University of Southern California Dissertations and Theses

Towards an optimal design of feedback strategies for structures controlled with smart dampers

(USC Thesis Other)

Towards an optimal design of feedback strategies for structures controlled with smart dampers

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content TOWARDS AN OPTIMAL DESIGN OF FEEDBACK STRATEGIES FOR
STRUCTURES CONTROLLED WITH SMART DAMPERS
by
Wael M. Elhaddad
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CIVIL AND ENVIRONMENTAL ENGINEERING)
December 2015
Copyright 2015 Wael M. Elhaddad
Contents
List of Tables v
List of Figures ix
Dedication xii
Acknowledgements xiii
Abstract xiv
1 Introduction 1
1.1 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . 3
2 Background 7
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Relevant literature . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Clipped-Optimal Strategy for Semiactive Structural Control . . . 19
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Optimal Semiactive Control for Deterministic Excitation 24
3.1 Optimal semiactive control . . . . . . . . . . . . . . . . . . . . 24
3.2 Application to semiactive control of a SDOF system . . . . . . . 27
4 Design of Semiactive Structural Control using Hybrid MPC 35
4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 Single Degree of Freedom System . . . . . . . . . . . . 38
4.2.2 Two Degrees of Freedom System . . . . . . . . . . . . 52
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
ii
5 Robustness of Controllable Damper Feedback Strategies 66
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Robustness Study of the SDOF System . . . . . . . . . . . . . . 67
5.3 Robustness Study of the 2DOF System . . . . . . . . . . . . . . 78
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Improved Linear Feedback Control 86
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Improved Linear Feedback Control using Robust Regression and
Hybrid MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Robust Regression . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Application to the SDOF system . . . . . . . . . . . . . . . . . 92
6.5 Application to the 2DOF system . . . . . . . . . . . . . . . . . 96
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7 On the Optimal Semiactive Control of Elevated Highway Bridges 98
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2 Simplified Elevated Highway Bridge Model . . . . . . . . . . . 99
7.3 Optimal Control Trajectories for the Bridge Model . . . . . . . 102
7.4 Results Comparison and Discussion . . . . . . . . . . . . . . . 104
7.5 Performance of Control Designed using Hybrid MPC . . . . . . 106
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8 Finite Element Model Updating of a Base Isolated Building 110
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.3 Initial Finite Element Modeling . . . . . . . . . . . . . . . . . . 116
8.4 Model Updating Methodology . . . . . . . . . . . . . . . . . . 117
8.4.1 Nelder-Mead Simplex Method . . . . . . . . . . . . . . 121
8.4.2 Model updating parameters . . . . . . . . . . . . . . . . 123
8.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 126
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9 Summary, Conclusions and Future Directions 134
9.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 137
Reference List 139
A Proof of the Certainty Equivalence Property 146
iii
B Effect of ODE Solver Tolerances on Simulation Accuracy 151
B.1 Accuracy of the SDOF system simulations . . . . . . . . . . . . 152
B.2 Accuracy of the 2DOF system simulations . . . . . . . . . . . . 156
iv
List of Tables
3.1 Comparison of peak responses, control force and cost values for
clipped LQR and optimal control using historical earthquakes . . 33
4.1 Minimum cost for various hybrid MPC prediction horizons . . . 42
4.2 Comparison of peak and root-mean-square responses, control force
and cost values using historical earthquakes . . . . . . . . . . . 46
4.3 Comparison of peak and root-mean-square responses, control force
and cost values using historical earthquakes for online and offline
HMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Root-mean-square response using Monte Carlo simulation with
1 million realizations . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Comparison of peak and root-mean-square responses, control force
and cost values for the 2DOF structure using historical earthquakes 59
4.6 Comparison between online and offline HMPC for 1940 El Cen-
tro and 1995 Kobe earthquakes . . . . . . . . . . . . . . . . . . 60
4.7 Root-mean-square responses of the 2DOF system using Monte
Carlo simulation with 40,000 samples . . . . . . . . . . . . . . 61
5.1 Response of the SDOF system with a perturbed stiffness when
subjected to El Centro earthquake using the clipped LQR control
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Response of the SDOF system with a perturbed stiffness when
subjected to El Centro earthquake using the hybrid MPC control
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Response of the SDOF system with a perturbed stiffness when
subjected to Kobe earthquake using the clipped LQR control design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Response of the SDOF system with a perturbed stiffness when
subjected to Kobe earthquake using the hybrid MPC control design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
v
5.5 Response of the SDOF system with a perturbed damping when
subjected to El Centro earthquake using the clipped LQR control
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6 Response of the SDOF system with a perturbed damping when
subjected to El Centro earthquake using the hybrid MPC control
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.7 Response of the SDOF system with a perturbed damping when
subjected to Kobe earthquake using the clipped LQR control design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.8 Response of the SDOF system with a perturbed damping when
subjected to Kobe earthquake using the hybrid MPC control design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1 Feedback gains resulting from linear robust regression of the non-
linear hybrid MPC control law for the SDOF system based on dif-
ferent sets of randomly generated state vectors (units of the gains
are [N=m;N s=m]) . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Comparison between clipped LQR and the improved linear feed-
back designs for peak and RMS responses, control force and cost
values using historical earthquakes . . . . . . . . . . . . . . . . 96
6.3 Comparison between clipped LQR and the improved linear feed-
back designs for peak responses, control force and cost values of
the 2DOF system using historical earthquakes . . . . . . . . . . 97
7.1 Comparison of peak pier responses and cost values for state feed-
back control and optimal control using El Centro earthquake . . 105
7.2 Comparison of control performance using clipped LQR and Hybrid
MPC with different prediction horizons . . . . . . . . . . . . . 108
7.3 Comparison of control performance using clipped LQR and Hybrid
MPC using different control force saturation limits . . . . . . . 108
8.1 Results of system identification using random excitation experi-
ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.2 Parameters chosen for model updating . . . . . . . . . . . . . . 123
8.3 Effective stiffness of isolators obtained from linear regression of
the force-displacement curves . . . . . . . . . . . . . . . . . . 126
8.4 Results of system identification using random excitation experi-
ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
vi
B.1 Response of the SDOF system when subjected to El Centro earth-
quake using the online implementation of the clipped LQR con-
trol design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.2 Response of the SDOF system when subjected to El Centro earth-
quake using the online implementation of the hybrid MPC control
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.3 Response of the SDOF system when subjected to El Centro earth-
quake using the offline implementation of the hybrid MPC con-
trol design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.4 Response of the SDOF system when subjected to Kobe earth-
quake using the online implementation of the clipped LQR con-
trol design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.5 Response of the SDOF system when subjected to Kobe earth-
quake using the online implementation of the hybrid MPC con-
trol design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.6 Response of the SDOF system when subjected to Kobe earth-
quake using the offline implementation of the hybrid MPC con-
trol design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.7 Response of the 2DOF system when subjected to El Centro earth-
quake using the online implementation of the hybrid MPC control
design computed using ODE45 solver . . . . . . . . . . . . . . 156
B.8 Response of the 2DOF system when subjected to El Centro earth-
quake using the offline-4 implementation of the hybrid MPC con-
trol design computed using ODE45 solver . . . . . . . . . . . . 157
B.9 Response of the 2DOF system when subjected to El Centro earth-
quake using the offline-8 implementation of the hybrid MPC con-
trol design computed using ODE45 solver . . . . . . . . . . . . 157
B.10 Response of the 2DOF system when subjected to El Centro earth-
quake using the online implementation of the hybrid MPC control
design computed using ODE5 solver . . . . . . . . . . . . . . 157
B.11 Response of the 2DOF system when subjected to El Centro earth-
quake using the offline-4 implementation of the hybrid MPC con-
trol design computed using ODE5 solver . . . . . . . . . . . . 158
B.12 Response of the 2DOF system when subjected to El Centro earth-
quake using the offline-8 implementation of the hybrid MPC con-
trol design computed using ODE5 solver . . . . . . . . . . . . 158
B.13 Response of the 2DOF system when subjected to Kobe earth-
quake using the online implementation of the hybrid MPC con-
trol design computed using ODE45 solver . . . . . . . . . . . . 158
vii
B.14 Response of the 2DOF system when subjected to Kobe earth-
quake using the offline-4 implementation of the hybrid MPC con-
trol design computed using ODE45 solver . . . . . . . . . . . . 159
B.15 Response of the 2DOF system when subjected to Kobe earth-
quake using the offline-8 implementation of the hybrid MPC con-
trol design computed using ODE45 solver . . . . . . . . . . . . 159
B.16 Response of the 2DOF system when subjected to Kobe earth-
quake using the online implementation of the hybrid MPC con-
trol design computed using ODE5 solver . . . . . . . . . . . . 160
B.17 Response of the 2DOF system when subjected to Kobe earth-
quake using the offline-4 implementation of the hybrid MPC con-
trol design computed using ODE5 solver . . . . . . . . . . . . 160
B.18 Response of the 2DOF system when subjected to Kobe earth-
quake using the offline-8 implementation of the hybrid MPC con-
trol design computed using ODE5 solver . . . . . . . . . . . . 160
viii
List of Figures
1.1 Feedback control for structural systems . . . . . . . . . . . . . 3
2.1 Semiactive control Strategies . . . . . . . . . . . . . . . . . . . 8
2.2 Experimental results for an Magnetorheological (MR) damper
under harmonic excitation with various constant voltages applied
to the electromagnet coil circuitry (Dyke et al., 1996) . . . . . . 14
2.3 pseudo-negative stiffness (PNS) control algorithm (Iemura and
Pradono, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Idealized model of the passivity constraint for controllable damp-
ing devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 SDOF structure . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Convergence of the cost functions iterations . . . . . . . . . . . 31
3.3 Comparison of clipped LQR and optimal control for El Centro
earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Control force vs. displacement hysteresis loop under the El Cen-
tro earthquake excitation during the time range [5.0,7.6] secs . . 33
4.1 Comparison of control input for SDOF system . . . . . . . . . . 43
4.2 Absolute acceleration response and control force for the first 10 s
of the 1940 El Centro earthquake . . . . . . . . . . . . . . . . . 45
4.3 Hybrid MPC control input for unit magnitude state vector . . . . 48
4.4 More realistic model for passivity constraint . . . . . . . . . . . 49
4.5 Mean square responses of MCS for SDOF system . . . . . . . . 50
4.6 2DOF structure . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.7 2DOF system control force at q = 0 as a function of velocity
vector v= ˙ q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.8 2DOF system control force at q=[0 1 cm]
T
as a function of
velocity vector v= ˙ q . . . . . . . . . . . . . . . . . . . . . . . 56
ix
4.9 Control force vs. displacement hysteretic loop during time range
[4.6,5.6] secs of the Kobe earthquake . . . . . . . . . . . . . . 58
4.10 Mean square responses of MCS for 2DOF system . . . . . . . . 63
5.1 Nominal SDOF system used to design control . . . . . . . . . . 68
5.2 SDOF system with perturbed stiffness . . . . . . . . . . . . . . 68
5.3 SDOF system with perturbed damping . . . . . . . . . . . . . . 69
5.4 Reduction in peak responses for the SDOF system when sub-
jected to the El Centro earthquake with perturbed stiffness . . . 74
5.5 Reduction in RMS responses for the SDOF system when sub-
jected to the El Centro earthquake with perturbed stiffness . . . 74
5.6 Reduction in peak responses for the SDOF system when sub-
jected to the Kobe earthquake with perturbed stifness . . . . . . 75
5.7 Reduction in RMS responses for the SDOF system when sub-
jected to the Kobe earthquake with perturbed stiffness . . . . . . 75
5.8 Reduction in peak responses for the SDOF system when sub-
jected to the El Centro earthquake with perturbed damping . . . 76
5.9 Reduction in RMS responses for the SDOF system when sub-
jected to the El Centro earthquake with perturbed damping . . . 76
5.10 Reduction in peak responses for the SDOF system when sub-
jected to the Kobe earthquake with perturbed damping . . . . . 77
5.11 Reduction in RMS responses for the SDOF system when sub-
jected to the Kobe earthquake with perturbed damping . . . . . 77
5.12 Nominal 2DOF system used to design control . . . . . . . . . . 78
5.13 2DOF system with perturbed stiffness . . . . . . . . . . . . . . 79
5.14 2DOF system with perturbed damping . . . . . . . . . . . . . . 79
5.15 Reduction in peak responses for the 2DOF system when sub-
jected to the El Centro earthquake with perturbed stiffness . . . 81
5.16 Reduction in RMS responses for the 2DOF system when sub-
jected to the El Centro earthquake with perturbed stiffness . . . 81
5.17 Reduction in peak responses for the 2DOF system when sub-
jected to the Kobe earthquake with perturbed stifness . . . . . . 82
5.18 Reduction in RMS responses for the 2DOF system when sub-
jected to the Kobe earthquake with perturbed stiffness . . . . . . 82
5.19 Reduction in peak responses for the 2DOF system when sub-
jected to the El Centro earthquake with perturbed damping . . . 83
5.20 Reduction in RMS responses for the 2DOF system when sub-
jected to the El Centro earthquake with perturbed damping . . . 83
x
5.21 Reduction in peak responses for the 2DOF system when sub-
jected to the Kobe earthquake with perturbed damping . . . . . 84
5.22 Reduction in RMS responses for the 2DOF system when sub-
jected to the Kobe earthquake with perturbed damping . . . . . 84
6.1 Linear fitting surface for hybrid MPC control law . . . . . . . . 93
6.2 Robust fitting surface for hybrid MPC control law . . . . . . . 94
7.1 Simplified bridge model . . . . . . . . . . . . . . . . . . . . . 100
7.2 Convergence of the cost functions during iterations . . . . . . . 104
7.3 Pier response during the first 10 seconds of El Centro earthquake 106
8.1 Photo of the base isolated building specimen at the E-defense
testing facility (photo courtesy E. A. Johnson) . . . . . . . . . . 112
8.2 FEM of the building . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3 Sensors locations at different floors (Sato and Sasaki, 2013) . . . 114
8.4 Elevation view of the building (Sato and Sasaki, 2013) . . . . . 117
8.5 Mode shapes and natural periods for the fixed-base model obtained
from OpenSEES . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.5 Force-Displacement curves of the isolators . . . . . . . . . . . . 128
8.6 Match between the FEM and the identified modes (values on the
axes are frequencies in Hz) . . . . . . . . . . . . . . . . . . . . 130
8.7 Minimization of cost function value in 500 iterations . . . . . . 131
xi
To my parents, Mostafa Elhaddad and Sabah Abdelhady
to whom I owe a lot.
To my brothers Ahmed and Tawfik and my sister Alaa for
their continuous support and encouragement.
xii
Acknowledgements
First, I would like to gratefully thank my advisor, Professor Erik A. Johnson, for
his guidance and support throughout my Ph.D. program. I would also like to
gratefully thank my committee members, Prof. Sami F. Masri, Prof. Ketan Savla
and Prof. Joe Qin, for their invaluable suggestions and advice on this research.
I would like to thank many colleagues who supported me a lot during my study
at USC including Mahmoud Kamalzare and Elham Hemmat Abiri. I would also
like to gratefully acknowledge the partial financial support of this research work
by the National Science Foundation (NSF) through the CMMI 08-26634 and 13-
44937 grants; and by the University of Southern California through Viterbi Ph.D.
fellowship, Gammel Ph.D. scholarship and Teaching Assistantship appointments.
xiii
Abstract
Structural control has attracted the attention of many researchers to study its
application to natural hazards mitigation with the ultimate goal of protecting
buildings and bridges against strong earthquakes and extreme winds. Signifi-
cant efforts have been devoted to realize structural control systems for real-life
applications, to protect structural systems and, consequently, human lives; such
an application is of great societal and economic importance. Several studies in
the area of optimal control of structures have been presented in this dissertation,
particularly structural control using smart dampers, such as controllable passive
devices. First, optimal control strategies for semiactive control is evaluated using
dynamic optimization, assuming the structure is subjected to a deterministic exci-
tation. The results of this study represent the motivation for seeking feedback
strategies for semiactive structural control with improved performance. Using
hybrid system representation to model the passivity constraints that characterize
semiactive control devices as on-off switches, it was shown that more efficient
control strategies can be obtained by employing a model predictive control (MPC)
xiv
design. The optimized control laws were found to be nonlinear functions of the
system states. Numerical simulations of simple structural models show that not
only the responses and cost were reduced significantly, but also the control forces
had significant reductions, indicating a more efficient use of the semiactive damp-
ing devices. Robustness of semiactive control feedback strategies against uncer-
tainties in the structural system parameters was studied for models with perturbed
stiffness and damping. In addition, an improved linear feedback control strategy
based on robust regression of the nonlinear hybrid MPC control laws was shown
to produce linear feedback control laws with significant performance improve-
ments compared to clipped LQR. The resulting controllers are more suitable for
real-time control applications in larger structures. Numerical applications to con-
trol design for simplified highway bridge models is also presented.
Additionally, this dissertation includes a finite element model (FEM) cali-
bration study of a base-isolated reinforced concrete building. The building is a
specimen tested at Japan’s E-Defense earthquake engineering research center in
2013 to evaluate the effects of seismic moat wall pounding and isolation during
long-period earthquakes. This full-scale four-story moment frame building, with
two reinforced concrete walls, sits on an isolation composed of rubber bearings,
elastic sliders, U-shaped steel dampers and oil dampers. Designing controllable
dampers for planned 2016 or 2017 tests requires a calibrated numerical model.
Modal analysis results are reported, based on random excitation responses dur-
ing the 2013 tests. The FEM is developed and updated to match the identified
xv
modal parameters. The superstructure FEM consists of line and shell elements
and (linear) spring isolation elements, resulting in 1757 nodes and 10,542 degrees
of freedom. A set of 21 parameters (various Young’s moduli, isolation layer stiff-
nesses, and point masses) was chosen and updated iteratively by the the Nelder-
Mead simplex method to minimize the frequency residuals and obtain a stronger
correlation between the mode shapes of the FEM and those identified from the
2013 random tests. The optimization converges to an updated FEM that provides
much better match with the experimental data, reducing the frequency and MAC
residuals by 60-90
xvi
Chapter 1
Introduction
Structural control has attracted the attention of many researchers to study its
application to natural hazards mitigation (Spencer and Nagarajaiah, 2003), with
the ultimate goal of protecting buildings and bridges against strong earthquakes
and extreme winds. Significant efforts have been devoted to realize structural
control systems for real-life applications, to protect structural systems and, conse-
quently, human lives; such an application is of great societal and economic impor-
tance. A variety of structural control systems are used for vibration reduction in
buildings and bridges around the world (Spencer and Sain, 1997). Applications of
different structural control strategies, such as passive, active, hybrid and semiac-
tive control, have been investigated extensively. Although passive control strate-
gies (Soong and Dargush, 1997) are the best established and most well-accepted
for structural vibration reduction, they are not the most efficient control technique
as they cannot adapt to different loading events and structural conditions. On the
other hand, active control strategies are more adaptive and efficient in vibration
reduction, though there are some concerns about their reliability (Spencer and
Nagarajaiah, 2003) since a poor design can render a structure unstable and they
1
can demand power that might not be available during an extreme loading event
such as an earthquake or severe wind exposure.
In the last two and a half decades, semiactive strategies based on controllable
passive devices have emerged as an alternative for vibration reduction in struc-
tures. Controllable passive devices, such as a controllable damper or control-
lable stiffener, are ones that exert forces through purely passive means but have
controllable properties that affect those forces. Some examples of controllable
passive devices are variable orifice dampers, variable friction devices, variable
stiffness devices and controllable fluid dampers (e.g., magnetorheological and
electrorheological fluid dampers) (Spencer and Nagarajaiah, 2003; Spencer and
Sain, 1997; Housner et al., 1997; Symans and Constantinou, 1999; Soong and
Spencer, 2002).
The main benefits of using semiactive control are (i) its inherent stability, as
it does not introduce energy into the controlled structure, (ii) its ability to focus
on multiple (and possibly changing) objectives, exerting a force that can depend
on non-local information, and (iii) the low power requirement that is critical in
the case of natural hazards like earthquakes (Dyke et al., 1996). In addition, it
has been shown that semiactive control is capable, in some cases, of achieving
performance comparable to that of a fully active system (Dyke et al., 1996). It is,
thus, beneficial to advance the understanding of the behavior of structural systems
controlled with such devices, to be able to take full advantage of their capabilities.
2
Controller
Measurements Control force
Excitation
Structural System
(building, bridge, etc.)
Response
(displacement,
velocity, etc.)
Figure 1.1: Feedback control for structural systems
Both active and semiactive control systems have the ability to adapt to differ-
ent operating conditions, relying on control algorithms to achieve this adaptabil-
ity. Both strategies make use of measurements and observations/estimations of
current structure state, as shown in Figure 1.1. The measurements are then pro-
vided as a feedback to the controllers that are responsible for making a decision
about how the device should adapt to the current state of the structure in order to
efficiently reduce its future vibrations.
1.1 Overview of the Dissertation
The organization of this dissertation can be summarized as follows:
• Chapter 2: This chapter reviews the literature of semiactive structural con-
trol and summarizes the current method used to design semiactive control
strategies. In addition, the chapter provides the motivation for seeking an
optimal design for such control strategies.
3
• Chapter 3: In this chapter, a gradient-based optimization procedure is used
to find the optimal control trajectories for a structure controlled with a
semiactive damper, assuming the excitation is deterministic. Although this
approach is not practical, it is useful as it represents an upper bound on the
performance of semiactive structural control.
• Chapter 4: In this chapter, a nonlinear model predictive control (MPC)
method is employed based on a hybrid system representation of the struc-
tural model. The method is illustrated using two different examples and is
shown to have the potential for improving semiactive control performance
compared to methods traditionally adopted in structural control design. The
computational expense of the method is also discussed and an offline imple-
mentation of the method is employed to enable fast computation of response
in Monte Carlo simulations, which were necessary for performance com-
parison.
• Chapter 5: The robustness of hybrid MPC controllable damper feedback
strategies against uncertainties in the structural system parameters is investi-
gated in this chapter. Systems with perturbed stiffness and damping are used
to evaluate and compare the performance of control design. This study is
necessary to validate that hybrid MPC control laws do not trade off robust-
ness for optimality.
4
• Chapter 6: In this chapter, an improved linear feedback for semiactive con-
trol is developed using robust regression of hybrid MPC control laws. The
method is applied to two numerical models for which control performance
is evaluated. The resulting control design may be more suitable for higher-
order structural system models because the online solution of optimization
problems will not be required to obtain the control force in real-time. The
resulting control also laws show improved response reduction, while being
more suitable for real-time control applications.
• Chapter 7: Optimal control of elevated highway bridges is investigated in
this chapter. Using a dynamic optimization technique, the optimal control
trajectories for a simplified bridge model are evaluated for both active and
semiactive control. Based on the results, it is shown that, for some control
objectives, such as reduction of bridge pier displacements, the effectiveness
of semiactive control is significantly lower than active control.
• Chapter 8: In this chapter, a finite element model of 4-story base-isolated
reinforced concrete building, tested at the E-defense shaking table in Japan,
is developed and calibrated based on experimental data. The building
employs different types of damping devices in its isolation layer: rubber
bearings, sliding bearings, U-shaped steel dampers and oil dampers. Using
vibration data from random excitation tests, identification of the first 6
5
mode shapes and their frequencies is presented. An initial 3D finite ele-
ment model is developed; using a Nelder-Mead simplex optimization, a set
of 21 parameters are updated in the model to match experimentally identi-
fied modal characteristics. The mode is updated in good agreement with the
experimental data, and the updated parameters show a significant increase
in the superstructure stiffness compared to the initially assumed values.
• Chapter 9: This chapter presents a summary of the research work and the
conclusions of this dissertation.
6
Chapter 2
Background
2.1 Motivation
The design process of semiactive control for vibration reduction of structures is
critical for ensuring an effective use of the control device. Different feedback
control strategies can be used to minimize vibrations in structures controlled with
semiactive devices (Jansen and Dyke, 2000). Some examples of these control
laws are neural/fuzzy control strategies (Sun and Goto, 1994), Lyapunov control
(Jansen and Dyke, 2000; Gavin, 2001; Wang and Gordaninejad, 2002), pseudo-
negative stiffness control (Iemura and Pradono, 2009), performance-guarantee
approaches (Scruggs et al., 2007a,b) and LMI control (Johnson and Erkus, 2007).
Many other applications of semiactive control are based on the clipped-optimal
control technique (Dyke et al., 1996), where the desired control input is deter-
mined by initially assuming an unconstrained device force model; then, any
infeasible force is clipped from the desired control, as shown in Figure 2.1a for
a semiactive controllable device, to ensure that the resulting force is realizable
by the control device. However, when the desired control is highly unrealizable
7
(i.e., very non-dissipative commands to a controllable damper), the frequent clip-
ping may render the clipped control strategy far from being optimal, as the design
methodology does not, and cannot, consider the semiactive control device’s inher-
ent passivity constraints — i.e., that it can only exert dissipative forces regardless
of the command it is given. For that reason, the design of control with such a
method is inconsistent in the minimization of the objective functions. The result
of such inconsistency is the need for heuristic approaches that usually involve
large scale parametric studies (Johnson et al., 2007; Ramallo et al., 2002) and
optimality is not guaranteed.
(a) Clipped LQR
(b) Hybrid MPC
Figure 2.1: Semiactive control Strategies
Model predictive control (MPC) is a control method that has been widely
adopted in industrial engineering in the past three decades (Qin and Badg-
well, 2003). The method is an iterative scheme that seeks to control the future
8
responses of a dynamical system by minimizing a metric of their values through-
out a finite time horizon in the future. Through manipulation of the future forces,
the MPC scheme attempts to optimize the future responses of the dynamical sys-
tem, given the current state of the system. When the optimal sequence of future
control forces is determined, only the first control force is commanded and the
whole process is repeated again to determine the optimal control forces for the
next time step. One of the main strengths of the MPC method for designing con-
trol is its ability to handle constraints, nonlinearities and time varying effects.
Today, the MPC method lends itself to different control applications, including
applications in automotive, aerospace and process industries.
In many applications where MPC has been employed for control design, the
objective is to minimize a quadratic cost function written in terms of the predicted
states and the control forces in future time steps. The number of future steps for
which the system states are predicted is called the prediction horizon, whereas the
number of steps over which the control force is optimized, in many cases equal
to the prediction horizon, is called the control horizon. Since the optimization
process is repeated at every time step, the prediction horizon is repeatedly shifted
forward in time, which is why the method is sometimes called in the literature the
receding horizon method. The resulting optimization is a quadratic programming
problem in terms of the control forces at each time step of the control horizon. As
this optimization process is repeated at each time step, the system analysis must
be performed fast enough so that the result can be obtained during the sampling
9
time and the control forces can be commanded in real time. Although this can be
readily implemented for simple models with a few states, it can be computation-
ally expensive for higher-order nonlinear systems or systems with high frequency
dynamics.
The application of MPC in structural control has been investigated previously
by Mei et al. (2001). In their study, the potential of MPC in structural control was
demonstrated on building models with active tendon systems with full state feed-
back, and later extended (Mei et al., 2002) to use acceleration feedback. Although
those implementations were suitable for simple structural systems (i.e., very few
degrees of freedom), they are not suitable for larger or highly detailed structural
models as the time required to run simulations that predict the structural response
might be orders of magnitude longer than the sampling time. If the computa-
tions can be done more efficiently, the MPC method can be applied to control
more practical structural models, which can enable the application of this control
method to multi-story buildings or more realistic bridge models. In addition, the
implementations of Mei et al. (2002) were only applicable for linear systems (i.e.,
linear structure controlled with an unconstrained active device), and are not suit-
able for structures controlled with “smart” dampers as they ignore the device’s
dissipative nature that leads to a controlled system with nonlinear behavior.
The on-off switching logic required to control semiactive devices is conve-
niently modeled as part of a hybrid system, which is a time evolving system
that exhibits mixed dynamics due to interacting physical laws, logical conditions
10
and/or different operating modes (Bemporad and Morari, 1999). Hybrid systems
can be represented in different forms, such as piecewise affine functions (Sontag,
1981), discrete hybrid automata (Alur et al., 1993) and mixed logical dynami-
cal (MLD) system models (Bemporad, 2002) — all mathematically equivalent
(Heemels et al., 2001). Herein, MLD systems are used to model structures con-
trolled with a semiactive device since MLD models can accurately reflect the
nonlinear constraints of the semiactive device by introducing auxiliary variables
into the system model. Hybrid MPC is an MPC scheme that finds optimal con-
trol strategies for hybrid system models. Optimization of this control results in
a mixed integer quadratic programming (MIQP) problem, which can be solved
numerically to find the optimal control input. The resulting control force satisfies
the passivity constraint and no longer needs clipping as shown in Figure 2.1b.
The online implementation of hybrid MPC must solve the optimization problem
during each sampling time of the system, which is a computationally challeng-
ing problem with a complex system model and/or anything but the shortest time
horizons. On the other hand, an offline implementation will carry out most of
the computations a priori, resulting in less demanding computation, such as a
table look-up, to be carried out in real time during the sampling time, but may
require significant memory to store the optimal control strategies. Depending on
the order of the system (i.e., number of states) and the type of nonlinearities, the
development of the look-up table for offline approach may or may not be compu-
tationally tractable.
11
An example of hybrid MPC for semiactive control of a simple (wheel plus
vehicle) quarter-car model is reported by Giorgetti et al. (2005), who minimize
a metric of vehicle handling and ride comfort by choosing an optimal control
law for the semiactive suspension. This hybrid MPC control law, a piece-wise
affine function (that could also be used in an offline control strategy), achieved
significant improvement in suspension performance compared to clipped LQR
control when the model undergoes free response from several initial conditions
as well as subjected to a Gaussian random road velocity.
As shown subsequently in the numerical examples in Chapter 4, hybrid MPC
is readily applied to typical structural system models with a few degrees-of-
freedom (DOFs). In contrast with control laws derived from unconstrained sys-
tem models (e.g., clipped LQR), hybrid MPC produces nonlinear state feedback
control laws, giving semiactive device forces that can achieve significantly better
performance for some control objectives (e.g., the reduction of absolute accel-
erations). Despite the nonlinear nature of the control laws, it is found that they
satisfy the homogeneity property (i.e., they scale linearly when the state vector is
scaled) when the structure model also exhibits homogeneity. This property can be
exploited to reduce the dimension of table look-ups. The resulting offline imple-
mentation of hybrid MPC is orders of magnitude faster than the online implemen-
tation, which enables rapid computation of response statistics from a Monte Carlo
simulation with stochastic excitation. Compared to clipped LQR, hybrid MPC is
found to be more consistent in the reduction of the structural responses, which
12
can help avoid using heuristic approaches and parametric studies in the struc-
tural control design. However, finding the optimal hybrid MPC force function
is computationally expensive for large structural systems with the current imple-
mentations and optimization tools; further research will be required to facilitate
application to more complex structures. Nevertheless, since many structures are
reasonably modeled with a few DOFs, it is demonstrated herein that controllable
dampers commanded with hybrid MPC can reduce seismic structural responses
by up to half relative to the conventional clipped-optimal strategy while using
comparable, or smaller, force levels.
2.2 Relevant literature
Magnetorheological (MR) fluid dampers are controllable damping devices that
make use of MR fluids. Spencer et al. (1997) investigated modeling the nonlinear
behavior of MR Dampers by a phenomenological model that has been validated
against an experimental test of a prototype MR damper. This model makes use
of the Bouc-Wen model, which is capable of representing the hysteretic behav-
ior of the MR damper (Wen, 1976). The same model was also verified against
experimental results for another MR damper in Dyke et al. (1996) and was used
in the simulation of a three-story test structure. The experimental results of the
MR damper in this study are shown in Figure 2.2. Yang et al. (2004) studied the
13
dynamic behavior of a 20-ton MR damper capable of providing semiactive damp-
ing for full-scale structural engineering applications. Jansen and Dyke (2000)
compared the application of different control strategies for a six-story building
model controlled with two MR dampers. The control strategies considered in this
comparative study were clipped-optimal control, control based on Lyapunouv
stability theory, maximum energy dissipation, decentralized bang-bang control
and modulated homogeneous friction control. This comparative study concluded
that Lyapunov controller, clipped-optimal and modulated homogeneous friction
algorithms all reduced structural response significantly.
Velocity [cm/sec] Displacement [cm]
Force [N]
−2 −1 0 1 2
−1500
−1000
−500
0
500
1000
1500
−10 −5 0 5 10
−1500
−1000
−500
0
500
1000
1500
0 V
0.75 V
1.5 V
2.25 V
Force [N]
Figure 2.2: Experimental results for an MR damper under harmonic excitation
with various constant voltages applied to the electromagnet coil circuitry (Dyke
et al., 1996)
Iemura and Pradono (2002) introduced the concept of a pseudo-negative stiff-
ness (PNS) control algorithm for semiactive structural control. The aim of this
control algorithm is to reduce the base shear and absolute accelerations of the
structure. In order to achieve that objective, the method seeks to form a “virtual
perfectly-plastic force-deformation characteristic” that is known to maximize the
14
energy dissipation of the system. Since the structural force-deformation relation-
ship is characterized as a positive stiffness, the damper force-displacement rela-
tion would have to resemble that of a negative stiffness, in order for the combined
force-displacement loops to become as rectangular as possible, as in Figure 2.3.
The method is applied for semiactive control of a benchmark bridge model and
was found to be effective in achieving the control objective.
= +
Damper
Force

Stroke
Member Stiffness PNS-controlled Damper
Combined Hysteretic Loop
Damper + Elastic
Force
Stroke Stroke
Elastic
Force
Figure 2.3: PNS control algorithm (Iemura and Pradono, 2003)
Gavin and Aldemir (2005) investigated optimal semiactive control of base
isolated structures subjected to ground motion. Their approach was based on a
gradient-based optimization algorithm. Results from a truly optimal semiactive
control were compared to passive control and psuedoskyhook control. While the
results from this optimization procedure provide the best performance in terms
of reduction of the structural response, this approach assumes that the ground
motion is known a priori. For that reason, this approach represents an upper
bound on the performance of causal semiactive control rules.
15
Alhan et al. (2006) used the same deterministic optimization approach to find
optimal semiactive control forces for a single-degree-of-freedom (SDOF) struc-
ture subjected to pulse-like ground motion. The paper investigated the effective-
ness of using semiactive control compared to passive viscous damping. Aldemir
and Gavin (2006) used a similar approach to study the effectiveness of different
semiactive control strategies for structural base isolation by comparing them to
the optimal semiactive control. The control algorithms investigated in this study
were pseudonegative stiffness, continuous pseudoskyhook and bang-bang pseu-
doskyhook control algorithms. Time history responses were obtained for a two-
degrees-of-freedom (2DOF) structural model subjected to sinusoidal and pulse-
like excitation. It was found that pseudoskyhook control performed the closest to
the optimal control for this structural model, which was capable of reducing the
isolator drift while maintaining the lowest superstructure accelerations.
Johnson and Erkus (2007) studied the dissipativity and performance of smart
damping structural control using linear matrix inequality (LMI) synthesis. In this
study, an index was introduced into control design in an attempt to constrain the
control force to be more dissipative. The proposed dissipativity indices resulted
in a modified LMI-based linear quadratic regulator (LQR) design, which pro-
duces linear state feedback control laws. The modified LQR design was applied
to a 2DOF building model, for which the proposed method was able to improve
the dissipative nature of the control design and the semiactive control perfor-
mance. The proposed method was also applied to a 2DOF bridge model, but was
16
not able to improve the semiactive control performance in this case. Erkus and
Johnson (2011) applied the same method to design the control for a semiactive
base-isolated benchmark structure. It was shown that the method can find a linear
controller that improves the performance of semiactive control with MR dampers.
Aly and Christenson (2008) investigated the efficacy of smart dampers in
vibration reduction of civil infrastructure. They used an energy-based approach
by using the probability of dissipative damping forces to find an equivalent lin-
ear damper that dissipates the same energy. The method was applied to a SDOF
building model controlled with a single damper. The results of this study showed
that, when the objective is to regulate the absolute accelerations of the structure,
the resulting primary controller from LQR is more prone to commanding non-
dissipative forces.
Tseng and Hedrick (1994) studied the optimality of semiactive control laws.
The study shows that the optimal control that minimizes a deterministic quadratic
performance index is a time varying solution (i.e., not a linear time-invariant state
feedback control law), that involves three related Riccati equations. This study
also investigates several constant-gain sub-optimal control strategies, concluding
that the clipped-optimal strategy is not optimal for semiactive suspensions.
Bemporad et al. (2002) developed an explicit linear quadratic regulator for
linear systems, with linear constraints on the states and the control input. The
method used multi-parametric quadratic programming to find the explicit solution
for the optimal control input, which in this case is described as a piecewise affine
17
and continuous feedback control law. The explicit control law can be described
as different feedback control gains defined over a set of polyhedral partitions
of the state space. This representation can be implemented offline using table
lookup, where a search method is used to find the polyhedral partition in which
the current state vector lies; then, the corresponding control gain is determined.
Bemporad et al. (2002) applied the method to a single-input single-output system
with constrained control input.
Bemporad and Morari (1999) introduced a method for modeling and control
of dynamical systems that involves operating constraints in the form of logical
conditions (e.g., On-Off switches). Those kinds of hybrid systems were mod-
eled as mixed logical dynamical (MLD) systems. The main idea is to trans-
form the logical conditions into linear inequalities involving integer and con-
tinuous variables; this results in linear dynamic equations that are subjected to
linear mixed-integer inequalities. This model can also represent nonlinear sys-
tems whose nonlinearities can be expressed by piecewise linear functions. Due
to the presence of integer variables, the resulting optimization problem is a mixed
integer quadratic programming (MIQP) problem. The paper used the developed
MLD system model to study the control of a gas supply system that included
five boilers which could be turned on or off; hence, the system was hybrid and
exhibited mixed dynamics. Borrelli et al. (2003) developed an efficient algorithm
to find the explicit representation of the optimal control laws for hybrid systems
by using multiparametric programming. The optimal control can be represented
18
as piece-wise affine (PWA) state feedback control law that can be implemented
off-line using table lookup. This offline implementation is more computationally
efficient compared to the online implementation of optimal control for hybrid
systems. Giorgetti et al. (2005) used the same model to find the explicit form of
the optimal control law for semiactive vehicle suspension.
2.3 Clipped-Optimal Strategy for Semiactive
Structural Control
Consider a structure with n degrees of freedom (DOF) that is subjected to an
external excitation and is controlled by some devices to reduce its response. The
equations of motion for such a structure can be written:
M¨ q(t)+ C˙ q(t)+ Kq(t)= B
w
w(t)+ B
u
u(t) (2.1)
where M, C and K are the mass, damping and stiffness matrices of the structure,
respectively; q is the displacement vector of the structure relative to the ground;
w is the external excitation (e.g., from ground acceleration or wind forces) with
influence matrix B
w
; u is a vector of the damping forces exerted by the control
devices with influence matrix B
u
.
19
These equations can also be written in continuous-time state-space form as
follows:
˙ x(t)= Ax(t)+ B
w
w(t)+ B
u
u(t) (2.2)
x=
8
>
<
>
:
q
˙ q
9
>
=
>
;
; A=
2
6
4
0 I
M
1
K M
1
C
3
7
5
; B
w
=
2
6
4
0
M
1
B
w
3
7
5
; B
u
=
2
6
4
0
M
1
B
u
3
7
5
where x is the state vector, A is the state matrix, B
w
is the influence matrix of the
excitation and B
u
is the influence matrix of the control force.
The optimal (unconstrained) control design for a linear system can be obtained
in a variety of ways; one example is using a linear quadratic regulator (LQR),
where the objective is to minimize an infinite-horizon quadratic performance
index of the form:
J=
Z
¥
0

x
T
Qx+ 2x
T
Nu+ u
T
Ru

dt (2.3)
in which Q, R and N are weighting matrices for the different objective terms. If
the external excitation is zero-mean white noise, the certainty equivalence prop-
erty (Van de Water and Willems, 1981; Chow, 1976) of stochastic control theory
would apply, and the stochastic optimal control is equivalent to that obtained from
deterministic analysis that assumes an initial condition and ignores the excita-
tion. The LQR control design is a quadratic programming problem, for which the
20
infinite-horizon optimal solution is a linear state feedback controller (Hespanha,
2009) with feedback gain K
LQR
:
u
des
(t)=K
LQR
x(t)=[R
1
(B
T
u
P+ N
T
)] x(t) (2.4)
where P is the positive definite symmetric solution of the algebraic Riccati equa-
tion given by:
(A B
u
R
1
N
T
)
T
P+ P(A B
u
R
1
N
T
) PB
u
R
1
B
T
u
P+ Q NR
1
N
T
= 0
(2.5)
Dissipative
Dissipative
Infeasible
u
des
v
rel
Infeasible
Figure 2.4: Idealized model of the passivity constraint for controllable damping
devices
Assuming the states of the structural system (i.e., displacements and veloci-
ties) can be measured or estimated, then equation (2.4) can be used to determine
the required control input, and the device can be commanded with the desired
21
damping force. However, semiactive dampers can only exert dissipative forces;
any commanded non-dissipative forces cannot be realized. Thus, the LQR control
design does not generally result in realizable control forces for controllable pas-
sive devices unless, of course, the desired control forces are purely dissipative.
To apply these forces to semiactive devices, the clipped-optimal control algo-
rithm (Dyke et al., 1996; Jansen and Dyke, 2000) employs a primary controller
obtained assuming unconstrained control device forces (i.e., ignoring the semiac-
tive device passivity constraints), and the device is commanded to only exert the
desired forces that are realizable by the device (e.g., only dissipative forces are
requested of a controllable damper). This behavior can be modeled in simulation
if a secondary controller exists that clips the unrealizable control forces.
Different semiactive devices have different constraints on the achievable
device forces; however, they all share a passivity constraint. For the sake of gen-
erality, an idealized model is adopted for describing the passivity constraint of a
controllable damper, as illustrated graphically in Figure 2.4. It is assumed that
all dissipative forces are realizable by the device and all nondissipative forces are
infeasible. The corresponding passivity constraint is a nonlinear constraint that
can be written as:
u
sa
v
rel
0 (2.6)
where u
sa
is the damper control force and v
rel
is the relative velocity across the
semiactive damper (chosen with sign convention such that a positive dissipative
22
force resists motion in the positive velocity direction). The resulting damper con-
trol force, that the semiactive damper will realize, can be calculated as follows:
u
sa
= u
des
H[u
des
v
rel
]=
8
>
>
<
>
>
:
u
des
; u
des
v
rel
0
0; u
des
v
rel
< 0
(2.7)
where u
des
is the desired damper control force determined by the clipped-optimal
strategy, and H[] is the Heaviside unit step function.
Note that some controllable dampers (Spencer et al., 1997; Yang et al., 2004)
have models with some minimum viscous damping and/or linear stiffness. Such
models can be recast in the idealized model discussed here if the linear stiffness
and/or damping is moved from the device model into the structure model.
2.4 Summary
This chapter presented a background and literature review on the design of feed-
back strategies for semiactive structural control. The clipped-optimal control
design, a method popular for the design of semactive control, was summarized
and explained. In the next chapter, optimal control trajectories for a structure con-
trolled with a semiactive damper will be presented. In the subsequent chapters,
it will be shown that clipped-optimal feedback strategies may not be optimal for
some control objectives, and a new feedback design will be developed and shown
to use the semiactive device much more efficiently.
23
Chapter 3
Optimal Semiactive Control for
Deterministic Excitation
This chapter studies the truly optimal semiactive control assuming the excita-
tion is known a priori. The chapter is divided into two section, the first section
summarizes the approach used by Gavin and Aldemir (2005) to find the truly
optimal semiactive control of a base isolated structure subject to an earthquake
ground motion. The second section applies the same approach to a SDOF struc-
ture which will be used in later sections to evaluate the efficacy of causal control
laws.
3.1 Optimal semiactive control
Consider a controlled dynamical system that is governed by the nonlinear equa-
tion:
˙ x= f(x(t);u(t);t) (3.1)
24
for which the control objective is to minimize the integral cost function:
J=
Z
t
f
0
L (x(t);u(t);t)dt (3.2)
whereL (x(t);u(t);t), the Lagrangian (i.e., integrand) of the cost function, is a
scalar function of the states and control inputs. The cost function J is to be min-
imized subject to the equality constraint represented by the system dynamics in
equation (3.1). The cost function can be augmented with the equality constraints
by using Lagrange multipliers so that the augmented cost function becomes:
J
A
= J+
Z
t
f
0
l l l
T
(t)[f(x;u;t) ˙ x(t)]dt (3.3)
where l l l(t) is a vector of Lagrange multipliers, also called the co-state vector
(Kirk, 2004), and the explicit dependence of x and u on time is omitted for nota-
tional clarity. By defining the Hamiltonian as:
H (x;u;l l l;t)=L (x;u;t)+l l l
T
(t)f(x;u;t) (3.4)
then the augmented cost function can be rewritten as:
J
A
=
Z
t
f
0
h
H (x;u;l l l;t)l l l
T
(t)˙ x(t)
i
dt (3.5)
25
The cost function can be minimized by setting the first variation of J
A
equal
to zero (Kirk, 2004), resulting in Euler-Lagrange equations that represent the
necessary optimality conditions as follows:
˙
l l l(t)=

¶H
¶x

T
=

¶f(x;u;t)
¶x

T
l l l(t)

¶L (x;u;t)
¶x

T
; l l l(t
f)
= 0
(3.6)
¶H
¶u
=l l l
T
(t)
¶f(x;u;t)
¶x
+
¶L (x;u;t)
¶u
= 0 (3.7)
The Euler-Lagrange equations can be solved numerically by using the steepest
descent method (Kirk, 2004). Starting with an initial guess for the control trajec-
tory, the control can be updated iteratively by using the Hamiltonian gradient. At
each iteration, the co-state dynamics in equation (3.6) can be solved backward in
time and the control trajectory can be updated as follows:
u
k+1
(t)= u
k
(t) s
k

¶H
¶u
(t)

T
(3.8)
where u
k
is the current control trajectory and u
k+1
is the updated control tra-
jectory after the current iteration. The scalar s
k
is the step size for the current
iteration. It has to be noted that the convergence rate of the optimization pro-
cess is highly dependent on the proper selection of the step size at each iteration.
A linear search can be carried out in the direction of the Hamiltonian gradient
to determine the optimum step size for each iteration. The iterative process is
repeated until the decrease in the cost function J is practically insignificant.
26
3.2 Application to semiactive control of a SDOF
system
An application to a SDOF structural model is presented in this section. Control
design was performed using clipped LQR and the optimal control was determined
assuming the excitation was deterministic. The resulting control from clipped
LQR is compared to the truly optimal one for two ground motion records. The
SDOF structural model considered here is shown in Figure 3.1 with equation of
motion:
m ¨ q+ c ˙ q+ kq=m ¨ q
g
u (3.9)
where ¨ q
g
is the ground acceleration.
m
q
k, c
Damper
Fixed Base
Ground Motion
Figure 3.1: SDOF structure
The model parameters used for this structure are m = 100 Mg, k =
3:948 MN=m and c= 62:833 kNs=m, which results in natural frequency w =
27
2p rad=s and damping ratioz = 0:05. The resulting state vector is x=[q ˙ q]
T
and
the state space matrices are
A=
2
6
4
0 1
w
2
2zw
3
7
5
; B
w
=
8
>
<
>
:
0
1
9
>
=
>
;
; and B
u
=
8
>
<
>
:
0
1=m
9
>
=
>
;
:
First, semiactive control is designed using the clipped LQR strategy discussed
previously in §2.3. The objective of the control design is the minimization of the
absolute acceleration of the structure; this is accomplished by using the following
weighting matrices for LQR control design:
Q=
2
6
4
w
4
2zw
3
2zw
3
4z
2
w
2
3
7
5
; N=
8
>
<
>
:
w
2
=m
2zw=m
9
>
=
>
;
and R= m
2
which results inL (x;u;t)=( ¨ q+ ¨ q
g
)
2
.
In order to find the optimal semiactive control, the control force u can be
rewritten as a function of a normalized control input u as follows:
u(t)= c
n
u(t)H[u(t)] ˙ q(t) (3.10)
where c
n
is some nominal damping constant of the control device, chosen here to
be 10 kNs=m. The state space equation can then be rewritten as:
˙ x= f(x;u;t)= Ax(t)+ B(x;u)+ B
w
¨ q
g
(3.11)
28
where
A=
2
6
4
0 1

k
m

c
m
3
7
5
; B(x;u)=
8
>
<
>
:
0

1
m
c
n
uH[u] ˙ q
9
>
=
>
;
and B
w
=
8
>
<
>
:
0
1
9
>
=
>
;
The Lagrangian function for the considered control objective becomes:
L (x;u;t)=[ ¨ q(t)+ ¨ q
g
(t)]
2
=

1
m
fkq+(c+ c
n
uH[u]) ˙ qg

2
(3.12)
In order to optimize the control trajectory using the steepest descent method dis-
cussed in the previous section, the following partial derivatives are evaluated:
¶f(x;u;t)
¶x
= A+
2
6
4
0 0
0
1
m
c
n
uH[u]
3
7
5
;
¶f(x;u;t)
¶u
=
8
>
<
>
:
0

1
m
c
n
H[u] ˙ q
9
>
=
>
;
;

¶L (x;u;t)
¶x

T
=
1
m
2
8
>
<
>
:
2k[kq+(c+ c
n
uH[u]) ˙ q]
2(c+ c
n
uH[u])[kq+(c+ c
n
uH[u]) ˙ q]
9
>
=
>
;
and
¶L (x;u;t)
¶u
=
1
m
2
(2c
n
H[u] ˙ q)[kq+(c+ c
n
uH[u]) ˙ q]
The iterative optimization process can be carried out to find the optimal semi-
active control. At each iteration, the previous derivatives are evaluated and a
gradient descent is used to update the current control input as in equation (3.8).
In order to improve the convergence rate, a line search is used at each iteration
29
to select a suitable step size. After examining the iterations in several trials, the
search domain for the step size was selected to be[2
0
;2
5
]. The initial trajectory
for normalized control u was selected to be equal to 1, which is equivalent to
a passive damper whose damping constant is equal to c
n
. The two earthquakes
considered in this analysis are the 1940 El Centro (S00E component of the Impe-
rial Valley Irrigation District substation record in El Centro, CA, of the 18 May
1940 Imperial Valley earthquake, PGA 0.3484g, duration 30 secs, sampled at
50 Hz) and 1995 Kobe (N90E component of the Kobe University record of the
17 January 1995 Hyogo-ken Nanbu Kobe earthquake, PGA 0.3105g, duration
31:99 secs, sampled at 100 Hz) ground motions. Figure 3.2 represents the change
in the cost value against the number of iterations. It is evident that 50 iterations
was sufficient for both earthquakes to achieve convergence by minimizing the
cost.
The truly optimal semiactive control can now be compared to the clipped LQR
control. Figure 3.3 shows a comparison of the displacement response and control
forces for clipped LQR against the optimal semiactive control. It is clear that the
pier displacement response based on the optimal control trajectories is reduced
significantly compared to clipped LQR response, and that the control force is
significantly different as well.
Figure 3.4 compares a hysteretic force-displacement loop of clipped LQR and
the optimal over the time period between 5 and 7.6 secs chosen to better visualize
30
0 10 20 30 40 50
0
10
20
Iteration number
Cost J [m
2
/s
3
]
(a) El Centro 1940 earthquake
0 10 20 30 40 50
0
5
10
Iteration number
Cost J [m
2
/s
3
]
(b) Kobe 1995 earthquake
Figure 3.2: Convergence of the cost functions iterations
the results. It is clear from this comparison that clipped LQR caused signifi-
cant control force clipping that resulted in the displacement magnitude becoming
larger. In addition, the force-displacement loop for optimal control is closer to
that of a PNS.
Table 3.1 compares the peak responses, control forces and cost values for
clipped LQR and the optimal control. For the El Centro earthquake, the optimal
control trajectories reduced the peak displacement and absolute acceleration, rel-
ative to clipped LQR, by approximately 40% and reduced the cost by 46% while
using control forces with peak only higher by 6%. For the Kobe earthquake,
31
0 2 4 6 8 10
−5
0
5
10
Time [sec.]
Displacement q [cm]

Optimal
Clipped LQR
(a) Displacement response
0 2 4 6 8 10
−400
−200
0
200
Time [sec.]
Force u [kN]

Optimal
Clipped LQR
(b) Control force
0 2 4 6 8 10
−4
−2
0
2
Time [sec.]
Abs. acceleration a [m/s
2
]

Optimal
Clipped LQR
(c) Absolute acceleration
Figure 3.3: Comparison of clipped LQR and optimal control for El Centro earth-
quake
32
−4 −2 0 2 4
−200
−100
0
100
200
Displacement q [cm]
Control force u [kN]

Optimal
Clipped LQR
Figure 3.4: Control force vs. displacement hysteresis loop under the El Centro
earthquake excitation during the time range [5.0,7.6] secs
the optimal control reduced the peak displacement and absolute acceleration by
approximately 60% and reduced the cost by 51% with peak control force that is
also lower by 35%.
Table 3.1: Comparison of peak responses, control force and cost values for
clipped LQR and optimal control using historical earthquakes
1940 El Centro 1995 Kobe
CLQR Optimal D [%] CLQR Optimal D [%]
q
max
[cm] 5.58 3.39 39 6.77 2.54 62
˙ q
max
[cm/s] 39.82 27.29 31 34.08 15.72 53
¨ q
abs
max
[cm/s
2
] 221.3 135.9 39 268.5 107.8 59
u
max
=mg [%] 22.13 23.43 +6 27.25 17.55 35
cost [m
2
/s
3
] 4.761 2.550 46 3.761 1.822 51
Note: D is the percent change from Clipped LQR to the
truly optimal control; negative numbers are improvements.
The results of the optimal control strategy shows that the performance of semi-
active control can be significantly improved over that of clipped LQR. Adaptive
control techniques that take into account measurements of ground motion might
have a potential to improve the control performance significantly. The optimal
semiactive control presented in this chapter represents an upper bound on the
33
performance of an adaptive control strategy. Moreover, it will be shown in the
next chapter that the difference in control performance between clipped LQR and
the truly optimal control is not solely the result of prior knowledge of the excita-
tion, but also due to the fact that clipped LQR was not an optimal control law for
this problem.
34
Chapter 4
Design of Semiactive Structural
Control using Hybrid MPC
In this chapter, the design of semiactive structural control using hybrid MPC is
studied. First, the formulation of the method is presented, then numerical exam-
ples are studied to demonstrate the potential of the method to improve control
design.
4.1 Formulation
Generally, a discrete time form of a mixed logical dynamical system can be writ-
ten as (Bemporad and Morari, 1999):
x
k+1
=
˜
Ax
k
+
˜
B
u
u
k
+
˜
B
d d d
d d d
k
+
˜
B
v
v
k
(4.1a)
E
d d d
d d d
k
+ E
v
v
k
E
u
u
k
+ E
x
x
k
+ E
0
(4.1b)
where x
k
is the state vector at time kDt for system sampling time Dt and
˜
A=
e
ADt
and
˜
B
u
=
R
Dt
0
e
At
B
u
dt = A
1
(
˜
A I)B
u
are the (zero-order hold) discrete-
time analogues of A and B
u
. d d d
k
and v
k
are auxiliary vectors that are binary
35
indicator (i.e., 0 or 1) and real vector functions, respectively, of the states and
inputs, and are governed by inequality (4.1b). It will be subsequently shown
in the numerical example that the nonlinear passivity constraint of controllable
dampers described in inequality (2.6) can be transformed to linear inequalities
(4.1b). Note that, similar to the previous section, the certainty equivalence is used,
i.e., the optimal stochastic control can be obtained from a deterministic analysis
that assumes a non-zero initial condition and ignores the zero-mean excitation
(proof of the equivalence is shown in Appendix A).
An MPC scheme based on hybrid model equation (4.1), can be used to deter-
mine the optimal input for semiactive control devices (Giorgetti et al., 2005).
Such a method is denoted, herein, a hybrid MPC (HMPC). The objective is to
minimize a quadratic cost function similar to the one used in equation (2.3) for
LQR design; however, this proposed approach considers the hybrid system model
in order to account for the semiactive control device’s passivity constraint(s).
Since the hybrid system used here is in discrete form, the cost function must be
written, instead of an integral, as a summation over p steps:
J(x x x;x
0
), x
T
p
Q
p
x
p
+
p1
å
k=1

x
T
k
Qx
k
+ 2x
T
k
Nu
k
+ u
T
k
Ru
k

(4.2)
where x x x = [u
T
0
;d d d
T
0
;v
T
0
;:::;u
T
p1
;d d d
T
p1
;v
T
p1
]
T
is a vector that concatenates the
optimization parameters together, including both the control forces and the aux-
iliary variables at all time steps within the horizon (Giorgetti et al., 2005); and
36
x
0
is the current state vector of the system. The first term in equation (4.2) is the
terminal cost term, which is added to emulate an infinite horizon cost function,
where Q
p
is the solution of the Riccati equation
˜
A
T
Q
p
˜
A Q
p
(
˜
A
T
Q
p
˜
B
u
+ N)(
˜
B
T
u
Q
p
˜
B
u
+ R)
1
(
˜
B
T
Q
p
˜
A+ N
T
)+ Q= 0
that is associated with the discrete LQR problem that minimizes the infinite hori-
zon quadratic cost
min
K
LQR
¥
å
k=1

x
T
k
Qx
k
+ 2x
T
k
Nu
k
+ u
T
k
Ru
k

(4.3)
Note that a horizon of p= 1 step results, then, in a control identical to clipped
LQR.
Cost function (4.2), which is quadratic, must be minimized subject to the
linear equality in state equation (4.1a) and to inequality constraints (4.1b), written
in terms of both binary and real variables. The resulting optimization problem
is a MIQP problem that, when solved, will provide the optimal control input
(Bemporad and Morari, 1999).
37
4.2 Numerical Examples
In this section, two numerical examples demonstrate the design of semiactive
structural control using hybrid MPC and show its performance improvement rel-
ative to a clipped-optimal strategy. This section also describes an offline imple-
mentation of hybrid MPC based on linear interpolation and table lookup that is
shown to be more efficient for rapid simulations.
4.2.1 Single Degree of Freedom System
The same SDOF structural system studied in §3.2 is used here to demonstrate the
potential of hybrid MPC for optimal semiactive control design. Control strategies
for a controllable damper are designed using both clipped LQR and hybrid MPC;
their respective control performances are compared for various ground motion
records and for a Gaussian white noise excitation.
First, semiactive damping control is designed using the clipped LQR strategy
described in §2.3. The objective of the damping control design is to minimize
the mean square of the structure absolute acceleration ¨ q
abs
= ¨ q+ ¨ q
g
— i.e., a
serviceability objective — which requires minimizing equation (2.3) using the
weighting matrices:
Q=
2
6
4
w
4
2zw
3
2zw
3
4z
2
w
2
3
7
5
; N=
8
>
<
>
:
w
2
=m
2zw=m
9
>
=
>
;
; R= m
2
38
Second, the proposed hybrid MPC is used for damping control design, incor-
porating into the hybrid MPC formulation the passivity constraint of an ideal
controllable damper (which is not possible with clipped-optimal control). The
optimization converges faster with limits on the responses and the control force
(even if those limits are inactive at the optimal solution). Together, these con-
straints result in three pairs of inequality constraints. (1) The nonlinear pas-
sivity constraint in this SDOF case is u ˙ q 0. Using auxiliary binary variables
d
x
k
d
˙ q
k
= H[ ˙ q
k
] andd
u
k
= H[u
k
], the passivity constraint can be expressed as a
single equality constraint d
˙ q
k
=d
u
k
or, to be consistent with the inequality con-
straints (4.1b), as the pair of inequalitiesd
˙ q
k
d
u
k
andd
˙ q
k
d
u
k
. (2) A constraint
on the velocity can be expressed as a single nonlinear inequalityj ˙ q
k
j v
max
or,
amenable for use in inequality (4.1b), a pair of linear inequalities ˙ q
k
v
max
d
˙ q
k
(constraining ˙ q
k
when it is positive) and ˙ q
k
v
max
(1d
˙ q
k
) (when ˙ q
k
is nega-
tive) using the auxiliary binary variables. (3) A constraint on the control force
can be expressed similar to the velocity: nonlinearju
k
j u
max
or the linear pair
u
k
u
max
d
u
k
and u
k
u
max
(1d
u
k
). In this example, v
max
= 10 m=s and
u
max
=m= 100 m=s
2
(which are chosen so large that they are inactive, far from
39
active, for any reasonable ground motion for the examples herein). The resulting
MLD system, similar to equation (4.1), is then:
x
k+1
=
˜
Ax
k
+
˜
B
u
u
k
(4.4a)
E
d d d
d d d
k
E
u
u
k
+ E
x
x
k
+ E
0
(4.4b)
where
d d d
k
=
8
>
<
>
:
d
˙ x
k
d
u
k
9
>
=
>
;
2f0;1g
2
; E
d d d
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
v
max
0
v
max
0
0 u
max
0 u
max
1 1
1 1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
; E
u
=
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
0
0
1
1
0
0
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
;
E
x
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 1
0 1
0 0
0 0
0 0
0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
and E
0
=
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
0
v
max
0
u
max
0
0
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
:
40
To design this hybrid MPC damper control, an optimization problem as in
equation (4.2) must be solved subject to equality constraint (4.4a) and inequality
constraint (4.4b).
The hybrid system model of the SDOF structure was generated in MATLAB
®
using a toolbox called YALMIP (L¨ ofberg, 2004), which can be used for fast pro-
totyping of optimization problems, and is used here to solve the resulting MIQP
problems. This toolbox relies on one of several supported external solvers to per-
form the optimization. In this study, two different solvers were used and found to
give the same solution for the MIQP problems required for the examples herein:
IBM CPLEX (IBM, 2012) and the Gurobi Optimizer (Gurobi, 2012).
To select a suitable prediction horizon, a preliminary parametric study was
performed, using the 1994 Northridge earthquake (S48W Rinaldi receiving sta-
tion record of the 17 January 1994 Northridge earthquake, PGA 0:8252g, dura-
tion 19:9sec, sampled at 100Hz but decimated to 50Hz); the resulting cost with a
hybrid MPC designed using various prediction horizons p are shown in Table 4.1.
The prediction horizon is selected to be p= 30 steps (of Dt = 0:02 secs each),
which provides a good compromise between near-optimality and computational
expense.
The damping control forces, as functions of the state vector, are shown in Fig-
ure 4.1 for both clipped LQR and hybrid MPC. The two damping control force
functions obtained from clipped LQR and hybrid MPC show drastic differences.
41
Table 4.1: Minimum cost for various hybrid MPC prediction horizons
p Cost
1 41.49
†
5 44.19
10 40.55
15 33.17
20 31.49
25 31.22
30 31.01
40 30.93
50 30.84
†
= CLQR
The control law obtained from LQR is a linear function of the states as in equa-
tion (2.4), but is clipped when the linear control law commands nondissipative
forces, as shown in Figure 4.1a. On the other hand, the proposed hybrid MPC
damping force function in Figure 4.1b is nonlinear and more often dissipative.
The differences suggest that the hybrid MPC strategy uses the damping device
more efficiently; however, in order to quantitatively compare both strategies, the
controlled structure must be simulated with some excitation.
Simulated responses of (the continuous-time model of) the SDOF structure to
several historical earthquakes with both the clipped (continuous-time) LQR and
hybrid MPC damper force functions were performed using MATLAB’s ode45;
the damper force control laws are computed at each time point, the clipped LQR
using the LQR state feedback control law and then clipped to ensure dissipativ-
ity, and the hybrid MPC by minimizing predicted response cost (4.2) over the
42
−2
−1
0
1
2
−20
−10
0
10
20
−10
0
10
displ. q [cm]
veloc. v [cm/s]
control force u/mg [%]
(a) Clipped LQR
−2
−1
0
1
2
−20
−10
0
10
20
−10
0
10
displ. q [cm]
veloc. v [cm/s]
control force u/mg [%]
(b) Hybrid MPC
Figure 4.1: Comparison of control input for SDOF system
p= 30 future discrete time steps ofDt= 0:02 secs. The excitations used are sev-
eral historical earthquakes, including the 1940 El Centro (S00E component of the
43
Imperial Valley Irrigation District substation record in El Centro, CA, of the 18
May 1940 Imperial Valley earthquake, PGA 0.3484g, duration 30 secs, sampled
at 50 Hz) and 1995 Kobe (N90E component of the Kobe University record of the
17 January 1995 Hyogo-ken Nanbu Kobe earthquake, PGA 0.3105g, duration
31:99 secs, sampled at 100 Hz) ground motions, linearly interpolated to the time
points at which ode45 evaluates state equation (2.2). The state responses are
computed byode45 at everyDt = 0:02 secs and used to evaluate response met-
rics. Simulations usingode45 showed that its default tolerances (absolute 10
6
,
relative 10
3
) do not result in accurate evaluation of peak absolute accelerations.
A relative tolerance of 10
6
and an absolute tolerance of 10
8
were found to be
small enough that the peak absolute acceleration metric has converged (Appendix
B). Figure 4.2 shows the trajectories of the absolute accelerations and control
forces, for clipped LQR and hybrid MPC, during the first 10 seconds of the 1940
El Centro earthquake.
The simulation results are compared in Table 4.2, including: peak and root-
mean-square (RMS) displacement, velocity and absolute acceleration responses;
peak and RMS controllable damper forces; and the values of the objective func-
tion. The results show that the hybrid MPC strategy results in peak and RMS
responses, as well as cost values, that are all below that of clipped LQR, except
for a slight and momentary increase in peak acceleration for the 1940 El Centro
response (minimizing the mean square does not guarantee minimizing the peak).
Although the control was designed to minimize the RMS absolute acceleration,
44
the results show that hybrid MPC can achieve significant reduction, compared
to clipped LQR, in the displacements and velocities as well, which suggests that
the clipped LQR strategy is not using the damping device effectively. Further,
the proposed hybrid MPC accomplishes these performance improvements using
control forces that are well below those of the clipped LQR control.
These simulations illustrate the efficacy of the proposed hybrid MPC for
reducing structural response to historical earthquakes. It is also beneficial to
examine the control efficacy when the structure is subjected to stochastic seismic
excitation. Such loads are sometimes assumed to be wide-band stochastic pro-
cess; without loss of generality, the stochastic excitation is approximated here as
a Gaussian pulse process (i.e., discrete-time band-limited white noise). To quan-
tify the control strategies’ efficacy for stochastic seismic response mitigation, a
−2
−1
0
1
2
Abs. Acc. [m/s
2
]
Hybrid MPC
Clipped LQR
0 1 2 3 4 5 6 7 8 9 10
−200
−100
0
100
200
Time [sec]
Control Force [kN]
Figure 4.2: Absolute acceleration response and control force for the first 10 s of
the 1940 El Centro earthquake
45
Table 4.2: Comparison of peak and root-mean-square responses, control force
and cost values using historical earthquakes
1940 El Centro 1995 Kobe
CLQR HMPC D [%] CLQR HMPC D [%]
q
rms
[cm] 1.574 1.131 28:1 1.457 1.005 31:1
˙ q
rms
[cm/s] 7.527 5.863 22:1 6.762 4.920 27:2
¨ q
abs
rms
[cm/s
2
] 39.66 37.02 6:7 35.64 30.50 14:4
u
rms
=mg [%] 4.805 3.933 18:2 4.536 3.524 22:3
q
max
[cm] 5.549 4.293 22:6 6.747 4.726 29:9
˙ q
max
[cm/s] 41.04 34.00 17:1 34.05 24.10 29:2
¨ q
abs
max
[cm/s
2
] 221.1 240.1 +8:6 267.7 265.5 0:8
u
max
=mg [%] 21.91 16.95 22:6 26.64 18.66 29:9
cost [m
2
/s
3
] 4.719 4.112 12:9 3.812 2.791 26:8
Note: D is the percent change from Clipped LQR to Hybrid MPC;
negative numbers are improvements.
Monte Carlo simulation is carried out where the controlled SDOF model was sub-
jected to 8 sec samples of zero-mean Gaussian white noise excitation, which was
a sufficient duration for the system statistics to reach a (near) stationary response.
While hybrid MPC can be implemented in real-time for the SDOF model, the
online implementation is not computationally practical for carrying out a Monte
Carlo simulation with 1 million realizations. To make Monte Carlo simulation
less computationally demanding, an offline implementation of hybrid MPC is
employed, which speeds up the computation by two orders of magnitude. The
implementation herein uses table look-up and linear interpolation and exploits
the property that the nonlinear control input in Figure 4.1b is a homogeneous
46
function of order 1 (i.e., it scales linearly when the state vector scales) and, hence,
can be written using radial coordinates of the form:
u
HMPC
(q; ˙ q)= u(r;q)= mr u(q); where q= r cosq and ˙ q= r sinq (4.5)
The nonlinear function u(q), which is mass-normalized and shown in Figure 4.3,
can be determined a priori (before simulating system response), so that most of
the computational effort is done offline and only limited computations are per-
formed during the simulations. The offline implementation used here is based on
a one-dimensional table lookup for the function u(q). This function is calculated
over a discretization grid of the angleq with a step size equal to 0:25

. The com-
putation time required to prepare this discretized function is 65.2 seconds and it
requires 8.04 KB to store the resulting data in memory, so that it can be used later
for table lookup and interpolation. The computing platform used herein is a Dell
inspiron 518 workstation, with an Intel quad core processor (Q8200) clocked at
2.33 GHz and 4 GB of RAM running Matlab R2011b on a 64-bit Windows 7
operating system.
Table 4.3 compares the results of the offline and online implementations of
HMPC. The offline implementation deviates slightly from the online due to the
linear table lookup interpolation; yet, the results still maintain significant superi-
ority over clipped LQR. Further, the offline implementation is significantly faster:
online implementation, which varies in its computational requirements as each
47
−180 −90 0 90 180
−30
−15
0
15
30
θ[
◦
]
u(θ)
Figure 4.3: Hybrid MPC control input for unit magnitude state vector
timestep’s optimization can take fewer or more iterations, takes 1–2.5 days; the
offline implementation reduces the simulation computation time down to about a
second, giving a speed-up factor of five orders of magnitudes for the two histori-
cal earthquake simulations. This indicates that the offline implementation might
be more suitable for computationally-efficient Monte Carlo simulation.
Table 4.3: Comparison of peak and root-mean-square responses, control force
and cost values using historical earthquakes for online and offline HMPC
1940 El Centro 1995 Kobe
online offline ratio online offline ratio
q
rms
[cm] 1.131 1.130 1:001 1.005 1.005 1:000
˙ q
rms
[cm/s] 5.863 5.864 1:000 4.920 4.923 0:999
¨ q
abs
rms
[cm/s
2
] 37.02 36.97 1:001 30.50 30.31 1:006
u
rms
=mg [%] 3.933 3.918 1:004 3.524 3.488 1:010
q
max
[cm] 4.293 4.282 1:003 4.726 4.720 1:001
˙ q
max
[cm/s] 34.00 34.04 0:999 24.10 24.12 0:999
¨ q
abs
max
[cm/s
2
] 240.1 236.6 1:015 265.5 261.4 1:016
u
max
=mg [%] 16.95 16.91 1:003 18.66 18.64 1:001
cost [m
2
/s
3
] 4.112 4.102 1:003 2.791 2.757 1:012
computation time [sec] 82568 1.009 81838 212224 1.019 208274
48
In addition to the idealized model portrayed in Figure 2.4 for the semiac-
tive device, another idealized model, shown in Figure 4.4, is also considered in
the Monte Carlo simulations. This model considers a minimum and maximum
energy dissipation in the damping device, which is a more realistic representa-
tion of a smart damper (e.g., magnetorheological fluid damper), and is helpful in
this study to illustrate the inconsistency in the clipped LQR design of semiac-
tive control. The model in Figure 2.4 is subsequently denoted “idealized model
I” and the model in Figure 4.4 is denoted “idealized model II.” For idealized
model II, it is assumed the maximum and minimum damping coefficients are
c
max
= 10
4
kN s=m and c
min
= 1 kN s=m, based on the experimental results for
the magnetorheological fluid damper in Spencer et al. (1997); these constraints
are implemented in a second hybrid MPC optimization using the two inequality
constraintsjuj> c
min
j ˙ qj andjuj< c
max
j ˙ qj.
Figure 4.4: More realistic model for passivity constraint
49
0 1 2 3 4
0
0.5
1
1.5
2
Time [sec]
E[q
2
] (cm)
2

Clipped LQR Hybrid MPC
(a) Mean square displacment
0 1 2 3 4
0
500
1000
1500
Time [sec]
E[¨ q
2
abs
] (cm/s
2
)
2

Clipped LQR Hybrid MPC
(b) Mean square absolute acceleration
0 1 2 3 4
0
10
20
30
40
Time [sec]
E[˙ q
2
] (cm/s)
2

Clipped LQR Hybrid MPC
(c) Mean square velocity
0 1 2 3 4
0
0.05
0.1
0.15
0.2
Time [sec]
E[(u/m)
2
] (m/s
2
)
2

Clipped LQR
Hybrid MPC
(d) Mean square normalized control
Figure 4.5: Mean square responses of MCS for SDOF system
50
The response statistics from Monte Carlo simulation for idealized model I
(Table 4.4a) show that hybrid MPC achieves an 11% reduction in the stationary
RMS absolute acceleration compared to clipped LQR. In addition, hybrid MPC
reduces RMS displacement by 24%, compared to clipped LQR. Not only does the
hybrid MPC strategy reduce the structural response, but it does so with a RMS
device force that is smaller by 15%, which suggests that hybrid MPC is, in fact,
making a more efficient use of the damping device. Results from idealized model
II (Table 4.4b) show a similar pattern in the achievable reduction of response:
compared to clipped LQR, hybrid MPC reduces RMS displacements and abso-
lute accelerations by 30% and 14%, respectively, while using a comparable RMS
damping force. Table 4.4b also shows the results for a passive-on case, which
assumes the damping device is a viscous damper with a damping constant c
max
;
this represents a semiactive device being operated at its maximum dissipative
power. Although the passive-on case is more efficient in reducing displacement
compared to the two semiactive control strategies, it must be noted that (a) the
main objective used in the control design in this example is the reduction of abso-
lute accelerations and (b) that the passive-on case uses RMS force that is 33%
larger. One of the most important observations in Table 4.4b is the fact that the
passive-on case performs better than clipped LQR strategy in reducing the abso-
lute acceleration. In fact, this is the main reason why the design of semiactive
control using clipped optimal strategy is performed using heuristic approaches
involving large parametric studies, because the clipped optimal method is not
51
consistent in minimizing the cost functions and the control optimality is not guar-
anteed for general performance objectives.
Table 4.4: Root-mean-square response using Monte Carlo simulation with 1 mil-
lion realizations
(a) Idealized model I
response CLQR HMPC D [%]
¨ q
abs
[cm/s
2
] 36.88 32.68 11
u=mg [%] 4.078 3.472 15
q [cm] 1.374 1.041 24
˙ q [cm/s] 5.679 4.312 24
(b) Idealized model II
response Passive-on CLQR HMPC D [%]
¨ q
abs
[cm/s
2
] 37.79 38.95 33.56 14
u=mg [%] 3.109 2.302 2.325 +1
q [cm] 0.488 1.215 0.849 30
˙ q [cm/s] 3.057 6.369 4.524 29
4.2.2 Two Degrees of Freedom System
A structural model with two degrees of freedom is considered for control design
as shown in Figure 4.6. The structure has a damping device between the first
mass and the ground. The model parameters used for this structure are m
1
= m
2
=
100 Mg, k
1
= k
2
= 15:79173 MN=m, c
1
= 112:5 kN s=m and c
2
= 0 — which
results in natural frequenciesw
1
u 7:769 rad=s andw
2
u 20:326 rad=s (approxi-
mately 1.237 and 3.235 Hz, respectively) and damping ratiosz
1
=z
2
u 0:02.
52
m
1
m
2
q
1
q
2
k
1
, c
1
k
2
, c
2
Damper
Fixed Base
Ground Motion
Figure 4.6: 2DOF structure
The objective of the control design in this problem is the minimization of the
following cost function:
J=
Z
¥
0
h
( ¨ q
abs
1
)
2
+( ¨ q
abs
2
)
2
+ ru
2
i
dt (4.6)
where r, the relative weight for the damper force term with respect to the absolute
acceleration terms, is chosen here to be 10
16
kg
2
. Cost function (4.6) can also
be written in matrix form in terms of the state vector and the control input as in
equation (2.3).
Control was designed using both clipped LQR and (usingDt= 0:025s) hybrid
MPC, and was found again to be significantly different. Since the control laws
53
are now functions of 4 states, they cannot be easily illustrated graphically; how-
ever, the control functions can be compared by taking different sections that slice
through them. In Figure 4.7, the control forces at q= 0 are shown as surface
functions of v= ˙ q. It is evident from the two surfaces that hybrid MPC produces
control forces that are clipped to zero in a smaller area of the(v
1
;v
2
) space than
with clipped LQR: roughly about 14% of this slice for hybrid MPC but two-and-
half times as likely with clipped LQR.
The control law obtained from hybrid MPC for this structural model is non-
linear; however, the nonlinearity is perhaps more obvious in slices that do not go
through the origin. In Figure 4.8, sections of the control laws at q=[0 1 cm]
T
are shown for clipped LQR and hybrid MPC. By comparing these sections, it is
evident that the control law obtained from hybrid MPC commands forces with
higher dissipativity and is a nonlinear function of the state vector, rather than a
clipped linear function as in the case of clipped LQR.
54
−15
−7.5
0
7.5
15
−10
−5
0
5
10
−200
−100
0
100
200
2
nd
floor veloc. v
2
[cm/s]
1
st
floor veloc. v
1
[cm/s]
control force u [kN]
(a) Clipped LQR
−15
−7.5
0
7.5
15
−10
−5
0
5
10
−200
−100
0
100
200
2
nd
floor veloc. v
2
[cm/s]
1
st
floor veloc. v
1
[cm/s]
control force u [kN]
(b) Hybrid MPC
Figure 4.7: 2DOF system control force at q= 0 as a function of velocity vector
v= ˙ q
55
−10
−5
0
5
10
−10
−5
0
5
10
−400
−300
−200
−100
0
100
200
veloc.v
2
[cm/s]
veloc.v
1
[cm/s]
e c r o f l o r t n o c u [kN]
(a) Clipped LQR
−10
−5
0
5
10
−10
−5
0
5
10
−400
−300
−200
−100
0
100
200
veloc.v
2
[cm/s]
veloc.v
1
[cm/s]
e c r o f l o r t n o c u [kN]
(b) Hybrid MPC
Figure 4.8: 2DOF system control force at q = [0 1 cm]
T
as a function of
velocity vector v= ˙ q
56
The 2DOF structure with a controllable damper was simulated using MAT-
LAB’s Simulink
®
subject to the 1940 El Centro and 1995 Kobe earthquake ground
motion records; the resulting performance metrics are given in Table 4.5. It is evi-
dent from these results that hybrid MPC damper control design is more effective
in reducing the peak and RMS structural responses and the cost function for both
earthquakes: the physical response metrics are decreased by hybrid MPC by 23–
51% relative to the clipped LQR for the El Centro earthquake, and by 3–55% for
the Kobe earthquake. Further, hybrid MPC accomplished these gains while using
peak control forces that are 49–52% smaller than those of clipped LQR and RMS
control forces that are 11–17% smaller than clipped LQR, clearly demonstrating
a more efficient use of the damping device.
Figure 4.9 shows the the hysteretic force-displacement behavior of the con-
trollable damper during the Kobe earthquake in the period between 4.6 and 5.6
secs. Hybrid MPC more frequently commands dissipative forces, resulting in
less clipping, thereby reducing the peak displacement (and the peak responses in
general) as well as the magnitude of the required control forces. It is interesting
to note the pseudo-negative stiffness (Iemura and Pradono, 2003) hysteresis loop
that, combined with the elastic forces produced in the structure, results in very
effective energy dissipation.
An offline implementation of hybrid MPC can also be used to speed up the
computations for Monte Carlo simulation. The control input is a function of the 4
states. By exploiting the homogeneity property, this function can be implemented
57
−4 −2 0 2 4 6
−1000
−800
−600
−400
−200
0
200
400
Displacement q
1
[cm]
Control force u [kN]

Clipped LQR
Hybrid MPC
Figure 4.9: Control force vs. displacement hysteretic loop during time range
[4.6,5.6] secs of the Kobe earthquake
using 3D table lookup. This can be illustrated by rewriting the control law using
spherical coordinates as follows:
u
HMPC
(x)= u(r;a;b;g)= r u(a;b;g) (4.7a)
where r=kxk
2
=
q
q
2
1
+ q
2
2
+ ˙ q
2
1
+ ˙ q
2
2
; (4.7b)
a = tan
1

q
2
q
1

2[p;p]; (4.7c)
b = tan
1
0
@
˙ q
1
q
q
2
1
+ q
2
2
1
A
2
h

p
2
;
p
2
i
; and (4.7d)
g = tan
1
0
@
˙ q
2
q
q
2
1
+ q
2
2
+ ˙ q
2
1
1
A
2
h

p
2
;
p
2
i
; (4.7e)
58
Table 4.5: Comparison of peak and root-mean-square responses, control force
and cost values for the 2DOF structure using historical earthquakes
1940 El Centro 1995 Kobe
CLQR HMPC D [%] CLQR HMPC D [%]
(q
1
)
rms
[cm] 1.533 0.891 41:9 1.116 0.712 36:2
(q
2
)
rms
[cm] 2.169 1.248 42:5 1.524 0.975 36:0
(q
2
q
1
)
rms
[cm] 0.722 0.468 35:2 0.510 0.350 31:4
( ¨ q
abs
1
)
rms
[cm/s
2
] 107.7 52.59 51:2 75.41 42.96 43:0
( ¨ q
abs
2
)
rms
[cm/s
2
] 114.0 73.92 35:2 80.61 55.32 31:4
u
rms
[kN] 107.1 95.50 10:8 90.70 75.46 16:8
(q
1
)
max
[cm] 5.834 3.359 42:4 5.529 3.516 36:4
(q
2
)
max
[cm] 9.739 5.086 47:8 9.356 5.063 45:9
(q
2
q
1
)
max
[cm] 3.906 2.478 36:6 3.827 1.697 55:7
( ¨ q
abs
1
)
max
[cm/s
2
] 630.8 483.3 23:4 528.7 510.5 3:44
( ¨ q
abs
2
)
max
[cm/s
2
] 616.8 391.4 36:5 604.4 268.0 55:7
u
max
[kN] 987.6 476.2 51:8 887.1 457.0 48:5
cost [m
2
/s
3
] 73.76 24.69 66:5 36.75 14.72 59:9
Note: D is the percent change from Clipped LQR to
Hybrid MPC; negative numbers are improvements.
where u(a;b;g) is the optimal control input using hybrid MPC for a state vector
on the unit hypersphere in the state space defined by the three anglesa,b andg.
The function u(a;b;g) is calculated over a discrete grid of the angles a, b and
g with step sizes equal to 4° and 8°; the resulting implementations are denoted
herein as offline-4 and offline-8, respectively. For offline-4, the computation time
to prepare this discretized function is 348 minutes; it requires 815 KB to store the
resulting data in memory. For offline-8, the computation time required to prepare
59
Table 4.6: Comparison between online and offline HMPC for 1940 El Centro and
1995 Kobe earthquakes
1940 El Centro 1995 Kobe
Online Offline-4 Offline-8 Online Offline-4 Offline-8
(q
1
)
rms
[cm] 0.89 0.89 (1.00) 0.87 (1.03) 0.71 0.71 (1.00) 0.69 (1.04)
(q
2
)
rms
[cm] 1.25 1.25 (1.00) 1.23 (0.98) 0.98 0.98 (1.00) 0.95 (1.03)
(q
2
q
1
)
rms
[cm] 0.47 0.47 (1.00) 0.47 (0.99) 0.35 0.35 (1.00) 0.35 (1.00)
( ¨ q
abs
1
)
rms
[cm/s
2
] 52.6 52.0 (1.01) 51.3 (1.03) 43.0 42.5 (1.01) 41.7 (1.03)
( ¨ q
abs
2
)
rms
[cm/s
2
] 73.9 74.2 (1.00) 74.9 (0.99) 55.3 55.5 (1.00) 55.5 (1.00)
u
rms
[kN] 95.5 95.0 (1.01) 92.9 (1.03) 75.5 74.9 (1.01) 71.0 (1.06)
(q
1
)
max
[cm] 3.36 3.34 (1.01) 3.26 (1.03) 3.52 3.54 (0.99) 3.42 (1.03)
(q
2
)
max
[cm] 5.09 5.13 (0.99) 5.17 (0.99) 5.06 5.08 (1.00) 4.95 (1.02)
(q
2
q
1
)
max
[cm] 2.48 2.49 (0.99) 2.58 (0.96) 1.70 1.68 (1.01) 1.69 (1.01)
( ¨ q
abs
1
)
max
[cm/s
2
] 483 343 (1.41) 322 (1.50) 510 429 (1.19) 430 (1.19)
( ¨ q
abs
2
)
max
[cm/s
2
] 391 394 (0.99) 407 (0.96) 268 265 (1.01) 267 (1.00)
u
max
[kN] 476 475 (1.00) 476 (1.00) 457 449 (1.02) 427 (1.07)
cost [m
2
/s
3
] 24.7 24.6 (1.00) 24.7 (1.00) 14.7 14.7 (1.00) 14.5 (1.01)
computation time [sec] 46880 206 (228) 196 (239) 39674 250 (159) 227 (175)
Note: values in parentheses represent the ratio of online HMPC value to offline HMPC
value.
this discretized function is 46 minutes; it requires 387 KB to store the resulting
data in memory.
Table 4.6 compares the results of the offline implementations of HMPC to
the online implementation. Offline-4 and offline-8 implementations provide very
similar performance and still maintain the superiority of HMPC over clipped
LQR design. In addition, it is clear that the offline implementation was sig-
nificantly faster, giving a speed-up factor of 228 for the El Centro earthquake
simulation and 159 for the Kobe earthquake.
60
The hybrid MPC also provides significant response reductions for the 2DOF
structure excited by a discrete-time approximation of Gaussian white noise;
a Monte Carlo simulation with 40,000 realization is used to compute RMS
responses and control forces as shown in Table 4.7. Simulations were carried out
using 8 sec samples of zero-mean Gaussian white noise excitations, which was a
sufficient duration for the system statistics to reach a (near) stationary response,
as shown in Figure 4.10. The hybrid MPC strategy reduces the RMS absolute
accelerations by a significant 25–30% compared to clipped LQR, reduces the
RMS displacements of both masses by 32% and the velocities by 32–37%, and
reduces the cost function by 10% — all while using a control force with RMS
27% lower than that with clipped LQR.
Table 4.7: Root-mean-square responses of the 2DOF system using Monte Carlo
simulation with 40,000 samples
q
1
q
2
˙ q
1
˙ q
2
¨ q
abs
1
¨ q
abs
2
cost u
[cm] [cm] [cm=s] [cm=s] [cm=s
2
] [cm=s
2
] [m=s
2
] [kN]
Clipped LQR 2.462 3.477 16.76 23.99 176.33 183.53 2.537 186.6
Hybrid MPC 1.789 2.517 11.45 17.44 134.35 144.71 1.976 171.9
D [%] –28 –28 –32 –28 –24 –21 –23 –8
Note:D is the percent change from Clipped LQR to Hybrid MPC; negative
numbers are improvements.
4.3 Conclusions
This chapter investigated the design of semiactive structural control using hybrid
MPC, a method capable of incorporating into the design process the inherent
passivity of semiactive devices. The performance of the resulting control design
61
0 2 4 6 8
0
2
4
6
8
Time [sec]
E[q
2
1
] (cm)
2

Clipped LQR Hybrid MPC
(a) Mean square displacment q
1
0 2 4 6 8
0
5
10
15
Time [sec]
E[q
2
2
] (cm)
2

Clipped LQR Hybrid MPC
(b) Mean square displacment q
2
0 2 4 6 8
0
1
2
3
4
Time [sec]
E[a
1
] (m/s
2
)
2

Clipped LQR Hybrid MPC
(c) Mean square absolute acceleration ¨ q
abs
1
62
0 2 4 6 8
0
1
2
3
4
Time [sec]
E[a
2
] (m/s
2
)
2

Clipped LQR Hybrid MPC
(d) Mean square absolute acceleration ¨ q
abs
2
0 2 4 6 8
0
2
4
6
8
Time [sec]
E[cost] (cm/s
2
)
2

Clipped LQR Hybrid MPC
(e) Mean cost
0 2 4 6 8
0
1
2
3
4
x 10
4
Time [sec]
E[u
2
] (kN)
2

Clipped LQR Hybrid MPC
(f) Mean square control force
Figure 4.10: Mean square responses of MCS for 2DOF system
63
for controllable dampers was compared to clipped-optimal control design for both
SDOF and 2DOF structural models. Based on the numerical results, the following
conclusions can be drawn:
1. Clipped-optimal control strategies are based on unconstrained force models,
which might not provide an optimal control performance for semiactive
structural control.
2. The nonlinear passivity constraints that characterize controllable dampers
can be taken into account by using a hybrid system model, resulting in a set
of linear constraints that makes optimization more tractable, which can be
employed by MPC for control design.
3. The optimal control laws for semiactive structural control are generally non-
linear and can achieve higher performance compared to clipped-optimal
control strategies for some control objectives.
4. Although better semiactive control designs can be achieved using hybrid
MPC, the method is computationally intensive if an online implementation
is used because MIQP problems must be solved during the sampling time
of the system. Offline implementations can be more practical for larger
structural systems, where most of the computations can be carried out a
priori, resulting in less demanding computational requirements during the
sampling time.
64
5. Further research is required to facilitate the application of hybrid MPC for
more complex structural models. Due to the nonlinearity of the optimal
control laws, learning-based control techniques might provide a suitable
approach for implementing control laws derived from hybrid MPC in larger
structural models.
6. For simple structural models, offline implementation of hybrid MPC can
achieve two orders of magnitude speed-up compared to the online one.
65
Chapter 5
Robustness of Controllable Damper
Feedback Strategies
5.1 Introduction
The models used in Chapter 4 assume that the systems parameters (mass, stiffness
and damping) are deterministic and are known a priori. In real structures, this is
generally not the case, and those parameters are uncertain (Spencer et al., 1994).
For instance, the mass of a real building may vary if the building use is changed.
Similarly, the stiffness of a real structure may vary over time due to strength
degradation or environmental conditions. Damping behaviors of real structures
are often very complex, as well as difficult to estimate; this usually results in
large uncertainties in any damping estimates. Given the complex nonlinear nature
of the control laws obtained by hybrid MPC, it is possible that those control
laws are fine-tuned to maximize the control performance for the specific system
parameters used to obtain them, but their performance may degrade if the system
parameters differ from the assumptions. In this chapter, the effect of uncertainties
66
in the system parameters on the control performance is investigated, for both the
clipped LQR and hybrid MPC control designs.
5.2 Robustness Study of the SDOF System
In order to study the robustness of control designs to changes in the system
parameters, it is assumed that a structure with nominal system parameters (m
0
,
k
0
and c
0
) is used to design the control system, as shown in Figure 5.1. For the
SDOF system used herein, the same system parameters used in Chapter 4 are con-
sidered to represent the nominal system. Two different systems with perturbed
parameters are used in this study: one with its stiffness perturbed byDk (Figure
5.2), and the other with its damping constant perturbed by Dc (Figure 5.3). To
compare the robustness of control designs when the system parameters changes,
the perturbed systems are used to simulate the system responses using control
design obtained from the nominal system. The magnitude of perturbations used
in the study are in the range of20% for stiffness perturbations and40% for the
damping coefficient perturbations. A response reduction ratio for some response
metric p is defined by:
Reduction Ratio r=
p
CLQR
p
HMPC
p
CLQR
67
The reduction ratio represents the relative of reduction of the system response
when hybrid MPC control is used instead of clipped LQR control design. If
this ratio is positive, then hybrid MPC control design is capable of reducing the
response. However, if the ratio is negative, then hybrid MPC control was less
effective in reducing the response compared to clipped LQR design.
Figure 5.1: Nominal SDOF system used to design control
Figure 5.2: SDOF system with perturbed stiffness
68
Figure 5.3: SDOF system with perturbed damping
Simulations of the systems with perturbed stiffness and damping were per-
formed for clipped LQR and hybrid MPC using both El Centro and Kobe earth-
quakes. Results for systems with perturbed stiffness are presented in Tables 5.1–
5.4 for both clipped LQR and hybrid MPC; results for perturbed damping are
presented in Tables 5.5–5.8.
Table 5.1: Response of the SDOF system with a perturbed stiffness when sub-
jected to El Centro earthquake using the clipped LQR control design
Dk=k –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
q
max
[cm] 6.603 6.227 5.320 5.463 5.549 5.589 5.592 5.661 5.863
˙ q
max
[cm/s] 34.26 34.96 36.30 38.70 41.04 43.37 45.65 47.90 49.94
¨ q
abs
max
[cm/s
2
] 210.4 210.4 190.8 207.0 221.1 233.4 243.9 258.0 278.5
u
max
=mg [%] 26.07 24.58 21.00 21.57 21.91 22.06 22.08 22.35 23.15
q
rms
[cm] 2.107 1.649 1.636 1.619 1.574 1.544 1.536 1.540 1.537
˙ q
rms
[cm/s] 6.626 6.862 7.061 7.274 7.527 7.901 8.209 8.547 8.865
¨ q
abs
rms
[cm/s
2
] 38.88 36.89 37.81 38.50 39.66 41.75 44.34 46.98 50.11
u
rms
=mg [%] 7.028 4.989 4.978 4.980 4.805 4.645 4.561 4.544 4.464
cost [m
2
/s
3
] 4.534 4.083 4.288 4.446 4.719 5.229 5.897 6.621 7.532
69
Table 5.2: Response of the SDOF system with a perturbed stiffness when sub-
jected to El Centro earthquake using the hybrid MPC control design
Dk=k –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
q
max
[cm] 4.610 4.724 4.632 4.478 4.282 4.050 3.853 3.747 3.774
˙ q
max
[cm/s] 30.36 31.14 32.04 33.01 34.04 35.11 36.15 37.17 38.03
¨ q
abs
max
[cm/s
2
] 213.5 236.2 239.1 240.5 236.6 231.4 226.5 232.0 240.9
u
max
=mg [%] 18.20 18.65 18.29 17.68 16.91 17.06 17.15 17.48 17.66
q
rms
[cm] 1.275 1.222 1.177 1.150 1.130 1.106 1.091 1.077 1.066
˙ q
rms
[cm/s] 5.495 5.566 5.641 5.750 5.864 5.969 6.076 6.182 6.285
¨ q
abs
rms
[cm/s
2
] 34.07 34.63 35.29 36.01 36.98 37.98 39.07 40.24 41.40
u
rms
=mg [%] 4.271 4.139 4.017 3.961 3.919 3.859 3.830 3.803 3.793
cost [m
2
/s
3
] 3.482 3.598 3.735 3.891 4.102 4.328 4.580 4.859 5.142
Table 5.3: Response of the SDOF system with a perturbed stiffness when sub-
jected to Kobe earthquake using the clipped LQR control design
Dk=k –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
q
max
[cm] 6.557 6.804 6.863 6.825 6.747 6.659 6.583 6.509 6.443
˙ q
max
[cm/s] 29.58 31.07 32.39 33.31 34.05 35.04 36.59 37.62 38.90
¨ q
abs
max
[cm/s
2
] 209.3 230.2 245.5 257.5 267.7 277.3 287.0 296.5 306.2
u
max
=mg [%] 25.89 26.86 27.09 26.94 26.64 26.29 25.99 25.70 25.44
q
rms
[cm] 1.669 1.552 1.458 1.423 1.457 1.514 1.549 1.530 1.467
˙ q
rms
[cm/s] 5.171 5.417 5.761 6.144 6.762 7.167 7.416 7.437 7.339
¨ q
abs
rms
[cm/s
2
] 31.68 30.04 29.78 31.55 35.64 39.29 42.75 45.49 46.63
u
rms
=mg [%] 5.483 5.119 4.767 4.565 4.536 4.681 4.745 4.603 4.350
cost [m
2
/s
3
] 3.011 2.708 2.661 2.986 3.812 4.632 5.484 6.208 6.522
In order to compare the results, the reduction ratios, computed from the tabu-
lar data, are presented in Figures 5.4–5.11. For El Centro earthquake simulation
of the system with perturbed stiffness, the reduction ratios for the peak response
and control force are shown in Figure 5.4, while the reduction ratios of RMS
responses, RMS control force and cost are presented in Figure 5.5. Except for the
peak absolute acceleration, all other response metrics maintain positive reduction
70
Table 5.4: Response of the SDOF system with a perturbed stiffness when sub-
jected to Kobe earthquake using the hybrid MPC control design
Dk=k –0.20 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 0.20
q
max
[cm] 4.509 4.628 4.590 4.693 4.720 4.702 4.649 4.572 4.490
˙ q
max
[cm/s] 22.25 22.00 22.45 23.35 24.12 24.83 26.06 26.93 27.50
¨ q
abs
max
[cm/s
2
] 215.7 230.1 235.5 251.6 261.4 268.2 275.2 280.0 284.0
u
max
=mg [%] 17.80 18.27 18.12 18.53 18.64 18.56 18.36 18.05 17.73
q
rms
[cm] 1.027 1.021 1.012 1.007 1.005 1.007 1.005 0.979 0.963
˙ q
rms
[cm/s] 4.340 4.545 4.681 4.808 4.923 5.025 5.085 5.060 5.056
¨ q
abs
rms
[cm/s
2
] 25.76 26.57 27.62 28.90 30.31 31.89 33.47 34.75 35.83
u
rms
=mg [%] 3.424 3.442 3.445 3.464 3.488 3.519 3.513 3.396 3.329
cost [m
2
/s
3
] 1.990 2.118 2.289 2.505 2.757 3.051 3.361 3.623 3.851
Table 5.5: Response of the SDOF system with a perturbed damping when sub-
jected to El Centro earthquake using the clipped LQR control design
Dc=c –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
q
max
[cm] 5.848 5.771 5.695 5.622 5.549 5.478 5.408 5.340 5.273
˙ q
max
[cm/s] 41.46 41.35 41.25 41.14 41.04 40.93 40.83 40.72 40.62
¨ q
abs
max
[cm/s
2
] 231.6 228.9 226.2 223.6 221.1 218.7 216.2 213.8 211.4
u
max
=mg [%] 23.09 22.78 22.49 22.19 21.91 21.63 21.35 21.08 20.82
q
rms
[cm] 1.648 1.629 1.610 1.592 1.574 1.556 1.538 1.521 1.504
˙ q
rms
[cm/s] 7.792 7.724 7.657 7.591 7.527 7.464 7.402 7.343 7.285
¨ q
abs
rms
[cm/s
2
] 40.33 40.12 39.98 39.78 39.66 39.51 39.38 39.25 39.15
u
rms
=mg [%] 5.055 4.993 4.929 4.869 4.805 4.745 4.685 4.628 4.568
cost [m
2
/s
3
] 4.879 4.829 4.796 4.748 4.719 4.682 4.653 4.622 4.599
ratios over the range of perturbations studied, indicating hybrid MPC provides
a more efficient control design even for systems with perturbed stiffness. It can
also be noted that the curve representing the reduction ratio of the cost is upward
concave, which means that not only does hybrid MPC maintain the cost reduc-
tion, but it does in fact provide better cost reduction for the perturbed system
compared to the nominal system. Similarly, the reduction ratios for the structure
71
Table 5.6: Response of the SDOF system with a perturbed damping when sub-
jected to El Centro earthquake using the hybrid MPC control design
Dc=c –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
q
max
[cm] 4.402 4.371 4.346 4.316 4.282 4.250 4.216 4.186 4.160
˙ q
max
[cm/s] 34.51 34.39 34.27 34.16 34.04 33.92 33.81 33.68 33.56
¨ q
abs
max
[cm/s
2
] 240.9 241.9 239.7 238.6 236.6 235.4 233.6 231.5 232.3
u
max
=mg [%] 17.38 17.26 17.16 17.04 16.91 16.78 16.65 16.53 16.42
q
rms
[cm] 1.165 1.157 1.149 1.140 1.130 1.121 1.111 1.107 1.100
˙ q
rms
[cm/s] 6.001 5.966 5.934 5.899 5.864 5.828 5.793 5.769 5.740
¨ q
abs
rms
[cm/s
2
] 36.79 36.88 36.95 36.98 36.98 37.01 37.03 37.16 37.20
u
rms
=mg [%] 4.039 4.011 3.984 3.952 3.919 3.887 3.854 3.838 3.813
cost [m
2
/s
3
] 4.060 4.081 4.095 4.102 4.102 4.109 4.114 4.142 4.152
Table 5.7: Response of the SDOF system with a perturbed damping when sub-
jected to Kobe earthquake using the clipped LQR control design
Dc=c –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
q
max
[cm] 7.099 7.008 6.920 6.832 6.747 6.662 6.579 6.497 6.417
˙ q
max
[cm/s] 35.32 35.00 34.68 34.36 34.05 33.75 33.44 33.14 32.84
¨ q
abs
max
[cm/s
2
] 280.7 277.4 274.1 270.9 267.7 264.6 261.5 258.7 255.8
u
max
=mg [%] 28.03 27.67 27.32 26.97 26.64 26.30 25.97 25.65 25.33
q
rms
[cm] 1.542 1.520 1.498 1.478 1.457 1.438 1.419 1.400 1.382
˙ q
rms
[cm/s] 7.073 6.993 6.914 6.837 6.762 6.688 6.616 6.546 6.478
¨ q
abs
rms
[cm/s
2
] 36.61 36.34 36.08 35.88 35.64 35.44 35.25 35.03 34.86
u
rms
=mg [%] 4.813 4.742 4.672 4.601 4.536 4.472 4.408 4.351 4.293
cost [m
2
/s
3
] 4.021 3.961 3.905 3.862 3.812 3.767 3.728 3.682 3.645
with perturbed stiffness subjected to Kobe earthquake are presented in Figures
5.6 and 5.7. In this case, all reduction ratios are positive expect for a small nega-
tive value of the peak absolute acceleration reduction with –20% perturbation in
the stiffness. The cost curve also shows an upward concave pattern, indicating
improved cost reductions for perturbed systems.
72
Table 5.8: Response of the SDOF system with a perturbed damping when sub-
jected to Kobe earthquake using the hybrid MPC control design
Dc=c –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40
q
max
[cm] 4.943 4.886 4.831 4.775 4.720 4.667 4.615 4.564 4.513
˙ q
max
[cm/s] 24.84 24.66 24.48 24.30 24.12 23.94 23.77 23.60 23.43
¨ q
abs
max
[cm/s
2
] 272.4 268.2 268.0 265.7 261.4 259.4 256.2 252.5 251.6
u
max
=mg [%] 19.52 19.29 19.07 18.85 18.64 18.43 18.22 18.02 17.82
q
rms
[cm] 1.045 1.035 1.025 1.015 1.005 0.996 0.986 0.977 0.968
˙ q
rms
[cm/s] 5.087 5.046 5.005 4.964 4.923 4.884 4.846 4.807 4.770
¨ q
abs
rms
[cm/s
2
] 30.39 30.38 30.36 30.35 30.31 30.30 30.31 30.31 30.30
u
rms
=mg [%] 3.627 3.592 3.557 3.522 3.488 3.455 3.422 3.390 3.358
cost [m
2
/s
3
] 2.770 2.768 2.765 2.764 2.757 2.755 2.757 2.755 2.754
For the system with perturbed damping, the reduction ratios when the struc-
ture is subjected to the El Centro earthquake are shown in Figures 5.8 and 5.9.
Despite using a wider range for damping perturbations (relative to the stiffness
perturbations), most of the reduction ratio curves in this case show patterns that
are almost flat; showing that the response of the system is less sensitive to damp-
ing perturbations compared to stiffness perturbations. Additionally, this indicates
that hybrid MPC control design maintains its control performance improvements
such as significant reductions in cost, response and control force, even when the
damping of the system is significantly perturbed. Similar argument can be made
for the results of Kobe earthquake simulation reduction ratios shown in Figures
5.10 and 5.11.
73
−20 −15 −10 −5 0 5 10 15 20
−40
−30
−20
−10
0
10
20
30
40
Δk/k [%]
Reduction [%]

Peak Disp.
Peak Vel.
Peak Abs. Acc.
Peak Control Force
Figure 5.4: Reduction in peak responses for the SDOF system when subjected to
the El Centro earthquake with perturbed stiffness
−20 −15 −10 −5 0 5 10 15 20
−40
−30
−20
−10
0
10
20
30
40
Δk/k [%]
Reduction [%]

Cost
RMS Disp.
RMS Vel.
RMS Abs. Acc.
RMS Control Force
Figure 5.5: Reduction in RMS responses for the SDOF system when subjected to
the El Centro earthquake with perturbed stiffness
74
−20 −15 −10 −5 0 5 10 15 20
−10
0
10
20
30
40
50
Δk/k [%]
Reduction [%]

Peak Disp.
Peak Vel.
Peak Abs. Acc.
Peak Control Force
Figure 5.6: Reduction in peak responses for the SDOF system when subjected to
the Kobe earthquake with perturbed stifness
−20 −15 −10 −5 0 5 10 15 20
−10
0
10
20
30
40
50
Δk/k [%]
Reduction [%]

Cost
RMS Disp.
RMS Vel.
RMS Abs. Acc.
RMS Control Force
Figure 5.7: Reduction in RMS responses for the SDOF system when subjected to
the Kobe earthquake with perturbed stiffness
75
−40 −30 −20 −10 0 10 20 30 40
−10
−5
0
5
10
15
20
25
30
Δc/c [%]
Reduction [%]

Peak Disp.
Peak Vel.
Peak Abs. Acc.
Peak Control Force
Figure 5.8: Reduction in peak responses for the SDOF system when subjected to
the El Centro earthquake with perturbed damping
−40 −30 −20 −10 0 10 20 30 40
−10
−5
0
5
10
15
20
25
30
Δc/c [%]
Reduction [%]

Cost
RMS Disp.
RMS Vel.
RMS Abs. Acc.
RMS Control Force
Figure 5.9: Reduction in RMS responses for the SDOF system when subjected to
the El Centro earthquake with perturbed damping
76
−40 −30 −20 −10 0 10 20 30 40
0
5
10
15
20
25
30
35
40
Δc/c [%]
Reduction [%]

Peak Disp.
Peak Vel.
Peak Abs. Acc.
Peak Control Force
Figure 5.10: Reduction in peak responses for the SDOF system when subjected
to the Kobe earthquake with perturbed damping
−40 −30 −20 −10 0 10 20 30 40
0
5
10
15
20
25
30
35
40
Δc/c [%]
Reduction [%]

Cost
RMS Disp.
RMS Vel.
RMS Abs. Acc.
RMS Control Force
Figure 5.11: Reduction in RMS responses for the SDOF system when subjected
to the Kobe earthquake with perturbed damping
77
5.3 Robustness Study of the 2DOF System
This section presents a robustness study for the control design of the 2DOF sys-
tem. Similar to the SDOF system, the system parameters are perturbed from their
nominal value (Figure 5.12) to study how this affects the performance of control
strategies designed using the nominal system parameters. Since the nominal sys-
tem has two equal stiffness values, one for each floor, it is assumed both stiffness
values are perturbed similarly (i.e., Dk
1
=k
1
=Dk
2
=k
2
=Dk
i
=k
i
), shown in Fig-
ure 5.13. On the other hand, the damping constant for the top floor was assumed
to be zero in the nominal system, so only the damping constant for the first floor
is perturbed, as in Figure 5.14.
Figure 5.12: Nominal 2DOF system used to design control
78
Figure 5.13: 2DOF system with perturbed stiffness
Figure 5.14: 2DOF system with perturbed damping
79
Response reduction ratios for the perturbed 2DOF system are presented in
Figures 5.15–5.22. For El Centro earthquake simulation of the 2DOF system with
perturbed stiffness values, the reduction ratios for the peak response and control
force are shown in Figure 5.15, and the reduction ratios of RMS responses, RMS
control force and cost are presented in Figure 5.16. Both the peaks and RMS val-
ues exhibit positive reduction ratios over the range of perturbations studied. Sim-
ilarly, the reduction ratios for the structure with perturbed stiffness subjected to
Kobe earthquake are presented in Figures 5.17 and 5.18. In this case, all reduction
ratios are positive expect for the peak control force which is increased up to 10%
with –20% perturbation in the stiffness. It is worth noting that the RMS control
force has maintained very significant reduction of magnitudes around 45–50%.
On the other hand, response reduction ratios of the system with perturbed
damping are shown in Figures 5.19–5.22. Similar to the SDOF system case, most
of the reduction ratio curves in this case show patterns that are almost flat, that
the response of the system is less sensitive to damping perturbations compared to
stiffness perturbations. It is important to note the all the reduction ratios values
are positive in this case; this indicates that hybrid MPC control design maintains
significant reductions in cost, response and control force, even when the damping
of the system is significantly perturbed.
80
−20 −15 −10 −5 0 5 10 15 20
0
10
20
30
40
50
60
Δk
i
/k
i
[%]
Reduction [%]

Peak Disp. 1
Peak Disp. 2
Peak Rel. Disp.
Peak Abs. Acc. 1
Peak Abs. Acc. 2
Peak Control Force
Figure 5.15: Reduction in peak responses for the 2DOF system when subjected
to the El Centro earthquake with perturbed stiffness
−20 −15 −10 −5 0 5 10 15 20
20
30
40
50
60
70
80
Δk
i
/k
i
[%]
Reduction [%]

RMS Disp. 1
RMS Disp. 2
RMS Rel. Disp.
RMS Abs. Acc. 1
RMS Abs. Acc. 2
RMS Control Force
Cost
Figure 5.16: Reduction in RMS responses for the 2DOF system when subjected
to the El Centro earthquake with perturbed stiffness
81
−20 −15 −10 −5 0 5 10 15 20
−20
−10
0
10
20
30
40
50
Δk
i
/k
i
[%]
Reduction [%]

Peak Disp. 1
Peak Disp. 2
Peak Rel. Disp.
Peak Abs. Acc. 1
Peak Abs. Acc. 2
Peak Control Force
Figure 5.17: Reduction in peak responses for the 2DOF system when subjected
to the Kobe earthquake with perturbed stifness
−20 −15 −10 −5 0 5 10 15 20
0
5
10
15
20
25
30
35
40
45
50
Δk
i
/k
i
[%]
Reduction [%]

RMS Disp. 1
RMS Disp. 2
RMS Rel. Disp.
RMS Abs. Acc. 1
RMS Abs. Acc. 2
RMS Control Force
Cost
Figure 5.18: Reduction in RMS responses for the 2DOF system when subjected
to the Kobe earthquake with perturbed stiffness
82
−20 −15 −10 −5 0 5 10 15 20
10
15
20
25
30
35
40
45
50
55
60
Δc
1
/c
1
[%]
Reduction [%]

Peak Disp. 1
Peak Disp. 2
Peak Rel. Disp.
Peak Abs. Acc. 1
Peak Abs. Acc. 2
Peak Control Force
Figure 5.19: Reduction in peak responses for the 2DOF system when subjected
to the El Centro earthquake with perturbed damping
−20 −15 −10 −5 0 5 10 15 20
30
35
40
45
50
55
60
65
70
Δc
1
/c
1
[%]
Reduction [%]

RMS Disp. 1
RMS Disp. 2
RMS Rel. Disp.
RMS Abs. Acc. 1
RMS Abs. Acc. 2
RMS Control Force
Cost
Figure 5.20: Reduction in RMS responses for the 2DOF system when subjected
to the El Centro earthquake with perturbed damping
83
−20 −15 −10 −5 0 5 10 15 20
20
25
30
35
40
45
50
Δc
1
/c
1
[%]
Reduction [%]

Peak Disp. 1
Peak Disp. 2
Peak Rel. Disp.
Peak Abs. Acc. 1
Peak Abs. Acc. 2
Peak Control Force
Figure 5.21: Reduction in peak responses for the 2DOF system when subjected
to the Kobe earthquake with perturbed damping
−20 −15 −10 −5 0 5 10 15 20
10
20
30
40
50
60
70
Δc
1
/c
1
[%]
Reduction [%]

RMS Disp. 1
RMS Disp. 2
RMS Rel. Disp.
RMS Abs. Acc. 1
RMS Abs. Acc. 2
RMS Control Force
Cost
Figure 5.22: Reduction in RMS responses for the 2DOF system when subjected
to the Kobe earthquake with perturbed damping
84
5.4 Conclusions
Based on the simulation results and the reduction ratios, the following conclu-
sions can be made:
1. Hybrid MPC control design for the SDOF system is more robust against
changes in the stiffness compared to clipped LQR, evidenced by the upward
concave patterns in the cost reduction curves.
2. The response and cost metrics are not very sensitive to changes in structure
damping, for both the SDOF and the 2DOF systems. Thus, hybrid MPC
control design maintains its performance advantage over clipped LQR when
the system damping is perturbed.
3. The nonlinear semiactive control laws obtained using hybrid MPC, for both
the SDOF and the 2DOF systems are not less robust against perturbations in
the structural system parameters compared to clipped LQR control design.
85
Chapter 6
Improved Linear Feedback Control
6.1 Introduction
In this chapter, an empirical method is developed to design an improved linear
feedback law for semiactive control based on the nonlinear feedback control laws
produced by hybrid MPC that were presented in Chapter 4. Despite the benefits
of the hybrid MPC control laws in terms of optimality (chapter 4) and robustness
(Chapter 5), the nonlinearity of the optimal control laws remains a challenge in
terms of both obtaining the feedback laws numerically and implementing them in
real-time for a large structures. It is postulated that an improved linear feedback
law exists, providing better performance compared to clipped LQR or a perfor-
mance close to that of a hybrid MPC nonlinear control law.
6.2 Improved Linear Feedback Control using
Robust Regression and Hybrid MPC
Normally, an improved linear feedback control law for semiactive control can
be obtained using either heuristics or parametric studies (Johnson et al., 2007;
86
Ramallo et al., 2002), which require a massive computational cost; however, such
an approach does not guarantee optimality. An empirical method is developed
to systematically obtain an improved linear feedback control law for semiactive
control using robust regression and hybrid MPC. The method is denoted here
as ILFC, which stands for Improved Linear Feedback Control. The nonlinear
control laws obtained from hybrid MPC using a discrete time system represen-
tation are, in fact, piece-wise affine functions (Bemporad et al., 2002). Thus,
this method tries to obtain an improved linear control by fitting a linear surface
through the hybrid MPC control. Since the nonlinear control law can be resolved
into affine functions over different zones in the state space, robust multilinear
regression is used so that the fitting is weighted towards fitting through more sig-
nificant linear pieces of the control law. The robust regression is implemented
using MATLAB’s robustfit function which uses an iteratively reweighted
least squares method to compute the regression (Holland and Welsch, 1977).
The method to obtain improved linear feedback law can be summarized in the
following steps:
1. Design a controller using hybrid MPC;
2. Generate a set of random state vectors to be used as input vectors for the
hybrid MPC controller;
3. Normalize the state vectors to unit vectors (having different magnitudes
of state vectors is not necessary because the control law obtained shall be
87
homogenous; however, obtaining vectors that lie in different locations along
the circumference of the unit hypersphere in the state space is necessary in
order correctly represent the hyrbid MPC control law);
4. Compute the control input for each randomly generated state vector using
the hybrid MPC control law;
5. Remove from the set all state vectors whose control input is zero (this step
is essential so that a linear control law is not fit through points that will turn
out to be clipped);
6. Duplicate the set of remaining state vectors and corresponding hybrid MPC
control by negating each (this ensures that fitting a linear surface will pro-
duce a homogeneous control law); and
7. Fit a linear feedback law through the resulting set of state vectors and cor-
responding control forces using robust regression
The result of this method is a linear control gain that represents an approxi-
mation of the hybrid MPC control law. It will be shown through numerical exam-
ples that this linear feedback law can retain many of the benefits of the nonlinear
hybrid MPC control law.
88
6.3 Robust Regression
Statistical regression methods are typically used to obtain a linear or nonlinear
relationship between a dependent variable and one or more independent variables,
based on a set of known observations. The fitted relationship can then be used to
estimate values of the dependent variable based on new values of the independent
variables. For linear fitting, the assumed relationship can be written in the form:
y=b b bX+e e e (6.1)
where y is a vector of dependent variable values for different observations, X
is a matrix whose rows represents the independent variables and the columns
corresponds to different observations,b b b is the set of parameters that linearly map
the independent variables to the dependent variable, and e e e is the error vector
assuming a linear relationship does not perfectly fit the observations. The linear
relationship can then be written as:
ˆ y= bX (6.2)
where ˆ y is the dependent variable and b is an estimate of the linear parameters
vector b b b. Consequently, the residual error in the ith data point estimate can be
written as:
e
i
= y
i
ˆ y
i
(6.3)
89
A quadratic error function can then be formed as the sum of square residual
errors:
e
s
=
n
å
i=1
e
2
i
=
n
å
i=1
(y
i
ˆ y
i
)
2
(6.4)
where n is the number of known observations. Then, the problem of finding the
best fit solution can be formulated as a linear least squares problem, where b is the
vector of parameters that minimizes the quadratic error. The linear least square
problem is typically solved using the Moore-Penrose pseudoinverse (Penrose,
1955). It is also possible to assign different weights to the residual by using a
weighting coefficient, such as:
e
w
=
n
å
i=1
w
i
(y
i
ˆ y
i
)
2
(6.5)
In robust regression, the weight are chosen to minimize the effect of the outliers
(Holland and Welsch, 1977). Normally, outliers produce larger-valued residual
errors in the cost function. The typical iterative procedure for a robust regression
uses the following steps:
1. Initially an ordinary least squares fit is computed without weighting (i.e., all
the observations are weighted equally);
2. Initial residuals are computed based on the ordinary least squares fit;
3. The weighting function is applied to the residuals, to estimate the weights
for next iteration of fitting;
90
4. A weighted least-squares is used to fit a linear function through the data
points;
5. Updated residuals are computed based on the resulting linear fit;
6. Steps 3–5 are repeated until the fitted function converges.
Thebisquare weighting function was used in the robust regression studies
included in this Chapter. This weighting function is defined in MATLAB in the
following form (The MathWorks Inc., 2015):
w
i
=
8
>
>
<
>
>
:
(1 ˆ e
2
i
)
2
; j ˆ e
i
j< 1
0; j ˆ e
i
j 1
(6.6)
where ˆ e
i
is an adjusted residual that is reduced for data points that have large
effect on the fitting solution (The MathWorks Inc., 2015). This adjusted residual
is obtained in MATLAB in two steps, first the residuals are scaled, such that:
e
h
i
=
e
i
p
1 h
i
(6.7)
where h
i
is the leverage of the data point whose residual is e
i
, a measure that indi-
cates how much the value of the independent variable for the given observation
91
differs from the rest of observations. The adjusted residual in MATLAB is then
computed from the previous scaled residual e
h
i
as follows:
e
h
i
=
e
h
i
c
t
s
(6.8)
where c
t
is a tuning constant whose default value in MATLAB is 4.685 and s is
a robust measure of scale and is given by d
ma
=0:6745 where d
ma
is the median
absolute deviation.
6.4 Application to the SDOF system
The method described in the previous two sections is applied herein for the SDOF
system previously discussed in section §3.2. Figures 6.1 and 6.2 show the result
of fitting a surface to the hybrid MPC control law with linear and robust regres-
sion, respectively. The figures show that robust regression causes the surface to
fit through the significant close-to-linear portion of the hybrid MPC control law.
Since the method proposed in this Chapter requires the generation of a ran-
dom set of state vectors, the fitted linear feedback control laws are expected to
vary depending on the random vectors generated each time. In order to asses the
sensitivity of the resulting control law to random state vector choices, the method
is used multiple times with different numbers of state vectors and using a differ-
ent seed for the random numbers generator. The resulting linear feedback gains
92
−1
−0.5
0
0.5
1
−1
0
1
−40
−20
0
20
40

Velocity [cm/s]
Displacement [cm]

Control Force u/mg [%]
(a) Linear fit surface
−1 −0.5 0 0.5 1
−1
0
1
−40
−20
0
20
40

Displacement [cm]
Velocity [cm/s]

Control Force u/mg [%]
(b) Linear fit side view (so plane is edge-on)
Figure 6.1: Linear fitting surface for hybrid MPC control law
are presented in Table 6.1. It can be seen that the feedback gains, particularly the
first of each pair (i.e., the displacement gain), are not very sensitive to the set of
randomly generated state vectors. It can also be seen that the sensitivity of the
gains is decreased as a larger number of state vectors is used.
Simulation results of the SDOF controlled system is compared in Table 6.2
for both clipped LQR and ILFC control designs. For ILFC, the feedback gains
93
−1
−0.5
0
0.5
1
−1
0
1
−40
−20
0
20
40

Velocity [cm/s]
Displacement [cm]

Control Force u/mg [%]
(a) Robust fit surface
−1 −0.5 0 0.5 1
−1
0
1
−40
−20
0
20
40

Velocity [cm/s]
Displacement [cm]

Control Force u/mg [%]
(b) Robust fit side view (so plane is edge-on)
Figure 6.2: Robust fitting surface for hybrid MPC control law
[38:47 N=m; 1:32 N s=m], which were obtained using 1000 random state
vectors and a random number generator seed of 0 were used. It is obvious
from the results comparison that the improved linear feedback is superior to
the clipped LQR design in terms of reduction of the cost function, the control
force and the different performance metrics. For the El Centro earthquake, the
cost was reduced by 15%, the peak responses were reduced by 13–14%, and
94
Table 6.1: Feedback gains resulting from linear robust regression of the nonlinear
hybrid MPC control law for the SDOF system based on different sets of randomly
generated state vectors (units of the gains are [N=m;N s=m])
Number of Random Sample Points
Seed 50 100 500 1000
0 [–38.59, 1.38] [–38.63, 1.20] [–38.41, 1.40] [–38.47, 1.32]
1 [–38.36, 1.23] [–38.45, 1.04] [–38.53, 1.21] [–38.45, 1.27]
2 [–38.36, 1.23] [–38.45, 1.04] [–38.53, 1.21] [–38.21, 1.61]
3 [–38.69, 1.24] [–38.54, 1.38] [–38.37, 1.53] [–38.34, 1.53]
4 [–38.33, 1.67] [–38.58, 1.26] [–38.48, 1.33] [–38.39, 1.40]
5 [–37.91, 2.22] [–38.53, 1.53] [–38.41, 1.68] [–38.31, 1.65]
the RMS responses were reduced by 8–15%; these control performance gains
were obtained while using smaller control forces: 15% smaller for the peak force
and 9% smaller for the RMS control force. For the Kobe earthquake, the cost
was reduced by 31%, the peak responses were reduced by 13–17%; and the
RMS responses were reduced by 16–30%; these control performance gains were
obtained while using smaller control forces: 14% smaller peak force and 10%
smaller RMS control force. The results indicates that the control device is being
used more efficiently with ILFC control gains. It can also be noted that the per-
formance of ILFC is very close to that of the hybrid MPC control, specifically in
terms of the cost value.
95
Table 6.2: Comparison between clipped LQR and the improved linear feedback
designs for peak and RMS responses, control force and cost values using histori-
cal earthquakes
El Centro 1940 Kobe 1995
CLQR HMPC ILFC D [%] CLQR HMPC ILFC D [%]
q
max
[cm] 5.549 1.293 4.824 –13.1 6.747 4.726 5.857 –13.2
˙ q
max
[cm/s] 41.04 34.00 35.96 –12.4 34.05 24.10 28.17 –17.3
¨ q
abs
max
[cm/s
2
] 221.1 240.1 191.1 –13.6 267.7 265.5 232.5 –13.1
u
max
=mg [%] 21.91 16.95 18.70 –14.7 26.64 18.66 22.84 –14.3
q
rms
[cm] 1.570 1.131 1.342 –14.5 1.457 1.005 1.021 –29.9
˙ q
rms
[cm/s] 7.527 5.863 6.546 –13.0 6.762 4.920 5.548 –18.0
¨ q
abs
rms
[cm/s
2
] 39.66 37.02 36.61 –7.69 35.64 30.50 30.04 –15.7
u
rms
=mg [%] 4.806 3.933 4.362 –9.24 4.536 3.524 4.090 –9.83
cost [m
2
/s
3
] 4.719 4.112 4.020 –14.8 3.912 2.791 2.707 –30.8
Note: D is the percent change from Clipped LQR to the improved linear feedback
design; negative numbers are improvements.
6.5 Application to the 2DOF system
Similar to the SDOF system, an improved linear feedback control design was
obtained for the 2DOF system, presented previously in §4.2.2, using robust
regression of the hybrid MPC control law. Comparison of the simulation
results is presented in Table 6.3. The control gains used for ILFC are 10
5

[175:26 N=m;54:11 N=m;13:23 N s=m; 4:53 N s=m]. Simulation results
with these ILFC gains show a significant improvement of the control performance
compared to clipped LQR. For the El Centro earthquake, the cost was reduced by
67%, the peak responses were reduced by 38–48% and the peak control force
was reduced by 52%. For the Kobe earthquake, the cost was reduced by 59%, the
96
peak responses were reduced by 32–54% and the peak control force was reduced
by 50%.
Table 6.3: Comparison between clipped LQR and the improved linear feedback
designs for peak responses, control force and cost values of the 2DOF system
using historical earthquakes
El Centro 1940 Kobe 1995
CLQR HMPC ILFC D [%] CLQR HMPC ILFC D [%]
(q
1
)
max
[cm] 5.834 3.359 3.380 –42.1 5.529 3.516 3.520 –36.3
(q
2
)
max
[cm] 9.739 5.086 5.016 –48.5 9.356 5.063 5.000 –46.6
(q
2
q
1
)
max
[cm] 3.906 2.478 2.414 –38.2 3.827 1.697 1.750 –54.3
( ¨ q
abs
1
)
max
[cm/s
2
] 630.8 483.3 318.9 –49.4 528.7 510.5 359.5 –32.0
( ¨ q
abs
2
)
max
[cm/s
2
] 616.8 391.4 381.2 –38.2 604.4 268.0 276.2 –54.3
u
max
[kN] 987.6 476.2 469.9 –52.4 887.1 457.0 441.9 –50.2
cost [m
2
/s
3
] 73.76 24.69 24.28 –67.1 36.75 14.72 14.95 –59.3
Note: D is the percent change from Clipped LQR to the improved linear feedback
design; negative numbers are improvements.
6.6 Conclusions
In this chapter, an empirical method was developed to obtain improved linear
feedback control laws for semiactive control using hybrid MPC and robust regres-
sion. Two numerical applications were presented for the same SDOF and 2DOF
structural systems studied in the previous chapters. As the simulation results
comparison showed, the method is capable of obtaining linear feedback control
laws, that retain the performance benefits of hybrid MPC, while providing a more
suitable alternative for real time control implementations.
97
Chapter 7
On the Optimal Semiactive Control
of Elevated Highway Bridges
7.1 Introduction
The effectiveness of semiactive control, using controllable damping devices, for
seismic protection of elevated highway bridges has been investigated previously.
In Erkus et al. (2002), a semiactive control based on the clipped LQR design was
employed and compared to optimally designed active and passive control tech-
niques for different control objectives. It was found that the semiactive control
can achieve performance similar to that of a fully active system when the control
objective was to reduce the bridge deck response. On the other hand, when the
control objective is to reduce pier response, semiactive control, designed with
clipped-optimal control, can only achieve performance similar to a passive con-
trol.
This Chapter focuses on the case where the control objective is the reduction
of pier response. The dynamic optimization approach, discussed in §3.1, is used
to obtain the optimal control assuming the excitation is deterministic (i.e., full
98
information of future excitations is available), through a numerical solution to the
Euler-Lagrange equations. A simplified bridge model is presented and optimal
control trajectories are evaluated for the bridge for both active and semiactive
control.
7.2 Simplified Elevated Highway Bridge Model
In this section, the steepest descent technique is used to find the optimal control
trajectories for a bridge model, similar to the one investigated by Erkus et al.
(2002). This simplified model assumes that the bridge can be represented as a
2DOF system, where the bridge pier and the bridge deck are each lumped into
a single mass. The control device is installed between the pier and the deck as
shown in Figure 7.1. The system parameters used for this model are m
1
= 100Mg,
m
2
= 500 Mg, k
1
= 15:791 MN=m, k
2
= 7:685 MN=m, c
1
= 125:6 kNs=m and
c
2
= 0, which results in natural frequenciesw
1
= 3:18 rad=s andw
2
= 15:5 rad=s
for the two modes, with damping ratiosz
1
= 0:45% andz
2
= 4:0%, respectively.
This 2DOF model can be represented in state space form (2.2) where:
M=
2
6
4
m
1
0
0 m
2
3
7
5
; C=
2
6
4
c
1
+ c
2
c
2
c
2
c
2
3
7
5
; K=
2
6
4
k
1
+ k
2
k
2
k
2
k
2
3
7
5
99
(a) Bridge damping system
(b) 2DOF model of the bridge
Figure 7.1: Simplified bridge model
B
w
=
2
6
4
1
1
3
7
5
and B
u
=
2
6
4
1
1
3
7
5
100
The main objective of control considered here is to minimize the pier response. In
order to design the control for that purpose, the following quadratic cost function
was selected (Erkus et al., 2002):
J=
Z
¥
0

r

1
2
k
1
q
2
1
+
1
2
m
1
˙ q
2
1

+

1
2
k
2
(q
2
q
1
)
2
+
1
2
m
2
( ˙ q
2
˙ q
1
)
2

+ Ru
2

dt
(7.1)
where r= 1000 is selected to reflect the higher importance of reducing the pier
response and R is selected to be 10
8
m=N, which was found in preliminary sim-
ulations to produce a controller with a high authority for vibration reduction. For
LQR control design, the same objective function can be rewritten as in (2.3), by
selecting the following coefficient matrices (Erkus et al., 2002):
Q=
1
2
2
6
6
6
6
6
6
6
6
4
rk
1
+ k
2
k
2
0 0
k
2
k
2
0 0
0 0 rm
1
+ m
2
m
2
0 0 m
2
m
2
3
7
7
7
7
7
7
7
7
5
and N= 0
The resulting control gain in this case becomes:
K
LQR
=

939:28 13:37 65:03 s 35:68 s

MN=m (7.2)
101
7.3 Optimal Control Trajectories for the Bridge
Model
In order to find the optimal control trajectories for an active control device using
the steepest descent approach, the following partial derivatives are evaluated:
¶f(x;u;t)
¶x
= A;
¶f(x;u;t)
¶u
= B;
¶L (x;u;t)
¶u
= 2Ru
and

¶L (x;u;t)
¶x

T
=
2
6
6
6
6
6
6
6
6
4
(rk
1
+ k
2
)q
1
k
2
q
2
k
2
q
2
k
2
q
1
(rm
1
+ m
2
) ˙ q
1
m
2
˙ q
2
m
2
˙ q
2
m
2
˙ q
1
3
7
7
7
7
7
7
7
7
5
For the semiactive control case, a clipped LQR strategy is used in the simula-
tion using the same control, by adding a secondary controller that clips the non-
dissipative control force. To facilitate finding the optimal semiactive control tra-
jectories using the steepest descent technique, the semiactive control force can be
rewritten as follows:
u= uH[u]( ˙ q
2
˙ q
1
) (7.3)
102
The partial derivatives involving u required for the steepest descent procedure
are:
¶f(x;u;t)
¶x
= A+ uH[u]
2
6
6
6
6
6
6
6
6
4
0 0 0 0
0 0 0 0
0 0
1
m
1
1
m
1
0 0
1
m
2
1
m
2
3
7
7
7
7
7
7
7
7
5
;
¶f(x;u;t)
¶u
=( ˙ q
2
˙ q
1
)H[u]
2
6
6
6
6
6
6
6
6
4
0
0
+
1
m
1

1
m
2
3
7
7
7
7
7
7
7
7
5
;

¶L (x;u;t)
¶x

T
=
2
6
6
6
6
6
6
6
6
4
(rk
1
+ k
2
)q
1
k
2
q
2
k
2
q
2
k
2
q
1
(rm
1
+ m
2
) ˙ q
1
m
2
˙ q
2
2Ru
2
( ˙ q
2
˙ q
1
)H[u]
m
2
˙ q
2
m
2
˙ q
1
+ 2Ru
2
( ˙ q
2
˙ q
1
)H[u]
3
7
7
7
7
7
7
7
7
5
and
¶L (x;u;t)
¶u
= 2Ru( ˙ q
2
˙ q
1
)
2
H[u]
103
0 10 20 30 40 50
0
5
10
15
x 10
7
Iteration number
Cost J

Active
Semiactive
Figure 7.2: Convergence of the cost functions during iterations
7.4 Results Comparison and Discussion
Simulations were carried out for the bridge model using state feedback control for
both active and semiactive control. In addition, the steepest descent optimization
approach was used to determine the optimal control trajectories for both cases.
The earthquake considered in this analysis is the 1940 El Centro record. A total
of 2000 iterations were carried out to ensure convergence of the optimization
process, though Figure 7.2 shows the cost function for the first 50 iterations only.
Table 7.1 shows the comparison of results for both active and semiactive control.
In this table, the part of the cost that represents the integral of the term x
T
Qx is
denoted by J
x
while that for the term u
T
Ru is denoted by J
u
.
Figure 7.3 shows the pier response during the first 10 seconds of the El Cen-
tro earthquake for both active and semiactive control. In both cases, the opti-
mal control trajectories can reduce the response significantly compared to the
state feedback control laws. It is also evident that active control performance is
104
Table 7.1: Comparison of peak pier responses and cost values for state feedback
control and optimal control using El Centro earthquake
Active control Semiactive control
LQR Optimal D [%] CLQR Optimal D [%]
q
max
1
[ cm] 1.25 0.39 69 4.50 2.56 43
J
x
[ MJ s] 3.72 0.587 84 25.17 12.37 51
J
u
[ MJ s] 0.029 0.009 69 0.012 0.005 63
J [ MJ s] 3.75 0.596 84 25.18 12.38 51
Note: D is the percent change from the LQR-based state feed-
back control to the truly optimal control; negative numbers are
improvements.
much better than semiactive control for the considered objective function; this
was shown previously by Erkus et al. (2002).
Although the values in Table 7.1 show that the optimal semiactive control can
reduce the pier displacement significantly to 2:56 cm (43% reduction compared
to clipped LQR), it also shows that this displacement is much higher than that
achieved by the active system (1:254 cm for LQR). This in fact means that even
a highly adaptive semiactive control laws may not be able to rise to the perfor-
mance of the active control for the considered control objective. It has to be noted
here that the performance of the optimal semiactive control trajectories is consid-
ered an upper bound for any adaptive semiactive control law that can incorporate
ground excitation information.
105
0 2 4 6 8 10
−2
−1
0
1
2
Time [sec.]
Pier Disp. q
1
[cm]

Optimal
LQR
(a) Active control
0 2 4 6 8 10
−5
0
5
Time [sec.]
Pier Disp. q
1
[cm]

Optimal
Clipped LQR
(b) Semiactive control
Figure 7.3: Pier response during the first 10 seconds of El Centro earthquake
7.5 Performance of Control Designed using Hybrid
MPC
Feedback control for this bridge model was designed using Hybrid MPC with
the purpose of improving the feedback control performance compared to clipped
LQR. The control objective used for this design is:
106
J= x
T
p
Q
p
x
p
+
p1
å
k=1

x
T
k
Qx
k
+ Ru
2
k

(7.4)
The value of R used for this study is 10

7 and Q is the same matrix defined
in section §7.2. Different prediction horizons were considered for hybrid MPC
as shown in Table 7.2. It was found that a very small prediction horizons (10–
20 milliseconds) is needed for this model to achieve significant reduction in the
response and the cost, compared to clipped LQR feedback design. For instance,
using 20 steps for the prediction horizon with 1 millisecond time steps, produces a
control design that reduces the pier displacement of the bridge model by 24% and
also reduces the cost by 27%. However, this improvement in cost and response
reduction comes at the cost of using significantly larger control forces, almost
four times the weight of the bridge, which is not achievable in practice. It is
important to note here that this might also indicate that the a more practical con-
trol design could consider imposing constraints on the maximum control forces,
in addition to the passivity constraint.
Table 7.3 shows the performance of both clipped LQR and hybrid MPC feed-
back strategies, with different saturation levels applied to the control force. The
Table shows simulation results with forces saturated at 50% and 20% of the
bridge weight, in addition to the case without saturation. It is clear from these
results that hybrid MPC control strategy is not able to reduce the bridge response
without significantly increasing the control force.
107
Table 7.2: Comparison of control performance using clipped LQR and Hybrid
MPC with different prediction horizons
CLQR Hybrid MPC (no. of steps step duration [ms]) D
Prediction horizon [ ms] 5 2 5 1 10 1 15 1 20 1
Pier disp. [ cm] 4.47 4.37 3.73 3.77 3.68 3.38 –24
Cost
x
[ MJ s] 25.4 22.6 21.5 19.85 17.5 18.4 –28
Cost
u
[ MJ s] 0.07 0.21 0.14 0.17 0.196 0.24 243
Cost [ MJ s] 25.5 22.8 21.6 20.02 17.69 18.65 –27
Peak Force [ kN] 9987 15452 13616 18687 22420 23623 137
Peak Force [ %weight] 170 263 231 317 381 401 137
Note: D is the percent change from Clipped LQR to Hybrid MPC with
20 1 ms prediction horizon; negative numbers are improvements. Predic-
tion horizons are listed by number of steps multiplied by the time step in
milliseconds.
Table 7.3: Comparison of control performance using clipped LQR and Hybrid
MPC using different control force saturation limits
Saturation limit
[ %weight]
¥ 50% 20%
CLQR HMPC CLQR HMPC CLQR HMPC
Pier disp. [ cm] 4.47 3.77 4.16 5.49 4.45 5.64
Cost
x
[ MJ s] 25.4 19.85 24.73 48.05 25.08 51.05
Cost
u
[ MJ s] 0.07 0.17 0.007 0.227 0.007 0.229
Cost [ MJ s] 25.5 20.02 24.73 48.27 25.09 51.27
Peak Force [ kN] 9987 18687 2943 2943 1177 1177
Peak Force [ %wt] 170 317 50 50 20 20
7.6 Conclusions
Based on the results discussed in the previous sections for the bridge model, the
following conclusions can be drawn:
108
1. Adaptive control strategies for elevated highway bridges have significant
room for improving the control performance, assuming additional informa-
tion about the ground excitation is available and can be incorporated in the
design of the control strategy.
2. The improvement of adaptive semiactive control strategies might be limited
for some control objectives, if compared to the improvement that adaptive
active control strategies can achieve.
3. For the minimization of bridge pier response, the performance of optimal
semiactive control cannot rise to the performance of optimal active control,
or even the state feedback active control based on LQR design.
4. Although a hybrid MPC design for the bridge model can improve the con-
trol performance in terms of reducing the response and the cost value, it
requires very large control forces that are not practically achievable.
109
Chapter 8
Finite Element Model Updating of a
Base Isolated Building
8.1 Introduction
Japan’s Hyogo Earthquake Engineering Research Center, commonly called E-
Defense, is a shake table testing facility constructed in the early 2000s in Miki
City, Japan, for performing full-scale earthquake experiments to better under-
stand the seismic behavior of building structures. The 20 m 15 m shake table is
capable of shaking test specimens in six degrees-of-freedom, thereby producing
velocities of up to2 m=s and displacements of1 m. For instance, a three-story
self-centering rocking steel frames structure was tested in 2009 (Ma et al., 2010)
to demonstrate the effectiveness of steel rocking frames with replaceable energy
dissipating devices (Deierlein et al., 2011). Another example of such studies is
the set of earthquake loading experiments carried out on a 5-story steel moment
frame building (Ryan et al., 2013); in these experiments, the structure was tested
both in a fixed-base configuration and a base-isolated configuration using triple
friction pendulum bearings and lead-rubber bearings. These tests also aimed at
110
understanding the behaviors of non-structural components, such as ceilings, pip-
ing systems and non-structural walls (Ryan et al., 2013).
8.2 Background
More recently, in 2013, a series of seismic shaking experiments (Sato et al., 2013)
were performed on a four-story base-isolated reinforced concrete (RC) building,
shown in Figure 8.1 The structure is supported against lateral loads using RC
moment frames along with two structural walls in one corner of the building. The
building is base isolated using four different kinds of passive isolation and damp-
ing devices: rubber bearings, elastic sliding bearings, U-shaped steel dampers and
oil dampers. In 2013, the building was subjected to a series of earthquake records
to evaluate the effectiveness of the passive isolation, particularly with regard to
pounding against the seismic moat (Sato et al., 2013). Additional tests using con-
trollable passive dampers are planned for 2016 or 2017. It is essential to obtain a
validated numerical model of the building in order to design the necessary control
strategies that will be used in future tests.
The building was equipped with different types of sensors, including 42
accelerometers for measuring structural accelerations, denoted SA01X – SA14Z,
as shown in Figure 8.3, 58 sensors measuring forces in the isolators and 4 sen-
sors measuring displacements in the isolation layer. The structure was subjected
to well-known earthquake records scaled to different magnitudes and applied in
111
different directions. At the beginning of the first day of testing, and subsequently
in between different earthquake records, a series of shaking table tests using ran-
dom excitations in different directions were performed to calibrate and evaluate
the extent of any damage, in both the structure and the isolators.
Figure 8.1: Photo of the base isolated building specimen at the E-defense testing
facility (photo courtesy E. A. Johnson)
A previous study reported modal analysis results (Brewick et al., 2015), based
on the 2013 initial tests with random excitations, using stochastic subspace iden-
tification (SSID) methods to obtain a set of stable mode shapes and natural fre-
quencies. In that study, two SSID algorithms were used and their results were
compared for validation. Specifically, the Numerical Algorithm for Subspace
112
(a) Disctretized view (b) Extruded view
Figure 8.2: FEM of the building
State-space System Identification (N4SID) (Van Overschee and De Moor, 1994)
and the Enhanced Canonical Correlation Analysis (ECCA) (Hong et al., 2012)
were chosen. The results of that study are summarized in Table 8.1 herein, as
they represent the target mode shapes and frequencies for the model updating
procedure presented in the next subsection. In the table, the frequencies resulting
from ECCA algorithm are denoted by f
ECCA
, while those obtained from N4SID
algorithm are denoted f
N4SID
. It is essential to note here the coupling between
modes 1 and 2, because the identified frequencies are very close. If these two
modal frequencies matched exactly, then their mode shapes would no longer be
113
(a) Base Floor
(b) Floors 2-4
(c) Roof
Figure 8.3: Sensors locations at different floors (Sato and Sasaki, 2013)
114
unique, and a system identification would identify one pair of an infinite num-
ber of valid mode shape pairs on a two-dimensional manifold. This may pose a
challenge for the model updating problem.
In this Chapter, a numerical finite element model (FEM) is developed and
updated to match the identified modal parameters. A FEM was developed for the
base-isolated building as shown in Figure 8.2. The model includes the columns,
beams, stairs, floors and structural walls. The resulting FEM is comprised of
1757 nodes and has 10542 degrees-of-freedom. Shell elements were used to
model floors, walls and stairs; line elements were used to model the columns and
beams, and linear spring elements were used to model the rubber bearings, elastic
bearings and steel dampers in the isolation layer.
Table 8.1: Results of system identification using random excitation experiments
Mode f
ECCA
[Hz] f
N4SID
[Hz] Error [%] MAC Mode Type
1 0.6533 0.6482 0.7778 0.7478 Translational
2 0.6556 0.6568 –0.1769 0.9744 Translational
3 0.7087 0.7065 0.3187 0.9854 Rotational
4 4.8563 4.8026 1.1048 0.7616 Rotational
5 5.2327 5.1258 2.0430 0.9676 Rotational
6 7.4716 7.2984 2.3180 0.9044 Rotational
7 10.4663 10.0556 3.9236 0.9193 Rotational
8 15.6796 16.0262 –2.2110 0.9010 Rotational
115
8.3 Initial Finite Element Modeling
Initially, a FEM was developed based on the structural drawings (Figure 8.4)
using OpenSEES (McKenna, 2011), an open source object-oriented software
framework that has been developed by the Pacific Earthquake Engineering
Research (PEER) Center. OpenSEES is highly popular among researchers in
the earthquake engineering community, in particular for its versatile nonlinear
element library that can be used for nonlinear pushover analysis of 2D structural
frames. The initial model was a fixed-base 3D model of the building, which can
represent the behavior of the superstructure. The non-structural elements, such
as non-structural walls or electrical and mechanical fittings, were not included in
the model. Also, steel framing elements that were used to fix different sensors to
the structural components were not included in the model because they typically
do not contribute significantly to the structural stiffness.
The initial fixed-base OpenSEES model was used to perform modal analysis
to asses the numerical representation of the super-structure; the resulting first six
modes (and their natural periods) are shown in Figure 8.5. The model was also
used to perform preliminary linear dynamic analysis, however the results were
found to diverge, even under a simple dynamic loading such as a sine wave. Fur-
ther investigation of these problematic results concluded that the OpenSEES 3D
shell element (ShellMITC4) computes distributed masses of that shell element
incorrectly, which was also verified to be a problem in the software with one the
116
Figure 8.4: Elevation view of the building (Sato and Sasaki, 2013)
its main developers. As a result, work with the OpenSEES model was discontin-
ued and a SAP2000 FEM was used instead to continue the analysis.
8.4 Model Updating Methodology
Although the finite element method has undergone significant advancements for
modeling complex linear and nonlinear structures, obtaining an FEM that accu-
rately represents the behavior of a real structure remains a challenging problem.
Often, the properties of an existing structure, such as those relating to its material
117
(a) Mode 1 (0.56 s): Transla-
tion in Y-direction + torsional
motion
(b) Mode 2 (0.31 s): Translation
in X-direction
(c) Mode 3 (0.18 s): Torsional
mode
(d) Mode 4 (0.31 s): Torsional
mode
(e) Mode 5 (0.11 s): Vertical
floor vibration
(f) Mode 6 (0.10 s): Vertical
floor vibration
Figure 8.5: Mode shapes and natural periods for the fixed-base model obtained
from OpenSEES
118
properties or geometry, are missing or inaccurate. In the case of missing prop-
erties, assumptions must be made, but these contribute to the inaccuracies in the
model (Mottershead et al., 2011). For these and other reasons, FEMs undergo an
updating process to reduce the discrepancies between the numerical FEM and its
corresponding real structure. Over the past two decades, several methods have
emerged for addressing the problem of FEM updating using a variety of tech-
niques (Mottershead and Friswell, 1993).
In this study, a modal-based approach is used for model updating, where dif-
ferent structural parameters are modified to match the modes shapes and fre-
quencies of the FEM to the target mode shapes and frequencies identified from
measured responses. It is assumed that the equation of motion of the structure is
as follows:
M¨ q+ C˙ q+ Kq= f (8.1)
where q is the displacement vector of the structure relative to the ground (shake
table), M, C and K are the mass, damping and stiffness matrices, respectively,
and f is the excitation force vector. For the undamped case, the previous equation
of motion becomes:
M¨ q+ Kq= f (8.2)
119
This equation can be transformed into the modal space by solving the generalized
eigenvalue problem, as follows:
KF F F=L L LMF F F (8.3)
whereF F F is the eigenvector matrix whose columns contain the mode shapes andL L L
is a diagonal matrix whose diagonal values are the eigenvalues, i.e., the squared
natural frequencies. Eq. 8.3 can be solved numerically from the FEM to obtain
the mode shapes and frequencies of the structural model. The mode shapes and
frequencies are often different from the target values obtained during system iden-
tification, so it is important to define metrics to evaluate the residuals between the
numerical values and those computed from measurements. The residual vector r,
defined for the natural frequencies, is used to quantify the difference between
computed and experimentally-determined frequencies:
r= f
m
f
c
(8.4)
In Eq. 8.4, f
m
is the vector of experimentally-determined target frequencies and
f
c
is the vector of computed frequencies (obtained from the eigenvalues). The
Modal Assurance Criteria (MAC) [11] is used to quantify the correlation between
120
mode shapesf f f
c
obtained from the FEM and those obtained during system iden-
tification, denotedf f f
m
, as follows:
MAC(f f f
m
;f f f
c
)=
jf f f

m
f f f
c
j
p
(f f f

m
f f f
m
)(f f f

c
f f f
c
)
(8.5)
For the purpose of FEM updating, an objective function based on residuals in
frequencies and the MAC values can be defined for a vector of parameters p
i
as
follows:
J(p
i
)= r
T
i
W
f
r
i
+ m
T
i
W
m
m
i
(8.6)
where r
i
is the residual in frequencies when the parameter vector p
i
is used in the
FEM. W
f
is a diagonal weighting matrix that can specify different weights for
the frequency residual values in different modes. m
i
is a vector whose elements
are one minus each of the diagonal elements of the MAC matrix, i.e., m
i
=
[1 MAC(f f f
m1
;f f f
i
c1
);1 MAC(f f f
m2
;f f f
i
c2
);:::etc] where f f f
i
c j
is the jth computed
mode from the FEM when the parameter vector p
i
is used. W
m
is a diagonal
matrix that is used to assign different weights for matching each mode shape.
8.4.1 Nelder-Mead Simplex Method
In this study, the Nelder-Mead Simplex method (Nelder and Mead, 1965) is
used to solve the optimization problem involved in the FEM updating procedure.
121
The method is implemented using MATLAB’s Optimization Toolbox (The Math-
Works Inc., 2015) that utilizes the algorithm developed by Lagarias et al. (1998).
The method is one of the most popular direct optimization techniques that does
not require the computation of gradients. The main procedure of the method can
be summarized as follows:
1. Generate an initial simplex with n+ 1 vertices (i.e., n+ 1 different p
i
),
where n is the dimension of the search space. For the purpose of model
updating, p
i
is the vector of parameters that are being updated, so n is the
number of elements in p
i
.
2. Compute the objective function at each vertex of the initial simplex.
3. Generate a new point by reflecting about the midpoint of the simplex the
point with highest function value.
4. Evaluate the objective function at the reflected point.
5. Change the simplex vertices coordinates using reflection, expansion or con-
traction operators, depending on how the function value at the reflected
point compares to the function values at the simplex vertices.
6. Iterate through steps 3–5 until a specified maximum number of iterations is
performed or convergence is achieved. Typically, convergence is achieved
when the function values at the vertices reach a specified tolerance or the
size of the simplex is reduced below a specified threshold.
122
8.4.2 Model updating parameters
The process of selecting the updating parameters is a very important step in the
model updating procedure. In many cases, this is done based on engineering
judgment as well as geometric and computational considerations. It is impor-
tant to note that, in many cases, the parameters selected for updating will end
up assuming non-realistic or non-physical values (Mottershead et al., 2011), e.g.,
extremely large/small stiffness, negative stiffness or negative mass. This can be
attributed to different sources of errors in the measurements, identification and
idealizations in the FEM, which manifest themselves in producing non-physical
parameter values. A proper choice of parameters may help reduce this draw-
back. After examining different sets of parameters, the final set of 21 parameters,
summarized in 8.2, were chosen.
Table 8.2: Parameters chosen for model updating
Parameters Locations
E
bx
: Young’s modulus of Beams in the X direc-
tion
123
E
by
: Young’s modulus of Beams in the Y direc-
tion
E
w
: Young’s modulus of Walls
E
c
: Young’s modulus of Columns
E
s
: Young’s modulus of Slabs
124
m
1
=m
2
=m
3
=m
4
: Four additional masses added
at the isolation level. These masses were added
to allow for correcting the mass distribution if it
is different in the model from the real structure
K
sdx1
=K
sdy1
=K
sdx2
=K
sdy2
: Stiffnesses of the two
steel dampers in X and Y directions, respectively;
K
rbx1
=K
rby1
=K
rbx2
=K
rby2
: Stiffnesses of the two
rubber bearings in X and Y directions, respec-
tively;
K
sbx1
=K
sby1
=K
sbx2
=K
sby2
: Stiffnesses of the two
slide bearings in X and Y directions, respectively
125
Using the measurements of forces in the bearings and steel dampers, along
with the displacements of the isolation layer, the force-displacement curves can
be obtained for the isolators (shown in Figure 8.5). The curves exhibit behaviors
that are dominated by a linear relationship, so linear regression was used to obtain
proper initial values for the linear stiffness of these elements. The fitted isolation
stiffness values are summarized in Table 8.3.
The initial values for the masses were chosen to be zero, while the initial val-
ues for the moduli of elasticity for the different structural elements were chosen
based on the formula provided by the ACI building code requirements for struc-
tural concrete (American Concrete Institute, 2014).
Table 8.3: Effective stiffness of isolators obtained from linear regression of the
force-displacement curves
Steel Dampers Rubber Bearings Sliding Bearings
4621 kN/m 1050 kN/m 1570 kN/m
8.5 Results and Discussion
The match between modal parameters obtained from the FEM and those identi-
fied from measurements must be examined prior to performing the model updat-
ing, as shown in Figure 8.6a for the first 6 modes. It is obvious from the figure
that the frequencies of the first 3 modes correlate well, even before updating the
model; this may be attributed to the fact that the initial values used for parameters
126
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
−80
−60
−40
−20
0
20
40
60
80
100
Displacement [cm]
Force [kN]
(a) Steel Dampers
−3 −2 −1 0 1 2 3
−40
−30
−20
−10
0
10
20
30
Displacement [cm]
Force [kN]
(b) Rubber Bearings
127
−3 −2 −1 0 1 2 3
−40
−30
−20
−10
0
10
20
30
40
Displacement [cm]
Force [kN]
(c) Sliding Bearings
Figure 8.5: Force-Displacement curves of the isolators
that represent the isolation layer (bearings and dampers stiffnesses) were obtained
from the experimental results. It was also found that the first 3 modes are ones in
which the dominant motion is the nearly rigid movement of the super-structure
above the isolation layer, which means that those modes are highly dependent on
the isolation layer properties and less dependent on the superstructure properties.
It can also be noted that mode shape 6 is well paired: mode 6 in the FEM has
strong correlation with the 6th identified mode, but it is not strongly correlated to
any other identified mode and vice versa. On the other hand, modes shapes 1, 2,
3, 4 and 5 are not well paired. For the first two modes, this might be attributed
128
to the fact that these two modes are coupled; however, the same argument cannot
be made for modes 3, 4 and 5.
The Nelder-Mead Simplex method was used to update the selected 21 param-
eters. The weighting matrix in the first term of the cost function is selected to
be W
f
= c[diag(f
m
)]
2
, which normalizes the error in frequency by dividing
it by the target frequency. The factor c is used to scale the first term in the cost
function so that it has a magnitude similar to that of second term for the first few
iterations. The weighting matrix of the second term is an identity matrix, which
assigns identical weight on each mode shape match.
The optimization procedure was carried out for 500 iterations; the resulting
convergence in the cost function is shown in Figure 8.7. The modal parameters
of the updated FEM match well with the identified mode shapes and frequency
as shown in Figure 8.6b. Table 8.4 compares the frequencies and MAC values
of the model before and after updating. It is important to note that the modes
1–3 in the updated model correlate better with the identified values compared to
modes 4–6. This can be shown by looking at both the MAC values and also the
errors in frequencies, which are lower by one order of magnitude for the first 3
modes. This can be attributed to the chosen weighting matrices and/or the fact
that the first 3 modes are highly dependent on the parameters in the isolation
layer, whose initial values were obtained from fitting the experimental results. In
a future study, different weighting might be investigated to achieve a better match
129
0.622829 0.648518 0.676421 3.63705 4.21808 5.44217
0.652443
0.65553
0.708034
4.8059
5.19214
7.16171
MAC values after updating
FE Model
ECCA

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) Initial FEM
0.654469 0.657994 0.708091 4.73054 5.45685 6.86431
0.652443
0.65553
0.708034
4.8059
5.19214
7.16171
MAC values after updating
FE Model
ECCA

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b) Updated FEM
Figure 8.6: Match between the FEM and the identified modes (values on the axes
are frequencies in Hz)
130
for the last 3 modes, in addition to introducing additional parameters to update
the masses in the superstructure.
0 50 100 150 200 250 300 350 400 450 500
0
1
2
3
4
Iteration
Cost Function, J
Figure 8.7: Minimization of cost function value in 500 iterations
Table 8.4: Results of system identification using random excitation experiments
Mode
Identified
Freq.
[Hz]
Initial FEM UpdatedFEM
Freq. [Hz] Error [%] MAC Freq. [Hz] Error [%] MAC
1 0.6524 0.6228 4.54 0.4941 0.6545 –0.32 0.9295
2 0.6555 0.6485 –1.07 0.6738 0.6580 –0.38 0.9487
3 0.7080 0.6764 4.46 0.7735 0.7081 –0.01 0.9271
4 4.8059 3.6371 24.32 0.7449 4.7305 1.57 0.8782
5 5.1921 4.2181 18.76 0.6021 5.4569 –5.10 0.8618
6 7.1617 5.4422 24.01 0.9553 6.8643 4.15 0.9669
It is also important to note that the final values of the updated moduli of elas-
ticity were all increased noticeably during the model updating, leading to a stiffer
model, with very significant increases for the beams and columns. Compared to
the initially assigned values, E
bx
increased by 104%, E
by
increased by 28%, E
c
increased by 71%, E
s
increased by 12% and E
w
increased by 10%. The signif-
icant increase in the beams’ elasticity moduli may be attributed to the effect of
beam offset from the floor, as projected RC beams often behave as a T-shaped
131
section (including adjacent zones from the slab), which results in a significantly
higher stiffness compared to that of a rectangular section. The increase in elastic-
ity moduli of the columns and walls may be resulting from the contribution of the
non-structural walls. In many cases, non-structural elements can lead to increase
in the vertical stiffness as well as the lateral stiffness of the structure as they pro-
vide additional coupling between other lateral load-resisting elements such as the
moment frames and the structural walls. On the other hand, the masses have only
slightly increased to result in a total additional mass of 18.2 tons, merely a 2.5%
increase in the total mass of the structure. It has to be noted that the additional
masses were not distributed uniformly on the 4 locations chosen in the isolation
layer, but rather have opposite signs on opposite sides of the building, leading to
a slight shift in the center of mass of the FEM.
8.6 Conclusions
In this chapter, a FEM was developed for the four-story base-isolated building
that was tested at the E-Defense facility in 2013. The FEM was updated using
the Nelder-Mead Simplex method to match the identified modes and frequen-
cies obtained from system identification. The FEM updating procedure showed a
good convergence in terms of minimization of the cost function and the updated
model shows good agreement with the experimental results. The first 3 modes
show stronger correlation with the experimental results compared to the last 3
132
modes, which is probably a result of the chosen weighting matrices and the set of
optimization parameters. The updated model is significantly stiffer compared to
the initial model, as evidenced by the significant increase in the moduli of elastic-
ity of all the structural elements; on the other hand, only a slight increase of mass
was observed. Future studies may investigate obtaining a stronger correlation for
the last 3 modes by introducing more optimization parameters in the superstruc-
ture and investigating the use of different weighting matrices in the optimization
procedure.
133
Chapter 9
Summary, Conclusions and Future
Directions
Several studies in the area of optimal control of structures have been presented
in this dissertation, particularly structural control using smart dampers, such as
controllable passive devices. First, optimal control strategies for semiactive con-
trol is evaluated using dynamic optimization, assuming the structure is subjected
to a deterministic excitation. The results of this study represent the motivation
for seeking feedback strategies for semiactive structural control with improved
performance. Using hybrid system models to model the passivity constraints
that characterize semiactive control devices as on-off switches, it was shown
that more efficient control strategies can be obtained using hybrid MPC design.
The optimized control laws were found to be nonlinear function of the system
states. Numerical simulations of simple structural models show that not only
the responses and cost were reduced significantly, but also the control forces had
significant reductions, indicating a more efficient use of the semiactive damping
devices.
134
Robustness of semiactive control feedback strategies against uncertainties in
the structural system parameters was studied for models with perturbed stiffness
and damping. This study is necessary to validate that hybrid MPC design does
not trade off robustness for optimality. It was shown that the nonlinear control
laws obtained using hybrid MPC are not less robust compared to clipped LQR
control design.
One of the challenges of the real-time application of a nonlinear control
method such as hybrid MPC is the computational burden necessary for evalu-
ating the optimal control force. An improved linear feedback control strategy
based on robust regression of the nonlinear hybrid MPC control laws was shown
to produce linear feedback control laws with significant performance improve-
ments compared to clipped LQR. The empirical procedure to find improved linear
controllers was presented and validated using two numerical applications. Once
found, the resulting controllers are more suitable for real-time control applica-
tions in large structures.
Optimal semiactive control of elevated highway bridges was investigated.
Using a simplified 2DOF bridge model, optimal control trajectories for active
and semiactive control were obtained using a dynamic optimization approach.
For the control objective of reducing the bridge pier response, results show that
highly adaptive control strategies for semiactive control may not rise to the perfor-
mance of optimal active control or even of active control based on LQR feedback
design.
135
A finite element model of a base-isolated reinforced concrete building was
developed and calibrated. The building is a specimen tested at Japan’s E-Defense
earthquake engineering research center in 2013 to evaluate the effects of seismic
moat wall pounding and isolation during long-period earthquakes. This full-scale
four-story moment frame building, with two reinforced concrete walls, sits on an
isolation composed of rubber bearings, elastic sliders, U-shaped steel dampers
and oil dampers. Designing controllable dampers for planned 2016 or 2017 tests
requires a calibrated numerical model. Modal analysis results are reported, based
on random excitation responses during the 2013 tests. The finite element model
(FEM) is developed and updated to match the identified modal parameters. The
superstructure FEM consists of line and shell elements and (linear) spring isola-
tion elements, resulting in 1757 nodes and 10,542 degrees of freedom. 21 param-
eters (various Youngs moduli, isolation layer stiffnesses, and point masses) were
updated iteratively using the Nelder-Mead simplex method using frequency resid-
uals and mode shape MAC values between FEM modes and those identified from
the 2013 random tests. The optimization converges to an updated FEM that pro-
vides much better match with the experimental data, reducing the frequency and
MAC residuals by 60-90%.
136
9.1 Future Directions
Studies in this dissertation have demonstrated that improved feedback design
can be obtained for structures controlled with smart dampers based on ideal-
ized damper models. While these results provides a good evidence that smart
dampers can be used more efficiently, it is still necessary to study more realis-
tic models of smart dampers, such as the Bouc-Wen model for MR dampers. In
addition, numerical applications to larger structural models using either improved
linear feedback, learning based control or decentralized control designs need to
be investigated. Moreover, experimental verification is needed to validate the
efficacy of Hybrid MPC design when applied to a structure with real life com-
plexities. In the near future, research efforts can be directed towards a series of
laboratory experiments, using a frame structure model and a small scale shake
table. In the longer term, large scale laboratory experiments are crucial to gain
better understanding of real structures controlled with smart dampers.
For the FEM updating study of the base isolated building, future research
shall focus on the nonlinear behaviour of the control devices used in the isolation
layer. Although, the random excitation records used in the building experiments
considered in this dissertation were of small magnitude, other experiments of
the same building specimen were carried out with larger magnitude excitations,
which might result in the damping devices exhibiting significant nonlinearities. It
137
is possible to obtain a more realistic numerical model of the base-isolated struc-
ture by calibrating nonlinear hysteretic models to the control devices; so that
the resulting calibrated nonlinear model can be used for future studies to design
feedback strategies for controllable passive dampers, to be used in future tests the
same building specimen.
138
Reference List
Aldemir, U., and Gavin, H. (2006). Optimal semiactive control of structures with
isolated base. International Applied Mechanics, 42:235–240.
Alhan, C., Gavin, H., and Aldemir, U. (2006). Optimal control: Basis for per-
formance comparison of passive and semiactive isolation systems. Journal of
Engineering Mechanics, 132(7):705–713.
Alur, R., Courcoubetis, C., Henzinger, T., and Ho, P. (1993). Hybrid automata:
An algorithmic approach to the specification and verification of hybrid sys-
tems. In Grossman, R., Nerode, A., Ravn, A., and Rischel, H., editors, Hybrid
Systems, volume 736 of Lecture Notes in Computer Science, pages 209–229.
Springer Berlin / Heidelberg. ISBN 978-3-540-57318-0.
Aly, A. M., and Christenson, R. E. (2008). On the evaluation of the efficacy of
a smart damper: a new equivalent energy-based probabilistic approach. Smart
Materials and Structures, 17(4):045008.
American Concrete Institute. (2014). Building Code Requirements for Structural
Concrete (318-14) and Commentary (318r-14).
Bemporad, A. (2002). An efficient technique for translating mixed logical
dynamical systems into piecewise affine systems. Proceedings of the 41st IEEE
Conference on Decision and Control, 2002, volume 2, pages 1970–1975. doi:
10.1109/CDC.2002.1184816.
Bemporad, A., and Morari, M. (1999). Control of systems integrating logic,
dynamics, and constraints. Automatica, 35(3):407–427. doi: 10.1016/
S0005-1098(98)00178-2.
Bemporad, A., Morari, M., Dua, V ., and Pistikopoulos, E. N. (2002). The explicit
linear quadratic regulator for constrained systems. Automatica, 38(1):3–20.
139
Borrelli, F., Baotic, M., Bemporad, A., and Morari, M. (2003). An efficient
algorithm for computing the state feedback optimal control law for discrete
time hybrid systems. American Control Conference, pages 4717–4722, Denver,
Colorado, USA.
Brewick, P., Barrios Hernandez, J., Elhaddad, W., Johnson, E., Christenson, R.,
Sato, E., and Sasaki, T. (2015). Modal Analysis of a Full-Scale Four-Story
Reinforced-Concrete Base-Isolated Building Subjected to Random and Sim-
ulated Earthquake Shake Table Excitations. Proceedings of the International
Workshop on Structural Health Monitoring, Palo Alto, CA, USA.
Chow, G. P. (1976). Analysis and Control of Dynamic Economic Systems. Wiley.
Deierlein, G., Krawinkler, H., Ma, X., Eatherton, M., Hajjar, J., Takeuchi, T.,
Kasai, K., and Midorikawa, M. (2011). Earthquake resilient steel braced frames
with controlled rocking and energy dissipating fuses. Steel Construction, 4(3):
171–175.
Dyke, S. J., Spencer, B. F., Sain, M. K., and Carlson, J. D. (1996). Modeling and
control of magnetorheological dampers for seismic response reduction. Smart
Materials and Structures, 5(5):565–575. doi: 10.1088/0964-1726/5/5/006.
Erkus, B., and Johnson, E. A. (2011). Dissipativity analysis of the base isolated
benchmark structure with magnetorheological fluid dampers. Smart Materials
and Structures, 20(10):105001.
Erkus, B., Ab´ e, M., and Fujino, Y . (2002). Investigation of semi-active control
for seismic protection of elevated highway bridges. Engineering Structures, 24
(3):281–293. doi: 10.1016/S0141-0296(01)00095-5.
Gavin, H., and Aldemir, U. (2005). Optimal control of earthquake response using
semiactive isolation. Journal of Engineering Mechanics, 131(8):769–776.
Gavin, H. (2001). Control of seismically excited vibration using electrorheo-
logical materials and Lyapunov methods. Control Systems Technology, IEEE
Transactions on, 9(1):27–36. doi: 10.1109/87.896743.
Giorgetti, N., Bemporad, A., Tseng, H. E., and Hrovat, D. (2005). Hybrid model
predictive control application towards optimal semi-active suspension. Pro-
ceedings of the IEEE International Symposium on Industrial Electronics, 2005.
ISIE 2005, volume 1, pages 391–398. doi: 10.1109/ISIE.2005.1528942.
140
Gurobi. (2012). Gurobi optimizer reference manual. http://www.gurobi.
com. Accessed on 05/2012.
Heemels, W. P. M. H., De Schutter, B., and Bemporad, A. (2001). On the equiv-
alence of classes of hybrid dynamical models. Proceedings of the 40th IEEE
Conference on Decision and Control, 2001, volume 1, pages 364–369. doi:
10.1109/.2001.980127.
Hespanha, J. P. (2009). Linear Systems Theory. Princeton Press, Princeton, New
Jersey.
Holland, P. W., and Welsch, R. E. (1977). Robust regression using iteratively
reweighted least-squares. Communications in Statistics - Theory and Methods,
6(9):813–827. doi: 10.1080/03610927708827533.
Hong, A., Ubertini, F., and Betti, R. (2012). New stochastic subspace approach
for system identification and its application to long-span bridges. Jour-
nal of Engineering Mechanics, 139(6):724–736. doi: 10.1061/(ASCE)EM.
1943-7889.0000524.
Housner, G. W., Bergman, L. A., Caughey, T. K., Chassiakos, A. G., Claus, R. O.,
Masri, S. F., Skelton, R. E., Soong, T. T., Spencer, B. F., Jr., and Yao, J. T. P.
(1997). Structural control: Past, present, and future. Journal of Engineering
Mechanics, 123(9):897–971.
IBM. (2012). CPLEX optimizer. http://www.ibm.com/software/
integration/optimization/cplex-optimizer. Accessed on
05/2012.
Iemura, H., and Pradono, M. H. (2002). Passive and semi-active seismic response
control of a cable-stayed bridge. Journal of Structural Control, 9(3):189–204.
Iemura, H., and Pradono, M. H. (2003). Application of pseudo-negative stiffness
control to the benchmark cable-stayed bridge. Journal of Structural Control,
10(3–4):187–203. doi: 10.1002/stc.25.
Iemura, H., and Pradono, M. H. (2009). Advances in the development of pseudo-
negative-stiffness dampers for seismic response control. Structural Control and
Health Monitoring, 16(7–8):784–799. doi: 10.1002/stc.345.
Jansen, L. M., and Dyke, S. J. (2000). Semiactive control strategies for MR
dampers: Comparative study. Journal of Engineering Mechanics, 126(8):795–
803. doi: 10.1061/(ASCE)0733-9399(2000)126:8(795).
141
Johnson, E. A., Baker, G. A., Spencer, B. F., Jr., and Fujino, Y . (2007). Semiactive
damping of stay cables. Journal of Engineering Mechanics, 133(1):1–11. doi:
10.1061/(ASCE)0733-9399(2007)133:1(1).
Johnson, E. A., and Erkus, B. (2007). Dissipativity and performance analysis of
smart dampers via LMI synthesis. Structural Control and Health Monitoring,
14(3):471–496. doi: 10.1002/stc.169.
Kirk, D. E. (2004). Optimal Control Theory: An Introduction. Dover Publica-
tions. ISBN 0486434842.
Lagarias, J. C., Reeds, J. A., Wright, M. H., and Wright, P. E. (1998). Conver-
gence properties of the Nelder-Mead simplex method in low dimensions. SIAM
Journal of Optimization, 9:112–147.
L¨ ofberg, J. (2004). YALMIP : a toolbox for modeling and optimization in MAT-
LAB. 2004 IEEE International Symposium on Computer Aided Control Sys-
tems Design, pages 284–289. doi: 10.1109/CACSD.2004.1393890.
Ma, X., Eatherton, M., Hajjar, J., Krawinkler, H., and Deierlein, G. (2010). Seis-
mic design and behavior of steel frames with controlled rocking – Part II: Large
scale shake table testing and system collapse analysis. Proceedings of Struc-
tures Congress, Orlando, FL, USA.
McKenna, F. (2011). Opensees: A framework for earthquake engineering simu-
lation. Computing in Science and Engineering, 13(4):58–66.
Mei, G., Kareem, A., and Kantor, J. C. (2001). Real-time model predictive con-
trol of structures under earthquakes. Earthquake Engineering & Structural
Dynamics, 30(7):995–1019. doi: 10.1002/eqe.49.
Mei, G., Kareem, A., and Kantor, J. C. (2002). Model predictive control of
structures under earthquakes using acceleration feedback. Journal of Engineer-
ing Mechanics, 128(5):574–585. doi: 10.1061/(ASCE)0733-9399(2002)128:
5(574).
Mottershead, J. E., and Friswell, M. I. (1993). Model updating in structural
dynamics: A survey. Journal of Sound and Vibration, 167(2):347–375. doi:
10.1006/jsvi.1993.1340.
Mottershead, J. E., Link, M., and Friswell, M. I. (2011). The sensitivity method
in finite element model updating: A tutorial. Mechanical Systems and Signal
Processing, 25(7):2275–2296. doi: 10.1016/j.ymssp.2010.10.012.
142
Nelder, J. A., and Mead, R. (1965). A simplex method for function minimization.
The Computer Journal, 7(4):308–313. doi: 10.1093/comjnl/7.4.308.
Penrose, R. (1955). A generalized inverse for matrices. Mathematical Pro-
ceedings of the Cambridge Philosophical Society, 51:406–413. doi: 10.1017/
S0305004100030401.
Qin, S. J., and Badgwell, T. A. (2003). A survey of industrial model predictive
control technology. Control Engineering Practice, 11(7):733–764.
Ramallo, J. C., Johnson, E. A., and Spencer, B. F., Jr. (2002). ‘Smart’ base iso-
lation systems. Journal of Engineering Mechanics, 128(10):1088–1099. doi:
10.1061/(ASCE)0733-9399(2002)128:10(1088).
Ryan, K. L., Sato, E., Sasaki, T., Okazaki, T., Guzman, J., Dao, N., Soroushian,
S., and Coria, C. (2013). Full scale 5-story building with triple pendulum
bearings at E-Defense. Network for Earthquake Engineering Simulation (dis-
tributor), Dataset, DOI:10.4231/D3X34MR7R. Dataset.
Sato, E., and Sasaki, T. (2013). Personal Communication of Design Drawings.
Sato, E., Sasaki, T., Fukuyama, K., Tahara, K., and Kajiwara, K. (2013). Devel-
opment of innovative base-isolation system based on E-Defense full-scale
shake table experiments – Part I: Outline of project research. AIJ Annual Meet-
ing, Hokkaido, Japan (in Japanese).
Scruggs, J. T., Taflanidis, A. A., and Iwan, W. D. (2007a). Non-linear stochastic
controllers for semiactive and regenerative systems with guaranteed quadratic
performance bounds - Part 1: State feedback control. Structural Control and
Health Monitoring, 14(8):1101–1120. doi: 10.1002/stc.196.
Scruggs, J. T., Taflanidis, A. A., and Iwan, W. D. (2007b). Non-linear stochastic
controllers for semiactive and regenerative systems with guaranteed quadratic
performance bounds - Part 2: Output feedback control. Structural Control and
Health Monitoring, 14(8):1121–1137. doi: 10.1002/stc.195.
Sontag, E. (1981). Nonlinear regulation: The piecewise linear approach. IEEE
Transactions on Automatic Control, 26(2):346–358. doi: 10.1109/TAC.1981.
1102596.
Soong, T. T., and Dargush, G. F. (1997). Passive Energy Dissipation Systems in
Structural Engineering. Wiley, Chichester.
143
Soong, T. T., and Spencer, B. F., Jr. (2002). Supplemental energy dissipation:
state-of-the-art and state-of-the-practice. Engineering Structures, 24(3):243–
259.
Spencer, B. F., Sain, M. K., Won, C.-H., Kaspari, D. C., and Sain, P. M. (1994).
Reliability-based measures of structural control robustness. Structural Safety:
Special Issue on Reliability on Special Structural Systems, 15(1):111 – 129.
doi: http://dx.doi.org/10.1016/0167-4730(94)90055-8.
Spencer, B. F., Jr., and Nagarajaiah, S. (2003). State of the art of structural
control. Journal of Structural Engineering, 129(7):845–856. doi: 10.1061/
(ASCE)0733-9445(2003)129:7(845).
Spencer, B. F., Jr., and Sain, M. K. (1997). Controlling buildings: a new frontier
in feedback. Control Systems, IEEE, 17(6):19–35. doi: 10.1109/37.642972.
Spencer, B. F., Jr., Dyke, S. J., Sain, M. K., and Carlson, J. D. (1997). Phe-
nomenological model for magnetorheological dampers. Journal of Engineer-
ing Mechanics, 123(3):230–238.
Sun, L., and Goto, Y . (1994). Application of fuzzy theory to variable dampers for
bridge vibration control. Proceedings of the 1st World Conference on Structural
Control, pages WP1:31–40, Los Angeles, California.
Symans, M. D., and Constantinou, M. C. (1999). Semi-active control systems for
seismic protection of structures: a state-of-the-art review. Engineering Struc-
tures, 21(6):469–487.
The MathWorks Inc. (2015). MATLAB and Optimization Toolbox Release
2013b.
Tseng, H., and Hedrick, J. (1994). Semi-active control laws - optimal and sub-
optimal. Vehicle System Dynamics, 23(1):545–569.
Van de Water, H., and Willems, J. (1981). The certainty equivalence property
in stochastic control theory. IEEE Transactions on Automatic Control, 26(5):
1080–1087. doi: 10.1109/TAC.1981.1102781.
Van Overschee, P., and De Moor, B. (1994). N4sid: Subspace Algorithms for the
Identification of Combined Deterministic-Stochastic Systems. Automatica, 30
(1):75–93.
144
Wang, X., and Gordaninejad, F. (2002). Lyapunov-based control of a bridge using
magneto-rheological fluid dampers. Journal of Intelligent Material Systems
and Structures, 13(7–8):415–419. doi: 10.1106/104538902025939.
Wen, Y .-K. (1976). Method for random vibration of hysteretic systems. Journal
of the Engineering Mechanics Division, 102(2):249–263.
Yang, G., Spencer, B., Jung, H., and Carlson, J. (2004). Dynamic modeling of
large-scale magnetorheological damper systems for civil engineering applica-
tions. Journal of Engineering Mechanics, 130(9):1107–1114.
145
Appendix A
Proof of the Certainty Equivalence
Property
This appendix shows the derivation of the equivalence between the MPC force
control law for a deterministic system with non-zero initial conditions and the
same system with zero initial condition but subjected to zero-mean random exci-
tation. Consider quadratic cost (4.2):
J, x
T
p
Q
p
x
p
+
p1
å
k=0

x
T
k
Qx
k
+ 2x
T
k
Nu
k
+ u
T
k
Ru
k

(A.1)
for the discrete time system
x
k+1
= Ax
k
+ Bu
k
+ Lw
k
(A.2)
with initial conditions x
0
, control force u
k
and stochastic excitation w
k
. Since
the excitation is not deterministic, the cost function J is not deterministic; thus,
instead of minimizing the cost, one can minimize its expected value.
146
E[J]=E
"
x
T
p
Q
p
x
p
+
p1
å
k=0

x
T
k
Qx
k
+ 2x
T
k
Nu
k
+ u
T
k
Ru
k

#
=E

x
T
p
Q
p
x
p

+
p1
å
k=0

E

x
T
k
Qx
k

+E

2x
T
k
Nu
k

+E

u
T
k
Ru
k

(A.3)
In equation (A.3), the terms with x
k
, for k > 0, are random. Consider the
sequence:
x
k
= A
k
x
0
+
k1
å
i=0
A
ki1
(Bu
i
+ Lw
i
)
= A
k
x
0
+
k1
å
i=0
A
ki1
Bu
i
+
k1
å
i=0
A
ki1
Lw
i
=[x
k
]
w0
+
k1
å
i=0
A
ki1
Lw
i
(A.4)
where[]
w0
denotes evaluation assuming the disturbance is nonexistent. In cost
(A.3), the term quadratic in x
k
is:
E

x
T
k
Qx
k

=E
"
e
J
Bu;Bu
k;Q
+
e
J
Lw;Lw
k;Q
+ 2
e
J
Bu;Lw
k;Q
+ x
T
0
A
T
k
QA
k
x
0
+ 2x
T
0
A
T
k
Q
k1
å
i=0
A
ki1
Bu
i
+ 2x
T
0
A
T
k
Q
k1
å
i=0
A
ki1
Lw
i
#
(A.5)
where
e
J
r;s
k;Q
=å
k1
i=0
å
k1
j=0
r
T
i
A
T
ki1
QA
k ji
s
j
. The quantities x
0
, A, Q, B and L
are deterministic; the control force sequencefu
0
;u
1
;:::;u
p1
g is chosen at time
t
0
, independent of future random excitations, so it is deterministic as well. If
147
excitation w
k
is zero-mean and uncorrelated with initial condition x
0
, then the
terms that are linear in the w’s will vanish: i.e.,E[w
k
]= 0 andE
h
e
J
Bu;Lw
k;Q
i
= 0.
Then, the quadratic term equation (A.5) becomes
E

x
T
k
Qx
k

=
e
J
Bu;Bu
k;Q
+E
h
e
J
Lw;Lw
k;Q
i
+ x
T
0
A
T
k
QA
k
x
0
+ 2x
T
0
A
T
k
Q
k1
å
i=0
A
ki1
Bu
i
=

x
T
k
Qx
k

w0
+E
h
e
J
Lw;Lw
k;Q
i
(A.6)
Further, since E[x
k
] = E

[x
k
]
w0
+å
k1
i=0
A
ki1
Lw
i

= [x
k
]
w0
+
å
k1
i=0
A
ki1
LE[w
i
] = [x
k
]
w0
for zero-mean excitation w, and since Nu
k
is deterministic, then the cross term in equation (A.3) is
E

2x
T
k
Nu
k

= 2E

x
T
k

Nu
k
= 2

x
T
k

w0
Nu
k
=

2x
T
k
Nu
k

w0
(A.7)
148
So, using equation (A.6) and equation (A.7), and the fact that u
T
k
Ru
k
is deter-
ministic, the expected cost becomes:
E[J]=

x
T
p
Q
p
x
p

w0
+E
h
e
J
Lw;Lw
p;Q
p
i
+
p1
å
k=0

x
T
k
Qx
k

w0
+E
h
e
J
Lw;Lw
k;Q
i
+

2x
T
k
Nu
k

w0
+ u
T
k
Ru
k

(A.8a)
=E
h
e
J
Lw;Lw
p;Q
p
i
+
p1
å
k=0
E
h
e
J
Lw;Lw
k;Q
i
+

x
T
p
Q
p
x
p

w0
+
p1
å
k=0

x
T
k
Qx
k

w0
+

2x
T
k
Nu
k

w0
+ u
T
k
Ru
k

(A.8b)
=E
h
e
J
Lw;Lw
p;Q
p
i
+
p1
å
k=0
E
h
e
J
Lw;Lw
k;Q
i
+
"
x
T
p
Q
p
x
p
+
p1
å
k=0

x
T
k
Qx
k
+ 2x
T
k
Nu
k
+ u
T
k
Ru
k

#
w0
(A.8c)
=E
h
e
J
Lw;Lw
p;Q
p
i
+
p1
å
k=0
E
h
e
J
Lw;Lw
k;Q
i
+[J]
w0
(A.8d)
The first two terms in expected cost (A.8d): (i) depend only on the determinis-
tic system and weighting matrices and the random excitation; (ii) are, thus, inde-
pendent of the control command sequencefu
0
;u
1
;:::;u
p1
g; and (iii) become
merely additive constants in the optimization choosing the control sequence that
minimizes expected costE[J]. Thus, the set of control inputs that minimize[J]
w0
are the same as those that minimizeE[J]; consequently, minimizing expected cost
149
E[J] with a zero-mean future excitation (that is uncorrelated with initial condition
x
0
) is the same as minimizing a deterministic cost J with no excitation at all.
150
Appendix B
Effect of ODE Solver Tolerances on
Simulation Accuracy
The state space equations of controlled structures considered in this dissertation
are ordinary differential equations (ODE) that can be solved numerically using
different methods, e.g., Runge-Kutta method. The MATLAB implementation of
the ODE solvers is used for numerical evaluation of the states of the structural
system considered in this dissertation. These solvers use either a constant time
step or a variable time step. Although variable time step solvers are usually more
efficient in obtaining accurate solutions without increasing the computational bur-
den, it is still necessary to validate the accuracy of the computations. ode45 is
a variable time step solver that uses Runge-Kutta algorithm. The solver evalu-
ates two solutions using the 4th order and 5th order Runge-Kutta methods; then,
based on how the two solutions compare, it adjusts the time step. For that pur-
pose, relative and absolute tolerance values are used to make a decision about
the accuracy of the solution and, consequently, the time step variation. In this
Appendix, simulation results using different tolerances are presented to validate
the accuracy of the results presented in Chapter 4 of the dissertation.
151
B.1 Accuracy of the SDOF system simulations
This section presents the validation of accuracy of the SDOF system simula-
tions. For that system, Tables B.1, B.2 and B.3 presents the results of El Cen-
tro earthquake simulations using clipped LQR, online hybrid MPC and offline
hybrid MPC implementations for different ode45 tolerances. Similarly, Kobe
earthquake simulations with different tolerances are presented in Tables B.4–B.6.
It can be observed that the only response metric that was highly sensitive the tol-
erances was the peak absolute acceleration. Based on the simulation results, a
relative tolerance of 1 10
6
and an absolute tolerance of 1 10
8
were found
to be sufficient to obtain accurate results.
Table B.1: Response of the SDOF system when subjected to El Centro earthquake
using the online implementation of the clipped LQR control design
Rel. Tol. 10
3
10
4
10
5
10
6
Abs. Tol. auto 10
6
10
7
10
8
q
max
[cm] 5.563 5.548 5.548 5.549
˙ q
max
[cm/s] 40.33 41.03 41.04 41.04
¨ q
abs
max
[cm/s
2
] 221.6 221.1 221.1 221.1
u
max
=mg [%] 21.93 21.90 21.90 21.91
q
rms
[cm] 1.569 1.574 1.574 1.574
˙ q
rms
[cm/s] 7.548 7.526 7.527 7.527
¨ q
abs
rms
[cm/s
2
] 39.69 39.66 39.66 39.66
u
rms
=mg [%] 4.782 4.805 4.805 4.805
cost [m
2
/s
3
] 4.725 4.719 4.720 4.719
152
Table B.2: Response of the SDOF system when subjected to El Centro earthquake
using the online implementation of the hybrid MPC control design
Rel. Tol. 10
3
10
4
10
5
10
6
Abs. Tol. auto 10
6
10
7
10
8
q
max
[cm] 4.229 4.322 4.290 4.293
˙ q
max
[cm/s] 34.38 33.97 34.05 34.00
¨ q
abs
max
[cm/s
2
] 180.6 232.3 238.1 240.1
u
max
=mg [%] 16.70 17.06 16.94 16.95
q
rms
[cm] 1.133 1.133 1.130 1.131
˙ q
rms
[cm/s] 5.909 5.872 5.866 5.863
¨ q
abs
rms
[cm/s
2
] 37.16 37.09 37.04 37.02
u
rms
=mg [%] 3.917 3.943 3.938 3.933
cost [m
2
/s
3
] 4.143 4.127 4.117 4.112
Table B.3: Response of the SDOF system when subjected to El Centro earthquake
using the offline implementation of the hybrid MPC control design
Rel. Tol. 10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
Abs. Tol. auto 10
6
10
7
10
8
10
9
10
10
10
11
10
12
q
max
[cm] 4.303 4.281 4.282 4.282 4.283 4.283 4.283 4.283
˙ q
max
[cm/s] 34.17 33.97 34.00 34.04 34.04 34.04 34.04 34.04
¨ q
abs
max
[cm/s
2
] 180.4 234.7 231.0 236.6 238.2 238.0 238.2 238.2
u
max
=mg [%] 16.98 16.90 16.91 16.91 16.91 16.91 16.91 16.91
q
rms
[cm] 1.120 1.130 1.130 1.130 1.130 1.130 1.130 1.130
˙ q
rms
[cm/s] 5.879 5.868 5.865 5.864 5.863 5.862 5.862 5.862
¨ q
abs
rms
[cm/s
2
] 37.06 36.97 37.01 36.98 36.97 36.96 36.96 36.96
u
rms
=mg [%] 3.865 3.915 3.918 3.919 3.917 3.917 3.917 3.917
cost [m
2
/s
3
] 4.121 4.100 4.110 4.102 4.099 4.099 4.099 4.099
153
Table B.4: Response of the SDOF system when subjected to Kobe earthquake
using the online implementation of the clipped LQR control design
Rel. Tol. 10
3
10
4
10
5
10
6
Abs. Tol. auto 10
6
10
7
10
8
q
max
[cm] 6.754 6.747 6.747 6.747
˙ q
max
[cm/s] 34.09 34.05 34.05 34.05
¨ q
abs
max
[cm/s
2
] 268.0 267.7 267.7 267.7
u
max
=mg [%] 26.56 26.63 26.64 26.64
q
rms
[cm] 1.465 1.457 1.457 1.457
˙ q
rms
[cm/s] 6.824 6.763 6.762 6.762
¨ q
abs
rms
[cm/s
2
] 36.55 35.65 35.66 35.64
u
rms
=mg [%] 4.504 4.537 4.535 4.536
cost [m
2
/s
3
] 4.007 3.812 3.815 3.812
Table B.5: Response of the SDOF system when subjected to Kobe earthquake
using the online implementation of the hybrid MPC control design
Rel. Tol. 10
3
10
4
10
5
10
6
Abs. Tol. auto 10
6
10
7
10
8
q
max
[cm] 4.753 4.710 4.729 4.726
˙ q
max
[cm/s] 24.07 24.10 24.11 24.10
¨ q
abs
max
[cm/s
2
] 201.9 252.0 265.3 265.5
u
max
=mg [%] 18.64 18.60 18.67 18.66
q
rms
[cm] 1.003 1.004 1.006 1.005
˙ q
rms
[cm/s] 4.934 4.917 4.923 4.920
¨ q
abs
rms
[cm/s
2
] 30.77 30.33 30.43 30.50
u
rms
=mg [%] 3.505 3.527 3.529 3.524
cost [m
2
/s
3
] 2.841 2.760 2.778 2.791
154
Table B.6: Response of the SDOF system when subjected to Kobe earthquake
using the offline implementation of the hybrid MPC control design
Rel. Tol. 10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
Abs. Tol. auto 10
6
10
7
10
8
10
9
10
10
10
11
10
12
q
max
[cm] 4.707 4.714 4.721 4.720 4.721 4.721 4.721 4.721
˙ q
max
[cm/s] 24.10 24.10 24.13 24.12 24.12 24.12 24.12 24.12
¨ q
abs
max
[cm/s
2
] 199.6 251.5 259.6 261.4 263.0 263.7 263.8 263.8
u
max
=mg [%] 18.46 18.61 18.64 18.64 18.64 18.64 18.64 18.64
q
rms
[cm] 0.999 1.005 1.005 1.005 1.005 1.005 1.005 1.005
˙ q
rms
[cm/s] 4.935 4.923 4.924 4.923 4.923 4.923 4.923 4.923
¨ q
abs
rms
[cm/s
2
] 30.67 30.26 30.32 30.31 30.32 30.32 30.32 30.32
u
rms
=mg [%] 3.471 3.485 3.486 3.488 3.488 3.488 3.488 3.488
cost [m
2
/s
3
] 2.821 2.747 2.758 2.757 2.758 2.758 2.758 2.758
155
B.2 Accuracy of the 2DOF system simulations
This section presents the validation of the simulation results for the 2DOF system
using both the variable step solverode45 and the 5th order Runge-Kutta solver
with constant time stepsode5. For this case, the variable step solver required a
very small tolerance that results in a very large computation time. As an alterna-
tive, the constant step solver was used and different simulations were carried out
to find a suitable step size that provides numerically accurate results. Tables B.7–
B.12 show the El Centro earthquake simulation results, while Tables B.13–B.18
show the Kobe earthquake simulation results. Based on the tabular data, the con-
stant step solver was chosen to report the results in Chapter 4 of the dissertation,
using a time step of 1/3200 second.
Table B.7: Response of the 2DOF system when subjected to El Centro earthquake
using the online implementation of the hybrid MPC control design computed
using ODE45 solver
Rel. Tol. 10
3
10
3
10
3
10
3
10
3
10
4
Abs. Tol. auto 10
3
10
4
10
5
10
6
10
7
(q
1
)
max
[cm] 3.40 3.37 3.38 3.36 3.34 3.36
(q
2
)
max
[cm] 5.06 5.15 5.08 5.05 5.09 5.07
(q
2
q
1
)
max
[cm] 2.47 2.52 2.50 2.48 2.50 2.46
( ¨ q
abs
1
)
max
[cm/s
2
] 305.11 309.96 313.85 320.73 311.55 488.63
( ¨ q
abs
2
)
max
[cm/s
2
] 390.75 398.55 394.25 391.68 394.84 388.72
u
max
[kN] 466.5 472.49 474.49 471.5 474.72 476.52
cost [m
2
/s
3
] 25.139 25.496 24.844 24.62 24.691 24.734
156
Table B.8: Response of the 2DOF system when subjected to El Centro earthquake
using the offline-4 implementation of the hybrid MPC control design computed
using ODE45 solver
Rel. Tol. 10
3
10
3
10
3
10
3
10
3
10
4
10
5
Abs. Tol. auto 10
3
10
4
10
5
10
6
10
7
10
8
(q
1
)
max
[cm] 3.32 3.35 3.33 3.33 3.33 3.34 3.34
(q
2
)
max
[cm] 5.12 5.13 5.17 5.17 5.17 5.13 5.12
(q
2
q
1
)
max
[cm] 2.48 2.48 2.50 2.50 2.50 2.48 2.49
( ¨ q
abs
1
)
max
[cm/s
2
] 284.87 285.34 305.25 307.09 307.70 316.42 312.82
( ¨ q
abs
2
)
max
[cm/s
2
] 391.61 391.36 394.2 394.78 394.73 392.01 393.63
u
max
[kN] 470.96 471.25 471.04 475.95 475.93 475.46 472.77
cost [m
2
/s
3
] 24.92 24.399 24.718 24.638 24.632 24.637 24.593
Table B.9: Response of the 2DOF system when subjected to El Centro earthquake
using the offline-8 implementation of the hybrid MPC control design computed
using ODE45 solver
Rel. Tol. 10
3
10
3
10
3
10
3
10
3
10
4
10
5
Abs. Tol. auto 10
3
10
4
10
5
10
6
10
7
10
8
(q
1
)
max
[cm] 3.24 3.28 3.24 3.24 3.24 3.25 3.26
(q
2
)
max
[cm] 5.13 5.14 5.14 5.15 5.15 5.17 5.17
(q
2
q
1
)
max
[cm] 2.55 2.56 2.56 2.56 2.56 2.57 2.57
( ¨ q
abs
1
)
max
[cm/s
2
] 267.50 278.04 288.32 292.54 292.78 295.46 317.04
( ¨ q
abs
2
)
max
[cm/s
2
] 403.34 404.02 403.66 404.67 404.44 405.56 406.24
u
max
[kN] 474.99 473.54 474.79 476.02 475.76 476.59 476.29
cost [m
2
/s
3
] 24.994 25.233 24.818 24.734 24.701 24.731 24.723
Table B.10: Response of the 2DOF system when subjected to El Centro earth-
quake using the online implementation of the hybrid MPC control design com-
puted using ODE5 solver
Time Step [sec.] 1/100 1/200 1/400 1/800 1/1600 1/3200
(q
1
)
max
[cm] 3.396 3.355 3.36 3.36 3.360 3.359
(q
2
)
max
[cm] 5.053 5.078 5.11 5.08 5.082 5.086
(q
2
q
1
)
max
[cm] 2.462 2.465 2.48 2.48 2.479 2.478
( ¨ q
abs
1
)
max
[cm/s
2
] 385.05 395.58 370.11 393.08 394.07 483.29
( ¨ q
abs
2
)
max
[cm/s
2
] 388.81 389.32 392.42 390.88 391.43 391.37
u
max
[kN] 465.50 466.43 473.99 475.37 475.86 476.19
cost [m
2
/s
3
] 25.071 24.997 24.898 24.777 24.692 24.687
157
Table B.11: Response of the 2DOF system when subjected to El Centro earth-
quake using the offline-4 implementation of the hybrid MPC control design com-
puted using ODE5 solver
Time Step [sec.] 1/100 1/200 1/400 1/800 1/1600 1/3200 1/6400
(q
1
)
max
[cm] 3.347 3.337 3.337 3.338 3.336 3.337 3.336
(q
2
)
max
[cm] 5.117 5.121 5.117 5.126 5.125 5.125 5.126
(q
2
q
1
)
max
[cm] 2.499 2.490 2.491 2.495 2.494 2.494 2.495
( ¨ q
abs
1
)
max
[cm/s
2
] 319.29 314.87 340.05 339.33 339.52 342.70 342.73
( ¨ q
abs
2
)
max
[cm/s
2
] 394.65 393.22 393.23 393.94 393.83 393.88 393.93
u
max
[kN] 472.70 473.99 474.09 475.01 474.25 474.80 475.09
cost [m
2
/s
3
] 25.095 24.811 24.630 24.627 24.615 24.609 24.600
Table B.12: Response of the 2DOF system when subjected to El Centro earth-
quake using the offline-8 implementation of the hybrid MPC control design com-
puted using ODE5 solver
Time Step [sec.] 1/100 1/200 1/400 1/800 1/1600 1/3200 1/6400
(q
1
)
max
[cm] 3.266 3.255 3.254 3.255 3.255 3.255 3.255
(q
2
)
max
[cm] 5.159 5.161 5.166 5.167 5.168 5.168 5.168
(q
2
q
1
)
max
[cm] 2.572 2.573 2.575 2.576 2.576 2.576 2.576
( ¨ q
abs
1
)
max
[cm/s
2
] 308.55 309.26 317.67 319.20 320.31 321.75 321.75
( ¨ q
abs
2
)
max
[cm/s
2
] 406.14 406.29 406.59 406.72 406.78 406.78 406.78
u
max
[kN] 473.80 476.17 476.05 476.30 476.33 476.36 476.36
cost [m
2
/s
3
] 24.072 24.821 24.772 24.752 24.715 24.699 24.691
Table B.13: Response of the 2DOF system when subjected to Kobe earthquake
using the online implementation of the hybrid MPC control design computed
using ODE45 solver
Rel. Tol. 10
3
10
3
10
3
10
3
10
4
Abs. Tol. 10
3
10
4
10
5
10
6
10
7
(q
1
)
max
[cm] 3.48 3.50 3.50 3.50 3.52
(q
2
)
max
[cm] 5.04 5.05 5.06 5.06 5.07
(q
2
q
1
)
max
[cm] 1.71 1.70 1.70 1.70 1.70
( ¨ q
abs
1
)
max
[cm/s
2
] 445.17 445.05 445.81 445.92 503.93
( ¨ q
abs
2
)
max
[cm/s
2
] 269.52 268.90 268.10 268.18 268.35
u
max
[kN] 442.52 449.98 449.81 445.92 454.20
cost [m
2
/s
3
] 15.0168 14.8869 14.7606 14.778 14.7505
158
Table B.14: Response of the 2DOF system when subjected to Kobe earthquake
using the offline-4 implementation of the hybrid MPC control design computed
using ODE45 solver
Rel. Tol. 10
3
10
3
10
3
10
3
10
4
10
5
Abs. Tol. 10
3
10
4
10
5
10
6
10
7
10
8
(q
1
)
max
[cm] 3.54 3.54 3.54 3.54 3.54 3.54
(q
2
)
max
[cm] 5.08 5.08 5.08 5.08 5.08 5.08
(q
2
q
1
)
max
[cm] 1.68 1.68 1.68 1.68 1.68 1.68
( ¨ q
abs
1
)
max
[cm/s
2
] 391.44 389.56 388.66 396.48 407.36 428.58
( ¨ q
abs
2
)
max
[cm/s
2
] 265.96 265.59 265.32 265.30 265.61 265.23
u
max
[kN] 446.10 446.74 449.02 449.05 449.42 448.96
cost [m
2
/s
3
] 15.1743 14.7023 14.6580 14.6480 14.664 14.6411
Table B.15: Response of the 2DOF system when subjected to Kobe earthquake
using the offline-8 implementation of the hybrid MPC control design computed
using ODE45 solver
Rel. Tol. 10
3
10
3
10
3
10
3
10
4
10
5
10
6
Abs. Tol. 10
3
10
4
10
5
10
6
10
7
10
8
10
9
(q
1
)
max
[cm] 3.42 3.42 3.42 3.42 3.42 3.42 3.42
(q
2
)
max
[cm] 4.96 4.96 4.96 4.96 4.95 4.95 4.95
(q
2
q
1
)
max
[cm] 1.69 1.69 1.69 1.69 1.69 1.69 1.69
( ¨ q
abs
1
)
max
[cm/s
2
] 402.55 402.75 402.87 402.79 402.92 420.96 430.26
( ¨ q
abs
2
)
max
[cm/s
2
] 267.07 266.66 266.66 266.66 266.45 266.93 266.91
u
max
[kN] 423.63 427.54 427.44 427.43 427.26 427.33 427.30
cost [m
2
/s
3
] 14.701 14.525 14.453 14.446 14.439 14.465 14.459
159
Table B.16: Response of the 2DOF system when subjected to Kobe earthquake
using the online implementation of the hybrid MPC control design computed
using ODE5 solver
Time Step [sec.] 1/100 1/200 1/400 1/800 1/1600 1/3200
(q
1
)
max
[cm] 3.483 3.528 3.519 3.517 3.516 3.516
(q
2
)
max
[cm] 5.037 5.076 5.067 5.066 5.062 5.063
(q
2
q
1
)
max
[cm] 1.707 1.698 1.703 1.698 1.698 1.697
( ¨ q
abs
1
)
max
[cm/s
2
] 445.08 459.49 497.16 495.57 510.53 510.00
( ¨ q
abs
2
)
max
[cm/s
2
] 269.57 268.11 268.98 268.09 268.17 268.02
u
max
[kN] 443.30 453.45 453.14 455.06 456.70 456.98
cost [m
2
/s
3
] 15.151 14.934 14.752 14.759 14.756 14.718
Table B.17: Response of the 2DOF system when subjected to Kobe earthquake
using the offline-4 implementation of the hybrid MPC control design computed
using ODE5 solver
Time Step [sec.] 1/100 1/200 1/400 1/800 1/1600 1/3200 1/6400
(q
1
)
max
[cm] 3.548 3.540 3.539 3.538 3.537 3.537 3.537
(q
2
)
max
[cm] 5.086 5.079 5.079 5.078 5.077 5.077 5.077
(q
2
q
1
)
max
[cm] 1.685 1.682 1.681 1.681 1.680 1.680 1.680
( ¨ q
abs
1
)
max
[cm/s
2
] 392.09 422.24 422.43 427.10 426.85 428.90 428.89
( ¨ q
abs
2
)
max
[cm/s
2
] 266.15 265.65 265.50 265.42 265.31 265.24 265.27
u
max
[kN] 446.25 446.67 447.47 447.39 448.11 448.53 448.79
cost [m
2
/s
3
] 15.309 14.777 14.680 14.701 14.763 14.654 14.68
Table B.18: Response of the 2DOF system when subjected to Kobe earthquake
using the offline-8 implementation of the hybrid MPC control design computed
using ODE5 solver
Time Step [sec.] 1/100 1/200 1/400 1/800 1/1600 1/3200 1/6400
(q
1
)
max
[cm] 3.420 3.416 3.416 3.416 3.416 3.416 3.416
(q
2
)
max
[cm] 4.958 4.954 4.954 4.954 4.954 4.954 4.954
(q
2
q
1
)
max
[cm] 1.691 1.690 1.690 1.690 1.690 1.690 1.690
( ¨ q
abs
1
)
max
[cm/s
2
] 402.19 416.31 424.29 426.25 429.66 429.65 429.84
( ¨ q
abs
2
)
max
[cm/s
2
] 267.07 266.85 266.90 266.90 266.91 266.91 266.91
u
max
[kN] 422.02 416.30 427.32 427.32 427.31 427.3 427.31
cost [m
2
/s
3
] 14.774 14.833 14.471 14.460 14.461 14.463 14.453
160

Abstract (if available)

Abstract Structural control has attracted the attention of many researchers to study its application to natural hazards mitigation with the ultimate goal of protecting buildings and bridges against strong earthquakes and extreme winds. Significant efforts have been devoted to realize structural control systems for real-life applications, to protect structural systems and, consequently, human lives

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Structural nonlinear control strategies to provide life safety and serviceability

PDF

Computationally efficient design of optimal strategies for passive and semiactive damping devices in smart structures

PDF

Optimal clipped linear strategies for controllable damping

PDF

Smart buildings: synergy in structural control, structural health monitoring and environmental systems

PDF

Studies into vibration-signature-based methods for system identification, damage detection and health monitoring of civil infrastructures

PDF

Novel multi-stage and CTLS-based model updating methods and real-time neural network-based semiactive model predictive control algorithms

PDF

Controlled substructure identification for shear structures

PDF

Elements of robustness and optimal control for infrastructure networks

PDF

Team decision theory and decentralized stochastic control

PDF

Studies into data-driven approaches for nonlinear system identification, condition assessment, and health monitoring

PDF

New approaches in modeling and control of dynamical systems

PDF

Characterization of environmental variability in identified dynamic properties of a soil-foundation-structure system

PDF

Uncertainty management for complex systems of systems: applications to the future smart grid

PDF

Nonlinear control of flexible rotating system with varying velocity

PDF

LQ feedback formulation for H∞ output feedback control

PDF

Analysis, design, and optimization of large-scale networks of dynamical systems

PDF

Routing problems for fuel efficient vehicles

PDF

Risk assessment, intrinsic interpolation and computationally efficient models for systems under uncertainty

PDF

A novel hybrid probabilistic framework for model validation

PDF

Adaptive control: transient response analysis and related problem formulations

Asset Metadata

Creator Elhaddad, Wael M. (author)

Core Title Towards an optimal design of feedback strategies for structures controlled with smart dampers

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 11/17/2015

Defense Date 10/02/2015

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

Tag base isolation,finite element model updating,hybrid systems,model predictive control,OAI-PMH Harvest,semi-active control,smart dampers,Structural dynamics

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Johnson, Erik A. (committee chair), Masri, Sami F. (committee member), Qin, S. Joe (committee member), Savla, Ketan (committee member)

Creator Email el7addad@gmail.com,welhadda@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-200227

Unique identifier UC11277469

Identifier etd-ElhaddadWa-4046.pdf (filename),usctheses-c40-200227 (legacy record id)

Legacy Identifier etd-ElhaddadWa-4046.pdf

Dmrecord 200227

Document Type Dissertation

Format application/pdf (imt)

Rights Elhaddad, Wael M.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA