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Structural nonlinear control strategies to provide life safety and serviceability

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Structural nonlinear control strategies to provide life safety and serviceability

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Content Structural Nonlinear Control Strategies to Provide
Life Safety and Serviceability
by
Elham Hemmat-Abiri
Advisor
Prof. Erik A. Johnson
A Thesis 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
May 2016
Copyright 2016 Elham Hemmat-Abiri
Dedication
Tomylovinghusbandandbelovedparents.
ii
Acknowledgements
I am deeply grateful to my adviser Prof. Erik A. Johnson for all his guidance and supports throughout
the last couple of years. I would also like to thank Prof. Steven F. Wojtkiewicz for his helpful
suggestions and guidance.
I am also thankful for the valuable advice from my other committee members: Prof. Sami F. Masri,
Prof. Ketan Savla, and Prof. Edmond A. Jonckheere.
I would like to have special thanks to my dear husband, Mahmoud Kamalzare, who has been my
best friend since our first year in undergraduate. I would have never made through this journey
without his enormous support, help and encouragement.
I do not know how to thank my family enough, especially my wonderful parents, Zahra Rezaei
and Naghi Hemmat-Abiri, and my amazing brother, Ehsan Hemmat-Abiri, who have been always
supportive and encouraging.
I would also like to thank my former and current labmates especially Dr. Mahmoud Kamalzare, Dr.
Tat Fu, Dr Dongyu Zhang, Dr. Charles Devore, Dr. Wael Elhaddad, Leonardo Velderrain Chavez
and Subhayan De for making these years so memorable.
Finally, the partial financial support of this research by the National Science Foundation (NSF)
through grants CMMI 13-44937, 14-36018, 14-36058; NEES and Hazard Mitigation and Structural
Engineering is gratefully acknowledged.
iii
Abstract
Various types of controllers have been studied and implemented to mitigate the effects of excitations
from natural hazards. Linear control laws are most often applied, for active devices as well as part
of the controller for semiactive devices, primarily due to their simple design and implementation.
Yet, linear structural control strategies generally cannot focus on different objectives in different
excitations, a goal that has real meaning in many structural control applications. For example,
minimizing structural drift is necessary during strong earthquakes to mitigate damage to the
structure, yet occupant comfort and safety of building contents in the much-more-frequent moderate
earthquakes demands reductions in absolute accelerations in the structure. Minimizing drift and
acceleration in different excitations are often somewhat competing objectives and cannot be achieved
with conventional linear strategies. Therefore, it is imperative to use nonlinear control strategies
instead. To design an optimal nonlinear controller for a linear structure, it is well established that
minimization of a nonquadratic cost function is required, which will lead to solving the Hamilton-
Jacobi-Bellman (HJB) partial differential equation. However, finding the exact analytical solution
of the HJB equation is very difficult, and may only have a solution when the cost function is cast in
a particular form.
The first part of this study presents a comparison of some of the analytical and numerical methods
for finding optimal nonlinear controllers, particularly for cost functions that are even-order powers
of the states and quadratic in the control. A scalar model (i.e., a scalar state-space equation) is used
for the comparisons since analytical solutions exist for some of the methods.
As most of the HJB-derived optimal nonlinear controllers proposed by other researchers are focused
on the single objective of reducing structural drift, the next part of this study presents an analytical
approach to solve the HJB equation and find the optimal nonlinear control law for a nonquadratic
cost function with higher-order polynomials. The resulting optimal nonlinear control law can be
formulated as the summation of a linear term, which is related to a linear quadratic regulator (LQR)
problem designed to be effective in reducing one performance metric in small excitations, and some
nonlinear terms that are dominant to reduce the other performance metric when the excitation level
is large. Thus, it can achieve significant improvements in both serviceability (i.e., acceleration
mitigation) in weak and moderate excitations and life safety goals (i.e., drift mitigation) in extreme
events – all in one general control law.
Next, the application of gain scheduling to achieve the two aforementioned objectives is investigated.
Two different optimal linear controls are designed, each to reduce a different response metric;
scheduling and switching between the two controllers, depending on the strength of earthquake, is
used to minimize both objectives. Numerical simulations show that this approach can successfully
achieve both life safety (drift reduction) and serviceability (acceleration reduction) objectives at
different excitation levels.
The next part of this study presents a nonlinear control strategy, based on model predictive control
(MPC), to minimize a nonquadratic cost function with higher-order polynomials. The nonquadratic
term in the cost function is defined such that it becomes negligible in weak excitations, resulting
the same performance as a LQR problem designed for acceleration reduction, but dominant in
iv
strong excitations, resulting in a superior drift reduction compared with that of corresponding LQR
problem.
For systems that exhibit dynamics that are most naturally expressed with a hybrid system model
containing both continuous and discrete states, such as structures controlled with smart dampers,
hybrid model predictive control is a useful design strategy. Further, this study investigates an offline
implementation of hybrid MPC that is based on neural networks, which may be generalizable
to controllable damping of more complex structures. The aim is to examine whether nonlinear
regression using neural networks is suitable for modeling the control laws obtained from hybrid
MPC. Intensive numerical simulations for both SDOF and 2DOF structural models show that a NN
with many neurons can indeed accurately replicate the MPC control function. However, a NN with
only a few neurons in the hidden layer can only grossly approximate the MPC control function;
nevertheless, the few-neuron NN is still capable of achieving a cost function similar to, and even
slightly better than, that provided by the MPC.
Finally, the last part of this dissertation presents concluding remarks on the various methods
developed in this dissertation for obtaining optimal nonlinear control strategies that can achieve
different objectives in different levels of excitation to provide, in one control strategy, both optimal
serviceability performance in the frequent weak-to-moderate events and optimal life safety in the
infrequent extreme events. It has been demonstrated via intensive numerical simulations that all
nonlinear control strategies studied in this research (i.e., HJB, GS and MPC) can successfully
achieve both life safety (drift reduction) and serviceability (acceleration reduction) objectives at
different excitation levels; however, it should be noted that, due to different optimization techniques
to find the nonlinear controller, their results are not identical. The last chapter also presents avenues
for future work and exploration of the subject, and highlights the areas needing further investigation
and study to improve and complete the results of this research. The first steps to extend this study
are to choose the weak and strong excitation levels and the desired building performance (e.g., drift,
acceleration, etc) based on characteristics of the building and its contents, the history of previous
excitations at the location, the building code and so forth. Moreover, all of the nonlinear control
approaches must be evaluated based on the ease of their application to the variety of models that
civil engineers use, so it is absolutely essential to verify the performance of the nonlinear control
strategy in a laboratory experiments. Considering more than one control device in different levels
of the structure is also necessary to extend the results of this study.
v
Table of Contents
Dedication ii
Acknowledgements iii
Abstract iv
List of Tables x
List of Figures xv
1 Introduction 1
1.1 Motivation 1
1.2 Structural Control 4
1.2.1 Passive Control 4
1.2.2 Active Control 5
1.2.3 Semiactive Control 6
1.3 HJB-based Control 6
1.4 Gain Scheduling Control 7
1.5 Model Predictive Control (MPC) 8
1.6 Neural Networks 9
1.7 Overview of This Dissertation 11
2 Optimal Nonlinear Control using a Non-Quadratic Cost Function for Scalar Sys-
tems 13
2.1 Introduction 13
2.2 Problem Formulation 14
2.2.1 HJB-based Methods and Test Bed Scalar System 15
2.2.2 Probability-based Method (Fokker-Planck-Kolmogorov (FPK) Equation) 16
2.2.3 Numerical Method (Monte Carlo (MC) Simulation) 17
2.2.4 Deterministic Method (Nonzero Initial Condition) 18
2.3 Numerical Example 18
2.3.1 Results Comparison 19
2.4 Conclusions 22
3 Preliminary Study on Control Design for Serviceability and Life Safety Goals 23
3.1 Introduction 23
3.2 Problem Formulation 24
3.3 Numerical Example 25
3.4 Conclusions 28
vi
4 HJB-based Optimal Nonlinear Control for Multiple Objectives in Different Levels
of Excitations 30
4.1 Introduction 30
4.2 Problem Formulation 30
4.3 Control Stability 33
4.4 Numerical Example 1: SDOF Building Model 34
4.4.1 Chooseh to Satisfy the Multiobjective Cost Function 35
4.4.2 Performance in Other Historical Earthquakes 38
4.4.3 Response to Synthetic Kanai-Tajimi Filter Excitation 40
4.5 Numerical Example 2: Base-isolated MDOF Model 41
4.5.1 Applying a Saturation Constraint on the Control Force 49
4.5.2 Semiactive Control Design 51
4.5.3 Robustness Study 54
4.5.3.1 Performance under several Kanai-Tajimi produced excitations 54
4.5.3.2 Error in base stiffness 55
4.5.3.3 Error in base damping 57
4.5.3.4 Error in base mass 58
4.5.4 Incorporating Kalman Filter 59
4.5.5 Incorporating Kalman Filter and Kanai-Tajimi Filter 61
4.6 Conclusions 65
5 Nonlinear Control for Life Safety and Serviceability using Gain Scheduling Strat-
egy 67
5.1 Introduction 67
5.2 Problem Formulation 67
5.3 Numerical Results 69
5.3.1 Approach I: Switching Between LQR
a
and LQR
d
without an Intermediate
Transition 69
5.3.2 Approach II: Gain Scheduled Controller with an Intermediate Transition
Function 75
5.3.2.1 Applying a saturation constraint on the control force 82
5.3.2.2 Multiple degree-of-freedom (MDOF) building model 82
5.3.3 Approach III: Online LQR Gain Scheduling 92
5.3.3.1 Control force saturation 94
5.3.3.2 Semiactive control design 95
5.4 Conclusions 96
6 Nonlinear Control for Life Safety and Serviceability using Model Predictive Con-
trol 98
6.1 Introduction 98
6.2 Problem Formulation 98
6.3 Numerical Example 100
6.3.1 Single degree-of-freedom (SDOF) Building Model 100
6.3.1.1 Quadratic cost function 101
6.3.1.2 Nonquadratic cost function 105
vii
6.3.1.3 Applying a saturation on the control force for nonquadratic MPC 108
6.3.1.4 Performance under other earthquake records 110
6.3.1.5 Quadratic objective function and applying constraints on states 110
6.3.2 Base-isolated multiple degree-of-freedom (MDOF) Building Model 112
6.3.2.1 Design LQR based on comparison to uncontrolled 113
6.3.3 Design Based on Comparison to Passive Viscous Damper 117
6.4 Conclusions 123
7 Learning-based Optimal Design of Smart Damping using Neural Network 125
7.1 Introduction 125
7.2 Problem Formulation 127
7.3 Numerical Example 1: single degree-of-freedom (SDOF) Model 129
7.3.1 Neural Network Implementation 132
7.3.2 Selecting NN Architecture and Features 135
7.3.2.1 Levenberg-Marquardt backpropagation training method 137
7.3.2.2 Bayesian Regularization training method 140
7.3.2.3 Resilient Backpropagation training algorithm 143
7.3.2.4 Summary of different training method results 143
7.3.3 Changing the Target Value atq = 0 145
7.3.4 Effects of Decimation of the MPC Data for NN Training 146
7.3.4.1 Decimation withDq = 5

146
7.3.4.2 Decimation withDq = 10

148
7.3.5 Summary of SDOF Results 149
7.4 Numerical Example 2: MDOF Model 150
7.4.1 NNs with One Hidden Layer 154
7.4.2 NNs with Two Hidden Layers 154
7.5 Conclusions 160
8 Concluding Remarks and Future Directions 162
Bibliography 167
viii
List of Tables
2.1 Free and MCS stochastic response statistics with the four nonlinear control laws 21
2.2 Earthquake response statistics with the four nonlinear control laws 22
3.1 Performance of time-varying control under scaled 1940 El Centro excitation 26
3.2 Performance of time-varying control under scaled 1994 Northridge excitation 27
4.1 Performance of HJB control under scaled 1940 El Centro excitation 37
4.2 Performance of HJB control under scaled 1994 Northridge excitation 40
4.3 Performance of HJB control under 2013 Pomona and 1995 Kobe excitations 40
4.4 Performance of HJB control under scaled Kanai-Tajimi filtered excitation 43
4.5 Parameters of a 6DOF Building Model Adapted from Kelly et al. (1972) 44
4.6 Performance of HJB under scaled 1940 El Centro excitation, 6DOF 51
4.7 Performance of saturated HJB under scaled 1940 El Centro excitation, 6DOF 52
4.8 Performance of semiactive controls under scaled 1940 El Centro excitation, 6DOF 54
4.9 Average Performance under 100 scaled Kanai-Tajimi produced excitations (with
standard deviation listed with notation below the corresponding mean value) 55
4.10 Effects of error in base stiffness on response in strong excitation 56
4.11 Robustness study, error in base stiffness, 6DOF 57
4.12 Robustness study, error in base damping, 6DOF 58
4.13 Robustness study, error in base mass, 6DOF 59
5.1 Performance of gain-scheduled control under scaled 1940 El Centro excitation 75
5.2 Performance of linear gain-scheduled control under scaled 1940 El Centro excitation 79
5.3 Performance of sigmoid gain-scheduled control under scaled 1940 El Centro excitation 81
5.4 Performance of saturated linear gain-scheduled control under scaled 1940 El Centro
excitation 83
5.5 Performance of linear and sigmoid gain-scheduled controls under scaled 1940 El
Centro excitation, 6DOF 91
5.6 Performance of saturated gain-scheduled controls under scaled 1940 El Centro
excitation, 6DOF 91
5.7 Performance of semiactive gain-scheduled controls under scaled 1940 El Centro
excitation, 6DOF 93
5.8 Performance of online LQR gain-scheduled controls under scaled 1940 El Centro
excitation, 6DOF 94
5.9 Performance of online LQR gain-scheduled controls under moderate excitation, 6DOF 95
5.10 Performance of online LQR gain-scheduled controls under other excitations, 6DOF 95
ix
5.11 Performance of saturated online LQR gain-scheduled controls under scaled 1940 El
Centro excitation, 6DOF 95
5.12 Performance of semiactive online LQR gain-scheduled controls under scaled 1940
El Centro excitation, 6DOF 96
6.1 MPC computation time [s] for quadratic problem, 2El Centro 104
6.2 Performance of nonquadratic MPC under scaled 1940 El Centro excitation,p=10,
Ipopt 107
6.3 Performance of saturated nonquadratic MPC under scaled El Centro excitation,
p=10, Ipopt 110
6.4 Performance of nonquadratic MPC under the 1994 Northridge earthquake,p=10, Ipopt110
6.5 Performance of two MPC controls under 2El Centro excitation 112
6.6 Performance of nonquadratic MPC under scaled 1940 El Centro excitation,p=15,
Ipopt, 6DOF 118
6.7 Performance of nonquadratic MPC under scaled 1940 El Centro excitation,p=40,
Ipopt, 6DOF 122
7.1 Performance comparison for MPC and CLQR under 1940 El Centro excitation 132
7.2 Comparison of 1940 El Centro response metrics using Levenberg-Marquardt training140
7.3 Performance comparison for MPC, CLQR and NN under 1940 El Centro excitation 141
7.4 Comparison of El Centro response metrics using Levenberg-Marquardt training 145
7.5 Comparison of El Centro response metrics using Levenberg-Marquardt training 149
7.6 Performance comparison under 1940 El Centro excitation,Dq=10

,ode5, different
time steps 153
7.7 Performance comparison under 1940 El Centro excitation,ode5,Dt= 1 10
2
sec 160
x
List of Figures
1.1 Cost comparison of building components, adapted from Taghavi and Miranda (2003) 2
1.2 Control block diagram 5
1.3 Model Predictive Control schematic strategy 8
1.4 NN schematic architecture 10
2.1 Cost function based on FPK equation 19
2.2 Cost function based on MC simulation for white noise excitation 20
2.3 Cost function based on free response 20
2.4 Cost function based on non-zero initial states 21
3.1 Controlled response comparison for 0.1El Centro, time-varying 26
3.2 Controlled response comparison for 3.0El Centro, time-varying 27
3.3 Controlled response comparison for 0.1Northridge, time-varying 28
3.4 Controlled response comparison for 1.0Northridge, time-varying 28
4.1 Variation of multiobjective cost relative toh 37
4.2 Controlled response comparison for 0.1El Centro for the besth, HJB 38
4.3 Controlled response comparison for 2.0El Centro for the besth, HJB 38
4.4 Controlled response comparison for 0.05Northridge, HJB 39
4.5 Controlled response comparison for 1.0Northridge, HJB 39
4.6 Frequency content of design earthquakes and Kanai-Tajimi shaping filter (adapted
from Ramallo et al. (2002)) 42
4.7 Controlled response comparison for 0.3 m/s
2
PGA Kanai-Tajimi, HJB 42
4.8 Controlled response comparison for 7 m/s
2
PGA Kanai-Tajimi, HJB 43
4.9 RMS response for 6DOF model with passive viscous dampers 45
4.10 RMS response reduction of LQR compared to Passive [%], El Centro, 6DOF 46
4.11 RMS required LQR control force [%W], El Centro, 6DOF 46
4.12 Peak response reduction of LQR compared to uncont. [%], El Centro, 6DOF 47
4.13 RMS response reduction of LQR compared to uncont. [%], El Centro, 6DOF 48
4.14 Peak required LQR control force [%W], El Centro, 6DOF 48
4.15 Variation of multiobjective cost relative toh, for the 6DOF problem 49
4.16 Controlled response comparison for 0.1El Centro, 6DOF, HJB 50
4.17 Controlled response comparison for 2El Centro, 6DOF, HJB 50
4.18 Controlled response comparison for 0.1El Centro, saturated, 6DOF, HJB 51
4.19 Controlled response comparison for 2El Centro, saturated, 6DOF, HJB 52
4.20 Active and Semiactive comparison, 6DOF, 2El Centro 53
4.21 Response ratio of 2El Centro over 0.1El Centro for varying base stiffness, 6DOF 57
xi
4.22 Response ratio of 2El Centro over 0.1El Centro for varying base damping, 6DOF 58
4.23 Response ratio of 2El Centro over 0.1El Centro for varying base mass, 6DOF 59
4.24 Actual and estimated roof abs. acceleration, 6DOF, 2El Centro 62
4.25 Actual and estimated base displacement, 6DOF, 2El Centro 62
4.26 Commanded force based on full state feedback and KF-estimated states, 6DOF,
2El Centro 63
4.27 Control diagram of a system augmented with Kanai-Tajimi and Kalman filters 63
4.28 Actual and estimated roof abs. acceleration, 6DOF, KT-produced excitation 64
4.29 Actual and estimated base displacement, 6DOF, KT-produced excitation 64
4.30 Commanded force based on full state feedback and KF-estimated states, 6DOF,
KT-produced excitation 65
5.1 Peak response reduction of LQR compared to uncont. [%], 2El Centro 71
5.2 Peak response reduction of LQR compared to uncont. [%], 2El Centro 71
5.3 Peak required LQR control force [% of the weight of the structure], 2El Centro 72
5.4 RMS response reduction of LQR compared to uncont. [%], 2El Centro 72
5.5 Simulink model used for Gain Scheduling 74
5.6 Response comparison for two optimal LQR and gain-scheduled controls, 0.1El
Centro 74
5.7 Response comparison for two optimal LQR and gain-scheduled controls, 2El Centro 75
5.8 Response comparison for two optimal LQR and gain-scheduled controls, 1El Centro 76
5.9 Simulink model used for Gain Scheduling with intermediate transition function 77
5.10 Response comparison for two optimal LQR and linear gain-scheduled controls,
0.1El Centro 78
5.11 Response comparison for two optimal LQR and linear gain-scheduled controls,
2El Centro 78
5.12 Response comparison for two optimal LQR and linear gain-scheduled controls,
1El Centro 79
5.13 Response comparison for two optimal LQR and sigmoid gain-scheduled controls,
0.1El Centro 80
5.14 Response comparison for two optimal LQR and sigmoid gain-scheduled controls,
2El Centro 80
5.15 Response comparison for two optimal LQR and sigmoid gain-scheduled controls,
1El Centro 81
5.16 g comparison for linear and sigmoid transition functions, 1El Centro 82
5.17 Response comparison for saturated linear gain-scheduled control, 2El Centro 83
5.18 RMS response reduction of LQR compared to Passive [%], El Centro, 6DOF 85
5.19 RMS required LQR control force [%W], El Centro, 6DOF 85
5.20 Peak required LQR control force [%W], El Centro, 6DOF 86
5.21 Peak response reduction of LQR compared to Passive [%], El Centro, 6DOF 86
5.22 RMS response reduction of LQR compared to uncont. [%], El Centro, 6DOF 87
5.23 Peak response reduction of LQR compared to uncont. [%], El Centro, 6DOF 88
5.24 Response comparison for linear gain-scheduled control, 0.1El Centro, 6DOF 88
5.25 Response comparison for linear gain-scheduled control, 2El Centro, 6DOF 89
5.26 Response comparison for linear gain-scheduled control, 1El Centro, 6DOF 89
xii
5.27 Response comparison for sigmoid gain-scheduled control, 0.1El Centro, 6DOF 90
5.28 Response comparison for sigmoid gain-scheduled control, 2El Centro, 6DOF 90
5.29 Response comparison for sigmoid gain-scheduled control, 1El Centro, 6DOF 91
5.30 Response comparison for saturated sigmoid gain-scheduled control, 2El Centro,
6DOF 92
5.31 Active and Semiactive comparison, 6DOF, 2El Centro 93
6.1 YALMIP and several corresponding external solvers 100
6.2 RMS response reduction of LQR compared to uncont. [%], El Centro, SDOF 102
6.3 RMS required LQR control force [%W], El Centro, SDOF 103
6.4 Peak response reduction of LQR compared to uncont. [%], El Centro, SDOF 103
6.5 Peak required LQR control force [%W], El Centro, SDOF 104
6.6 Quadratic MPC and LQR response comparison for 2El Centro 105
6.7 RMS Response reduction of MPC
NQ
compared to LQR for 2El Centro andp=10 106
6.8 Effects of prediction horizon on nonquadratic MPC computation time and cost,
2El Centro, Ipopt 107
6.9 Response comparison for nonquadratic MPC, 0.1El Centro and prediction horizon
p=10, Ipopt 108
6.10 Response comparison for nonquadratic MPC, 2El Centro and prediction horizon
p=10, Ipopt 108
6.11 Response comparison for saturated nonquadratic MPC, 2El Centro, prediction
horizonp=10, Ipopt 109
6.12 Response comparison for nonquadratic MPC, 0.1Northridge and prediction hori-
zonp=10, Ipopt 111
6.13 Response comparison for nonquadratic MPC, 1Northridge and prediction horizon
p=10, Ipopt 111
6.14 Peak response reduction of LQR compared to uncont. [%], El Centro, 6DOF 114
6.15 Peak required LQR control force [%W], El Centro, 6DOF 114
6.16 RMS response reduction of LQR compared to uncont. [%], El Centro, 6DOF 115
6.17 RMS required LQR control force [%W], El Centro, 6DOF 115
6.18 RMS response reduction of MPC
NQ
compared to LQR, 2El Centro,p=10 116
6.19 Effects of prediction horizon on nonquadratic MPC computation time and cost,
2El Centro, Ipopt 117
6.20 Response comparison for nonquadratic MPC, 0.1El Centro, 6DOF, prediction
horizonp=15, solver:Ipopt 117
6.21 Response comparison for nonquadratic MPC, 2.0El Centro, 6DOF, prediction
horizonp=15, solver:Ipopt 118
6.22 RMS response reduction of LQR compared to Passive [%], El Centro, 6DOF 119
6.23 RMS required LQR control force [%W], El Centro, 6DOF 120
6.24 Peak response reduction of LQR compared to Passive [%], El Centro, 6DOF 120
6.25 Peak required LQR control force [%W], El Centro, 6DOF 121
6.26 RMS Response reduction of MPC
NQ
compared to LQR for 2El Centro andp=10 121
6.27 Effects of prediction horizon on nonquadratic MPC computation time and cost,
2El Centro, solver:Ipopt 122
xiii
7.1 CLQR and MPC control strategies 126
7.2 Passivity constraint for controllable damping devices 128
7.3 MPC control force for SDOF model 130
7.4 CLQR control force for SDOF model 130
7.5 Simulink model to calculate the CLQR response of a SDOF structure 131
7.6 Simulink model to calculate the online MPC response of a SDOF structure 131
7.7 NN architecture when input isfdisplacement,velocityg and output isfcontrol forceg133
7.8 Control force on a unit circle around the origin 133
7.9 Unit circle around the origin in the state space 134
7.10 Control force on a unit circle around origin in the state space on scaled axes 134
7.11 NN architecture when input isq and output is the control force 135
7.12 Simulink model to calculate the offline MPC response of a SDOF structure 135
7.13 Simulink model to calculate the NN response of a SDOF structure 136
7.14 Error between NN outputs trained with Levenberg-Marquardt and the target MPC
results 138
7.15 Cost value of different NNs trained with Levenberg-Marquardt 139
7.16 Control force comparison for Levenberg-Marquardt-trained NNs 139
7.17 Response comparison of the two Levenberg-Marquardt-trained NNs 140
7.18 Error between NN outputs trained with Bayesian and the target MPC results 141
7.19 Cost value of different NNs trained with Bayesian 142
7.20 Control force comparison for Bayesian-trained NNs 142
7.21 Error between NN outputs trained with Resilient and the target MPC results 143
7.22 Cost value of different NNs trained with Resilient 144
7.23 Control force comparison for Resilient-trained NN 144
7.24 Error between NN outputs trained with Levenberg-Marquardt and the target MPC
results 146
7.25 Cost value of different NNs trained with Levenberg-Marquardt 147
7.26 Control force comparison for Levenberg-Marquardt-trained NNs 147
7.27 Error between Levenberg-Marquardt-trained NN outputs and the target MPC results
decimated withDq = 5

148
7.28 Cost value of different NNs trained with Levenberg-Marquardt, decimated with
Dq = 5

149
7.29 Control force comparison for Levenberg-Marquardt-trained NNs decimated with
Dq = 5

150
7.30 Error between Levenberg-Marquardt-trained NN outputs and the target MPC results
decimated withDq = 10

151
7.31 Cost value of different NNs trained with Levenberg-Marquardt, decimated with
Dq = 10

152
7.32 Control force comparison for Levenberg-Marquardt-trained NNs decimated with
Dq = 10

153
7.33 NN architecture for 2DOF problem 154
7.34 Error between Levenberg-Marquardt-trained NN outputs and MPC target values,
2DOF 155
7.35 Cost value of different NNs trained with Levenberg-Marquardt, 2DOF 155
xiv
7.36 Error between Levenberg-Marquardt-trained NN (with 2 hidden layers) outputs and
MPC target values, 2DOF 156
7.37 Cost value of different Levenberg-Marquardt-trained NNs with two hidden layers,
2DOF 157
7.38 Levenberg-Marquardt-trained NN with 200 neurons in the hidden layer and seed 3,
x
1
=x
2
= 0 158
7.39 Levenberg-Marquardt-trained NN with 40 neurons in each of 2 hidden layers and
seed 7,x
1
=x
2
= 0 158
7.40 CLQR-determined control force surface,x
1
=x
2
= 0 159
7.41 MPC-determined control force surface,x
1
=x
2
= 0 159
xv
Chapter 1
Introduction
1.1 Motivation
Each year, excessive loading produced by extreme events, such as earthquakes and hurricanes,
causes damage and failure of many buildings and infrastructures, with losses to public and private
building owners in terms of actual structural damage and its economic impacts as well as injury
and loss of life. Moreover, moderate-to-strong winds and weak-to-moderate earthquakes occur
much more frequently, affecting human comfort and causing damage to nonstructural components
and building contents in addition to loss of occupancy. “Over the last decade, China, the United
States, the Philippines, India and Indonesia [have been] the five countries ... most frequently hit
by natural disasters” (Guha-Sapir et al., 2011). Quantitative estimates of losses in any disaster
can be divided into two major categories: (1) deaths, injuries and disabilities, costs associated
with repair and replacement of damaged buildings and properties, loss of fundamental function
and restoration time for critical facilities (e.g., hospitals, transportation and utilities); and (2)
psychosocial effects, health effects and long-term impact. “The annualized loss from earthquakes
nationwide is estimated to be $5.3 billion per year, with California, Oregon, and Washington
accounting for $4.1 billion, or 77%” of the U.S. total (FEMA, 2008). Moreover, damage to building
nonstructural components and other building contents plays a very important role in the required
repair cost and time; together, these represent the vast majority of the cost of commercial structures,
from 65–85% in many cases as shown in Figure 1.1, which is adapted from Taghavi and Miranda
(2003). Nonstructural components are vulnerable to large accelerations, with damage initiating at
low levels of external excitations; even though a weak earthquake may cause little (if any) structural
damage due to the small induced structural drifts, the accelerations can easily result in significant
1
Structure
Controller
Excitation
(wind, earthquake, etc)

Response
(drift, accel., etc.)
sensor
Office Hotel Hospital
0
10
20
30
40
50
60
70
80
Contents
Nonstructural
Structural
% Cost of the building
Figure 1.1: Cost comparison of building components, adapted from Taghavi and Miranda (2003)
damage to nonstructural components such as ceilings, piping and so forth (Taghavi and Miranda,
2003). Similarly, common wind-induced vibration may not approach a level to affect structural
safety, but it may still cause significant discomfort to the occupants and nonstructural effects; a
common reminder is the 60-story John Hancock Tower in Boston, which experienced acceleration
levels far beyond those intended for occupant comfort, and resulting in falling panels of glass from
many windows on the tower (Christenson, 2002, Williams and Kareem, 2003, Schwartz, 2004).
Hence, it is imperative for the civil engineering community to develop and implement strategies to
mitigate structural responses in both weak and strong excitations, reducing the absolute acceleration
responses in weak-to-moderate excitations to prevent nonstructural and building content damage and
ensure occupant comfort, and decreasing the drift responses in strong excitations to avoid structural
damage and possible structural collapse — for both newly designed structures and retrofitting of
existing structures.
In the last few decades, in addition to conventional means of changing structural design, various
types of structural control devices have been studied and implemented to mitigate the effects
of external forces on structures. These devices, and their corresponding strategies for designing
them and commanding their forces, can be divided into three primary categories: passive devices,
active control strategies, and controllable passive (“semiactive”) strategies. Passive control devices
dissipate energy induced by external excitations, usually by converting the kinetic energy of motion
and potential energy of strain into heat or inelastic behavior of the device, or change the amount of
energy allowed into the structure. Passive devices are attractive since they are relatively simple and
require no external power source; however, their properties cannot be changed once designed, and
the forces they provide are functions only of local deformations and not of global responses, so their
effectiveness can be limited. Active strategies collect data by instrumentation and require external
power to operate actuators to apply forces, both dissipative and nondissipative, to the structure. Such
2
strategies have been shown to significantly reduce responses, but their implementation has been very
modest due to the high levels of external power necessary for significant performance — power that
may not be available during strong events due to electricity outage — and concerns about stability if
the control system is not able to well-predict the response of the structure. Researchers have turned
to controllable passive devices, also called “semiactive” because they combine properties of passive
and active systems. Controllable passive devices are those that dissipate energy through passive
means, but with controllable properties so that device forces can be modified and adjusted based on
sensor measurements, feedback of excitation, changing objectives and so forth. Common strategies
for commanding the forces in these devices involve designing a primary controller such as for an
active device, but then using a secondary controller to eliminate from the command any unrealizable
control forces (Housner et al., 1997). Thus, these devices provide the reliability of passive devices
and the adaptability of active systems, requiring only battery-strength power levels.
For active and controllable passive devices, the real challenge is developing appropriate control
strategies to command them in such a way that they can accomplish the twin objectives of mitigating
drift-induced damage in the rare extreme events but also sustain serviceability in the much more
frequent moderate events. Most of the research, and the limited implementation, using these devices
focus on one objective (or a linear combination of several objectives) for a particular “design-level”
excitation; the performance in other excitations, whether more moderate or more extreme, and/or
performance measured by different metrics are dictated by the control strategy and not optimized —
often resulting in significantly lower performance than if the same type of control strategy were
designed and optimized for this other excitation level and/or other objective. Different objectives for
different design levels cannot be achieved by linear strategies. Further, it is simply not cost-effective
to focus the design on extreme events that are rather unlikely to occur during a structure’s expected
lifetime when it means sacrificing performance in the more moderate, but still significant, events
that are quite likely to occur in that lifetime and cause significant costs for repair of nonstructural
components, occupancy loss and so forth; similarly, focusing on the moderate events may mean
suboptimal performance in the extreme event, which could be extremely costly with extensive
damage or failure if that extreme event actually comes to pass.
Thus, this research seeks to study optimal nonlinear control strategies that can achieve different
objectives in different levels of excitation to provide, in one control strategy, both optimal service-
ability performance in the frequent weak-to-moderate events and optimal life safety in the infrequent
extreme events. The main goal of this study is to develop nonlinear strategies that will provide
safer and more comfortable structures that are less likely to require expensive repairs and retrofits.
The goal is to develop methodologies for designing, analyzing and implementing nonlinear control
strategies that are optimal in mitigating different structural responses in different magnitudes of
3
excitation in order to reduce structural damage and maintain occupant comfort and serviceability.
Beyond civil engineering, the research results should be generalizable to other structural control
problems, such as those in mechanical and aerospace engineering, where the objectives may be
slightly different, but the same principle of different objectives for different excitations often still
applies.
1.2 Structural Control
Structural control had its roots primarily in the aerospace field, but it was rapidly introduced into
the civil engineering arena for the protection of buildings and bridges against extreme loads of
earthquakes and winds (Housner et al., 1997). Over the last few decades, many researchers have
investigated a variety of means for mitigating building dynamic response to natural hazards, both for
new structures and retrofit of existing ones (Datta, 2003). This has led to major accomplishments
in structural control techniques and devices which can enhance the performance of structures and
reduce damages and its associated costs. The interested reader is directed to detailed reviews of the
state-of-the-art and state-of-the-practice of structural control strategies (Housner et al., 1997, Soong
and Spencer, 2002).
Structural control methods can be divided into three major categories: passive, active and control-
lable passive (“semiactive”) strategies. A brief description of each of these control strategies is
discussed in the following paragraphs.
1.2.1 Passive Control
Passive control devices — such as viscous fluid dampers (Makris et al., 1993, Terenzi, 1999, Pekcan
et al., 1999, Main and Jones, 2002, Lin and Chopra, 2003), tuned mass dampers (Den Hartog, 1947,
Villaverde and Martin, 1995, Hrovat et al., 1983), metallic yielding devices (Kelly et al., 1972,
Skinner et al., 1975, 1980), and so forth — dissipate the mechanical energy of structural motion
that is induced by the external excitation; base isolation systems, which may explicitly incorporate
passive energy dissipators, are another passive strategy to isolate the structure from the ground,
reducing the flow of energy into the structure. An advantage of passive devices is that they require
no external power source to operate. On the other hand, they are unable to adapt to structural
and excitation variation since they do not have feedback capability; the only information that they
“know” is local deformation, with the resulting forces being a predetermined function of the local
motion.
4
1.2.2 Active Control
In an initial concept study, Yao (1972) proposed the idea of applying an additional force to reduce
structural responses, thus beginning research into active control of civil structures. Active and
semiactive controllers, also known as smart control strategies, can adjust and tune their properties
to adapt themselves based on any change in structural behavior, resulting in a better applicability
and performance.
Active control strategies, such as active bracing systems, hydraulic actuators, and active mass
dampers (AMDs), collect data via instrumentation installed on the structure, feed this information
into a control computer that then command various actuators to exert forces on, or within, the
structure. The actuators require significant external power to exert the forces necessary to move the
large mass of a civil structure; further, they can apply both dissipative and nondissipative forces
to the structure. However, the external power requirements may not be available during strong
events due to electricity outage; further, as actuators can add energy to the structural/mechanical
system, they have the potential to cause further problems if the controller has a poor model of
structural behavior or faulty sensors give erroneous information. Finally, installing and maintaining
an active actuator and sensor network can be costly. As a result, active control strategies have had
limited impact at full-scale since the first demonstration in 1989 (Kobori et al., 1991a,b, Soong and
Reinhorn, 1993, Sakamoto et al., 1994). Figure 1.2 shows a typical active control block diagram of
the flow of signals and information within a structural control system. The control strategy will
determine the control force based on the feedback data of structural response and feedforward of
excitation data. The interested reader is directed to the papers by Housner et al. (1997), Spencer
and Sain (1997), Nishitani and Inoue (2001) for a comprehensive literature review of active control
strategies.
Excitation
States
Structure
Controller Sensors
Responses
Figure 1.2: Control block diagram
5
1.2.3 Semiactive Control
Controllable passive strategies combine the properties of passive and active systems, so they are
often denoted “semiactive” systems. At the heart of these systems are devices that are inherently
passive but with controllable properties, such as variable orifice dampers (which can act as a variable
damping device or an on/off variable stiffness element), variable friction dampers, and controllable
fluid dampers such as electrorheological and magnetorheological (MR) dampers. The algorithms
for commanding the controllable properties may take several forms; one common approach is to
design a primary controller assuming the device were linear and fully active, but coupled with a
secondary controller to eliminate the nondissipative forces that the device cannot exert, or to change
the command via pulse-width modulation. Semiactive strategies offer the reliability of passive
devices and the adaptability of active systems, requiring only battery power to operate (Housner
et al., 1997).
The interested reader is directed to the Housner et al. (1997) and Spencer and Sain (1997) reviews
on structural control, a state-of-the-art review on semiactive control systems in Symans and Con-
stantinou (1997), and state-of-the-art and state-of-the practice in Soong and Spencer (2002) as well
as the wealth of papers in the literature: friction dampers (Kannan et al., 1995), electrorheologi-
cal (ER) dampers (Gavin and Hanson, 1994, Ehrgott and Masri, 1992, Masri et al., 1995), active
variable stiffness devices (Kobori et al., 1993), magnetorheological (MR) dampers (Dyke et al.,
1996, Spencer et al., 1997), large-scale MR fluid dampers (Yang et al., 2002), controllable friction
dampers (Fujita et al., 1994), active and hybrid structural control systems (Soong and Reinhorn,
1993) and seismically isolated structures (Naeim and Kelly, 1999).
A common challenge for all of these strategies is how to develop optimal designs to achieve the
control objectives. Among common control strategies, linear control laws are most often applied
for active devices and in the primary controller for semiactive devices, mostly due to their simple
design and implementation, but they may not provide the best performance in some cases. Thus,
some researchers have investigated nonlinear control techniques to address these challenges.
1.3 Hamilton-Jacobi-Bellman (HJB)-based Control
While there are a multitude of accepted techniques for designing linear control strategies for mit-
igating the response of structures due to external loads and excitations, there are no elegant and
widely-applied nonlinear methods due to the unique nature of each nonlinear system or control.
6
Some researchers have used nonlinear control to directly address nonlinearities in structural models
(Suhardjo et al., 1992, Yang et al., 1995). Nonlinear control of linear structure-like systems was
first proposed by Rekasius (1964), using the HJB equation for a class of non-quadratic costs; others
(Speyer, 1976, Bernstein, 1993, Tomasula et al., 1994, Wu et al., 1994, 1995, Agrawal and Yang,
1996) have proposed similar approaches. Results of these studies show some performance improve-
ments relative to that with linear controllers, particularly in the presence of strong earthquakes. It is
well established that designing an optimal control strategy for a nonlinear system will lead to the
HJB partial differential equation (Bryson and Ho, 1975, Primbs et al., 1999). The HJB equation is a
partial differential equation which is central to optimal control theory. The “value function”, which
is the solution of the HJB equation, gives the optimal cost-to-go for a given dynamical system with
a corresponding cost function. This equation is derived from the theory of dynamic programming
by Richard Bellman and coworkers in the 1950s (Bellman, 1957). Further, Bellman’s equation is
the corresponding discrete-time equation. In continuous time, the result is the extension of earlier
work on the Hamilton-Jacobi equation by William Rowan Hamilton and Carl Gustav Jacob Jacobi.
Finding the exact analytical solution of the HJB equation is very difficult, so many researchers have
found other approximate methods more attractive (Primbs et al., 1999), such as: control Lyapunov
function (CLF) (Sontag, 1983); receding horizon control (RHC) (Kwon and Pearson, 1977); fuzzy
logic control (Feng, 2006); etc.
1.4 Gain Scheduling Control
Gain scheduling is a successful control methodology in many engineering applications. In control
theory, gain scheduling strategy is one approach mostly applied to control non-linear systems and
systems whose dynamics vary with time or operating condition. Basically, it linearizes the system
around different operating/equilibruim points of the system and finds appropriate optimal linear
controllers for each linearized region of the system response space. At each time step(s), scheduling
variables, also known as observable variables, determine which linear controller should be applied
based on the current operating region of the system (Leith and Leithead, 1998). Gain scheduling
has proven to be most suitable when the scheduling variables vary slowly compared to the control
bandwidth, and is most challenging for fast-varying scheduling variables. Theoretical analysis is
required to investigate stability and robustness and global performance of the design (Shamma and
Athans, 1990).
7
1.5 Model Predictive Control (MPC)
Model predictive control (MPC) is a control method that was popularized in the 1970s for petroleum
refinery operations. Since then, model predictive control (MPC) has become the benchmark for
complex constrained multivariable control problems in industrial engineering (Qin and Badgwell,
2003, Bemporad et al., 2004). Since MPC has the ability to handle constraints, nonlinearities and
time varying effects, it is frequently applied in different control applications including automotive,
aerospace and process industries. As shown in Figure 1.3, given the current state of the system,
MPC manipulates candidate future control inputs to optimize the predicted future outputs of the
dynamical system. The optimal sequence of future control inputs is determined by minimization of
a cost function written in terms of the predicted states and the control inputs in future time steps;
however, only the first input is actually commanded and the whole set of calculations is repeated
again at each subsequent future time step. The number of future steps over which states of the
system are predicted is called the prediction horizon, whereas the number of steps over which a
control input is considered, in many cases equal to the prediction horizon, is called the control
horizon. MPC is also called Receding Horizon Control since the prediction horizon keeps being
shifted forward.
Since MPC performs the optimization for each time step — a computationally expensive process
k k+1 k+N k+p
Figure 1.3: Model Predictive Control schematic strategy
8
— its application for complex and high-order dynamical systems is quite challenging unless the
optimization can be performed rapidly in real time. To overcome this challenge, one can preperform
the optimization, determine control forces for a large set of state vectors, store them in a look-up table
and implement the appropriate control force based on the current states in real-time computation.
This approach, called Offline MPC (Bemporad et al., 2002), is quite fast in realtime computations,
but it can be challenging to precompute control forces for all possible states, especially for more
complex systems.
Mei and colleagues (Mei et al., 2001) explored the application of MPC in structural control. They
investigated building models with active tendon systems with full state feedback. This work was
later extended for acceleration feedback (Mei et al., 2002); however, their approach was suitable only
for linear structures controlled with unconstrained control devices and not suitable for structures
with smart dampers, constrained control forces with nonlinear behaviors.
1.6 Neural Networks
Nonlinearity, uncertainty, and time-varying structural properties have led researchers to seek some
promising control techniques other than conventional optimal linear control strategies. Artificial
neural networks (ANN) have been studied widely in civil engineering and especially structural
control to mitigate undesired vibration. Initial studies by Chen et al. (1995) and Ghaboussi and
Joghataie (1995) showed that neural network based control can be applied successfully to control
vibration of civil structures (Kim et al., 2000). Since a neural network can learn very complex and
nonlinear relations, its application in structural control has been intensively investigated by many
researchers (e.g., Bani-Hani and Ghaboussi, 1998, Wen et al., 1995). A comprehensive review of
neural networks for feedback control is presented in Lewis (1999).
A general schematic architecture of a NN is shown in Figure 1.4. The output of each neuron can be
presented as
O=b+
n
å
i=1
w
i
I
i
(1.1)
whereO is the output of a neuron,b is the bias,w
i
is the weight corresponding to the link connecting
input I
i
to the neuron, and n is the total number of connections coming to the neuron (Dreyfus,
2005). NN topologies can be divided into two major classes: Feedforward, also known as multilayer
Perceptron, and feedback NN, also known as recurrent NN. Feedforward NN has been applied
as a function approximator in variety of applications including curve fitting (Tang et al., 2007).
Feedback NN has some loop connections between nodes, resulting in slow convergence compared
9
Input Layer
Hidden Layer
Output Layer
Figure 1.4: NN schematic architecture
to feedforward NN. Training techniques can be divided into two major categories: supervised and
unsupervised learning. In supervised learning, a NN adjusts its weights and biases so that its outputs
become close to the provided target values. In unsupervised learning, a NN adjusts its weights and
biases so that it can describe the hidden relation between provided unlabeled data.
A NN (supervised or unsupervised) should be trained such that it learns the general behavior of
the training data; however, some strategy should be taken in order to prevent overfitting, where
the NN learns every specific behavior of the training data, but performs poorly for any other data
not included in the training process. Using early stopping is one approach to prevent overfitting.
In this method, data is divided into three subsets, namely: training, validation and test data sets.
The training set is used for computing the gradient and updating the network weights and biases
(Beale et al., 2014). While the NN is being trained with the training data, it is simultaneously being
monitored and validated with validation data error. Overfitting is indicated and the training will
stop if the validation error begins to increase continuously while the training error is still decreasing
(Brown et al., 2005). Performance of the trained NN can be examined with the test data that was
excluded from both the training and validation data.
In the Neural Network toolbox (Beale et al., 2014) in MATLAB, there are several functions to divide
data into training, validation and test subsets: dividing the data randomly (dividerand), dividing the
data into contiguous blocks (divideblock), dividing the data using an interleaved selection (divideint)
and dividing the data by index (divideind) (Beale et al., 2014). Depending on how data is distributed
(sorted vs. randomly distributed), one or some of these methods may be more appropriate (e.g., if
the data is sorted in ascending or descending order, dividerand is a good choice, while divideblock
10
would give poor results because it is essential that the NN be provided training data that spans the
range of input and output data).
There are several training algorithms available in the Neural Network toolbox in MATLAB:
Levenberg-Marquardt, Bayesian Regularization, Resilient Backpropagation, etc. The fastest training
algorithm is generally Levenberg-Marquardt, which is the default training function for feedforward
NNs in the Neural Network toolbox in MATLAB (Beale et al., 2014). The Levenberg-Marquardt
algorithm, developed by Levenberg (1944) and Marquardt (1963), provides a numerical solution to
the problem of minimizing a nonlinear function. It is stable and is the fastest algorithm suitable
for training small and medium-sized NN problems (Demuth and Beale, 1995). The Levenberg-
Marquardt algorithm has the stability of the steepest descent method (Rumelhart et al., 1986),
and also the fast convergence of the Gauss-Newton algorithm, named after the mathematicians
Carl Friedrich Gauss and Isaac Newton. In fact, the Levenberg-Marquardt algorithm assumes a
quadratic curvature approximation, when possible, to significantly speed up the convergence; for
more complex curvatures it switches to the steepest descent algorithm (Yu and Wilamowski, 2011).
In Bayesian regularization, a linear combination of squared errors and weights is minimized.
Moreover, the linear combination is modified such that the resulting network has good generalization
qualities after training (MacKay, 1992). In this method, the weights and biases are assumed to be
random variables with specified distributions. Then, the regularization parameters are related to the
unknown variances associated with these distributions. Further, statistical techniques are used to
estimate these parameters (Beale et al., 2014).
1.7 Overview of This Dissertation
This dissertation consists of the following parts:
First, Chapter 2 presents a comparison of some of the analytical and numerical methods for finding
optimal nonlinear controllers, particularly for cost functions that are even order powers of the states
and quadratic in the control. A scalar model (i.e., a scalar state-space equation) is used for the
comparisons since analytical solutions exist for some of the methods.
Next, Chapter 3 presents an example of a control strategy that can achieve both service-level and life
safety-level objectives with one general control law. This particular control strategy is not intended
to be optimal, but to show that a strategy exists to reduce drift in the rare extreme events and reduce
absolute acceleration in the more frequent weak-to-moderate ground motions.
11
Further, in Chapter 4, this dissertation develops optimal nonlinear control strategies specifically
to address the bifurcation in control objectives between moderate events and extreme excitations:
maintaining serviceability in the more common moderate motions, with low accelerations so that
occupants are not disturbed, nonstructural elements and building contents are not damaged, and
occupancy is not interrupted; and focus on life safety in the rare extreme events to ensure that
structural damage is minimized and the likelihood of failure is small. HJB based methods are used
to design these optimal nonlinear control strategies analytically.
Next, Chapter 5 investigates the application of gain scheduling to achieve the two aforementioned
objectives. Scheduling of the resulting controller and switching between the two controllers is used
depending on the strength of earthquake.
Further, Chapter 6 presents a nonlinear control strategy based on a MPC method that minimizes
a nonquadratic cost function with higher-order polynomials of the states. The nonquadratic term
in the cost function is defined such that it becomes negligible in weak excitations, resulting in
the same performance as an acceleration-reducing LQR control, whereas it becomes dominant in
strong excitations, resulting in better drift reduction compared with that of the corresponding LQR
problem.
Finally, in Chapter 7, this dissertation investigates and develops optimal damping strategies based
on neural networks that can be trained using the results from the MPC methodology, presented
in Elhaddad and Johnson (2015), to represent the optimal control law of a semiactively-damped
structural model.
The last chapter of this dissertation (Chapter 8) presents a list of key conclusions and also some of
the areas that need further investigation.
12
Chapter 2
Optimal Nonlinear Control using a
Non-Quadratic Cost Function for Scalar
Systems
1
2.1 Introduction
Linear control laws have been applied for designing all types of structural control strategies due
to the simplicity of linear feedback. However, linear control is not necessarily the best approach
for achieving the diverse objectives of structural control. This chapter presents an investigation of
several aspects of the optimal nonlinear control problem; the focus herein is on scalar systems as
analytical solutions and results exist (not generally the case for more complex systems). The cost
function is assumed in a specific form with non-quadratic terms such that an exact solution exists for
the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Four different approaches for finding
the optimal control law are studied. First, solving the HJB equation for the non-quadratic cost
function, the optimal control law can be obtained as a polynomial in the state for the scalar system.
Second, for a Gaussian white noise excitation, the Fokker-Planck-Kolmogorov (FPK) equation can
be solved exactly for a scalar system, giving a direct computation of response moments that can
be used in the non-quadratic cost function to determine an optimal nonlinear control. Third, a
Monte Carlo (MC) simulation is performed with Gaussian white noise excitation for a range of
control gains to numerically approximate the optimal gains in a nonlinear control law. Fourth and
finally, the cost function is computed for the same system with non-zero initial condition but no
1
Hemmat-Abiri and Johnson (2013b) reports a subset of this chapter.
13
excitation by analytically solving the state equation for the free response; the optimal control gains
can be found by setting equal to zero the derivatives of the cost function. Because this system
has non-Gaussian responses and a non-quadratic cost function, the certainty equivalence property
(van de Water and Willems, 1981, Chow, 1976) does not apply —i.e., in contrast with linear control
of linear systems with quadratic cost functions, the HJB-derived “optimal” solution for the optimal
nonlinear control of a system without excitation but non-zero initial condition may be different from
the optimal nonlinear control for a Gaussian white noise excitation.
2.2 Problem Formulation
This section will first formulate the problem for an arbitrary linear system, and then briefly sum-
marize the HJB-based methods for finding the optimal nonlinear control. Based on one such
method, one particular scalar system and a corresponding non-quadratic cost function will both be
defined and used as a test bed for all nonlinear control strategies discussed herein (MC simulation,
probability-based FPK formulation, and the optimal nonlinear control for the system without forcing
but a non-zero initial condition.) The MC simulation approach is expected to give an approximate
solution similar to the FPK result.
Consider a linear time-invariant system subjected to unknown disturbance vector w and control
signal vector u; for example, this may be cast in state space as
˙ q= Aq+ Bu+ Ew (2.1)
with initial condition q(0)= q
0
. For structural systems, state vector q will contain generalized
displacements and velocities, whereas state matrix A, control influence matrix B and excitation in-
fluence matrix E depend on the usual structural system matrices, the force locations and orientations.
LetL
Q
(t)= q
T
Qq+ 2q
T
Nu+ u
T
Ru denote a quadratic function of the states and control at some
timet, with weighting matrices Q= Q
T
0, R= R
T
> 0 and N such that

[Q N]
T
[N
T
R]

0.
Then, the control law that minimizes quadratic cost functions of the forms (depending on the nature
of excitation w and the initial condition q
0
)
J
Q
=
Z
¥
0
L
Q
(t)dt or J
Q
= lim
t
f
!¥
1
t
f
Z
t
f
0
L
Q
(t)dt or J
Q
=
E
[L
Q
(t)] (2.2)
are linear and can be designed using thelqr command in MATLAB
R
.
As quadratic cost functions do not capture the trade offs in control objectives for some problems, a
14
cost function that is non-quadratic in the states (and/or control) can be useful. For example, Wu
et al. (1995) proposed non-quadratic cost
J=
1
2
Z
t
f
0
[q
T
Qq(1+ q
T
Pq)+ u
T
Ru+(q
T
Pq)q
T
PBR
1
B
T
Pq(1+ q
T
Pq)]dt (2.3)
where P is the positive definite symmetric solution to the algebraic Riccati equation of the linear
control
PA+ A
T
P PBR
1
B
T
P+ Q= 0 (2.4)
Similarly, Agrawal and Yang (1996) proposed cost function
J=
Z
¥
0
[q
T
Qq+ u
T
Ru+(q
T
M
2
q)(q
T
Q
2
q)+(q
T
M
2
q)
2
q
T
M
2
BR
1
B
T
M
2
q]dt (2.5a)
where
M
2
(A BR
1
B
T
P)+(A BR
1
B
T
P)
T
M
2
+ Q
2
= 0 (2.5b)
in which M
2
= M
T
2
> 0; and matrix Q
2
can be chosen arbitrarily. It can be verified that costs (2.3)
and (2.5) are identical for a specific choice of Q
2
= Q+ PBR
1
B
T
P which results in M
2
= P.
Although the proposed nonlinear control laws by Wu et al. (1995) and Agrawal and Yang (1996)
are described here, their approaches are similar to what has been reported previously (Rekasius,
1964, Speyer, 1976, Bernstein, 1993, Tomasula et al., 1994).
2.2.1 HJB-based Methods and Test Bed Scalar System
It is well established that designing an optimal control strategy for a nonlinear system will lead
to the HJB partial differential equation (Bryson and Ho, 1975, Primbs et al., 1999), which can be
expressed as
¶V
¶t
=H

q,u
opt
,
¶V
¶q
,t

(2.6)
in whichV is the value function (the cost to go),H =L+(¶V /¶q) ˙ q is the Hamiltonian function,
and u
opt
is the optimal control found by solving¶H/¶u= 0. However, finding the HJB analytical
solution is very difficult unless the non-quadratic cost function is defined in a specific form. The
cost functions in (2.3) and (2.5) have solutions, respectively,
u
(2.3)
opt
=R
1
B
T
Pq(1+ q
T
Pq) (2.7a)
u
(2.5)
opt
=R
1
B
T
Pq R
1
B
T
(q
T
M
2
q)M
2
q (2.7b)
15
ScalarSystem: A test bed structural system model must be chosen so that numerical comparisons
can be made between the HJB-based approaches and others. As the FPK-based approach discussed
in the next section can be analytically solved for a scalar system, the test bed system equation of
motion is
˙ x=ax+bu+w (2.8)
The form of the cost function of Agrawal and Yang (1996) is used here in scalar form
J=
Z
¥
0
h
qx
2
+ru
2
+m
2
q
2
x
4
+m
4
2
b
2
r
1
x
6
i
dt (2.9)
whereq,r,m
2
andq
2
are the scalar forms of Q, R, M
2
and Q
2
, respectively, in (2.5a). Solving the
corresponding HJB problem ignoring disturbance w, the optimal control is
u=r
1
bpxr
1
bm
2
2
x
3
(2.10)
where m
2
= q
2
/2
p
ˆ a and ˆ a = a
2
+qb
2
r
1
, with p = r(a+
p
ˆ a)b
2
as the scalar form of P.
Alternately, if the optimal control is assumed of the formu=k
L
xk
NL
x
3
, wherek
L
andk
NL
are
control gains, one can rewrite cost (2.9), in a form that will be used in the subsequent sections to
find optimal control laws with different excitation and/or initial condition assumptions, as
J=
Z
¥
0
h
(q+rk
2
L
)x
2
+(m
2
q
2
+ 2rk
L
k
NL
)x
4
+(m
4
2
r
1
b
2
+rk
2
NL
)x
6
i
dt (2.11)
2.2.2 Probability-based Method (FPK Equation)
The FPK equation (Soong and Grigoriu, 1993) is a partial differential equation in the transition
probability density function of a dynamical system. Assume that the excitation w is a stationary
zero-mean Gaussian white noise process with unit intensity (i.e.,
E
[w(t)w(t+t)]=d(t)). The
drift term in (2.8) is defined asd(x)=ax+bu. The interest here is in the stationary solution, so the
time derivative terms of the FPK equation are set to zero, leaving
d
dx
[2d(x)f(x)]=
d
2
dx
2
f(x) (2.12)
where f(x) is the stationary probability density function ofx. With a cubic control law as previously
discussed, the drift becomes d(x)=(a+bk
L
)xbk
NL
x
3
. Substituting this form of the drift
16
into (2.12) and solving for density function f(x) gives
f(x)= exp

(bk
L
a)
2
4bk
NL

1
K
1/4

[bk
L
a]
2
4bk
NL

r
2bk
NL
bk
L
a
exp

(bk
L
a)x
2

bk
NL
x
4
2

(2.13)
whereK
n
(z) is the modified Bessel function of the second kind defined as
K
n
(z)=
p[I
n
(z)I
n
(z)]
2sin(pn)
(2.14)
in whichI
n
(z) is the modified Bessel function of the first kind as
I
n
(z)=
1
p
Z
p
0
e
zcosq
cos(nq)dq
sin(np)
p
Z
¥
0
e
zcoshtat
dt (2.15)
The solution (2.13) for the density function can be used to compute the even-ordered moments
E
[x
n
]=
Z
¥
¥
x
n
f(x)dx, n= 2,4,6 (2.16)
in cost function (2.9)
J=(q+rk
2
L
)
E
[x
2
]+(m
2
q
2
+ 2rk
L
k
NL
)
E
[x
4
]+(m
4
2
b
2
r
+rk
2
NL
)
E
[x
6
] (2.17)
The optimal control gains are then found by setting equal to zero the partial derivatives ofJ with
respect tok
L
andk
NL
.
2.2.3 Numerical Method (MC Simulation)
A numerical MC simulation approach should be able to also find the optimal gains k
L
and k
NL
through a parameter study. Using a discrete-time approximation of zero-mean unit-intensity
Gaussian white noise — a Gaussian pulse process with variance 1/Dt for time stepDt — the cost
(2.17) can be approximated usingN realizations, simulated from quiescent initial conditions until
the mean square statistics converge (here, mean square responses over[0,t
f
] are averaged overN
realizations to approximate the statistics). A response surface, built from the parameter study, is
then minimized to find optimal control gains.
17
2.2.4 Deterministic Method (Nonzero Initial Condition)
The HJB approach effectively ignores the excitation, so the optimal nonlinear control from HJB
should be the same as the control that minimizes the same cost function as the system decays from
non-zero initial conditions. With the cubic control law, the system equation of motion becomes
˙ x=ax+bu=(bk
L
a)xbk
NL
x
3
(2.18)
which can be solved analytically to find the response
x(t)=
p
bk
L
a

bk
NL
+e
2(bk
L
a)t

bk
NL
+
bk
L
a
x
0
2

1/2
(2.19)
where x(0)= x
0
is the non-zero initial condition. Even powers of free response (2.19) can be
integrated analytically to compute cost (2.11). The result can be differentiated with respect to gains
k
L
andk
NL
to find their optimal values.
2.3 Numerical Example
In this section, a numerical example is presented, solving for the optimal controlu=k
L
xk
NL
x
3
of the scalar system with each of the four methods presented in the previous section. For the
purposes of comparison, the values of the parameters are chosen to be those used by Agrawal and
Yang (1996):
a=0.025, b= 1, r= 1, q= 1.5 and q
2
= 1.5 (2.20a)
) ˆ a= 1.5006, m
2
= 0.6122 and p= 1.2 (2.20b)
HJB: The HJB solution is found by simply substituting the parameters (2.20) into optimal control
(2.10) to findk
HJB
L
= 1.2 andk
HJB
NL
= 0.3748.
FPK: Using the parameter values, the control gains that minimize the expected cost (2.17) are
k
FPK
L
= 1.52926 andk
FPK
NL
= 0.329258. The cost surface, shown in Figure 2.1 for the control gain
ranges[0.8,2.0][0.1,1.0], is convex.
MonteCarlo: Simulations, using MATLAB’s Simulink
R
, of 10000 realizations of the system over
a time interval [0,50] s, with a time step equal to 0.01 s, are performed with the discrete-time
18
Printed by Mathematica for Students
Figure 2.1: Cost function based on FPK equation
approximation to Gaussian white noise for a grid of control gains(k
L
,k
NL
)2f0.8,0.9, ,2.0g
f0.1,0.2, ,1.0g. The expected cost (2.17) is estimated from these realizations, as shown in
Figure 2.2. Minimizing the resulting response surface gives approximate optimal control gains
k
MCS
L
= 1.53 andk
MCS
NL
= 0.317.
Freeresponse: Using free response (2.19) with the parameters (2.20), the cost of the free response
can be computed as a function of the control gains. For small initial conditions, the cost function
is merely a function ofk
L
whereas, in large initial conditions, it is a function ofk
NL
. Figures 2.3a
and 2.3b show the variation of the cost function relative to k
L
and k
NL
for large and small initial
conditions, respectively. For initial condition x
0
= 1 the optimal control gains are k
FR
L
= 1.2 and
k
FR
NL
= 0.3748, the same as the HJB result.
Cost surface (2.9) is shown in Figure 2.4 for the case with x
0
= 1 over the control gain ranges
[0.8,2.0][0.1,1.0].
2.3.1 Results Comparison
Different methods resulted in different optimal control laws. As expected, the HJB-derived optimal
control law is exactly the same as the optimal nonlinear control for a system without excitation but
non-zero initial condition. Also the optimal control laws for a Gaussian white noise excitation found
19
Printed by Mathematica for Students
Figure 2.2: Cost function based on MC simulation for white noise excitation
Printed by Mathematica for Students
(a) x
0
=10
Printed by Mathematica for Students
(b)x
0
=0.1
Figure 2.3: Cost function based on free response
by FPK and Monte Carlo simulation (MCS) methods are almost the same. But, the HJB-derived
and FPK-derived controls are not identical.
To explore the comparison further, the system is simulated using each of the four “optimal” control
laws for (a) a MC simulation in Simulink (all simulation parameters as described previously for
MCS), (b) a deterministic free response from a non-zero initial condition ofx
0
= 1, (c) the response
20
Printed by Mathematica for Students
Figure 2.4: Cost function based on non-zero initial states
Table 2.1: Free and MCS stochastic response statistics with the four nonlinear control laws
Methods
Optimal Gains MCS statistics Free Response
k
L
k
NL
maxx rmsx maxu rmsu J J
HJB 1.20 0.375 1.7436 0.5579 4.1271 0.8597 1.5529 1.3874
FPK 1.53 0.329 1.6638 0.5180 4.1016 0.9272 1.5270 1.4150
MCS 1.53 0.317 1.6687 0.5190 4.0660 0.9249 1.5269 1.4142
FR 1.20 0.375 1.7436 0.5579 4.1271 0.8597 1.5529 1.3874
to the 1940 El Centro earthquake and (d) the response to the 1994 Northridge earthquake. Tables
2.1 and 2.2 show the results of this analysis. It should be noted the El Centro record used in this
dissertation is the N-S Imperial Valley Irrigation District substation record of the 1940 Imperial
Valley earthquake with PGA= 0.348g, and the Northridge record used in this section is the E-W
motion of the 1994 Northridge earthquake recorded at the USC 90003 station at 17645 Saticoy st.
with PGA= 0.368g. Not surprisingly, the FPK/MCS optimal control gains better reduce the cost
computed from the MC simulations with somewhat smaller response statistics and slightly larger
control force, whereas the HJB / free response optimal control better reduce the cost of the free
response. However, the difference in costs is not large, only on the order of 2–3%.
Table 2.2 shows the responses and costs when the excitation is the North-South component of the
1940 El Centro and of the 1994 Northridge earthquakes. These results show that HJB and free
response derived control laws are able to reduce the cost function — slightly better than MC and
FPK derived controls — by exerting smaller control force but resulting in very slightly larger system
response than with the MCS and FPK derived controlled systems.
21
Table 2.2: Earthquake response statistics with the four nonlinear control laws
Methods
Optimal Gains 1940 El Centro 1994 Northridge
k
L
k
NL
maxx rmsx maxu rmsu J maxx rmsx maxu rmsu J
HJB 1.20 0.375 0.3212 0.0549 0.3979 0.0665 0.0090 0.2976 0.0809 0.3670 0.0979 0.0195
FPK 1.53 0.329 0.3116 0.0528 0.4864 0.0813 0.0109 0.2963 0.0782 0.4616 0.1203 0.0238
MCS 1.53 0.317 0.3116 0.0528 0.4863 0.0813 0.0109 0.2963 0.0782 0.4615 0.1203 0.0238
FR 1.20 0.375 0.3212 0.0549 0.3979 0.0665 0.0090 0.2976 0.0809 0.3670 0.0979 0.0195
Uncont. — — 0.3798 0.0638 — — — 0.2990 0.0875 — — —
2.4 Conclusions
Different “optimal” control laws were obtained by different methods and were compared to each
other. Interestingly, for the scalar problem investigated here, the HJB-derived “optimal” control law,
which is the same as the optimal nonlinear control for free response of a system without excitation
but non-zero initial condition, is different from the optimal nonlinear control for a Gaussian white
noise excitation found from FPK and MCS results. The certainty equivalence property clearly does
not hold, likely because the system is non-Gaussian with a non-quadratic cost function. Although
the control laws are different, the resulting costs found by these methods are almost the same.
It is not clear from this study whether the HJB optimal control or the FPK optimal control is more
appropriate for seismically-excited structures. The difference here is relatively small — on the
order of a few percent — but that may not be the case for other structure models or other control
objectives. Obviously, the solutions for simple, but real, structure models must be studied, as well
as more complex structure models, before general conclusions can be drawn.
22
Chapter 3
Preliminary Study on Control Design for
Serviceability and Life Safety Goals
1
3.1 Introduction
The most critical building response in the case of severe excitation — such as strong earthquakes,
hurricanes and other extreme dynamic loads — is structural drift; exceeding drift beyond design
values may result in severe damage to the structure and its occupants and contents. In more moderate
excitation, which occurs much more frequently, the absolute accelerations are more likely to directly
affect serviceability, both in terms of occupant comfort and the safety and continued operation of
building contents. Many structural control systems implemented to date are designed more for
safety because it is often difficult, if not impossible, to simultaneously design for both events.
This chapter gives an example of a control strategy that can achieve both service-level and life
safety-level objectives with one general control law. This particular control strategy is not intended
to be optimal, but to show that a strategy exists to reduce drift in the rare extreme events and reduce
absolute acceleration in the more frequent weak-to-moderate ground motions.
1
Hemmat-Abiri and Johnson (2013a) reports a subset of this chapter.
23
3.2 Problem Formulation
Consider an n degree-of-freedom (DOF) linear system subjected to external disturbance w. The
equations of motion, and those governing sensor measurements and responses, can be expressed in
state-space form
˙ q= Aq+ Bu+ Ew (3.1a)
y= C
y
q+ D
y
u+ F
y
w+ v (3.1b)
z= C
z
q+ D
z
u+ F
z
w (3.1c)
where state vector q contains generalized displacements and velocities; sensor measurement vector
y may contain absolute accelerations, relative drifts and/or strains; z is a vector of outputs to be
regulated (typically interstory drifts, absolute accelerations and internal forces); u is a vector of
control forces; v is the sensor noise vector; state matrix A, input matrices B and E, output matrices
C
y
and C
z
, and feedthrough matrices D
y
, D
z
, F
y
and F
z
all depend on the usual structural system
matrices, the force locations and orientations, etc.
To achieve the two different objectives, drift reduction and absolute acceleration reduction, in
different levels of excitations, define the following two cost functions that should be minimized:
J
drift
=
Z
t
f
0
[z
T
Q
d
z+ u
T
d
Ru
d
]dt and J
accel
=
Z
t
f
0
[z
T
Q
a
z+ u
T
a
Ru
a
]dt
in which output vector z=[Dx
T
(¨ x
a
)
T
]
T
contains driftDx and absolute accelerations ¨ x
a
; weighting
matrices Q
d
=

[q
d
0]
T
[0 0]
T

and Q
a
=

[0 0]
T
[0 q
a
]
T

force the cost functions to focus
on drift and acceleration, respectively; and q
d
, q
a
and R are symmetric weighting coefficients on
structural drift, absolute accelerations and control forces.
Linear control minimizing one or the other of the two cost functions J
drift
and J
accel
, which are
quadratic in states and the control force (the latter is necessary for the optimization to be well-posed),
can be accomplished for an infinite time horizon by using MATLAB’slqr command. Each of these
objective functions will result in a different optimal linear control law, namely u
d
=K
d
q and
u
a
=K
a
q in the optimal linear control gains K
d
and K
a
. However, as u
d
is designed to minimize
the drift, it may result in increased accelerations, which would be undesirable in the moderate
excitations. Similarly, u
a
will achieve reductions in acceleration, but is not optimal in reducing drift
in the case of strong excitation.
There are a multitude of possible control laws that may satisfy the two objectives in their corre-
24
sponding excitation levels. The point of this chapter is to show that there does indeed exist a control
strategy that satisfies the goals. This specific control strategy uses a simple combination of the two
optimal linear control forces u
a
and u
d
:
u=g(t,q,w)u
d
+[1g(t,q,w)]u
a
=fK
d
g(t,q,w) K
a
[1g(t,q,w)]gq (3.2)
The weighting coefficientg, which may depend on time, structural response and excitation, should be
determined appropriately such that the nonlinear control in (3.2) approaches u
d
when the excitation
is strong but closely tracks u
a
under weak/moderate excitation. There are many possible forms forg
that can satisfy these objectives. One isg =G
4
/[G
4
+a], wherea is a parameter that adjusts the
transition from u
a
to u
d
, and
G(t)= max
tt
kw(t)k (3.3)
which is the running peak of some norm of the excitation; for example, in an earthquake,G(t) is the
peak ground acceleration seen up through timet ifkw(t)k=j ¨ x
g
(t)j for ground acceleration ¨ x
g
(t).
It should be noted thatG increases, or stays the same, over the duration of the excitation. Then, (3.2)
can be rewritten
u=[K
d
G
4
(t)+ K
a
a]q/[G
4
(t)+a] (3.4)
3.3 Numerical Example
To demonstrate the potential of a control strategy to achieve both goals, a numerical simulation
is performed using control law (3.4) for a single degree-of-freedom (SDOF) building structure
with parameters m= 29485 kg, c= 23710 Ns/m and k = 1.1640 MN/m, which are adapted
from Ramallo et al. (2002) (who based their SDOF model on a 5DOF model from Kelly et al.
(1987)), but with the stiffness reduced so that the structure has a 1 Hz natural frequency (the
resulting damping is 6.4%). The weighting coefficients inJ
drift
andJ
accel
are selected such that the
cost functions are dimensionless: q
d
= 10 m
2
,q
a
= 10(m/k)
2
= 0.0064 s
4
/m
2
andR= 1/k
2
=
0.73804(MN)
2
. The strong and weak excitations are chosen to be 3.0- and 0.1-scaled versions of
the N-S component of the 1940 El Centro earthquake, with peak ground accelerations (PGAs) of
10.251m/s
2
and 0.3417m/s
2
, respectively. A preliminary parameter study identifieda= 50m
2
/s
4
met the objectives. The control gains K
a
and K
d
for the linear strategies are computed using
MATLAB’s lqr command. The responses were computed, using MATLAB’s Simulink, for the
time-varying control law and for linear control laws denoted LQR
d
and LQR
a
. The responses to the
weak excitation are shown in Figure 3.1, and to the strong excitation in Figure 3.2; peak and root
25
Table 3.1: Performance of time-varying control under scaled 1940 El Centro excitation
0.1El Centro 3El Centro
PGA = 0.3417 m/s
2
PGA = 10.251 m/s
2
LQR
d
LQR
a
Time-Var. LQR
d
LQR
a
Time-Var.
Peak displ. [cm] 0.1938 0.5501 0.5497 5.8131 16.5019 5.8444
RMS displ. [cm] 0.0250 0.1014 0.1013 0.7489 3.0432 0.8083
Peak abs. accel. [m/s
2
] 0.3896 0.1236 0.1237 11.6885 3.7077 11.7483
RMS abs. accel. [m/s
2
] 0.0468 0.0228 0.0228 1.4047 0.6850 1.3982
Peak control force /mg 0.0327 0.0179 0.0179 0.9798 0.5382 0.9855
RMS control force /mg 0.0040 0.0033 0.0033 0.1191 0.0979 0.1182
−0.5
0
0.5
displ [cm]

Time−Varying
LQR
d
LQR
a
−0.2
0
0.2
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
Figure 3.1: Controlled response comparison for 0.1El Centro, time-varying
mean square (RMS) metrics of drift, acceleration and weight-normalized control force are listed for
both excitations in Table 3.1.
Clearly, the time-varying control strategy closely tracks the LQR
d
design in the case of strong
excitation and it performs almost the same as LQR
a
in the weak excitation, demonstrating that the
same time-varying control law can achieve both of the competing objectives. Table 3.1 shows that, in
the weak excitation (0.1El Centro), the maximum absolute acceleration response controlled with
the time-varying strategy is equal to 0.1236 m/s
2
, which is nearly identical to the corresponding
0.1237 m/s
2
value with the LQR
a
design; but the drift-focused LQR
d
has much larger acceleration
of 0.3896 m/s
2
. On the other hand, for the large excitation (3.0El Centro), the maximum drifts
with both the time-varying and LQR
d
strategies are about 5.8 cm, whereas the drift with the LQR
a
control is much larger at 16.50 cm. The same trend exists in the maximum and RMS values of
control force and other metrics.
26
−10
0
10
displ [cm]

Time−Varying
LQR
d
LQR
a
−5
0
5
10
accel [m/s
2
]
0 5 10 15 20
−50
0
50
time [s]
force [%W]
Figure 3.2: Controlled response comparison for 3.0El Centro, time-varying
Table 3.2: Performance of time-varying control under scaled 1994 Northridge excitation
0.1Northridge 1Northridge
PGA = 0.8268 m/s
2
PGA = 8.2676 m/s
2
LQR
d
LQR
a
Time-Var. LQR
d
LQR
a
Time-Var.
Peak displ. [cm] 0.4022 1.7965 1.7921 4.0221 17.9651 4.0688
RMS displ. [cm] 0.0695 0.4189 0.4058 0.6949 4.1893 0.7545
Peak abs. accel. [m/s
2
] 0.8134 0.4466 0.4596 8.1335 4.4656 8.2764
RMS abs. accel. [m/s
2
] 0.1289 0.0819 0.0833 1.2886 0.8192 1.3005
Peak control force /mg 0.0695 0.0624 0.0619 0.6946 0.6242 0.7072
RMS control force /mg 0.0109 0.0129 0.0123 0.1089 0.1292 0.1089
The control law is optimized for scaled El Centro excitations, however it should perform well
under other excitations as well. Figures 3.3 and 3.4 and Table 3.2 show the performance of the
time-varying control along with the two optimal linear control laws for 0.1 and 1.0-scaled 1994
Northridge excitations. It should be noted that the Northridge record used in this and the following
chapters is the N-S motion of the 1994 Northridge earthquake with PGA of 8.2676 m/s
2
, recorded
at the Sylmar County Hospital parking lot. Clearly, the time-varying control focuses on acceleration
reduction for weak excitation 0.1-scaled Northridge and it reduces drift under strong excitation
1.0-scaled Northridge. It must be noted that force saturation was not introduced into this example
for the sake of simplicity; any real implementation would, of course, need to use lower force levels,
perhaps on the order of 1015% of structure weight, possibly by introducing a saturation in the
commanded force to accommodate control device capacity, distribution of the load into the structure
and so forth.
27
−1
0
1
displ [cm]

Time−Varying
LQR
d
LQR
a
−0.5
0
0.5
accel [m/s
2
]
0 5 10 15 20
−5
0
5
time [s]
force [%W]
Figure 3.3: Controlled response comparison for 0.1Northridge, time-varying
−10
0
10
displ [cm]

Time−Varying
LQR
d
LQR
a
−5
0
5
accel [m/s
2
]
0 5 10 15 20
−50
0
50
time [s]
force [%W]
Figure 3.4: Controlled response comparison for 1.0Northridge, time-varying
3.4 Conclusions
This chapter presents an example of a control strategy that can achieve both service-level and life
safety-level objectives with one general control law. This particular control strategy is not intended
to be optimal, but to show that a strategy exists to reduce drift in the rare extreme events and reduce
absolute acceleration in the more frequent weak to moderate ground motions.
28
The performance of the time-varying control strategy is demonstrated by intensive numerical
simulations for a SDOF building model subjected to different excitations, showing that it can
successfully achieve both life safety (drift reduction) and serviceability (acceleration reduction)
objectives at different excitation levels. However, it is a simple unoptimized example. In the
following chapters, procedures for deriving and optimizing the control strategies that achieve both
objectives will be explored and presented.
29
Chapter 4
HJB-based Optimal Nonlinear Control for
Multiple Objectives in Different Levels of
Excitations
1
4.1 Introduction
In this chapter, a nonquadratic cost function is introduced such that a solution exists to the
Hamilton-Jacobi-Bellman (HJB) equation, and leads to an optimal nonlinear control strategy. While
HJB-driven nonlinear control laws proposed by other researchers are designed just to reduce drift
in case of strong excitation, the intended advantage of the control law proposed herein is that it
performs, in a weak excitation, as an optimal linear controller that reduces acceleration, whereas it
is fully nonlinear to reduce drift response in a strong excitation. This load adaptivity characteristic
of the proposed controller can reduce the damage and dynamic effects of external excitation with a
wide range of magnitudes and will provide safer and more comfortable structures that are less likely
to require expensive repairs and retrofits.
4.2 Problem Formulation
Consider an n degree-of-freedom (DOF) linear system subjected to external disturbance w. The
equations of motion, and those governing sensor measurements and responses are the same as those
1
Hemmat-Abiri and Johnson (2014a) presents a subset of this chapter.
30
presented in Section 3.2. For quadratic cost kernels like
L
Q
=
1
2
[q
T
Qq+ 2q
T
Nu+ u
T
Ru], or (4.1a)
L
Q
=
1
2
[z
T
Qz+ 2z
T
Nu+ u
T
Ru], (4.1b)
with weighting matrices Q= Q
T
0, R= R
T
0 and N such that

[Q N]
T
[N
T
R]
T

0,
minimizing quadratic cost functions of the forms
J
Q
=
Z
¥
0
L
Q
(t)dt, J
Q
= lim
t
f
!¥
1
t
f
Z
t
f
0
L
Q
(t)dt, J
Q
=
E
[L
Q
(t)] (4.2)
(depending on the nature of excitation w and the initial condition q
0
) results in linear control laws,
which are easily designed using thelqr orlqry commands in MATLAB
R
.
As quadratic cost functions do not capture the trade offs in control objectives for some problems, a
cost function that is non-quadratic in the states (and/or control) can be useful. For the structural
control problem, where acceleration reduction is the priority in smaller excitations and drift reduction
is necessary in larger excitations, a higher-than-quadratic function of the drift can enable this shift
in objective for different excitation (and response) levels. It is well established that designing an
optimal control strategy for a nonlinear system will lead to the HJB PDE (Bryson and Ho, 1975,
Primbs et al., 1999), which can be expressed
¶V
¶t
=H

q,u
opt
,
¶V
¶q
,t

(4.3)
in whichH =L+(¶V /¶q) ˙ q is the Hamiltonian function,LL(q,u) is the integrand in the cost
function,V is the value function (the cost to go), and u
opt
is the optimal control found by solving
¶H/¶u= 0.
Assuming a general nonquadratic cost function
J=
1
2
Z
¥
0
[q
T
Qq+ u
T
Ru+ 2q
T
Nu+h(q)]dt, (4.4)
with nonquadratic termh(q), the Hamiltonian is
H =L+

¶V
¶q

˙ q (4.5)
=
1
2
[q
T
Qq+ u
T
Ru+ 2q
T
Nu+h(q)]+l l l
T
(Aq+ Bu),
where l l l
T
= ¶V /¶q. The optimal nonlinear control law is found by setting ¶H/¶u to 0 and
31
solving, resulting in
u=R
1
B
T
l l l R
1
N
T
q (4.6)
If the cost were quadratic —i.e.,h(q) 0 — thenl l l would be Pq, and the optimal control becomes
the familiar linear law u=R
1
(B
T
P+ N
T
)q, where P is the positive semidefinite symmetric
solution to the algebraic Riccati equation (ARE)
PA+ A
T
P+ Q(PB+ N)R
1
(PB+ N)
T
= 0 (4.7)
Rather than the linear form ofl l l in the quadratic cost case, it is assumed here thatl l l is cubic in the
states, with the form
l l l =(1+hq
T
Pq)Pq, (4.8)
whereh is an additional weighting coefficient for tuning the control. The resulting optimal control
law is
u=R
1
[B
T
(1+hq
T
Pq)P+ N
T
]q (4.9)
Taking the partial derivatives of both sides of HJB PDE (4.3) with respect to q results in
1
¶t
(
¶V
¶q
)
T
=
(
¶H
¶q
)
T
; substituting¶V /¶q=l l l
T
gives
˙
l l l =

¶H
¶q

T
=(1+hq
T
Pq)A
T
Pq Qq Nu
1
2

¶h
¶q

T
(4.10)
where u is as defined in (4.9). Since the only dependence ofl l l on time is implicitly through q,
˙
l l l is
also calculated as
˙
l l l =
¶l l l
¶q
˙ q=

(1+hq
T
Pq)P+ 2hPqq
T
P

˙ q (4.11)
Equating the right-hand sides of (4.10) and (4.11) gives, using state equation (3.1a) without excita-
tion w and the ARE (4.7),
0=
1
2

¶h
¶q

T
+ g(q) (4.12a)
g(q)=h(q
T
Pq)Qqh(q
T
Qq)Pqh[q
T
(PBR
1
B
T
P)q]Pq (4.12b)
h(q
T
Pq)PBR
1
B
T
Pq 2h
2
(q
T
Pq)(q
T
PBR
1
B
T
Pq)Pq
h
2
(q
T
Pq)
2
PBR
1
B
T
Pq+h(q
T
Pq)NR
1
N
T
q+h[q
T
(NR
1
N
T
)q]Pq
32
It can be verified that a nonquadratich(q) of the form
h(q)=h(q
T
Pq)(q
T
[Q+ PBR
1
B
T
P NR
1
N
T
]q)+h
2
(q
T
Pq)
2
(q
T
PBR
1
B
T
Pq) (4.13)
will satisfy (4.12). Therefore, the cost function (4.4) becomes
J=
1
2
Z
¥
0

q
T
Qq+ u
T
Ru+ 2q
T
Nu (4.14)
+h(q
T
Pq)(q
T
[Q+ PBR
1
B
T
P NR
1
N
T
]q)
+h
2
(q
T
Pq)
2
(q
T
PBR
1
B
T
Pq)

dt
for which the optimal nonlinear control law is shown in (4.9). The linear quadratic regulator (LQR)
weighting matrices (Q, R and N) are chosen to minimize one objective (reducing structural accel-
erations) when excitation and response is smaller; P then follows from ARE (4.7); the weighting
coefficient h will be chosen to optimize a separate objective in stronger excitations (explained
further in the following section). In contrast, the HJB-driven nonlinear control laws proposed by
other researchers had a single objective focus.
4.3 Control Stability
The proposed nonlinear control strategies must be evaluated carefully to ensure that their application
to a real structure is appropriate, safe and robust. The stability of the nonlinear controller is,
obviously, of paramount importance (certainly if the control device is an active one that can inject
energy into the structural/mechanical system; controllable semiactive passive devices, on the other
hand, can guarantee energy dissipation even if commanded suboptimally, though the performance
may suffer significantly.)
The stability of the closed-loop system with the nonlinear control (4.9) can be examined through
several tests (Slotine and Li, 1991). Lyapunov’s direct method, a useful procedure for investigating
the stability of controlled systems, is used in this section. This method states that stability is
guaranteed if
˙
V 0,i.e., if the time derivative of the value function is nonpositive.
For a quadratic cost function of the form (4.2) with kernel (4.1a), and the corresponding optimal
linear control u=R
1
(B
T
P+N
T
)q, the value function isV =
1
2
q
T
Pq 0 where P is the solution
33
of (4.7). So, stability is ensured by
˙
V =

¶V
¶q

˙ q= q
T
P[A BR
1
(PB+ N)
T
]q 0 (4.15)
In other words, since P is positive semidefinite, stability is guaranteed for the linearly controlled
system if
L
L
= A BR
1
(PB+ N)
T
0 (4.16)
Now consider the nonlinear control law (4.9) in whichl l l =(¶V /¶q)
T
=(1+hq
T
Pq)Pq, where
the value function is
V =
1
2
(q
T
Pq)+
h
4
(q
T
Pq)
2
, (4.17)
which is nonnegative ifh 0, and is equal to zero at q= 0. Applying the Lyapunov theorem of
stability
˙
V =l l l
T
˙ q (4.18)
=

(1+hq
T
Pq)Pq

T
[Aq+ Bu]
=[(1+hq
T
Pq)q
T
P]
h
(A BR
1
[(1+hq
T
Pq)PB+ N]
T
)q
i
If(1+hq
T
Pq) is positive, then (4.18) is nonpositive if
L
NL
= A BR
1
[(1+hq
T
Pq)PB+ N]
T
0 (4.19)
Comparing (4.19) with (4.16), one can verify that L
NL
= L
L
hBR
1
B
T
P(q
T
Pq). One can
conclude that, ifh 0, L
NL
is in fact negative semidefinite due to the positive semidefiniteness of
BR
1
B
T
P and P. Therefore, the stability of the system with the optimal nonlinear control law (4.9)
is guaranteed withh 0.
4.4 Numerical Example 1: SDOF Building Model
To achieve the two different objectives, drift reduction and absolute acceleration reduction, in
different levels of excitations, define the following two cost functions that should be minimized:
J
d
=
Z
¥
0
[z
T
Q
d
z+ u
T
d
Ru
d
]dt (4.20a)
J
a
=
Z
¥
0
[z
T
Q
a
z+ u
T
a
Ru
a
]dt (4.20b)
34
in which output vector z=[Dx
T
¨ x
T
a
]
T
contains structure story driftsDx and absolute accelerations
¨ x
a
; weighting matrices Q
d
=

[q
d
0]
T
[0 0]
T

and Q
a
=

[0 0]
T
[0 q
a
]
T

force the cost
functions to focus on drift and acceleration, respectively; and q
d
, q
a
and R are square symmetric
weighting matrices on structural drift, absolute accelerations and control forces, respectively. Linear
control minimizing one or the other of the two cost functions J
d
and J
a
, which are quadratic in
states and the control force (the latter is necessary for the optimization to be well-posed), can be
accomplished for an infinite time horizon by using MATLAB’s lqry command. Each of these
objective functions will result in a different optimal linear control law, namely u
d
=K
d
q and
u
a
=K
a
q in the optimal linear control gains K
d
and K
a
. However, as u
d
is designed to minimize
the drift, it may result in increased accelerations, which would be undesirable in the moderate
excitations. Similarly, u
a
will achieve reductions in acceleration, but is not optimal in reducing drift
in the case of strong excitation.
To demonstrate the potential of the proposed controller (4.9) to achieve both goals, a numerical
simulation is performed for the same single degree-of-freedom (SDOF) ground-excited building
structure presented in Section 3.3 (with massm= 29485 kg, damping coefficientc= 23710 Ns/m
and stiffnessk= 1.1640 MN/m), giving state matrices:
A=
"
0 1

k
m

c
m
#
, B=
(
0
1
)
, (4.21a)
C
z
=
"
1 0

k
m

c
m
#
, D
z
=
(
0

1
m
)
, F
z
= 0. (4.21b)
The weighting coefficients inJ
d
andJ
a
are selected such that all terms in the cost function kernels
are dimensionless and, for a fair comparison, the peak control force is almost the same in a strong
historical ground motion for both strategies: q
d
= 1 m
2
, q
a
= 30(m/k)
2
= 0.0192 s
4
/m
2
and
R= 0.8/k
2
= 0.5905(MN)
2
.
4.4.1 Chooseh to Satisfy the Multiobjective Cost Function
The strong and weak excitations for this problem are considered to be 2.0- and 0.1-scaled versions
of the North-South component of the 1940 El Centro earthquake, with peak ground accelerations
(PGAs) of 6.8340 m/s
2
and 0.3417 m/s
2
, respectively. The control gains K
a
and K
d
for the linear
strategies are computed using thelqry command in MATLAB. The responses of this system were
computed, using MATLAB’s Simulink, for the nonlinear control law and for linear control laws
denoted LQR
d
and LQR
a
.
35
It is desired that the HJB-based optimal nonlinear control in (4.9) approaches u
d
when the excitation
is strong and closely tracks u
a
under weak/moderate excitation. This goal is achieved with an
“optimal” selection of weighting coefficient h, found herein using the MATLAB Optimization
Toolbox functionfminsearch.
For the SDOF building, the nonquadratic cost function is
J
NQ
=
1
2
Z
¥
0

q
a
¨ x
2
a
+Ru
2
+h(q)

dt (4.22)
where ¨ x
a
is the absolute acceleration of the structure; h(q) contains the nonquadratic terms as
defined in (4.13). Therefore, (4.22) can be rewritten as
J
NQ
=
1
2
Z
¥
0
[q
T
Q
a
q+ 2q
T
N
a
u+u
T
R
a
u+h(q)]dt (4.23)
in which
Q
a
=
q
a
m
2
"
k
2
ck
ck c
2
#
, N
a
=
q
a
m
2
(
k
c
)
, R
a
=R+
q
a
m
2
It should be pointed out that the quadratic terms in the cost function are defined to reduce acceleration.
h must be selected such that the nonlinear terms in the control (4.9) result in a significant drift
reduction in strong excitation; in weak excitation, the control reverts to an LQR problem that
reduces acceleration.
While there are a multitude of approaches to find the best h, one is as follows. The objective
function to be minimized in fminsearch is considered to be composed of two parts: (1) the
relative difference between the root mean square (RMS) of the nonlinear controlled drift response
and that of optimal linear control law LQR
d
in strong excitation (2.0El Centro); and (2) the
relative difference between the RMS of the nonlinear controlled acceleration response and that of
the optimal linear control law LQR
a
in weak excitation (0.1El Centro).
This objective function can be expressed
f = f
d
+ f
a
=

1
kx
NL
k
kx
LQR
d
k

Strong
+

1
k ¨ x
NL
k
k ¨ x
LQR
a
k

Weak
(4.24)
wherejjjj denotes the RMS of(). Figure 4.1 shows the variation of f
a
and f
d
relative toh for
weak and strong excitation, respectively. The unique choice ofh = 1328.0 minimizing f is found
byfminsearch.
36
10
3
10
4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
η

f
a
f
d
f = f
a
+ f
d
Figure 4.1: Variation of multiobjective cost relative toh
Table 4.1: Performance of HJB control under scaled 1940 El Centro excitation
0.1 El Centro 2 El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak displ.[cm] 1.154 0.614 0.320 0.598 22.686 12.508 6.386 4.466
RMS displ.[cm] 0.300 0.194 0.085 0.190 5.907 3.905 1.694 1.694
Peak abs. accel.[m/s
2
] 0.460 0.105 0.320 0.113 9.046 2.117 6.393 6.883
RMS abs. accel.[m/s
2
] 0.120 0.028 0.066 0.029 2.353 0.562 1.316 1.031
Peak cont. force [% bldg.weight] — 2.232 2.266 2.173 — 45.440 45.296 64.759
RMS cont. force [% bldg.weight] — 0.682 0.416 0.669 — 13.730 8.300 10.345
The response of the structure is computed for the resulting optimal nonlinear control along with
that of the two linear controls. Responses in the weak excitation are shown in Figure 4.2, whereas
Figure 4.3 shows the responses to the strong excitation; the results are also presented in Table
4.1. Clearly, the nonlinear control strategy closely tracks the LQR
d
design in the case of strong
excitation and it performs almost the same as LQR
a
in the weak excitation, demonstrating that the
same nonlinear control law can achieve both of the competing objectives.
37
−0.5
0
0.5
displ [cm]

Nonlinear
LQR
d
LQR
a
−0.2
0
0.2
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
Figure 4.2: Controlled response comparison for 0.1El Centro for the besth, HJB
−10
0
10
displ [cm]

Nonlinear
LQR
d
LQR
a
−4
−2
0
2
4
6
accel [m/s
2
]
0 5 10 15 20
−40
−20
0
20
40
60
time [s]
force [%W]
Figure 4.3: Controlled response comparison for 2.0El Centro for the besth, HJB
4.4.2 Performance in Other Historical Earthquakes
The nonlinear control law is optimized for scaled El Centro excitations, but should be evaluated
with other excitations as well. Table 4.2 shows the performance of the nonlinear control, along
with the two linear control laws, for the 1994 Northridge earthquake recorded at Sylmar, CA (PGA
0.8268g) scaled by 0.05 and 1.0. Also, to investigate the performance of nonlinear control under
38
moderate to strong excitation, the system is subjected to the 2013 Pomona (PGA 0.0205g) and 1995
Kobe (PGA 0.8337g) excitations; the results are shown in Table 4.3.
Clearly, the nonlinear control focuses on acceleration reduction for weak excitations (0.05-scaled
Northridge and Pomona) and, for strong excitations (1.0-scaled Northridge and Kobe), it reduces
drift. In fact, the drift reduction provided by the nonlinear control is quite superior to that of LQR
d
−0.5
0
0.5
1
displ [cm]

Nonlinear
LQR
d
LQR
a
−0.2
0
0.2
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
Figure 4.4: Controlled response comparison for 0.05Northridge, HJB
−10
0
10
20
displ [cm]

Nonlinear
LQR
d
LQR
a
−5
0
5
accel [m/s
2
]
0 5 10 15 20
−50
0
50
time [s]
force [%W]
Figure 4.5: Controlled response comparison for 1.0Northridge, HJB
39
for the Kobe earthquake excitation as the nonlinear control exerts a larger control force. However, it
must be noted that force saturation was not introduced into this problem for the sake of simplicity;
any real implementation would, of course, need to use lower force levels, on the order of 10–15%
of structure weight, possibly by introducing a saturation in the commanded force to accommodate
control device capacity, distribution of the load into the structure and so forth — a task that will be
examined in a subsequent study.
4.4.3 Response to Synthetic Kanai-Tajimi Filter Excitation
Finally, the performance of the nonlinear control strategy is evaluated with a filtered Gaussian
white noise excitation. The Kanai-Tajimi (KT) filter (Soong and Grigoriu, 1993) is a second-order
low-pass filter, representing the transfer function from bedrock motion to the surface ground motion,
given by
F(s)=
2z
g
w
g
s+w
2
g
s
2
+ 2z
g
w
g
s+w
2
g
(4.25)
Table 4.2: Performance of HJB control under scaled 1994 Northridge excitation
0.05 Northridge 1 Northridge
PGA= 0.4134 m/s
2
PGA= 8.2676 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak displ.[cm] 1.029 1.080 0.408 1.018 20.455 21.692 8.165 5.629
RMS displ.[cm] 0.222 0.265 0.075 0.249 4.425 5.282 1.491 1.424
Peak abs. accel.[m/s
2
] 0.408 0.175 0.334 0.197 8.110 3.508 6.643 7.606
RMS abs. accel.[m/s
2
] 0.088 0.034 0.055 0.035 1.760 0.671 1.099 1.029
Peak cont. force [% bldg.weight] — 3.933 2.451 3.730 — 48.891 78.883 76.694
RMS cont. force [% bldg.weight] — 0.922 0.328 0.861 — 18.379 6.552 10.242
Table 4.3: Performance of HJB control under 2013 Pomona and 1995 Kobe excitations
2013 Pomona 1995 Kobe
PGA= 0.2007 m/s
2
PGA= 8.1782 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak displ.[cm] 0.060 0.051 0.038 0.052 35.965 11.524 11.712 5.423
RMS displ.[cm] 0.013 0.005 0.006 0.005 8.425 3.578 2.208 1.408
Peak abs. accel.[m/s
2
] 0.023 0.023 0.055 0.023 14.339 2.702 9.408 8.539
RMS abs. accel.[m/s
2
] 0.005 0.002 0.006 0.002 3.356 0.652 1.722 1.370
Peak cont. force [% bldg.weight] — 0.297 0.500 0.302 — 45.373 58.970 87.990
RMS cont. force [% bldg.weight] — 0.026 0.048 0.026 — 12.957 10.913 13.494
40
or as the response of a SDOF system as
¨ q
g
(t)+ 2z
g
w
g
˙ q
g
(t)+w
2
g
q
g
(t)=w(t) (4.26a)
¨ x
g
(t)=w
2
g
q
g
(t) 2z
g
w
g
˙ q
g
(t) (4.26b)
where parameters z
g
and w
g
are the damping ratio and characteristic frequency of the ground
motion; w(t) is a zero-mean Gaussian white noise; and q
g
and ¨ x
g
are the KT state and produced
ground excitation, respectively. Herein, the parameters are selected asw
g
= 17 rad/s andz
g
= 0.3
to approximate the frequency content of four historical design earthquakes (Ramallo et al., 2002),
as shown in Figure 4.6: i.e., the spectral content of two moderate earthquakes (1940 El Centro and
1968 Hachinohe) and two severe earthquakes (1994 Northridge and 1995 Kobe).
(4.26) can be rewritten in state-space format as
˙ q
g
= A
KT
q
g
+ B
KT
w (4.27a)
¨ x
g
= C
KT
q
g
+ D
KT
w (4.27b)
and then augmented to the state-space equations of the original system, presented previously in
(3.1). The resulting combined equation can be presented in the form of (3.1a) where
A=
"
A
str
EC
KT
0 A
KT
#
, B=
"
B
str
0
#
, E=
"
E
str
D
KT
B
KT
#
(4.28)
and the new states are states of the original system plus two KT states. To obtain specific magnitude
or PGA of the filtered noise, the Gaussian white noise input to the filter is scaled appropriately. For
example, a filtered excitation of appropriate duration with PGA approximately 3 m/s
2
has standard
deviation approximately equal to 1 m/s
2
, so the intensity of the Gaussian white noise should be
scaled accordingly. Figures 4.7 and 4.8 show the performance of the controlled system under filtered
white noise excitation with PGA approximately 0.3 m/s
2
(weak excitation) and 7 m/s
2
(strong
excitation). Clearly, the nonlinear controller performs similar to LQR
a
in weak excitation, and
tracks LQR
d
in strong excitation.
4.5 Numerical Example 2: Base-isolated MDOF Model
In this section, a base-isolated multiple degree-of-freedom (MDOF) building structure with a
control device at the base level is investigated to demonstrate the potential of the HJB strategy to
41
achieve both goals. Numerical simulations are performed for a base-isolated building structure
with parameters, shown in Table 4.5, which are adapted from Kelly et al. (1987). To achieve the
10
ï 1
10
0
10
1
10
ï 4
10
ï 3
10
ï 2
10
ï 1
10
0
Kanai ï Tajimi
Magnitude
Frequency [Hz]
Figure 4.6: Frequency content of design earthquakes and Kanai-Tajimi shaping filter (adapted from
Ramallo et al. (2002))
−0.5
0
0.5
displ [cm]

Nonlinear
LQR
d
LQR
a
−0.2
−0.1
0
0.1
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
Figure 4.7: Controlled response comparison for 0.3 m/s
2
PGA Kanai-Tajimi, HJB
42
−10
0
10
20
displ [cm]

Nonlinear
LQR
d
LQR
a
−5
0
5
accel [m/s
2
]
0 5 10 15 20
−50
0
50
time [s]
force [%W]
Figure 4.8: Controlled response comparison for 7 m/s
2
PGA Kanai-Tajimi, HJB
Table 4.4: Performance of HJB control under scaled Kanai-Tajimi filtered excitation
Weak Strong
PGA= 0.3 m/s
2
PGA= 7 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak displ.[cm] 0.648 0.901 0.276 0.858 14.929 20.941 6.471 5.248
RMS displ.[cm] 0.238 0.339 0.093 0.327 5.559 7.834 2.171 2.198
Peak abs. accel.[m/s
2
] 0.258 0.124 0.208 0.130 5.948 2.882 4.856 6.485
RMS abs. accel.[m/s
2
] 0.095 0.040 0.076 0.040 2.215 0.920 1.775 1.685
Peak cont. force [% bldg.weight] — 3.160 1.315 3.008 — 73.461 30.707 69.052
RMS cont. force [% bldg.weight] — 1.173 0.502 1.128 — 27.084 11.731 16.215
two different objectives, drift reduction and absolute acceleration reduction, in different levels of
excitations, define the following two quadratic cost functions:
J
d
=
1
2
R
¥
0

q
x
d
x
2
base
+q
a
d
¨ x
2
a
roof
+Ru
2

dt (4.29a)
J
a
=
1
2
R
¥
0

q
x
a
x
2
base
+q
a
a
¨ x
2
a
roof
+Ru
2

dt (4.29b)
and the nonquadratic cost function:
J
NQ
=
1
2
Z
¥
0

q
x
a
x
2
base
+q
a
a
¨ x
2
a
roof
+Ru
2
+h(q)

dt (4.30)
where x
base
is the base drift, ¨ x
a
roof
is the absolute acceleration of the roof and h(q) contains the
nonquadratic terms as defined in (4.13).
43
Therefore, (4.30) can be rewritten as
J
NQ
=
1
2
Z
¥
0
[q
T
Q
a
q+ 2q
T
N
a
u+u
T
R
a
u+h(q)]dt (4.31)
Assuming a constant value ofR= 10
4
/(m
1
g)
2
= 2.988110
6
N
2
, one should obtain two desired
pairs of (q
x
d
, q
a
d
) and (q
x
a
, q
a
a
) for designing LQR
d
and LQR
a
. It is assumed there is one control
device in the isolation layer of the structure. Further, it should be noted that the two quadratic
objective functions in (4.29) are cast in a general form to show that HJB approach is not limited
toq
a
d
= 0 andq
x
a
= 0 to achieve the two objectives. Moreover, in order to investigate the effect of
weighting coefficients in the cost function, here it is assumed that LQR
a
and LQR
d
have the same
control force and absolute acceleration weights, and their very different behavior is only the result
of their different drift weights. However, one may choose any desired coefficients to have a different
performance.
The nonlinear control must be designed such that it performs similar to LQR
a
and LQR
d
in weak
and strong excitations, respectively. Consideringq
x
=a 10
8
/H
2
andq
a
=b 10
8
(m
1
/Hk
1
)
2
,
in whichH = 4 m is the height of each story of the structure, one can choose the desired nondimen-
sional (a,b) pairs by observing how changinga andb affects the resulting LQR-controlled drift
and absolute acceleration compared with the designed optimal passive viscous damper and with the
corresponding uncontrolled response. In this section, the North-South component of the first 20 sec
of the 1940 El Centro earthquake is considered as the external excitation.
LQR design for weak excitation: LQR
a
First, it is assumed that there is a passive viscous damper in the isolation layer. A family of different
passive viscous dampers, u=c
p
˙ x
b
, where ˙ x
b
is the velocity across the device, are considered
for this isolation layer withc
p
2[10
2
, 10
6
] N s/m. Figure 4.9 shows the variation of RMS base
Table 4.5: Parameters of a 6DOF Building Model Adapted from Kelly et al. (1972)
Floor No. Mass Stiffness Damping
[ kg] [ MN/m] [ kNs/m]
5 5897 19.059 38
4 5897 24.954 50
3 5897 28.621 57
2 5897 29.093 58
1 5897 33.732 67
base 6800 0.232 3.74
44
10
2
10
3
10
4
10
5
10
6
0
0.5
1
RMS of Response
10
2
10
3
10
4
10
5
10
6
0
5
10
RMS Force
C
p
[N.s/m]

RMS force [% weight]
RMS base drift [m]
RMS roof abs. accel. [m/s
2
]
Figure 4.9: RMS response for 6DOF model with passive viscous dampers
drift and roof acceleration for this family of controllers. Clearly, the maximum RMS absolute
acceleration reduction occurs at c
p
= 6 10
4
N s/m. Figure 4.10 and Figure 4.11 show the
percent reduction of the RMS responses produced by an LQR control compared to the optimal
passive system, and RMS of the required LQR control force, respectively. In order to have a good
reference to compare the performance of different LQR controls and observe the switching behavior
between drift and acceleration reduction, here it is assumed that LQR
a
should perform at least as
well in its acceleration reduction as an optimal passive design (with the maximum acceleration
reduction); i.e., it is assumed that there is a good passive design for acceleration reduction and
LQR
a
is desired to provide similar performance. Therefore, one approach is to picka andb such
that the RMS base drift and roof absolute accelerations are the same as those of optimal passive
design. From Figure 4.10,a = 1.02 10
2
andb = 4.8 10
3
are selected based on this criterion,
resulting inq
x
a
= 6.3710
4
m
2
andq
a
a
= 9.1710
2
s
4
/m
2
. The corresponding peak base drift and
roof absolute acceleration reductions (improvements) compared to the uncontrolled responses are
66.50% and 45.00%, respectively. The peak control force is 5.88% of the weight of the structure for
the El Centro earthquake. Moreover, the corresponding RMS base drift and absolute acceleration
reductions are 75.67% and 64.62%, respectively, relative to uncontrolled responses. The RMS
control force is 1.72% of the weight of the structure.
45
LQR design for strong excitation: LQR
d
Several approaches can be used to design LQR
d
for strong excitation; herein, b is taken to be
the same as the corresponding value used in LQR
a
; however, additional base drift reduction is
obtained by selecting a larger value ofa. Further,a for LQR
d
is chosen to maximize base drift
−120
−100 −100
−80
−80
−60
−60
−60
−40
−40
−40
−20
−20
−20
−20
0
0
0
20
20
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
RMS base drift improv.[%]
RMS roof accel. improv.[%]
−60
−40
−20
0
20
40
60
Figure 4.10: RMS response reduction of LQR compared to Passive [%], El Centro, 6DOF
1.5
2
2
2.5
2.5
3
3
3.5
3.5
4
4
4.5
4.5
5
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 4.11: RMS required LQR control force [%W], El Centro, 6DOF
46
reduction subject to the peak absolute acceleration not be larger than the corresponding uncontrolled
value. Figure 4.12 and Figure 4.13 show the percent reduction of the Peak and RMS responses of
LQR compared to the uncontrolled system. The corresponding maximum required LQR control
forces are shown in Figure 4.14. From Figure 4.12, a = 2.1 10
1
is selected, resulting in
q
x
d
= 1.13 10
6
m
2
and q
a
d
= 9.17 10
2
s
4
/m
2
. The corresponding peak base drift and roof
absolute acceleration reductions improvements compared to uncontrolled responses are 84.39%
and 1.07%, respectively. The peak control force is 13.08% of the weight of the structure for the
El Centro earthquake. Moreover, the corresponding RMS base drift and absolute acceleration
reductions are 89.19% and 44.06% , respectively. The RMS of the control force is 3.69% of the
weight of the structure.
Chooseh to satisfy the multi-objective problem
It is desired that (4.30) approaches LQR
d
and LQR
a
— designed in the previous paragraphs —
in strong and weak excitation, respectively. An approach similar to that in Section 4.4.1 is used
here to find the best h that satisfies the multi-objective problem. While there are a multitude
of approaches to find the best h, one is as follows. The objective function to be minimized in
fminsearch is considered to be composed of two parts: (1) the relative difference between the
RMS of the nonlinear controlled base drift response and that of optimal linear control law LQR
d
in
−60
−60
−40
−40
−20
−20
0
0
0
20
20
20
40
40
40 40
60
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
Peak base drift improv.[%]
Peak roof accel. improv.[%]
45
50
55
60
65
70
75
80
85
90
Figure 4.12: Peak response reduction of LQR compared to uncont. [%], El Centro, 6DOF
47
strong excitation (2.0El Centro); and (2) the relative difference between the RMS of the nonlinear
controlled roof ansolute acceleration response and that of the optimal linear control law LQR
a
in
20
30
30
40
40
40
50
50
50
60
60
60 60
70
70
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
RMS base drift improv.[%]
RMS roof accel. improv.[%]
60
65
70
75
80
85
90
Figure 4.13: RMS response reduction of LQR compared to uncont. [%], El Centro, 6DOF
5
10
10
15
15
20
25
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
5
10
15
20
25
Figure 4.14: Peak required LQR control force [%W], El Centro, 6DOF
48
weak excitation (0.1El Centro). This objective function can be expressed
f = f
d
+ f
a
=

1
kx
base
NL
k
kx
base
LQR
d
k

Strong
+

1
k ¨ x
roof
NL
k
k ¨ x
roof
LQR
a
k

Weak
(4.32)
Figure 4.15 shows the variation of f
a
and f
d
relative toh for weak and strong excitation, respectively.
Clearly, the minimum of f occurs ath = 1.8 10
2
. The response of the structure is computed
for the optimal nonlinear control along with that of the two linear controls. Responses in the
weak excitation are shown in Figure 4.16, whereas Figure 4.17 shows the responses to the strong
excitation, The results are also presented in Table 4.6. Clearly, the nonlinear control strategy closely
tracks the LQR
d
design in the case of strong excitation and it performs almost the same as LQR
a
in the weak excitation, demonstrating that the same nonlinear control law can achieve both of the
competing objectives.
4.5.1 Applying a Saturation Constraint on the Control Force
The results shown so far in this study present a very promising performance for the proposed
nonlinear control under both weak and strong excitation; however, the required control force in
strong excitation is very large. Any real implementation would, of course, need to use lower
10
-3
10
-2
10
-1
10
0
10
-3
10
-2
10
-1
10
0
η

f
a
f
d
f = f
a
+ f
d
Figure 4.15: Variation of multiobjective cost relative toh, for the 6DOF problem
49
−0.5
0
0.5
1
b. drift
[cm]

Nonlinear
LQR
d
LQR
a
−0.1
0
0.1
0.2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−1
0
1
time [s]
force
[%W]
Figure 4.16: Controlled response comparison for 0.1El Centro, 6DOF, HJB
−10
0
10
20
b. drift
[cm]

Nonlinear
LQR
d
LQR
a
−4
−2
0
2
4
6
r. abs. accel
[m/s
2
]
0 5 10 15 20
−20
0
20
40
time [s]
force
[%W]
Figure 4.17: Controlled response comparison for 2El Centro, 6DOF, HJB
force levels, with peak on the order of 10–15% of structure weight. Therefore, a saturation in the
commanded force may be introduced into the problem to accommodate control device capacity.
The control force saturation does not affect the performance of the designed controllers under weak
excitation; therefore, the response of the structure is the same with or without presence of the
control force saturation under weak excitation. It is worth mentioning thath is not recalculated
here; rather the same h = 1.8 10
2
obtained from the unsaturated control forces will be used
50
Table 4.6: Performance of HJB under scaled 1940 El Centro excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak b. drift[cm] 3.182 1.067 0.499 0.995 63.374 21.358 9.961 9.406
RMS b. drift[cm] 1.292 0.315 0.140 0.301 25.694 6.306 2.803 2.841
Peak r. abs. accel.[m/s
2
] 0.208 0.115 0.206 0.121 4.135 2.270 4.0646 4.124
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.030 1.659 0.592 0.937 1.260
Peak cont. force [% bldg.weight] — 0.587 1.310 0.660 — 11.778 26.191 45.934
RMS cont. force [% bldg.weight] — 0.172 0.369 0.181 — 3.451 7.388 10.483
−0.5
0
0.5
1
b. drift
[cm]

Nonlinear
LQR
d
LQR
a
−0.1
0
0.1
0.2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−1
0
1
time [s]
force
[%W]
Figure 4.18: Controlled response comparison for 0.1El Centro, saturated, 6DOF, HJB
herein. Figure 4.18 and Figure 4.19 show the response of the nonlinear control, along with the
two linear controls, under weak and strong excitations, respectively when 15% of structure weight
is considered as the control force saturation. The numerical values are also shown in Table 4.7.
Clearly, the performance of the nonlinear control is still similar to LQR
d
in the strong excitation:
saturated nonlinear control results in 3.234 cm RMS base drift, which is only 5% different from the
corresponding value provided by the saturated LQR
d
(3.077 cm), while 95% different from that of
saturated LQR
a
(6.306 cm).
4.5.2 Semiactive Control Design
Researchers have turned to controllable passive devices, also called semiactive because they combine
the characteristics of passive and active systems. Semiactive devices are those that dissipate energy
51
−20
0
20
b. drift
[cm]

Nonlinear
LQR
d
LQR
a
−2
0
2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−15
0
15
time [s]
force
[%W]
Figure 4.19: Controlled response comparison for 2El Centro, saturated, 6DOF, HJB
Table 4.7: Performance of saturated HJB under scaled 1940 El Centro excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak b. drift[cm] 3.182 1.067 0.499 0.995 63.374 21.358 12.185 11.609
RMS b. drift[cm] 1.292 0.315 0.140 0.301 25.694 6.306 3.077 3.234
Peak r. abs. accel.[m/s
2
] 0.208 0.115 0.206 0.121 4.135 2.270 3.326 3.572
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.030 1.659 0.592 0.879 0.870
Peak cont. force [% bldg.weight] — 0.587 1.310 0.660 — 11.778 15.000 15.000
RMS cont. force [% bldg.weight] — 0.172 0.369 0.181 — 3.451 6.613 6.631
through passive means, but with controllable properties so that device forces can be modified
and adjusted based on sensor measurements, feedback of excitation, changing objectives and so
forth. Common strategies for commanding the forces in these devices involve designing a primary
controller such as for an active device, but then use a secondary controller to eliminate from the
command any nondissipative control forces (Housner et al., 1997). The idealized passivity constraint
can be presented as
u
active
v
device
< 0 (4.33)
i.e., if the desired active control forceu
active
and velocityv
device
across the device are the same sign,
then the force is nondissipative and will be eliminated by the secondary controller. Thus,
u
semiactive
=u
active
H(u
active
v
device
) (4.34)
52
where H() is the Heaviside unit step function. If the desired active control is determined from
optimal LQR, the resulting semiactive control is often called clipped LQR (CLQR).
The forces commanded by the (unsaturated) HJB nonlinear control and the corresponding clipped
semiactive control, for the MDOF building subjected to 2El Centro excitation, are shown in
Figure 4.20 as functions of the velocity across the device (i.e., the base velocity). By comparing
these two graphs, one can verify that the semiactive control force is always dissipative, as it is always
in the opposite direction of the velocity across the device, while the active control force may have
nondissipative components (shown with black dots in Figure 4.20). The resulting response of the
structure to scaled El Centro excitation is presented in Table 4.8. The semiactive nonlinear control
tracks the semiactive CLQR
a
and semiactive CLQR
d
in weak and strong excitations, respectively. It
should be noted theh value used here is the sameh = 1.8 10
2
obtained for active control force
(one could recalculate the optimal value ofh based on the semiactive LQR control forces, which is
likely to provide even better performance).
-0.5 0 0.5
-40
-30
-20
-10
0
10
20
30
40
50
Velocity across device [m/s]
Semiactive control force [%weight]
-0.5 0 0.5
-40
-30
-20
-10
0
10
20
30
40
50
Velocity across device [m/s]
Active control force [%weight]
Figure 4.20: Active and Semiactive comparison, 6DOF, 2El Centro
53
Table 4.8: Performance of semiactive controls under scaled 1940 El Centro excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. CLQR
a
CLQR
d
Nonlinear Uncont. CLQR
a
CLQR
d
Nonlinear
Peak b. drift[cm] 3.182 1.067 0.486 0.996 63.374 21.345 9.751 9.039
RMS b. drift[cm] 1.292 0.316 0.141 0.302 25.694 6.305 2.816 2.709
Peak r. abs. accel.[m/s
2
] 0.208 0.113 0.190 0.121 4.135 2.320 3.805 6.910
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.048 0.030 1.659 0.591 0.951 1.198
Peak cont. force [% bldg.weight] — 0.591 1.291 0.664 — 11.818 25.819 42.815
RMS cont. force [% bldg.weight] — 0.173 0.354 0.182 — 3.450 7.077 8.981
4.5.3 Robustness Study
The analysis and promising results presented so far in this study assumed that all parameters of the
system —e.g., mass, stiffness and damping — are known. It was also assumed that the excitation
is a scaled version of the El Centro earthquake. However, it should be investigated whether these
results are still promising when some of these assumptions are violated; i.e., while the original
analysis is based on some specific parameters, what will happen if the real parameters are different
from those values? If the results are still promising, the robustness of the proposed method is
confirmed.
4.5.3.1 Performance under several Kanai-Tajimi produced excitations
Assuming all analysis and calculations are performed based on 0.1 and 2-scaled versions of the El
Centro earthquake as the weak and strong excitations, respectively, the 6DOF building model is
now subjected to different earthquake records to investigate if the proposed nonlinear controller can
still switch its behavior under other weak and strong excitations. It should be noted that the value of
h used in this section is the value obtained previously based on the El Centro excitation; further,
this robustness study is for unsaturated active control.
A KT filter — presented previously in Section 4.4.3 — which filters Gaussian white noise to produce
earthquake-like excitations, is incorporated into the problem herein. Assuming the Gaussian white
noisew(t) is scaled such that the KT output signal has RMS and PGA similar to the 0.1El Centro
and 2El Centro earthquake, the response of the system subjected to these excitations can be
calculated. The duration of all excitation realizations is considered to be 20 sec to be consistent
with the length of the El Centro record used in the analysis. The mean and standard deviation of
the responses of the system subjected to 100 different earthquake-like records, produced by the
KT filter, are calculated and presented in Table 4.9. Results confirm that, although the nonlinear
54
Table 4.9: Average Performance under 100 scaled Kanai-Tajimi produced excitations (with standard
deviation listed with notation below the corresponding mean value)
0.1 Kanai-Tajimi 2 Kanai-Tajimi
PGA= 0.34 m/s
2
PGA= 6.8 m/s
2
Uncont. LQR
a
LQR
d
Nonlinear Uncont. LQR
a
LQR
d
Nonlinear
Peak b. drift[cm] 3.969 1.464 0.645 1.370 79.409 29.283 12.901 13.556
1.188 0.253 0.098 0.213 23.782 5.063 1.950 2.192
RMS b. drift[cm] 1.693 0.543 0.230 0.515 33.860 10.865 4.595 4.729
0.581 0.080 0.023 0.070 11.629 1.598 0.460 0.628
Peak r. abs. accel.[m/s
2
] 0.259 0.139 0.220 0.151 5.175 2.780 4.407 5.426
0.077 0.019 0.024 0.023 1.538 0.383 0.472 0.688
RMS r. abs. accel.[m/s
2
] 0.109 0.048 0.072 0.048 2.188 0.955 1.434 1.689
0.038 0.005 0.003 0.005 0.750 0.107 0.066 0.110
Peak cont. force [% bldg.weight] — 0.737 1.615 0.880 — 14.744 32.293 43.583
0.106 0.186 0.156 2.114 3.721 6.220
RMS cont. force [% bldg.weight] — 0.251 0.537 0.268 — 5.021 10.736 13.572
0.026 0.031 0.029 0.513 0.619 1.035
control was designed for El Centro excitation, it still can achieve both drift reduction and absolute
acceleration reduction under other strong and weak excitations, respectively.
4.5.3.2 Error in base stiffness
Assuming the nonlinear controller is designed considering some specific value for base stiffness,
it is necessary to investigate whether it can still switch between base drift reduction and roof
absolute accreditation reduction in strong and weak excitations, respectively, when the assumed
base stiffness is not quite accurate. While there are many possible metrics to observe and evaluate
the performance of the nonlinear controller for any possible error in base stiffness estimation, herein,
two of these are investigated and presented: (1) change in peak and RMS response and (2) changes
in nonlinear controller capability to switch between drift and acceleration reduction in strong and
weak excitations, respectively. The latter is shown using the ratio of the response in strong excitation,
2El Centro, over the corresponding value in weak excitation, 0.1El Centro:
Response ratio=
Response to 2 El Centro
Response to 0.1 El Centro
(4.35)
For linear systems and linear controllers, if the excitation scales by some number — 2/0.1= 20,
herein — all responses will be scaled by the same factor, 20. However, the proposed nonlinear
controller was shown to be capable of scaling base drift and absolute acceleration differently. If it
can be shown that the nonlinear controller still result in different scaling of base drift and absolute
acceleration, even if it is designed for a different base stiffness, robustness of its performance with
regard to error in base stiffness is confirmed. On the other hand, if the scaling factors are not
55
Table 4.10: Effects of error in base stiffness on response in strong excitation
Error in base stiffness,Dk
base
[%]
–20% –15% –10% –5% 0% 5% 10% 15% 20%
Peak b. drift
Uncont. 73.563 65.691 62.139 63.279 63.374 62.858 61.550 59.531 57.330
LQR
a
19.489 20.134 20.591 20.978 21.358 21.696 21.900 21.995 22.099
[cm] LQR
d
9.853 9.867 9.923 9.923 9.961 9.977 10.003 10.022 10.047
Nonlinear 9.330 9.351 9.371 9.389 9.406 9.421 9.434 9.445 9.455
RMS b. drift
Uncont. 39.157 36.816 32.858 28.940 25.694 23.582 22.634 22.486 22.681
LQR
a
6.294 6.328 6.328 6.313 6.306 6.287 6.237 6.194 6.151
[cm] LQR
d
2.770 2.776 2.786 2.792 2.803 2.810 2.816 2.825 2.833
Nonlinear 2.819 2.828 2.835 2.839 2.841 2.841 2.838 2.834 2.828
Peak r. abs. accel.
Uncont. 3.808 3.646 3.658 3.887 4.135 4.347 4.440 4.497 4.482
LQR
a
2.172 2.204 2.237 2.254 2.270 2.320 2.378 2.464 2.458
[m/s
2
] LQR
d
4.195 4.178 4.163 4.148 4.124 4.123 4.096 4.081 4.090
Nonlinear 6.375 6.315 6.268 6.232 6.207 6.192 6.184 6.184 6.191
RMS r. abs. accel.
Uncont. 2.022 2.019 1.908 1.775 1.659 1.600 1.610 1.672 1.761
LQR
a
0.550 0.561 0.572 0.581 0.592 0.603 0.612 0.622 0.632
[m/s
2
] LQR
d
0.937 0.936 0.936 0.936 0.937 0.937 0.937 0.937 0.939
Nonlinear 1.286 1.278 1.270 1.265 1.260 1.257 1.254 1.253 1.252
Peak cont. force
LQR
a
12.515 12.409 12.235 12.013 11.778 11.545 11.307 11.039 10.793
[% bldg.weight]
LQR
d
26.850 26.627 26.498 26.274 26.191 26.076 25.854 25.729 25.745
Nonlinear 48.519 47.740 47.054 46.456 45.934 45.483 45.095 44.764 44.484
RMS cont. force
LQR
a
3.497 3.487 3.472 3.457 3.451 3.446 3.438 3.434 3.433
[% bldg.weight]
LQR
d
7.567 7.516 7.473 7.424 7.388 7.347 7.299 7.261 7.227
Nonlinear 10.903 10.778 10.667 10.569 10.483 10.407 10.342 10.285 10.236
different or approaching the same value, 20 herein, it can be concluded that the nonlinear controller
is no longer achieving the two different objectives, and it is not robust to base stiffness assumption
error.
Assuming the real value of base stiffness is varied between 80% to 120% of the value used to
design the control, the response of the system with no controller, and with two linear and nonlinear
controllers subjected to strong excitation are obtained and presented in Table 4.10. It is clear that
inaccurate assumption of base stiffness does not significantly affect the nonlinear controlled peak
and RMS responses. In fact, the changes in the nonlinearly controlled responses are even smaller
than the corresponding values using the LQR
d
controller.
To investigate the robustness of the switching property of the nonlinear controller, the responses of
the system with the nonlinear controller, subjected to weak and strong excitations, are obtained and
used to calculate four response ratios, namely peak base drift, RMS base drift, peak roof absolute
acceleration and RMS roof absolute acceleration, computed by (4.35). The results are shown in
Table 4.11 and Figure 4.21. Clearly, assuming an inaccurate base stiffness does not affect peak and
RMS base drift scaling and they remain constant. It is also clear that, if the nonlinear control is
56
Table 4.11: Robustness study, error in base stiffness, 6DOF
Error in base stiffness,Dk
base
[%]
–20% –15% –10% –5% 0% 5% 10% 15% 20%
Peak b. drift ratio 10.199 9.961 9.758 9.587 9.450 9.342 9.250 9.187 9.152
RMS b. drift ratio 9.377 9.379 9.390 9.408 9.434 9.466 9.499 9.539 9.581
Peak r. abs. accel. ratio 53.928 53.063 52.337 51.745 51.279 50.917 50.690 49.912 49.221
RMS r. abs. accel. ratio 45.914 44.857 43.877 42.972 42.139 41.342 40.727 40.022 39.426
0.8 0.9 1 1.1 1.2 1.3
5
10
15
20
25
30
35
40
45
50
55
Response ratio, strong/weak
Stiffness ratio, k
new
/k
original

Peak b. drift ratio
RMS b. drift ratio
Peak r. abs. accel. ratio
RMS r. abs. accel. ratio
Figure 4.21: Response ratio of 2El Centro over 0.1El Centro for varying base stiffness, 6DOF
designed for a base stiffness that is larger than its real value, it can result in a larger increase in roof
absolute acceleration when subjected to strong excitation. However, the level of increase in roof
absolute acceleration is small, less than 10% of the corresponding values of the original design.
4.5.3.3 Error in base damping
Assuming the nonlinear controller is designed considering some specific value for base damping,
this section investigates whether the nonlinear controller can still switch between base drift reduction
and roof absolute accreditation reduction in strong and weak excitations, respectively, when the
assumed base damping is not quite accurate.
Assuming the real value of base damping is varied between 80% to 120% of the value used in
57
Table 4.12: Robustness study, error in base damping, 6DOF
Error in base damping,Dc
base
[%]
–20% –15% –10% –5% 0% 5% 10% 15% 20%
Peak b. drift ratio 9.468 9.464 9.459 9.455 9.450 9.446 9.441 9.437 9.432
RMS b. drift ratio 9.383 9.395 9.408 9.421 9.434 9.447 9.459 9.472 9.485
Peak r. abs. accel. ratio 51.876 51.726 51.577 51.428 51.279 51.130 50.982 50.833 50.684
RMS r. abs. accel. ratio 42.425 42.354 42.282 42.210 42.139 42.068 41.996 41.925 41.854
0.8 0.9 1 1.1 1.2 1.3
5
10
15
20
25
30
35
40
45
50
55
Damping ratio, c
new
/c
original

Response ratio, strong/weak

Peak b. drift ratio
RMS b. drift ratio
Peak r. abs. accel. ratio
RMS r. abs. accel. ratio
Figure 4.22: Response ratio of 2El Centro over 0.1El Centro for varying base damping, 6DOF
design, the responses of the system with the nonlinear controller, subjected to weak and strong
excitations, are obtained and used to calculate the four response ratios. The results are shown in
Table 4.12 and Figure 4.22. Clearly, assuming an inaccurate base damping does not significantly
affect any of the response ratios, as they all remain almost constant. These results confirm the robust
performance of the nonlinear control to an inaccurate assumption of the base damping.
4.5.3.4 Error in base mass
In this section, it is assumed that the real value of base mass is varied between 80% to 120% of that
used for design; the responses of the system with the nonlinear controller, subjected to weak and
strong excitations, are obtained and used to calculate the four response ratios. The results are shown
in Table 4.13 and Figure 4.23. Clearly, assuming an inaccurate base mass does not affect peak
58
Table 4.13: Robustness study, error in base mass, 6DOF
Error in base mass,Dm
base
[%]
–20% –15% –10% –5% 0% 5% 10% 15% 20%
Peak b. drift ratio 9.354 9.381 9.405 9.427 9.450 9.473 9.496 9.520 9.544
RMS b. drift ratio 9.573 9.539 9.505 9.470 9.434 9.398 9.362 9.325 9.289
Peak r. abs. accel. ratio 50.754 50.869 50.997 51.134 51.279 51.436 51.603 51.778 51.962
RMS r. abs. accel. ratio 40.917 41.222 41.527 41.833 42.139 42.446 42.754 43.063 43.373
0.8 0.9 1 1.1 1.2 1.3
5
10
15
20
25
30
35
40
45
50
55
Mass ratio, m
new
/m
original

Response ratio, strong/weak

Peak b. drift ratio
RMS b. drift ratio
Peak r. abs. accel. ratio
RMS r. abs. accel. ratio
Figure 4.23: Response ratio of 2El Centro over 0.1El Centro for varying base mass, 6DOF
and RMS base drift scaling as they remain constant. It is also clear that, if the nonlinear control is
designed for a base mass that is smaller than its real value, it can result in a larger increase in roof
absolute acceleration when subjected to strong excitation. However, the level of increase in roof
absolute acceleration is very small, less than 3% of the corresponding values of the original design.
4.5.4 Incorporating Kalman Filter
So far in this study, it was assumed that full-state feedback is available to the controller. However,
in real-world application, the full-state feedback is rarely available, but only a few noisy sensor
measurements (e.g., from accelerometers) are available to the controller. To overcome this challenge
and minimize the effect of noise for accurate estimation of full states, the Kalman filter (KF)
(Kalman, 1960, Kalman and Bucy, 1961, Kalman, 1963) can be incorporated into the problem. In
59
this section, it is desired to show that the proposed nonlinear control design is also applicable to
the more realistic cases when only a few noisy response measurements are available, instead of all
accurate states. Therefore, a KF estimator will be incorporated into the system in order to estimate
the states from measurements; the continous-time KF is given in state-space form as:
b y = C
y
b q+ D
y
u (4.36a)
˙
b q = Ab q+ Bu+ K
k
(yb y) (4.36b)
= Ab q+ Bu+ K
k
(y C
y
b q D
y
u)
= (A K
k
C
y
)b q+(B K
k
D
y
)u+ K
k
y
= (A K
k
C
y
)b q+(B K
k
D
y
)u+ K
k
(C
y
q+ D
y
u+ F
y
w+ v)
= (A K
k
C
y
)b q+ K
k
C
y
q+ Bu+ K
k
F
y
w+ K
k
v
whereb y is the output estimate;b q is the optimal estimate of the state vector; K
k
is the Kalman gain
matrix that can be calculated as
K
k
=(P
v
C
T
y
+ N)R
1
(4.37)
where P
v
is the error covariance matrix for the state estimate that can be determined by solving
the algebraic Riccati equation AP
v
+ P
v
A
T
+ Q(P
v
C
T
y
+ N)R
1
n
(C
y
P
v
+ N
T
)= 0 where Q=
EQ
n
E
T
; R= R
n
+ F
y
N
n
+ N
T
n
F
T
y
+ F
y
Q
n
F
T
y
and N= E(Q
n
F
T
y
+ N
n
). w(t) and v(t) are assumed
to be zero-mean stationary Gaussian noises (i.e.,
E
[w(t)]= 0,
E
[v(t)]= 0) with
E
[w(t)w
T
(t)]= Q
n
,
E
[v(t)v
T
(t)]= R
n
, and
E
[w(t)v
T
(t)]= N
n
= 0 (4.38)
Considering q=[q
T
b q
T
]
T
as the generalized state vector; and w=[w
T
v
T
]
T
as the generalized
excitation vector, (4.36b) can be rewritten as
˙
q= Aq+ Bu+ Ew (4.39)
where
A=
"
A 0
K
k
C
y
A K
k
C
y
#
, B=
"
B
B
#
, E=
"
E 0
K
k
F
y
K
k
#
(4.40)
Herein, it is assumed that the structure is instrumented with identical accelerometers on all floors,
and also one on the ground to record the ground excitation; the measurement noise is a Gaussian
process with an RMS of approximately 10% of the RMS of the roof absolute acceleration.
60
The main objective of this section is to show that the HJB-based control force can be obtained even
if only a few noisy output measurement are available, whereas previous sections assumed full state
feedback was available.
Assuming the system is subjected to a 2El Centro excitation, Figure 4.24 shows the actual
and estimated roof absolute acceleration along with the measurement noise. The corresponding
actual and estimated base drift and HJB-based control force are shown in Figures 4.25 and 4.26,
respectively; the results appear visually to be almost identical, verifying that the KF estimates
the response quite accurately. The resulting RMS estimated base drift is 2.80 cm which is only
1.2% different from the actual base drift, 2.84 cm. Further, estimated roof absolute acceleration,
1.23 m/s
2
, is only 2.4% different from the actual value, 1.26 m/s
2
. Also, the difference between
the commanded force based on the KF-estimated states, 36.418kN, and the commanded force based
on the full state feedback, 37.197 kN, is only 2.09%. It should be noted that the aforementioned
errors are based on the following metric:
Error of RMS=

RMS(exact) RMS(estimated)
RMS(exact)

(4.41)
However, an alternate error assessment can also be obtained from the following metric:
RMS of Error=

RMS(exactestimated)
RMS(exact)

(4.42)
The corresponding RMS errors between the exact and estimated values, based on (4.42), are 13.49%
for base drift, 5.50% for roof absolute acceleration and 8.42% for the commanded control force.
4.5.5 Incorporating Kalman Filter and Kanai-Tajimi Filter
The KF provides an optimal estimate of the original signal when system is linear subjected to
white Gaussian noise. In this section, the KT filter — presented previously in Section 4.4.3 — is
incorporated into the problem. As shown in Figure 4.27, the KT filter converts Gaussian white
noise into an earthquake-like excitation. The KT states will be augmented into (4.39);i.e., q should
be changed to [q
T
,q
T
g
]
T
, where q
g
contains two KT states. Assuming the Gaussian white noise
w(t) is scaled such that the KT output signal ¨ x
g
(t) has RMS and PGA similar to the 2El Centro
earthquake, and the measurement noise is the same as previously presented in Section 4.5.4,
the estimated response of the system subjected to this excitation can be calculated. Figure 4.28
shows the actual and estimated roof absolute acceleration along with the measurement noise. The
corresponding actual and estimated base drift and HJB-based control force are shown in Figures
61
0 5 10 15 20
−6
−4
−2
0
2
4
6
8
time [s]
roof abs. accel [m/s
2
]

Estimated roof abs. accel
Actual roof abs. accel
Noise
Figure 4.24: Actual and estimated roof abs. acceleration, 6DOF, 2El Centro
0 5 10 15 20
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
time [s]
base displ [m]

Estimated base displ
Actual base displ
Figure 4.25: Actual and estimated base displacement, 6DOF, 2El Centro
4.29 and 4.30, respectively. The resulting RMS estimated base drift is 4.80 cm which is only
1.4% different (computed using (4.41)) from the actual base drift 4.86 cm. Further, estimated roof
62
0 2 4 6 8 10 12 14 16 18 20
-50
-40
-30
-20
-10
0
10
20
30
40
50
time [s]
control force [%W]

based on KF-estimated states
based on full state feedback
Figure 4.26: Commanded force based on full state feedback and KF-estimated states, 6DOF, 2El
Centro
Excitation
States
CLQR
Control Force
Excitation
States
MPC
Control Force
LQR
Control gain
MPC
(YALMIP)
System
Clipping
System
White noise
Kanai-Tajimi
filter
Kalman filter
Actual control
force
LQR
Control gain
Clipping
System
Excitation
+
Meausrement
noise
Output
Estimated
states
Measurement
Velocity across the device
Desired
control force
White noise
Kanai-Tajimi
filter
Kalman filter
control force
Controller
System
Excitation
+
Meausrement
noise
Output
Estimated
states
Measurement
Figure 4.27: Control diagram of a system augmented with Kanai-Tajimi and Kalman filters
absolute acceleration, 1.79 m/s
2
, is only 0.88% different from the actual value, 1.80 m/s
2
. Also,
the difference between the commanded force based on the KF-estimated states, 52.11 kN, and the
commanded force based on the full state feedback, 52.18 kN, is only 0.13%. The corresponding
errors based on (4.42) are 10.29% for base drift, 5.04% for roof absolute acceleration and 7.77%
for the commanded control force. Clearly, the almost identical results verify that KF estimates the
response very accurately.
63
0 5 10 15 20
−5
−4
−3
−2
−1
0
1
2
3
4
5
time [s]
roof abs. accel [m/s
2
]

Estimated roof abs. accel
Actual roof abs. accel
Noise
Figure 4.28: Actual and estimated roof abs. acceleration, 6DOF, KT-produced excitation
0 5 10 15 20
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
time [s]
base displ [m]

Estimated base displ
Actual base displ
Figure 4.29: Actual and estimated base displacement, 6DOF, KT-produced excitation
64
0 5 10 15 20
-40
-30
-20
-10
0
10
20
30
40
time [s]
Control force [%W]

based on KF-estimated states
based on full state feedback
Figure 4.30: Commanded force based on full state feedback and KF-estimated states, 6DOF,
KT-produced excitation
4.6 Conclusions
A new “optimal” nonlinear control law that can achieve different objectives in different levels of
excitations is derived analytically by solving the HJB equation. The cost function is a 6
th
order
polynomial of the states and quadratic in the control force such that corresponding HJB equation
has a solution that results in a cubic optimal control law: the sum of a linear term that is related to
an LQR problem that is effective in small excitations, and a cubic nonlinear term that is dominant
when the excitation level is large. The main advantage of this nonlinear controller is that it reduces
acceleration response to weak excitation and focuses on drift reduction in strong excitation. In one
control law, both objectives are satisfied. The linear control gain is obtained by solving an algebraic
Riccati equation; to find the optimal cubic gain, an optimization is performed using an algorithm
like MATLAB’sfminsearch. For a zero nonlinear control gain, both the control law and the cost
function revert back to the optimal LQR control problem.
The performance of the nonlinear control strategy is demonstrated by numerical simulations for
SDOF and a base-isolated MDOF building models subjected to different excitations, showing that it
can successfully achieve both life safety (drift reduction) and serviceability (acceleration reduction)
objectives at different excitation levels. Moreover, it is shown that the promising performance is
65
still valid when semiactive control force is applied with or without control force saturation. Also,
the robustness of the proposed controller is verified by showing that the promising performance
remains in the presence of inaccurate mass, stiffness and damping and in a variety of excitations.
The KT filter is implemented to produce several synthetic earthquake-like excitations to be applied
to the structure. Finally, the KF is incorporated into the problem and it is shown that it is successful
in predicting the response when some measurement noise exists.
66
Chapter 5
Nonlinear Control for Life Safety and
Serviceability using Gain Scheduling
Strategy
1
5.1 Introduction
In this chapter, two optimal linear control laws are obtained for weak and strong magnitude of
earthquakes to provide serviceability and safety, respectively. Gain Scheduling is introduced into
the problem to find a control strategy which alters its performance and remains “optimal” as the
magnitude of earthquake changes.
5.2 Problem Formulation
Consider an n degree-of-freedom (DOF) linear system subjected to external disturbance w. The
equations of motion, and those governing sensor measurements and responses are the same as those
presented in Section 3.2 but are cast in discrete time where, for example, discrete-time q(k) is a
1
Hemmat-Abiri and Johnson (2015) reports a subset of this chapter.
67
zero-order hold (ZOH) approximation of the continuous time q(k4t):
q(k+ 1)= A
d
q(k)+ B
d
u(k)+ E
d
w(k) (5.1a)
z(k)= C
z
q(k)+ D
z
u(k)+ F
z
w(k) (5.1b)
To obtain the optimal linear control, one may ignore the system disturbance w(k) and minimize
a cost function that is quadratic in the states and the control force (the latter is necessary for the
optimization to be well-posed), cast here in discrete-time as
J=
¥
å
k=1

q
T
Qq+ 2q
T
Nu+ u
T
Ru

(5.2)
in which(k) is dropped for the sake of clarity; Q, N and R are matrices of appropriate dimensions
for the discrete-time problem. It is also possible to construct the cost function in terms of outputs as
J=
¥
å
k=1

z
T
Q
z
z+ 2z
T
N
z
u+ u
T
R
z
u

(5.3)
In this case, the corresponding Q, N and R can be obtained via the following transformation:
"
Q N
N
T
R
#
=
"
C
T
z
0
D
T
z
I
#"
Q
z
N
z
N
T
z
R
z
#"
C
z
D
z
0 I
#
Therefore
Q= C
T
z
Q
z
C
z
N
d
= C
T
z
Q
z
D
z
R= D
T
z
Q
z
D
z
+ R
z
(5.4)
To achieve the two different objectives, drift reduction and absolute acceleration reduction, in
different levels of excitations, two quadratic cost functionsJ
d
andJ
a
are constructed as defined in
(5.2) with two different sets of weighing matrices Q, N and R. Each of these objective functions
will result in a different optimal linear control law, namely u
d
=K
d
q and u
a
=K
a
q where K
d
and K
a
are the optimal linear control gains, found using MATLAB’slqr command. However, as
u
d
is designed to focus more on drift reduction in strong excitations, it may result in increased
accelerations, which would be undesirable in the moderate excitations. Similarly, u
a
will achieve
further reductions in acceleration, but is not optimal in reducing drift in the case of strong excitation.
In this study,gainscheduling is applied to find a control strategy that satisfies the two objectives
in their corresponding excitation levels. The point of this study is to design a controller such that,
in the strong excitation, it closely tracks u
d
and approaches u
a
when the excitation is weak, and
interpolates between the two for moderate excitations. The switching criterion, which may depend
68
on time, structural response and excitation, should be defined appropriately such that both objectives
are obtained. A preliminary gain scheduling was presented in Chapter 3.
To design a gain scheduling controller for the purpose of this study, three approaches are investigated
and presented herein; all three approaches use the two optimal linear laws given by gains K
a
and
K
d
for weak and strong excitations. The first gain scheduling approach switches between the two
linear quadratic regulator (LQR) controls as the observable parameter, the running peak ground
acceleration (PGA), changes. It is demonstrated that, since there is no intermediate transition
function in this method, it does not perform optimally for moderate excitations. In the second
approach, gain-scheduled controls with linear and sigmoid transition functions are designed such
that switching between the two designed LQR controls depends on some metrics of the PGA. In the
third approach, it is assumed that the cost coefficients in the LQR design are not constant and depend
on some metric of the PGA; the resulting optimal linear control varies as the level of excitation
changes.
5.3 Numerical Results
In this section, to demonstrate the potential of the gain scheduling-based controller to achieve both
goals, a numerical simulation is performed for several building structure models. The performance
improvement of the gain scheduling-based nonlinear control strategy is presented and compared
with that of optimal linear control laws found by LQR for the model subjected to different levels of
excitations. The strong and weak excitation for this problem are considered to be 2.0- and 0.1-scaled
versions of the 20 sec record of the North-South component of the 1940 El Centro earthquake, with
PGAs of 6.8340 m/s
2
and 0.3417 m/s
2
, respectively.
5.3.1 Approach I: Switching Between LQR
a
and LQR
d
without an Interme-
diate Transition
Considering the same single degree-of-freedom (SDOF) building structure presented in Section 3.3
(with massm= 29485kg, damping coefficientc= 23710 Ns/m and stiffnessk= 1.1640MN/m),
the infinite-horizon discrete-time quadratic cost function can be presented as
J=
¥
å
k=1
[q
x
x
2
(k)+q
a
¨ x
2
abs
(k)+ru
2
(k)] (5.5)
69
where ¨ x
abs
=kx/mc ˙ x/m+u/m is the absolute acceleration of the structure and k, c, and m
are stiffness, damping and mass of the structure, respectively.
Considering equation (5.1) for a SDOF system, where q=[x ˙ x]
T
, and z=[x ¨ x
abs
]
T
, one can
rewrite (5.5) as (5.3) where
Q
z
=
"
q
x
0
0 q
a
#
, N
z
=
"
0
0
#
, R
z
=r
(5.5) can be also written as (5.2) where
Q=
q
a
m
2
"
k
2
+
m
2
q
x
q
a
ck
ck c
2
#
, N=
q
a
m
2
"
k
c
#
, R=r+
q
a
m
2
The optimal control gains K
a
and K
d
can be simply obtained using thelqry command in MAT-
LAB
R
.
Assuming r= 10
4
/(mg)
2
= 1.1953 10
7
N
2
, one can find two desired pairs of (q
a
, q
x
) for
weak and strong excitations. Consideringq
x
=a 10
8
/H
2
andq
a
=b 10
8
(m/Hk)
2
in which
H = 4 m is the height of the structure, one can choose the desired nondimensional a and b by
observing how changinga andb affect the resulting LQR-controlled drift and absolute acceleration
compared with the uncontrolled response of the same system and excitation. In this section, scaled
versions of the North-South component of the first 20 sec of the 1940 El Centro earthquake record
are considered as the external excitation. Further,Dt is considered the same as time steps in the
earthquake record, 0.02 sec. Figure 5.1 and Figure 5.2 show the percent reduction of peak responses
provided by LQR compared to the uncontrolled system. For various a and b, the filled surface
and contours show (kx
uncontrolled
kkx
lqr
k)/kx
uncontrolled
k 100%, which is the drift reduction
percentage provided by LQR compared to the uncontrolled response; the mesh surface and dashed
contours show the absolute acceleration reduction percentage provided by LQR compared to the
uncontrolled response. Figure 5.3 shows the corresponding peak required control force in terms of
percentage of the weight of the structure.
LQR design for weak excitation, LQR
a
There are many possible choices ofa andb (depending on the desired performance, maximum
allowable response, maximum required control force, etc.) for weak excitation. For example, one
could choosea = 0 and then choose ab value that gives an appropriate trade off of acceleration
reduction versus the required control forces (such asb about 0.5); then, to shift to drift reduction,a
70
Figure 5.1: Peak response reduction of LQR compared to uncont. [%], 2El Centro
−20
0
20 20
20
40
40
40
40
60
60
60 60
80
80
80
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
0
10
3
10
−3
10
0
10
3
Peak drift improv.[%]
Peak accel. improv.[%]
0
10
20
30
40
50
60
70
80
90
Figure 5.2: Peak response reduction of LQR compared to uncont. [%], 2El Centro
could be increased. Instead, as seen in Figures 5.1–5.4,a = 1 andb = 30 are used herein, resulting
inq
x
= 6.25 10
6
m
2
andq
a
= 1.203 10
5
s
4
/m
2
, which provides performance at a similar level
(then, for drift reduction, b will be reduced to shift the optimization). The corresponding peak
drift and absolute acceleration reductions improvements compared to uncontrolled responses are
71
20
30
30
40
40
40
50
50
50
60
60
60
40
40
70
70
50
50
60
60
70
70
80
80
90
90
100
110
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
0
10
3
10
−3
10
0
10
3
20
30
40
50
60
70
80
90
100
110
Figure 5.3: Peak required LQR control force [% of the weight of the structure], 2El Centro
30
30
40
40
40
40
50
50
50
50
60
60
60
60
70
70
70
80
80
80
90
90
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
0
10
3
10
−3
10
0
10
3
RMS drift improv.[%]
RMS accel. improv.[%]
−60
−40
−20
0
20
40
60
80
Figure 5.4: RMS response reduction of LQR compared to uncont. [%], 2El Centro
50% and 76%, respectively. The peak control force is 41% of the weight of the structure for 2El
Centro. The corresponding root mean square (RMS) drift and absolute acceleration reductions
improvements compared to uncontrolled responses are 41% and 76%, respectively. The RMS of
the control force is 12% of the weight of the structure.
72
LQR design for strong excitation, LQR
d
a and b are selected such that drift reduction is more significant. Hence, a = 1 and b = 1.35
are used with resultingq
x
= 6.25 10
6
m
2
andq
a
= 5.41 10
3
s
4
/m
2
. The corresponding peak
drift and absolute acceleration reductions relative to uncontrolled responses are 76% and 42%,
respectively. The peak control force is 41% of the weight of the structure for 2El Centro. The
RMS drift and absolute acceleration reductions compared to uncontrolled responses are 71% and
56%, respectively. The RMS of the control force is 8% of the weight of the structure.
It should be noted thata andb are selected such that: (1) the maximum required control force for
both designs are the same for 2El Centro excitation; (2) the resulting LQR
a
and LQR
d
designs are
more focused on absolute acceleration and drift reduction, respectively; and (3)a is assumed to be
the same for both LQR
a
and LQR
d
, to illustrate the effect ofb in switching the focus of control
between drift and absolute acceleration reduction; therefore,a = 0 is not assumed in LQR
a
.
Gain scheduling control
In this method, the switching (observable) parameter, which determines which linear controller
should be applied based on the current operating region of the system, is considered to be the
running maximum drift over a particular time window t
max
, chosen to be 2 sec herein; i.e., d =
max
0tt
max
kx(tt)k. If this value is smaller than some predefined threshold, 2 cm herein, the
earthquake is considered as weak, and the optimal control u
a
for LQR
a
is applied to the system. On
the other hand, if the maximum drift over the last two seconds is larger than the threshold, then u
d
is applied.
gain scheduling control=
(
LQR
a
:d threshold
LQR
d
:d > threshold
It should be noted that selecting the appropriate values for time window length and the drift
threshold is somewhat problem-specific and can depend on the structure, the excitation and the
desired performance of the controlled structure. Figure 5.5 shows the MATLAB Simulink model
used for gain scheduling simulation with this approach. The responses in the weak excitation are
shown in Figure 5.6, whereas Figure 5.7 shows the responses to the strong excitation; peak and RMS
metrics of drift and acceleration responses and of the control force are listed for both excitations in
Table 5.1. Clearly, the gain scheduled control strategy closely tracks the LQR
d
design in the case
of strong excitation and it performs the same as LQR
a
in the weak excitation, demonstrating that
the switching control is indeed switching between the two optimal controls, achieving both of the
competing objectives.
The designed gain scheduling control performs optimally for weak and strong excitation but
73
its evaluation is not complete until its performance for moderate earthquakes is also confirmed.
Figure 5.8 shows the responses to 1El Centro excitation, a moderate earthquake. Clearly, the
designed controller performs very similar u
a
, therefore not optimally for this moderate strength
excitation. In fact, it performs as either u
d
(when the observable parameter is smaller than the
predefined threshold) oru
a
(when the observable parameter exceeds the predefined threshold) for
any magnitude excitation; i.e., there is no intermediate control force transition for different range of
w
u
u control force
q
states
w
Excitation
Drift
Velocity
Discrete State-Space
Zero-Order
Hold
Abs
Transport
Delay
max
KLQR_a KLQR_d
u_d u_a
δ
Switch if
δ > threshold
Figure 5.5: Simulink model used for Gain Scheduling
−0.5
0
0.5
displ [cm]

−0.1
0
0.1
0.2
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.6: Response comparison for two optimal LQR and gain-scheduled controls, 0.1El Centro
74
Table 5.1: Performance of gain-scheduled control under scaled 1940 El Centro excitation
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS Uncont. LQR
a
LQR
d
GS
Peak displ.[cm] 1.140 0.573 0.276 0.573 22.803 11.452 5.524 5.524
RMS displ.[cm] 0.298 0.176 0.085 0.176 5.960 3.530 1.710 1.841
Peak abs. accel.[m/s
2
] 0.454 0.108 0.262 0.108 9.085 2.161 5.241 5.239
RMS abs. accel.[m/s
2
] 0.119 0.028 0.052 0.028 2.374 0.565 1.033 1.027
Peak cont. force [% weight] — 2.069 2.059 2.069 — 41.374 41.166 41.157
RMS cont. force [% weight] — 0.615 0.406 0.615 — 0.123 0.081 0.083
excitation. To overcome this problem, a different type of gain scheduled control with an intermediate
transition function is investigated in the following.
5.3.2 Approach II: Gain Scheduled Controller with an Intermediate Transi-
tion Function
In this section, LQR
a
and LQR
d
designs are considered to be the same as previous section. To
guarantee the desired performance of the gain scheduled control in the presence of moderate
earthquakes, a transition function can be introduced as following:
u=g(t,q,w)u
d
+[1g(t,q,w)]u
a
=fK
d
g(t,q,w)+ K
a
[1g(t,q,w)]gq (5.6)
−10
0
10
displ [cm]

−2
0
2
4
accel [m/s
2
]
0 5 10 15 20
−40
−20
0
20
40
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.7: Response comparison for two optimal LQR and gain-scheduled controls, 2El Centro
75
−5
0
5
displ [cm]

−1
0
1
2
accel [m/s
2
]
0 5 10 15 20
−20
0
20
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.8: Response comparison for two optimal LQR and gain-scheduled controls, 1El Centro
where transition coefficientg, which may depend on time, structural response and excitation, should
be determined appropriately such that the gain scheduled control in (5.6) approaches u
d
when the
excitation is strong, i.e., g! 1, and closely tracks u
a
under weak excitation, i.e., g! 0. There
are many possible linear and nonlinear forms forg that can satisfy these objectives. It should be
noted that (5.6) is the same equation used in Chapter 3; however, in this section, different transition
functions are studied. In this study, two different transition functions, namely Linear and Sigmoid
functions are investigated. The transition function should be constructed in terms of some switching
parameter. In this section, the running peak of the earthquake ground acceleration over the last one
second is considered as the switching parameter. It should be noted that one second is selected
to have a desired transition between the two optimal controls, i.e., for a very large time window,
there is no transition and there might be very rapid transitions if a small time window is selected.
Figure 5.9 shows MATLAB’s Simulink model used for this gain scheduling approach.
Case 1: Linear transition:
A linear transition function is considered as the first candidate. Assume
g(t)=
PGA(t)
PGA
Expected
(5.7)
where PGA
Expected
is the maximum peak ground acceleration expected at the region for which u
d
is
76
w
q
u
u
Control Force
Discrete State-Space
Zero-Order
Hold
Abs max
Multiply

KLQR_d
Strong
KLQR_a
Weak
1
1
w Excitation
Transport
Delay
γ (PGA)
floating PGA
γ
min
Figure 5.9: Simulink model used for Gain Scheduling with intermediate transition function
designed, and
PGA(t)= max
0tt
max
kw(tt)k (5.8)
which is the running peak of ground acceleration seen over the lastt seconds up through timet.
Note that for this studykw(t)k=j ¨ x
g
(t)j where ¨ x
g
(t) is the ground acceleration, and alsot
max
is
considered to be 1 sec. In this part of the study, PGA
Expected
is assumed to be 6.8340 m/s
2
which is
the PGA of 2El Centro. It should be noted that PGA(t) is assumed to not exceed PGA
Expected
,
therefore 0g(t) 1. If it is desired to expect larger value for PGA(t), one can adjust PGA
Expected
accordingly, or replaceg(t) by maxf0,[1g(t)]g to prevent negative values of[1g(t)].
The response of the structure is computed for the resulting gain scheduled control along with that
of the two linear controls. Responses in the weak excitation are shown in Figure 5.10, whereas
Figure 5.11 shows the responses to the strong excitation, the results are also presented in Table
5.2. Clearly, the gain scheduled control closely tracks the LQR
a
and LQR
d
in weak and strong
excitations, respectively.
As explained in the previous section, it is essential for the gain-scheduled controller to perform
optimally in the presence of weak and strong excitation as well as moderate excitations. Figure 5.12
shows the responses to a 1El Centro excitation, which is considered as a moderate excitation since
its PGA is between those of the weak and the strong excitations. Clearly the designed controller
performs much better than what was shown in Figure 5.8. The resulting responses and control force
lie between those of LQR
a
and LQR
d
, as expected.
77
−0.5
0
0.5
displ [cm]

−0.1
0
0.1
0.2
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.10: Response comparison for two optimal LQR and linear gain-scheduled controls, 0.1El
Centro
−10
0
10
displ [cm]

−2
0
2
4
accel [m/s
2
]
0 5 10 15 20
−40
−20
0
20
40
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.11: Response comparison for two optimal LQR and linear gain-scheduled controls, 2El
Centro
Case 2: Sigmoid transition
In this case, a sigmoid transition function is used to transition from weak to moderate excitation. A
wide variety of sigmoid functions including hyperbolic tangent and logistic functions can be used
for this purpose.
78
Table 5.2: Performance of linear gain-scheduled control under scaled 1940 El Centro excitation
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS Uncont. LQR
a
LQR
d
GS
Peak displ.[cm] 1.140 0.573 0.276 0.567 22.803 11.452 5.524 6.985
RMS displ.[cm] 0.298 0.176 0.085 0.172 5.960 3.530 1.710 2.174
Peak abs. accel.[m/s
2
] 0.454 0.108 0.262 0.113 9.085 2.161 5.241 4.807
RMS abs. accel.[m/s
2
] 0.119 0.028 0.052 0.029 2.374 0.565 1.033 0.894
Peak cont. force [% weight] — 2.069 2.058 2.020 — 41.374 41.166 40.160
RMS cont. force [% weight] — 0.615 0.406 0.591 — 12.307 8.124 8.080
−0.5
0
0.5
displ [cm]

−1
0
1
2
accel [m/s
2
]
0 5 10 15 20
−20
0
20
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.12: Response comparison for two optimal LQR and linear gain-scheduled controls, 1El
Centro
Considering the definition of PGA(t) in(5.8) and a hyperbolic tangent equation
g(t)=x

z+ tanh[hf
PGA(t)
PGA
Expected
0.5g]

(5.9)
several scaling and adjustments are needed to make the resulting functiong(t)! 0 andg(t)! 1
for PGA(t)! 0 and PGA(t)! PGA
Expected
, respectively. One such an equation can be obtained
as
g(t)= 0.5

1+ tanh[PGA(t) s
2
/m 0.5PGA
Expected
s
2
/m]

(5.10)
(i.e., x = 0.5, z = 1 and h = PGA
Expected
( s
2
/m)). It is also assumed that g(t)! 0.5 for
PGA(t)! 0.5PGA
Expected
. The resulting responses of the structure to weak and strong excitations
are shown in Figure 5.13 and Figure 5.14, respectively. The results are also presented in Table
79
−0.5
0
0.5
displ [cm]

−0.1
0
0.1
0.2
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.13: Response comparison for two optimal LQR and sigmoid gain-scheduled controls,
0.1El Centro
−10
0
10
displ [cm]

−2
0
2
4
accel [m/s
2
]
0 5 10 15 20
−40
−20
0
20
40
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.14: Response comparison for two optimal LQR and sigmoid gain-scheduled controls,
2El Centro
5.3. Similar to the linear transition, the gain scheduled control closely tracks the LQR
a
and LQR
d
in weak and strong excitations, respectively. Figure 5.15 shows the responses to 1El Centro
excitation, a moderate excitation. It is clear that the designed gain-scheduled controller switches
between LQR
a
and LQR
d
depending on the value of g(t), which changes based on PGA(t) as
80
Table 5.3: Performance of sigmoid gain-scheduled control under scaled 1940 El Centro excitation
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS Uncont. LQR
a
LQR
d
GS
Peak displ.[cm] 1.140 0.573 0.276 0.572 22.803 11.452 5.524 6.985
RMS displ.[cm] 0.298 0.176 0.085 0.176 5.960 3.530 1.710 2.289
Peak abs. accel.[m/s
2
] 0.454 0.108 0.262 0.108 9.085 2.161 5.241 4.885
RMS abs. accel.[m/s
2
] 0.119 0.028 0.052 0.028 2.374 0.565 1.033 0.910
Peak cont. force / bldg.weight — 2.069 2.058 2.066 — 41.374 41.166 37.759
RMS cont. force / bldg.weight — 0.615 0.406 0.614 — 12.307 8.124 8.871
−5
0
5
displ [cm]

−1
0
1
2
accel [m/s
2
]
0 5 10 15 20
−20
0
20
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.15: Response comparison for two optimal LQR and sigmoid gain-scheduled controls,
1El Centro
in (5.8). It should be noted that although both linear and sigmoid transition functions result in a
good performance in different magnitude of excitations, their results, shown in Figure 5.12 and
Figure 5.15, are not identical especially for moderate excitations. This is due to the fact that for
each PGA(t), linear and sigmoid functions result in different values ofg(t) as shown in Figure 5.16.
The plot shows the variation ofg(t) for both linear (5.7) and sigmoid (5.10) transition functions,
along with the floating PGA defined in (5.8) fort
max
= 1 sec. Different performance of sigmoid
transition function can be achieved by different adjusting of parameters in (5.10).
81
0 5 10 15 20
0
0.2
0.4
0.6
γ

0 5 10 15 20
0
2
4
6
Floating PGA [m/s
2
]
Time [sec]
γ(t), linear transition
γ(t), sigmoid transition
PGA(t) with τ
max
=1 s
Figure 5.16: g comparison for linear and sigmoid transition functions, 1El Centro
5.3.2.1 Applying a saturation constraint on the control force
The results shown so far in this study present a very promising performance for a gain scheduled
control for both weak and strong excitation; however, the required control force in strong excitation
is again very large. Any real implementation would, of course, need to use lower force levels, with
peak on the order of 10 15% of structure weight. Therefore, a saturation in the commanded force
must be introduced into the problem to accommodate control device capacity. Figure 5.17 shows the
response of linear gain-scheduled control along with the two linear controls, when 15% of structure
weight is considered as the control force saturation level. The numerical values are also shown in
Table 5.4. Clearly, these results are different from what was shown in Figure 5.11 and Table 5.2, of
course due to control force saturation; however, the performance of gain scheduled control is still
similar to LQR
d
in the strong excitation.
5.3.2.2 Multiple degree-of-freedom (MDOF) building model
In this section, a base-isolated multiple degree-of-freedom (MDOF) building structure is investigated
to demonstrate the potential of the designed gain scheduled controller to achieve both goals, focusing
on drift reduction in strong earthquakes and on absolute acceleration reduction in weak to moderate
earthquakes. Numerical simulations are performed for a base-isolated building structure with
82
−10
0
10
displ [cm]

−4
−2
0
2
4
accel [m/s
2
]
0 5 10 15 20
−10
0
10
time [s]
force [%W]
GS
LQR
d
LQR
a
Figure 5.17: Response comparison for saturated linear gain-scheduled control, 2El Centro
Table 5.4: Performance of saturated linear gain-scheduled control under scaled 1940 El Centro
excitation
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS Uncont. LQR
a
LQR
d
GS
Peak displ.[cm] 1.140 0.573 0.276 0.567 22.803 12.900 8.357 7.543
RMS displ.[cm] 0.298 0.176 0.085 0.172 5.960 3.942 1.085 2.174
Peak abs. accel.[m/s
2
] 0.454 0.108 0.262 0.113 9.085 3.652 4.473 3.957
RMS abs. accel.[m/s
2
] 0.119 0.028 0.052 0.029 2.374 0.939 1.085 0.894
Peak cont. force [% weight] — 2.069 2.058 2.020 — 15.000 15.000 15.000
RMS cont. force [% weight] — 0.615 0.406 0.591 — 9.122 7.087 7.118
parameters shown in Table 4.5, which is adapted from Kelly et al. (1987). The infinite horizon
discrete-time quadratic cost function can be presented as
J=
¥
å
k=1
[q
x
x
2
b
(k)+q
a
¨ x
2
a
roof
(k)+ru
2
(k)] (5.11)
wherex
b
is the base drift and ¨ x
a
roof
is the roof absolute acceleration. Assuming a constant value of
r= 10
4
/(m
1
g)
2
= 2.9881 10
6
N
2
, one should obtain two desired pairs (q
x
,q
a
) for designing
LQR
a
and LQR
d
. It is assumed there is one control device located in the isolation layer of the
structure. Further, it is assumedDt= 0.02 sec, the same as the time discretization of the El Centro
earthquake record.
The gain scheduled control must be designed such that it performs similar to LQR
a
and LQR
d
in
83
weak and strong excitations, respectively. Moreover, the transition functiong must be defined such
that the performance of the resulting gain scheduled controller in moderate excitations are also
“optimal”.
Considering q
x
=a 10
8
/H
2
and q
a
=b 10
8
(m
1
/Hk
1
)
2
in which H = 4 m is the height of
each story of the structure, one can choose the desireda andb by observing how changinga andb
affects the resulting LQR-controlled drift and absolute acceleration compared with the uncontrolled
response of the same system and excitation. In this section, the North-South component of the first
20 sec of 1940 El Centro earthquake record is considered as the external excitation.
LQR design for weak excitation, LQR
a
It was shown in Section 4.5 that a passive viscous damper,u=c
p
˙ x
b
, has maximum RMS absolute
acceleration reduction atc
p
= 6 10
4
N s/m.
Figure 5.18 and Figure 5.21 show the percent reduction of RMS and Peak responses of LQR
compared to the designed passive system. The filled contours show the percent base drift reduction
provided by LQR compared to the passive damper, while contour lines show the corresponding
absolute acceleration reduction percentage. Further, RMS and Peak required control force are
shown in Figure 5.19 and Figure 5.20, respectively. It should be noted that these graphs are similar
to those shown in Chapter 4; however, they are not identical due to slight differences between
continuous-time and discrete-time responses.
As discussed in Section 4.5, one way to design LQR
a
is to match or exceed the acceleration reduction
provided by a passive viscous damper that best reduces acceleration. Thus, one approach is to
picka andb such that the RMS base drift and roof absolute acceleration of LQR
a
are the same as
those of the “optimal” passive controller with c
p
= 6 10
4
N s/m (with maximum acceleration
reduction). From Figure 5.18, a = 1.17 10
2
and b = 4.11 10
3
are selected based on this
criterion, resulting inq
x
= 7.3210
4
m
2
andq
a
= 7.8510
2
s
4
/m
2
. The corresponding peak base
drift and roof absolute acceleration reductions improvements compared to uncontrolled responses
are 66.15% and 43.10%, respectively. The peak control force is 6.02% of the weight of the structure
for El Centro earthquake. The corresponding RMS base drift and absolute acceleration reductions
are 75.79% and 63.27% , respectively. The RMS of the control force is 1.77% of the weight of the
structure.
LQR design for strong excitation, LQR
d
Figure 5.22 and Figure 5.23 show the percent reduction of RMS and Peak responses of LQR
compared to the uncontrolled system (similar to Figure 4.13 and Figure 4.12, respectively, but
84
−120
−100 −100
−80
−80
−60
−60
−60
−40
−40
−40
−20
−20
−20
−20
0
0
0
20
20
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
RMS base drift improv.[%]
RMS roof accel. improv.[%]
−60
−40
−20
0
20
40
60
Figure 5.18: RMS response reduction of LQR compared to Passive [%], El Centro, 6DOF
1.5
2
2
2.5
2.5
3
3
3.5
3.5
4
4
4.5
4.5
5
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 5.19: RMS required LQR control force [%W], El Centro, 6DOF
slightly different due to small changes going from continuous- to discrete-time). The filled contours
show the percent base drift reduction provided by LQR compared to the uncontrolled system, while
contour lines show the corresponding absolute acceleration reduction percentage.
For strong excitation,a andb are selected such that the LQR
d
results in the same peak roof absolute
85
5
10
10
15
15
20
25
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
5
10
15
20
25
Figure 5.20: Peak required LQR control force [%W], El Centro, 6DOF
−150
−150
−100
−100
−50
−50
−50
0
0 0
0
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
Peak base drift improv.[%]
Peak roof accel. improv.[%]
−60
−40
−20
0
20
40
60
Figure 5.21: Peak response reduction of LQR compared to Passive [%], El Centro, 6DOF
acceleration as the uncontrolled response, and the peak base drift is not larger than some threshold
value depending on the design goals and failure criteria (chosen herein as 10 cm). It should be noted
that this value could be interpreted as the required seismic clearance between adjacent buildings
and can be determined based on building code (e.g., ASCE 7-10) depending on the structure and its
86
20
30
30
40
40
40
50
50
50
60
60
60 60
70
70
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
RMS base drift improv.[%]
RMS roof accel. improv.[%]
60
65
70
75
80
85
90
Figure 5.22: RMS response reduction of LQR compared to uncont. [%], El Centro, 6DOF
location seismic properties. Hence, from Figure 5.23,a = 0.18 andb = 4.11 10
3
are selected
resulting inq
x
= 1.11 10
6
m
2
andq
a
= 7.85 10
2
s
4
/m
2
. Note thatq
x
is the same value used
for designing LQR
a
.
The corresponding peak drift reduction compared to uncontrolled response is 82.91%. The peak
control force is 12.83% of the weight of the structure for El Centro earthquake. The corresponding
RMS drift and absolute acceleration reductions are 88.23% and 43.58%, respectively. The RMS of
the control force is 3.63% of the weight of the structure.
Case 1: Linear transition:
First, a linear transition function the same as (5.7) is considered for the MDOF problem. The
resulting responses of the structure to weak and strong excitations are shown in Figure 5.24 and
Figure 5.25, respectively. The results are also presented in Table 5.5. Similar to the linear transition
for the SDOF problem, the gain scheduled control closely tracks the LQR
a
and LQR
d
in weak and
strong excitations, respectively. The response to a moderate excitation, 1El Centro, is shown in
Figure 5.26.
Case 2: Sigmoid transition:
In this case, the transition function is assumed as in (5.10). The resulting responses of the structure
to weak and strong excitations are shown in Figure 5.27 and Figure 5.28, respectively. The results
are also presented in Table 5.5. Similar to the linear transition for the SDOF structure, the gain
87
−60
−60
−40
−40
−20
−20
0
0
0
20
20
20
40
40
40 40
60
LQR
a
LQR
d
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
Peak base drift improv.[%]
Peak roof accel. improv.[%]
45
50
55
60
65
70
75
80
85
90
Figure 5.23: Peak response reduction of LQR compared to uncont. [%], El Centro, 6DOF
−0.5
0
0.5
1
b. drift
[cm]

−0.1
0
0.1
0.2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−1
0
1
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.24: Response comparison for linear gain-scheduled control, 0.1El Centro, 6DOF
scheduled control closely tracks the LQR
a
and LQR
d
in weak and strong excitations, respectively.
The response to a moderate excitation, 1El Centro, is shown in Figure 5.29.
As was shown in Figure 5.16, due to different behavior ofg in the linear and sigmoid transition
functions, the performance of the corresponding gain scheduled control designs shown in Figure 5.26
and Figure 5.29 are different.
88
−10
0
10
20
b. drift
[cm]

−2
0
2
4
r. abs. accel
[m/s
2
]
0 5 10 15 20
−20
0
20
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.25: Response comparison for linear gain-scheduled control, 2El Centro, 6DOF
−5
0
5
10
b. drift
[cm]

−1
0
1
2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−10
0
10
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.26: Response comparison for linear gain-scheduled control, 1El Centro, 6DOF
5.3.2.2.1 Applying a saturation constraint on the control force
Assume control force saturation is applied in order to limit the maximum control force to 15%
weight of the structure. Figure 5.30 shows the similar performance of LQR
d
and gain-scheduled
control design with control force saturation under strong excitation 2El Centro. Results are also
presented in Table 5.6.
89
−0.5
0
0.5
1
b. drift
[cm]

−0.1
0
0.1
0.2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−1
0
1
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.27: Response comparison for sigmoid gain-scheduled control, 0.1El Centro, 6DOF
−10
0
10
20
b. drift
[cm]

−2
0
2
4
r. abs. accel
[m/s
2
]
0 5 10 15 20
−20
0
20
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.28: Response comparison for sigmoid gain-scheduled control, 2El Centro, 6DOF
5.3.2.2.2 Semiactive Control Design
This study, so far, was focused on active control strategies. This is intentional because imposing
a dissipativity constraint is a limitation on the forces that a control strategy may wish to exert,
resulting in performance that is “suboptimal” relative to the active system. Thus, the first step has
been to prove that active control performance can be significantly improved by focusing on the twin
90
Table 5.5: Performance of linear and sigmoid gain-scheduled controls under scaled 1940 El Centro
excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 3.176 1.075 0.543 1.048 1.074 63.523 21.502 10.853 12.683 12.154
RMS b. drift[cm] 1.291 0.312 0.152 0.301 0.312 25.810 6.244 3.037 3.812 3.951
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.206 0.117 0.117 4.123 2.346 4.123 3.825 3.855
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.031 0.031 1.667 0.612 0.940 0.818 0.840
Peak cont. force [% weight] — 0.603 1.284 0.626 0.604 — 12.053 25.676 20.697 21.771
RMS cont. force [% weight] — 0.177 0.363 0.184 0.178 — 3.546 7.258 6.026 6.259
−5
0
5
10
b. drift
[cm]

−1
0
1
2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−10
0
10
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.29: Response comparison for sigmoid gain-scheduled control, 1El Centro, 6DOF
Table 5.6: Performance of saturated gain-scheduled controls under scaled 1940 El Centro excitation,
6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 3.176 1.075 0.543 1.048 1.074 63.523 21.502 12.849 13.951 13.226
RMS b. drift[cm] 1.291 0.312 0.152 0.301 0.312 25.810 6.244 3.268 3.956 4.134
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.206 0.117 0.117 4.123 2.346 3.302 3.310 3.299
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.031 0.031 1.667 0.612 0.885 0.791 0.802
Peak cont. force [% weight] — 0.603 1.284 0.626 0.604 — 12.053 15.000 15.000 15.000
RMS cont. force [% weight] — 0.177 0.363 0.184 0.178 — 3.546 6.538 5.686 5.713
91
−10
0
10
20
b. drift
[cm]

−2
0
2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−10
0
10
time [s]
force
[%W]
GS
LQR
d
LQR
a
Figure 5.30: Response comparison for saturated sigmoid gain-scheduled control, 2El Centro,
6DOF
objectives in weak and extreme events. This section, then, begins to explore the effectiveness of the
designed gain scheduled controls when using semiactive devices.
Here, a clipped-optimal strategy, which eliminates all nondissipative forces and presented precious
chapter in Section 4.5.2, is used for the same MDOF building model defined previously, with one
semiactive control device in the isolation layer. The (unsaturated) active gain-scheduled control
forces, with both linear and sigmoid transitions, and the corresponding semactive control forces
for the MDOF building subjected to 2El Centro excitation are shown in Figure 5.31 as functions
of the base velocity (i.e., the velocity across the device). By comparing these two graphs, one can
verify that the semiactive control force is always dissipative, as it is always in the opposite direction
of the velocity across the device. The resulting response of the structure to 2El Centro excitation
is presented in Table 5.7. Clearly, the gain scheduled control closely tracks the CLQR
a
and CLQR
d
in weak and strong excitations, respectively.
5.3.3 Approach III: Online LQR Gain Scheduling
So far in this study, switching control was defined based on a linear combination of the two LQR
control gains. In this part of the study, a different approach, based on a time-varying drift weight in
the LQR cost function, is investigated.
92
-0.5 0 0.5
-20
-15
-10
-5
0
5
10
15
20
Velocity across device [m/s]
Semiactive control force [%weight]
-0.5 0 0.5
-20
-15
-10
-5
0
5
10
15
20
Velocity across device [m/s]
Active control force [%weight]
Figure 5.31: Active and Semiactive comparison, 6DOF, 2El Centro
Table 5.7: Performance of semiactive gain-scheduled controls under scaled 1940 El Centro excita-
tion, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. CLQR
a
CLQR
d
GS
Lin
GS
Sig
Uncont. CLQR
a
CLQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 3.176 1.073 0.536 1.045 1.072 63.523 21.461 10.718 12.316 12.112
RMS b. drift[cm] 1.291 0.312 0.151 0.301 0.312 25.810 6.243 3.028 3.792 3.951
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.97 0.119 0.117 4.123 2.346 3.936 3.932 3.950
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.031 0.031 1.667 0.610 0.949 0.817 0.836
Peak cont. force [% weight] — 0.605 1.297 0.628 0.606 — 12.095 25.932 21.332 22.258
RMS cont. force [% weight] — 0.177 0.350 0.184 0.177 — 3.542 7.010 5.823 6.036
LQR designs for weak and strong excitations, LQR
a
and LQR
d
, are considered to be the same as
those previously designed in this study and shown in Figure 5.18. Moreover, the linear and sigmoid
transition equations are the same as in (5.7) and (5.10) with PGA(t) defined in (5.8). However, the
force equation (5.6) is no longer implemented, but is replaced by
a =(1g)a
a
+ga
d
(5.12)
93
Table 5.8: Performance of online LQR gain-scheduled controls under scaled 1940 El Centro
excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 3.176 1.075 0.543 0.991 1.070 63.523 21.502 10.853 11.875 11.488
RMS b. drift[cm] 1.291 0.312 0.152 0.286 0.311 25.810 6.244 3.037 3.515 3.754
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.206 0.120 0.117 4.123 2.346 4.123 3.929 3.920
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.031 0.031 1.667 0.612 0.940 0.853 0.859
Peak cont. force [% weight] — 0.603 1.284 0.654 0.605 — 12.053 25.676 22.538 22.889
RMS cont. force [% weight] — 0.177 0.363 0.193 0.178 — 3.546 7.258 6.426 6.475
wherea
a
anda
d
are scaled drift weights in the cost function defined in (5.11) for weak and strong
excitations, respectively. a is the time-varying drift weight that changes as the intensity of the
excitation is changing. b andr are considered the same for LQR
a
and LQR
d
and also for the gain
scheduling control. In order to calculate the optimal gain scheduling control, one approach is to
precalculate the optimal control gain for some possiblea values, store them in a lookup table, and
interpolate to obtain the appropriate values for anya. The disadvantage of this method is that it
may result in suboptimality or even loss of stability due to interpolation if not enough appropriate
values ofa are predicted in the construction of the lookup table. A better approach, implemented
herein, is online calculation of the optimal control gain at each time instant for the current value of
a, and therefore obtaining the optimal control force at each time instant.
For both linear and sigmoid transitions forg, the resulting responses of the structure subjected to
weak and strong excitations are calculated and presented in Table 5.8. The designed gain scheduling
control performs optimally for weak and strong excitation but it is also essential that its optimal
performance be confirmed for all levels of excitation including moderate earthquakes. Moreover,
gain scheduling should still perform well for earthquake records other than the design record El
Centro. However, it should be noted that LQR
a
and LQR
d
are designed based on the El Centro
earthquake and might not be optimal for other earthquakes. Table 5.9 shows the responses to 1El
Centro excitation and Table 5.10 shows the response to two different earthquakes, the Northridge
and Kobe earthquake records. Clearly the online LQR control closely tracks u
d
for other strong
earthquake excitations, and interpolates between u
a
and u
d
for the moderate excitation.
5.3.3.1 Control force saturation
Assume control force saturation is applied in order to limit the maximum control force to 15%
weight of the structure. Table 5.11 shows the similar performance of LQR
d
and the two gain-
94
Table 5.9: Performance of online LQR gain-scheduled controls under moderate excitation, 6DOF
1El Centro
PGA= 3.417 m/s
2
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 31.762 10.751 5.427 7.019 8.529
RMS b. drift[cm] 12.899 3.122 1.518 2.049 2.544
Peak r. abs. accel.[m/s
2
] 2.061 1.173 2.061 1.563 1.516
RMS r. abs. accel.[m/s
2
] 0.832 0.306 0.470 0.379 0.338
Peak cont. force [% weight] — 6.026 12.838 8.741 8.785
RMS cont. force [% weight] — 1.773 3.629 2.737 2.242
Table 5.10: Performance of online LQR gain-scheduled controls under other excitations, 6DOF
1 Northridge 1 Kobe
PGA= 8.27 m/s
2
PGA= 8.18 m/s
2
LQR
a
LQR
d
GS
Lin
GS
Sig
LQR
a
LQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 33.694 17.993 19.669 18.894 18.927 13.026 13.018 12.886
RMS b. drift[cm] 8.909 5.575 4.492 4.661 5.139 3.510 3.642 3.751
Peak r. abs. accel.[m/s
2
] 3.536 4.046 5.643 5.615 2.338 5.706 5.676 5.735
RMS r. abs. accel.[m/s
2
] 0.816 1.018 0.983 0.992 0.613 1.181 1.121 1.130
Peak cont. force [% weight] 22.970 45.457 46.660 46.008 16.603 45.491 45.222 45.639
RMS cont. force [% weight] 4.616 7.842 7.474 7.540 8.920 9.436 8.920 9.001
Table 5.11: Performance of saturated online LQR gain-scheduled controls under scaled 1940 El
Centro excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Uncont. LQR
a
LQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 3.176 1.075 0.543 0.991 1.070 63.523 21.502 12.849 13.42 13.059
RMS b. drift[cm] 1.291 0.312 0.152 0.286 0.311 25.810 6.244 3.268 3.687 3.935
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.206 0.120 0.117 4.123 2.346 3.302 3.287 3.295
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.031 0.031 1.667 0.612 0.885 0.816 0.816
Peak cont. force [% weight] — 0.603 1.284 0.653 0.605 — 12.053 15.000 15.000 15.000
RMS cont. force [% weight] — 0.177 0.363 0.193 0.178 — 3.546 6.538 5.947 5.863
scheduled control designs with linear and sigmoid transitions with control force saturation, under
strong excitation 2El Centro.
5.3.3.2 Semiactive control design
Consider the clipped-optimal strategy to investigate semiactive control performance. The resulting
response of the structure subjected to weak and strong excitations for both linear controls and
gain-scheduled semiactive controls with linear and sigmoid transitions, are calculated and presented
in Table 5.12.
95
Table 5.12: Performance of semiactive online LQR gain-scheduled controls under scaled 1940 El
Centro excitation, 6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. CLQR
a
CLQR
d
GS
Lin
GS
Sig
Uncont. CLQR
a
CLQR
d
GS
Lin
GS
Sig
Peak b. drift[cm] 3.176 1.073 0.536 0.990 1.068 63.523 21.461 10.718 11.433 11.458
RMS b. drift[cm] 1.291 0.312 0.151 0.287 0.310 25.810 6.243 3.028 3.501 3.744
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.97 0.120 0.117 4.123 2.346 3.936 4.014 4.005
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.047 0.031 0.031 1.667 0.610 0.949 0.854 0.861
Peak cont. force [% weight] — 0.605 1.297 0.657 0.608 — 12.095 25.932 22.949 23.144
RMS cont. force [% weight] — 0.177 0.350 0.193 0.178 — 3.542 7.010 6.213 6.237
5.4 Conclusions
It is imperative for the civil engineering community to develop and implement strategies to mitigate
structural responses in both weak and strong excitations, reducing the absolute acceleration re-
sponses in weak-to-moderate excitations to prevent nonstructural and building content damage and
ensure occupant comfort, and decreasing the drift responses in strong excitations to avoid structural
damage and possible structural collapse — for both newly designed structures and retrofitting of
existing structures.
Optimal linear control laws are the most common applied control strategies, designed for a specific
quadratic cost functions with a design excitation level; therefore their performance might not be
optimal in the presence of different magnitude excitation.
In this study, two optimal linear control laws are obtained for two different levels of excitation,
namely, LQR
d
for strong excitation (2El Centro), and LQR
a
in weak excitation (0.1El Centro).
Gain-scheduled control strategies are applied to design a control with performance similar to LQR
a
and LQR
d
in the presence of weak and strong excitations, respectively. Moreover, several transition
functions in gain-scheduled control designs are investigated to achieve a desired performance in the
presence of moderate excitations.
The performance of the designed gain-scheduled control strategy is demonstrated by intensive
numerical simulations for a SDOF building model and a base-isolated MDOF building model
subjected to different magnitude excitations. Both active and semiactive control strategies are studied
with and without considering control force saturation. Results indicate that it can successfully
achieve both life safety (drift reduction) and serviceability (acceleration reduction) objectives at
different excitation levels. Although the results presented in this study are promising, for some
96
specific LQR designs or as the result of rapid switching between different controls, gain scheduling
control might become unstable. Further study is needed to investigate the stability of these control
strategies.
97
Chapter 6
Nonlinear Control for Life Safety and
Serviceability using Model Predictive
Control
1
6.1 Introduction
This chapter presents a nonlinear control strategy, based on the model predictive control (MPC)
method, that minimizes a nonquadratic cost function with higher-order polynomials in states. The
nonquadratic term in the cost function is defined such that it becomes negligible in weak excitations,
resulting in performance similar to that of a linear quadratic regulator (LQR), whereas in strong
excitations in which it becomes dominant, resulting in a better drift reduction compared with that of
corresponding LQR problem. Hence, the resulting control can achieve both service-level and life
safety-level objectives in different level of excitations.
6.2 Problem Formulation
Since MPC is a discrete time control method, this section formulates the problem for an arbitrary
discrete time linear system using a nonquadratic cost function in states. Consider ann degree-of-
freedom (DOF) linear system subjected to external disturbance w. The equations of motion, and
1
Hemmat-Abiri and Johnson (2014b) presents a subset of this chapter.
98
those governing responses, can be expressed in state-space form
q(k+ 1)= A
d
q(k)+ B
d
u(k)+ E
d
w(k) (6.1a)
z(k)= C
z
q(k)+ D
z
u(k)+ F
z
w(k) (6.1b)
where state vector q(k) q(kDt) contains generalized displacements and velocities; w(k) is a
vector of external disturbances; z(k) is a vector of outputs to be regulated (typically interstory drifts,
absolute accelerations and internal forces); u(k) is a vector of control forces; discrete state matrix
A
d
, input matrices B
d
and E
d
, output matric C
z
and feedthrough matrices D
z
and F
z
all depend on
the usual structural system matrices, the force locations and orientations, etc. It should be noted
that zero-order hold (ZOH) is assumed for excitation to produce these discrete-time equation of
motions. To achieve the two different objectives, drift reduction and absolute acceleration reduction
in different levels of excitations, a nonquadratic cost function can be defined as
J=
¥
å
k=1

q
T
Qq+ 2q
T
Nu+ u
T
Ru+h(q)

(6.2)
in which dependence on(k) is omitted for clarity. Q, N and R are positive definite symmetric ma-
trices of appropriate dimensions for the discrete time problem, andh(q) represents all nonquadratic
terms in the cost function.
The goal is to minimize the nonquadratic cost function (6.2) using MPC and also a similar quadratic
cost function withh(q)= 0, similar to (5.2)
J=
¥
å
k=1

q
T
Qq+ 2q
T
Nu+ u
T
Ru

(6.3)
to compare the performance of corresponding control forces in different level of excitations. The
optimal linear control law that minimizes the quadratic cost function (5.2) can be written as
u(k)=K
LQR
q(k) (6.4)
where K
LQR
is the optimal control gain that can be obtained simply by using typical solvers like
MATLAB’s dlqr command.
To apply MPC and emulate an infinite horizon quadratic cost function, one can rewrite the quadratic
cost function as
J= q
T
(p)Pq(p)+
p1
å
k=1

q
T
Qq+ 2q
T
Nu+ u
T
Ru

(6.5)
99
Matlab
(SIMULINK)
YALMIP
QUADPROG
GUROBI
SNOPT
IPOPT
Figure 6.1: YALMIP and several corresponding external solvers
where the first term is the terminal cost term and P is the solution of the Riccati equation associated
with the discrete LQR problem that minimizes an infinite horizon quadratic cost function as (5.2)
and p is the prediction and control horizon shown in Figure 1.3.
To find the optimal control using MPC, the YALMIP toolbox (L¨ ofberg, 2004) in MATLAB is used.
As shown in Figure 6.1, YALMIP relies on one of several supported external solvers to perform the
optimization. In this study, several different solvers, includingquadprog,Ipopt (W¨ achter and
Biegler, 2006) and the Gurobi optimizer (Gurobi Optimization, Inc., 2014), are investigated.
6.3 Numerical Example
In this section, to demonstrate the potential of the MPC-based nonlinear controller to achieve both
goals, a numerical simulation is performed for several building structure models. The performance
improvement of the MPC-based nonlinear control strategy is presented and compared with that of
an optimal linear control found by LQR for the model subjected to different levels of excitation.
The strong and weak excitation for this problem is considered to be 2.0- and 0.1-scaled versions of
the 20 sec record of North-South component of the 1940 El Centro earthquake, with peak ground
accelerations (PGAs)) of 6.8340 m/s
2
and 0.3417 m/s
2
, respectively.
6.3.1 Single degree-of-freedom (SDOF) Building Model
Consider the same single degree-of-freedom (SDOF) building structure presented in Section 3.3
(with massm= 29485 kg, damping coefficientc= 23710 Ns/m and stiffnessk= 1.1640 MN/m).
100
6.3.1.1 Quadratic cost function
The infinite horizon discrete-time quadratic cost function can be presented similar to (5.5)
J=
¥
å
k=1
[q
x
x
2
(k)+q
a
¨ x
2
abs
(k)+ru
2
(k)] (6.6)
where ¨ x
abs
=kx/mc ˙ x/m+u/m is the absolute acceleration of the structure and k, c, and m
are stiffness, damping and mass of the structure, respectively. Considering equation (6.1) where
q=[x ˙ x]
T
, and z=[x ¨ x
abs
]
T
, it is shown in Chapter 5 that (5.5) can be rewritten as
J=
¥
å
k=1

z
T
Q
z
z+ 2z
T
N
z
u+ u
T
R
z
u

(6.7)
where
Q
z
=
"
q
x
0
0 q
a
#
, N
z
=
"
0
0
#
, R
z
=r
(5.5) can be also written as (5.2) in which Q, N and R can be obtained via the following transforma-
tion:
"
Q N
N
T
R
#
=
"
C
T
z
0
D
T
z
I
#"
Q
z
N
z
N
T
z
R
z
#"
C
z
D
z
0 I
#
Therefore
Q= C
T
z
Q
z
C
z
N= C
T
z
Q
z
D
z
R= D
T
z
Q
z
D
z
+ R
z
(6.8)
Q=
q
a
m
2
"
k
2
+
m
2
q
x
q
a
ck
ck c
2
#
, N=
q
a
m
2
"
k
c
#
, R=r+
q
a
m
2
The optimal linear control law that minimizes the quadratic cost function (5.2) can be obtained as
(7.1).
Assuming a constant control force weighting coefficientr= 10
4
/(mg)
2
= 1.1953 10
7
N
2
, one
should find a desired pair ofq
x
andq
a
by consideringq
x
=a10
8
/H
2
andq
a
=b10
8
(m/Hk)
2
in whichH = 4 m is the height of the structure. One can choose the desireda andb by observing
how changinga andb affects the resulting LQR-controlled drift and absolute acceleration compared
with the uncontrolled response of the same system and excitation. Figures 6.2 and 6.3 show the
percent reduction of root mean square (RMS) responses of LQR compared to the uncontrolled
system and the corresponding required control force in terms of percentage of the weight of the
structure. Similarly, Figures 6.4 and 6.5 show the percentage improvements of peak response
101
45
50
50
55
55
60
60
60
65
65
65
70
70
70
70
75
75
75
80
80
85
85
LQR
α (drift weight)
β (accel. weight)

10
−1
10
0
10
1
10
0
10
1
10
2
RMS displ. improv.[%]
RMS accel. improv.[%]
10
20
30
40
50
60
70
80
Figure 6.2: RMS response reduction of LQR compared to uncont. [%], El Centro, SDOF
provided by LQR compared to the uncontrolled, and the peak of the required LQR control force in
terms of the percentage of the weight of the structure, respectively.
Assuming it is desired to achieve 50% peak displacement reduction and about 75% peak and
RMS absolute acceleration reductions in weak to moderate excitations, a = 1 and b = 30 are
selected resulting in q
x
= 6.25 10
6
m
2
and q
a
= 1.20 10
5
s
4
/m
2
. The corresponding RMS
drift reduction is 41%. Also, the peak and RMS of the required control force are 20.7% and 6.1%
of the weight of the structure.
The main goal of this part of the study is to use MPC to minimize a nonquadratic cost function in
order to show that the resulting performance is better than that of minimization of a corresponding
quadratic cost function (LQR) in different level of excitations. However, before this can be done, it
is essential to show that both MPC and LQR give the same result for the same system and quadratic
cost function. For this reason, an approach similar to (6.5) should be taken for the infinite horizon
cost function (5.2). For a fair comparison, both responses are calculated using MATLAB’s Simulink,
for the same system subjected to the scaled version of El Centro earthquake which is discretized
using Zero-Order Hold block.
In this study, to find the optimal control using MPC, the YALMIP toolbox with different supported
external solvers including quadprog, Ipopt and the Gurobi optimizer are used to perform
the optimization. It should be noted that all solvers were found to give the same solution for
102
4
4
4
5
5
5
6
6
7
7
4
4
8
8
9
5
10
LQR
α (drift weight)
β (accel. weight)

10
−1
10
0
10
1
10
0
10
1
10
2
4
5
6
7
8
9
10
11
Figure 6.3: RMS required LQR control force [%W], El Centro, SDOF
20
30
30
40
40
50
50
50
60
60
60
70
70
70
70
80
80
LQR
α (drift weight)
β (accel. weight)

10
−1
10
0
10
1
10
0
10
1
10
2
RMS displ. improv.[%]
RMS accel. improv.[%]
30
40
50
60
70
80
Figure 6.4: Peak response reduction of LQR compared to uncont. [%], El Centro, SDOF
the quadratic problem. However, the required computation time is different for different solvers.
Moreover, changing the prediction horizon p affects the computation time — as expected. Table
6.1 presents the computation time in seconds for the SDOF model defined earlier, subjected to the
103
16
18
18
18
20
20
18
18
22
22
20
20
22
22
24
24
26
28
26
30
28
16
16
LQR
α (drift weight)
β (accel. weight)

10
−1
10
0
10
1
10
0
10
1
10
2
16
18
20
22
24
26
28
30
32
Figure 6.5: Peak required LQR control force [%W], El Centro, SDOF
2.0-scaled version of the North-South component of the 1940 El Centro earthquake, for different
solvers and prediction horizons. The results show thatIpopt, which is based on the interior point
method optimization, is the slowest solver among those considered in this study. However, faster
solvers likequadprog and Gurobi might not be able to solve more complicated problems like
the nonquadratic optimization we study herein. It should be also noted that, for the quadratic cost
function, the resulting control force and the corresponding structural response are the same for all
of the solvers, prediction horizons and scales of the excitation studied here. Figure 6.6 shows the
responses to the strong excitation (2.0El Centro). Clearly the performance of LQR and MPC are
the same, suggesting that the MPC method is indeed providing the optimal linear control for the
quadratic problem.
Table 6.1: MPC computation time [s] for quadratic problem, 2El Centro
p = 1 p = 10 p = 50 p = 100
quadprog 4.66 5.21 16.40 27.83
Gurobi 4.44 6.29 10.36 22.57
Ipopt 31.01 34.12 64.46 283.67
104
−20
0
20
displ [cm]

−5
0
5
accel [m/s
2
]
0 5 10 15 20
−40
−20
0
20
40
time [s]
force [%W]
Quad. MPC
LQR
Uncontrolled
Figure 6.6: Quadratic MPC and LQR response comparison for 2El Centro
6.3.1.2 Nonquadratic cost function
In the previous section, it was shown that MPC results in the optimal control design for a quadratic
cost function. In this section, MPC is used to obtain a nonlinear control minimizing a nonquadratic
cost function. Considering a nonquadratic cost function as defined in (6.2) with the same quadratic
terms as used in (5.5); and a nonquadratic term h(q)= q
nq
x
4
, q
nq
should be selected such that
the nonquadratic termq
nq
x
4
becomes dominant when the excitation level is large (2El Centro),
resulting in a better drift reduction compared to the LQR control, whereas, in the weak excitation
(0.1El Centro), the nonquadratic term becomes negligible compared to the quadratic terms,
resulting in the same performance provided by the LQR control. It should be noted that selecting a
larger nonquadratic term will result in a larger drift reduction in case of strong excitation; however,
it also increases acceleration which is not desired.
Selecting nonquadratic weighting coefficientq
nq
:
Depending on the desired displacement reduction and the acceptable level of increase in acceleration,
the nonquadratic term can be selected. Figure 6.7 shows the percentage RMS displacement and
absolute acceleration reduction of the nonquadratic MPC design, compared to LQR for the SDOF
problem subjected to the 2El Centro, prediction horizon equal to 10 and Ipopt solver as a
function of nonquadratic coefficientq
nq
. Assuming it is desired to achieve an RMS displacement
response reduction of 20% relative to that provided by LQR,q
nq
is selected as 1.5 10
10
m
4
for
the SDOF problem considered in this study.
105
10
8
10
9
10
10
10
11
-100
-80
-60
-40
-20
0
20
40
q
nq
(NQ term Coeff.)
MPC
NQ
improvements compared to LQR [%]

displ. reduction
accel. reduction
control force reduction
Figure 6.7: RMS Response reduction of MPC
NQ
compared to LQR for 2El Centro andp=10
Selecting prediction horizon p:
A longer prediction horizon means a higher-dimensional optimization search space, which results in
larger computation time; however, a very long prediction horizon may not necessarily result in much
better performance. i.e., it might be possible to get a desired performance with a smaller prediction
horizon and computation time. Figure 6.8 shows the variation of computation time and objective
function value versus prediction horizon forIpopt for the 2.0-scaled version of the El Centro
earthquake. Clearly, for longer prediction horizon the performance will still improve, however the
rate of its change is slower for p 8; moreover, the computation time increases significantly for
longer prediction horizons. Therefore, prediction horizon p= 10 is selected for simulation. If one
can obtain the desired performance based on this prediction horizon, even better performance gains
are possible using longer prediction horizon if larger computation time can be accommodated. Now,
considering p= 10 andq
nq
= 1.5 10
10
m
4
, the response of the SDOF model can be calculated.
The responses in the weak excitation (0.1El Centro) are shown in Figure 6.9, whereas Figure 6.10
shows the responses to the strong excitation (2.0El Centro). The results are also presented in
Table 6.2.
Table 6.2 shows that, in the weak excitation (0.1El Centro), the nonquadratic MPC control strategy
closely tracks the optimal LQR control. On the other hand, for the large excitation (2El Centro),
106
2 4 6 8 10 12 14 16
0
50
100
150
200
250
300
Prediction Horizon
Computation Time (s)

2 4 6 8 10 12 14 16
2000
2200
2400
2600
2800
3000
3200
Objective Function
Computation Time
Objective Function
Figure 6.8: Effects of prediction horizon on nonquadratic MPC computation time and cost, 2El
Centro, Ipopt
Table 6.2: Performance of nonquadratic MPC under scaled 1940 El Centro excitation,p=10, Ipopt
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR MPC
nq
Uncont. LQR MPC
nq
Peak displ.[cm] 1.140 0.573 0.572 22.803 11.452 8.695
RMS displ.[cm] 0.298 0.176 0.176 5.960 3.529 2.820
Peak abs. accel.[m/s
2
] 0.454 0.108 0.109 9.085 2.161 3.739
RMS abs. accel.[m/s
2
] 0.119 0.028 0.028 2.374 0.565 0.784
Peak cont. force [% bldg.weight] — 2.069 2.065 — 41.374 31.207
RMS cont. force [% bldg.weight] — 0.615 0.614 — 12.302 8.812
the maximum drift of the nonquadratic MPC control strategy is 8.69 cm which is 24% smaller
than 11.45 cm, the maximum drift of the LQR design. The same trend exists in the RMS values of
drifts. It should be noted that the maximum required control force for MPC in this case is equal
to 31% of the weight of the structure, while LQR needs 41% of the weight of the structure as the
maximum required control force. Clearly the nonquadratic MPC control strategy can focus on
different objectives in different level of excitations with smaller required control force compared to
LQR. i.e., it reduces drift and absolute acceleration similar to LQR design in weak excitation, and
focuses on drift reduction in large excitation.
107
−1
−0.5
0
0.5
displ [cm]

−0.2
0
0.2
0.4
accel [m/s
2
]
0 5 10 15 20
−2
0
2
time [s]
force [%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.9: Response comparison for nonquadratic MPC, 0.1El Centro and prediction horizon
p=10, Ipopt
−20
0
20
displ [cm]

−5
0
5
accel [m/s
2
]
0 5 10 15 20
−40
−20
0
20
40
time [s]
force [%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.10: Response comparison for nonquadratic MPC, 2El Centro and prediction horizon
p=10, Ipopt
6.3.1.3 Applying a saturation on the control force for nonquadratic MPC
It must be noted that, so far in this chapter, force saturation was not introduced into the problem
being investigated for the sake of simplicity; any real implementation would, of course, need to use
108
−20
0
20
displ [cm]

−5
0
5
accel [m/s
2
]
0 5 10 15 20
−10
0
10
time [s]
force [%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.11: Response comparison for saturated nonquadratic MPC, 2El Centro, prediction
horizonp=10, Ipopt
lower force levels, perhaps on the order of 10 15% of structure weight. In this section, 15% of
structure weight is considered as the maximum allowable control force. MPC takes this constraint
into optimization process; for LQR, a saturation block is introduced in the command force.
The responses in the weak excitation (0.1El Centro) will be remain the same since the control
force is much smaller than 15% of structure weight. Figure 6.11 shows the responses to the strong
excitation (2.0El Centro). The results are also presented in Table 6.3. It shows that, in the
weak excitation, there is no saturation and the nonquadratic MPC control strategy closely tracks
the optimal LQR control. On the other hand, for the large excitation, the maximum drift of the
nonquadratic MPC control strategy is 8.65 cm which is 33% smaller than 12.90 cm, the maximum
drift of the LQR design. A similar trend exists in the RMS values of drifts. It should be noted that,
although the maximum required control force for MPC and LQR are set to be equal to 15% of the
weight of the structure, the corresponding RMS value for MPC is equal to 7.79% of the weight of
the structure, while LQR needs an RMS control force that is 9.12% of the weight of the structure.
Clearly, the nonquadratic MPC saturated control can reduce drift better than saturated LQR control,
while also requiring smaller control force.
109
Table 6.3: Performance of saturated nonquadratic MPC under scaled El Centro excitation,p=10,
Ipopt
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR MPC
nq
Uncont. LQR MPC
nq
Peak displ.[cm] 1.140 0.573 0.572 22.803 12.900 8.650
RMS displ.[cm] 0.298 0.176 0.176 5.960 3.940 3.083
Peak abs. accel.[m/s
2
] 0.454 0.108 0.109 9.085 3.652 4.293
RMS abs. accel.[m/s
2
] 0.119 0.028 0.028 2.374 0.938 0.952
Peak cont. force[% bldg.weight] — 2.069 2.065 — 15.00 15.00
RMS cont. force [% bldg.weight] — 0.615 0.614 — 9.120 7.794
Table 6.4: Performance of nonquadratic MPC under the 1994 Northridge earthquake,p=10, Ipopt
0.1Northridge 1 Northridge
PGA= 0.8267 m/s
2
PGA= 8.2676 m/s
2
Uncont. LQR MPC
nq
Uncont. LQR MPC
nq
Peak displ.[cm] 2.059 2.050 2.006 20.594 20.495 12.826
RMS displ.[cm] 0.444 0.478 0.471 4.439 4.779 2.892
Peak abs. accel.[m/s
2
] 0.817 0.354 0.377 8.166 3.537 7.202
RMS abs. accel.[m/s
2
] 0.177 0.066 0.068 1.765 0.660 1.032
Peak cont. force [% bldg.weight] — 7.444 7.254 — 74.442 45.530
RMS cont. force [% bldg.weight] — 1.640 1.596 — 16.400 8.603
6.3.1.4 Performance under other earthquake records
To investigate the performance of the nonquadratic MPC compared to LQR, another strong earth-
quake record is used in this section. The 20 sec record of the North-South component of the 1994
Northridge earthquake, with PGA 8.2676 m/s
2
, is considered here. The responses in the weak
excitation (0.1Northridge) are shown in Figure 6.12, whereas Figure 6.13 shows the responses
to the strong excitation (1.0Northrdge). The results are also presented in Table 6.4. Clearly, the
nonquadratic MPC control strategy closely tracks the optimal LQR in the weak excitation control.
On the other hand, for the large excitation, the maximum drift of the nonquadratic MPC control
strategy is 12.83 cm, which is 37% smaller than 20.50 cm, the maximum drift of the LQR design.
The required control force for the MPC strategy is also much smaller than that of LQR; in fact, the
nonquadratic MPC control force is about half that of the LQR.
6.3.1.5 Quadratic objective function and applying constraints on states
In this part of the study, constrained optimization is investigated. For this purpose, a quadratic cost
function, similar to that in previous sections for the SDOF, is considered; the MPC method is used
to minimize this cost function such that the maximum states — drift and absolute acceleration —
cannot be larger than some specific values. It is expected that the performance of the constrained
110
MPC is identical to LQR when the excitation is small (i.e., the sates are smaller than the specified
values). On the other hand, in strong excitation, LQR responses become larger than the limit values,
while MPC forces the states to stay within the allowable range by applying larger control force.
−2
0
2
displ [cm]

−0.5
0
0.5
accel [m/s
2
]
0 5 10 15 20
−5
0
5
time [s]
force [%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.12: Response comparison for nonquadratic MPC, 0.1Northridge and prediction horizon
p=10, Ipopt
−20
0
20
displ [cm]

−5
0
5
accel [m/s
2
]
0 5 10 15 20
−50
0
50
time [s]
force [%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.13: Response comparison for nonquadratic MPC, 1Northridge and prediction horizon
p=10, Ipopt
111
Table 6.5: Performance of two MPC controls under 2El Centro excitation
2El Centro
PGA= 6.8340 m/s
2
Uncont. LQR MPC
const
MPC
nq
Peak displ.[cm] 22.803 11.452 8.810 8.695
RMS displ.[cm] 5.960 3.529 3.222 2.820
Peak abs. accel.[m/s
2
] 9.085 2.161 7.480 3.739
RMS abs. accel.[m/s
2
] 2.374 0.565 0.780 0.784
Peak cont. force [% bldg.weight] — 41.374 41.619 31.207
RMS cont. force [% bldg.weight] — 12.302 11.142 8.812
First, assume the constraint is only applied on drift (i.e., there is no constraint on absolute accelera-
tion values).
J=
¥
å
k=1
[q
x
x
2
+q
a
¨ x
2
abs
+ru
2
] (6.9)
s. t.jxjx
max
Assume the maximum allowable displacement is x
max
= 8.7 cm for the SDOF problem being
investigated here. Note that 8.7 cm is chosen based on the maximum drift of nonquadratic MPC in
Table 6.2. The responses in the weak and strong excitation are computed and presented in Table 6.5
along with the results previously found from nonquadratic MPC. Results show that the constrained
quadratic MPC control strategy can reduce drift close to the desired value in strong excitation;
however, it increases the peak absolute acceleration significantly, more than nonquadratic MPC
studied in the previous section. This is due to the sudden increase in the control force to prevent the
displacement from passing the desired 8.7 cm limit. Clearly, the performance of the constrained
quadratic MPC control strategy is not acceptable compared to nonqaudratic MPC. Hence, the
former will not be pursued further in this study.
6.3.2 Base-isolated multiple degree-of-freedom (MDOF) Building Model
In this section, a base-isolated MDOF building structure is investigated to demonstrate the potential
of nonquadratic MPC strategy to achieve both goals. Numerical simulations are performed for the
base-isolated building structure with parameters shown in Section 4.5, which are adapted from
Kelly et al. (1987).
The cost function includes base drift and roof absolute acceleration; the control force is applied at
the base in the isolation layer.
112
6.3.2.1 Design LQR based on comparison to uncontrolled
The infinite horizon discrete-time quadratic cost function can be presented as
J=
¥
å
k=1
[q
x
x
2
base
(k)+q
a
¨ x
2
a
roof
(k)+ru
2
(k)] (6.10)
Assuming r = 10
4
/(m
1
g)
2
= 2.98 10
6
N
2
, one should find a desired pair of q
x
and q
a
by
consideringq
x
=a 10
8
/H
2
andq
a
=b 10
8
(m
1
/Hk
1
)
2
in whichH = 4 m is the height of the
structure. One can choose the desired nondimensionala andb by observing how changinga andb
affects the resulting LQR-controlled drift and absolute acceleration compared with the uncontrolled
response of the same system and excitation. The North-South component of the first 20 sec of
the 1940 El Centro earthquake record is considered as the external excitation. Figure 6.14 and
Figure 6.15 show the percent reduction of peak responses provided by LQR compared to those of
the uncontrolled system and the corresponding required control force in terms of percentage of the
weight of the structure. Similarly, Figure 6.16 and Figure 6.17 show the percentage improvements
of RMS responses provided by LQR compared to uncontrolled, and the RMS of the required LQR
control force in terms of the percentage of the weight of the structure for the El Centro excitation.
Assuming it is desired to achieve 70% and 50% reductions in peak base drift and roof absolute
acceleration reductions for weak to moderate excitations,a = 0.0043 andb = 14384 are selected,
resulting inq
x
= 2.6758 10
4
m
2
andq
a
= 2.7476 10
3
s
4
/m
2
. The corresponding peak control
force is 5.5% of the weight of the structure. It should be noted that, one could select differenta and
b based on any other desired performance (for example,a = 0 andb 20000 would give similar
performance with slightly different trade off of its drift, acceleration and control force metrics).
Selecting nonquadratic term:
Considering a nonquadratic cost function as
J=
¥
å
k=1
[q
x
x
2
base
(k)+q
a
¨ x
2
a
roof
(k)+ru
2
(k)+q
nq
x
4
base
(k)] (6.11)
Figure 6.18 shows, as a function ofq
nq
, the percentage RMS displacement and absolute acceleration
reduction of the nonquadratic MPC design compared to LQR for the MDOF problem subjected to
2El Centro, prediction horizon equal to 10 andIpopt solver. Depending on the desired base drift
reduction and the acceptable level of increase in roof absolute acceleration, the nonquadratic term
can be selected. Assuming it is desired to achieve 20% RMS displacement reduction relative to that
provided by LQR,q
nq
is selected as 1.65 10
8
m
4
for the MDOF problem considered here.
113
−60 −60
−40
−40
−20
−20
0
0 0
20
20
20
40
40
40
40
60
60
LQR
α (drift weight)
β (accel. weight)

10
−4
10
−2
10
0
10
3
10
4
10
5
Peak base drift improv.[%]
Peak roof accel. improv.[%]
40
50
60
70
80
90
Figure 6.14: Peak response reduction of LQR compared to uncont. [%], El Centro, 6DOF
5
5
10
10
15
15
20
25
LQR
α (drift weight)
β (accel. weight)

10
−4
10
−2
10
0
10
3
10
4
10
5
5
10
15
20
25
Figure 6.15: Peak required LQR control force [%W], El Centro, 6DOF
Selecting prediction horizon p:
As mentioned earlier, a longer prediction horizon does not necessarily result in significantly better
performance,i.e., it might be possible to get a desired performance with a shorter prediction horizon
114
20
30
30
40 40
40
50
50
50
50
60
60
60
60
60
70
70
LQR
α (drift weight)
β (accel. weight)

10
−4
10
−2
10
0
10
3
10
4
10
5
RMS base drift improv.[%]
RMS roof accel. improv.[%]
50
55
60
65
70
75
80
85
90
Figure 6.16: RMS response reduction of LQR compared to uncont. [%], El Centro, 6DOF
1
1.5
1.5
2
2
2.5
2.5
3
3
3.5
3.5
4
4
4.5
4.5
5
LQR
α (drift weight)
β (accel. weight)

10
−4
10
−2
10
0
10
3
10
4
10
5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 6.17: RMS required LQR control force [%W], El Centro, 6DOF
and less computation time. Figure 6.19 shows the variation of computation time and objective
function value versus prediction horizon forIpopt for the 2.0-scaled version of the El Centro
earthquake. Clearly, for longer prediction horizon, the performance will still improve, though the
115
10
6
10
7
10
8
10
9
-160
-140
-120
-100
-80
-60
-40
-20
0
20
40
q
nq
(NQ term Coeff.)
MPC
NQ
improvements compared to LQR [%]

displ. reduction
accel. reduction
force reduction
Figure 6.18: RMS response reduction of MPC
NQ
compared to LQR, 2El Centro,p=10
rate of improvement is modest for p 12; moreover, the computation time increases significantly
for longer prediction horizon. Therefore, a prediction horizon p= 15 is selected for simulation
here for the MDOF structure; slightly better performance is possible with longer prediction horizon
if larger computation time is permitted. Now, considering p= 15 and q
nq
= 1.65 10
8
m
4
, the
response of the MDOF model can be calculated. The responses to the weak excitation (0.1El
Centro) are shown in Figure 6.20, whereas Figure 6.21 shows the responses to the strong excitation
(2.0El Centro). The results are also presented in Table 6.6.
Table 6.6 shows that, in the weak excitation (0.1El Centro), the nonquadratic MPC control
strategy closely tracks the optimal LQR control. On the other hand, for the large excitation (2El
Centro), the RMS base drift of the nonquadratic MPC control strategy is 4.67 cm which is 32%
smaller than 6.87 cm, the corresponding value of LQR design. The same trend is exhibited by the
peak base drifts. Clearly, the nonquadratic MPC control strategy can focus on different objectives
in different level of excitations;i.e., it reduces drift and absolute acceleration similar to LQR design
in weak excitation, and focuses on drift reduction in large excitation.
116
2 4 6 8 10 12 14 16 18
0
100
200
300
Prediction Horizon
Computation Time (s)

2 4 6 8 10 12 14 16 18
1
2
3
4
x 10
5
Objective Function
Computation Time
Objective Function
Figure 6.19: Effects of prediction horizon on nonquadratic MPC computation time and cost, 2El
Centro, Ipopt
−2
0
2
b. drift
[cm]

−0.1
0
0.1
r. abs. accel
[m/s
2
]

0 5 10 15 20
−0.4
0
0.4
time [s]
force
[%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.20: Response comparison for nonquadratic MPC, 0.1El Centro, 6DOF, prediction horizon
p=15, solver:Ipopt
6.3.3 Design Based on Comparison to Passive Viscous Damper
In the previous section, the LQR design was performed based on comparison with the uncontrolled
responses of the system. In a different approach in this section, the LQR design will be based on
117
−50
0
50
b. drift
[cm]

−2
0
2
r. abs. accel
[m/s
2
]
0 5 10 15 20
−20
0
20
time [s]
force
[%W]
NonQuad. MPC
LQR
Uncontrolled
Figure 6.21: Response comparison for nonquadratic MPC, 2.0El Centro, 6DOF, prediction horizon
p=15, solver:Ipopt
Table 6.6: Performance of nonquadratic MPC under scaled 1940 El Centro excitation,p=15, Ipopt,
6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR MPC
nq
Uncont. LQR MPC
nq
Peak b. drift.[cm] 3.176 0.980 0.986 63.523 19.600 14.621
RMS b. drift.[cm] 1.291 0.342 0.342 25.810 6.870 4.672
Peak r. abs. accel.[m/s
2
] 0.206 0.103 0.105 4.123 2.065 3.834
RMS r. abs. accel.[m/s
2
] 0.083 0.027 0.027 1.667 0.534 0.822
Peak cont. force [% bldg.weight] — 0.548 0.548 — 10.963 23.631
RMS cont. force [% bldg.weight] — 0.165 0.166 — 3.306 5.901
response comparison with an “optimal” passive viscous damper, which was presented in Section
4.5. It was shown in Figure 4.9 that the optimal viscous damping coefficient for the purpose of this
problem isc
p
= 6 10
4
N s/m. Now, similar to Section 6.3.2.1, it is assumedr= 10
4
/(m
1
g)
2
=
2.98 10
6
N
2
, q
x
= a 10
8
/H
2
and q
a
= b 10
8
(m
1
/Hk
1
)
2
; however, a and b will be
selected by observing how changinga andb affects the resulting LQR-controlled drift and absolute
acceleration compared with the passive design response. Figure 6.22 and Figure 6.24 show the
percent reduction of RMS and Peak responses provided by LQR compared to the designed passive
system. The filled contours show the percent base drift reduction provided by LQR compared to
the passive damper, while line contours show the corresponding absolute acceleration reduction
percentage. As discussed in Section 4.5 and Section 5.3.2.2, the LQR
a
design here is chosen
with acceleration comparable to, or better than, that provided by a passive viscous damper that
118
−120
−100 −100
−80
−80
−60 −60
−60
−40
−40
−40
−20
−20 −20
0
0
0
20
20
LQR
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
RMS base drift improv.[%]
RMS roof accel. improv.[%]
−60
−40
−20
0
20
40
60
Figure 6.22: RMS response reduction of LQR compared to Passive [%], El Centro, 6DOF
best reduces acceleration. Thus, one approach is to pick a and b such that the RMS base drift
and roof absolute acceleration of LQR
a
are the same as those of the designed “optimal” passive
controller withc
p
= 6 10
4
N s/m. From Figure 5.18,a = 1.17 10
2
andb = 4.11 10
3
are
selected based on this criterion, resulting inq
x
= 7.32 10
4
m
2
andq
a
= 7.85 10
2
s
4
/m
2
. The
corresponding peak base drift and roof absolute acceleration reductions improvements compared
to uncontrolled responses are 66.15% and 43.10%, respectively. The peak control force is 6.02%
of the weight of the structure for El Centro earthquake. The corresponding RMS base drift and
absolute acceleration reductions are 75.79% and 63.27% , respectively. The RMS of the control
force is 1.77% of the weight of the structure.
It should be noted that this LQR design is chosen to be identical to the LQR
a
design presented in
Section 5.3.1.
Selecting nonquadratic term:
Figure 6.26 shows, as a function of nonquadratic coefficient q
nq
, the percentage RMS base drift
and roof absolute acceleration reduction of the nonquadratic MPC design compared to LQR for the
MDOF problem subjected to the 2El Centro, prediction horizon equal to 10 andIpopt solver.
Depending on the desired base drift reduction and the acceptable level of increase in roof absolute
acceleration, the nonquadratic term can be selected. Assuming that it is desired to achieve a base
drift reduction that is 30% beyond that with LQR,q
nq
is selected as 3.1 10
8
m
4
for the MDOF
problem considered in this study.
119
1.5
2
2
2.5
2.5
3
3
3.5
3.5
4
4
4.5
4.5
5
LQR
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 6.23: RMS required LQR control force [%W], El Centro, 6DOF
−150
−150
−100
−100
−50
−50 −50
0
0 0
0
LQR
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
Peak base drift improv.[%]
Peak roof accel. improv.[%]
−60
−40
−20
0
20
40
60
Figure 6.24: Peak response reduction of LQR compared to Passive [%], El Centro, 6DOF
Selecting prediction horizon p:
Figure 6.27 shows the variation of computation time and objective function value versus prediction
horizon for Ipopt for the 2.0-scaled version of the El Centro earthquake. Clearly, for longer
120
5
10
10
15
15
20
25
LQR
α (drift weight)
β (accel. weight)

10
−3
10
−2
10
−1
10
0
10
3
10
4
10
5
5
10
15
20
25
Figure 6.25: Peak required LQR control force [%W], El Centro, 6DOF
10
7
10
8
10
9
-200
-150
-100
-50
0
50
q
nq
(NQ term Coeff.)
MPC
NQ
improvements compared to LQR [%]

base drift reduction
roof accel. reduction
control force reduction
Figure 6.26: RMS Response reduction of MPC
NQ
compared to LQR for 2El Centro andp=10
prediction horizon, the performance will still improve, though the rate of its change diminishes
for p 10; moreover, the computation time increases significantly for longer prediction horizon.
Therefore, prediction horizon p= 10 is selected for simulation. Now, considering p= 10 and
121
2 4 6 8 10 12 14 16 18
0
200
400
600
Prediction Horizon
Computation Time (s)

2 4 6 8 10 12 14 16 18
0
2
4
6
x 10
5
Objective Function
Computation Time
Objective Function
Figure 6.27: Effects of prediction horizon on nonquadratic MPC computation time and cost, 2El
Centro, solver:Ipopt
q
nq
= 3.1 10
8
m
4
, the response of the MDOF model can be calculated. The responses in the
weak excitation (0.1El Centro) and the strong excitation (2.0El Centro) are calculated and
presented in Table 6.7. In the weak excitation (0.1El Centro), the nonquadratic MPC control
strategy closely tracks the optimal LQR control. On the other hand, for the large excitation (2El
Centro), the RMS base drift of the nonquadratic MPC control strategy is 4.345 cm, which is 30.4%
smaller than 6.244 cm, the corresponding value provided by the LQR design. The same trend exists
in the peak values of base drifts.
Table 6.7: Performance of nonquadratic MPC under scaled 1940 El Centro excitation,p=40, Ipopt,
6DOF
0.1El Centro 2El Centro
PGA= 0.3417 m/s
2
PGA= 6.8340 m/s
2
Uncont. LQR MPC
nq
Uncont. LQR MPC
nq
Peak b. drift.[cm] 3.176 1.075 1.073 63.523 21.502 12.529
RMS b. drift.[cm] 1.291 0.312 0.311 25.810 6.244 4.345
Peak r. abs. accel.[m/s
2
] 0.206 0.117 0.119 4.123 2.346 4.295
RMS r. abs. accel.[m/s
2
] 0.083 0.030 0.031 1.667 0.612 0.950
Peak cont. force [%weight] — 0.603 0.606 — 12.053 26.075
RMS cont. force [%weight] — 0.177 0.178 — 3.546 6.756
122
6.4 Conclusions
Minimizing structural drift is necessary during strong earthquakes to mitigate damage to the
structure, yet occupant comfort and safety of building contents in the much-more-frequent moderate
earthquakes demands reductions in absolute accelerations in the structure. Drift and acceleration
reduction in different excitations are often somewhat competing objectives and cannot be achieved
with conventional linear strategies.
This study presents the application of an MPC strategy to find the “optimal” nonlinear control
that minimizes a nonquadratic cost function and can achieve significant improvements in both
serviceability (i.e., acceleration mitigation) in weak to moderate excitations and life safety goals
(i.e., drift mitigation) in extreme events. The performance of this control strategy is demonstrated by
extensive numerical simulations for a SDOF and a base-isolated MDOF building model subjected to
different scaled version of the El Centro earthquake. For the SDOF problem studied in this research,
results show that the nonquadratic MPC control strategy closely tracks the optimal LQR control
in the weak excitation, while results in a 20% drift reduction relative to that provided by the LQR
design in the strong excitation. Moreover, the maximum required control force for MPC in this case
is equal to 31% of the weight of the structure, while LQR uses 41% of the weight of the structure.
Clearly, the nonquadratic MPC control strategy can focus on different objectives in different levels
of excitation, with control force even smaller than LQR. Applying other historic earthquakes, like
the 1994 Northridge earthquake also shows promising performance.
Introducing control force saturation at 15% of the structure weight to accommodate control device
capacity also shows that the nonquadratic MPC saturated control can still achieve both objectives in
different levels of excitations;i.e., nonquadratic MPC saturated control can reduce drift further than
saturated LQR control and does so with smaller RMS control force.
In a different approach, a quadratic cost function with some constraints on maximum allowable
drift is studied. Results show that, although the constrained MPC control can better reduce drift
compared to constrained LQR, it also causes the absolute acceleration to be significantly larger than
both the LQR and nonquadratic MPC designs. This suggests that the nonquadratic MPC strategy
outperforms the constrained quadratic MPC strategy studied here.
To extend the results and investigate if this MPC approach can be applied to more complicated
and realistic building models, a base-isolated MDOF building model is studied in this research.
Moreover, two different approaches — based on uncontrolled responses and an optimal passive
controller — are used to design the LQR controller to compare with the nonquadratic MPC. Results
123
show that the nonquadratic MPC control can successfully achieve both base drift reduction and
roof absolute acceleration reduction objectives at different excitation levels. The nonquadratic term
in the cost function in this study is selected as q
nq
x
4
base
, which becomes dominant in the strong
excitation and negligible in weak excitation; however, further study is needed to investigate other
possible choices of nonquadratic term(s) and their effect on the results.
124
Chapter 7
Learning-based Optimal Design of Smart
Damping using Neural Network
1
7.1 Introduction
Researchers have turned to controllable passive devices, also called “semiactive” because they
combine the characteristics of passive and active systems. Controllable passive devices are those
that dissipate energy through passive means, but with controllable properties so that device forces
can be modified and adjusted based on sensor measurements, feedback of excitation, changing
objectives and so forth. Common strategies for commanding the forces in these devices, as shown
in Figure 7.1a, involve designing a primary controller such as for an active device, but then use a
secondary controller to eliminate from the command any nondissipative control forces (Housner
et al., 1997). Thus, these devices provide the reliability of passive devices and the adaptability of
active systems, requiring only battery-strength power levels. In the case when the desired control
force is very non-dissipative and cannot be applied by the semiactive control device, it is possible
that the clipped control force can no longer be considered optimal. The use of large heuristic large
parametric studies is one approach to overcome this challenge (Ramallo et al., 2002, Johnson et al.,
2007).
The main advantage of model predictive control (MPC) is its ability to handle nonlinearities and
constraints — unlike the clipped LQR (CLQR) control strategy that requires designing a secondary
control or clipping. This difference is evident by comparing Figure 7.1b with Figure 7.1a. This
feature of MPC is exactly what needed for controllable damping strategies.
1
Hemmat-Abiri et al. (2015) reports a subset of this chapter.
125

(a) CLQR

(b) MPC
Figure 7.1: CLQR and MPC control strategies
Recent work by Elhaddad and Johnson (2015) proposed an MPC-based approach for designing
dissipative controllable dampers. Since the dissipativity constraint is considered inside the MPC op-
timization process, it is demonstrated that MPC can reduce the cost function more significantly than
the conventional CLQR control strategy that applies the constraint after optimization is performed.
Results also show that the required MPC control force is far smaller than the corresponding CLQR
control forces.
Implementing offline MPC and performing interpolation necessary for using table look-up is
practical for small systems; however, it might not be suitable for higher-order systems with greater
complexity. To overcome this challenge, this study will investigate the implementation of neural
networks (NN) to find the optimal control force (Pich´ e et al., 2000, Parisini and Zoppoli, 1995,
˚
Akesson et al., 2005). The goal is to train the NN based on the MPC results in order to estimate the
optimal control force for all possible states — even those states not included in the training. The
strength of a NN approach is that it can “learn” the very nonlinear regression of the optimal control
forces, which are complex nonlinear functions of the structure displacements and velocities. It is
also capable of capturing the “approximate” behavior of MPC with a few neurons and for small
number of available data points, inevitable for more complex structures, and then be generalized
and implemented for controllable damping of earthquake-excited structures. It is desired to train a
NN that is quite fast and computationally efficient for real-time application.
Training and testing datasets will be obtained by solving the MPC problem over a displacemen-
t/velocity grid for several testbed problems, for both single degree-of-freedom (SDOF) and two
degree-of-freedom (2DOF) models. Elhaddad and Johnson (2015) showed that MPC control forces
scale linearly when the state vector is scaled; i.e., they satisfy the homogeneity property. This
property will be exploited to reduce the dimension of data in the NN analysis.
126
Different neural network techniques and formations will be studied to find the most suitable NN
architecture (e.g., number of hidden layers, number of neurons in the hidden layer, etc.), and the
best training algorithm. This study is focused on feedforward techniques; further, since the target
values are known in this study, it is a supervised learning. Finally, to evaluate the fidelity of the
neural network results, the performance will be compared with MPC and CLQR.
7.2 Problem Formulation
Consider ann degree-of-freedom (DOF) linear system subjected to external disturbance w. Since
MPC is a discrete time control method, the discrete-time equation of motion can be expressed in
the form presented in Section 6.2.
Consider a quadratic cost function of the form of (5.2)
J=
¥
å
k=1

q
T
Q
d
q+ 2q
T
N
d
u+ u
T
R
d
u

in which dependence on term (k) is omitted for clarity, where Q
d
, N
d
and R
d
are matrices of
appropriate dimensions for the discrete time problem. The optimal linear control law that minimizes
the quadratic cost function (5.2) can be written as
u
LQR
(k)=K
LQR
q(k) (7.1)
where K
LQR
is the optimal control gain simply obtained using typical solvers like MATLAB’slqr
command. However, since the linear quadratic regulator (LQR) does not consider the dissipativity
constraint in the optimization, it may result in a force that is non-dissipative, and therefore not
realizable with controllable passive devices. One approach to address this challenge is to use the
clipped-optimal control algorithm (Dyke et al., 1996) that omits all non-dissipative force commands
using a secondary control algorithm, as shown in Figure 7.1a.
Herein, as shown in Figure 7.2, it is assumed that all dissipative forces are realized and all nondissi-
pative forces are infeasible. This idealized passivity constraint can be presented as
u
LQR
v
device
> 0 (7.2)
whereu
LQR
=K
LQR
q is the desired control force (with a sign convention, opposite that assumed
in (4.33), that the force is positive when resisting motion with positive velocity) andv
device
is the
127
Dissipative Non-dissipative
Non-dissipative Dissipative
v
device
u
LQR
Figure 7.2: Passivity constraint for controllable damping devices
relative velocity across the damper;i.e., the force is dissipative — and therefore realizable — if the
desired control force and velocity across the device are the same sign. Otherwise, the force will be
clipped. To apply MPC and emulate an infinite horizon quadratic cost function, one can rewrite the
quadratic cost function as
J= q
T
p
Pq
p
+
p1
å
k=1

q
T
Q
d
q+ 2q
T
N
d
u+ u
T
R
d
u

(7.3)
where the first term is the terminal cost, q
p
= q(p) and P is the solution of the algebraic Riccati
equation, P= A
T
PA(A
T
PB)(R+ B
T
PB)
1
(B
T
PA)+ Q, associated with the discrete LQR
problem that minimizes the infinite horizon quadratic cost function in (5.2), and p is the prediction
and control horizon shown in Figure 1.3.
To find the optimal control using MPC, the YALMIP toolbox (L¨ ofberg, 2004) in MATLAB is used.
As shown in Figure 6.1, YALMIP relies on one of several supported external solvers to perform
the optimization. In this study, the Gurobi Optimizer (Gurobi Optimization, Inc., 2014) will be
used. Note that dissipativity constraint (7.2) will be considered as a constraint provided to the MPC
optimization, so there is no need to post-apply clipping to the resulting control force. The goal is
to use the control force data obtained from MPC to train different neural networks to produce a
control force similar to MPC. Once the neural network has been trained to fit the data, it creates a
generalized relationship from the input (structure states) to the output (control force) and can be
used to generate outputs for new inputs within the range of its training input data.
128
To investigate which feedforward NN suits best for this specific problem, intensive analysis and
parameter studies will be carried out: several training algorithms — including Levenberg-Marquardt
backpropagation (Marquardt, 1963, Levenberg, 1944), Bayesian Regularization (MacKay, 1992) and
Resilient Backpropagation (Riedmiller and Braun, 1993) training algorithms — will be investigated;
several NNs with different numbers of neurons will be studied to determine the optimal number of
neurons in the hidden layer. Using more neurons might be suitable for more complex problems;
however, more neurons require more computation, and overfitting may also result with a large
number of neurons. It should be noted that a neural network may result in a different solution
each time it is trained due to different initial weight and bias values and different divisions of data
into training, validation, and test subsets. To overcome this challenge, the analysis in this study is
performed over several initial weight and bias values, selecting the one leading to the best result.
Moreover, for a fair comparison of different NN analysis, the data sets considered as training, test
and validation are held unchanged — usingdivideint (interleaved training, test and validation data
points) instead ofdividerand (random arrangements of training, test and validation data).
7.3 Numerical Example 1: SDOF Model
First, consider a SDOF structure model with parameters m = 100 ton, k = 3.948 MN/m and
c= 62.833 KN s/m, adapted from Elhaddad and Johnson (2015). (It should be noted that, since
MPC is a discrete-time process, the system considered in this section is a discrete system, whereas
Elhaddad and Johnson (2015) used a continuous-time system.)
The objective is to minimize the mean square of the absolute acceleration of the structure:
J=
¥
å
k=1
¨ x
2
abs
(k) (7.4)
where ¨ x
abs
=
k
m
x
c
m
˙ x+
1
m
u, where x, ˙ x and u are displacement,velocity and control force,
respectively. To find the CLQR control force, first the optimal LQR control force is calculated using
u=K
LQR
[x ˙ x]
T
and then passivity constraint (7.2) will be applied. To find the MPC control force,
optimization is performed to minimize (7.3) subject to passivity constraint (7.2). The prediction
horizon and control horizon are chosen to be 30 time steps of 0.02 sec each, identical to that in
Elhaddad and Johnson (2015).
Figure 7.3 shows the optimal controllable damping force computed with MPC for the SDOF
structure model. The clipped LQR control force for the same model is presented in Figure 7.4. It
129
should be noted that these figures are similar, but not identical to those in Elhaddad and Johnson
(2015) due to different system assumption (discrete-time vs. continuous-time, 20-sec vs. 30-sec
excitation, and different time-step duration). Clearly, the MPC control force differs significantly
from the CLQR control force.
−0.1
−0.05
0
0.05
0.1
−1
−0.5
0
0.5
1
−0.5
0
0.5
Displacement [m]
Velocity [m/s]
Control signal, u
MPC
[MN]
Figure 7.3: MPC control force for SDOF model
−0.1
−0.05
0
0.05
0.1
−1
−0.5
0
0.5
1
−0.5
0
0.5
Displacement [m]
Velocity [m/s]
Control signal, u
CLQR
[MN]
Figure 7.4: CLQR control force for SDOF model
130
w
u
u control force
Discrete State-Space
Displ
Velocity
w Excitation
Zero-Order
Hold
K_LQR
Gain
Multiply
max
Clipping
0
q
Figure 7.5: Simulink model to calculate the CLQR response of a SDOF structure
w
u
u control force
Discrete State-Space
Displ
Velocity
w Excitation
Zero-Order
Hold
Interpreted
MATLAB Fcn
OnlineMPC
q
Figure 7.6: Simulink model to calculate the online MPC response of a SDOF structure
To compare the response of the model equipped with these controllers, the model is subjected to the
1940 El Centro earthquake excitation. Figures 7.5 and 7.6 show the Simulink models used to find
the CLQR and online MPC response of the SDOF structure model, respectively. It should be noted
that the predefined controller command in YALMIP is used in the Interpreted MATLAB Fcn in
Figure 7.6 to find the optimal MPC control command. As shown in Table 7.1, the responses of the
system with the two controllers are significantly different; MPC results in better performance and,
therefore, a smaller cost than CLQR. It should be noted that, due to different system assumptions,
the results presented herein are not exactly identical — but still similar — to those in Elhaddad and
Johnson (2015).
131
Table 7.1: Performance comparison for MPC and CLQR under 1940 El Centro excitation
1940 El Centro
PGA= 3.41 m/s
2
Uncont. CLQR Online MPC 4

[%]
Peak displ.[m] 0.1281 0.0552 0.0413 25.18
RMS displ.[m] 0.0330 0.0172 0.0127 26.16
Peak abs. accel.[m/s
2
] 5.0859 2.2036 1.6638 24.49
RMS abs. accel.[m/s
2
] 1.3103 0.4610 0.4321 6.26
Peak cont. force [% bldg.weight] — 22.03 16.62 24.55
RMS cont. force [% bldg.weight] — 5.12 4.49 12.30
Cost [m
2
/s
3
] 34.3358 4.2508 3.7341 12.15

4=jOnline MPC CLQRj/CLQR 100%
7.3.1 Neural Network Implementation
The main goal of this study is to use the data obtained from MPC (Figure 7.3) to train a Neural
Network using various training methods and network architectures to investigate if the same
nonlinear control, or even a simpler control force with similar performance, can be obtained from
the trained Neural Network. The first approach could be to consider displacement and velocity as
input values and the control force as the target value for the NN as shown in Figure 7.7. However,
the control force is linear with respect to the distance from the origin in the phase space, as shown in
Figure 7.3. Therefore, the control force shown in Figure 7.3 can be obtained by scaling the control
force on a unit circle around the origin. Moreover, there is some symmetric behavior in the control
force: the control force in the third and fourth quadrants (i.e., negative velocity) are exactly the
opposite of the corresponding values in the first and second quadrants, respectively. Considering
these properties will significantly reduce the size of the data set required for training the desired
NN, resulting in a simpler NN that requires less time to train and simulate. This will be explored in
more detail in the following section. Consider a unit circle (r= 1) around the origin as shown in
Figure 7.9a, where r=
p
q
2
+ ˙ q
2
andq = tan
1
(
˙ q
q
); one can find the MPC control force for all
q= cos(q) and ˙ q= sin(q) whereq =[p,p]. Moreover, considering the symmetric behavior of
control force, the rangeq =[0,p] is sufficient to find the control force on the perimeter on the unit
circle. The resulting MPC and CLQR control forces are shown in Figure 7.8. It should be noted
that the control force at a different radius can be simply calculated by scaling the corresponding
value on the unit circle.
Since displacement and velocity have different magnitudes (on the order of the natural frequency
w of the system), a scaling factor could be introduced in calculation of r andq to eliminate any
undesired effect of different magnitudes. Therefore, the unit circle can be considered on q and ˙ q/w
as shown in Figure 7.9b (as opposed to q and ˙ q shown in Figure 7.9a). Here, r=
q
q
2
+(
˙ q
w
)
2
,
132
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer

) (  u



) , , (    u
) , ( q q u 
q
q 
Figure 7.7: NN architecture when input isfdisplacement,velocityg and output isfcontrol forceg
-180 -90 0 90 180
-40
-30
-20
-10
0
10
20
30
40
θ [deg]
control force / mass [m/s
2
]

MPC
CLQR
Figure 7.8: Control force on a unit circle around the origin
q= cos(q), ˙ q=w sin(q) and q = tan
1
(
˙ q/w
q
). The resulting scaled MPC and CLQR on the
scaled unit circle are shown in Figure 7.10 as a function ofq.
As mentioned previously, the MPC data corresponding toq =[0,p] is sufficient to calculate the
entire control force. Therefore, q =[0,p] is considered as inputs and the corresponding scaled
MPC control force is considered as the target values for the NN problem shown in Figure 7.11.
133
It should be noted that a gridq = 0

,0.5

,:::,180

is used in the discrete simulation herein. The
Simulink models to calculate offline MPC and the NN response are shown in Figures 7.12 and 7.13,
respectively.

(a) on original axes

(b) on scaled axes
Figure 7.9: Unit circle around the origin in the state space
-180 -90 0 90 180
-40
-30
-20
-10
0
10
20
30
40
θ [deg]
control force / mass [m/s
2
]

scaled MPC
CLQR
Figure 7.10: Control force on a unit circle around origin in the state space on scaled axes
134
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer

) (  u



) , , (    u
) , ( q q u 
q
q 
Figure 7.11: NN architecture when input isq and output is the control force
w
u
u control force
Discrete State-Space
Displ
Velocity
w Excitation
Zero-Order
Hold
Multiply
max
Clipping
0
θ=atan2[Vel/(k/m)
0.5
, Displ]
*180/pi
Of fline MPC
u = r. MPC(θ, r=1)
q
r=[Displ
2
+Vel
2
/(k/m)]
0.5
Figure 7.12: Simulink model to calculate the offline MPC response of a SDOF structure
7.3.2 Selecting NN Architecture and Features
To find a NN that performs well for this problem and results in the desired responses, extensive
investigation will be carried out over various possible parameters, architectures and algorithms in
this section.
Since one hidden layer is usually enough for any engineering problem (Cybenko, 1989), only one
hidden layer is assumed in this study; however, the number of neurons in the hidden layer is very
problem specific: more neurons in the hidden layer can emulate more complex behavior, but using
135
w
q
u
u control force
Discrete State-Space
Displ
Velocity
w Excitation
Zero-Order
Hold
Multiply
max
Clipping
0
r=[Displ
2
+Vel
2
/(k/m)]
0.5
θ=atan2[Vel/(k/m)
0.5
, Displ]
*180/pi
u = r. NN(θ, r=1)
NN(θ) -NN( θ+180)
θ
if( θ>=0) else
switch block
Figure 7.13: Simulink model to calculate the NN response of a SDOF structure
many neurons may also result in overfitting the data. Simulation is performed for different numbers
of neurons, and their results will be compared to find a number of neurons that performs well for
this problem.
Another NN feature to be determined is how to divide the whole data into Training, Validation
and Testing subsets. A common fraction considered in the literature is 70%, 15% and 15% for
Training, Validation and Testing subsets, respectively (Beale et al., 2014). There are also different
possible methods for this data division; while randomly dividing data (thedividerand command in
MATLAB) is the most common method in the literature, a fair comparison between different NN
results demands that same data division be used in each method. Therefore, data division using
interleaved indices (thedivideint command in MATLAB) has been applied here.
NN weight and bias values play the main role in NN performance. These values are initialized
randomly in the beginning of training process; each training may result in a different performance,
depending on the specific initial set of weights and biases. Hence, to find the best performance for
any specific problem, the training should be repeated multiple times and the best initial weight and
biases will be determined based on their performance. In this section, 10 different initial weight and
bias sets are considered using a Random Number Generator (rng command in MATLAB).
Finally, there are various algorithms for training the NN; depending on the complexity of the
problem, the number of data points, the number of weights and biases in the network, etc., one or
multiple of these training algorithms will perform well for a given problem. In this study, the NN
is being used for function approximation (regression); three different training algorithms will be
studied here: Levenberg-Marquardt backpropagation (trainlm command in MATLAB), Bayesian
136
Regulation backpropagation (trainbr command in MATLAB) and Resilient Backpropagation (trainrp
command in MATLAB).
7.3.2.1 Levenberg-Marquardt backpropagation training method
First, consider Levenberg-Marquardt backpropagation as the training algorithm, with only one
hidden layer. To find the best number of neurons in the hidden layer, one can consider a small-sized
NN and observe how increasing the size of the network will affect the performance. NNs with
1–250 neurons in the hidden layer are considered herein. Moreover, as mentioned previously, the
values of initial weights and biases will affect the performance of NN; therefore, 10 different sets
for initial weights and biases (usingrng(356seed) whereseed= 1,2,:::10) are considered here
to find the best initial weight and biases that result in the best performance (the 356 is completely
arbitrary, but any such integer will suffice).
Figure 7.14 shows the mean square error (MSE) between the output of the trained NNs and the
target values corresponding to the scaled MPC results shown in Figure 7.10; this MSE is obtained
using theperformance command in MATLAB’s NN toolbox for all data.
The corresponding cost values (J=
R
¨ q
2
dt) for the SDOF building model subjected to 20 sec of the
1940 El Centro earthquake are shown in Figure 7.15. It is clear from Figure 7.14 that the MSE is at
its largest value when there is only one neuron in the hidden layer and generally decreases as the
number of neurons in the hidden layer increases — but up to the point where there is a good match
between the target values and the output of the NN — after that, the MSE will increase again due
to overfitting. The cost values, shown in Figure 7.15, differ slightly as the number of neurons or
seed changes, but there is clear overfitting when using large numbers of neurons in the hidden layer,
leading to larger cost values.
To compare the response and performance of the trained NN, two particular NNs are selected: the
first NN has only 2 neurons in the hidden layer and seed 2 is used for initial weighting; the second
NN uses 70 neurons and seed 9. According to Figure 7.14, it is expected that the NN will better
track the target values with 70 neurons (NN
H=70
) compared to using only 2 neurons (NN
H=2
);
however, it will be seen that the cost difference is not significant. Figure 7.16 shows the outputs of
the two NNs along with the target MPC results. As expected, NN
H=70
tracks MPC very closely,
while NN
H=2
ignores the details of MPC and only follows its general path.
Figure 7.17 shows the response of the SDOF model controlled by NN
H=2
and NN
H=70
subjected
to 1940 El Centro earthquake, computed using the Simulink model shown in Figure 7.13. The
137
1 2 5 10 30 50 70 90 110 130 150 170 190 210 230 250
0
10
20
30
40
50
60
Error, MSE(NN - MPC) [m
2
/s
4
]

number of neurons in the hidden layer
seed=1
seed=2
seed=3
seed=4
seed=5
seed=6
seed=7
seed=8
seed=9
seed=10
Figure 7.14: Error between NN outputs trained with Levenberg-Marquardt and the target MPC
results
results are also presented in Table 7.2. It is clear from Figure 7.17 that the responses of the two
NNs are very similar. This suggests that the jagged lines in MPC do not play an important role in
its significant outperformance of CLQR. In fact, these jagged lines may result in some undesired
low-amplitude high-frequency (on and off) control force, as shown in Figure 7.17. The results
in Table 7.2 verify that the performance of NN
H=2
and NN
H=70
are very similar to each other;
both have costs slightly smaller than MPC, which itself significantly outperforms CLQR. It should
be noted that the cost value corresponding to NN
H=70
is smaller than that of NN
H=2
, but the
difference is on the order of 1%; therefore, it is more reasonable to use the simpler NN with only
two neurons in the hidden layer. As mentioned previously, Levenberg-Marquardt training method is
one of the fastest algorithms suitable for function approximation and curve fitting. In the following
sections, two other training algorithms will be examined to show that they do not outperform the
Levenberg-Marquardt method.
138
1 2 5 10 30 50 70 90 110 130 150 170 190 210 230 250
3.4
3.5
3.6
3.7
3.8
3.9
4
number of neurons in the hidden layer
cost [m
2
/s
3
]

Figure 7.15: Cost value of different NNs trained with Levenberg-Marquardt
0 20 40 60 80 100 120 140 160 180
-40
-30
-20
-10
0
10
20
30
40
θ [deg]
control force / mass [m/s
2
]

MPC
H=2
H=70
Figure 7.16: Control force comparison for Levenberg-Marquardt-trained NNs
139
-4
0
4
displ [cm]

-1
0
1
accel [m/s
2
]
0 2 4 6 8 10 12 14 16 18 20
-10
0
10
time [s]
force [%W]
H=70
H=2
Figure 7.17: Response comparison of the two Levenberg-Marquardt-trained NNs
Table 7.2: Comparison of 1940 El Centro response metrics using Levenberg-Marquardt training
1940 El Centro
PGA= 3.41 m/s
2
Uncont. CLQR MPC NN
H=2
NN
H=70
Peak displ.[m] 0.1281 0.0552 0.0414 0.0418 0.0408
RMS displ.[m] 0.0330 0.0172 0.0128 0.0127 0.0126
Peak abs. accel.[m/s
2
] 5.0859 2.2036 1.6710 1.7114 1.6534
RMS abs. accel.[m/s
2
] 1.3103 0.4610 0.4328 0.4306 0.4284
Peak cont. force / bldg.weight — 0.2203 0.1666 0.1629 0.1642
RMS cont. force / bldg.weight — 0.0512 0.0449 0.0444 0.0444
Cost [m
2
/s
3
] 34.3358 4.2508 3.7452 3.7081 3.6696
7.3.2.2 Bayesian Regularization training method
For the Bayesian Regularization training algorithm, the procedure explained in the previous section
will be repeated here. A preliminary study shows that the performance of the NN does not improve
beyond 150 neurons. Therefore, NNs with 1–150 neurons in the hidden layer will be studied here.
Figure 7.18 shows the error between the output of the trained NN and the target MPC results. The
corresponding costs are shown in Figure 7.19. The results are similar to what was seen in the
previous section; however, Bayesian Regularization training convergence is much slower compared
with Levenberg-Marquardt method. To compare the response and performance of the trained NNs,
two particular NNs are selected: the first NN has only 2 neurons in the hidden layer and seed 2 is
used for initial weighting; the second NN uses 50 neurons and seed 7. According to Figure 7.18, it
140
1 2 5 10 30 50 70 90 110 130 150
0
10
20
30
40
50
60
Error, MSE(NN - MPC) [m
2
/s
4
]
number of neurons in the hidden layer

seed1
seed 2
seed 3
seed 4
seed 5
seed 6
seed 7
seed 8
seed 9
seed10
Figure 7.18: Error between NN outputs trained with Bayesian and the target MPC results
Table 7.3: Performance comparison for MPC, CLQR and NN under 1940 El Centro excitation
1940 El Centro
PGA= 3.41 m/s
2
Uncont. CLQR MPC NN
H=2
NN
H=50
Peak displ.[m] 0.1281 0.0552 0.0414 0.0418 0.0426
RMS displ.[m] 0.0330 0.0172 0.0128 0.0127 0.0128
Peak abs. accel.[m/s
2
] 5.0859 2.2036 1.6710 1.7138 1.8916
RMS abs. accel.[m/s
2
] 1.3103 0.4610 0.4328 0.4315 0.4292
Peak cont. force / bldg.weight — 0.2203 0.1666 0.1629 0.1715
RMS cont. force / bldg.weight — 0.0512 0.0449 0.0444 0.0452
Cost [m
2
/s
3
] 34.3358 4.2508 3.7452 3.7227 3.6828
is expected that the response of the model with NN
H=50
control will be much closer to the MPC
results compared to that of NN
H=2
; however, the cost difference is not significant. Figure 7.20
shows the outputs of the two NNs along with the target values. As expected, NN
H=50
tracks MPC
very closely, while NN
H=2
ignores the details of the MPC and only follows its general path.
Table 7.3 shows the response of the SDOF model subjected to El Centro earthquake. Results verify
that the performance of NN
H=2
and NN
H=50
are very similar to each other; both are better than
MPC, which itself outperforms CLQR. It should be noted that, similar to the previous section, the
cost value corresponding to NN
H=50
is smaller than that of NN
H=2
, but the difference is on the
order of 1%; therefore, it is more reasonable to use the simpler NN with only two neurons in the
hidden layer.
141
1 2 5 10 30 50 70 90 110 130 150
3.4
3.5
3.6
3.7
3.8
3.9
Cost [m
2
/s
3
]
number of neurons in the hidden layer
Figure 7.19: Cost value of different NNs trained with Bayesian
0 20 40 60 80 100 120 140 160 180
-40
-30
-20
-10
0
10
20
30
40
θ [deg]
control force / mass [m/s
2
]

MPC
H=2
H=50
Figure 7.20: Control force comparison for Bayesian-trained NNs
142
1 2 5 10 30 50 70 90 110 130 150 170 190 210 230 250
0
20
40
60
80
100
120
140
160
180
200
Error, MSE(NN - MPC) [m
2
/s
4
]
number of neurons in the hidden layer
Figure 7.21: Error between NN outputs trained with Resilient and the target MPC results
7.3.2.3 Resilient Backpropagation training algorithm
The last training method to be considered in this study is the Resilient Backpropagation training
algorithm. Figure 7.21 shows the error between the output of the trained NN and the MPC target
values. The corresponding cost values are shown in Figure 7.22. It should be noted that the
minimum MSE found in this method is 5.1 m
2
/s
4
while the corresponding minimum values of
Levenberg and Bayesian methods are 0.753m
2
/s
4
and 0.225m
2
/s
4
, respectively. This suggests that
the Resilient Backpropagation training algorithm is not able to produce outputs similar to the target
values in this problem. Figure 7.23 shows the outputs of the Resilient-trained NN with the smallest
MSE — NN with 90 neurons and seed 2 — along with the target values. Clearly, this NN cannot
track MPC very closely; therefore, it is not suitable for the purpose of this study.
7.3.2.4 Summary of different training method results
It has been shown that both Levenberg-Marquardt and Bayesian training algorithms can produce
outputs very similar to the target values considered in this study. Moreover, it was shown that
using only two neurons in the hidden layer can lead to a smaller cost compared to MPC. Since
143
1 2 5 10 30 50 70 90 110 130 150 170 190 210 230 250
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
cost [m
2
/s
3
]
number of neurons in the hidden layer
Figure 7.22: Cost value of different NNs trained with Resilient
0 20 40 60 80 100 120 140 160 180
-40
-30
-20
-10
0
10
20
30
40
θ [deg]
control force / mass [m/s
2
]

MPC
H=90
Figure 7.23: Control force comparison for Resilient-trained NN
Levenberg-Marquardt is usually faster than the Bayesian Regularization algorithm, the former is
used in the rest of this study.
144
Table 7.4: Comparison of El Centro response metrics using Levenberg-Marquardt training
1940 El Centro
PGA= 3.41 m/s
2
Uncont. CLQR MPC NN
H=2
NN
H=130
Peak displ.[m] 0.1281 0.0552 0.0414 0.0419 0.0410
RMS displ.[m] 0.0330 0.0172 0.0128 0.0127 0.0126
Peak abs. accel.[m/s
2
] 5.0859 2.2036 1.6710 1.7491 1.6409
RMS abs. accel.[m/s
2
] 1.3103 0.4610 0.4328 0.4302 0.4273
Peak cont. force / bldg.weight — 0.2203 0.1666 0.1635 0.1650
RMS cont. force / bldg.weight — 0.0512 0.0449 0.0445 0.0444
Cost [m
2
/s
3
] 34.3358 4.2508 3.7452 3.6999 3.6519
7.3.3 Changing the Target Value atq = 0
As mentioned previously, the main goal of this study is to investigate if a NN can be used for more
complicated multiple degree-of-freedom (MDOF) models, when MPC results can be calculated
only for a very small number of data points. Therefore, in the following section, the MPC data will
be decimated and used to train a NN, whose performance will be compared with that of the original
MPC. However, since the NN usually does not perform well when it must extrapolate,q = 0 and
q = 180

must be included in the down-sampled data. However, as shown in Figure 7.23, the
target value (MPC) atq = 0 is very different from the rest of data points, leading to inaccurate NN
outputs, particularly when data is decimated. To overcome this challenge, this section considers that
u(q = 0)=u(q = 0
+
)= 0 instead ofu(q = 0)=u(q = 0

)

=40.
Figure 7.24 shows the error between the output of the Levenberg-Marquardt-trained NN and the
target MPC values. The corresponding cost values are shown in Figure 7.25. By comparing
Figure 7.24 and Figure 7.14, it is clear that changing the target value atq = 0 has decreased the
error when there is only one neuron in the hidden layer — from 55 to 7. To compare the response
and performance of the trained NNs, two particular NNs are selected: the first NN has only 2
neurons in the hidden layer and uses seed 8 for initial weighting; the second NN with 130 neurons
and seed 8. Figure 7.26 shows the outputs of the two NNs along with the target MPC values.
NN
H=130
tracks MPC very closely, while NN
H=2
ignores the details of MPC and only follows its
general path.
Table 7.4 shows the response of the SDOF model controlled by NN
H=2
and NN
H=130
subjected to
El Centro earthquake. The results verify that the performance of NN
H=2
and NN
H=130
are very
similar to each other; both have slightly smaller costs compared to MPC, which itself significantly
outperforms CLQR.
145
1 2 5 10 30 50 70 90 110 130 150 170 190 210 230 250
0
2
4
6
8
10
12
Error, MSE(NN - MPC) [m
2
/s
4
]
number of neurons in the hidden layer
Figure 7.24: Error between NN outputs trained with Levenberg-Marquardt and the target MPC
results
7.3.4 Effects of Decimation of the MPC Data for NN Training
The objective of this section is to investigate the effects of data decimation on the performance of
trained NNs.
7.3.4.1 Decimation withDq = 5

First assume that the MPC results are only available for anglesq = 0

,5

,:::180

— as opposed
toq = 0

,0.5

,:::180

in the previous sections. Figure 7.27 shows the error between the output
of the trained NN and the decimated target values. The corresponding cost values are shown in
Figure 7.28.
To compare the response and performance of the trained NN, a NN with the smallest cost is selected:
with 5 neurons in the hidden layer and using seed 9 for initial weighting. Figure 7.29 shows the
output of this NN along with the original and decimated target MPC results.
146
1 2 5 10 30 50 70 90 110 130 150 170 190 210 230 250
3.3
3.4
3.5
3.6
3.7
3.8
3.9
cost [m
2
/s
3
]
number of neurons in the hidden layer
Figure 7.25: Cost value of different NNs trained with Levenberg-Marquardt
0 20 40 60 80 100 120 140 160 180
-5
0
5
10
15
20
25
30
35
40
θ [deg]
Control force / mass [m/s
2
]

MPC
H=2
H=130
Figure 7.26: Control force comparison for Levenberg-Marquardt-trained NNs
147
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 30 40 50
0
20
40
60
80
100
120
140
160
180
Error, MSE(NN - MPC) [m
2
/s
4
]
Number of neurons in the hidden layer
Figure 7.27: Error between Levenberg-Marquardt-trained NN outputs and the target MPC results
decimated withDq = 5

7.3.4.2 Decimation withDq = 10

Assuming that the MPC data is decimated withDq = 10

, Figure 7.30 shows the error between the
output of the trained NN and the decimated target values. The corresponding cost values are shown
in Figure 7.31. To compare the response and performance of the trained NN, a NN with a small
cost and error is selected: with 6 neurons in the hidden layer and the seed 5 for initial weighting.
Figure 7.32 shows the output of this NN along with the original and decimated target MPC results.
Table 7.5 shows the response of the SDOF model controlled by two decimated NNs, decimated
MPC, the original MPC, the original CLQR and the uncontrolled response, subjected to the 1940 El
Centro earthquake. Numerical results show that, for downsampling with steps ofDq = 5

, the NN
cost is 17.10% smaller than the cost of CLQR, and even 10.89% better than decimated MPC. In
the case of downsampling with steps ofDq = 10

, the corresponding cost reductions are 13.83%
and 9.13%. This confirms the potential use of NN for more complex models when only a few data
points can be obtained using MPC.
148
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 30 40 50
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
cost [m
2
/s
3
]
number of neurons in the hidden layer
Figure 7.28: Cost value of different NNs trained with Levenberg-Marquardt, decimated with
Dq = 5

Table 7.5: Comparison of El Centro response metrics using Levenberg-Marquardt training
1940 El Centro
PGA= 3.41 m/s
2
Uncont. CLQR MPC MPC
Dq=5
NN
Dq=5,H=5
MPC
Dq=10
NN
Dq=10,H=6
Peak displ.[m] 0.1281 0.0552 0.0414 0.0415 0.0444 0.0419 0.0426
RMS displ.[m] 0.0330 0.0172 0.0127 0.0128 0.0132 0.0127 0.0129
Peak abs. accel.[m/s
2
] 5.0859 2.2036 1.6710 2.2594 1.7679 2.0696 1.7142
RMS abs. accel.[m/s
2
] 1.3103 0.4610 0.4328 0.4447 0.4198 0.4449 0.4280
Peak cont. force / bldg.weight — 0.2203 0.1634 0.1648 0.1832 0.1686 0.1689
RMS cont. force / bldg.weight — 0.0512 0.0453 0.0446 0.0463 0.0448 0.0452
Cost [m
2
/s
3
] 34.3358 4.2508 3.7452 3.9545 3.5241 3.9586 3.6629
7.3.5 Summary of SDOF Results
It has been shown that using homogeneity property of MPC control forces reduces the dimension of
data in the NN analysis. Different neural network techniques and formations are studied for the
SDOF problem. The results show that both Levenberg-Marquardt and Bayesian training algorithms
can produce outputs very similar to the target MPC values. Moreover, it was shown that using only
two neurons in the hidden layer can lead to a cost smaller than that of MPC. Further, the effects of
decimation are investigated to confirm the potential use of NNs for more complex models when
149
0 20 40 60 80 100 120 140 160 180
-5
0
5
10
15
20
25
30
35
40
45
θ [deg]
Control force / mass [m/s
2
]

decimated MPC
NN
MPC
Figure 7.29: Control force comparison for Levenberg-Marquardt-trained NNs decimated with
Dq = 5

only a few data points can be obtained using MPC. The results show that decimated NNs lead to
smaller cost compared to that of CLQR.
7.4 Numerical Example 2: MDOF Model
Consider a 2DOF shear structure with a smart damper between the ground and the first floor
mass with model parameters adapted from Elhaddad and Johnson (2015): m
1
= m
2
= 100 ton,
k
1
= k
2
= 15.792 MN/m, c
1
= 112.5 KN s/m and c
2
= 0, resulting in modal frequencies 1.24
and 3.23 Hz (0.81 and 0.31 sec periods) and 2% modal damping. This model is subjected to a
30 sec record of the 1940 El Centro earthquake. In this section, the objective of the controllable
damper design is to minimize mean square absolute accelerations of both stories as:
J=
Z
¥
0
[( ¨ x
abs
1
)
2
+( ¨ x
abs
2
)
2
+ru
2
]dt (7.5)
where ¨ x
abs
1
and ¨ x
abs
2
are the absolute accelerations of the first and second stories, respectively;u is
the control force and relative weight r is chosen so that the damper force is not too large, equal
to 10
16
N
2
herein. It should be noted that, in this section, continuous-time system equations
150
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
10
20
30
40
50
60
70
80
90
100
110
Error, MSE(NN - MPC) [m
2
/s
4
]
Number of neurons in the hidden layer
Figure 7.30: Error between Levenberg-Marquardt-trained NN outputs and the target MPC results
decimated withDq = 10

are used — unlike the SDOF analysis in the previous section — to show that the method is valid
regardless of the system choice and to be consistent with the results shown in Elhaddad and Johnson
(2015).
It is shown in Elhaddad and Johnson (2015) that the control input, a function of the 4 states, can be
implemented using 3D table lookup by exploiting the homogeneity property. This can be illustrated
by rewriting the control law using spherical coordinates as follows:
u
MPC
=u(r,a,b,g)=r ¯ u(a,b,g) (7.6a)
r=
q
x
2
1
+x
2
2
+ ˙ x
2
1
+ ˙ x
2
2
(7.6b)
a = tan
1
(
x
2
x
1
), a2[p,p] (7.6c)
b = tan
1
0
@
˙ x
1
q
x
2
1
+x
2
2
1
A
, b2[
p
2
,
p
2
] (7.6d)
g = tan
1
0
@
˙ x
2
q
x
2
1
+x
2
2
+ ˙ x
2
1
1
A
, g2(
p
2
,
p
2
) (7.6e)
151
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
5.4
cost [m
2
/s
3
]
number of neurons in the hidden layer
Figure 7.31: Cost value of different NNs trained with Levenberg-Marquardt, decimated with
Dq = 10

where ¯ u(a,b,g) is the optimal MPC control force for a state vector on the unit hypersphere in the
state space defined by the three anglesa,b andg. It should be noted that one can express states in
terms ofa,b,g andr as
x
1
=r cosg cosa cosb (7.7a)
x
2
=r cosg sina cosb (7.7b)
˙ x
1
=r cosg sinb (7.7c)
˙ x
2
=r sing (7.7d)
The optimal control force using offline MPC is computed on a unit hypersphere,r= 1, considering
a discrete grid ofa=180

,170

,:::,180

,b = 0

,10

,:::,180

andg= 0

,10

,:::,180

. For
all states not on this hypersphere, linear scaling will be applied as explained earlier. Considering
homogeneity and state transformation will significantly reduce both MPC and NN computational
effort. The responses of the model subjected to the first 30 sec of the 1940 El Centro earthquake
excitation, using fixed-step discrete solverode5 in MATLAB with different time steps, are calculated
and presented in Table 7.6. It is clear that the results of the three time step durations do not differ
significantly; therefore,Dt= 0.01 sec will be used for all subsequent simulations in this section as
152
0 20 40 60 80 100 120 140 160 180
-5
0
5
10
15
20
25
30
35
40
θ [deg]
Control force / mass [m/s
2
]

decimated MPC
NN
MPC
Figure 7.32: Control force comparison for Levenberg-Marquardt-trained NNs decimated with
Dq = 10

Table 7.6: Performance comparison under 1940 El Centro excitation,Dq=10

, ode5, different
time steps
1940 El Centro
PGA= 3.41 m/s
2
Off.MPC
Dt=3.12510
4 Off.MPC
Dt=510
3 Off.MPC
Dt=110
2
Peak displ. 1
st
fl. [m] 0.0333 0.0332 0.0332
RMS displ. 1
st
fl.[m] 0.0085 0.0085 0.0085
Peak displ. 2
nd
fl. [m] 0.0504 0.0503 0.0504
RMS displ. 2
nd
fl.[m] 0.0120 0.0120 0.0120
Peak drift. 2
nd
fl. [m] 0.0253 0.0253 0.0253
RMS drift. 2
nd
fl. [m] 0.0047 0.0047 0.0047
Peak abs. accel. 1
st
fl. [m/s
2
] 3.0484 2.9916 2.9962
RMS abs. accel. 1
st
fl. [m/s
2
] 0.5048 0.5073 0.5195
Peak abs. accel. 2
nd
fl. [m/s
2
] 3.9969 3.9920 3.9997
RMS abs. accel. 2
nd
fl. [m/s
2
] 0.7356 0.7367 0.7372
Peak cont. force / bldg.weight 0.2411 0.2405 0.2396
RMS cont. force / bldg.weight 0.0481 0.0480 0.0476
Cost [m
2
/s
3
] 23.88 24.00 23.88
it is less computation demanding compared to smaller time steps. Consideringa,b andg as input
values and the optimal control force obtained from offline MPC as target values, a NN is constructed
as shown in Figure 7.33. In order to decide which NN is the best for the purpose of this section —
resulting in a small cost defined in (7.5) — the best training algorithm, the best number of hidden
153
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer
Input Layer
Hidden Layer
Output Layer

) (  u



) , , (    u
) , ( q q u 
q
q 
Figure 7.33: NN architecture for 2DOF problem
layers, the best number of neurons in each hidden layer and the best initial weighting should be
determined. As shown for the SDOF simulation, Levenberg-Marquardt backpropagation training
method is the fastest and the most suitable method for curve fitting and function approximation;
therefore, this section is focused on using this method.
7.4.1 NNs with One Hidden Layer
First, it is assumed there is only one hidden layer, and the number of neurons in the hidden layer
varies from 10 to 700. For each of these NNs, 5 different initial weights and biases are applied,
similar to the process for the SDOF example in the previous section. The NN-controlled response
of the 2DOF model, subjected to the first 30 sec of the 1940 El Centro earthquake, is calculated.
Figure 7.34 shows the error between the output of each trained NN and the target MPC results. The
corresponding costs are presented in Figure 7.35. Clearly, the smallest number of neurons that leads
to the smallest cost and error is 200 neurons and seed 3.
7.4.2 NNs with Two Hidden Layers
Since using more than one hidden layer is very time consuming and might not improve the
performance for small and simple problems (Beale et al., 2014), it was not studied for the SDOF
problem. However, since the input-output relation of the 2DOF model is much more complex than
154
10 50 100 200 300 400 500 600 700
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Error, MSE(NN - MPC) [N
2
]
number of neurons in the hidden layer

seed=1
seed=2
seed=3
seed=4
seed=5
Figure 7.34: Error between Levenberg-Marquardt-trained NN outputs and MPC target values, 2DOF
10 50 100 200 300 400 500 600 700
10
0
10
1
10
2
cost [m
2
/s
3
]
number of neurons in the hidden layer
Figure 7.35: Cost value of different NNs trained with Levenberg-Marquardt, 2DOF
155
10 20 30 40 50
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Error, MSE(NN - MPC) [N
2
]
number of neurons in each hidden layer

Figure 7.36: Error between Levenberg-Marquardt-trained NN (with 2 hidden layers) outputs and
MPC target values, 2DOF
the SDOF problem, it is possible that using two hidden layers will result in smaller error and better
performance. To investigate this possibility, in this section, it is assumed that there are two hidden
layers and that the number of neurons in both hidden layer are the same and vary from 10 to 50.
It should be noted that fewer than 10 neurons results in large costs, so they are not shown here.
Moreover, 10 different seeds are considered for initial weighting of each NN. Figure 7.36 shows the
error between the output of each trained NN and the target MPC results. The corresponding costs
are presented in Figure 7.37. Clearly, for 40 neurons in each hidden layer and seed 7 leads to the
smallest cost and also a relatively small error. Since the resulting control force is now a function of
4 states — displacement and velocity of two stories — it is difficult to illustrate its behavior relative
to all states; however, assuming a particular value for two of these states, the control force can be
shown realtive to the other two states;e.g., assuming the displacement of both stories are equal to
zero, the control force can be plotted as a surface function of the velocities of each story. Figure 7.38
shows the control force of the best NN with one hidden layer, while Figure 7.39 represents the
corresponding control force of the best NN with two hidden layers. As expected, the latter shows
more detail and more significantly nonlinearities in the surface.
156
10 20 30 40 50
10
0
10
1
10
2
10
3
cost [m
2
/s
3
]
number of neurons in each hidden layer
Figure 7.37: Cost value of different Levenberg-Marquardt-trained NNs with two hidden layers,
2DOF
To compare both of these graphs with their target graph, the offline MPC control force is shown in
Figure 7.41. Moreover, the corresponding CLQR control force is presented in Figure 7.40. One
can verify that the control designed by both NNs and MPC are very similar to each other, and are
significantly different from CLQR control force.
157
−0.1
−0.05
0
0.05
0.1
−0.1
0
0.1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
velocity
1
[m/s]
velocity
2
[m/s]
control force [MN]
Figure 7.38: Levenberg-Marquardt-trained NN with 200 neurons in the hidden layer and seed 3,
x
1
=x
2
= 0
−0.1
−0.05
0.1
0.05
0.1
−0.1
0
0.1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
velocity
1
[m/s]
velocity
2
[m/s]
control force [MN]
Figure 7.39: Levenberg-Marquardt-trained NN with 40 neurons in each of 2 hidden layers and
seed 7,x
1
=x
2
= 0
158
−0.1
−0.05
0
0.05
0.1
−0.1
0
0.1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
velocity
1
[m/s]
velocity
2
[m/s]
control force [MN]
Figure 7.40: CLQR-determined control force surface,x
1
=x
2
= 0
−0.1
−0.05
0
0.05
0.1
−0.1
0
0.1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
velocity
1
[m/s]
velocity
2
[m/s]
control force [MN] Figure 7.41: MPC-determined control force surface,x
1
=x
2
= 0
To compare the performance of these control designs, the controlled response of the 2DOF model
subjected to the 1940 El Centro earthquake excitation is computed. As presented in Table 7.7, both
online and offline implementations of MPC provide very similar performance and all maintain the
159
superiority of MPC over clipped LQR design. Moreover, the results of both NN designs are similar
to each other and to the MPC results.
Table 7.7: Performance comparison under 1940 El Centro excitation,ode5,Dt= 1 10
2
sec
1940 El Centro
PGA= 3.41 m/s
2
CLQR On.MPC Off.MPC NN
H=200
NN
H=40,H=40
Peak displ. 1
st
fl. [m] 0.0590 0.0340 0.0332 0.0314 0.0354
RMS displ. 1
st
fl.[m] 0.0154 0.0089 0.0085 0.0077 0.0087
Peak displ. 2
nd
fl. [m] 0.0972 0.0505 0.0504 0.0515 0.0549
RMS displ. 2
nd
fl.[m] 0.0217 0.0125 0.0120 0.0111 0.0123
Peak drift. 2
nd
fl. [m] 0.0382 0.0246 0.0253 0.0230 0.0256
RMS drift. 2
nd
fl. [m] 0.0072 0.0047 0.0047 0.0047 0.0048
Peak abs. accel. 1
st
fl. [m/s
2
] 6.3718 3.8505 2.9962 2.9431 3.0926
RMS abs. accel. 1
st
fl. [m/s
2
] 1.0822 0.5363 0.5195 0.5396 0.5219
Peak abs. accel. 2
nd
fl. [m/s
2
] 6.0397 3.8881 3.9997 3.6366 4.0373
RMS abs. accel. 2
nd
fl. [m/s
2
] 1.1374 0.7405 0.7372 0.7409 0.7514
Peak cont. force / bldg.weight 0.3290 0.2373 0.2396 0.2699 0.2596
RMS cont. force / bldg.weight 0.0506 0.0483 0.0476 0.0496 0.0478
Cost [m
2
/s
3
] 73.9452 25.0800 23.8800 25.2011 25.1086
7.5 Conclusions
NN-based semiactive control design, trained with the data drawn from MPC-based design, is
proposed in this study. The advantage of this approach is that a NN can learn the very nonlinear
regression of the optimal controllable damping forces, which are complex nonlinear functions of
the structure displacements and velocities. It is also capable of capturing the approximate behavior
of MPC with a few neurons.
Different training algorithms and network parameters are investigated in order to find the optimal
NN architecture and learning method that results in a control force similar to that of MPC and
reduces the objective function (root mean square (RMS) absolute acceleration). The performance of
the resulting NN control design is demonstrated by intensive numerical simulations for both SDOF
and 2DOF structural models subjected to earthquake excitation. NN results are also compared with
those of CLQR and MPC designs.
For the SDOF model, it is shown that a NN with only two neurons in the hidden layer only grossly
160
approximates the MPC control function, but it is capable of achieving a cost function similar to, and
even slightly better than, that of MPC. It is also shown that adding more neurons in the hidden layer
makes the output of NN more similar to the very nonlinear MPC control. Moreover, decimation is
performed on the MPC data for the SDOF to investigate the potential use of NNs for more complex
models when only a few data points can be obtained using MPC. The numerical results show that
NN cost is 17% smaller than the cost of CLQR, 10% smaller than decimated MPC and even 2 5%
smaller than non-decimated MPC.
For the 2DOF model, due to a larger number of inputs and greater complexity, more neurons are
needed to achieve a desired performance. In this case, using 200 neurons in one hidden layer or 40
neurons in each of two hidden layers result in a cost that is about 65% smaller than that of CLQR,
while the peak required control force is also about 25% smaller than that of CLQR.
The resulting NNs for both SDOF and 2DOF models are found to be very promising in the reduction
of the objective function and can even outperform the MPC target; however, further studies are
required to demonstrate the performance of the proposed method for more complex and realistic
building models subjected to different excitations.
161
Chapter 8
Concluding Remarks and Future Directions
It is imperative for the civil engineering community to develop and implement strategies to miti-
gate structural responses in both weak and strong excitations, reducing the absolute acceleration
responses in weak-to-moderate excitations to prevent nonstructural and building content damage
and ensure occupant comfort, and decreasing the drift responses in strong excitations to avoid
structural damage and possible structural collapse — for both newly designed structures and the
retrofit of existing structures. Optimal linear control laws are the most common applied control
strategies, designed for a specific quadratic cost functions with a design excitation level; therefore,
their performance might not be optimal in the presence of different magnitude excitation.
This dissertation starts with comparing different “optimal” control laws obtained by different
methods. Interestingly, for the scalar problem investigated here, the Hamilton-Jacobi-Bellman (HJB)
derived “optimal” control law, which is the same as the optimal nonlinear control for free response
of a system without excitation but non-zero initial condition, is different from the optimal nonlinear
control for a Gaussian white noise excitation found from the Fokker-Planck-Kolmogorov (FPK)
and Monte Carlo simulation (MCS) results. The certainty equivalence property clearly does not
hold, likely because the system is non-Gaussian with a non-quadratic cost function. Although the
control laws are different, the resulting costs found by these methods are almost the same.
Next, an example of a control strategy that can achieve both service-level and life safety-level
objectives with one general control law is presented. This particular control strategy is not intended
to be optimal, but to show that a strategy exists to reduce drift in the rare extreme events and reduce
absolute acceleration in the more frequent weak-to-moderate ground motions. The performance of
the time-varying control strategy is demonstrated by intensive numerical simulations for a single
degree-of-freedom (SDOF) building model subjected to different excitations, showing that it can
162
successfully achieve both life safety (drift reduction) and serviceability (acceleration reduction)
objectives at different excitation levels. However, since it is a simple unoptimized example,
procedures for deriving and optimizing the control strategies that achieve both objectives are
investigated in the rest of this dissertation.
In the next part of this dissertation, a new “optimal” nonlinear control law that can achieve different
objectives in different levels of excitations is derived analytically by solving the HJB equation. The
cost function is a 6
th
-order polynomial of the states and quadratic in the control force such that
corresponding HJB equation has a solution that results in a cubic optimal control law: the sum
of a linear term that is related to a linear quadratic regulator (LQR) problem and is effective in
small excitations, and a cubic nonlinear term that is dominant when the excitation level is large.
The main advantage of this nonlinear controller is that it reduces acceleration response to weak
excitation and focuses on drift reduction in strong excitation. In one control law, both objectives are
satisfied. The linear control gain is obtained by solving an algebraic Riccati equation; to find the
optimal cubic gain, an optimization is performed using an algorithm like MATLAB’sfminsearch.
For a zero nonlinear control gain, both the control law and the cost function revert back to the
optimal LQR control problem. The performance of the nonlinear control strategy is demonstrated by
numerical simulations of SDOF and a base-isolated multiple degree-of-freedom (MDOF) building
models subjected to different excitations, showing that it can successfully achieve both life safety
(drift reduction) and serviceability (acceleration reduction) objectives at different excitation levels.
Further, it is shown that the promising performance is still valid when semiactive control force is
applied with or without control force saturation. Also, the robustness of the proposed controller is
verified by showing the promising performance in the presence of inaccurate assumptions of mass,
stiffness and damping of the model, as well as in various excitations. The Kanai-Tajimi (KT) filter is
implemented to produce several synthetic earthquake-like excitations to be applied to the structure.
Finally, the Kalman filter (KF) is implemented into the problem to show that it is successful in
predicting the response when some measurement noise exists. Moreover, to assess the optimality of
the proposed method and compare it with other possible methods, several other control strategies
(e.g., gain scheduling and model predictive control) are investigated and the results are demonstrated
in this dissertation.
To investigate the application of gain scheduling to find a controller that can achieve the two
aforementioned objectives, two optimal linear control laws are obtained for two different levels of
excitation, namely LQR
d
for strong excitation (2El Centro), and LQR
a
for weak excitation (0.1El
Centro). Gain-scheduled control strategies are applied to design a control with a performance similar
to those of LQR
a
and LQR
d
in the presence of weak and strong excitations, respectively. Moreover,
several transition functions in gain-scheduled control designs are investigated to guarantee a desired
163
performance in the presence of moderate excitations. The performance of the designed gain-
scheduled control strategy is demonstrated by intensive numerical simulations for a SDOF building
model and a base-isolated MDOF building model subjected to different magnitude excitations. Both
active and semiactive control strategies are studied with and without control force saturation. These
results indicate that it can successfully achieve both life safety (drift reduction) and serviceability
(acceleration reduction) objectives at different excitation levels.
Next, this dissertation presents the application of a model predictive control (MPC) strategy to
find the “optimal” nonlinear control that minimizes a nonquadratic cost function and can achieve
significant improvements in both serviceability (i.e., acceleration mitigation) in weak-to-moderate
excitations and life safety goals (i.e., drift mitigation) in extreme events. The performance of this
control strategy is demonstrated by extensive numerical simulations for SDOF and base-isolated
MDOF building models subjected to different scaled versions of the El Centro earthquake. For the
SDOF problem studied in this research, the results show that the nonquadratic MPC control strategy
closely tracks the optimal LQR control in the weak excitation, while providing an additional 20%
reduction in drift relative to the LQR design in the strong excitation. Moreover, the maximum
required control force for MPC in this case is equal to 31% of the weight of the structure, while
LQR needs 41% of the weight of the structure as the maximum required control force. Clearly, the
nonquadratic MPC control strategy can focus on different objectives in different level of excitations
with smaller required control force compared to LQR. Applying other historical earthquakes, like
the 1994 Northridge earthquake, also shows promising performance. Introducing a control force
saturation, at 15% of building weight, to accommodate control device capacity for the LQR problem;
and defining a constraint on the control force for the MPC problem also shows that the nonquadratic
MPC saturated control can still achieve both objectives in different level of excitations; i.e., the
nonquadratic MPC saturated control can provide drift reduction superior to saturated LQR control
and requires smaller root mean square (RMS) control force. In a different approach, a quadratic
cost function with some constraints on maximum allowable drift is studied. The results show
that, although the constrained MPC control can reduce drift better than constrained LQR, it also
causes the absolute acceleration to be significantly larger than both LQR and the nonquadratic MPC
designs. This shows that the nonquadratic MPC strategy outperforms the constrained quadratic
MPC strategy studied herein. To extend the results and investigate if it can be applied to more
complicated and realistic building models, a base-isolated MDOF building model is studied. The
results show that the nonquadratic MPC control can successfully achieve both base drift reduction
and roof absolute acceleration reduction objectives at different excitation levels. However, further
study is needed to investigate other possible choices of nonquadratic term(s) and their effects on the
results.
164
It has been demonstrated via intensive numerical results that all nonlinear control strategies studied
in this research (e.g., HJB, GS and MPC) can successfully achieve both life safety (drift reduction)
and serviceability (acceleration reduction) objectives at different excitation levels; however, it
should be noted that their results are not identical: for any particular problem, depending on the
desired performance metric, one or some of these methods could be applied. Moreover, experimental
verification of these results is necessary to guarantee that they can be applied to real-world structures.
Additionally, there are still many necessary aspects of this work that need to be studied, properties
to be investigated, and numerous open questions to be answered in order to confirm the concept
for realistic structure models and a wider array of excitations and applications. First, the weak and
strong excitation levels should be determined based on the characteristics of the building and its
contents, the history of previous excitations at the location, the local building code and so forth.
Further, the desired performance in weak and strong excitations should be determined based on
building code (e.g., maximum allowable drift could be obtained from ASCE 7-10 for any specific
building based on its importance factor, seismic risk category, etc.). Further, real structures are
complex, often modeled with finite element approaches with thousands of degrees of freedom.
Therefore, all of the nonlinear control approaches must be evaluated to determine whether they
can be easily applied to the variety of models that civil engineers use. Also, in this dissertation, it
is assumed there is only one control device at the base level; however, it should be investigated
whether more control devices in different levels will improve the performance. Moreover, stability
of GS control strategy should be investigated, particularly for rapidly changing dynamic systems.
There are also some equations cast in specific forms (e.g., (4.32) to find the tuning parameter in
HJB control, (5.9) for the transition between two optimal LQR controls in gain scheduling control,
(6.11) for the nonquadratic objective function for MPC, etc.) for which it should be investigated if
there are other possible forms that could improve the performance.
Finally, the last part of this dissertation presents a NN-based semiactive control design trained
with the data drawn from MPC-based design. The advantage of this approach is that a NN can
learn the very nonlinear regression of the optimal controllable damping forces, which are complex
nonlinear functions of the structure displacements and velocities. It is also capable of capturing the
approximate behavior of MPC with only a few neurons. Different training algorithms and network
parameters are investigated in order to find the optimal NN architecture and learning method
that results in a control force similar to that of MPC and reduces the objective function (RMS
absolute acceleration). The performance of the resulting NN control design is demonstrated by
intensive numerical simulations for both SDOF and 2DOF structural models subjected to earthquake
excitation. The NN results are also compared with those of clipped LQR (CLQR) and MPC designs.
For the SDOF model, it is shown that a NN with only two neurons in the hidden layer only grossly
165
approximates the MPC control function, but is nevertheless capable of achieving a cost function
similar to, and even slightly better than, that of MPC. It is also shown that adding more neurons in
the hidden layer makes the output of NN more similar to the very nonlinear MPC control. Moreover,
decimation is performed to MPC data for the SDOF to investigate the potential use of NNs for more
complex models when only a few data points can be obtained using MPC. The numerical results
show that the NN cost is smaller than the cost of CLQR, decimated MPC and even non-decimated
MPC. For the 2DOF model, due to the larger number of inputs and greater complexity, more neurons
are needed to achieve a desired performance. In this case, using 200 neurons in one hidden layer
or 40 neurons in each of two hidden layers result in a cost that is about 65% smaller than that of
CLQR, while the peak required control force is also about 25% smaller than that of CLQR. The
resulting NNs for both SDOF and 2DOF models are found to be very promising in the reduction
of the objective function and can even outperform the MPC target; however, further studies are
required to demonstrate the performance of the proposed method for more complex and realistic
building models subjected to different excitations.
166
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Abstract (if available)

Abstract Various types of controllers have been studied and implemented to mitigate the effects of excitations from natural hazards. Linear control laws are most often applied, for active devices as well as part of the controller for semiactive devices, primarily due to their simple design and implementation. Yet, linear structural control strategies generally cannot focus on different objectives in different excitations, a goal that has real meaning in many structural control applications. For example, minimizing structural drift is necessary during strong earthquakes to mitigate damage to the structure, yet occupant comfort and safety of building contents in the much-more-frequent moderate earthquakes demands reductions in absolute accelerations in the structure. Minimizing drift and acceleration in different excitations are often somewhat competing objectives and cannot be achieved with conventional linear strategies. Therefore, it is imperative to use nonlinear control strategies instead. To design an optimal nonlinear controller for a linear structure, it is well established that minimization of a nonquadratic cost function is required, which will lead to solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation. However, finding the exact analytical solution of the HJB equation is very difficult, and may only have a solution when the cost function is cast in a particular form. ❧ The first part of this study presents a comparison of some of the analytical and numerical methods for finding optimal nonlinear controllers, particularly for cost functions that are even-order powers of the states and quadratic in the control. A scalar model (i.e., a scalar state-space equation) is used for the comparisons since analytical solutions exist for some of the methods. ❧ As most of the HJB-derived optimal nonlinear controllers proposed by other researchers are focused on the single objective of reducing structural drift, the next part of this study presents an analytical approach to solve the HJB equation and find the optimal nonlinear control law for a nonquadratic cost function with higher-order polynomials. The resulting optimal nonlinear control law can be formulated as the summation of a linear term, which is related to a linear quadratic regulator (LQR) problem designed to be effective in reducing one performance metric in small excitations, and some nonlinear terms that are dominant to reduce the other performance metric when the excitation level is large. Thus, it can achieve significant improvements in both serviceability (i.e., acceleration mitigation) in weak and moderate excitations and life safety goals (i.e., drift mitigation) in extreme events—all in one general control law. ❧ Next, the application of gain scheduling to achieve the two aforementioned objectives is investigated. Two different optimal linear controls are designed, each to reduce a different response metric

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Creator Hemmat-Abiri, Elham (author)

Core Title Structural nonlinear control strategies to provide life safety and serviceability

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering (Structural Engineering)

Defense Date 10/21/2015

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

Tag gain scheduling,Hamilton-Jacobi-Bellman,life safety and serviceability,model predictive control,multi-objective control,neural network,non-quadratic cost function,OAI-PMH Harvest,semiactive control,structural nonlinear control

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