4776 –
CLQR 1.3362(13) – (–) 0.9312 (38) – (–) 13.7772 (69) – (–) 9.0376 (40) – (–)
4p
Uncontrol 1.4624 – 2.4693 – 8.5723 – 5.9521 –
OCLC 1.2739 – 0.9515 – 3.1164 – 2.2852 –
CLQR 1.4171(11) – (–) 1.9046(100) – (–) 6.4749 (108) – (–) 4.6027 (101) – (–)
6p
Uncontrol 1.5299 – 1.8823 – 3.2745 – 2.0370 –
OCLC 1.2886 – 1.3437 – 1.5257 – 1.4333 –
CLQR 1.3538 (5) – (–) 1.4468 (8) – (–) 2.5650 (68) – (–) 1.5640 (9) – (–)
128
Statistics of response to non-design excitation: The performance of OCLC is evaluated for an
excitation other than its design excitation through both numerical simulation and RTHS. Tables
6.22 to 6.25 show the 2DOF building models’ cost metric, mean square and peak absolute acceler-
ation and inter-story drift responses from numerical simulations and RTHS using OCLC designed
for GWN but evaluated with the El Centro earthquake and vice versa. Both numerical simulation
and RTHS results demonstrate that, for the 2DOF systems, OCLC designed for one specific exci-
tation achieves very similar performance improvements for the same structure evaluated with the
other non-design excitation. Note that this improvement is more significant and consistent than it
was for the SDOF system.
When r
a
= 10
13
kg
–2
(Tables 6.22 and 6.23), for the case when OCLC is designed for a
GWN excitation but the structure is excited by the El Centro earthquake, CLQR’s cost metric, first
and second floor mean square absolute accelerations are up to twice those of OCLC in numerical
simulation, and up to two-and-a-half times those of OCLC for first floor mean square absolute
acceleration in RTHS tests. Further, CLQR has a mean square first story drift that is up to four
times that of OCLC. The OCLC can simultaneously decrease peak responses, as shown in Table
6.23; e.g., CLQR has first story peak drift and second floor peak absolute acceleration that are up
to 103% and 115% larger, respectively, than those of OCLC. When the OCLC designed for El
Centro is evaluated with a GWN excitation, CLQR has a cost metric that is 76% larger than that of
OCLC when the building fundamental frequencyw
1
= 4p rad=s in numerical simulation; although
the improvement in RTHS tests is not as significant as in numerical simulation, CLQR still has a
cost metric that is up to 31% larger than that of OCLC. Notably, CLQR’s first story mean square
drift is seven times that of OCLC in numerical simulation and more than five times larger in RTHS
tests. Further, the first story peak drift and second floor peak absolute acceleration with CLQR are
up to 149% and 97% larger, respectively, than those with OCLC.
When the control weight r
a
is decreased to 10
16
kg
–2
(Tables 6.24 to 6.25), the OCLCs can
provide similar performance in reducing cost metric as well as the mean square and peak structural
responses. Remarkably, when the OCLC designed for GWN is used to control the structure excited
129
Table 6.22: Mean square response statistics for 2DOF buildings subjected to a non-design excitation forz
1
= 0:025 andr
a
= 10
13
kg
–2
;
D is the percent change of CLQR relative to OCLC.
J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Design Evaluation w 1 Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation [ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN El Centro
2p
OCLC 0.0370 0.0374 0.0116 0.0121 0.0255 0.0253 3.2119 2.1034 2.3409 2.3250 0.1581 0.5083 1.9641 3.5280
CLQR 0.0727 (96) 0.0480 (22) 0.0245(112) 0.0195 (61) 0.0482 (89) 0.0285 (12) 10.8827 (239) 7.2076 (243) 4.4669 (91) 2.6214 (13) 0.1011 (36) 0.3214 (37) 1.0320 (–47) 2.0064 (–43)
4p
OCLC 0.0883 0.0569 0.0318 0.0208 0.0565 0.0361 0.6256 0.3526 0.3259 0.2073 0.1904 0.7809 3.0938 5.2090
CLQR 0.1854 (110) 0.1347 (58) 0.0641(102) 0.0532(156) 0.1213(115) 0.0815 (126) 1.8873 (202) 1.4773 (319) 0.7045 (116) 0.4716 (128) 0.1070 (44) 0.3614 (54) 1.1839 (–62) 2.6401 (–49)
6p
OCLC 0.0900 0.0652 0.0332 0.0224 0.0568 0.0428 0.1334 0.0891 0.0650 0.0487 0.1173 0.7978 1.5444 3.8000
CLQR 0.1392 (55) 0.1074 (39) 0.0472 (42) 0.0417 (86) 0.0920 (62) 0.0657 (54) 0.2646 (98) 0.2252 (153) 0.1058 (63) 0.0752 (55) 0.0734 (37) 0.3783 (53) 0.5387 (–65) 1.5642 (–59)
El Centro GWN
2p
OCLC 0.1143 0.1164 0.0485 0.0582 0.0658 0.0582 13.1245 11.5244 6.0583 5.5073 0.2275 0.8158 6.3537 10.2942
CLQR 0.1780 (56) 0.1410 (17) 0.0691 (43) 0.0648 (11) 0.1088 (65) 0.0762 (31) 27.5624 (110) 21.8495 (90) 10.0932 (67) 7.0334 (28) 0.1170 (49) 0.5746 (30) 2.4920 (–61) 5.8464 (–43)
4p
OCLC 0.2864 0.2769 0.1229 0.1384 0.1635 0.1384 0.7243 0.7018 0.9446 0.8642 0.4211 0.9654 17.0527 22.4748
CLQR 0.5044 (76) 0.3572 (22) 0.1870 (52) 0.1479 (7) 0.3174 (94) 0.2094 (51) 5.1532 (611) 3.8296 (446) 1.8445 (95) 1.2124 (40) 0.1618 (62) 0.5745 (40) 3.5391 (–79) 7.3996 (–67)
6p
OCLC 0.4056 0.3641 0.1645 0.1821 0.2411 0.1821 0.3022 0.2735 0.2754 0.2471 0.4251 0.9616 17.7022 21.9552
CLQR 0.7005 (73) 0.5291 (31) 0.2539 (54) 0.2105 (16) 0.4466 (85) 0.3186 (75) 1.3604 (350) 1.0610 (288) 0.5136 (86) 0.3653 (48) 0.1711 (60) 0.5896 (39) 3.0676 (–83) 6.3203 (–71)
Table 6.23: Peak response statistics for 2DOF buildings subjected to a non-design excitation for z
1
= 0:025 and r
a
= 10
13
kg
–2
;D is
the percent change of CLQR relative to OCLC.
( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Design Evaluation w
1
Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation [ rad=s] Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN El Centro
2p
OCLC 0.9455 0.6438 0.7052 0.7169 7.1895 6.3866 6.7524 6.8789
CLQR 0.9643 (2) 0.8443(31) 1.0017(42) 0.8418 (17) 14.5744(103) 11.8200 (85) 9.7268 (44) 8.1430 (18)
4p
OCLC 1.2522 0.9418 1.0602 0.9213 4.7174 3.6454 2.5328 2.2028
CLQR 1.5009 (20) 1.5401(64) 1.9715(86) 1.9786(115) 6.7851 (44) 6.5108 (79) 4.7665 (88) 4.7890 (117)
6p
OCLC 1.0327 0.9046 1.2561 1.2815 1.6998 1.9148 1.3349 1.3544
CLQR 1.3764 (33) 1.3819(53) 1.5003(19) 1.5351 (20) 2.6604 (57) 2.5595 (34) 1.6239 (22) 1.6579 (22)
El Centro GWN
2p
OCLC 1.1099 1.1907 0.9047 1.1907 13.7620 12.9615 8.7817 8.0341
CLQR 1.2408 (12) 1.1584(–3) 1.3031(44) 1.1170 (–6) 18.8930 (37) 15.6456 (21) 12.6326 (44) 10.8328 (35)
4p
OCLC 1.3788 1.3834 1.2952 1.3834 3.7383 3.7608 3.1104 2.9881
CLQR 2.0222 (47) 2.0998(52) 2.5516(97) 2.4780 (79) 9.3122 (149) 9.3801 (149) 6.1859 (99) 6.0113 (101)
6p
OCLC 1.5471 1.6759 1.7637 1.6759 2.2892 2.1833 1.8828 1.7446
CLQR 2.1626 (40) 2.1633(29) 2.7033(53) 2.3401 (40) 4.2459 (85) 3.8056 (74) 2.9086 (54) 2.5263 (45)
130
Table 6.24: Mean square response statistics for 2DOF buildings subjected to a non-design excitation forz
1
= 0:025 andr
a
= 10
16
;D
is the percent change of CLQR relative to OCLC.
J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Design Evaluation w 1 Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation [ rad=s] Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN El Centro
2p
OCLC 0.0371 0.0374 0.0116 0.0120 0.0255 0.0253 3.2418 2.1028 2.3421 2.3271 0.1565 0.5232 1.9513 3.5246
CLQR 0.0749 (102) 0.0501(25) 0.0249(115) 0.0205 (70) 0.0500 (96) 0.0296 (17) 11.2057 (246) 7.6364 (263) 4.6344 (98) 2.7265 (17) 0.1064 (32) 0.3200 (39) 1.0592 (–46) 2.0142 (–43)
4p
OCLC 0.0878 0.0568 0.0316 0.0207 0.0562 0.0361 0.6138 0.3429 0.3241 0.2073 0.1875 0.7820 3.1274 5.2073
CLQR 0.1881 (114) 0.1390(59) 0.0648(105) 0.0545(163) 0.1233(119) 0.0845 (134) 1.9134 (212) 1.5126 (341) 0.7161 (121) 0.4889 (136) 0.1031 (45) 0.3571 (54) 1.1280 (–64) 2.4917 (–52)
6p
OCLC 0.0908 0.0657 0.0336 0.0227 0.0572 0.0430 0.1360 0.0914 0.0655 0.0489 0.1179 0.7962 1.4944 3.8292
CLQR 0.1406 (55) 0.1096(40) 0.0475 (42) 0.0424 (87) 0.0931 (63) 0.0672 (56) 0.2673 (97) 0.2291 (151) 0.1070 (63) 0.0770 (57) 0.0701 (41) 0.3758 (53) 0.5412 (–64) 1.4849 (–61)
El Centro GWN
2p
OCLC 0.1140 0.1167 0.0484 0.0584 0.0656 0.0584 12.5627 10.9639 6.0387 5.5810 0.2352 0.8250 6.6715 10.6156
CLQR 0.1809 (59) 0.1430(18) 0.0698 (44) 0.0650 (11) 0.1111 (69) 0.0779 (34) 28.0518 (123) 22.2785 (103) 10.3087 (71) 7.1989 (29) 0.1127 (52) 0.5630 (32) 2.3603 (–65) 5.6236 (–47)
4p
OCLC 0.2871 0.2761 0.1223 0.1381 0.1647 0.1381 0.7194 0.6988 0.9519 0.8682 0.4232 0.9648 17.1250 22.4597
CLQR 0.5165 (80) 0.3682(25) 0.1901 (55) 0.1507 (9) 0.3264 (98) 0.2174 (58) 5.2776 (634) 3.9386 (464) 1.8971 (99) 1.2596 (45) 0.1558 (63) 0.5593 (42) 3.3847 (–80) 7.1133 (–68)
6p
OCLC 0.4051 0.3626 0.1637 0.1813 0.2414 0.1813 0.3022 0.2717 0.2757 0.2471 0.4232 0.9601 17.6058 22.0115
CLQR 0.7119 (76) 0.5469(34) 0.2568 (57) 0.2152 (19) 0.4552 (89) 0.3317 (83) 1.3823 (357) 1.0908 (301) 0.5235 (90) 0.3806 (54) 0.1666 (61) 0.5685 (41) 2.9688 (–83) 5.8325 (–74)
Table 6.25: Peak response statistics for 2DOF buildings subjected to a non-design excitation for z
1
= 0:025 and r
a
= 10
16
kg
–2
;D is
the percent change of CLQR relative to OCLC.
( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Design Evaluation w
1
Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation [ rad=s] Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN El Centro
2p
OCLC 0.9705 0.6351 0.7047 0.7193 7.2835 6.4683 6.7474 6.9011
CLQR 0.9495 (–2) 0.8776(38) 1.0117 (44) 0.8729 (21) 14.8321 (104) 11.8946 (84) 9.8241 (46) 8.4458 (22)
4p
OCLC 1.2397 0.9978 1.0542 0.9262 4.6821 3.6200 2.5191 2.2146
CLQR 1.4769 (19) 1.5399(54) 1.9707 (87) 1.9974(116) 6.8007 (45) 6.5908 (82) 4.7636 (89) 4.8351 (118)
6p
OCLC 1.0880 0.9260 1.2549 1.2951 1.7431 1.9719 1.3332 1.3683
CLQR 1.3845 (27) 1.3837(49) 1.5054 (20) 1.5392 (19) 2.6792 (54) 2.5715 (30) 1.6293 (22) 1.6626 (22)
El Centro GWN
2p
OCLC 1.0668 1.2166 0.8917 1.2166 13.7684 12.9405 8.6571 7.8530
CLQR 1.2477 (17) 1.1678(–4) 1.3357 (50) 1.1425 (–6) 19.1625 (39) 15.9030 (23) 12.9640 (50) 11.0767 (41)
4p
OCLC 1.2958 1.3848 1.2995 1.3848 3.7366 3.7737 3.1211 2.9901
CLQR 2.0157 (56) 2.0622(49) 2.5709(98) 2.5060 (81) 9.3213 (149) 9.3918 (149) 6.2292 (100) 6.0741 (103)
6p
OCLC 1.5291 1.6801 1.7668 1.6801 2.2944 2.1874 1.8864 1.7595
CLQR 2.1931 (43) 1.9615(17) 2.7473 (55) 2.4071 (43) 4.2667 (86) 3.8501 (76) 2.9564 (57) 2.6012 (48)
131
by the El Centro earthquake, CLQR has cost metric, first and second floor mean square absolute
accelerations that are up to twice those of OCLC in numerical simulation, and up to two-and-a-
half times that of OCLC for first floor mean square absolute acceleration in RTHS tests. Moreover,
CLQR has a first story peak drift that is 104% larger than that of OCLC. When the system is
excited by GWN but using the OCLC designed for the El Centro earthquake, CLQR has a cost
metric that is up to 80% larger, with the first story mean square and peak drifts up to seven and
two-and-a-half times those of OCLC, respectively. In all, OCLC designed for one excitation can
still provide performance superior to CLQR when the 2DOF building is subjected to the other
non-design excitation.
6.3.5.2 Bridge deck model
Selected time histories: Figures 6.15 to 6.18 provide a means to compare the absolute accelera-
tion, drift and control force time histories from numerical simulation and RTHS tests for the bridge
deck model when the control weight r
a
= 10
13
kg
–2
. Notably, with both CLQR and OCLC, the
bridge pier absolute accelerations oscillate at a higher frequency than the bridge deck, and the
pier drifts are smaller than the deck drifts: both of their behaviors are expected given the relative
stiffness of the pier and the relative flexibility of the pier-deck connection. OCLC provides better
performance in reducing absolute acceleration and drift responses than CLQR. RTHS experimen-
tal absolute accelerations and drifts are in excellent agreement with numerical simulated results,
though there are some notable differences in the control forces. The peak drifts of numerical simu-
lation in each cycle are slightly larger with less than a half millimeter difference; the peak absolute
accelerations of numerical simulation in each cycle are overestimated, with a difference on the
order of 0.02 m=s
2
for a signal that has a peak of about 0.2 m=s
2
.
Statistics of response to OCLC design excitation: Tables 6.26 to 6.29 show the mean square and
peak bridge deck model responses and control forces from numerical simulation and RTHS tests
with different control force weights (r
a
= 10
13
kg
–2
for Tables 6.26 and 6.27 andr
a
= 10
20
kg
–2
for Tables 6.28 and 6.29). The pier’s mean square and peak absolute accelerations and drifts are
132
0 2 4 6 8 10
Time [s]
0
0.04
0.08
0.12
experiment
num. sim.
(a) Pier absolute acceleration
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
2
2.5
experiment
num. sim.
(b) Pier drift
0 2 4 6 8 10
Time [s]
0
0.02
0.04
0.06
experiment
num. sim.
(c) Deck absolute acceleration
0 2 4 6 8 10
Time [s]
0
1
2
3
4
experiment
num. sim.
(d) Deck drift
20 22 24 26 28 30
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(e) Control force
Figure 6.15. Numerical simulation and RTHS bridge deck GWN response time histories with
CLQR with control weightr
a
= 10
13
kg
–2
.
133
0 2 4 6 8 10
Time [s]
0
0.04
0.08
0.12
experiment
num. sim.
(a) Pier absolute acceleration
0 2 4 6 8 10
Time [s]
0
0.5
1
1.5
2
2.5
experiment
num. sim.
(b) Pier drift
0 2 4 6 8 10
Time [s]
0
0.02
0.04
0.06
experiment
num. sim.
(c) Deck absolute acceleration
0 2 4 6 8 10
Time [s]
0
1
2
3
4
experiment
num. sim.
(d) Deck drift
20 22 24 26 28 30
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(e) Control force
Figure 6.16. Numerical simulation and RTHS bridge deck GWN response time histories with
OCLC with control weightr
a
= 10
13
kg
–2
.
134
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
0.2
0.25
experiment
num. sim.
(a) Pier absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(b) Pier drift
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(c) Deck absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
8
experiment
num. sim.
(d) Deck drift
10 12 14 16 18 20
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(e) Control force
Figure 6.17. Numerical simulation and RTHS bridge deck El Centro response time histories with
CLQR with control weightr
a
= 10
13
kg
–2
.
135
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
0.2
0.25
experiment
num. sim.
(a) Pier absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
experiment
num. sim.
(b) Pier drift
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(c) Deck absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
8
experiment
num. sim.
(d) Deck drift
10 12 14 16 18 20
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(e) Control force
Figure 6.18. Numerical simulation and RTHS bridge deck El Centro response time histories with
OCLC with control weightr
a
= 10
13
kg
–2
.
136
smaller than those of the deck when the external excitation is a GWN excitation. OCLC always
supplies performance comparable to or, for most cases, superior to that of CLQR in minimizing
the cost metric, mean square and peak absolute accelerations and drifts. Although the experimental
responses are a little smaller than the numerical simulation responses, the performance improve-
ments provided by OCLC over CLQR, by decreasing structural responses, are almost identical; in
fact, the OCLC improvements in the RTHS tests are even larger than those of pure simulation.
Whenr
a
= 10
13
kg
–2
(Tables 6.26 and 6.27), OCLC has more significantly performance im-
provements over CLQR in reducing structural responses when designed for and subjected to GWN
excitation. Notably, CLQR’s cost metric, bridge and deck mean square absolute accelerations and
drifts are 28%–45% larger than those of OCLC for both numerical simulation and RTHS tests,
with a 47% smallerE[I
c
] value (though the mean square control force is 84% smaller than OCLC).
Further, the CLQR case exhibits peak absolute accelerations and drifts that are 12%–17% larger
than those of OCLC. OCLC’s performance improvements in the El Centro earthquake are similar
to those with the Kobe earthquake. The cost metric and mean square absolute accelerations and
drifts with CLQR are 14%–34% larger, with a 55% smallerE[I
c
] value. Nevertheless, OCLC per-
forms comparably to CLQR when decreasing the peak responses for both the El Centro and Kobe
earthquakes.
When control weightr
a
= 10
20
kg
–2
(Tables 6.28 to 6.29), the superior performance of OCLC
is maintained, particularly for the GWN and El Centro design/evaluation excitation. Remarkably,
CLQR provides 32%–78% larger cost metric, mean square absolute accelerations and drifts, and
15%–32% peak values, when subjected to a GWN excitation. When subjected to the Kobe earth-
quake, CLQR has a 21% larger mean square pier absolute acceleration, but slightly smaller mean
square deck absolute acceleration and pier and deck drifts. Further, CLQR provides modestly
larger peak responses for the El Centro earthquake, but slightly smaller peak deck absolute accel-
eration and pier and deck peak drifts for the Kobe earthquake.
Statistics of response to non-design excitation: Furthermore, Tables 6.30 to 6.33 tabulate the
cost metric, mean square and peak absolute acceleration and drift responses to an excitation other
137
Table 6.26: Mean square response statistics for the bridge deck model with control weightr
a
= 10
13
kg
–2
;D is the percent change of
CLQR relative to OCLC.
Design and J
a
E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q
2
q
1
)
2
] E[I
c
] E[u
2
]
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [–] [%] [–] [%] [(%m
1
g)
2
] [%] [(%m
1
g)
2
] [%]
GWN
Uncontrol 0.0102 – 0.0027 – 0.0075 – 8.6101 – 31.7637 – – – – –
OCLC 0.0067 0.0047 0.0019 0.0014 0.0048 0.0033 5.4983 3.7932 19.8402 13.8398 0.7723 0.9112 0.0041 0.0078
CLQR 0.0093(28) 0.0065 (28) 0.0026(32) 0.0019(30) 0.0067 (41) 0.0047 (42) 7.7220 (40) 5.3724 (42) 28.4153 (43) 20.0367 (45) 0.4069 (47) 0.4988 (45) 0.0007 (–84) 0.0038 (–51)
El Centro
Uncontrol 0.0086 – 0.0037 – 0.0049 – 5.5937 – 20.6392 – – – – –
OCLC 0.0069 0.0054 0.0029 0.0022 0.0039 0.0031 4.5175 3.5812 16.3606 13.1857 0.5257 0.8327 0.0032 0.0072
CLQR 0.0082(16) 0.0067 (20) 0.0037(26) 0.0030(34) 0.0045 (14) 0.0037 (19) 5.1588 (14) 4.2720 (19) 18.9499 (16) 15.9864 (21) 0.2373 (55) 0.5327 (36) 0.0008 (–74) 0.0042 (–41)
Kobe
Uncontrol 0.0048 – 0.0034 – 0.0015 – 1.6923 – 6.2078 – – – – –
OCLC 0.0041 – 0.0029 – 0.0011 – 1.3012 – 4.6146 – 0.4030 – 0.0028 –
CLQR 0.0051(20) – (–) 0.0037(27) – (–) 0.0013 (19) – (–) 1.5625 (20) – (–) 5.6592 (23) – (–) 0.4030 (60) – (–) 0.0007 (–74) – (–)
Table 6.27: Peak response statistics for the bridge deck model with control weight r
a
= 10
13
kg
–2
;D is the percent change of CLQR
relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m/s
2
] [%] [m/s
2
][%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN
Uncontrol 0.1754 – 0.2258 – 7.9000 – 14.6907 –
OCLC 0.1549 0.1357 0.1829 0.1487 6.5229 5.2267 11.8873 9.5869
CLQR 0.1729(12) 0.1461 (8) 0.2138(17) 0.1732 (16) 7.5360 (16) 6.0342 (15) 13.9060 (17) 11.2597 (17)
El Centro
Uncontrol 0.2070 – 0.1577 – 5.3457 – 10.2622 –
OCLC 0.2127 0.1960 0.1517 0.1417 5.0108 4.7280 9.8601 9.2260
CLQR 0.2156 (1) 0.1973 (1) 0.1554 (2) 0.1467 (4) 5.2639 (5) 5.0003 (6) 10.0973 (2) 9.5540 (4)
Kobe
Uncontrol 0.3245 – 0.1136 – 3.7018 – 7.3906 –
OCLC 0.3222 – 0.1128 – 3.5460 – 7.2379 –
CLQR 0.3498 (9) – (–) 0.1119(–1) – (–) 3.5855 (1) – (–) 7.2779 (1) – (–)
138
Table 6.28: Mean square response statistics for the bridge deck model with control weightr
a
= 10
20
kg
–2
;D is the percent change of
CLQR relative to OCLC.
Design and J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q
2
q
1
)
2
] E[I c] E[u
2
]
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m
2
/s
4
][%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m
1
g)
2
] [%] [(%m
1
g)
2
] [%]
GWN
Uncontrol 0.0102 – 0.0027 – 0.0075 – 8.6101 – 31.7637 – – – – –
OCLC 0.0067 0.0047 0.0019 0.0014 0.0048 0.0033 5.5491 3.8253 20.0392 13.9675 0.7632 0.8975 0.0040 0.0077
CLQR 0.0098(32) 0.0079(41) 0.0027 (41) 0.0021(52) 0.0072 (49) 0.0058 (75) 8.2362 (48) 6.6981 (75) 30.2770 (51) 24.8627 (78) 0.3356 (56) 0.3276 (63) 0.0006 (–85) 0.0019 (–75)
El Centro
Uncontrol 0.0086 – 0.0037 – 0.0049 – 5.5937 – 20.6392 – – – – –
OCLC 0.0071 0.0055 0.0031 0.0024 0.0039 0.0031 4.5285 3.5065 16.4699 12.9814 0.4629 0.8465 0.0028 0.0075
CLQR 0.0085(17) 0.0074(26) 0.0038 (22) 0.0031(30) 0.0047 (18) 0.0043 (40) 5.3710 (19) 4.9343 (41) 19.6982 (20) 18.3330 (41) 0.2577 (44) 0.3425 (60) 0.0007 (–74) 0.0019 (–74)
Kobe
Uncontrol 0.0048 – 0.0034 – 0.0015 – 1.6923 – 6.2078 – – – – –
OCLC 0.0050 – 0.0033 – 0.0017 – 1.9889 – 7.1262 – 0.4446 – 0.0017 –
CLQR 0.0055 (9) – (–) 0.0040 (21) – (–) 0.0015 (–12) – (–) 1.7682 (–11) – (–) 6.3705 (–11) – (–) 0.4446 (29) – (–) 0.0013 (–27) – (–)
Table 6.29: Peak response statistics for the bridge deck model with control weight r
a
= 10
20
kg
–2
;D is the percent change of CLQR
relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN
Uncontrol 0.1754 – 0.2258 – 7.9000 – 14.6907 –
OCLC 0.1533 0.1319 0.1833 0.1494 6.5326 5.2246 11.9175 9.6343
CLQR 0.1767(15) 0.1661 (26) 0.2236 (22) 0.1978 (32) 7.8472 (20) 6.8907 (32) 14.5485 (22) 12.8590 (33)
El Centro
Uncontrol 0.2070 – 0.1577 – 5.3457 – 10.2622 –
OCLC 0.2115 0.1957 0.1515 0.1405 5.0114 4.6765 9.8460 9.1388
CLQR 0.2160 (2) 0.1994 (2) 0.1561 (3) 0.1535 (9) 5.3032 (6) 5.1979 (11) 10.1446 (3) 9.9967 (9)
Kobe
Uncontrol 0.3245 – 0.1136 – 3.7018 – 7.3906 –
OCLC 0.3280 – 0.1146 – 3.6704 – 7.3858 –
CLQR 0.3581 (9) – (–) 0.1125 (–2) – (–) 3.6262 (–1) – (–) 7.3136 (–1) – (–)
139
than the OCLC design earthquake for the two control weights r
a
. OCLC designed for a specific
excitation can still provide performance superior to CLQR when the structure is subjected to the
other excitation. The largest performance improvements of OCLC over CLQR are provided by the
OCLC designed for the El Centro earthquake when evaluated by a GWN excitation with control
weight r
a
= 10
20
kg
–2
: CLQR provides a 38%–40% larger cost metric, 38%–70% larger pier
and deck mean square absolute accelerations and drifts, and 19%–31% larger pier and deck peak
absolute accelerations and drifts. The OCLC designed for a GWN excitation provides moderate
performance improvements when the structure is subjected to the El Centro earthquake, though
CLQR still provides up to 46% larger cost metric and mean square and peak responses relative to
OCLC. In all cases, the improvements provided by OCLC, relative to CLQR, in the RTHS tests
are quite similar to those in numerical simulations.
6.3.5.3 Base isolated structure
Selected time histories: Figures 6.19 to 6.22 display the absolute acceleration, drift and control
force time histories of the base-isolated structure when the control weight r
a
= 10
13
kg
–2
. The
maximum magnitude of the superstructure drift is about 2% of the base drift for both CLQR and
OCLC strategies when subjected to either GWN or El Centro earthquake — as expected since
the purpose of the isolation is to mitigate superstructure drift and absolute acceleration responses.
The numerical simulation and RTHS results are in the same phase, though have relatively larger
differences than the 2DOF building and bridge deck models. In the RTHS tests, the base absolute
accelerations exhibit significant high frequency content that is induced by the corresponding high
frequency content of the control force due to sensor noise. To verify that removing this high
frequency content provides results closer to numerical simulation, the RTHS measured control
force is filtered with an 8
th
order Butterworth filter with the half-power frequency 25 Hz using
MATLAB’s filtfilt function (which filters forward and then backward in time so this is no
phase distortion). Figure 6.23 compares the time history and PSD of control force measured at
RTHS, those filtered from RTHS, and those from numerical simulation. The filtered control force
140
Table 6.30: Mean square bridge deck responses to a non-design excitation for control weightr
a
= 10
13
kg
–2
;D is the percent change
of CLQR relative to OCLC.
J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN El Centro
OCLC 0.0070 0.0053 0.0032 0.0024 0.0038 0.0029 4.3518 3.3717 15.7105 12.3831 0.5708 0.9153 0.0037 0.0083
CLQR 0.0082 (17) 0.0067(21) 0.0037 (16) 0.0030(26) 0.0045(18) 0.0037 (26) 5.1589 (19) 4.2720 (27) 18.9499 (21) 15.9864 (29) 0.2373 (58) 0.5327 (42) 0.0008 (–78) 0.0042 (–49)
El Centro GWN
OCLC 0.0068 0.0049 0.0017 0.0013 0.0051 0.0036 5.8319 4.1364 21.1173 15.1454 0.5545 0.8339 0.0035 0.0066
CLQR 0.0093 (37) 0.0065(25) 0.0026 (49) 0.0019(42) 0.0067(33) 0.0047 (30) 7.7225 (32) 5.3724 (30) 28.4153 (35) 20.0367 (32) 0.2354 (58) 0.4988 (40) 0.0007 (–81) 0.0038 (–43)
Table 6.31: Peak bridge deck responses to a non-design excitation for control weightr
a
= 10
13
kg
–2
;D is the percent change of CLQR
relative to OCLC.
( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN El Centro
OCLC 0.2117 0.1948 0.1503 0.1392 4.9913 4.6533 9.7722 9.0599
CLQR 0.2156 (2) 0.1973 (1) 0.1554 (3) 0.1467 (5) 5.2639 (5) 5.0003 (7) 10.0973 (3) 9.5540 (5)
El Centro GWN
OCLC 0.1435 0.1276 0.1887 0.1559 6.6323 5.3882 12.2038 10.0589
CLQR 0.1729 (20) 0.1461(14) 0.2138(13) 0.1732(11) 7.5360(14) 6.0342 (12) 13.9060 (14) 11.2597 (12)
Table 6.32: Mean square bridge deck responses to a non-design excitation for control weightr
a
= 10
20
kg
–2
;D is the percent change
of CLQR relative to OCLC.
J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q
2
q
1
)
2
] E[I c] E[u
2
]
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m
1
g)
2
] [%] [(%m
1
g)
2
] [%]
GWN El Centro
OCLC 0.0069 0.0053 0.0031 0.0024 0.0038 0.0030 4.3885 3.4058 15.8530 12.5143 0.5704 0.9028 0.0036 0.0080
CLQR 0.0085 (22) 0.0074(28) 0.0038(23) 0.0031 (32) 0.0047(22) 0.0043 (45) 5.3711 (22) 4.9343 (45) 19.6982 (24) 18.3330 (46) 0.2577 (55) 0.3425 (62) 0.0007 (–80) 0.0019 (–76)
El Centro GWN
OCLC 0.0070 0.0049 0.0019 0.0014 0.0051 0.0035 5.8695 3.9793 21.3609 14.6645 0.4688 0.8264 0.0030 0.0070
CLQR 0.0098 (40) 0.0079(38) 0.0027(38) 0.0021 (48) 0.0072(40) 0.0058 (69) 8.2367 (40) 6.6981 (68) 30.2770 (42) 24.8627 (70) 0.2145 (54) 0.3276 (60) 0.0006 (–80) 0.0019 (–72)
Table 6.33: Peak bridge deck responses to a non-design excitation for control weightr
a
= 10
20
kg
–2
;D is the percent change of CLQR
relative to OCLC.
( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN El Centro
OCLC 0.2120 0.1945 0.1507 0.1396 4.9953 4.6631 9.7942 9.0912
CLQR 0.2160 (2) 0.1994 (3) 0.1561 (4) 0.1535(10) 5.3032 (6) 5.1979 (11) 10.1446 (4) 9.9967 (10)
El Centro GWN
OCLC 0.1581 0.1407 0.1885 0.1518 6.6389 5.2518 12.2592 9.8621
CLQR 0.1767 (12) 0.1661 (18) 0.2236 (19) 0.1978(30) 7.8472 (18) 6.8907 (31) 14.5485 (19) 12.8590 (30)
141
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(a) Base absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
experiment
num. sim.
(b) Base drift
0 2 4 6 8 10
Time [s]
0
0.02
0.04
0.06
experiment
num. sim.
(c) Superstructure absolute acceleration
0 2 4 6 8 10
Time [s]
0
0.04
0.08
0.12
experiment
num. sim.
(d) Superstructure drift
0 2 4 6 8 10
Time [s]
0
0.4
0.8
1.2
1.6
experiment
num. sim.
(e) Control force
Figure 6.19. Numerical simulation and RTHS base-isolated structure response time histories to a
GWN excitation with CLQR with control weightr
a
= 10
13
kg
–2
.
142
no longer contains the high-frequency content and matches well with numerical simulation results.
Therefore, it may be concluded that the high-frequency content in the RTHS measured absolute
acceleration and control force is due to the significant high-frequency noise of the sensing system at
RTHS. The OCLC superstructure absolute acceleration exhibits similar behavior in the RTHS tests.
The spikes in the base absolute acceleration (e.g., in Figures 6.19a and 6.21a and Figures 6.20a and
6.22a) and corresponding control force, again, are due to the hysteresis parametersaz in the MR
damper representation equation Eq. (6.2) and the bang-bang control law.
Statistics of response to OCLC design excitation: The mean square and peak absolute accelera-
tion and drift and control forces for the base-isolated structure with different control force weights
r
a
are tabulated in Tables 6.34 to 6.37. Both CLQR and OCLC provide good control performance
in reducing structural responses. Notably, the uncontrolled mean square base drift is thirty-three
times that of OCLC when subjected to the GWN excitation. The base drifts and absolute ac-
celerations are larger than those of superstructure, as expected. CLQR clips more frequently than
OCLC, providingE[I
c
] up to 38% smaller (worse) than that of OCLC, though the mean square con-
trol force is smaller than OCLC. The numerical simulation results indicate dramatic performance
improvements for OCLC strategies in reducing mean square responses.
When control weightr
a
= 10
13
kg
–2
(Tables 6.34 to 6.35), CLQR’s cost metric is 52% larger
and its mean square absolute accelerations and drifts are 106%–468% larger than those provided
by OCLC when subjected to a GWN excitation in simulation; the experimental performance im-
provements of OCLC are not as significant: although the mean square base drift with CLQR is
more than five times that of OCLC in RTHS, CLQR’s cost metric, mean square absolute accel-
erations and superstructure mean square drift are slightly larger or smaller than those of OCLC.
Specifically, when the external excitation is GWN, CLQR provides a 44% smaller cost metric and
62% smaller mean square superstructure absolute acceleration than those of OCLC, which might
be due to sensor noise at RTHS and imperfection of the MR damper numerical model. The Kobe
response statistics, again, are very similar to those of El Centro in numerical simulation; CLQR
provides 198% larger mean square base drift and 96% larger mean square superstructure absolute
143
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(a) Base absolute acceleration
0 2 4 6 8 10
Time [s]
0
2
4
6
experiment
num. sim.
(b) Base drift
0 2 4 6 8 10
Time [s]
0
0.02
0.04
0.06
experiment
num. sim.
(c) Superstructure absolute acceleration
0 2 4 6 8 10
Time [s]
0
0.04
0.08
0.12
experiment
num. sim.
(d) Superstructure drift
0 2 4 6 8 10
Time [s]
0
0.4
0.8
1.2
1.6
experiment
num. sim.
(e) Control force
Figure 6.20. Numerical simulation and RTHS base-isolated structure response time histories to a
GWN excitation with OCLC with control weightr
a
= 10
13
kg
–2
.
144
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(a) Base absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
10
15
experiment
num. sim.
(b) Base drift
0 2 4 6 8 10
Time [s]
0
0.02
0.04
0.06
0.08
0.1
experiment
num. sim.
(c) Superstructure absolute acceleration
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
0.2
0.25
experiment
num. sim.
(d) Superstructure drift
0 2 4 6 8 10
Time [s]
0
1
2
experiment
num. sim.
(e) Control force
Figure 6.21. Numerical simulation and RTHS base-isolated structure response time histories to an
El Centro excitation with CLQR with control weightr
a
= 10
13
kg
–2
.
145
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
experiment
num. sim.
(a) Base absolute acceleration
0 2 4 6 8 10
Time [s]
0
5
10
15
experiment
num. sim.
(b) Base drift
0 2 4 6 8 10
Time [s]
0
0.02
0.04
0.06
0.08
0.1
experiment
num. sim.
(c) Superstructure absolute acceleration
0 2 4 6 8 10
Time [s]
0
0.05
0.1
0.15
0.2
0.25
experiment
num. sim.
(d) Superstructure drift
0 2 4 6 8 10
Time [s]
0
1
2
experiment
num. sim.
(e) Control force
Figure 6.22. Numerical simulation and RTHS base-isolated structure response time histories to an
El Centro excitation with OCLC with control weightr
a
= 10
13
kg
–2
.
146
4 4.5 5 5.5 6 6.5 7 7.5 8
Time [s]
0
500
1000
1500
exper. meas.
exper. filtered
num. sim.
10 10
0
10
1
10
2
Frequency [Hz]
10
0
Figure 6.23. Time history and PSD of control force measured and filtered in RTHS tests and
numerical simulation, when the base-isolated structure is subjected to a GWN excitation with
OCLC with control weightr
a
= 10
13
kg
–2
.
acceleration. Nevertheless, both numerical simulation and RTHS tests indicate that OCLC can bet-
ter reduce peak superstructure absolute acceleration and base and superstructure drifts than CLQR
at the cost of comparable or slightly larger peak base acceleration due to the on-off switching of the
MR damper (the modified bang-bang control may help to mitigate these base acceleration spikes).
Specifically, the peak base drift with CLQR is 144% larger than that of OCLC in numerical simu-
lation when subjected to GWN, and 57% larger in RTHS tests. Further, the performance improve-
ments of OCLC is similar when the control weight r
a
decreases from 10
13
kg
–2
to 10
20
kg
–2
.
When the control weight r
a
= 10
20
kg
–2
(Tables 6.36 to 6.37), in numerical simulation, CLQR
provides up to 54% and 122% larger cost metric and mean square absolute accelerations, respec-
tively, and up to 122% larger superstructure drift, and even up to six times those of OCLC for
mean square base drift; though OCLC does not perform as well as in numerical simulation, CLQR
provides up to 19% and 31% larger cost metric and mean square absolute accelerations, and nearly
three times that of OCLC for mean square base drift in RTHS tests.
Although OCLC can decrease the mean square base drifts better than CLQR, OCLC cannot
achieve the same level of performance in minimizing the cost metric, the mean square absolute
accelerations and the superstructure drift in the RTHS tests, especially when the external excitation
is GWN, which might be due to, again, the imperfection of the MR damper numerical model.
147
Further, OCLC supplies better performance in reducing peak superstructure absolute acceleration
and base and superstructure drift. Notably, CLQR provides up to a 152% larger peak base drift than
that of OCLC when subjected to GWN in numerical simulation, and 64% larger in RTHS tests.
However, OCLC has slightly poorer performance in reducing peak base absolute acceleration than
CLQR in RTHS.
Statistics of response to non-design excitation: Tables 6.38 to 6.41 show the cost metric, mean
square and peak absolute acceleration and the OCLC drift responses to an excitation different
from that used to design. Both pure simulation and RTHS results demonstrate that, for the base
isolated structure, OCLC designed for one specific excitation achieves very similar performance
improvements when the structure is subjected to the other excitation.
When the control weightr
a
= 10
13
kg
–2
(Tables 6.38 and 6.39), for numerical simulation, the
cost metric and mean square absolute accelerations with CLQR are 76%–87% larger than those
provided by OCLC designed for the El Centro earthquake when the structure is excited by GWN,
and 5%–14% larger in RTHS tests. The mean square base drift with CLQR in RTHS tests is nearly
three times, and more than five times in numerical simulation, that of OCLC designed for GWN
when the external excitation is the El Centro earthquake. OCLC can provide numerically simulated
and RTHS performance superior to CLQR in minimizing peak superstructure absolute acceleration
and base and superstructure drift (Table 6.39); e.g., the numerically simulated peak base drift of
CLQR is twice that of OCLC designed for GWN when the structure is subjected to the El Centro
earthquake; however, CLQR results in almost identical or smaller peak base absolute acceleration
compared with those of OCLC for numerical simulation and RTHS tests.
When the control weightr
a
= 10
20
kg
–2
, the performance improvements of OCLC for certain
metrics are even larger compared with those whenr
a
= 10
13
kg
–2
. For example, as shown in the
mean square statistics Table 6.40, the numerically simulated cost metric and mean square absolute
accelerations with CLQR are 93%–96% larger than those of OCLC designed for El Centro when
the structure is excited by GWN, and 8%–20% larger in RTHS tests. The mean square base drift
with CLQR is more than three and five times that of OCLC designed for GWN in RTHS tests and
148
Table 6.34: Mean square response statistics for the base isolated structure with control weightr
a
= 10
13
kg
–2
;D is the percent change
of CLQR relative to OCLC.
Design and J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q
2
q
1
)
2
] E[I c] E[u
2
]
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [–] [%] [–] [%] [(%m
1
g)
2
] [%] [(%m
1
g)
2
] [%]
GWN
Uncontrol 0.0054 – 0.0026 – 0.0027 – 66.2888 – 0.0167 – – – – –
OCLC 0.0005 0.0007 0.0003 0.0004 0.0003 0.0004 2.1812 1.7649 0.0016 0.0008 0.2010 0.9122 0.2171 0.1655
CLQR 0.0011(52) 0.0005(–44) 0.0006(106) 0.0004 (1) 0.0005 (112) 0.0001 (–62) 12.3787 (468) 3.9423 (123) 0.0034 (112) 0.0008 (5) 0.1495 (26) 0.6036 (34) 0.0262 (–88) 0.1195 (–28)
El Centro
Uncontrol 0.0023 – 0.0011 – 0.0012 – 28.4806 – 0.0072 – – – – –
OCLC 0.0005 0.0004 0.0003 0.0003 0.0002 0.0001 3.0280 1.4449 0.0015 0.0007 0.1555 0.2565 0.1379 0.2130
CLQR 0.0009(45) 0.0005 (21) 0.0005 (73) 0.0003 (29) 0.0005 (93) 0.0001 (21) 10.6720 (252) 4.3202 (199) 0.0028 (93) 0.0009 (21) 0.1066 (31) 0.1835 (28) 0.0346 (–75) 0.1256 (–41)
Kobe
Uncontrol 0.0020 – 0.0010 – 0.0010 – 24.2945 – 0.0061 – – – – –
OCLC 0.0003 – 0.0001 – 0.0001 – 1.8786 – 0.0008 – 0.0909 – 0.0691 –
CLQR 0.0005(48) – (–) 0.0003 (89) – (–) 0.0003 (96) – (–) 5.5989 (198) – (–) 0.0015 (96) – (–) 0.0909 (13) – (–) 0.0277 (–60) – (–)
Table 6.35: Peak response statistics for the base isolated structure with control weightr
a
= 10
13
kg
–2
;D is the percent change of CLQR
relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN
Uncontrol 0.1401 – 0.1429 – 22.1952 – 0.3538 –
OCLC 0.1546 0.0790 0.0511 0.0790 4.1131 3.7613 0.1263 0.0818
CLQR 0.1048(–32) 0.0778 (–1) 0.0677 (33) 0.0410(–48) 10.0449 (144) 5.8970 (57) 0.1676 (33) 0.1012 (24)
El Centro
Uncontrol 0.1013 – 0.1019 – 15.8694 – 0.2523 –
OCLC 0.1615 0.0917 0.0668 0.0398 7.7676 4.1469 0.1654 0.0990
CLQR 0.1478 (–9) 0.0767(–16) 0.0834(25) 0.0523 (32) 12.6098 (62) 7.5273 (82) 0.2064 (25) 0.1291 (30)
Kobe
Uncontrol 0.0790 – 0.0789 – 12.2695 – 0.1953 –
OCLC 0.1186 – 0.0536 – 5.8716 – 0.1326 –
CLQR 0.1338 (13) – (–) 0.0582 (9) – (–) 8.7330 (49) – (–) 0.1442 (9) – (–)
149
Table 6.36: Mean square response statistics for the base isolated structure with control weightr
a
= 10
20
kg
–2
;D is the percent change
of CLQR relative to OCLC.
Design and J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q
2
q
1
)
2
] E[I c] E[u
2
]
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m
1
g)
2
] [%] [(%m
1
g)
2
] [%]
GWN
Uncontrol 0.0054 – 0.0026 – 0.0027 – 66.2888 – 0.0167 – – – – –
OCLC 0.0005 0.0007 0.0003 0.0004 0.0003 0.0004 2.2420 1.7679 0.0016 0.0008 0.2016 0.9072 0.2156 0.1661
CLQR 0.0012(54) 0.0005(–37) 0.0006(115) 0.0004 (5) 0.0006 (122) 0.0001 (–59) 13.1159 (485) 4.2918 (143) 0.0035 (122) 0.0009 (12) 0.1308 (35) 0.5659 (38) 0.0248 (–89) 0.1166 (–30)
El Centro
Uncontrol 0.0023 – 0.0011 – 0.0012 – 28.4806 – 0.0072 – – – – –
OCLC 0.0005 0.0004 0.0002 0.0003 0.0002 0.0001 3.1451 1.8436 0.0015 0.0007 0.1494 0.2642 0.1242 0.1665
CLQR 0.0010(50) 0.0005 (19) 0.0005 (96) 0.0004(21) 0.0005 (106) 0.0002 (31) 11.5179 (266) 4.8773 (165) 0.0030 (106) 0.0010 (31) 0.1038 (30) 0.1667 (37) 0.0304 (–76) 0.1201 (–28)
Kobe
Uncontrol 0.0020 – 0.0010 – 0.0010 – 24.2945 – 0.0061 – – – – –
OCLC 0.0003 – 0.0001 – 0.0001 – 1.6257 – 0.0008 – 0.1001 – 0.0756 –
CLQR 0.0006(53) – (–) 0.0003(108) – (–) 0.0003 (119) – (–) 6.3581 (291) – (–) 0.0017 (119) – (–) 0.1001 (21) – (–) 0.0245 (–68) – (–)
Table 6.37: Peak response statistics for the base isolated structure with control weightr
a
= 10
20
kg
–2
;D is the percent change of CLQR
relative to OCLC.
Design and ( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN
Uncontrol 0.1401 – 0.1429 – 22.1952 – 0.3538 –
OCLC 0.1525 0.0790 0.0518 0.0790 4.1577 3.7603 0.1282 0.0832
CLQR 0.1008(–34) 0.0787 (–0) 0.0701(35) 0.0420(–47) 10.4684 (152) 6.1625 (64) 0.1736 (35) 0.1037 (25)
El Centro
Uncontrol 0.1013 – 0.1019 – 15.8694 – 0.2523 –
OCLC 0.1566 0.0914 0.0641 0.0427 7.8675 5.2104 0.1587 0.1051
CLQR 0.0897(–43) 0.0862(–6) 0.0871(36) 0.0540 (26) 12.9761 (65) 7.9638 (53) 0.2156 (36) 0.1331 (27)
Kobe
Uncontrol 0.0790 – 0.0789 – 12.2695 – 0.1953 –
OCLC 0.1313 – 0.0547 – 5.6628 – 0.1355 –
CLQR 0.0659(–50) – (–) 0.0601(10) – (–) 8.9865 (59) – (–) 0.1487 (10) – (–)
150
numerical simulation, respectively, when the structure is subjected to the El Centro earthquake.
CLQR’s numerically simulated peak base drift is twice that of OCLC designed for GWN when
the structure is subjected to the El Centro earthquake, though with smaller peak base absolute
acceleration than that of OCLC in both numerical simulation and RTHS tests, as listed in Table
6.41. It is noted that OCLC has mixed performance — OCLC performs better than CLQR in RTHS
tests in some metrics, but worse in others — but was similar to what was already observed when
the structure is subjected to the OCLC design excitation.
6.4 Summary
This chapter validated the effectiveness of the novel semiactive control strategy OCLC through
both numerical simulation and RTHS tests. A suite of structure models were tested, including a
series of SDOF and 2DOF structure models with different structural properties, a 2DOF elevated
highway bridge deck and a 2DOF base-isolated shear building structure. Both numerical simu-
lation and RTHS test results prove variable levels of superiority of OCLC over CLQR, further
verifying the effectiveness and advantages of OCLC strategies, though OCLC uses more control
effort than CLQR (this is not an issue in this study since the purpose of this study is to demonstrate
that OCLC is superior to CLQR in decreasing structural responses; if the control force is a concern
in practical implementation of OCLC, one could consider adjusting the control weightr
a
value in
the cost metric). Further, OCLC does not clip as often as CLQR, thereby making more efficient use
of the controllable damper. RTHS test results match well with the numerical simulation results in
most cases. Furthermore, the OCLC designed for one specific excitation was evaluated by different
external excitations and behaved well in reducing structural responses.
151
Table 6.38: Mean square base-isolated structure responses to a non-design excitation for control weightr
a
= 10
13
kg
–2
;D is the percent
change of CLQR relative to OCLC.
J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN El Centro
OCLC 0.0005 0.0004 0.0003 0.0003 0.0003 0.0001 2.0883 1.5285 0.0016 0.0008 0.2095 0.9097 0.2240 0.1907
CLQR 0.0009 (69) 0.0005 (9) 0.0005(64) 0.0003 (11) 0.0005(75) 0.0001 (9) 10.6721 (411) 4.3202 (183) 0.0028 (75) 0.0009 (9) 0.1066 (49) 0.6117 (33) 0.0346 (–85) 0.1256 (–34)
El Centro GWN
OCLC 0.0006 0.0005 0.0003 0.0003 0.0003 0.0001 3.8030 2.0516 0.0018 0.0007 0.1787 0.8150 0.1694 0.6000
CLQR 0.0011 (82) 0.0005 (7) 0.0006(76) 0.0004 (5) 0.0006(87) 0.0001 (14) 12.3996 (226) 3.9423 (92) 0.0034 (87) 0.0008 (15) 0.1495 (16) 0.6036 (26) 0.0262 (–85) 0.3982 (–34)
Table 6.39: Peak base-isolated structure responses to a non-design excitation for control weightr
a
= 10
13
kg
–2
;D is the percent change
of CLQR relative to OCLC.
( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN El Centro
OCLC 0.1644 0.0898 0.0669 0.0474 6.4798 5.0023 0.1656 0.1178
CLQR 0.1478 (–10) 0.0767(–15) 0.0834(25) 0.0523(10) 12.6098(95) 7.5273 (50) 0.2064 (25) 0.1291 (10)
El Centro GWN
OCLC 0.1654 0.0762 0.0540 0.0306 5.6083 3.8901 0.1336 0.0757
CLQR 0.1048 (–37) 0.0778 (2) 0.0677(25) 0.0410(34) 10.0449(79) 5.8970 (52) 0.1676 (25) 0.1012 (34)
Table 6.40: Mean square base-isolated structure responses to a non-design excitation for control weightr
a
= 10
20
kg
–2
;D is the percent
change of CLQR relative to OCLC.
J a E[ ¨ q
a
1
2
] E[ ¨ q
a
2
2
] E[q
2
1
] E[(q 2 q 1)
2
] E[I c] E[u
2
]
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [m
2
/s
4
][%] [m
2
/s
4
] [%] [m
2
/s
4
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] [mm
2
] [%] – [%] – [%] [(%m 1 g)
2
] [%] [(%m 1 g)
2
] [%]
GWN El Centro
OCLC 0.0005 0.0004 0.0003 0.0003 0.0003 0.0001 2.1132 1.4775 0.0016 0.0008 0.2032 0.9115 0.2198 0.1993
CLQR 0.0010 (81) 0.0005 (16) 0.0005 (73) 0.0004(20) 0.0005 (89) 0.0002 (18) 11.5179 (445) 4.8773 (230) 0.0030 (89) 0.0010 (18) 0.1038 (49) 0.5555 (39) 0.0304 (–86) 0.1201 (–40)
El Centro GWN
OCLC 0.0006 0.0005 0.0003 0.0003 0.0003 0.0001 3.9865 2.1708 0.0018 0.0008 0.1753 0.8497 0.1471 0.4989
CLQR 0.0012 (94) 0.0005 (10) 0.0006 (93) 0.0004 (8) 0.0006 (96) 0.0001 (16) 13.1373 (230) 4.2918 (98) 0.0035 (96) 0.0009 (16) 0.1308 (25) 0.5659 (33) 0.0248 (–83) 0.3887 (–22)
Table 6.41: Peak base-isolated structure responses to a non-design excitation for control weightr
a
= 10
20
kg
–2
;D is the percent change
of CLQR relative to OCLC.
( ¨ q
a
1
)
max
( ¨ q
a
2
)
max
q
1
max
(q
2
q
1
)
max
Design Evaluation Control Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D) Num. (D) Exper. (D)
Excitation Excitation Strategy [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [m/s
2
] [%] [mm] [%] [mm] [%] [mm] [%] [mm] [%]
GWN El Centro
OCLC 0.1515 0.1601 0.0655 0.0483 6.5475 4.7006 0.1620 0.1197
CLQR 0.0897 (–41) 0.0862(–46) 0.0871(33) 0.0540(12) 12.9761(98) 7.9638 (69) 0.2156 (33) 0.1331 (11)
El Centro GWN
OCLC 0.1651 0.0778 0.0515 0.0325 5.6097 4.1656 0.1274 0.0805
CLQR 0.1008 (–39) 0.0787 (1) 0.0701(36) 0.0420(29) 10.4684(87) 6.1625 (48) 0.1736 (36) 0.1037 (29)
152
Chapter 7
E-Defense Shake Table Experiments and Real-time Hybrid
Simulation Tests of Controllable Damping Strategies
The previous chapter explored OCLC in RTHS tests with a physical controllable damper but a
numerically simulated structure. This chapter presents the results of shake table experiments at
Japan’s NIED “E-Defense” laboratory, along with a series of corresponding RTHS tests at Kobe
University, using several controllable damping strategies designed to mitigate the responses of a
full-size base-isolated structure specimen with an MR fluid damper installed in the isolation layer.
Section one provides some literature review of E-Defense shake table tests; the second section
describes the E-Defense experimental set-up; numerical simulation and RTHS tests procedures are
presented, respectively, in sections three and four; the fifth section summarizes and compares the
shake table experimental results with RTHS and numerical simulation results; section six summa-
rizes the research work.
7.1 E-Defense shake table
Experimental testing is a supplement to numerical simulation to verify innovative concepts across
all branches of engineering. Full-scale physical experiments are practically meaningful, though
expensive and difficult to realize in certain cases.
153
E-Defense is the largest three dimensional 6-DOF shake table in the world, established by the
National Research Institute for Earth Science and Disaster Resilience (NIED) based on the lessons
learned from the destructive 1995 Kobe earthquake. The E-Defense shake table is 20 m long and
15 m wide, with a payload up to 12 MN (1,200 tonf). The maximum table acceleration, velocity
and displacement are 900cm=s
2
, 200cm=s and100cm, respectively, for the horizontal directions,
and 1500 cm=s
2
, 70 cm=s and50 cm for the vertical direction [80]. The E-Defense shake table
is capable of conducting physical experiments for true-size infrastructure subjected to short- or
long-period ground motions [21].
The E-Defense shake table has been utilized to excite many full-size structural models with
strong ground motions over the past decade-and-a-half, providing experimental results that have
advanced scientific knowledge in earthquake engineering [61]. For instance, the collapse of a
full-scale four-story moment frame was performed on the E-Defense shake table in 2007; the
modeling techniques, estimation of damping and stiffness, and collapse simulation results were
reported [79, 102]. Tests of a full-scale four story reinforced concrete building and a prestressed
concrete building using precast post-tensioned concrete members were conducted using the E-
Defense shake table in 2010, providing the opportunity to compare their dynamic responses [76,
110]. E-Defense shake table tests on 3-D piping system models were facilitated in 2011, indicating
the seismic safety capacity of piping systems with wall thinning [77]. In 2013, a 686-ton full-scale
four-story base-isolated RC frame building was tested on the E-Defense shake table to investigate
moat wall pounding effects and the structural responses with different isolation components to
long-period and long duration ground motions [124]. A recent E-Defense experiment of a two-
story base-isolated structure processed in 2019, using various semiactive control strategies with a
large-scale MR damper, is the main focus of this study.
154
7.2 E-Defense experimental setup
In July 2019, four days of testing of a base-isolated frame specimen at Japan’s NIED E-Defense
laboratory were conducted. The following paragraphs describe the structural specimen and the
controllable damper used to mitigate structural response.
7.2.1 Structural test specimen
The test specimen is a 14.9 ton full-scale two-story steel-frame nearly-rigid superstructure on an
isolation layer, which is placed at the center of the shake table, as shown in Figure 7.1 with x and y
directions indicated with red arrows. The 2.3 m-high superstructure frame has a span of 6 m in the
y direction and 4.5 m in the x direction. Column and beam sections of the superstructure frame are
H300 mm 300 mm 10 mm 15 mm and H600 mm 200 mm 11 mm 17 mm, respectively.
Each framing elevation (xz- and yz-) has a V-shaped torsional bracing system, which is composed of
two H200mm200mm8mm12mm braces intersected in the middle of the first-story beams.
The isolation layer has: four sliding bearings, one on each corner (allowing the superstructure to
slide, relative to the shake table, only along the y axis), which are secured to a steel plate connected
to the frame above and the shake table underneath; a single low-damping rubber bearing under the
center of the x-z vertical frame at y3 m to provide the conventional stiffness (rubber bearings
in each of the four corners — the more usual configuration — would have provided too much
stiffness for the frame’s mass); and a 10 kN MR damper (further described in the next subsection)
installed under the center of the x-z frame at y+3 m to provide adaptability.
The structure was instrumented with: five tri-directional accelerometers on the shake table
(one near the table center and the others at the four corners of the table) and four near the four
columns on each level of the superstructure; force transducers below each isolation-layer compo-
nent; LVDT and laser displacement sensors across the isolation layer; and a number of cameras to
video components during the experiments. Various load cells measured the forces in the isolation
layer devices: a three-component load cell below the natural rubber bearing; four three-component
155
load cells below the linear guides; and a load cell at the tip of the MR damper rod. For the MR
damper, a magnetostrictive displacement meter was also used to measure both the displacement
and velocity of the MR damper rod during testing. Further details of the specimen are provided in
Sato et al. [90].
y
x
Figure 7.1. E-Defense Test specimen [90].
7.2.2 MR damper
The MR damper used herein is custom made by the Sanwa Tekki Corporation, with a mass of
76.19 kg, a stroke limit of300 mm, a maximum control force of 10 kN, and a maximum current
of 5 A. Figure 7.2 displays the design schematic of the MR damper. A Bingham-viscoplastic
model, portrayed in Figure 7.3, is calibrated to model the MR damper [47].
The force generated by the Bingham-viscoplastic model is expressed as:
u= f
c
(I)sgn ˙ q+ c
MR
(I) ˙ q (7.1)
156
Figure 7.2. MR damper design schematic [32].
Coulomb friction element
Viscous damping element
Figure 7.3. Bingham-viscoplastic model of MR damper.
where f
c
(I) is the Coulomb friction force, c
MR
(I) is the damping coefficient of MR damper, and
˙ q is the velocity across the MR damper. To identify the parameters of the model, preliminary tests
[47] were conducted at Kobe University to measure the displacement, velocity and damper force
under different loading conditions. The actuator was driven by a sinusoidal displacement time
history with a given amplitude and frequency while the electric current applied to the MR damper
was held at a constant level. The loading conditions and drive currents are listed in Table 7.1.
All signals were sampled at 1 kHz. Based on the force-displacement and force-velocity curves
generated from the sinusoidal excitation experiments, the damping coefficient c
MR
was calculated
from a slope of the force-velocity relationship for each case in Table 7.1 and then averaged over
different frequencies and amplitudes at each current level to develop an approximate relationship
with respect to current I [47]:
c
MR
(I)= 0:501 kNsm
–1
A
–1
I+ 0:859 kNsm
–1
(7.2)
157
Table 7.1: Sinusoidal excitation and electric current of preliminary tests to calibrate the Bingham-
viscoplastic MR damper model.
Amplitude [mm] Frequency [Hz] Electric current [A]
100
1
0, 0:25,:::, 1:5, 2, 2:5, 3
0:5
0:33
0:2
200
0:5
0:33
0:2
The Coulomb friction force f
c
was determined in a similar way after subtracting the viscous damp-
ing force c
MR
(I), and fit as a quadratic function of current I:
f
c
(I)=0:171 kNA
–2
I
2
+ 2:522 kNA
–1
I+ 0:247 kN (7.3)
The actual command from the control computer to the current driver amplifier is a voltage in
the10 V range; the current driver settings use a 1 A = 2 V conversion factor, so this command
voltage range corresponds to a commanded current of5 A. This voltage command signal is sent
from the control computer’s SIMULINK [107] model through a D/A converter to the current driver
amplifier. Because the D/A converter uses integer quantization, an input that is very slightly above
+10V (or below10V) is wrapped to10V (to+10V); slight variations such as these can occur
(e.g., overshoot) in the SIMULINK model of the controller, resulting in unintentional and sudden
changes in the command voltage and the resulting commanded current. As a result, while the peak
current considered safe for this MR damper is 5 A, a 4 A limit is imposed to ensure that this D/A
overflow does not occur.
7.3 Numerical simulation
The base-isolated structure is modeled as a single-degree-of-freedom (SDOF) system, since the
superstructure is nearly rigid, with mass m = 14:9 tons, stiffness k = 38:5 kN=m and damping
coefficient c= 2:4 kN s=m, resulting in natural frequency w = 1:6849 rads=s= 0:2682 Hz and
158
damping ratioz = 0:05. The ground-relative displacement q(t), excited by base acceleration ¨ q
g
, is
given by equation of motion:
¨ q+ 2zw ˙ q+w
2
q= ¨ q
g

u
m

F
fric
m
(7.4)
where u is the control force exerted by the MR damper (actual force, not mass-normalized), F
fric
is the Coulomb friction force provided by the sliders under the superstructure columns and and is
represented as:
F
fric
= f
fric
sgn ˙ q (7.5)
where f
fric
is estimated to be 3.75 kN based upon preliminary test results.
7.3.1 Cost metric J
In this study, control strategies were designed for two different, but related, non-dimensional cost
metrics:
J
1
= kN
2
Z
¥
0

a
d
q
2
+a
v
˙ q
2
+a
a
¨ q
2
a
+ u
2

dt a
d
0; a
v
0; a
a
0
J
2
=
s
4
cm
2
Z
¥
0

aw
4
q
2
+(1 a) ¨ q
2
a
+r

u
m

2

dt 0< a< 1; r > 0
(7.6)
where ¨ q
a
= ¨ q+ ¨ q
g
is the structure absolute acceleration. For cost metric J
1
, the weights are further
simplified asa
d
= 10
b
N
2
=m
2
,a
v
= 10
b
N
2
s
2
=m
2
, anda
d
= 10
b
N
2
s
4
=m
2
; herein,b = 7 is used
to trade off the drift, velocity and absolute acceleration responses relative to the control force; these
weights place greater emphasis on reducing absolute acceleration. For cost metric J
2
, controllers
are designed with a= 0:7 andr = 0:1, which places greater emphasis on base drift reduction.
7.3.2 Bang-bang control law
A bang-bang secondary control algorithm [24] is used to select the current applied to the current
driver amplifier based on the desired control force because it is infeasible to directly command
159
the MR damper force. The desired current is the maximum current I
max
when the desired control
force u
d
is the same sign as, and of larger magnitude than, the MR damper measured control force
u
m
(for numerical simulation, u
m
u, where u is the actual control force generated by the MR
damper); the desired current is otherwise zero. This relationship can be described with
I
d
= I
max
H
h
(u
d
u
m
)u
m
i
(7.7)
where H[] is the Heaviside unit step function. As discussed previously, while the maximum allow-
able current for the MR damper used in this study is 5A, I
max
is set to be 4A to ensure no overflow
in the D/A converter.
7.3.3 Current driver amplifier model
Preliminary experiments with this MR damper indicated that the current amplifier driving the MR
damper has a limit on the rate of change of current. Hence, the actual current I in the MR damper
coil is a rate-limited version of the desired current I
d
. I
df
is a first-order low-pass filtered version of
I
d
. The first-order low-pass filter was estimated using the commanded and measured currents data
from E-Defense test when the commanded current is not saturated at all, e.g., when commanded
current is smaller than 0:24A, and is given by 1500s
–1
=(s+1500s
–1
). Figure 7.4 shows a compar-
ison of the filtered current I
df
with the commanded and measured currents when the commanded
current is smaller than 0:24 A over the time duration, when the structure is subjected to a random
excitation (nominal peak 100cm=s
2
, frequency range 0.1–30Hz) at E-Defense using random 25Hz
square wave current control.
160
3.32 3.34 3.36 3.38 3.4 3.42 3.44
time [s]
0
0.5
1
1.5
2
2.5
3
3.5
4
current [A]
commanded
filtered
measured
Figure 7.4. E-Defense commanded and measured currents, and low-pass-filtered current I
df
(the
structure is subjected to a random excitation [nominal peak 100cm=s
2
, frequency range 0.1–30Hz]
using random 25 Hz square wave current control).
The current rate limiter can be described by:
I(t)= I(tDt)+Dt sat
˙
I
max
˙
I
min
(e(t))
where sat
˙
I
max
˙
I
min
(e)= min

max(e;
˙
I
min
);
˙
I
max

=
8
>
>
>
>
<
>
>
>
>
:
˙
I
max
; e
˙
I
max
e;
˙
I
min
e
˙
I
max
˙
I
min
; e
˙
I
min
e(t)=
1
Dt
[I
df
(t) I(tDt)]
Let t
+
(t)= H[e(t)> 0](t maxftj0t t; e(t) 0g)
Let t

(t)= H[e(t)< 0](t maxftj0t t; e(t) 0g)
˙
I
max
(t)=(240+ 110H[t
+
(t) 3 ms]) A=s
˙
I
min
(t)=(800 150H[t

(t) 3 ms]) A=s
(7.8)
whereDt = 0:001 s is the sampling time, and
˙
I
min
and
˙
I
max
are minimum and maximum current
rates; experimental data showed that these limits are approximately800 and 240 A=s, respec-
tively, though the current driver amplifier can sustain a wider range[950;350]A=s for up to three
milliseconds. Figure 7.5 shows the commanded and measured currents, along with the current
161
predicted by Eq. (7.8), using random 25 Hz square wave current control when the structure is sub-
jected to a random excitation (nominal peak 100 cm=s
2
, frequency range 0.1–30 Hz) at E-Defense.
The top subfigure (Figure 7.5a) indicates the E-Defense commanded and measured currents and
the current predicted by the filtered multilevel rate-limiter; the second subfigure (Figure 7.5b) plots
the corresponding current rates e(t). Figures 7.5c and 7.5e illustrate the time duration when the
current rises with current ratee(t) larger than 240 A=s and decreases with current ratee(t) smaller
than800 A=s, respectively. Figures 7.5d and 7.5f show the corresponding maximum and mini-
mum current rates based on Eq. (7.8).
162
0
1
2
3
4
5
commanded
measured
rate limiter model
0
1000
350 A/s
240 A/s
0 ms
5 ms
10 ms
3 ms threshold
240
350
0 ms
5 ms
10 ms
3 ms threshold
52.38 52.4 52.42 52.44 52.46 52.48 52.5 52.52 52.54 52.56 52.58
time [s]
Figure 7.5. Commanded, measured and modeled MR damper current time histories over a short du-
ration using random 25Hz square wave current control when the structure is subjected to a random
excitation (nominal peak 100 cm=s
2
, frequency range 0.1–30 Hz): (a) E-Defense commanded and
measured currents and the current predicted by the filtered multilevel rate-limiter; (b) correspond-
ing current rates e(t); (c) time duration when the current increases with current rate larger than
240 A=s; (d) corresponding maximum current rate; (e) time duration when the current decreases
with current rate smaller than800 A=s; (f) corresponding minimum current rate.
163
A graphical representation of saturator function sat
˙
I
max
˙
I
min
(e) is shown in Figure 7.6. This current
e =
I
df
(t)I(tDt)
Dt
sat
˙
I
max
˙
I
min
(e)
˙
I
max
˙
I
min
Figure 7.6. Graphical representation of saturator function sat
˙
I
max
˙
I
min
(e).
driver amplifier model is built using SIMULINK [107] and depicted in Figure 7.7.
I
d
(t)
1500
s+1500
Dt
1
sat
˙
I
max
˙
I
min
() Dt
I(t)
z
1
I
df
+
I(tDt)
Figure 7.7. Current driver amplifier model.
One might consider an alternate simpler rate limiter given by:
I(t)= I
df
(tDt)+Dt sat
˙
I
max
˙
I
min

I
df
(t) I
df
(tDt)
Dt

(7.9)
However, this simpler rate limiter cannot capture the behavior of the current driver amplifier as
well as the multi-level rate limiter, as shown in Figure 7.8.
Further, the effect of current driver amplifier with its rate limiter on the actual (measured)
current I can be approximated as a first-order low-pass filter of the desired (commanded) current
I
d
. This first-order low-pass filter is f
p
=(s+ f
p
), where f
p
is determined based upon the behavior of
the multilevel rate limiter. Figure 7.9 compares the E-Defense commanded and measured currents,
current modeled with the multilevel rate limiter, and modeled with a simple first-order low-pass
filter f
p
=(s+ f
p
) of the commanded current for f
p
= 50, 100, or 150 s
–1
, using random 25 Hz
164
52.496 52.498 52.5 52.502 52.504 52.506
time [s]
1
1.5
2
2.5
3
3.5
current [A]
commanded
measured
multilevel rate-limiter
simpler rate-limiter
Figure 7.8. E-Defense commanded and measured currents compared to two models of the current:
a simpler rate-limiter and the multi-level rate-limiter (the structure is subjected to a random ex-
citation (nominal peak 100 cm=s
2
, frequency range 0.1–30 Hz) using random 25 Hz square wave
current control).
square wave current control when the structure is subjected to a random excitation (nominal peak
100 cm=s
2
, frequency range 0.1–30 Hz) at E-Defense. The first-order low pass filter with f
p
=
150 s
–1
best captures the behavior of the measured control force. Therefore, the first-order low
pass filter is determined to be approximately 150 s
–1
=(s+ 150 s
–1
).
7.3.4 Semiactive controllers
Five state feedback controllers, listed in Table 7.2, are designed numerically with three scaled
historical earthquake excitations:
• 95% of the 1940 El Centro earthquake (the N-S component of the Imperial Valley earth-
quake, recorded at the Imperial Valley Irrigation District substation in El Centro, CA; origi-
nal PGA 0.3484g);
• 32% of the 1995 Kobe earthquake (the N-S component of the Hyogo-ken Nanbu Kobe earth-
quake, recorded at the JR Takatori station in Kobe, Japan; original PGA 0.6174g);
• 50% of the 1994 Northridge earthquake (the N-S component of the Northridge earthquake,
recorded at the Sylmar County Hospital parking lot in Sylmar, CA; original PGA 0.8431g).
165
11 11.05 11.1 11.15 11.2
time [s]
0
1
2
3
4
5
current [A]
E-Defense commanded
E-Defense measured
Multilevel rate-limiter
First-order filter (f
p
=50s
-1
)
First-order filter (f
p
=100s
-1
)
First-order filter (f
p
=150s
-1
)
Figure 7.9. E-Defense commanded and measured currents, current modeled with the multilevel
rate limiter, and modeled with a simple first-order low-pass filter f
p
=(s+ f
p
) of the commanded
current for f
p
= 50, 100, or 150s
–1
(the structure is subjected to a random excitation (nominal peak
100 cm=s
2
, frequency range 0.1–30 Hz), using random 25 Hz square wave current control).
and then tested with the specimen at E-Defense. To differentiate the five controllers designed by
the author from other controllers designed by colleagues at Kobe University, the University of Con-
necticut and NIED in this project, the five controllers are called USCi (where i= 1; 2; 3; 4 or 5).
The first controller is a standard CLQR designed to minimize J
1
; the remaining ones are OCLC
with particular design earthquakes with different scale factors, cost functions and states. The
OCLC designs are determined through simulating the response, to a design earthquake, of the
closed-loop system composed of the structure Eq. (7.4), a candidate CLC gain, the secondary con-
troller bang-bang clipping Eq. (7.7), optionally the current driver amplifier model Eq. (7.8), and the
MR damper model Eqs. (7.1) to (7.3). The first two OCLCs are designed to minimize cost metric
J
1
; the final two are OCLCs designed with the rate limiter to minimize J
2
. The first four controllers
use the conventional states: structure displacement (i.e., base drift) and the corresponding velocity.
While the secondary controller’s bang-bang clipping Eq. (7.7), the current driver amplifier with
its rate limiter, and the MR damper — combined — are nonlinear, the control-amplifier-damper-
structure system can be approximated with a linear system in which the MR damper force u is
a first-order low-pass filtered version of the desired force u
d
, as shown in Figure 7.10; the first-
order low-pass filter is approximately 150 s
–1
=(s+ 150 s
–1
), with the actual control force u as the
166
Table 7.2: Different state-feedback controllers designed numerically and then tested at E-Defense.
Label Strategy Cost Design EQ Current Rate Limiter States
USC1 CLQR J
1
ideal GWN no q; ˙ q
USC2 OCLC J
1
95% El Centro no q; ˙ q
USC3 OCLC J
1
32% Northridge no q; ˙ q
USC4 OCLC J
2
50% Northridge yes q; ˙ q
USC5 OCLC J
2
50% Northridge yes q; ˙ q;u
state. Thus, a three-state LQR control gain
¯
K
LQR
designed for this combined system is used
to parameterize a three-state CLC that is subsequently optimized (to minimize cost J
2
) using the
full closed-loop system, including the current driver amplifier model, to obtain the final controller
denoted USC5.
¨ q
g
150
s+150
Structure

¯
K
LQR
M
U
X
y
q
˙ q
u u
d
Figure 7.10. Approximated three-state linear system with a third state, the actual control force u.
7.4 RTHS
Subsequent to the July 2019 E-Defense tests, RTHS tests were performed in November 2019 at
Kobe University using the same MR damper (and accompanying amplifier, data acquisition and
control system), as depicted in Figure 7.11. These RTHS tests were conducted to: (1) validate the
accuracy of the MR damper model used in numerical simulation; (2) reproduce the E-Defense tests
and validate the capability of RTHS tests to predict the performance of several OCLC strategies,
relative to that of a conventional CLQR strategy; (3) compare the performance in reducing struc-
tural responses for many additional scenarios (different structural models, control strategies, etc.)
167
Figure 7.11. RTHS setup.
at low cost. The structural model (i.e., simulated component) in these RTHS tests is modified from
original numerical model used to design the controllers at E-Defense:
1. Kobe University colleagues [47] found that the E-Defense test specimen has larger damp-
ing than previously modeled; to quantify this additional damping, damping coefficient c=
2:9 kN s=m is considered, resulting in an increased damping ratio: z = 0:0578;
2. Kobe University colleagues conducted a series of preliminary passive-off RTHS tests and
found that the constant in the model of Coulomb friction force provided by the sliders under
the superstructure columns should be modified to be f
fric
= 2 kN to have better agreement
with E-Defense results [47].
During a RTHS test, first, the seismic isolation specimen model is uploaded to a control com-
puter. Subjected to a ground motion, the model’s dynamical responses are achieved using the same
iBIS digital signal processor (DSP) as E-Defense experiments in real time at a sampling frequency
of 500Hz. Structural response commands are provided to a second DSP to calculate the semiactive
control force. Corresponding current commands are computed and then applied, through the cur-
rent driver amplifier, to the physical MR damper mounted between a reaction frame and a shake
table used as an actuator. The displacement of the MR damper is measured by a displacement
transducer. The damper restoring force is measured by a load cell. The measured restoring force
168
is fed back into the specimen model in the computer to complete the RTHS feedback loop. All
data is collected by an EDX Data Acquisition System. An outline of the RTHS tests is presented
in Figure 7.12 [47].
Figure 7.12. RTHS outline [47].
7.5 Experimental and simulation results
7.5.1 E-Defense results
The E-Defense shake table experiments using the control strategies detailed herein, as well as those
designed by partners in this project [33], were conducted with a variety of excitations in July 2019.
Three scaled historical earthquake excitations were applied to evaluate the performance of USC
controllers:
• 150% of the 1940 El Centro earthquake (the N-S component of the Imperial Valley earth-
quake, recorded at the Imperial Valley Irrigation District substation in El Centro, CA; origi-
nal PGA 0.3484g);
169
• 40% of the 1995 Kobe earthquake (the N-S component of the Hyogo-ken Nanbu Kobe earth-
quake, recorded at the JR Takatori station in Kobe, Japan; original PGA 0.6174g);
• 50% of the 1994 Northridge earthquake (the N-S component of the Northridge earthquake,
recorded at the Sylmar County Hospital parking lot in Sylmar, CA; original PGA 0.8431g).
Mean square statistics of the responses with the control strategies USC1–USC4 and “passive on”
control, subjected to the three scaled earthquake ground motions, are listed in Tables 7.3 to 7.5.
It is worth noting that E-Defense results of USC5 were not reported. The reason is E-Defense
commanded and measured currents for USC5 remain at or about 4 A over the entire time duration,
as shown in Figure 7.13 when the system is subjected to the Kobe excitation whereas their numer-
ically simulated counterparts — using the actual E-Defense measured displacement, velocity and
control force — are quite different. This indicates that some other controller (perhaps the “passive
on” case) was mistakenly used in this E-Defense test instead of the USC5 controller. The same ob-
servation was noticed in the other two earthquakes (El Centro and Northridge). Therefore, Tables
7.3 to 7.5 reported a “passive on” case instead of USC5.
170
0
1
2
3
4
5
command current [A]
0
1
2
3
4
5
command current [A]
15 20 25
time [s]
0
1
2
3
4
5
measured current [A]
(a) E-Defense
15 20 25
time [s]
0
1
2
3
4
5
measured current [A]
(b) Numerical Simulation
Figure 7.13. (a) USC5 controller’s commanded and measured currents in the E-Defense experi-
ment when the structure is subjected to the Kobe excitation, and (b) the corresponding currents in
a pure SIMULINK numerical simulation using the actual E-Defense measured displacement, veloc-
ity and control force to compute the commanded current and the MR damper model to compute
the “measured” current. The two commanded currents — E-Defense and simulated — should be
identical; as they are not, it is suspected that some other controller (perhaps the “passive on” case)
was mistakenly used in this E-Defense test instead of the USC5 controller.
An indicator function of commanded current, denoted I
c
, is defined as:
I
c
(t)=
8
>
<
>
:
1; commanded current I
d
(t) is on
0; otherwise
(7.10)
171
In previous chapters, D was defined to be how much a response increased with some control
strategy relative to an OCLC baseline. Here, because multiple OCLC designs are considered
but just one CLQR design, D is defined differently as the change relative to a baseline CLQR;
i.e.,D(p
CLQR
p
OCLC
)=p
CLQR
, where p is some response metric: for response metrics, pos-
itive D means that the corresponding OCLC design is more effective than CLQR; for the mean
square control force metric, positiveD means OCLC uses less control effort than CLQR; for the
metricE[I
c
],D> 0 indicates OCLC clips less frequently than CLQR (D is omitted for the J
1
met-
ric for USC4 and USC5 since they are intentionally designed for the other cost metric J
2
, resulting
in a comparison with the J
1
value of CLQR having little, if any, meaning).
The results validate that OCLC is superior to CLQR for certain metrics when subjected to dif-
ferent earthquake excitations. Comparing USC1–USC3 (as stated previously, USC1 is a standard
CLQR designed to minimize J
1
; USC2–3 are OCLCs designed for cost metric J
1
using conven-
tional states), USC2 and USC3 have comparable performance, both with a J
1
cost that is about
35% to 43% below that of CLQR; the mean square absolute acceleration is also around 33% to
44% smaller than CLQR, with a mean square control force that is roughly half that of CLQR,
indicating that OCLC achieves considerable absolute acceleration response reduction with much
less control effort compared with CLQR; theE[I
c
] value is smaller than CLQR in general (except
for the case when the structure is controlled by USC3 and subjected to the Northridge earthquake),
meaning that USC2 and USC3 clip more frequently than CLQR (however, USC2 and USC3 still
outperform CLQR in reducing the cost metric). There is a modest increase in the mean square
drift and velocity for all three excitations for both USC2 and USC3 relative to the baseline USC1.
As may be expected, OCLC designed for a specific excitation performs best in minimizing the
same cost metric when evaluated by that excitation (though different scaling factors); e.g., OCLC
designed for the 95% El Centro excitation (USC2) has a smaller cost metric than that of OCLC de-
signed for the 32% Northridge excitation (USC3) when the system is subjected to 150% El Centro
ground motion. Nevertheless, OCLC designed for one specific excitation still provides significant
172
Table 7.3: Statistics of the E-Defense structural response to 150% of the 1940 El Centro earthquake
for different controllers USC1 to USC5. D denotes reductions (i.e., improvements) relative to the
baseline CLQR strategy designed using cost J
1
; bold values denote the best performance in each
column; gray text denotes incomparable quantities: USC1/2/3 are designed for J
1
; USC4/5 are
designed for J
2
.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 1.6037 (–) 386.9966 0.1097 (–) 84.3192 (–) 11.9805 (–) 0.4102 (–) 0.0656 (–)
USC2 (OCLC/ElCntr.) 0.9195 (43) 249.4796 0.0613 (44) 91.1936 (–8) 13.8343 (–15) 0.2014 (51) 0.0224 (–66)
USC3 (OCLC/Northrdg.) 0.9533 (41) 254.8838 0.0637 (42) 88.0796 (–4) 13.4326 (–12) 0.2147 (48) 0.0389 (–41)
USC4 (OCLC/Northrdg.) 2.7860 (–) 520.1900 0.1590 (–45) 69.1147 (18) 8.1917 (32) 1.1191 (–173) 0.3017 (360)
Passive on 11.5842 (–) 400.6630 0.1093 (0) 39.5521 (53) 5.5149 (54) 10.4464 (–2446) 1.0000 (1424)
Table 7.4: Statistics of the E-Defense structural response to 40% of the 1995 Kobe earthquake for
different controllers USC1 to USC5.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 1.7480 (–) 415.5522 0.1212 (–) 98.1715 (–) 10.7342 (–) 0.4275 (–) 0.0745 (–)
USC2 (OCLC/ElCntr.) 1.0138 (42) 256.3490 0.0673 (44) 99.9250 (–2) 11.4346 (–7) 0.2295 (46) 0.0292 (–61)
USC3 (OCLC/Northrdg.) 1.0640 (39) 266.6155 0.0711 (41) 98.2483 (–0) 11.1599 (–4) 0.2433 (43) 0.0549 (–26)
USC4 (OCLC/Northrdg.) 3.0357 (–) 554.0098 0.1701 (–40) 85.3471 (13) 8.1704 (24) 1.2414 (–190) 0.4390 (489)
Passive on 7.3869 (–) 275.1014 0.0725 (40) 50.2691 (49) 5.9342 (45) 6.6054 (–1445) 1.0000 (1242)
Table 7.5: Statistics of the E-Defense structural response to 50% of the 1994 Northridge earthquake
for different controllers USC1 to USC5.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.9670 (–) 231.4708 0.0627 (–) 57.3021 (–) 9.0232 (–) 0.2738 (–) 0.0415 (–)
USC2 (OCLC/ElCntr.) 0.6244 (35) 177.7629 0.0421 (33) 62.4143 (–9) 10.8763 (–21) 0.1300 (53) 0.0273 (–34)
USC3 (OCLC/Northrdg.) 0.6208 (36) 173.1842 0.0409 (35) 61.2443 (–7) 10.6650 (–18) 0.1399 (49) 0.0527 (27)
USC4 (OCLC/Northrdg.) 1.3423 (–) 246.6826 0.0716 (–14) 42.4128 (26) 6.2490 (31) 0.5773 (–111) 0.4018 (868)
Passive on 2.6093 (–) 110.1773 0.0282 (55) 21.0327 (63) 3.2675 (64) 2.3032 (–741) 1.0000 (2308)
performance improvements compared to CLQR when the structure is subjected to other excita-
tions. For example, OCLC designed to minimize J
1
in the Northridge earthquake (USC2) has a
cost and mean square absolute acceleration that are 42% and 44% smaller, respectively, using a
46% lower control force level, than those provided by CLQR when the structure is excited by the
Kobe earthquake.
USC4, which is designed with the rate limiter to minimize J
2
for the Northridge excitation, can
reduce the mean square drift and velocity by up to 32% and 26%, respectively, though at the ex-
pense of increasing the mean square absolute acceleration by up to 45%; the mean square control
173
force is up to three times that of CLQR; theE[I
c
] value is up to ten times that of CLQR, indicating
that USC4 clips less frequently than CLQR. Notably, USC4 provides best performance when eval-
uated by the Northridge excitation compared with the other two excitations, as it decreases drift
and velocity by 31% and 26%, respectively, while only increasing the absolute acceleration by
14%. Again, although USC4 is designed for the Northridge excitation, it still outperforms CLQR
in reducing mean square drift and velocity when the structure is subjected to the El Centro and
Kobe earthquakes.
Finally, the “passive on” case has a smaller J
2
cost than USC4 for all earthquake excitations;
it can mitigate not only the mean square absolute acceleration by up to 55%, but also the mean
square drift and velocity by up to 63% and 64%, respectively, relative to CLQR, though at the
expense of a much larger mean square control force level than others; theE[I
c
] value is always
one, as expected, since the commanded current remains at 4 A.
7.5.2 RTHS results
RTHS tests were conducted using the USC1 (CLQR) and USC4 (OCLC/Northrdg./J2) controllers
when the structure was subjected to each of the three earthquake excitations at E-Defense, and
using the USC2 (OCLC/Northrdg./J1), USC3 (OCLC/ElCntr.) and USC5 (OCLCnorlim3) con-
trollers only for the Kobe earthquake (when running USC2, USC3 and USC5 for the other two
earthquake excitations, RTHS tests hit one of the safety stop criteria: the measured shake table
acceleration spiked beyond the safe region, as shown in the acceleration in Figure 7.14 when the
structure, controlled by USC2, is subjected to the Northridge excitation; the difference between
measured table drift and real-time hybrid simulated structural drift [i.e., commanded table drift] is
probably due to the spike of the measured table acceleration).
Tables 7.6 to 7.8 summarize the RTHS results for different earthquakes and corresponding
control strategies. As explained previously, USC1 is a standard CLQR designed to minimize J
1
;
USC2–3 are OCLCs designed to minimize cost metric J
1
in the El Centro and Northridge earth-
quake, respectively; USC4–5 are OCLCs designed with the rate limiter to minimize J
2
. Comparing
174
0 5 10 15
0
0.1
0.2
drift [m]
table drift (meas)
structure drift (sim)
0 5 10 15
time [s]
0
10
accel. [m/s
2
]
table accel. (meas)
structure rel. accel. (sim)
Figure 7.14. Shake table and structure drift and acceleration using USC2 when the structure is
subjected to the Northridge excitation.
USC1–3 in Table 7.7 when the structure is subjected to the Kobe earthquake, USC2 and USC3 can
provide up to 40% and 16% cost function and mean square absolute acceleration reduction, re-
spectively, and use up to 62% smaller mean square control force; USC5 even uses 81% lower
control effort to achieve 21% mean square absolute acceleration reduction while increasing the
mean square drift only 15% compared to CLQR. Notably, comparing USC2, USC3 and USC5
when subjected to the Kobe excitation, USC5 provides better absolute acceleration reduction with
comparable mean square drift and velocity, but with a mean square control force that is roughly half
that of USC2 and USC3, most likely due to USC5 having additional information of the measured
MR damper force as one of its states. Comparing all three tables, USC4 performs best among all
controllers for reducing the mean square drift and velocity, by up to 58% and 41%, respectively,
when the structure is subjected to the Northridge earthquake, though does so with larger accel-
erations (due to the larger damping force switching on and off); it is expected that USC4 should
provide drift and velocity response reduction superior to USC2 and USC3 since USC4’s design
cost metric J
2
emphasizes drift reduction whereas USC2 and USC3 use J
1
that emphasizes abso-
lute acceleration reduction. Further, OCLC designed for one specific excitation also performs well
when subjected to other excitations. Comparing theE[I
c
] value of all RTHS test results (Tables 7.6
to 7.8), USC4 and USC5 clip much less frequently than USC1 (CLQR).
175
Table 7.6: RTHS results using modified friction and specimen model subjected to the El Centro
earthquake for controllers USC1 and USC4.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.7722 (–) 84.2852 0.0275 (–) 71.0765 (–) 8.7641 (–) 0.4177 (–) 0.0893 (–)
USC4 (OCLC/Northrdg.) 3.9292 (–) 163.7615 0.0495 (–80) 50.0907 (30) 3.6830 (58) 3.3803 (–709) 0.3054 (242)
Table 7.7: RTHS results using modified friction and specimen model subjected to the Kobe earth-
quake for different controllers USC1 to USC5.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.8469 (–) 78.6855 0.0255 (–) 88.3671 (–) 9.2234 (–) 0.4945 (–) 0.1231 (–)
USC2 (OCLC/ElCntr.) 0.5098 (40) 66.2818 0.0218 (14) 93.7523 (–6) 10.4522 (–13) 0.1874 (62) 0.0457 (–63)
USC3 (OCLC/Northrdg.) 0.5179 (39) 64.9494 0.0213 (16) 92.1442 (–4) 10.2315 (–11) 0.2020 (59) 0.0861 (–30)
USC4 (OCLC/Northrdg.) 4.8391 (–) 184.3706 0.0551 (–116) 70.8174 (20) 6.2993 (32) 4.2106 (–752) 0.4108 (234)
USC5 (OCLCnorlim3) 0.4018 (–) 61.1299 0.0202 (21) 93.5212 (–6) 10.6312 (–15) 0.0953 (81) 0.2507 (104)
Table 7.8: RTHS results using modified friction and specimen model subjected to the Northridge
earthquake for controllers USC1 and USC4.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.4677 (–) 44.4018 0.0144 (–) 46.9899 (–) 7.2431 (–) 0.2695 (–) 0.0766 (–)
USC4 (OCLC/Northrdg.) 1.6830 (–) 69.2124 0.0209 (–45) 27.8449 (41) 3.0781 (58) 1.4430 (–435) 0.4890 (538)
7.5.3 Numerical simulation revisited to better calibrate MR damper and
friction models
Comparing numerical simulation based on the original MR damper numerical model in Eq. (7.1)
against the RTHS test results indicate that the original damper model developed by Kobe University
imperfectly predicts the MR damper force, leading to non-negligible differences between RTHS
and numerical simulation results. For example, Figure 7.15 displays the actual control force u
measured in the RTHS test using USC2 to control the structure response to the Kobe earthquake,
and the corresponding numerically simulated MR damper force using sgn ˙ q. An alternative is to
replace the signum function with a smoother hyperbolic tangent function tanha ˙ q
u= f
c
(I)tanha ˙ q+ c
MR
(I) ˙ q (7.11)
176
where a is the slope of the hyperbolic tangent function when ˙ q= 0. This MR damper numerical
model with a hyperbolic tangent function can better capture the control force at RTHS than that
with a signum function, especially when the control force is small, because the RTHS control
force does not change as fast as the numerically simulated control force with a signum function.
When the control force is larger, the models with hyperbolic tangent and signum functions perform
comparably, and cannot capture the peak value of the actual control force measured at RTHS. The
0
2
4
6
actual control force [kN]
(a)
18 18.5 19 19.5 20
time [s]
0
2
4
6
actual control force [kN]
(c)
0
0.2
0.4
actual control force [kN]
(b)
4 4.5 5 5.5 6
time [s]
0
0.2
0.4
actual control force [kN]
(d)
Figure 7.15. Actual control force u measured in the RTHS test subjected to the Kobe excitation
and controlled by USC2, and the corresponding MR damper control force modeled with sgn ˙ q
and tanha ˙ q with a= 3000 s=m: (a) & (b) RTHS control force and numerically simulated MR
damper force using numerically simulated current and velocity; (c) & (d) RTHS control force and
numerically simulated MR damper force using measured current and velocity from this RTHS test.
value of a is determined via a parametric study over value ranging in 500–10000 s=m; for each
candidate a value, the MR damper force is computed using Eq. (7.11) with I(t) and ˙ q(t) from
numerical simulation. The resulting MR damper force time histories u(t;a) are compared with the
177
corresponding RTHS results; a= 3000s=m leads to better track of the real MR damper force when
the control force is small, with the peak value difference in each cycle on the order of 0:2 kN.
Similarly, comparing the numerical simulation of the column-supported sliders’ friction using
the original numerical model in Eq. (7.5) and the corresponding E-Defense responses suggested
that the sgn friction force is imperfect as well. Therefore, the friction force model is also modified
with a hyperbolic tangent function:
F
fric
= f
fric
tanhb ˙ q (7.12)
A preliminary study found that b= 700 s=m was sufficiently steep to match the E-Defense mea-
sured friction force with damper velocity data but not so steep that simulations required extremely
small time steps to accurately capture the hyperbolic tangent behavior. Figure 7.16 shows the
friction force F
fric
measured directly from load cells in the E-Defense test when the structure is
controlled by USC1 when subjected to the Kobe excitation, and the corresponding model predic-
tions using sgn in Eq. (7.5) and tanh in Eq. (7.12). The friction force numerical model with a
hyperbolic tangent function matches better the friction force at E-Defense than that with a signum
function, with the peak value difference in each cycle on the order of up to 2 kN for a signal that
has a peak of about 6 kN.
The numerical simulation responses using the modified MR damper and friction force F
fric
models are computed using SIMULINK and summarized in Tables 7.9 to 7.11. The results have
good agreement with the RTHS results for all controllers as the response and response reduction
values are similar to the RTHS results; specifically, the mean square drift difference is on the or-
der of 0.03–2 cm
2
for a mean square drift value of up to 12 cm
2
. USC2, USC3 and USC5 have
comparable performance and have smallerE[I
c
] values than USC1 (CLQR), but USC5 uses mod-
estly less control effort than USC2 and USC3. When the structure is subjected to the El Centro
and Northridge earthquake excitations, USC4 has a smaller J
2
cost compared with USC5; when
the structure is subjected to the Northridge earthquake, USC4 not only provides 44% and 39%
178
0
2
4
6
8
friction force [kN]
(a)
20 20.5 21 21.5 22 22.5 23 23.5 24
time [s]
0
2
4
6
8
friction force [kN]
(c)
0
1
2
3
friction force [kN]
(b)
32.35 32.4 32.45 32.5 32.55
time [s]
0
1
2
3
friction force [kN]
(d)
Figure 7.16. Friction force F
fric
measured in the E-Defense test subjected to the Kobe excitation
and controlled by USC1, and the corresponding friction force modeled with sgn ˙ q and tanhb ˙ q
with b= 700 s=m: (a) & (b) E-Defense friction force and numerically simulated friction force
using numerically simulated velocity; (c) & (d) E-Defense friction force and numerically simulated
friction force using measured velocity from this E-Defense test.
reduction, respectively, in mean square drift and velocity, but also reduces the mean square abso-
lute acceleration by 5% compared to CLQR; USC4 achieves the response reductions with much
less frequent clipping than CLQR, evidenced by USC4’sE[I
c
] value being up to five times that
of CLQR. It is worth noting that numerical models used herein have different structural damping
ratios, MR damper and friction force models than the numerical models used to design the con-
trollers; thus, the designed controllers might not be fully optimal. For example, USC5 does not
perform well when the structure is subjected to the Northridge earthquake, though still holds a
much lower control force level than CLQR.
179
Table 7.9: Numerical simulation results using modified MR damper and friction force models
when the structure is subjected to the El Centro earthquake and mitigated by controllers USC1 to
USC5.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.5310 (–) 120.8532 0.0235 (–) 65.1482 (–) 10.5793 (–) 0.2205 (–) 0.0201 (–)
USC2 (OCLC/ElCntr.) 0.3979 (25) 129.0138 0.0231 (2) 74.2614 (–14) 12.7003 (–20) 0.0799 (64) 0.0032 (–84)
USC3 (OCLC/Northrdg.) 0.4042 (24) 128.1255 0.0230 (2) 73.3233 (–13) 12.5710 (–19) 0.0884 (60) 0.0100 (–50)
USC4 (OCLC/Northrdg.) 1.4336 (–) 114.7219 0.0274 (–17) 45.7641 (30) 5.8647 (45) 1.1076 (–402) 0.1026 (412)
USC5 (OCLCnorlim3) 0.3852 (–) 123.7079 0.0229 (3) 76.4177 (–17) 11.7267 (–11) 0.0684 (69) 0.0041 (–80)
Table 7.10: Numerical simulation results using modified MR damper and friction force models
when the structure is subjected to the Kobe earthquake and mitigated by controllers USC1 to
USC5.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.5499 (–) 102.1308 0.0202 (–) 79.3640 (–) 8.6277 (–) 0.2598 (–) 0.0217 (–)
USC2 (OCLC/ElCntr.) 0.3489 (37) 102.3264 0.0182 (10) 85.9065 (–8) 10.1185 (–17) 0.0704 (73) 0.0023 (–89)
USC3 (OCLC/Northrdg.) 0.3541 (36) 100.9497 0.0180 (11) 84.9900 (–7) 9.9405 (–15) 0.0788 (70) 0.0073 (–66)
USC4 (OCLC/Northrdg.) 1.5439 (–) 104.5283 0.0241 (–19) 57.3392 (28) 5.6738 (34) 1.2395 (–377) 0.0925 (326)
USC5 (OCLCnorlim3) 0.3343 (–) 98.0066 0.0173 (14) 85.8003 (–8) 9.7785 (–13) 0.0654 (75) 0.0066 (–70)
Table 7.11: Numerical simulation results using modified MR damper and friction force models
when the structure is subjected to the Northridge earthquake and mitigated by controllers USC1 to
USC5.
Control J
1
(D) J
2
E[ ¨ q
2
a
] (D) E[ ˙ q
2
] (D) E[q
2
] (D) E[u
2
] (D) E[I
c
] (D)
strategy [–] [%] [–] [m
2
/s
4
] [%] [cm
2
/s
2
] [%] [cm
2
] [%] [kN
2
] [%] [–] [%]
USC1 (CLQR) 0.3109 (–) 70.6829 0.0120 (–) 43.1909 (–) 7.2715 (–) 0.1402 (–) 0.0177 (–)
USC2 (OCLC/ElCntr.) 0.2300 (26) 77.7154 0.0122 (–2) 50.9738 (–18) 8.7243 (–20) 0.0479 (66) 0.0036 (–80)
USC3 (OCLC/Northrdg.) 0.2385 (23) 76.2495 0.0120 (0) 49.9043 (–16) 8.5659 (–18) 0.0602 (57) 0.0138 (–22)
USC4 (OCLC/Northrdg.) 0.7369 (–) 56.1474 0.0114 (5) 26.3864 (39) 4.0941 (44) 0.5919 (–322) 0.0330 (87)
USC5 (OCLCnorlim3) 0.2310 (–) 80.3668 0.0121 (–1) 53.6958 (–24) 9.3924 (–29) 0.0470 (66) 0.0095 (–46)
7.5.4 Results comparison
The time histories of E-Defense structural responses and control force are compared with cor-
responding RTHS and numerical simulation results using the modified models of specimen, MR
damper and friction force provided by the sliders under the four columns. It is worth noting that the
semiactive controllers applied in the E-Defense are designed using the original numerical model
(original structure, MR damper and friction force models) without modification. Each graph in
Figures 7.17 to 7.19 depicts a comparison of a response as measured in an E-Defense test, as
180
computed in the virtual structure model of the corresponding RTHS test, as computed in a pure
numerical simulation, and (for absolute acceleration only) as calculated using the SDOF model
¨ q
a
=2zw ˙ qw
2
q
u
m

F
fric
m
(7.13)
using time histories of responses measured during the E-Defense test: base drift q(t), measured
base velocity ˙ q(t), measured MR damper force u(t), and the combined sliders’ forces (each mea-
sured from the load cells under each of the structure’s four columns). Figure 7.17 shows this
comparison for the structural absolute acceleration, velocity and drift responses to the three earth-
quakes (150% El Centro, 40% Kobe, and 50% Northridge) when the structure is controlled by
USC1. Figure 7.19 shows the same but using the USC4 controller. As only the Kobe earthquake
was used for the other controllers, Figure 7.18 shows the Kobe earthquake responses using con-
trollers USC2, USC3 and USC5.
181
15 20 25
time [s]
0
5
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(a) 1940 El Centro
15 20 25
time [s]
0
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(b) 1995 Kobe
15 20 25
time [s]
0
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(c) 1994 Northridge
Figure 7.17. Structural responses measured or computed when the structure is controlled by USC1
and subjected to the one of three scaled earthquake excitations.
182
15 20 25
time [s]
0
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(a) USC2
15 20 25
time [s]
0
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(b) USC3
15 20 25
time [s]
0
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
RTHS Num. Sim.
(c) USC5
Figure 7.18. Structural responses measured or computed when the structure is controlled by USC2,
USC3 & USC5 and subjected to the Kobe earthquake.
183
15 20 25
time [s]
0
5
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(a) 1940 El Centro
15 20 25
time [s]
0
10
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(b) 1995 Kobe
15 20 25
time [s]
0
5
drift [cm]
0
50
vel [cm/s]
0
4
abs. accel. [m/s
2
]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
(c) 1994 Northridge
Figure 7.19. Structural responses measured or computed when the structure is controlled by USC4
and subjected to the one of three scaled earthquake excitations.
184
RTHS and numerical simulation results match quite well for drift, velocity and absolute accel-
eration responses for USC1 to USC5, indicating that the modified MR damper model matches the
actual damper’s behavior very well. However, many of the drift and control force time histories
show that the RTHS results have a bias, though not necessarily the same bias in all RTHS tests.
The exact reason for the bias is unknown, but the best guess is miscalibration of the displace-
ment transducer. The E-Defense responses match those of RTHS and numerical simulation, but
contain high-frequency content in the velocity, calculated absolute acceleration, and, especially,
measured absolute acceleration responses. The moderate high-frequency content in the velocity
and calculated absolute acceleration is mainly from the sensor noise. To evaluate why the mea-
sured absolute acceleration at E-Defense contains significant high-frequency content, Figure 7.20
shows the power spectral densities (PSD) of absolute acceleration measured from E-Defense, cal-
culated using E-Defense data, computed in the virtual structure in the RTHS test and from nu-
merical simulation, when the controller is USC1 (CLQR) and the structure is subjected to the El
Centro excitation. As expected, the measured E-Defense absolute acceleration has larger PSD
values at high frequency. To verify that removing this high frequency content provides results
closer to RTHS, the E-Defense measured structure absolute acceleration response is filtered with
an 8
th
order Butterworth filter with the half-power frequency 25 Hz using MATLAB’s filtfilt
function (which filters forward and then backward in time so this is no phase distortion). Fig-
ure 7.21 compares the time history and PSD of the absolute accelerations measured at E-Defense,
those filtered from E-Defense, and those computed from RTHS using the USC1 (CLQR) con-
troller and the El Centro external excitation. The filtered absolute acceleration no longer contains
the high-frequency content and matches quite well with RTHS results. Therefore, it may be con-
cluded that the high-frequency content in the E-Defense measured absolute acceleration is due to
the significant high-frequency noise of the sensing system at E-Defense (the reason why the noise
is multiplicative but not additive is unknown). To further validate that the velocity feedback sen-
185
10 10
0
10
1
10
2
Frequency [Hz]
10
10
10
10
10
10
10
10
10
10
0
abs. accel. PSD [m
2
/s
4
/Hz]
E-Defense meas.
E-Defense calc.
RTHS
Num. Sim.
Figure 7.20. Power spectral densities (PSD) of absolute acceleration measured from E-Defense,
calculated using E-Defense data, computed in the virtual structure in the RTHS test and from
numerical simulation, when the controller is USC1 (CLQR) and the structure is subjected to the El
Centro excitation.
sor causes the high frequency content in the E-Defense measured absolute acceleration, the base
velocity is computed as the central difference approximation of the E-Defense laser displacement
˙ q(t)=
q(t+Dt) q(tDt)
2Dt
(7.14)
(it should be noted that the noise in the laser displacement gets amplified in the central difference
approximated velocity), and is compared with the E-Defense measured damper velocity and the
velocity computed in the virtual structure model of the corresponding RTHS test. Figure 7.22
compares the time history and PSD of the damper velocity measured at E-Defense, central dif-
ference approximated velocity using E-Defense laser displacement data, and velocity computed
from RTHS, and, for the PSD comparison only, the laser displacement PSD times w
2
, using the
186
15 20 25
time [s]
0
2
4
abs. accel. [m/s
2
]
10 10
0
10
1
10
2
Frequency [Hz]
10
10
10
0
abs. accel. PSD [m
2
/s
4
/Hz]
E-Defense meas.
E-Defense filtered
RTHS
Figure 7.21. Time history and PSD of the absolute accelerations measured at E-Defense, those
filtered from E-Defense, and those computed from RTHS using the USC1 (CLQR) controller and
the El Centro external excitation.
USC1 (CLQR) controller and passive on control when the structure is subjected to the El Cen-
tro earthquake excitation. For USC1 (CLQR) control which uses the velocity feedback signal in
the controller, the central difference approximated velocity has much smaller oscillations than the
damper velocity and matches quite well with RTHS results; for passive on control, the damper
velocity does not contain high frequency content at all since passive on does not require any state
feedback signals. Therefore, it may be concluded that the high-frequency content in the E-Defense
measured absolute acceleration is due to the noisy velocity feedback sensing system at E-Defense.
Further, the RTHS and numerically simulated responses match quite well when the structure is
controlled by USC5 and subjected to Kobe earthquake.
Figures 7.23 to 7.25 compare the actual control force measured in the E-Defense test, the mea-
sured force in the RTHS test and the numerically simulated force. The E-Defense data, again,
187
17 17.2 17.4 17.6 17.8 18
time [s]
0
10
20
30
vel [cm/s]
E-Defense damper
E-Defense laser
RTHS
10 10
0
10
1
10
2
Frequency [Hz]
10
10
10
0
10
5
vel PSD [cm
2
/s
2
/Hz]
E-Defense damper
E-Defense laser
2
*E-Defense laser displ
RTHS
(a) USC1 (CLQR)
17 17.2 17.4 17.6 17.8 18
time [s]
0
10
20
vel [cm/s]
E-Defense damper
E-Defense laser
RTHS
10 10
0
10
1
10
2
Frequency [Hz]
10
10
0
vel PSD [cm
2
/s
2
/Hz]
E-Defense damper
E-Defense laser
2
*E-Defense laser displ
RTHS
(b) passive on
Figure 7.22. Time history and PSD of the damper velocity and central difference approximated
velocity at E-Defense, and velocity computed from RTHS, and, for the PSD comparison only, the
laser displacement PSD timesw
2
, using the USC1 (CLQR) controller and passive on control, and
the El Centro external excitation.
exhibits high-frequency content. To verify that removing this high frequency content provides re-
sults closer to RTHS, the E-Defense measured control force is filtered with an 8
th
order Butterworth
filter with the half-power frequency 25 Hz using MATLAB’sfiltfilt function. Figure 7.26 com-
pares the time history and PSD of the control forces measured at E-Defense, those filtered from
E-Defense, and those measured from RTHS using the USC1 (CLQR) controller and the Northridge
earthquake excitation. The filtered control force no longer contains the high-frequency content and
matches quite well with RTHS results. Therefore, it may be concluded that the high-frequency
content in the E-Defense measured control force is due to the significant high-frequency sensor
noise at E-Defense.
188
15 20 25
time [s]
0
2
4
6
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(a) 1940 El Centro
15 20 25
time [s]
0
2
4
6
8
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(b) 1995 Kobe
15 20 25
time [s]
0
2
4
6
8
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(c) 1994 Northridge
Figure 7.23. Actual control force u measured in the E-Defense test, measured in the RTHS test and
numerically simulated when the structure is controlled by USC1.
189
15 20 25
time [s]
0
2
4
6
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(a) USC2
15 20 25
time [s]
0
1
2
3
4
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(b) USC3
15 20 25
time [s]
0
0.5
1
1.5
actual control force [kN]
RTHS
Num. Sim.
(c) USC5
Figure 7.24. Actual control force u measured in the E-Defense test, measured in the RTHS test and
numerically simulated when the structure is controlled by USC2, USC3 & USC5 and subjected to
the Kobe earthquake.
190
15 20 25
time [s]
0
2
4
6
8
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(a) 1940 El Centro
15 20 25
time [s]
0
5
10
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(b) 1995 Kobe
15 20 25
time [s]
0
2
4
6
8
actual control force [kN]
E-Defense
RTHS
Num. Sim.
(c) 1994 Northridge
Figure 7.25. Actual control force u measured in the E-Defense test, measured in the RTHS test and
numerically simulated when the structure is controlled by USC4.
For USC1, E-Defense results are in phase with those of RTHS and numerical simulation,
though contain high-frequency content and spikes when the control force level is higher; E-Defense
results match quite well with those of RTHS and numerical simulation when the control force level
is lower, e.g., when the external excitation is Northridge, E-Defense results only have significant
high-frequency content during 12–18 s, whereas they match quite well with those of RTHS and
numerical simulation when the control force level is lower before 12 s or after 18 s. For the struc-
ture controlled by USC2 and USC3 and subjected to the Kobe excitation, the E-Defense results
are also in phase with those of RTHS and numerical simulation, though still with high-frequency
191
10 15 20 25
time [s]
0
5
control force [kN]
10 10
0
10
1
10
2
Frequency [Hz]
10
10
10
0
control force PSD [kN
2
/Hz]
E-Defense meas.
E-Defense filtered
RTHS
Figure 7.26. Time history and PSD of the control forces measured at E-Defense, those filtered from
E-Defense, and those measured from RTHS using the USC1 (CLQR) controller and the Northridge
earthquake excitation.
content and spikes. For USC5 control of the structure subjected to the Kobe excitation, RTHS and
numerical simulation results match quite well. For USC4, the E-Defense control forces also have
high-frequency content and spikes when subjected to all the three earthquake excitations. When
the structure is subjected to the Northridge excitation, again, the E-Defense control forces only
have significant high-frequency content during 12–18 s, and match quite well with those of RTHS
and numerical simulation when the control forces are much smaller.
7.6 Summary
This chapter discussed a series of E-Defense shake table experiments using several controllable
damping strategies designed to mitigate the responses of a full-size base-isolated structure spec-
imen with an MR fluid damper installed in the isolation layer, as well as corresponding RTHS
192
and numerical simulation results. The experimental results showed that OCLC can provide per-
formance superior to CLQR, either in minimizing absolute acceleration while not significantly
increasing ground-relative displacement, or in minimizing ground-relative displacement and ve-
locity while not significantly increasing absolute acceleration for the base-isolated structure. Fur-
ther, OCLC designed with one specific excitation also performs well when subjected to a different
excitation. The E-Defense, RTHS and numerical simulation results match well for drift, velocity
and absolute acceleration responses for USC1 to USC4, though the E-Defense data contains sig-
nificant high-frequency content due to sensor noise. Further, the RTHS and numerically simulated
results for USC5 match well also.
193
Chapter 8
Solutions of the Fokker-Planck-Kolmogorov Equation
Associated with Ideal Optimal Clipped Linear Control of a
SDOF System Excited by Gaussian White Noise
The response of structural systems subjected to random excitations, such as Gaussian white noise
(GWN), can be determined through random vibration methods, e.g., investigating the response
joint probability density function. The response of such systems may be expressed in terms of
the transient probability density function, which satisfies the Fokker-Planck-Kolmogorov (FPK)
equation. The FPK equation associated with different dynamical systems can be easily derived;
however, exact solutions of the FPK equation only exist for some simple systems. This chapter
investigates numerical solutions to the FPK equation associated with the ideal OCLC of a SDOF
system excited by GWN. The organization of this chapter is as follows: section one briefly intro-
duces the FPK equation and finite difference simulation of partial differential equations (PDEs);
the second section conducts a case study of a linear oscillator problem, providing possible choices
of parameters and solution methods of the finite difference scheme for the nonlinear semiactive
system; in section three, the numerical solutions of the FPK equation for a SDOF system with
OCLC are discussed; section four explores the design of OCLC using FPK solutions; the last
section summarizes this study.
194
8.1 Background
In the past several decades, extensive research has focused on numerical solutions of FPK equa-
tions associated with various dynamical systems. Finite element and finite difference methods have
been applied to solve FPK equations. Spencer and Bergman [96] applied the finite element method
to the solution of the transient FPK equation for several nonlinear stochastic systems giving, for
the first time, the joint probability density function of the response. Wojtkiewicz et al. [113, 114]
studied numerical solutions of the FPK equation of linear and nonlinear problems using finite
element and finite difference methods, including solutions to the stationary FPK equation using
high-order finite difference methods. A finite difference scheme with an improved discretization
of the standard FPK operator-splitting method was proposed by Xie et al. [116] to solve the FPK
equation associated with the Duffing oscillator driven by colored noise. Kumar et al. [63] solved
the stationary FPK equation associated with nonlinear dynamical systems using a local numerical
technique based on the meshless partition of the unity finite element method (PUFEM). Pichler et
al. [83] compared different finite element and difference methods with a novel stabilized multi-
scale finite element method that reduces the number of elements while maintaining accuracy and
is, thus, more computationally efficient. Yan and Cui [118] applied the finite difference scheme to
the solution of a time-fractional FPK equation with the external force and source term depending
on space and time. Compared to the finite element method, the finite difference method is more
easily extended to higher order schemes [113]; therefore, this study will apply the finite difference
method to solve the resulting FPK equation.
8.1.1 Fokker-Planck-Kolmogorov equation
Consider a typical n-dimensional vector stochastic differential equation (SDE) of the form:
˙ x(t)=
˜
b(x)+s(x)w(t) (8.1)
195
where x is anR
n
-valued process; w denotes a Gaussian white noise vector process with zero mean,
i.e.,E[w]= 0, and intensityE[w(t)w
T
(t+t)]= Dd(t); and
˜
b(x) ands(x) are the drift vector and
diffusion matrix, respectively. Defining the transition probability density function (PDF) p(x;tjx
0
)
for the process x, the Fokker-Planck-Kolmogorov equation (or the forward Kolmogorov equation)
associated with the SDE Eq. (8.1) can be derived from the Itˆ o form:
dx(t)= b(x)dt+s(x)dB(t) (8.2)
where B(t) denotes a vector Wiener process (standard Brownian motion) and b(x) is the Wong-and
Zakai-corrected drift vector [28]; the resulting FPK equation is given by
¶ p
¶t
=
n
å
i=1
¶
¶x
i
[b
i
(x)p]+
1
2
n
å
i=1
n
å
j=1
¶
2
¶x
i
¶x
j
[h
i j
(x)p] (8.3)
where h(x)=s(x)Ds
T
(x). The PDF p has a normalization condition
Z
R
n
p(x;tjx
0
)dx= 1 (8.4)
and either a deterministic initial condition
lim
t!0
p(x;tjx
0
)=d(x x
0
) (8.5)
or random initial condition
lim
t!0
p(x;t)= p
x
0
(x) (8.6)
where p
x
0
(x) is the density function of the initial condition.
196
8.1.2 Finite difference method
Let p
i; j
denote the density function at discrete locations relative to a candidate point as shown in
Figure 8.1 and given by:
p
i; j
= p(q+ iDq; ˙ q+ jD ˙ q) (8.7)
Considering the q dimension herein for simplicity, the density function at p
1;0
= p(q+Dq; ˙ q) and
p
3;0
p
0;3
p
2;0
p
0;2
p
1;0
p
0;1
p
1;0
p
0;1
p
2;0
p
0;2
p
3;0
p
0;3
p
0;0
Dq
D ˙ q
Figure 8.1. A typical finite difference stencil [113].
p
1;0
= p(qDq; ˙ q) can each be expressed as a 2nd order Taylor series
p
1;0
= p
0;0
+ p
0
0;0
Dq+
1
2
p
00
0;0
Dq
2
+ O(Dq
3
)
p
1;0
= p
0;0
p
0
0;0
Dq+
1
2
p
00
0;0
Dq
2
+ O(Dq
3
)
(8.8)
where ()
0
denotes a derivative with respect to q. The solutions for the derivative terms at the
candidate point are:
p
0
0;0
=
¶ p(q; ˙ q)
¶q
=
p
1;0
p
1;0
2Dq
+ O(Dq
3
)
p
00
0;0
=
¶
2
p(q; ˙ q)
¶q
2
=
p
1;0
2p
0;0
+ p
1;0
Dq
2
+ O(Dq
3
)
(8.9)
197
Derivatives with respect to ˙ q can be formulated in the same manner. By using higher-order Tay-
lor series expansions and more discretization points, higher-order schemes can be achieved. For
example, the 10th order difference approximations are [113]:
¶ p
¶q

2100d
q
1
600d
q
2
+ 150d
q
3
25d
q
4
+ 2d
q
5
2520Dq
; d
q
k
= p
k;0
p
k;0
¶ p
¶ ˙ q

2100d
˙ q
1
600d
˙ q
2
+ 150d
˙ q
3
25d
˙ q
4
+ 2d
˙ q
5
2520D ˙ q
; d
˙ q
k
= p
0;k
p
0;k
¶
2
p
¶ ˙ q
2

42000S
1
6000S
2
+ 1000S
3
125S
4
+ 8S
5
36883S
0
25200D ˙ q
2
; S
k
= p
0;k
+ p
0;k
(8.10)
Substituting Eq. (8.10) into the FPK equation, one equation in the nodal PDF values can be formu-
lated for each of the MN nodes, resulting in a set of MN discrete equations that can be written in
matrix form as
˙ p Kp= 0 (8.11)
where p is a column vector consisting of the MN nodal PDF values. Figure 8.2 displays a
typical finite difference discretization. The point in the ith column and jth row should be the
[(i 1) M+ j]
th
element in p. Moreover, sparse matrix techniques are applied to assemble and
q
˙ q
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
i = 1, 2, 3...,N
j = 1, 2, 3...,M
Δq
Δ ˙ q
one FD 2nd-order stencil
Figure 8.2. Finite difference discretization.
198
store the coefficient matrix K to conserve memory; it is assumed that the PDF values are zero
everywhere outside the domain. Then, Eq. (8.11) is discretized in time using the standard Crank-
Nicolson method [22]:
[I
Dt
2
K]p(t+Dt)=[I+
Dt
2
K]p(t) (8.12)
or
˜
Cp(t+Dt)=
˜
Kp(t) (8.13)
where I is the identity matrix andDt is the time step. The change of PDF values during time step
is defined as:
Dp(t)[
˜
C
1
˜
K I]p(t) (8.14)
Further, the PDF value at the center point is defined as p
c
p(t;0;0).
One might consider some simple methodology for solving the stationary problem, ˙ p= Kp= 0;
e.g., compute the null space of K, or set p(0;0)= 1 and solve the resulting equation. However,
these methods would introduce large errors because (1) approximations are used in the spatial dis-
cretization, and (2) round-off errors exist when multiplying K and p together since the K matrix
has both very small and large values, and (3) when the grid density is sufficiently fine, i.e., Dq
or D ˙ q! 0, the spatial discretization approximation in Eq. (8.10) or Eq. (8.11) will become nu-
merically unstable due to near-zero values in both numerator and denominator. In summary, the
coefficient matrix K is ill-conditioned; thus, even if Kp= 0 is solved directly, it does not mean
that p is the exact solution. An alternate approach is to use a pseudo time stepping approach [113],
starting from a distribution p
0
near stationarity provided by a coarse Monte Carlo simulation of the
system, or any guessed distribution; the system is then allowed to evolve from the initial distribu-
tion to a converged numerical solution. This approach is generalized in Figure 8.3.
199
initialize: t= 0, p(0)= p
0
solve Eq. (8.14) forDp(t)
set t= t+Dt
kDp(t)k
¥
?
< tol
p(t+Dt)=
p(t)+Dp(t)
p
c
(t)+Dp
c
(t)
no
yes
Figure 8.3. Pseudo time stepping approach for the solution of stationary FPK equation.
8.2 Case study: solutions of stationary FPK equation for a linear
oscillator
A linear oscillator has been studied via finite difference solutions [113, 114]. Since an exact so-
lution already exists, the accuracy of finite difference solutions can be evaluated so that the pa-
rameters of the method (e.g., grid density, method for solving Eq. (8.13), etc.) can be tuned in
preparation for applying the finite difference scheme to the nonlinear semiactive system.
8.2.1 Problem formulation
Consider a linear SDOF oscillator excited by additive, zero-mean Gaussian white noise, given by
the state equations:
˙ x= Ax+ B
w
w; A=
2
6
4
0 1
1 2z
3
7
5
; B
w
=
8
>
<
>
:
0
1
9
>
=
>
;
(8.15)
The explicit dependence of x and w on time is omitted for notational clarity. The state vector
x=[q ˙ q]
T
, where q is the displacement (note that for convenience, q is assumed non-dimensional)
and ˙ q is the velocity,z is the damping ratio, and w(t) is the zero-mean Gaussian white noise, with
200
intensityE[w(t)w(t+t)]= Dd(t), where d() is the Dirac delta function. Note that Eq. (8.15)
may be considered a time scaled ordinary differential equation (ODE) such that the frequency is
unity. To facilitate comparisons with studies in Wojtkiewicz et al. [113], the same parameters are
chosen herein: D= 4z = 0:8. The FPK equation associated with Eq. (8.15) is given by:
¶ p
¶t
=
¶( ˙ qp)
¶q

¶[(2z ˙ q q)p]
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
(8.16)
The stationary FPK equation (i.e.,¶ p=¶t= 0) is:
0=
¶( ˙ qp)
¶q

¶[(2z ˙ q q)p]
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
(8.17)
The exact stationary solution is given by:
p
stat
(q; ˙ q)=
z
2pD
exp

z
2D
(q
2
+ ˙ q
2
)

(8.18)
8.2.2 Parametric study for accuracy and computation time
This section investigates the accuracies and computation times of finite different solutions with
various parameters, e.g., order of finite difference scheme, time stepDt, grid spacing and different
methods for solving Eq. (8.13), as well as different discretization methods, to select the most
appropriate parameters and methods to solve the two-state linear oscillator problem. The results
will provide some indications of how to choose the corresponding parameters and discretization
and solution methods for the nonlinear semiactive system.
Different orders of finite difference schemes are utilized to compute the stationary probability
density. The maximum absolute nodal error (i.e., infinity norm) in the joint and two marginal
201
density functions between center-point normalized numerical solutions and exact solutions are
evaluated as metrics to compare the various solutions’ accuracies:
e
joint
=

p(q; ˙ q)
p
stat
(q; ˙ q)
p
stat
(0;0)

¥
e
q
=

Z

p(q; ˙ q)
p
stat
(q; ˙ q)
p
stat
(0;0)

d ˙ q

¥
e
˙ q
=

Z

p(q; ˙ q)
p
stat
(q; ˙ q)
p
stat
(0;0)

dq

¥
(8.19)
Different parameters are tuned to compare the results’ accuracies and computation time, includ-
ing grid spacingDq andD ˙ q in the spatial discretization stencil, the span of the domain, i.e.,jqj
max
andj ˙ qj
max
, time stepDt and tolerance tol to stop the pseudo time stepping method. Four methods
are explored for solving Eq. (8.13):
M1) the LU factorization of
˜
C gives
˜
C= LU; then p(t+Dt)= U
1
L
1
˜
Kp(t);
M2) direct inverse: p(t+Dt)=
˜
C
1
˜
Kp(t) using MATLAB’s backslash operator;
M3) the GMRES iteration method [89] with tolerance the same as tol; and
M4) the GMRES iteration method using incomplete LU factorization of
˜
C as a preconditioner
with tolerance the same as tol [114].
Moreover, two discretization approaches are explored [67, 108]:
Ma) Conservative discretization: the direct discretization of the right-hand side of Eq. (8.17). For
example, using a second order scheme, the discretization can be written as:

¶( ˙ qp)
¶q

¶[(2z ˙ q q)p]
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
=
˙ q
1;0
p
1;0
˙ q
1;0
p
1;0
2Dq

(2z ˙ q
0;1
q
0;1
)p
0;1
(2z ˙ q
0;1
q
0;1
)p
0;1
2D ˙ q
+
D
2
p
0;1
2p
0;0
+ p
0;1
D ˙ q
2
(8.20)
202
Mb) Non-conservative discretization: applying the chain rule — i.e., expanding out the drift terms
of the right hand side of Eq. (8.17) — and then spatially discretize the resulting equation.
For example, using a second order scheme, the discretization is given by:

¶( ˙ qp)
¶q

¶[(2z ˙ q q)p]
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
=
˙ q
¶ p
¶q
+ 2z p+(2z ˙ q+ q)
¶ p
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
=
˙ q
0;0
p
1;0
p
1;0
2Dq
+ 2z p
0;0
+(2z ˙ q
0;0
+ q
0;0
)
p
0;1
p
0;1
2D ˙ q
+
D
2
p
0;1
2p
0;0
+ p
0;1
D ˙ q
2
(8.21)
The two approaches are different. Ma (conservative discretization) conserves the displacement
and velocity “fluxes” of the original FPK equation, which locates the physical quantities more
accurately; however, formulations that are conservative purely in a mathematical sense but do not
make physical sense might produce wrong solutions [108]. Mb (non-conservative discretization)
includes fewer quantities; however, non-conservative schemes might not converge to the correct
solution if discontinuities, such as shock waves, exist in the solution [108].
All simulations are run in MATLAB on a computer with a 2.9 GHz Intel core i5 processor and
8 GB of random-access memory. Comparisons of the maximum absolute nodal error in the joint
and two marginal density functions, as well as the computation time (mainly consisting of the
time to assemble the coefficient matrix K and the pseudo time stepping) determined by MATLAB’s
tic and toc timing commands, for different orders of finite difference schemes, with M1 (LU
decomposition) and Mb (non-conservative discretization), but variousjqj
max
,j ˙ qj
max
, Dq, D ˙ q, Dt
and tol, are shown in Table 8.1. The levels of error decrease and the computation times increase,
as expected, with increasing stencil order. Notably, the tenth-order scheme gives a joint error on the
order of 10
11
. Further, expanding the domain bounds from[8;8] to[10;10] has negligible
effect on the accuracy of numerical solutions, which means the smaller domain is sufficiently
broad, and is less computationally expensive than broader domains. The levels of error decrease
but the computation times increase asDq andD ˙ q decrease. DecreasingDt from 0.02s to 0.005s has
203
Table 8.1: Comparisons of maximum absolute nodal error in the joint and two marginal density
functions between normalized numerical and exact solutions and computation time using 2nd, 4th,
6th, 8th and 10th order finite difference schemes, with M1 (LU decomposition) and Mb (non-
conservative discretization), for various domain boundary, grid density, time step and tolerance,
for the stationary FPK equation associated with a linear oscillator subjected to Gaussian white
noise.
Order jqj
max
j ˙ qj
max
Dq D ˙ q Dt tol e
joint
e
q
e
˙ q
Computation Time
[s] [s]
2 8 8 0.1 0.1 0.02 110
14
1.6110
3
6.1710
3
5.8710
3
135
4 8 8 0.1 0.1 0.02 110
14
1.1710
5
3.6410
5
3.1910
5
230
6 8 8 0.1 0.1 0.02 110
14
1.4210
7
3.9810
7
3.2310
7
295
8 8 8 0.1 0.1 0.02 110
14
2.3910
9
6.3510
9
4.8610
9
409
10 8 8 0.1 0.1 0.02 110
14
5.2910
11
1.3810
10
1.0010
10
492
2 8 8 0.2 0.2 0.02 110
14
6.5210
3
2.4810
2
2.3610
2
32
4 8 8 0.2 0.2 0.02 110
14
1.8410
4
5.7310
4
5.0210
4
52
6 8 8 0.2 0.2 0.02 110
14
8.6410
6
2.4410
5
1.9810
5
58
8 8 8 0.2 0.2 0.02 110
14
5.6110
7
1.5010
6
1.1510
6
66
10 8 8 0.2 0.2 0.02 110
14
4.6510
8
1.2110
7
8.8910
8
83
2 8 8 0.1 0.1 0.005 110
14
1.6110
3
6.1710
3
5.8710
3
597
4 8 8 0.1 0.1 0.005 110
14
1.1710
5
3.6410
5
3.1910
5
825
6 8 8 0.1 0.1 0.005 110
14
1.4210
7
3.9810
7
3.2310
7
1077
8 8 8 0.1 0.1 0.005 110
14
2.4010
9
6.3710
9
4.8710
9
1455
10 8 8 0.1 0.1 0.005 110
14
6.0010
11
1.6010
10
1.0310
10
1882
2 8 8 0.1 0.1 0.02 110
8
1.6110
3
6.1810
3
5.8710
3
78
4 8 8 0.1 0.1 0.02 110
8
1.3410
5
4.3210
5
3.3810
5
140
6 8 8 0.1 0.1 0.02 110
8
2.9110
6
7.1810
6
2.2510
6
167
8 8 8 0.1 0.1 0.02 110
8
2.8210
6
6.8110
6
1.9610
6
242
10 8 8 0.1 0.1 0.02 110
8
2.8210
6
6.8210
6
1.9710
6
336
2 10 10 0.2 0.2 0.02 110
14
6.5210
3
2.4810
2
2.3610
2
43
4 10 10 0.2 0.2 0.02 110
14
1.8410
4
5.7310
4
5.0210
4
74
6 10 10 0.2 0.2 0.02 110
14
8.6410
6
2.4410
5
1.9810
5
95
8 10 10 0.2 0.2 0.02 110
14
5.6110
7
1.5010
6
1.1510
6
115
10 10 10 0.2 0.2 0.02 110
14
4.6510
8
1.2110
7
8.8910
8
141
no effect on accuracy for lower-order schemes and very small effect for 10th order scheme, but has
an O(1=Dt) effect on computation time. Smaller tol has little effect on the less-accurate low-order
discretization schemes but a clear positive impact on the accuracy of the higher-order schemes,
with a little longer computation time. Thus, it is suggested to use the domain8 q; ˙ q 8,Dq=
D ˙ q= 0:1,Dt = 0:02 s and tol = 1 10
14
for the solution of the stationary problem considering
the trade-off of result accuracy and computation time.
Moreover, to evaluate the results of different solution methods M1 to M4, Table 8.2 provides
comparisons of maximum absolute nodal error in the joint and two marginal density functions, as
204
Table 8.2: Comparisons of maximum absolute nodal error in the joint and two marginal density
functions between normalized numerical and exact solutions and computation time using different
orders of finite difference schemes, withjqj
max
=j ˙ qj
max
= 8,Dq=D ˙ q= 0:1,Dt = 0:02 and tol=
1 10
14
, and different methods M1 to M4 to solve Eq. (8.13), for the stationary FPK equation
associated with a linear oscillator subjected to Gaussian white noise.
Method Order e
joint
e
q
e
˙ q
Computation Time
[s]
M1 (LU)
2 1.6110
3
6.1710
3
5.8710
3
135
4 1.1710
5
3.6410
5
3.1910
5
230
6 1.4210
7
3.9810
7
3.2310
7
295
8 2.3910
9
6.3510
9
4.8610
9
409
10 5.2910
11
1.3810
10
1.0010
10
492
M2 (n)
2 1.6110
3
6.1710
3
5.8710
3
923
4 1.1710
5
3.6410
5
3.1910
5
1919
6 1.4210
7
3.9810
7
3.2310
7
3739
8 2.3910
9
6.3510
9
4.8710
9
6309
10 5.4910
11
1.4110
10
1.0110
10
9128
M3 (GMRES)
2 1.6110
3
6.1710
3
5.8710
3
111
4 1.1710
5
3.6410
5
3.1910
5
129
6 1.4210
7
3.9810
7
3.2310
7
159
8 2.3910
9
6.3410
9
4.8710
9
201
10 5.1910
11
1.3510
10
1.0410
10
253
M4 (GMRES/prec.)
2 1.6110
3
6.1710
3
5.8710
3
2262
4 1.1710
5
3.6410
5
3.1910
5
3342
6 1.4210
7
3.9810
7
3.2310
7
5619
8 2.3910
9
6.3510
9
4.8710
9
10399
10 5.5010
11
1.3810
10
1.0410
10
15982
well as computation time, for different orders of finite difference schemes (with8 q; ˙ q 8,
Dq=D ˙ q= 0:1, Dt = 0:02 and tol = 1 10
14
) using M1 to M4. All discretization orders have
the same levels of error for different solution methods. However, M1 (LU) and M3 (GMRES)
have the similar levels of computation time; M2 (n) and M4 (GMRES/prec.) have higher levels
of computation time than M1 and M3. As expected, for each solution method, the computation
time increases with discretization order. It is worth noting that M3 might exhibit stagnation for the
iterative solver when the coefficient matrix is indefinite [114], resulting in inaccurate solutions and
more computational effort. Therefore, it is recommended to use M1 to solve the problem.
Finally, to compare the results with different discretization methods Ma and Mb, the maximum
absolute nodal error in the joint and two marginal density functions as well as computation time
for different orders of finite difference schemes (with8 q; ˙ q 8, Dq=D ˙ q= 0:1, Dt = 0:02,
205
Table 8.3: Maximum absolute nodal error in the joint and two marginal density functions between
normalized numerical and exact solutions and computation time using different orders of finite
difference schemes, withjqj
max
= 8,j ˙ qj
max
= 8,Dq= 0:1,D ˙ q= 0:1,Dt= 0:02 and tol= 110
14
,
with M1 (LU), and Ma (conservative) and Mb (non-conservative), for the stationary FPK equation
associated with a linear oscillator subjected to Gaussian white noise.
Method Order e
joint
e
q
e
˙ q
Computation Time
[s]
Ma (conservative)
2 8.3210
4
2.6210
3
2.1110
3
162
4 8.8710
6
2.3410
5
1.5110
5
211
6 1.2110
7
2.8010
7
1.5610
7
336
8 2.1010
9
4.4010
9
2.2710
9
478
10 4.6610
11
8.7110
11
4.6710
11
584
Mb (non-conservative)
2 1.6110
3
6.1710
3
5.8710
3
142
4 1.1710
5
3.6410
5
3.1910
5
194
6 1.4210
7
3.9810
7
3.2310
7
261
8 2.3910
9
6.3510
9
4.8610
9
443
10 5.2910
11
1.3810
10
1.0010
10
502
tol = 1 10
14
) are shown in Table 8.3 for both conservative (Ma) and non-conservative (Mb)
spatial discretization. In all, the conservative discretization Ma solutions have smaller error than
the non-conservative Mb, up to about three times smaller for the lower-order schemes’ marginal
errors, but have slightly increased computation time.
8.2.3 Summary
In conclusion, different parameters and methods have variable effects on the accuracy of results
and computation time, summarized as follows. The levels of error decrease and the computation
time increases with increasing stencil order. The smaller domain [8;8]
2
is sufficiently broad,
and is less computationally expensive than broader domains. The levels of error decrease but the
computation times increase asDq andD ˙ q decrease. DecreasingDt from 0.02 s to 0.005 s has an
O(1=Dt) effect on computation time. Smaller tol has little effect on the low-order discretization
schemes but clear positive impact on the accuracy of the higher-order schemes, with a little longer
computation time. M2 (n) and M4 (GMRES/prec.) have higher levels of computation time than
206
M1 (LU) and M3 (GMRES), but all four solution methods have the same levels of accuracy. Ma
(conservative) solutions have smaller error than Mb (non-conservative), but have slightly increased
computation time. One must consider the trade-off between accuracy and computation cost, and
select the “best” algorithm. It is suggested that, for the solution of the FPK equation associated with
this linear oscillator, one use M1 (LU) and Ma (conservative), with8 q; ˙ q 8,Dq=D ˙ q= 0:1,
Dt= 0:02 s and tol= 1 10
14
, and a higher order finite difference scheme.
8.3 Solutions of stationary FPK equation for a SDOF system
with a clipped linear control
8.3.1 FPK equation for a SDOF system with CLC
Consider the SDOF system with the same parameters used in Chapter 3. The ground-relative
displacement q(t) (assumed non-dimensional for convenience), excited by base acceleration ¨ q
g
, is
given by equation of motion:
¨ q(t)+ 2zw ˙ q(t)+w
2
q(t)= ¨ q
g
(t) u(t) (8.22)
u(t) is a mass-normalized optimal clipped controllable damping force: u= u
d
H[u
d
˙ q]; for a SDOF
system with OCLC, u
d
=k
1
qk
2
˙ q=k
LQR
1
(1a)qk
LQR
2
(1b) ˙ q. The initial conditions are
assumed quiescent, and the ground excitation ¨ q
g
is a zero mean stochastic Gaussian white noise
(GWN) process, with intensity D= 0:02; i.e., E[ ¨ q
g
(t) ¨ q
g
(t+t)]= Dd(t). The stationary FPK
equation can be represented by:
0=
¶(m
1
p)
¶q

¶(m
2
p)
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
(8.23)
where m
1
(q; ˙ q)= ˙ q, m
2
(q; ˙ q)=2zw ˙ qw
2
q u
d
H[u
d
˙ q].
207
For the uncontrolled linear oscillator studied previously with parametersz = 0:2,w = 1 rad=s,
and D= 0:8 inE[w(t)w(t+t)]= Dd(t) for the GWN excitation, the standard deviations of dis-
placement and velocity are [9]:
s
uc
q
=
s
D
4zw
3
= 1
s
uc
˙ q
=
s
D
4zw
= 1
(8.24)
The[8;8] domain — i.e., 8 standard deviation — was found to be sufficiently broad to achieve
converged results using finite difference methods. To determine the domain for the OCLC con-
trolled SDOF system studied herein, a coarse Monte Carlo simulation of the OCLC-controlled
SDOF oscillator, subjected to GWN with intensity D= 0:02, is performed and the standard devia-
tions of displacement and velocity of the resulting PDF values are computed:
s
c
q
= 0:0095
s
c
˙ q
= 0:04
(8.25)
The domain should be at least 8 standard deviation, i.e., [0:076;0:32], for the semiactive sys-
tem. The whole domain is then chosen to be0:2 q 0:2 and0:4 ˙ q 0:4 for the semiactive
system; this larger domain will more strongly support the assumption that the PDF values are zero
everywhere outside the domain, but not negatively affect the result accuracy based on the uncon-
trolled linear oscillator’s study.
8.3.2 Finite difference scheme with variable mesh density
When the CLC-controlle oscillator approaches and crosses the ˙ q= 0 boundary, the system jumps
between the dissipative and non-dissipative regions, resulting in a sharp gradient at zero velocity;
however, this jump would require special handling of the nodes at and near the ˙ q= 0 boundary.
Instead, as discussed subsequently in Section 8.3.4, this jump will be approximated with a smooth,
208
though still somewhat sharp, function; to ensure the gradients from node to node are not too large,
the grid density must be finer where this approximation changes swiftly near the ˙ q= 0 boundary.
Thus, this subsection explores a finite difference scheme with variable mesh density to achieve the
coefficients of derivatives¶ p=¶q,¶ p=¶ ˙ q and¶
2
p=¶ ˙ q
2
. For an n
th
-order finite difference scheme,
let p( ¯ x
j
) be the density function at an unsorted set of points ¯ x
j
, j = 0;1;:::;n; then the Taylor
series expansion of p( ¯ x
j
) about a candidate point ¯ x
0
= 0 is given by:
p( ¯ x
j
)
n
å
i=0
1
i!
p
(i)
0
¯ x
i
j
(8.26)
where p
(i)
0
=(¶
i
p=¶ ¯ x
i
)
¯ x=0
. Similarly, a polynomial fit problem can be rewritten as:
p( ¯ x
j
)=
n
å
i=0
a
i
¯ x
i
j
(8.27)
For these to be the same, a
i
= p
(i)
0
=i!. The polynomial fit problem is to determine the unknown
coefficients a
i
that fit the n+ 1 points

¯ x
j
; p( ¯ x
j
)

, j = 0;1;:::;n, which can be described in the
matrix form:
V
z }| {
2
6
6
6
6
6
6
6
4
1 0 0 0
1 ¯ x
1
¯ x
2
1
¯ x
n
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 ¯ x
n
¯ x
2
n
¯ x
n
n
3
7
7
7
7
7
7
7
5
a
z }| {
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
a
0
a
1
.
.
.
a
n
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
=
p
z }| {
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
p( ¯ x
0
)
p( ¯ x
1
)
.
.
.
p( ¯ x
n
)
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
(8.28)
where V is a V ondermonde matrix. The Vandermonde matrix and its inverse have known LU
decompositions [81, 120], leading to:
a= V
1
p= U
1
L
1
p (8.29)
which is computationally affordable using MATLAB for high-order finite difference schemes with
variableDq andD ˙ q. Finally, p
(1)
0
= a
1
and p
(2)
0
= 2a
2
are the derivatives used in the finite difference
method.
209
8.3.2.1 Computation of inverse Vondermonde matrix
The inverse V ondermonde matrix can be computed symbolically or numerically. A symbolic ver-
sion first creates a symbolic vector ¯ x
j
, j= 0;1;:::;n, representing the n+ 1 locations for an n
th
order finite difference stencil, then Crout’s LU factorization of the V ondermonde matrix satisfies:
V
1
= U
1
L
1
(U
1
)
i; j
=(1)
i+ j
s
ji
( ¯ x
0
;:::; ¯ x
j1
); 0 i j n
(L
1
)
i; j
=
1
Õ
i
k=0;k6= j
( ¯ x
j
¯ x
k
)
; 0 j i n
(8.30)
wheres
k
denotes the k
th
elementary symmetric function [81, 120].
However, for a higher-order spatial discretization, i.e., larger n, the symbolic method is com-
putational costly. A more computationally efficient numeric method is also explored. The numeric
approach first creates a Hankel matrix of ¯ x values, using sliding windows of ¯ x to represent the
n+ 1 locations for an n
th
order finite difference stencil; then the same equation Eq. (8.30) is used
to compute the inverse V ondermonde matrix. It is worth noting that, for the numeric method, com-
putation of the inverse of the Vandermonde matrix scales down the resulting ¯ x by 1.5 times the
standard deviation of ¯ x for better numerical conditioning.
While the numeric approach is much faster, subsequent tests found that it introduced small
but significant errors in the matrix K form Eq. (8.11) that made it even more numerically ill-
conditioned. Shifting to the slower but more accurate symbolic computation of the V ondermonde
matrix and its LU decomposition and their inverses improve the conditioning of K. Therefore, the
symbolic V ondermonde is used starting in Section 8.3.5 and through the remaining of the chapter.
8.3.3 Calculation of statistics using a polynomial quadrature
When using a high-order finite difference scheme to solve the FPK equation, a low-order quadra-
ture method (e.g., trapezoid rule) to calculate the probability mass and marginal probability densi-
ties, as well as the statistics (mean, mean square, etc.), will destroy any accuracy gained by using
210
a high-order spatial discretization. Therefore, a high-order polynomial quadrature is explored,
which can be used in conjunction with a high-order spatial discretization to conserve the accuracy
of finite difference solutions.
For an n
th
order finite difference method, an (n+ 1)
th
order polynomial is fit using n+ 2 dis-
cretization points( ¯ x
j
; p( ¯ x
j
)), j= 0;1;:::;n+ 1 around the candidate quadrature interval:
p( ¯ x)=
n+1
å
i=0
a
i
¯ x
i
(8.31)
Figure 8.4 shows a polynomial fit stencil example when n= 10, in which six points on each side
of the quadrature interval [ ¯ x
5
; ¯ x
6
] are selected as the fit points ( ¯ x
j
; p( ¯ x
j
)), j = 0;1;:::;11. To
¯ x
0
¯ x
1
¯ x
2
¯ x
3
¯ x
4
¯ x
5
¯ x
6
¯ x
7
¯ x
8
¯ x
9
¯ x
10
¯ x
11
quadrature interval d ¯ x
Figure 8.4. Polynomial fit stencil.
be consistent with the assumption made in the spatial discretization, it is, again, assumed that
the PDF values are zero everywhere outside the domain, i.e., when the fit points are outside the
domain, the corresponding PDF values are zero. After the polynomial is fit (i.e., the coefficients
a
i
, i= 0;1;:::;11 are determined), the integral of the quadrature interval is computed as
Z
¯ x
6
¯ x
5
p( ¯ x)d ¯ x (8.32)
The polynomial quadrature can be applied to calculate the probability mass and marginal PDF
values given the discrete PDF values across the whole domain, e.g., the displacement marginal
PDF value at q= q
j
, j= 1;2;:::;n
1
is given by:
p
q
(q
j
)=
n
2
1
å
i=1
Z
˙ q
i+1
˙ q
i
p
i
(q
j
; ˙ q)d ˙ q (8.33)
where n
1
and n
2
are the number of grid points, respectively, along the displacement and velocity
directions; p
i
(q
j
; ˙ q) is the fit polynomial for one interval [ ˙ q
i
; ˙ q
i+1
]. Further, since the GWN and
211
responses all are zero mean, the expected value of some function g(q; ˙ q) of q and ˙ q can be written
as:
E[g(q; ˙ q)]=
Z
q
Z
˙ q
g(q; ˙ q)p(q; ˙ q)dqd ˙ q (8.34)
Therefore, the mean and mean square values of q and ˙ q can be computed; e.g., the mean displace-
ment is:
E[q]=
n
1
1
å
j=1
Z
q
j+1
q
j
qp
q
(q)dq (8.35)
where p
q
(q) is the fit polynomial using corresponding displacement marginal PDFs.
Instead of point-wise computation of the polynomial quadrature, a more computational efficient
method is to use matrix computation. The integration of a function p( ¯ x) over points ( ¯ x
1
; p
1
),
( ¯ x
2
; p
2
),:::,( ¯ x
r
; p
r
) — where r is the number of points — can be computed as a linear combination
of the function values p
1
; :::; p
r
, and written as a row vector H
¯ x
(that depends only on polynomial
order and the ¯ x values ¯ x
1
,:::, ¯ x
r
) times a column vector of the function values p=[p
1
p
2
::: p
r
]
T
.
A row vector can be computed a priori for displacement q and velocity ˙ q grid data, respectively,
for different integration functions: H
qi
to approximate
R
()q
i
dq and H
˙ q j
to approximate
R
() ˙ q
j
d ˙ q.
Thus, if P
MN
is the matrix of PDF values across the whole domain (i.e., the MN 1 vector p
reshaped into a matrix), where M and N are the numbers of ˙ q and q grid points, respectively, then
the probability mass can be computed very fast with H
˙ q0
PH
T
q0
using simple matrix multiplication
that can be performed very fast; in a similar manner, means and mean squares can be computed
withE[q
i
˙ q
j
]= H
˙ q j
PH
T
qi
.
212
0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
Figure 8.5. Approximation function
˜
H(z), z2[0:4;0:4], for differente values.
8.3.4 Approximation of Heaviside function
The Heaviside function H[z] introduces nonlinearity and discontinuity to the FPK equation, making
the equation difficult to solve numerically using finite difference schemes. A smooth approxima-
tion function
˜
H[z] of the Heaviside function H[z] is used to approximate the nonlinearity and, thus,
simplify the problem of solving the FPK equation for the semiactive system:
˜
H[z]=
1
2
+
1
2
tanh
z
e
(8.36)
where z= u
d
˙ q and e is some positive constant (smaller e gives better approximation of the ex-
act Heaviside function). Figure 8.5 shows the approximated function
˜
H(z), z2[0:4;0:4], for
differente values.
It is worth noting that this approximation might introduce some errors to the FPK solutions:
when u
d
˙ q is negative but very close to zero, u
d
˜
H[u
d
˙ q] will be non-dissipative instead of zero. One
may consider other approximations, e.g.,
˜
H
1
[z]=(z=e)(H[z] H[ze])+ H(ze) which would
ensure dissipativity, or
˜
H
2
[z]=(z+e)(H[z+e] H[ze])=(2e)+ H(ze) which is symmetric
but not always enforcing dissipativity control force. These alternative approximations are worth
exploration for future study. The hyperbolic tangent
˜
H[z] is used herein because it is smooth and
without sharp corners.
213
Further, Section 8.3.10 will explore whether using H[u
d
˙ q] or the equivalent H[ ˙ q sgnu
d
], and
their approximation using
˜
H[], provide superior results.
8.3.5 Parametric study for convergence
To determine the most appropriate parameters and solution methods for the finite difference schemes
used to solve the FPK equation associated with an ideal OCLC of the SDOF system, a paramet-
ric study is conducted, computing solution convergence metrics for each case. The OCLC design
found in Chapter 3 — a = 0:3498 and b = 7:8759 — is used for this parametric study (subse-
quent optimization of a and b using the FPK solution will be presented in Section 8.4). Various
e values approximating the Heaviside function, along with various mesh densities in both q and ˙ q
directions for eache, are evaluated to determine how the PDF solutions converge. The PDF values
are center-point normalized for each iteration step. The mesh is always symmetric about the lines
q= 0 and ˙ q= 0. Based on the prior study of a linear system in Section 8.2, in this section, M1
(LU factorization) solution and Ma (conservative, drift terms not expanded before) discretization
are applied to solve the FPK equation. For the stationary solution in this section (8.3.5), the initial
PDF values p
0
are provided by a coarse Monte Carlo simulation of the system.
8.3.5.1 Effect of grid points on convergence
The finite difference results are very sensitive to the choice of grid points, and some choices can
result in numerical instability (solution tends to infinity or zero everywhere), whereas an overly-
coarse mesh may give erroneous results or cause instability. This instability is due to the interaction
of the discretization method with mesh density, leading to an ill-conditioned matrix K. The dis-
cretization results, as explored in Eq. (8.20) or Eq. (8.21), highly depend on the mesh density,
because the coefficients of the derivative terms at each candidate point are different due to variable
grid spacing. Therefore, the effect of grid points on the solution’s convergence is worth explo-
ration.
214
0 50 100 150
10
10
0
10
5
10
10
(a) Peak trailing change during time step
0 50 100 150
10
10
0
10
5
(b) Peak leading change during time step
0 50 100 150
10
10
0
10
5
10
10
(c) Peak difference from final PDF (at 300 s)
Figure 8.6. Peak trailing change during time stepkp(t) p(tDt)k
¥
, peak leading change during
time stepkDp(t)k
¥
and peak difference from final PDFkp(t) p(300 s)k
¥
for the first 1500 iter-
ation steps (150 s) with different grids,e = 1:25 10
3
,Dt = 0:1 s, finite difference order n= 10;
the solid black curve (147 195 mesh) has grid points displayed in Figure 8.7.
215
Figure 8.6 displays, as a function of time t, the infinity normkp(t)p(tDt)k
¥
of the trailing
difference between the PDF values at two successive iteration steps, the infinity normkDp(t)k
¥
of the leading difference between the PDF values at two successive iteration steps, and the infinity
normkp(t) p(t
f
)k
¥
between the PDF values and those at the end of the simulation for different
grid densities whene = 1:25 10
3
, finite difference order n= 10, time stepDt = 0:1 s, and final
time t
f
= 300 s; to see the details of the evolution of these values more clearly, only the first 150 s
are shown. The legend shows the mesh used for each case. The dash-dotted red (49 97 mesh)
and solid black (147 195 mesh) curves are converging to the smallest value, respectively, for
kp(t) p(tDt)k
¥
andkDp(t)k
¥
, likely because they have the densest grids in the ˙ q direction.
Remarkably, when the q axis has a coarse grid, the results still converge with sufficiently fine grid
along the ˙ q direction. The solid black, dash-dotted red and dotted black (49 195 mesh) curves
have converging differences with the final step on the order of 10
4
, as shown in Figure 8.6c.
Figure 8.7 portrays the finite difference mesh and normalized grid location corresponding to
the solid black curve in Figure 8.6, with a 147 195 mesh with variable density in bothDq andD ˙ q
whene = 1:25 10
3
. The curves in Figure 8.7b show that both grids are of variable density, but
the ˙ q-direction grid is very fine near ˙ q= 0 (i.e., grid location 0). Figure 8.8 displays the approxi-
mate Heaviside function
˜
H[u
d
˙ q] with different magnitudes of u
d
ranging from 0 to max
q
ju
d
(q;0)j
for the finite difference mesh in Figure 8.7; the grid density seems fine for u
d
= 1 m=s
2
and
e = 1:25 10
3
, but probably too coarse when u
d
is the maximum 5:1337 m=s
2
.
In summary, the mesh should be sufficiently fine in both q and ˙ q directions; specifically, the grid
spacing between two points can be constant or variable along the q direction, but should be variable
along the ˙ q direction because, as stated previously in Section 8.3.2, the low gradient area, which
is close to domain boundaries, does not require mesh as dense as that in the high gradient area ( ˙ q
close to zero). When the SDOF system hits the ˙ q= 0 boundary, it will jump between the dissipative
and non-dissipative regions; due to the approximate Heaviside function, the region when ˙ q is close
to zero (not exactly ˙ q = 0) has a sharp gradient; overly-coarse meshing in this region may be
characterized by waves being “shed” from the evolving density function. Further, the grid spacing
216
0 0.1 0.2
0
0.2
0.4
(a) Finite difference mesh
0 50 100
0
0.5
1
(b) Normalized grid location
Figure 8.7. One set of the finite difference mesh and normalized grid location when e = 1:25
10
3
, with a 147 195 mesh, and variableDq andD ˙ q; much finer grid points are defined when ˙ q
is close to zero.
should decrease smoothly from boundaries to the origin to avoid instability; the grid distribution
used in this study is a logarithmically spaced grid from the boundaries to a small value determined
bye (the number of grid points and the small value are not fixed for a specifice; one just needs to
ensure that the overall mesh is sufficiently fine, i.e., has a small peak change of PDF values during
a time step, and the area near ˙ q= 0 should have much finer mesh), and then linearly spaced grids
from the small value to zero, with linear grid spacing equal to the last (smallest) logarithmically
spacing. The MATLAB code to generate the grid data along the ˙ q direction is listed herein:
1 ep =1.25 e −3; % n e p s i l o n v a l u e
2 y =( l o g s p a c e ( log10 ( ep ) , 0 , 8 1 ) ) . ˆ . 7 ;
3 y =[ y l i n s p a c e ( − y ( 1 ) , 0 , round ( y ( 1 ) / d i f f ( y ( 1 : 2 ) ) ) +1) ] ;
4 y= u ni qu e ([ − y y ] ) ’
*
. 4 ;
8.3.5.2 Effects of discretization order n and time stepDt on convergence
This section investigates the effects of discretization order n and time step Dt on the solutions’
convergence value and rate for the semiactive system. Figure 8.9 displays the infinity norms of the
leading differencekDp(t)k
¥
between two successive iteration steps, and the infinity normkp(t)
217
0 0.2 0.4
0
0.5
1
1.5
(a) u
d
= 0 m=s
2
0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
(b) u
d
= 1 m=s
2
0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
(c) u
d
= 2 m=s
2
0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
(d) u
d
= 3 m=s
2
0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
(e) u
d
= 4 m=s
2
0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
(f) u
d
= max
q
ju
d
(q;0)j= 5:1337 m=s
2
Figure 8.8. Approximated Heaviside function
˜
H[u
d
˙ q] with different magnitudes of u
d
ranging
from 0 to max
q
ju
d
(q;0)j for the finite difference mesh in Figure 8.7, whene = 1:25 10
3
with a
147 195 mesh.
218
0 50 100 150 200 250 300
10
10
10
10
10
10
0
(a) Peak leading change during time step
0 50 100 150 200 250 300
10
10
10
0
(b) Peak difference from final PDF (at 300 s)
Figure 8.9. Peak leading change during time stepkDp(t)k
¥
and peak difference from final PDF
kp(t) p(300 s)k
¥
with different orders of finite difference method n and time step Dt using a
147 195 mesh whene = 1:25 10
3
.
219
p(t
f
)k
¥
between the PDF values and those at the end of the simulation (t
f
= 300 s) with different
orders n of finite difference discretization and two values of time step Dt, all using a 147 195
mesh when e = 1:25 10
3
. All cases seem to be converging. The dash-dotted red curve with
Dt = 0:1 s and n= 10 converges much more slowly than the other cases with smaller time step
Dt, including those with much lower order finite difference methods, even when accounting for
the expected two-orders-of-magnitude effect of the time integration method’s O(Dt
2
) accuracy
[22]. With Dt = 0:01 s, the dash-dotted blue (n= 6) and dashed magenta (n= 4) curves have a
small difference on the order of 10
9
. The best guess for a high-order finite difference scheme
underperforming a lower-order scheme is that the coefficient matrix K is very ill-conditioned;
a higher order scheme considers more points in the stencil, perhaps resulting in larger spatial
discretization errors, i.e., overfitting, resulting in less accurate results than lower-order difference
methods.
The stationary FPK equation with the exact solution p
stat
is given by:
˙ p
stat
= Kp
stat
= 0 (8.37)
Then the error of the final PDF values p(t
f
) and the exact solutions p
stat
may be represented by:
˙ p(t
f
)= Kp(t
f
) Kp
stat
= Kp(t
f
) (8.38)
Figure 8.10 shows the surface plot of the absolute errorjKp(t
f
)j as a function of q and ˙ q for
the four cases (Dt; n) in Figure 8.9; Table 8.4 lists the corresponding 1- and infinity-norms of
absolute error (kKp(t
f
)k
1
andkKp(t
f
)k
¥
) and relative error (k˙ p(t
f
) p(t
f
)k
1
andk˙ p(t
f
) p(t
f
)k
¥
,
where is a Hadamard division, i.e., element-wise division [42]) for the four cases in Figure 8.9
when e = 1:25 10
3
; note that the relative errors are dominated by the density values in the
tails where the accuracy is rather smaller than in the high probability regions. The smaller time
step Dt and spatial discretization order n result in smaller change of PDF values from step to
step; specifically, the case when n= 4 and Dt = 0:01 s has the maximum absolute error on the
220
(a)jKp(t
f
)j forDt= 0:1 s and n= 10 (b)jKp(t
f
)j forDt= 0:01 s and n= 10
(c)jKp(t
f
)j forDt= 0:01 s and n= 6 (d)jKp(t
f
)j forDt= 0:01 s and n= 4
Figure 8.10. Surface plot of absolute errorjKp(t
f
)j as a function of q and ˙ q for the four cases
(Dt; n) in Figure 8.9 when e = 1:25 10
3
and t
f
= 300 s. Note that the scales differ in the
z-direction.
order of 10
5
. However, though the errors are smaller with lower-order finite difference schemes,
decreasing the order n does not affect the accuracy of the results as much as decreasingDt. For
example, the cases when n= 4 and 6, and Dt = 0:1 s have the same level of absolute error (not
shown here) as that when n= 10 andDt= 0:1 s.
Some caveats must be noted. The examples shown herein use a relatively largee value, and the
stiffness K matrix is not very ill-conditioned. If K is more ill-conditioned, then even if Kp(t
f
) is
221
Table 8.4: Norms of absolute and relative errors between final and exact PDF values for the four
cases(Dt; n) in Figure 8.9 whene = 1:25 10
3
and t
f
= 300 s.
n Dt [s] kKp(t
f
)k
1
kKp(t
f
)k
¥
k˙ p(t
f
) p(t
f
)k
1
k˙ p(t
f
) p(t
f
)k
¥
10 0.10 591.69 3.61 3.6710
4
5.5310
3
10 0.01 0.07 4.4510
4
4.49 1.6410
1
6 0.01 0.02 1.3410
4
1.32 5.1310
2
4 0.01 0.01 3.3810
5
1.08 1.8210
1
222
small (with some norm), it does not mean p(t
f
) is accurate or close to the exact solution, and vice
versa; if K or p is perturbed by a small valueDK orDp, then K(p+Dp)Kp and(K+DK)pKp
may be much larger (with some norm) thanDp andDK when K is ill-conditioned.
8.3.6 Improved initial guess p
0
via matrix partition
To better approximate the Heaviside function and, thus, provide more accurate FPK solutions,
a smaller e value is desirable. However, when e is very small, the peak PDF time step change
Dp(t) is still relatively large after a long computation time using LU decomposition, as displayed
in Figure 8.11 when using the initial PDF values p
0
provided by a coarse Monte Carlo simulation
of the system, e = 1:25 10
8
, time step Dt = 0:1 s, spatial discretization order n= 10, and a
135425 mesh. The time step changeDp is on the order of 10
3
after 150s (1500 iteration steps);
the difference between the PDF values of each step and the final step oscillates and is on the order
of 10
3
as well. Even for lower order n= 4 and much smaller time stepDt = 0:01 s, the time step
changeDp is still on the order of 10
4
after 150 s (15,000 iteration steps).
One solution to improve convergence is to use a solver that is more accurate; this will be
explored in the next section (Section 8.3.7). The other solution is to use a different initial guess
p
0
, one that is much closer to the stationary solution than that provided by the coarse Monte Carlo
simulation. The FPK equation Eq. (8.11) can be rewritten in terms of a subdivided PDF vector
p=[p
T
1
p
c
p
T
2
]
T
, where p
c
is the element of p associated with the center point (i.e., q= ˙ q= 0),
and then a similarly partitioned coefficient matrix K:
2
6
6
6
6
4
K
11
K
1c
K
12
K
c1
k
cc
K
c2
K
21
K
2c
K
22
3
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:
p
1
p
c
p
2
9
>
>
>
>
=
>
>
>
>
;
=
8
>
>
>
>
<
>
>
>
>
:
˙ p
1
˙ p
c
˙ p
2
9
>
>
>
>
=
>
>
>
>
;
= ˙ p (8.39)
223
For the stationary solution ˙ p= 0, let p
c
= 1 — i.e., the solution will be center-point normalized —
and substitute p
c
into Eq. (8.39); then the stationary PDF can be represented by:
8
>
<
>
:
p
1
p
2
9
>
=
>
;
˙ p=0
=
2
6
4
K
11
K
12
K
21
K
22
3
7
5
1
8
>
<
>
:
K
1c
K
2c
9
>
=
>
;
or p
0
=
8
>
>
>
>
<
>
>
>
>
:
0
1
0
9
>
>
>
>
=
>
>
>
>
;

2
6
6
6
6
4
I 0
0 0
0 I
3
7
7
7
7
5
2
6
4
K
11
K
12
K
21
K
22
3
7
5
1
8
>
<
>
:
K
1c
K
2c
9
>
=
>
;
(8.40)
Notably, the resulting p
0
will likely have some negative values in the low-probability area, espe-
cially near the boundary, because the coefficient matrix K is ill-conditioned. To make the initial
guess more reasonable, the negative values are set to be zero (alternately, one could use the ab-
solute of the negative values). With this new initial guess p
0
, the time marching in Eq. (8.13)
proceeds. The resulting norm of the changeDp(t) in PDF values in each iteration step, as well as
the norm of each time step’s PDF difference from the final time PDF, are shown in Figure 8.12.
The norm values converge much faster than using the initial guess based on the coarse Monte Carlo
simulation. The changeDp(t) in PDF values in each time step is now on the order of 10
6
, though
with mild oscillations through most of the simulation time duration.
8.3.7 Methods to solve the time marching Eq. (8.13)
As noted in the linear case study (Section 8.2), four methods are explored to solve the time march-
ing Eq. (8.13). Thus far, all results shown for the clipped linear system (Sections 8.3.5 to 8.3.6) use
method M1 (LU decomposition). As shown in the linear system case study, M2 (MATLAB’s back-
slash operator) provided no accuracy benefit and was computationally more expensive than the
other methods; it is not evaluated further. Solution methods M3 and M4 use the iterative method
GMRES [89]; M3 uses GMRES directly, whereas M4 uses a preconditioner that may avoid possi-
ble stagnation of the iterative solver [114]. The convergence using M3 and M4 for the semiactive
system is explored in this section for smalle values. Herein, the M4 preconditioner is a column-
sum modified Crout version of incomplete LU factorization of
˜
C in Eq. (8.13) [88], with some level
224
of drop tolerance. The drop tolerance droptol determines the incompleteness of LU factorization
[107]. The nonzero entries of U satisfy:
jU
i; j
j droptolk˜ c
j
k
2
; (8.41)
with the exception of the diagonal entries, which are retained regardless of satisfying the criterion;
where ˜ c
j
is the j
th
column of
˜
C. For nonzero entries in L:
jL
i; j
j droptol
k˜ c
j
k
2
U
j; j
(8.42)
where()
i; j
denotes entry of a matrix in row i and column j. An appropriate level of tolerance for
a specific
˜
C will reduce the computational effort; a 0.1 drop tolerance is utilized herein [88].
Figure 8.13 plots the peak nodal change Dp(t) (Figure 8.13a) and the peak nodal difference
between the PDF at time t and at t
f
(Figure 8.13b) when e = 1:25 10
8
, time stepDt = 0:01 s,
spatial discretization order n= 4, and a 135 425 mesh, using the new p
0
computed assuming
p
c
= 1, using both M3 (GMRES) and M4 (GMRES with a preconditioner) iteration methods to
solve Eq. (8.13). M3 has a much faster initial convergence rate than M4. Comparing against the
M1 results in Figure 8.12, M4 performs comparably to M1; in contrast, M3 converges much faster
decreasing to 10
10
at the beginning, and then has very small time step changes. Further, the
difference from the final PDF reduces quickly below 10
7
and eventually 10
10
. The M3 and M4
methods also exhibit some oscillations in the solutions due to the ill-conditioned stiffness matrix
K, early and late in the simulation, though much less so than M1.
8.3.8 Exploit symmetry of p to refine the mesh without increasing computational
cost
The iterative process of solving Eq. (8.13) is computationally costly when e approximating the
Heaviside function is close to zero (i.e., closer to the Heaviside function), as much finer grid points
225
0 50 100 150
10
10
10
0
(a) Peak change during time step
0 50 100 150
10
10
10
0
(b) Peak difference from final PDF (at 148.3 s)
Figure 8.11. Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 148:3 s) with p
0
provided by Monte Carlo simulation and solved using the M1
(LU factorization) solution method, whene = 1:25 10
8
,Dt = 0:1 s, spatial discretization order
n= 10, and a 135 425 mesh.
0 2 4 6 8 10
10
10
10
(a) Peak change during time step
0 2 4 6 8 10
10
10
10
10
10
10
(b) Peak difference from final PDF (at 10 s)
Figure 8.12. Peak changekDp(t)k
¥
during time step and peak differencekp(t)p(t
f
)k
¥
from final
PDF (t
f
= 10 s) with p
0
computed assuming p
c
= 1 and solved using the M1 (LU factorization)
solution method, when e = 1:25 10
8
, Dt = 0:01 s, spatial discretization order n = 4, and a
135 425 mesh.
0 2 4 6 8 10
10
10
10
10
M3 (GMRES)
M4 (GMRES with preconditioner)
(a) Peak change during time step
0 2 4 6 8 10
10
10
10
10
10
10
M3 (GMRES)
M4 (GMRES with preconditioner)
(b) Peak difference from final PDF (at 10 s)
Figure 8.13. Peak changekDp(t)k
¥
during time step and peak differencekp(t)p(t
f
)k
¥
from final
PDF (t
f
= 10 s) with p
0
computed assuming p
c
= 1 and solved using the M3 (GMRES) and M4
(GMRES with a preconditioner) solution methods, when e = 1:25 10
8
, time stepDt = 0:01 s,
spatial discretization order n= 4, and a 135 425 mesh.
226
are needed, resulting in huge singular and ill-conditioned coefficient K and
˜
C matrices. To increase
the computational efficiency, the symmetry of p is exploited: as noted in Section 8.3.6, p can be
written as p=[p
T
1
p
c
p
T
2
]
T
, where p
c
is the PDF value at the center point (i.e., q= ˙ q= 0), p
1
and
p
2
are the PDF nodal values before and after the center point, respectively. Using a column-wise
arrangement of the nodes in Figure 8.2, as suggested in Section 8.1.2, p
1
will contain the nodal
values for q< 0 and for q= 0 and ˙ q< 0, and p
2
for q> 0 and q= 0 and ˙ q> 0. However, because
of symmetry in the FPK equation, p(q; ˙ q) p(q; ˙ q), so the nodal values are symmetric about
the origin, i.e., the PDF nodal value in i
th
column and j
th
row of Figure 8.2 is equal to the nodal
value in the [N i+ 1]
th
column and [M j+ 1]
th
row; thus, p
1
and p
2
are symmetric about p
c
,
i.e., p
2
= Jp
1
, where J is an exchange matrix defined by J
i; j
=d
i;n j+1
, whered
i; j
is the Kronecker
delta; Jp
1
has the same effect as MATLAB’sflipud(p
1
). Replacing p
2
with Jp
1
in Eq. (8.39):
2
6
4
K
11
+ K
12
J K
1c
K
c1
+ K
c2
J k
cc
3
7
5
8
>
<
>
:
p
1
p
c
9
>
=
>
;
=
8
>
<
>
:
˙ p
1
˙ p
c
9
>
=
>
;
(8.43)
where XJ has the effect of swapping the order of X’s columns (like MATLAB’s fliplr() com-
mand), or equivalently
˙
¯ p
¯
K¯ p= 0 (8.44)
or, time-discretizing using the standard Crank-Nicolson method:
˜
¯
C¯ p(t+Dt)=
˜
¯
K¯ p(t) (8.45)
where
˜
¯
C= I(Dt=2)
¯
K and
˜
¯
K= I+(Dt=2)
¯
K. Herein the dimensions of
¯
K are each just over half
those of K. It is worth noting that, when exploiting the symmetry of p, the numeric method of com-
puting the finite difference stencils, as stated in Section 8.3.2.1, will result in a coefficient matrix
K not anti-diagonally symmetric, leading to unstable solutions with higher spatial discretization
order n when solving Eq. (8.45). Therefore, though the numeric method is more computational
efficient, symbolic computation of the finite difference stencil is utilized, through the remainder of
this study.
227
The computation time is much shorter when exploiting the symmetry of p, e.g., when using p
0
computed assuming p
c
= 1 and solution method M1 (LU decomposition), with e = 1:25 10
8
,
time stepDt = 0:01 s, spatial discretization order n= 4, and a 135 425 mesh, the computation
time is decreased by about 60% compared with not using the symmetry of p.
This reduced computational cost can then enable a finer grid along q, previously near-uniform
across the domain, to ensure more accurate results. Figures 8.14 to 8.16 show the values of the peak
per-step PDF changekDp(t)k
¥
and the differencekp(t)p(t
f
)k
¥
for t
f
= 10s using three different
solutions methods: an improved solution method M1
0
(LU factorization of
˜
¯
C= DP
1
LUQ
1
using
permutation matrices P and Q and row-scaling matrix D, which leads to a sparser and more stable
factorization and requires less computational effort [107]), as well as M3 (GMRES, tolerance
10
10
) and M4 (GMRES with a preconditioner, tolerance 10
10
), respectively, to solve Eq. (8.45)
for different time step durationsDt and spatial discretization orders n, whene= 1:2510
8
, using
a 401 425 mesh, and the initial guess p
0
computed assuming p
c
= 1.
All four(Dt;n) cases exhibit oscillations using improved LU decomposition (M1
0
, Figure 8.14);
order n= 4 has a slower convergence level than orders n= 6 and 10 when the time step duration
Dt is the same; when the order is the same, decreasingDt only decreases the peak per-step PDF
change approximately one order of magnitude. Notably, whenDt = 0:01 s, n= 6 and 10 perform
comparably as they both have a fastest convergence level down to the order of 10
8
. Further, the
peak difference from final PDF at 10 skp(t) p(t
f
)k
¥
reaches the order of 10
7
for all cases.
GMRES (M3, Figure 8.15) solution method converges faster than M1
0
, i.e., the peak per-step PDF
change decreases quickly to the order of 10
10
at the beginning; the peak per-step PDF change, as
well as the peak difference from final step can reach to the order of 10
12
and 10
11
, respectively.
Interestingly, when order n= 10, decreasing time stepDt does not affect the convergence level at
all, since the two cases (n= 10,Dt = 0:1 s and 0.01 s) both have the smallest peak per-step PDF
change; whenDt is the same, n= 4 performs less well than the cases of n= 6 and 10 — actually,
the cases of n= 6 and 10 andDt = 0:01 s have a similar level of peak per-step PDF change on the
order of 10
11
, though with mild oscillations; further, the peak per-step PDF changekDp(t)k
¥
rises
228
slowly after the first 5 s when n= 6 and 10 andDt = 0:01 s. Notably, when n= 4 andDt = 0:01 s,
the peak PDF change is relatively large (but still on the order of 10
11
), but oscillates less than
the other cases. Figure 8.16 uses the M4 (GMRES with preconditioner) solution method. When
the spatial discretization order n= 10 and time stepDt = 0:1 s, the system is unstable. WhenDt
is 0.01 s, n= 4 has a much larger peak per-step PDF change than n= 6 and 10. Again, the cases
of n= 6 and 10 and Dt = 0:01 s perform best, evidenced by a much smaller peak per-step PDF
change on the order of 10
11
and the peak difference from final step at 10 s is the smallest.
A comparison of Figures 8.14 to 8.16 shows that M3 (GMRES, Figure 8.15) and M4 (GM-
RES/prec., Figure 8.16) exhibit fewer solution oscillations and smaller peak per-step PDF change.
The cases of order n= 6 and 10 and time stepDt= 0:01 s perform best for all three solution meth-
ods. It is also worth noting that M1
0
(improved LU decomposition) and M4 (GMRES/prec.) are
both about eight times faster than M3 (GMRES). For example, when n= 6 andDt = 0:01 s, M1
0
and M4 require around 300 s for the iterative process to arrive at t= 10 s, but about 2600 s for M3.
Therefore, considering the convergence level and computational effort whene is small, n= 6 and
Dt= 0:01 s are most reasonable choices, and M4 is preferable to both M1
0
and M3.
8.3.9 FPK solutions of a passively-controlled system
This section investigates the FPK solutions of a passively-controlled system with a damping level
similar to that of (i.e., similar standard deviations as) the semiactive system. An exact solution
exists for the passively-controlled system; thus, the errors between finite difference solutions and
exact solutions will be computed and the accuracy of the finite difference scheme with selected
parameters (mesh density, spatial discretization order n and time step durationDt, etc.) and solution
method will be evaluated. If the parameters and solution method do not cause instability for the
passive system, it can be hypothesized that errors for the nonlinear system are likely not caused by
those parameters and solution method.
229
0 2 4 6 8 10
10
10
10
(a) Peak change during time step
0 2 4 6 8 10
10
10
10
0
10
5
10
10
(b) Peak difference from final PDF (at 10 s)
Figure 8.14. Peak changekDp(t)k
¥
during time step and peak differencekp(t)p(t
f
)k
¥
from final
PDF (t
f
= 10 s) with Eq. (8.45) solved by M1
0
(improved LU decomposition) and p
0
computed
assuming p
c
= 1, for different time step durationDt and spatial discretization order n, when e =
1:25 10
8
using a 401 425 mesh.
0 2 4 6 8 10
10
10
10
(a) Peak change during time step
0 2 4 6 8 10
10
10
10
0
10
5
10
10
(b) Peak difference from final PDF (at 10 s)
Figure 8.15. Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) with Eq. (8.45) solved by M3 (GMRES) and p
0
computed assuming p
c
= 1,
for different time step durationDt and spatial discretization order n, whene = 1:25 10
8
using a
401 425 mesh.
0 2 4 6 8 10
10
10
10
(a) Peak change during time step
0 2 4 6 8 10
10
10
10
0
10
5
10
10
(b) Peak difference from final PDF (at 10 s)
Figure 8.16. Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10s) with Eq. (8.45) solved by M4 (GMRES with preconditioner) and p
0
computed
assuming p
c
= 1, for different time step durationDt and spatial discretization order n, when e =
1:25 10
8
using a 401 425 mesh. 230
From Eq. (8.22), the closed-loop passively-controlled equation of motion is then
¨ q(t)+(2zw+ c
d
) ˙ q(t)+w
2
q(t)= ¨ q
g
(t) (8.46)
wherew is the same as the semiactive system to conserve the SDOF system’s dynamic character-
istics; c
d
is the mass-normalized damping coefficient and u= c
d
˙ q is the mass-normalized passive
control force. The same GWN as the semiactive system is used to excite the passive system. The
stationary FPK equation of the passively-controlled system is given by:
0=
¶( ˙ qp)
¶q

¶[(2zw+ c
d
) ˙ q q]p
¶ ˙ q
+
D
2
¶
2
p
¶ ˙ q
2
(8.47)
The standard deviations of displacement and velocity for the passively-controlled system should
be similar to the expected standard derivations of the FPK solutions for the semiactive system:
s
p
q
=
s
D
4(z+
c
d
2w
)w
3
0:0095
s
p
˙ q
=
s
D
4(z+
c
d
2w
)w
0:043 s
1
(8.48)
Solving for c
d
results in mass-normalized passive damping coefficients 2:1784s
1
and 5:6217s
1
,
respectively. A passive damping coefficient between these values — c
d
= 3:5 s
1
, which results in
s
p
q
= 0:0078 ands
p
˙ q
= 0:049 s
1
— is selected for study. The exact solution of the FPK equation
associated with the passive system is:
p
stat
(q; ˙ q)=
1
2ps
p
q
s
p
˙ q
exp
"

1
2

q
2
s
p
q
2
+
˙ q
2
s
p
˙ q
2
!#
(8.49)
To solve the time marching Eq. (8.45) for this passive system, the discretization parameters used for
the semiactive system whene= 1:2510
8
will be used again: Eq. (8.45) solved by M4 (GMRES
with preconditioner), p
0
computed assuming p
c
= 1, time stepDt = 0:01 s, spatial discretization
order n= 6, and a 401 425 mesh. Figure 8.17 shows the values of the peak nodal difference
231
from step to step, the peak nodal difference between the PDF at time t and at t
f
= 10 s, and the
peak error of the finite difference solution from the exact solution. The peak nodal difference from
step to step is on the order of 10
12
. The peak nodal difference between the PDF at time t and at
t
f
is on the order of 10
11
. The error between the finite difference solution and the exact solution
increases during the first 1 s and then stays on the order of 10
6
.
In conclusion, the results for the passive system are converging to within 10
6
of the exact
solutions, which provides an indication that the ill-posedness of semiactive system is not due to
the parameters used to solve the FPK solution (mesh,Dt, n, solver, etc.); rather, the ill-posedness
may be because of the nonlinearity of the semiactive system.
8.3.10 Finite difference solutions for the semiactive system
Given the parametric studies in the previous sections, the semiactive FPK equation is now solved
withDt= 0:01s, t
f
= 10s (1,000 iteration steps), n= 6, p
0
computed assuming p
c
= 1, and method
M4 (GMRES with a preconditioner) for differente values to solve Eq. (8.45). An appropriate mesh
401 433 is selected for eache value to provide a small peak per-step PDF changekDp(t)k
¥
.
Figures 8.18a and 8.18b display the resulting PDF when ˙ q= 0 and q= 0 (probability mass
normalized) for several e values, as well as a corresponding standard multivariate Gaussian dis-
tribution with the same covariance matrix as the FPK solution with e = 10
6
. Figures 8.18c and
8.18d show the same result but with the axis limits focused to show the peaks around q= 0 and
˙ q= 0. Figures 8.18e and 8.18f show the same results as Figures 8.18a and 8.18b, but on a logarith-
mic scale to show the behavior in the low-probability tails (note that the finite difference solution
has negative PDF values, which cannot be plotted logarithmically). It is worth noting that smaller
e values provide closer approximations to the exact Heaviside function. The solutions are, as ex-
pected, symmetric about the origin. When velocity ˙ q= 0, the PDF at peak is wider and lower than
Gaussian, and noisy in the tail due to numerical simulation errors; notably, whene 10
6
, the tail
of p(q;0) is much less noisy than the other cases whene < 10
6
. When displacement q= 0, the
PDF at peak is narrower and lower than Gaussian, but wider in the low probability “tail” region,
232
0 2 4 6 8 10
10
10
10
10
(a) Peak change during time step
0 2 4 6 8 10
10
10
10
(b) Peak difference from final PDF (at 10 s)
0 2 4 6 8 10
4.335
4.34
4.345
4.35
4.355
4.36
10
-6
(c) Peak error from exact solution
Figure 8.17. Peak changekDp(t)k
¥
during time step and peak differencekp(t) p(t
f
)k
¥
from
final PDF (t
f
= 10 s) and error from the exact solutionkp(t) p
stat
k
¥
for the passively-controlled
system with similar level of damping as semiactive system, with Eq. (8.45) solved by M4 (GM-
RES with preconditioner) and p
0
computed assuming p
c
= 1, when time stepDt = 0:01 s, spatial
discretization order n= 6, and a 401 425 mesh.
233
0 0.1 0.2
0
50
100
150
200
250
300
350
400
(a) PDF when velocity ˙ q= 0
0 0.2 0.4
0
50
100
150
200
250
300
350
400
(b) PDF when displacement q= 0
0 0.005 0.01
320
330
340
350
360
370
380
390
400
(c) Scaled peak PDF when ˙ q= 0
0 0.01 0.02
320
330
340
350
360
370
380
390
400
(d) Scaled peak PDF when q= 0
0 0.1 0.2
10
10
10
10
10
10
0
10
5
(e) Log-scale PDF when ˙ q= 0
0 0.2 0.4
10
10
10
10
10
10
0
10
5
(f) Log-scale PDF when q= 0
Figure 8.18. PDF when velocity ˙ q= 0 and displacement q= 0 (probability mass normalized) for
various e values and a corresponding standard multivariate Gaussian distribution with the same
covariance matrix as the FPK solution withe = 10
6
.
234
Table 8.5: Comparison of descriptive statistics for the OCLC PDF p(q;0) (probability mass nor-
malized) when ˙ q= 0 withe = 10
6
along with those of a standard multivariate Gaussian distribu-
tion with the same covariance matrix.
data mean median mode max std. dev. skewness kurtosis
FPK sol. 2.0510
19
0 –0.003 359.77 0.0100 6.5710
17
2.6691
Std. Gaussian 0 0 0 390.37 0.0095 0 3.0000
Table 8.6: Comparison of descriptive statistics for the OCLC PDF p(0; ˙ q) (probability mass nor-
malized) when q= 0 withe = 10
6
along with those of a standard multivariate Gaussian distribu-
tion with the same covariance matrix.
data mean median mode max std. dev. skewness kurtosis
FPK sol. –7.9210
19
0 0 356.42 0.0437 2.8110
17
2.9426
Std. Gaussian 0 0 0 390.37 0.0430 0 3.0000
where finite difference PDF decays more slowly than Gaussian for both p(q;0) and p(0; ˙ q), though
it is not clear if this is a real feature of the solution or discretization effects. Considering result
accuracy, it is determined to use e = 10
6
in the approximate Heaviside function for solving the
FPK equation. Tables 8.5 and 8.6 compare descriptive statistics for the OCLC PDF when ˙ q= 0
and q= 0, respectively, withe = 10
6
and those of a standard multivariate Gaussian distribution
with the same covariance matrix. The mean values are, as expected, very close to zero for both
˙ q= 0 and q= 0; the maximum PDF value for the FPK solution is smaller than that of Gaussian;
the standard deviations of the FPK solution and the Gaussian are almost identical; the skewness for
both the FPK solution and the Gaussian are essentially zero, meaning both PDF values are sym-
metric; the kurtosis for the FPK solution is smaller than, but very close to, that for the Gaussian
(equal to 3), indicating the FPK solution is less heavy-tailed than the Gaussian.
Figure 8.19 show the marginal PDFs of q and ˙ q (probability mass normalized), and a corre-
sponding standard multivariate Gaussian distribution with the same covariance matrix as the FPK
solution withe= 10
6
. The q marginal PDF at peak is lower than Gaussian, but a bit broader, con-
sistent with a larger standard deviation, and noisy in the tail due to numerical simulation errors; the
˙ q marginal PDF at peak is higher and narrower than Gaussian, interestingly and unexpectedly with
235
Table 8.7: Comparison of descriptive statistics for the OCLC marginal PDF of displacement q
(probability mass normalized) withe = 10
6
along with those of a standard multivariate Gaussian
distribution with the same covariance matrix.
data mean median mode max std. dev. skewness kurtosis
FPK sol. 2.0310
20
0 0 39.44 0.0095 2.8110
17
2.8457
Std. Gaussian 0 0 0 42.05 0.0095 0 3.0000
Table 8.8: Comparison of descriptive statistics for the OCLC marginal PDF of velocity ˙ q (proba-
bility mass normalized) withe = 10
6
along with those of a standard multivariate Gaussian distri-
bution with the same covariance matrix.
data mean median mode max std. dev. skewness kurtosis
FPK sol. –2.3610
18
0 0 10.33 0.0430 –4.0010
18
3.0757
Std. Gaussian 0 0 0 9.28 0.0430 0 3.0000
linear decrease from the peak outward, and wider in the tail; again, the PDF values decay more
slowly than Gaussian. Tables 8.7 and 8.8 compare descriptive statistics for the OCLC marginal
PDFs of displacement q and velocity ˙ q, respectively, withe = 10
6
along with those of a standard
multivariate Gaussian distribution with the same covariance matrix. The mean, median and mode
values are, as expected, essentially zero for both marginal displacement and velocity. The maxi-
mum PDF value for the FPK solution is smaller than that of Gaussian for marginal displacement,
but larger for marginal velocity. The standard deviations of the FPK solution and the Gaussian are
identical. The skewness for both the FPK solution and the Gaussian are essentially zero, meaning
both PDF values are symmetric. The kurtosis for the FPK marginal displacement is smaller than,
but close to, that for the Gaussian, indicating the FPK solution is slightly less heavy-tailed than the
Gaussian; the kurtosis for the FPK marginal velocity is slightly larger than that for the Gaussian,
meaning the FPK solution is slightly more heavy-tailed.
Further, according to Tables 8.7 and 8.8, the standard deviations of displacement q and velocity ˙ q
of the FPK solutions whene = 10
6
are:
s
OCLC
q
= 0:0095
s
OCLC
˙ q
= 0:043 s
1
(8.50)
236
(This is consistent with the 0.0095m value reported in Table 3.1 from the Monte Carlo simulation;
recall that Chapter 3 assumed q is in units of length, but this chapter assumes q is non-dimensional).
The equivalent natural frequencyw

and damping ratioz

of a linear system that, in a least squares
sense, has standard deviations close to those of FPK solutions are computed using Eq. (8.51)
s
equiv
q
=
s
D
4z

w
3
s
OCLC
q
= 0:0095
s
equiv
˙ q
=
s
D
4z

w

s
OCLC
˙ q
= 0:043 s
1
(8.51)
resulting in w

= 4:53 rad=s and z

= 60%. In addition, the correlation of displacement q and
velocity ˙ q for the FPK solution is also computed: corr(q; ˙ q)=E[q ˙ q]=(s
OCLC
q
s
OCLC
˙ q
)=1:21
10
7
, which corresponds to a correlation coefficient of about0:0296%; note that for a linear
system with Gaussian distribution, the correlation of q and ˙ q should be exactly zero.
Figure 8.20 shows surface and contour graphs of the OCLC PDF (probability mass normalized)
whene = 10
6
. The contour graph depicts 20 contour lines logarithmically spaced from the curve
p(q; ˙ q)= 0:3598 to the maximum value p
max
= 359:78. Again, the results are non-Gaussian. The
contour is symmetric about origin, but is elliptical and somewhat skewed near ˙ q.
8.3.10.1 Finite difference solution with Heaviside function H[ ˙ q sgnu
d
]
In the previous sections, the Heaviside function H[u
d
˙ q] is considered and its approximation
˜
H[u
d
˙ q]
is used in the finite difference solution. The Heaviside function H[u
d
˙ q]= 1 when u
d
and ˙ q are the
same sign, and 0 otherwise. An equivalent Heaviside function is H[ ˙ q sgnu
d
], which has a different
transition from 0 to 1 that does not depend on the magnitude of u
d
. This section investigates
whether H[ ˙ q sgnu
d
] provides solutions of similar accuracy to H[u
d
˙ q]. The resulting FPK equation
with the approximation
˜
H[ ˙ q sgnu
d
] is solved withe = 10
6
,Dt = 0:01 s, t
f
= 10 s (1,000 iteration
steps), n= 6, and p
0
computed assuming p
c
= 1 and the M4 (GMRES with a preconditioner)
solution method to solve Eq. (8.45).
237
Table 8.9: Comparison of descriptive statistics for the OCLC PDF p(q;0) (probability mass nor-
malized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] withe= 10
6
and those of a standard multivariate Gaussian
distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution.
data mean median mode max std. dev. skewness kurtosis
˜
H[u
d
˙ q] 2.0510
19
0 –0.003 359.77 0.0100 6.5710
17
2.6691
˜
H[ ˙ q sgnu
d
] 1.8410
19
0 –0.003 359.78 0.0100 6.0710
17
2.6691
Std. Gaussian 0 0 0 390.37 0.0095 0 3.0000
Table 8.10: Comparison of descriptive statistics for the OCLC PDF p(0; ˙ q) (probability mass nor-
malized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] withe= 10
6
and those of a standard multivariate Gaussian
distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution.
data mean median mode max std. dev. skewness kurtosis
˜
H[u
d
˙ q] –7.9210
19
0 0 356.42 0.0437 2.8110
17
2.9426
˜
H[ ˙ q sgnu
d
] –7.4810
19
0 0 356.43 0.0437 –1.6210
17
2.9426
Std. Gaussian 0 0 0 390.37 0.0430 0 3.0000
Figure 8.21 depicts the PDF when ˙ q= 0 and q= 0 (probability mass normalized) using
˜
H[u
d
˙ q]
and
˜
H[ ˙ q sgnu
d
], and a corresponding standard multivariate Gaussian distribution with the same
covariance matrix as the FPK solution using
˜
H[ ˙ q sgnu
d
]. The solutions using
˜
H[ ˙ q sgnu
d
] are in
good agreement with those using
˜
H[u
d
˙ q], with the maximum absolute nodal differencekp
u
d
˙ q

p
˙ q sgnu
dk
¥
0:01 for PDFs that have a peak of about 360. Tables 8.9 and 8.10 compare descriptive
statistics for the OCLC PDF when ˙ q= 0 and q= 0, respectively, using H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
]
withe= 10
6
and those of a standard multivariate Gaussian distribution with the same covariance
matrix as the
˜
H[ ˙ q sgnu
d
] solution. The statistics for
˜
H[ ˙ q sgnu
d
] are very close to those for
˜
H[u
d
˙ q].
The resulting standard deviations of displacement q and velocity ˙ q of the FPK solutions when using
˜
H[ ˙ q sgnu
d
], and the natural frequencyw

and damping ratioz

of a linear system with equivalent
mean square responses, are essentially identical to those when using
˜
H[u
d
˙ q].
Figure 8.22 illustrates the marginal PDF of displacement q and velocity ˙ q (probability mass
normalized) for
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] and a corresponding standard multivariate Gaussian dis-
tribution with the same covariance matrix as the FPK solution using
˜
H[ ˙ q sgnu
d
]. The marginal
PDF values of q and ˙ q for
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] are, again, nearly identical. Tables 8.11 and
238
Table 8.11: Comparison of descriptive statistics for the marginal PDF of displacement q (prob-
ability mass normalized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] with e = 10
6
and those of a standard
multivariate Gaussian distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution.
data mean median mode max std. dev. skewness kurtosis
˜
H[u
d
˙ q] 2.0310
20
0 0 39.44 0.0095 2.8110
17
2.8457
˜
H[ ˙ q sgnu
d
] –7.4510
20
0 0 39.44 0.0095 3.3910
17
2.8444
Std. Gaussian 0 0 0 42.05 0.0095 0 3.0000
Table 8.12: Comparison of descriptive statistics for the marginal PDF of velocity ˙ q (probability
mass normalized) using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] withe= 10
6
and those of a standard multivariate
Gaussian distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution.
data mean median mode max std. dev. skewness kurtosis
˜
H[u
d
˙ q] –2.3610
18
0 0 10.33 0.0430 –4.0010
18
3.0757
˜
H[ ˙ q sgnu
d
] 1.2710
18
0 0 10.33 0.0430 1.1610
16
3.0756
Std. Gaussian 0 0 0 9.28 0.0430 0 3.0000
8.12 compare descriptive statistics for the marginal PDFs of displacement q and velocity ˙ q, respec-
tively, using
˜
H[u
d
˙ q] and
˜
H[ ˙ q sgnu
d
] withe = 10
6
and those of a standard multivariate Gaussian
distribution with the same covariance matrix as the
˜
H[ ˙ q sgnu
d
] solution. The statistics are, again,
very close to those for
˜
H[u
d
˙ q].
Further, Figure 8.23 shows the corresponding surface and contour graphs of the OCLC PDF
(probability mass normalized) using
˜
H[ ˙ q sgnu
d
] when e = 10
6
. The contour graph depicts 20
contour lines logarithmically spaced from the curve p(q; ˙ q)= 0:3598 to the maximum value p
max
=
359:78. The graphs are nearly identical to those when using
˜
H[u
d
˙ q].
8.3.10.2 Discussion
The FPK solutions for this nonlinear semiactive system are, as expected, not an exact Gaussian
distribution. Specifically, the OCLC solution has smaller PDF values at the peak and decays much
more slowly in the tails compared to a Gaussian distribution (though that may be an artifact of
the numerical solution). The FPK solution’s non-Gaussian characteristics are important when ac-
counting for structural safety, i.e., likelihood of failure. When a nonlinear semiactive system is
239
0 0.1 0.2
0
10
20
30
40
50
(a) Marginal PDF of displacement q
0 0.2 0.4
0
2
4
6
8
10
12
(b) Marginal PDF of velocity ˙ q
0 0.005 0.01
30
35
40
45
(c) Scaled peak q marginal PDF
0 0.01 0.02
8
8.5
9
9.5
10
10.5
11
(d) Scaled peak ˙ q marginal PDF
0 0.1 0.2
10
10
10
0
(e) Log-scale q marginal PDF
0 0.2 0.4
10
10
10
10
0
10
5
(f) Log-scale ˙ q marginal PDF
Figure 8.19. Marginal PDFs of displacement q and velocity ˙ q (probability mass normalized) for
various e values and a corresponding standard multivariate Gaussian distribution with the same
covariance matrix as the FPK solution withe = 10
6
.
240
(a) Surface of OCLC PDF (b) Contour of OCLC PDF
Figure 8.20. Surface and contour graphs of the OCLC PDF (probability mass normalized) when
e = 10
6
.
241
0 0.1 0.2
0
50
100
150
200
250
300
350
400
(a) PDF when velocity ˙ q= 0
0 0.2 0.4
0
50
100
150
200
250
300
350
400
(b) PDF when displacement q= 0
0 0.005 0.01
300
320
340
360
380
400
420
(c) Scaled peak PDF when ˙ q= 0
0 0.01 0.02 0.03
300
320
340
360
380
400
420
(d) Scaled peak PDF when q= 0
0 0.1 0.2
10
10
10
10
10
10
0
10
5
(e) Log-scale PDF when ˙ q= 0
0 0.2 0.4
10
10
10
10
10
10
0
10
5
(f) Log-scale PDF when q= 0
Figure 8.21. PDF when velocity ˙ q= 0 and displacement q= 0 (probability mass normalized) when
approximating H[u
d
˙ q] and H[ ˙ q sgnu
d
] (solved with e = 10
6
, Dt = 0:01 s, t
f
= 10 s, n= 6, and
p
0
computed assuming p
c
= 1 and the M4 [GMRES with a preconditioner] solution method to
solve Eq. (8.45)), and a corresponding standard multivariate Gaussian distribution with the same
covariance matrix as the FPK solution using the
˜
H[ ˙ q sgnu
d
].
242
0 0.1 0.2
0
10
20
30
40
50
(a) Marginal PDF of displacement q
0 0.2 0.4
0
2
4
6
8
10
12
(b) Marginal PDF of velocity ˙ q
0 0.005 0.01
30
35
40
45
(c) Scaled peak q marginal PDF
0 0.01 0.02
8
8.5
9
9.5
10
10.5
11
(d) Scaled peak ˙ q marginal PDF
0 0.1 0.2
10
10
10
10
0
10
5
(e) Log-scale q marginal PDF
0 0.2 0.4
10
10
10
10
0
10
5
(f) Log-scale ˙ q marginal PDF
Figure 8.22. Marginal PDFs of displacement q and velocity ˙ q (probability mass normalized) when
approximating H[u
d
˙ q] and H[ ˙ q sgnu
d
] (solved with e = 10
6
, Dt = 0:01 s, t
f
= 10 s, n= 6, and
p
0
computed assuming p
c
= 1 and the M4 [GMRES with a preconditioner] solution method to
solve Eq. (8.45)), and a corresponding standard multivariate Gaussian distribution with the same
covariance matrix as the FPK solution using the
˜
H[ ˙ q sgnu
d
].
243
(a) Surface of OCLC PDF (b) Contour of OCLC PDF
Figure 8.23. Surface and contour graphs of the OCLC PDF (probability mass normalized) using
˜
H[ ˙ q sgnu
d
] whene = 10
6
.
244
subjected to GWN, if the displacement/velocity distribution decays slower than Gaussian, a Gaus-
sian assumption will underestimate the likelihood of failure. Therefore, one should be careful when
evaluating structural safety for a GWN-driven nonlinear system that has non-Gaussian probability
density function of the response.
8.4 Implication of OCLC
For the SDOF system with OCLC, the cost metric J
a
minimized in Chapter 3 was the mean square
absolute acceleration (which may be viewed as a serviceability cost metric [26]):
J
a
=E[ ¨ q
a2
]
=E[(2zw ˙ qw
2
q u
d
H[u
d
˙ q])
2
]
=w
4
E[q
2
]+ 4z
2
w
2
E[ ˙ q
2
]+E[(u
d
H)
2
]+ 4zw
3
E[q ˙ q]+ 4zwE[ ˙ qu
d
H]+ 2w
2
E[qu
d
H]
(8.52)
where H here is short for H[u
d
˙ q]. With the solutions p(q; ˙ q) of the FPK equation, the statistics in
Eq. (8.52) can be computed using a quadrature method, as stated in Section 8.3.3. Then, J
a
can be
calculated using the solutions of the stationary FPK equation for a set of CLC parameters(a;b);
the optimal(a

;b

) corresponds to the minimum of J
a
(a;b) and provides the parameters for the
OCLC.
8.4.1 Optimization results
The numerical FPK solution and the resulting cost metric J
a
are computed over a grid of clipped
linear strategies with (a;b)2[0;10]
2
(the range is chosen based on the previous Monte Carlo
simulation results in Chapter 3, which indicated that the optimal a and b for the SDOF system
are both positive for GWN excitation), using e = 1:25 10
8
, n= 6,Dt = 0:001 s, t
f
= 10 s and
a 135 425 mesh, with Eq. (8.45) solved by method M4 (GMRES with preconditioner) and p
0
computed assuming p
c
= 1. The final time t
f
= 10 s is chosen because this shorter duration with a
smallerDt is sufficient for a fairly close evaluation of the cost metric contour based on the previous
245
study. The 135 425 mesh is used because it has been well studied for the OCLC semiactive
system and is less computational costly than the denser mesh in the previous section. The contours
of cost metric J
a
as a function of parametersa andb are shown in Figure 8.24.
minimum of surface
0 2 4 6 8 10
stiffness factor
0
2
4
6
8
10
damping factor
-2
-1.5
-1
-0.5
Figure 8.24. Contours of cost metric J
a
as a function of parametersa andb using FPK solutions
for the SDOF system subjected to GWN.
To find the global optimal (a

, b

) that minimizes J
a
(a;b), MATLAB’s optimization toolbox
function fminsearch [107] is used to optimize the system using as a starting guess the minimum
point found via the contour graph. Table 8.13 summarizes the comparison of optimization results
found via Monte Carlo simulation excited by GWN with t
f
= 10000 s (Section 3.1.5) and those
found via FPK solution usinge= 1:2510
8
, spatial discretization order n= 6, time step duration
Dt = 0:01 s, a 401 425 mesh and t
f
= 20 s (chosen because the optimization with fminsearch
does not require as much computational effort as the contour plot with the same time duration),
and Eq. (8.45) solved by method M4 (GMRES with preconditioner) and p
0
computed assuming
p
c
= 1. The optimization results of the two methods are very close, though the minimum cost
achieved from the FPK equation is very slightly larger than that of the Monte Carlo simulation
(however, this difference does not mean that FPK results are inaccurate: Monte Carlo simulation
uses a sampled discrete-time band-limited Gaussian white noise with sampling timeDt = 0:02 s;
the mean square responses are approximated by averaging over the duration of the simulation;
therefore, the Monte Carlo simulation results are not exact solutions either).
246
Table 8.13: SDOF OCLC optimization results found via Monte Carlo simulation excited by GWN
with t
f
= 10000 s (Section 3.1.5) and those found via FPK solution usinge = 1:25 10
8
, spatial
discretization order n= 6, time step duration Dt = 0:01 s, a 401 425 mesh and t
f
= 20 s, and
Eq. (8.45) solved by M4 (GMRES with preconditioner) and p
0
computed assuming p
c
= 1. Note
that theD column denotes percent change relative to Monte Carlo Simulation result.
Method (a

;b

) J
a
(D)
[m
2
/s
4
] [%]
Monte Carlo Sim. (0.3498,7.8759) 0.09799
FPK solution (0.3345,7.8380) 0.09948 (1)
8.5 Summary
This chapter studied the solution of the FPK equation associated with an ideal OCLC of a SDOF
system excited by GWN. A case study of an uncontrolled linear oscillator system indicated that
LU factorization and conservative discretization of the FPK equation can maintain the accuracy of
solutions and are computational efficient for the uncontrolled system. Next, the finite difference
method was applied to solve the FPK equation of a SDOF system with OCLC excited by GWN.
Some special considerations were presented: the Heaviside function was approximated by a hyper-
bolic tangent function; variable grid spacing was utilized for the finite difference discretization; a
high-order polynomial quadrature method was proposed to calculate response statistics. Further, it
was determined that the finite difference mesh should be denser when the velocity is close to zero
because this region has a sharp gradient and overly-coarse meshing may be characterized by waves
being “shed” from the evolving density function. A smaller time step durationDt and a medium-
order finite difference scheme was found to improve the convergence rate. Different solvers for the
resulting system linear in PDF nodal values were tested to select the most reasonable one: solver
M4 (GMRES with a preconditioner) demonstrated fast convergence but reasonable computation
time. Moreover, an improved initial guess for the iterative process was computed assuming a unit
solution at the origin, and the symmetry of solutions was exploited to further reduce the compu-
tational effort and increase the convergence rate. Differente values in the approximate Heaviside
247
function were tested. The finite difference solutions are non-Gaussian. Finally, the OCLC opti-
mal factors (a

;b

) could be achieved using an optimization of solutions of the stationary FPK
equation.
248
Chapter 9
Summary, Conclusion and Future Directions
A novel optimal clipped linear control (OCLC) approach has been proposed for determining op-
timal controllable damping strategies and has been evaluated through numerical simulation and
physical experiments in this Dissertation. The OCLC is optimized for one realization of the exci-
tation using a quadratic performance metric similar to LQR control but does so accounting for the
passivity constraint of a controllable damper — in contrast with the CLQR approach that designs
the control assuming a hypothetical linear actuator and may result in much poorer performance
than expected for some objectives.
The optimality of OCLC over CLQR and an optimal passive linear viscous damper was first
demonstrated using a SDOF structure model. The results showed that OCLC always provides
performance superior to both CLQR and a passive linear viscous damper for a stochastic Gaussian
white noise excitation; OCLC also outperforms CLQR in a set of historical earthquakes and a suite
of Kanai-Tajimi filtered excitations:
• CLQR and optimal passive linear damping have cost metric values that are 50% and 30%
higher, respectively, than that with the OCLC for GWN;
• CLQR’s cost metric value is 34% and 81% larger, respectively, than that with the OCLC for
1940 El Centro and 1995 Kobe;
• OCLC always provides a consistent performance improvement over CLQR in reducing the
cost metric when the frequency content of the excitation changes.
249
Further, the performance of OCLC designed for one excitation was cross validated when subjected
to a different excitation (GWN, 1940 El Centro or 1995 Kobe) for this SDOF system, indicating
that OCLC designed for one specific excitation always outperforms CLQR regardless of the eval-
uation excitation. OCLC was also evaluated for 20 ground motions chosen from large-magnitude
events in the PEER NGA database, further validating the superiority of OCLC. An optimal control
gain was selected from among 25 control gain candidates through a comprehensive analysis. Fi-
nally, the relationship between OCLC control gain and structural characteristics of SDOF systems
was explored, showing that the stiffness factora remains unchanged with respect to structural nat-
ural frequency and damping ratio, whereas the damping factor b is inversely proportional to the
uncontrolled structure’s damping ratio.
Then, the OCLC of a 2DOF model, excited by a Gaussian white noise excitation, was deter-
mined by optimization over a four-dimensional parameter space. The OCLC performance was
compared to that of CLQR as well as of an optimal passive viscous damping strategy; OCLC again
demonstrated its optimality for a 2DOF structure model:
• When the controllable damper is in the first story, CLQR’s cost metric, as well as mean
square accelerations and drifts, are two to two-and-a-half times larger than OCLC for mini-
mizing absolute accelerations;
• When the controllable damper is in the second story, CLQR has a cost value about four times
that of OCLC, and the CLQR drifts, velocities and accelerations, are two to three-and-a-half
times larger than OCLC when minimizing absolute accelerations.
Moreover, the evaluation of an optimal active control for the 2DOF system suggested that LQR
is not the “best” active control strategy if the excitation is anything other than ideal GWN; thus,
using a design excitation to determine an OCLC is similar to what would be done to truly evaluate
and design the optimal active system. Next, a robustness study for both SDOF and 2DOF models
indicated that the improvements provided by the OCLC, relative to CLQR, do not degrade much
at all when the structure parameters are perturbed from those used to design the OCLC strategy.
250
The effectiveness of OCLC was then evaluated through real-time hybrid simulation (RTHS)
tests for a suite of structure models, including a series of SDOF and 2DOF structure models with
different structural properties, a 2DOF model of elevated highway bridge deck, and a 2DOF model
of base-isolated shear building structure. OCLC provides variable levels of performance improve-
ments over CLQR for different structures for numerical simulation and RTHS tests:
• SDOF systems: CLQR has mean square absolute acceleration and drift metrics up to 37%
and 117% larger, respectively, than those of OCLC and 59% and 57% for peak absolute
acceleration and drift metrics, though the drift is not included in the cost metric;
• 2DOF systems: CLQR’s cost metric is up to 58% larger than that of OCLC for 2DOF build-
ing models; CLQR has 32%–78% larger cost metric, mean square absolute accelerations and
drifts for the bridge deck model; CLQR has a 54% larger cost metric for the base-isolated
structure.
The RTHS test results match well with the numerical simulation results in most cases. Further-
more, the OCLC control designed for one specific excitation was evaluated for different external
excitations through RTHS and numerical simulation, further validating the superiority of OCLC.
Next, shake table experiments were conducted at Japan’s NIED “E-Defense” laboratory using
OCLC and CLQR control strategies subjected to three scaled earthquake excitations. The results
demonstrated that OCLC is superior to CLQR for certain metrics:
• OCLC has mean square absolute accelerations 33% to 44% smaller than CLQR, with a mean
square control force that is roughly half that of CLQR;
• OCLC can reduce the mean square drift and velocity by up to 32% and 26%, respectively,
without significant increase in mean square absolute acceleration.
The E-Defense experimental results also match RTHS and pure simulation results for drift, velocity
and absolute acceleration responses for USC1–4, though the E-Defense data was found to contain
high-frequency components; for USC5, E-Defense results are far different than RTHS and pure
simulation results, because E-Defense used a different controller other than USC5.
251
Finally, the solutions of the FPK equation associated with an ideal OCLC of a SDOF system ex-
cited by GWN were investigated via high order finite difference methods. The nonlinear Heaviside
function was approximated by a smooth hyperbolic tangent function. Variable grid spacing was
explored using a mesh that is much finer when the velocity is close to zero. A high order quadra-
ture method was proposed to calculate the marginal PDF and response statistics. A parametric
study was conducted to select the most reasonable parameters (e.g., time step, finite difference
order, solver, etc.) for the solutions. The symmetry of solutions was utilized to further reduce the
computational effort and increase the convergence rate. The final results were also compared with
a standard Gaussian distribution. Further, the cost function of the SDOF system, i.e., mean square
absolute accelerations, can be calculated using the solutions of the FPK equation; the optimal
factors of OCLC can be determined by minimizing the resulting cost function values.
9.1 Future directions
Studies in this Dissertation suggest that the OCLC strategy has significant promise for use in
semiactive control applications, particularly those with a need to improve a serviceability cost
metric. Further study of OCLC is still needed.
• For the solutions of the FPK equation study, research efforts can be directed towards OCLC
design for higher order structure models using numerical solutions of the FPK equation;
more computational efficient methods are worth exploration; a combination of high-order
and low-order finite difference methods is also worth exploration.
• OCLC is designed with a prior knowledge of external excitations, which may be unknown
in practical engineering projects. This limitation has been treated appropriately in the Dis-
sertation. Chapter 3 and Chapter 6 cross validated the superiority of OCLC designed for
one specific excitation when subjected to a different excitation; Chapter 3 also proposed one
solution that is to design the OCLC controller to a suite of expected excitations and select
252
the final optimal control gain through a comprehensive study. However, the relationship be-
tween excitation characteristics and the optimal control gain is still worth exploring. Another
research direction is to design OCLC adaptively based upon the online response contents.
• OCLC requires greater computational effort when the system order is higher due to the
optimization procedure; though high-order complex systems may be simplified to low-order
ones and OCLC can be applied to the reduced-order systems, future research may focus
on efficient computation methods for OCLC, e.g., utilizing the nonlinear V olterra integral
equations. For more complex models with a much higher-dimensional parameter space,
computational efficiency may be further enhanced by exploiting the localized nature of the
dynamical characteristics of systems.
Nevertheless, this Dissertation develops a novel OCLC strategy which is easy to implement and
shows significant performance improvements compared with CLQR and an optimal passive linear
damping for simple structures.
253
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Abstract (if available)

Abstract In the past few decades, structural control research has focused mostly on passive and semiactive vibration mitigation strategies. Among the latter, the clipped-optimal control paradigm is one of the most commonly used. With this control strategy, a primary controller is designed using an optimal linear feedback assuming the control device is a linear actuator that can exert any force. Then, a secondary controller is commanded to exert the force generated by the primary controller if it is dissipative and minimal force if it is non-dissipative. However, if the optimization to design the optimal control law cannot consider the inherent passivity constraints of controllable dampers, then the commanded forces may often be non-dissipative for some structures and some performance objectives, causing frequent clipping; the result effectively deactivates the controllable damper for most of the duration of the response. ❧ This study presents an alternate clipped linear control strategy. The clipped LQR (CLQR) strategy is just one of a family of clipped linear strategies; herein, the optimal choice, considering the dissipative nature of the controllable damper, from among this family is denoted the “Optimal Clipped Linear Control” (OCLC). This OCLC strategy provides performance superior to all other clipped linear strategies and can be far superior to CLQR when CLQR exhibits frequent clipping. For convenience and for physical interpretation, the closed-loop OCLC system is parameterized relative to a CLQR solution, and the parameters are chosen to minimize a response metric to a particular excitation. ❧ This proposed approach is first applied to a single degree-of-freedom (SDOF) structure model subjected to a Gaussian white noise (GWN) excitation. To illustrate the optimality of OCLC, it is compared to both CLQR and an optimal passive linear viscous damper. Then, the SDOF model, with the OCLC and CLQR strategies, is excited by multiple historical earthquakes and Kanai-Tajimi filtered excitations. OCLC designed for one specific excitation is also evaluated with other excitations. A strategy to select the best control gain parameters for OCLC for a SDOF system is proposed and validated. Moreover, the relationship between OCLC control gain parameters and structural characteristics for SDOF systems is explored. ❧ Next, the effectiveness of OCLC and CLQR strategies in reducing three different response metrics (absolute floor accelerations, inter-story drifts and ground-relative floor velocities) is studied for a two-degree-of-freedom (2DOF) system with a control device either in the first or second story. With a GWN ground acceleration, OCLC again reduces the cost metric and structural response better than either the corresponding CLQR or an optimal passive linear viscous damper. Then, the robustness of the proposed control strategy for both SDOF and 2DOF models is explored through evaluating the controllable damper performance when the structure model differs from the nominal ones used to design the OCLC strategy. ❧ The proposed strategy is also tested for a physical magnetorheological (MR) damper. Real-time hybrid simulation (RTHS) tests are conducted for a set of different structural systems and OCLC shows variable levels of performance improvements over CLQR for different structures for both numerical simulation and RTHS tests. The performance of OCLC designed for one specific excitation is evaluated when subjected to other excitations through RTHS. ❧ Next, shake table experiments are conducted at Japan's NIED “E-Defense” laboratory, using several controllable damping strategies designed to mitigate the responses of a full-sized base-isolated structure specimen, with an MR fluid damper installed in the isolation layer. The experimental results show that OCLC can provide performance superior to CLQR in minimizing absolute acceleration while not increasing ground relative displacement significantly for the base-isolated structure, and vice versa. E-Defense results are also compared with RTHS and pure simulation results, and they have good correspondence. ❧ Finally, numerical solutions to the Fokker-Planck-Kolmogorov (FPK) equation associated with ideal OCLC of a SDOF system excited by GWN are investigated using finite difference methods. Some special considerations are presented: the Heaviside function is approximated by a hyperbolic tangent function; variable grid spacing is applied for the finite difference discretization; a high order polynomial quadrature method is proposed to calculate the statistics. The results indicated that the mesh should be sufficiently fine for a convergent solution; a smaller time step and a lower order finite difference scheme may increase the convergence rate for the semiactive system. Moreover, a new choice of the initial guess for solving the time discretization equation is proposed and the symmetry of solutions is utilized to reduce the computational effort. Further, the implications for optimal clipped linear strategies using FPK solutions are indicated.

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Creator Fang, Qian (author)

Core Title Optimal clipped linear strategies for controllable damping

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Degree Conferral Date 2021-08

Publication Date 07/13/2021

Defense Date 05/03/2021

Publisher Los Angeles (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag controllable damping,dissipativity,Fokker-Planck-Kolmogorov (FPK) equation,OAI-PMH Harvest,optimization,structural control

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Language English

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Advisor Johnson, Erik A. (committee chair), Ghanem, Roger G. (committee member), Jovanovic, Mihailo R. (committee member), Masri, Sami F. (committee member), Savla, Ketan D. (committee member)

Creator Email qianfang.monica@gmail.com,qianfang@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15491150

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