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

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

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Content Computationally Efficient Design of Optimal Strategies
for Passive and Semiactive Damping Devices in Smart Structures
by
Mahmoud Kamalzare
Advisors
Prof. Erik A. Johnson
Prof. Steven F. Wojtkiewicz
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)
December 2014
Copyright 2014 Mahmoud Kamalzare
Dedication
To my loving wife and beloved parents.
ii
Acknowledgements
The author acknowledges the partial financial support of this research by the National Science Foundation
(NSF) through various grants: CMMI 08-26634, 11-00528, 11-33023, and 13-44937 and the support of a
Viterbi Doctoral Fellowship at University of Southern California (USC).
I would like to express my deepest thanks to my advisors, Profs. Erik A. Johnson and Steven F. Wojtkiewicz,
for their support, guidance, and friendship throughout my PhD education. I have been always impressed with
their knowledge, mentorship, and their very kind personalities. I learned a lot from them.
I would like to gratefully thank my committee members, Profs. Sami F. Masri, Ketan Savla, and Edmond A.
Jonckheere, for their invaluable suggestions and advices on this research.
I would like to have special thanks to my dear wife, Elham Hemmat Abiri, for her very kind love, help,
support, and encouragement during this PhD journey. I share this achievement with her.
I would like to thank my former and current colleagues especially (Drs. and soon-to-be Drs.) Tat S. Fu,
Charles Devore, Elham Hemmat Abiri, Wael Elhaddad, Leonardo Velderrain Chavez and Subhayan De for
making these years so memorable.
Finally, I would like to express my sincere gratitude to my family, especially my parents, Nematollah
Kamalzare and Maryam Elmi, who provided me with all the opportunities possible. I would have never made
it through this journey without their encouragement, love and support.
iii
Abstract
In recent years, significant improvements in memory capacity and processing speed of computers have
provided the ability of modeling and analyzing large and complex dynamical systems. These systems
usually consist of many elements, of which some have nonlinear properties. Standard nonlinear solvers
ignore the localized nature of the nonlinearities when computing responses, which can result in a very
time-consuming process. However, since the nonlinearities are often limited to only a few of the many
degrees of freedom (DOFs), an alternate method has been developed in which the nonlinear perturbation
dynamics are excluded from the nominal linear system and evaluated based on the response of the nominal
system. This reduces the high-order system to a much lower-order system of nonlinear V olterra integral
equations (NVIEs), which provides a very computationally efficient solution. The total response of the system
can be then easily calculated using superposition.
This study adapts the methodology to provide a fast and computationally inexpensive method for designing
control strategies implemented in — but not limited to — smart building structures. The development of
control strategies for controllable passive dampers, i.e., “semiactive” damping devices, is complicated by
the nonlinear and dissipative nature of the devices and the nonlinear nature of the closed-loop system with
any feedback control. Control design for nonlinear systems is often achieved by designing a control for
a linearized model since strategies for linear systems are straightforward. One such approach is clipped-
optimal control in which the desired damper forces are determined from an optimal controller (e.g., linear
quadratic regulator (LQR), linear quadratic Gaussian (LQG), H
2
, etc.), which is designed assuming that the
damping devices are fully linear actuators that can exert any forces (dissipative or non-dissipative), and a
secondary bang-bang controller commands the controllable damper to exert forces as close as possible to the
desired forces. However, designs using any linearized model generally results in suboptimal (and sometimes
lousy) performance because the linear actuator assumption differs from the actual implementation with a
dissipative damping device. Thus, one must generally resort to a large-scale parameter study (or performing
an optimization algorithm) in which the nonlinear system is simulated many times to determine control
strategies that are actually optimal for the nonlinear controlled closed-loop system. Herein, it is demonstrated
how the proposed approach can significantly decrease the computational burden of a complex control design
study for controllable dampers.
Next, this study expands the applicability of the proposed method by demonstrating that the approach can also
be adapted to accommodate the more realistic cases when, instead of full-state feedback, only a limited set of
noisy response measurements are available to the controller, which requires incorporating a Kalman filter
estimator, which is linear, into the nominal linear model. Furthermore, since the primary controller is rarely
designed using a high-order model (because it is impractical due to numerical difficulties, as well as often
unnecessary since high-order models, such as complex finite element structure models, have high frequency
dynamics that remain mostly unexcited by an external disturbance), to bring the method to full maturity, a
reduced-order model for control design is incorporated with the full model to simulate semiactively controlled
structural responses using the proposed NVIE approach. Finally, it is explained briefly how the proposed
approach can be implemented when uncertainties are involved in the system.
This dissertation provides a broad and comprehensive methodology for designing control strategies for smart
structures using the proposed computationally efficient method. Numerical results confirm the accuracy,
stability, and computational efficiency of the proposed simulation methodology and specifically show about
two orders of magnitude speed up relative to the conventional solvers for the typical semiactive design
parameter studies.
iv
Table of Contents
Dedication ii
Acknowledgements iii
Abstract iv
List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Structural Control 1
1.2 Passive Control 3
1.3 Active Control 5
1.4 Semiactive Control 6
1.5 Magnetorheological Fluid Damper: a Semiactive Control Device 9
1.6 Computational Burden of Semiactive Control Design 10
1.7 Fast Analysis of Systems with Local Modifications 11
1.8 Overview of this Dissertation 13
2 Fast Analysis of Linear Systems with Local Nonlinearities/Modifications 15
2.1 Introduction 15
2.2 Efficient Response Computation 16
2.2.1 Nominal System 16
2.2.2 Modified System 16
2.2.3 Response Computation 16
2.2.4 Solution for NVIEs 17
2.2.5 Fast Solution of NVIEs using Fast Fourier Transform (FFT) 20
2.3 Sensitivity of Responses 22
3 Efficient Optimal Design of Passive Structural Control with Application to Isolator Design 23
3.1 Introduction 23
3.1.1 Background in Optimal Base Isolation Design 24
3.2 Methodology: Optimization Procedure 25
3.3 Numerical Example: Optimal Design of Isolation Hysteresis for Isolated Building 26
3.3.1 Model Description 26
3.3.2 Baseline Lead Rubber Bearing (LRB) Design 30
3.3.3 Sensitivity Formulation 31
3.3.4 Design of the Optimal Base Isolation 33
3.3.5 Preliminary Optimization over Q
y
and k
pre
34
3.3.6 Optimization over Q
y
, k
pre
and k
post
35
3.3.7 Timing and Accuracy of the Proposed Approach 37
3.3.8 Investigation of the Transitional Region of the Bouc-Wen Model 38
v
3.3.9 Case Study: Convergence Rate of the Nonlinear V olterra Integral Equation (NVIE)
Approach for a Strictly Bilinear Hysteretic Model 40
3.4 Conclusions 42
4 Computationally Efficient Design of Optimal Strategies for Semiactive Control Devices 43
4.1 Introduction 43
4.2 Methodology 45
4.2.1 Clipped-Optimal Control 45
4.2.2 Semiactive Control Design Parameter Study 47
4.3 Numerical Example 49
4.3.1 Model Description: Base Isolated Frame Structure 49
4.3.2 Baseline Performance: lead rubber bearing (LRB) System 51
4.3.3 Control with Ideal Smart Controllable Dampers 52
4.3.4 Parameter Study: Design of the Optimal Controller 53
4.3.5 Timing and Accuracy of the Proposed Approach 54
4.3.6 Design an Optimal Semiactive Control Strategy Using an Optimization Algorithm 56
4.3.7 Semiactive Performance Compared to a Linear Passive Viscous Damper 61
4.4 Conclusions 64
5 Computationally Efficient Design of Semiactive Structural Control in Presence of Measure-
ment Noise 66
5.1 Introduction 66
5.2 Methodology 67
5.2.1 Equations of Motion, Sensor Measurements, and Regulated Output 67
5.2.2 Incorporating Kalman Filter 68
5.3 Numerical Examples 70
5.3.1 Ground Excitation 70
5.3.2 State-Space Model 72
5.3.3 Augmented System with Kalman Filter 73
5.3.4 Accuracy and Computational Cost of the Semiactive Control Design 74
5.3.5 Effects of Sensor Noise Magnitude and Kalman Filter Tuning on Estimated Outputs 79
5.4 Conclusions 81
Appendix 5.A Kalman Gain and its Corresponding Riccati Equation 82
Appendix 5.B Impulse Response of the System Augmented by Kalman Filter 85
6 Rapid Controllable Damper Design for Complex Structures with a Hybrid Reduced-Order
Modeling/Simulation Approach 87
6.1 Introduction 87
6.2 Methodology 90
6.2.1 Model Reduction Techniques 90
6.2.1.1 Balanced Reduction 91
6.2.1.2 Guyan Reduction 92
6.2.1.3 Modal Truncation 93
6.2.2 Incorporating a Kalman Filter to Estimate the Reduced-Order Model States 94
6.3 Numerical Examples 94
6.3.1 100-DOF Base-Isolated Structure Example 95
6.3.1.1 Model Description and Ground Excitation 95
6.3.1.2 Effectiveness of Controllable Damping Designed using a Reduced Model 95
vi
6.3.1.3 Design Point, Accuracy, and Computation Time 98
6.3.1.4 Investigation of the System Reliability 100
6.3.2 High-Order Structure Example with Three Roof-Mounted Controllably-Damped
Tuned Mass Dampers (TMDs) 103
6.3.2.1 Model Description 103
6.3.2.2 Wind Excitation 105
6.3.2.3 Baseline Performance: a Linear Passive Viscous Damper 105
6.3.2.4 Reduced-Order Model and Kalman Filter Design 106
6.3.2.5 Design Point, Accuracy, and Computation Time 107
6.4 Conclusions 110
7 Computationally Efficient Control Design for a Smart Structure with Uncertainty 112
7.1 Introduction 112
7.2 Methodology 113
7.2.1 System of Equations along with Uncertainty 113
7.3 Numerical Example 114
7.3.1 Model Description 114
7.3.2 Model Uncertainty 115
7.3.3 Computational Cost 116
7.4 Conclusions 118
8 Concluding Remarks and Future Directions 120
Bibliography 123
vii
List of Tables
1.1 Passive, active, and semiactive control comparison (adapted from Erkus (2006)) 8
3.1 Design points for different exponent n values for the El Centro earthquake. 39
3.2 root mean square (RMS) base drift and roof acceleration changes (%) when the design points
are evaluated at different exponent n values. 40
4.1 Cost ratio comparison. 56
4.2 Design points at different RMS force levels under the El Centro excitation. 59
4.3 Design points at different RMS force levels under the Northridge excitation. 60
4.4 Design points at different RMS force levels under the El Centro excitation using the passive
linear viscous damper as a baseline. 64
5.1 RMS responses comparison. 75
5.2 Design points at different RMS force levels. 76
5.3 Cost ratio comparison. 79
5.4 Base drift error mean [%] over 10,000 realizations. 80
5.5 Roof absolute acceleration error mean [%] over 10,000 realizations. 80
5.6 Device force error mean [%] over 10,000 realizations. 81
6.1 Computation (CPU) time and memory usage for solving the primary control gain Riccati
equation. 89
6.2 Computational cost comparison for 1000 simulations (averages of 5–10 runs since individual
timings vary roughly up to 5%). 102
6.3 Computational cost comparison for 1000 simulations (averages of 5–10 runs since individual
timings vary up to about 5%). 110
7.1 RMS and peak of base drift and roof acceleration of a Monte Carlo simulation 116
7.2 Cost ratio comparison. 118
viii
List of Figures
1.1 A passive control block diagram 4
1.2 An active control block diagram 5
1.3 A semiactive control block diagram 7
1.4 An magnetorheological (MR) fluid damper (adapted from Soong and Spencer (2002)) 9
1.5 MR fluid in absence and presence of a magnetic field (adapted from Erkus (2006)) 10
1.6 Force-velocity relation of an MR damper at different voltage level (adapted from Erkus (2006)) 10
2.1 Trapezoidal rule 18
2.2 Newton’s method 19
2.3 Subdivision of the convolution space: squares by fast Fourier transform (FFT) and triangles
by Trapezoidal rule (adapted from Gaurav et al. (2011)) 21
3.1 100-DOF base-isolated structure. 27
3.2 Bilinear and Bouc-Wen models to represent the hysteretic behavior of materials. 28
3.3 Base drift response sensitivity to the design parameters when subjected to the 1940 El Centro
ground motion at the design point. 32
3.4 Cost contours as a function of two design parameters for two historical earthquakes. 35
3.5 Contour lines of the cost as a function of the design parameters for two earthquakes. The
vertex of the cutout shows the optimal design location. 36
3.6 Structural responses to the El Centro earthquake using the design point isolation. 37
3.7 Single hysteresis loop using the optimal base isolator found for different exponents n when
building is subjected to the El Centro earthquake. 39
3.8 Newton’s method convergence for a bilinear function. 41
4.1 Clipping algorithm for “ideal” semiactive damper. 46
4.2 Semiactive control block diagram. 47
4.3 Dimension of optimization space. 48
4.4 11-story base-isolated frame model. 50
4.5 Schematic view of an LRB device. 51
4.6 Modified cost function. 53
4.7 Parameter study procedure. 53
4.8 RMS of the semiactive responses relative to those with the LRB. 54
4.9 Base drift (left) and absolute roof acceleration (right) responses to the 1940 El Centro
earthquake using the design point controller 55
4.10 Design points at different RMS force levels under the El Centro excitation 59
4.11 Design points at different RMS force levels under the Northridge excitation 60
4.12 Base drift (relative to LRB) and corresponding peak force (relative to building weight) under
El Centro and Northridge excitations for different weight-normalized RMS force levels 61
ix
4.13 Responses and device force of the system with a linear viscous damper under El Centro
excitation 62
4.14 Design points and response improvements compared to the viscous damper performance
under the El Centro excitation. 63
5.1 Frequency content of design earthquakes and Kanai-Tajimi shaping filter (adapted from
Ramallo et al. (2002)). 71
5.2 Time history of the applied excitation 71
5.3 Frequency content of the applied excitation 72
5.4 Combined filter/structure model for semiactive control design using clipped-optimal controller. 74
5.5 RMS semiactive responses relative to those with the LRB. 76
5.6 Actual and estimated roof absolute acceleration response and the measurement noise signal. 77
5.7 Estimated base drift response to the generated excitation. 78
5.8 Commanded device force. 78
6.1 Block diagram of a semiactive design 90
6.2 Hankel singular values of the system. 96
6.3 Relative base drift error for various reduction techniques. 96
6.4 Relative roof acceleration error for various reduction techniques. 97
6.5 Base drift with the full and reduced-order design models. 99
6.6 Roof absolute acceleration: full and reduced-order designs. 99
6.7 Base drift calculated at the design point using the proposed approach andode45. 100
6.8 Roof absolute acceleration calculated at the design point using the proposed approach and
ode45. 101
6.9 Reliability contours of the controlled system 103
6.10 1623-DOF structural model with three controllably-damped roof-mounted Tuned mass
dampers (TMDs) (adapted from Wojtkiewicz and Johnson (2014)) 104
6.11 Hankel singular values of the 1623-DOF structural model. 107
6.12 RMS controllable damping responses relative to those with the passive viscous damper 108
6.13 Roof-center TMD (x-direction) stroke with the full and reduced-order models. 108
6.14 Top-floor acceleration at the southwest corner in the x-direction with the full and reduced-
order models. 109
7.1 Uncertainty analysis using a traditional solver. 117
7.2 Uncertainty analysis using the proposed approach. 118
x
Chapter 1
Introduction
1.1 Structural Control
Civil structures are often exposed to strong excitations due to wind gusts, earthquakes, tsunamis and other
sources and are prone to severe damage, or even failure, if their responses are not controlled and mitigated
properly. Examples of such behavior include: the collapse of Tacoma Narrows bridge under 40 mile-per-
hour winds in November 1940; the severe damage of some buildings and bridges in Northridge and Kobe
earthquakes occurred in January of 1994 and 1995, respectively; and the Fukushima Daiichi nuclear power
plant disaster of March 2011 when a tsunami triggered by the Tohoku earthquake reached the facility resulting
in a meltdown of several of the plant’s nuclear reactors.
Structural control is an active research topic in civil engineering with the aim of devising methods to
mitigate the response of structural and nonstructural components of a structural system that is subjected
to strong excitations such as winds or earthquakes. Designing control strategies for structures is generally
accomplished by modifying mass, stiffness, or damping of some structural components or installing some
devices (such as passive, active, semiactive, and hybrid devices) to apply forces into the structure and disperse
the structural/mechanical energy.
One of the pioneers in structural control is John Milne, a professor of engineering in Japan, who installed ball
bearings under a small house (and basically made the first base-isolated building 100 years ago) to prevent the
1
earthquake energy from being transferred into the building (Housner et al., 1997). There was little progress in
the structural control field from the beginning of the twentieth century until World War II when many new
concepts were developed and implemented arising from military applications. The application of control
concepts started to emerge in civil engineering a few decades later in the 1960s and 1970s. Since then, the
structural control field has been rapidly evolved and expanded.
Thus, the objective of structural control (SC) is to minimize structural damage and maintain the safety of
occupants by reducing the effect of external forces (such as wind, earthquake, and so on) on the structure.
The direction, magnitude, and duration of these external excitations vary from one event to another — most
often with random characteristics. Structural control provides methods and algorithms to protect the structure
against those possibly devastating events. Structural control can be utilized for a wide variety of applications
such as mitigating the effect of natural hazards (e.g., severe winds or earthquakes) on building structures;
preventing damage or possible failure of infrastructures such as dams or bridges; increasing the safety of the
critical structures such as hospitals or nuclear plants; and increasing and insuring the robustness of aircraft
components.
The author encourages the interested reader to see the two excellent papers by Housner et al. (1997) and Soong
and Spencer (2002), which provide very detailed reviews of the state-of-the-art and state-of-the-practice of
SC systems.
A typical way of classifying control strategies is to categorize them into passive, active, and semiactive
methods. Each of these has its own advantages and disadvantages. Characteristics of the system to be
controlled determine which strategy is the “best” candidate to mitigate the response of the system. A brief
description of each control strategy is discussed in the following sections. It is worth noting that some people
also consider a fourth group of controllers, hybrid control, when two or more strategies are combined into one
system (for example, as noted by Housner et al. (1997), an active control system can be added to a passive
control system to improve the performance of the control strategy or, alternatively, a secondary passive system
can be augmented to an active system to reduce external energy requirement).
2
1.2 Passive Control
Passive strategies use the most well-known devices for protecting structures. These passive elements include
linear/nonlinear isolators and dampers installed in the system to mitigate structural vibrations. Various types
of passive systems have been designed and implemented on real structures, including metallic yielding
elements (Kelly et al., 1972, Skinner et al., 1975, 1980, Whittaker et al., 1991, Skinner et al., 1993) which
dissipate energy through inelastic deformation; friction dampers (Pall and Marsh, 1982, Filiatrault and Cherry,
1987, 1990) that convert energy to heat; viscous fluid dampers (Pekcan et al., 1999, Terenzi, 1999, Main and
Jones, 2002, Krenk and Høgsberg, 2005, Lin and Chopra, 2003, Makris et al., 1993a,b); tuned mass dampers
(Den Hartog, 1947, Villaverde and Martin, 1995); and tuned liquid dampers (Fujino et al., 1992, Sun et al.,
1989).
Passive strategies are designed with two main approaches. The first tries to isolate the structure from the
base (or other source of excitations) and prevents the energy from being transferred into the system. The
first modern implementation of base isolated systems was performed in New Zealand in the mid-1970s
for a rail bridge across the South Rangitikei River (Buckle, 2000). Presently, isolation systems are widely
used for various types of structures and “include elastomeric and sliding bearings with/without damping
mechanisms” (Buckle, 2000). Base isolation systems (Skinner et al., 1993) are among the most acknowledged
and implemented methods for passive control of civil structures. In these systems, most of the ground
vibration energy is reflected; the energy that is transmitted to the structure is absorbed primarily by the base
layer of the structure using nonlinear elements such as lead rubber bearing (LRB) and friction pendulum
bearings (Skinner et al., 1993, Naeim and Kelly, 1999); since the superstructure is uncoupled from the ground,
it will experience minimum internal motion. This mitigates the probability of damage occurrence in structural
and non-structural components of the superstructure.
The second group of passive devices dissipates the kinetic energy either by converting it to thermal energy or
by transferring the energy to some noncritical mode vibrations. Some of those devices which directly dissipate
energy are displacement-dependent devices and have hysteretic characteristics; e.g., yielding and deforming
nonlinear metals (U-shape steel dampers, metallic dampers) and sliding elements (friction dampers). Some
others are velocity-dependent devices and have viscoelastic properties; e.g., viscous fluid dampers, fluid
3
orifice dampers, etc. (Buckle, 2000). Tuned mass dampers (TMDs) and tuned sloshing dampers are among
those which transfer the kinetic energy to less important modes of vibration. The John Hancock Tower built in
1976 in Boston is one of the first real-world structures to use a TMD system (Fu, 2009); subsequently, TMDs
have become one of the most common passive systems, effectively implemented widely in many structures.
However, it is worth noting that, although TMDs are very promising in damping the structural vibration of
high-rise buildings and towers due to wind loads, they are not very effective in seismic applications (Buckle,
2000).
A block diagram of a passive control system is shown in Figure 1.1. Some of the main advantages of passive
controllers can be summarized as:
Passive controllers are simple to design;
The response of passively controlled systems is usually very robust and predictable;
Passive systems can not inject energy into the system and, therefore, never make a stable system
unstable;
Passive devices do not require an external source of energy, resulting in minimum operational cost. In
addition, the passive controller keeps working even if electricity outage happens and no external power
is available. Also, in the absence of a sensor network, maintenance cost is negligible.
Structure
+
Passive Device
Excitation Outputs
Structure
Excitation Outputs
Control Force Measurements
State
Estimator
Control
Strategy
Actuator(s)
States Desired Signal
Structure
+
Semiactive Device
Excitation Outputs
Measurements
State
Estimator
Dissipative
Control
Strategy
States
Desired Signal to Modify
Device Characteristics
Passive Control
Active Control
Semiactive Control
Figure 1.1: A passive control block diagram
Despite all of these advantages, their performance is limited. When a passive device is designed for some
specific criteria and installed in the system, its mechanical properties, such as stiffness and damping, can not
generally be tuned or modified; whereas, active and semiactive controllers, known as “smart” controllers,
have the capability to change their properties to adapt themselves to the behavior of the structure, which can
result in a significantly better performance and applicability.
4
1.3 Active Control
Active control systems consist of a network of sensors, one or more computer(s), and one or more actuator(s).
The sensors installed in the structure collect the global response of the structure and feed them back to the
main computer(s). The sensors’ information is then analyzed by the computer(s) and, based on the design
control law, a command signal regarding the magnitude and direction of the required control force(s) is
sent to the actuator(s). The control strategy is designed beforehand based on some control objectives and
is often obtained through the solution of an optimization problem of minimizing a linear or nonlinear cost
function involving the structural response and the required control energy. Actuators, the last elements of
this control cycle, are generally large devices, powered by an external electrical source, which apply the
desired control force to reduce the effect of the external excitations such as wind or earthquake and mitigate
these potentially destructive responses. Examples include active bracing systems, hydraulic actuators, and
active mass dampers (AMDs). The Kyobashi Seiwa building in Japan is the first real-world building with an
AMD device installed (Kobori, 1990, 1996, Sakamoto and Kobori, 1995). Subsequently, many active control
systems have been tested and implemented in high-rise buildings and bridges — more than 50 buildings in
Japan have been equipped with AMDs (Yamamoto and Sone, 2014).
A block diagram of a typical active control system is shown in Figure 1.2. The interested reader is encouraged
to see the papers by Housner et al. (1997), Spencer and Sain (1997), Nishitani and Inoue (2001), and Spencer
and Nagarajaiah (2003), which provide a comprehensive literature review of active control systems.
Structure
+
Passive Device
Excitation Outputs
Structure
Excitation Outputs
Control Force Measurements
State
Estimator
Control
Strategy
Actuator(s)
States Desired Signal
Structure
+
Semiactive Device
Excitation Outputs
Measurements
State
Estimator
Dissipative
Control
Strategy
States
Desired Signal to Modify
Device Characteristics
Passive Control
Active Control
Semiactive Control
Figure 1.2: An active control block diagram
5
In theory, an active system is the best controller because it is capable of both adding/dissipating energy to/from
the system and, therefore, can cancel the effect of any external excitation and prevent the superstructure from
any destructive movement. However, since active systems can insert energy to the system, they are sensitive
to uncertainties, such as signal and model errors, and can easily cause instability (specifically, when structures
are subjected to seismic excitations where the external force has a very random and unpredictable nature and
changes rapidly). Also, an active control device depends on an external power source to work and, in case of
electricity outage, which is very common in major events, the actuator will stop working. Furthermore, to
apply a possibly large control force to the structure, the actuator requires a huge amount of energy, which is
not always practical and cost-effective. Finally, it should also be noted that installing an active actuator and
the regular maintenance of a sensor network are both very costly. As a result, very few structure owners are
attracted to the use of active systems for protection.
1.4 Semiactive Control
Semiactive controllers, including variable orifice dampers (Feng and Shinozuka, 1990), variable friction
dampers (Kannan et al., 1995), controllable tuned liquid dampers (Haroun et al., 1994, Chang et al., 1998,
Taflanidis et al., 2005, 2007), controllable fluid dampers such as electrorheological (ER) dampers (Ehrgott and
Masri, 1992, 1994, Masri et al., 1995, Gavin et al., 1996, Makris et al., 1996) and magnetorheological (MR)
dampers (Dyke et al., 1996, Spencer et al., 1997), and semiactive impact dampers (Caughey and Karyeaclis,
1989, Masri et al., 1989, 1994) are of great interest since they combine the best features of both passive and
active controllers. Similar to passive controllers, semiactive systems apply only dissipative forces, which
guarantees the stability of the system; however, since their parameters, such as stiffness and damping, are
controllable and can be adapted based on the external excitation, they can achieve most of the performance
benefits of an active system. Figure 1.3 represents a semiactive block diagram. Some of the semiactive
advantages over the passive and active controllers are listed in the following lines and also summarized in
Table 1.1.
6
Structure
+
Passive Device
Excitation Outputs
Structure
Excitation Outputs
Control Force Measurements
State
Estimator
Control
Strategy
Actuator(s)
States Desired Signal
Structure
+
Semiactive Device
Excitation Outputs
Measurements
State
Estimator
Dissipative
Control
Strategy
States
Desired Signal to Modify
Device Characteristics
Passive Control
Active Control
Semiactive Control
Figure 1.3: A semiactive control block diagram
Characteristics of semiactive controllers compared to passive ones are:
They perform significantly better. Semiactive controllers can usually mitigate the structural responses
more successfully than a passive controller. In examples discussed in the following chapters, it is
shown that a great reduction in both base drift and roof absolute acceleration responses can be achieved
when a semiactive device is implemented instead of a typical passive controller.
They are adaptive to changing objectives. Unlike passive controllers that retain the same character-
istics and performance as initially designed, the objective function of a semiactive controller can be
modified easily at any time without requiring any physical modifications in the actual installed device.
They can incorporate global information. The behavior of passive controllers are based on the
responses of the points to which the controllers are directly connected (i.e., it depends on local
information); whereas, semiactive controllers use global response information collected by sensors to
determine the magnitude of the applied force(s).
The characteristics of semiactive controllers compared to active ones are:
Stability is guaranteed. Since active strategies can exert any desired force, they may add energy to
the structure and, therefore, the system instability is always possible. This is the main reason that
stakeholders have less interest in implementing active devices in real-world structures. In contrast,
semiactive devices can only exert dissipative forces on or within the structure and, as a result, stability
is always guaranteed.
7
They require minimal external energy. Active actuators rely on large supply of external energy for
applying the desired force on the structure; this is a serious drawback since, during strong excitations,
there is a significant possibility of a power outage, which will result in a nonfunctional controller. Semi-
active devices, on the other hand, require minimal energy to adjust their properties and, interestingly,
most of them can even function on battery power alone.
If they fail, they act as passive control devices. If, because of electrical or other failure, an active
actuator is disabled and the structure will behave as an uncontrolled one; in contrast, if a semiactive
controller stops working, at least it acts as a passive device and still achieves some energy dissipation.
Table 1.1: Passive, active, and semiactive control comparison (adapted from Erkus (2006))
Characteristics Passive Active Semiactive
Energy Dissipates energy Adds or dissipates energy Dissipates energy
Controllability No Fully controllable
Some characteristic
changes available
Robustness Good
Critical for extremely
Good
nonlinear behavior
Performance Low High Moderate
Operation Cost Negligible High Low
Although semiactive control strategies show great promise, they were not applied to civil engineering
problems until recently. Feng and Shinozuka (1990), Patten et al. (1994), Masri et al. (1995) and Dyke et al.
(1996) were among the first who demonstrated numerically and experimentally that a semiactive controller
can provide a performance comparable to that of an active controller and discussed its advantages. Afterwards,
many researchers (Housner et al., 1997, Spencer and Sain, 1997, Symans and Constantinou, 1999, Erkus
et al., 2002, Ramallo et al., 2002, Soong and Spencer, 2002, Johnson et al., 2007, Kamalzare et al., 2014b)
have studied the possibility of implementing semiactive systems for mitigating the response of various types
of structures under different external excitation scenarios. Also, semiactive devices have been installed in
many real-world structures (such as the Kajima Shizuoka Building (Kurata et al., 1999), the I-35 Highway
Bridge (Patten, 1998), the V olgograd Bridge in Russia (Weber et al., 2013), etc.) in the last two decades and
have shown very promising performance.
8
1.5 Magnetorheological Fluid Damper: a Semiactive Control Device
A semiactive (smart) damper is a device with controllable damping characteristics. One type of semiactive
control device is an magnetorheological (MR) damper which basically consists of a cylinder filled with an
MR fluid and a piston that moves through this cylinder as shown in Figure 1.4. The superiority of an MR
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! & $ L
<
)
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J 3 # # R % : ) K D 3 . 5 C % & : " . ! " " . % 3 " # . & 5 1 5 : 2 % 3 3 ! 1 ! 0 1 # " 5
$ # / % # 3 ! < < 4 5 < 4 % ! " # ( 5 4 % / % 1 # & : % & # # 4 % & : ! < < 1 % ! " % 5 & 3 ) Y "
$ # 3 % : & / # 1 5 % " % # 3 D " . # $ 2 & ! 9 % 4 ! & : # 5 ( ( 5 4 # 3 < 4 5 $ 8 # $
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< 5 C # 4 4 # M 8 % 4 # $ 0 2 " . # $ # / % # % 3 5 & 1 2 ? V ? T ) 6 5 4 # =
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_ # 1 $ 3 * [ - )
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4 # 3 8 1 " 3 4 # : ! 4 $ % & : $ # 3 % : & D 5 & 3 " 4 8 " % 5 & ! & $ 5 9 9 # 4 % ! 1
< 4 5 $ 8 " % 5 & 5 ( 1 ! 4 : # 3 ! 1 # 6 ’ $ ! 9 < # 4 3 D C . % . 3 . 5 8 1 $
: 4 # ! " 1 2 ! # 1 # 4 ! " # " . # % & " 4 5 $ 8 " % 5 & 5 ( " . % 3 " # . & 5 1 5 : 2
% & " 5 < 4 ! " % # )
b c d e f g h i j k f l m n o p m q r
Y & ! " " # 9 < " . ! 3 0 # # & 9 ! $ # % & " . % 3 < ! < # 4 " 5 % & " 4 5 $ 8 #
" . # 0 ! 3 % 5 & # < " 3 5 ( < ! 3 3 % / # ! & $ ! " % / # 3 " 4 8 " 8 4 ! 1 5 & =
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! & $ 3 " 4 8 " 8 4 ! 1 ! < < 1 % ! " % 5 & 3 % & " . % 3 # ] % " % & : ! & $ ( ! 3 "
# ] < ! & $ % & : _ # 1 $ ) T . % 1 # 3 % : & % _ ! & " 3 " 4 % $ # 3 . ! / # 0 # # &
9 ! $ # % & " # 4 9 3 5 ( % 9 < 1 # 9 # & " ! " % 5 & 5 ( " . # 3 # 5 & # < " 3 " 5
3 " 4 8 " 8 4 ! 1 $ # 3 % : & ! & $ 4 # " 4 5 _ " D % " 3 . 5 8 1 $ 0 # # 9 < . ! 3 % H # $
" . ! " " . % 3 # & " % 4 # " # . & 5 1 5 : 2 % 3 3 " % 1 1 # / 5 1 / % & : ) a % : & % _ ! & "
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< 4 5 # $ 8 4 # 3 C % 1 1 # 4 " ! % & 1 2 5 & " % & 8 # ( 5 4 ! & 8 9 0 # 4 5 ( 2 # ! 4 3
" 5 5 9 # )
G . # ! # < " ! & # 5 ( % & & 5 / ! " % / # 3 2 3 " # 9 3 % & 3 " 4 8 " 8 4 ! 1
# & : % & # # 4 % & : % 3 0 ! 3 # $ 5 & ! 5 9 0 % & ! " % 5 & 5 ( < # 4 ( 5 4 9 ! & #
# & . ! & # 9 # & " / # 4 3 8 3 5 & 3 " 4 8 " % 5 & 5 3 " 3 ! & $ 1 5 & : = " # 4 9
# ( ( # " 3 ) B 5 & " % & 8 % & : # ( ( 5 4 " 3 ! 4 # & # # $ # $ % & 5 4 $ # 4 " 5 ( ! % 1 % =
" ! " # C % $ # 4 ! & $ 3 < # # $ % # 4 % 9 < 1 # 9 # & " ! " % 5 & ) G . # 3 # % & 1 8 $ #
# ( ( # " % / # 3 2 3 " # 9 % & " # : 4 ! " % 5 & ! & $ ( 8 4 " . # 4 $ # / # 1 5 < 9 # & " 5 (
! & ! 1 2 " % ! 1 ! & $ # ] < # 4 % 9 # & " ! 1 " # . & % M 8 # 3 0 2 C . % . < # 4 =
( 5 4 9 ! & # 3 5 ( " . # 3 # 3 2 3 " # 9 3 ! & 0 # 4 # ! 1 % 3 " % ! 1 1 2 ! 3 3 # 3 3 # $ )
a " 4 8 " 8 4 ! 1 3 2 3 " # 9 3 ! 4 # 5 9 < 1 # ] 5 9 0 % & ! " % 5 & 3 5 ( % & $ % / % $ =
8 ! 1 3 " 4 8 " 8 4 ! 1 5 9 < 5 & # & " 3 ) s # C % & & 5 / ! " % / # $ # / % # 3 & # # $
" 5 0 # % & " # : 4 ! " # $ % & " 5 " . # 3 # 5 9 < 1 # ] 3 2 3 " # 9 3 D C % " .
4 # ! 1 % 3 " % # / ! 1 8 ! " % 5 & 5 ( " . # % 4 < # 4 ( 5 4 9 ! & # ! & $ % 9 < ! " 5 &
" . # 3 " 4 8 " 8 4 ! 1 3 2 3 " # 9 D ! 3 C # 1 1 ! 3 / # 4 % _ ! " % 5 & 5 ( " . # % 4
! 0 % 1 % " 2 ( 5 4 1 5 & : = " # 4 9 5 < # 4 ! " % 5 & )
Figure 1.4: An MR fluid damper (adapted from Soong and Spencer (2002))
damper comes from the unique characteristic of the MR fluid that its yield strength can be modified almost
instantly in couple of milliseconds. In the absence of a magnetic field, an MR fluid behaves as a typical
Newtonian fluid but its particles are polarized in the presence of a magnetic field and start making chain-like
structures in the fluid as shown in Figure 1.5. Since the amount of polarization depends on the magnitude of
the magnetic field, the characteristics of an MR fluid can be easily modified. It is also very desirable that the
polarization and depolarization happen almost instantly and don’t leave any residual effect (See Figure 1.6).
9
MR fluid
Polarizable micron-size particles
Magnetic field
(a) Without magnetic field
MR fluid
Polarizable micron-size particles
Magnetic field
(b) With magnetic field
Figure 1.5: MR fluid in absence and presence of a magnetic field (adapted from Erkus (2006))
Velocity
Damper
force
Velocity
Damper
force
Velocity
Damper
force
(a) Minimum voltage
Velocity
Damper
force
Velocity
Damper
force
Velocity
Damper
force
(b) Maximum voltage
Velocity
Damper
force
Velocity
Damper
force
Velocity
Damper
force
(c) Varying voltage
Figure 1.6: Force-velocity relation of an MR damper at different voltage level (adapted from Erkus
(2006))
1.6 Computational Burden of Semiactive Control Design
While implementing a semiactive device is of great interest, the development of control strategies for a
controllable passive damper, i.e., a “semiactive” damping device, is complicated by the nonlinear and
dissipative nature of the device and the nonlinear nature of the closed-loop system with any feedback control.
Control design for nonlinear systems is often achieved by designing a control for a linearized model since
strategies for linear systems are straightforward. One such approach is clipped-optimal control (Dyke et al.,
1996, Ramallo et al., 2002, Johnson et al., 2007) in which the desired damper forces are determined from an
optimal controller (e.g., linear quadratic regulator (LQR), linear quadratic Gaussian (LQG), H
2
, etc.), which
10
is designed assuming that the damping devices are fully linear actuators (basically an active control) that
can exert any forces (dissipative or non-dissipative), and a secondary bang-bang controller commands the
controllable damper to exert dissipative forces as close as possible to the desired forces commanded by the
active control. However, designs using any linearized model generally result in suboptimal (and sometimes
poor) performance because the linear actuator assumption differs from the actual implementation with a
dissipative damping device. Thus, one must generally resort to a large-scale parameter study in which the
nonlinear system is simulated many times to determine control strategies that are actually optimal for the
nonlinear semi-actively controlled closed-loop system. Therefore, designing semiactive control strategies is
very computationally intensive. For example, performing optimal semiactive control design of a single cable
of a cable-stayed bridge required 9 CPU-months in 2005-06 (Johnson et al., 2007).
1.7 Fast Analysis of Systems with Local Modifications
Designing control strategies for large and complex structures requires analyzing and accurately predicting
the dynamical behavior of the structural systems, which is often very time consuming. For many decades
researchers have developed many methods to reduce the computational cost of such analyses/simulations.
Most of these methods provide simplistic models that can capture the main characteristics of the corresponding
systems but require a significantly lower computation cost. Although these methods are very useful, they
suffer from at least two drawbacks. First, finding and generating an appropriate simple model that mimics the
main properties of a corresponding complex system is often not an easy and straightforward task and may
itself require an intensive computational effort. Second, it is often infeasible to predict the accurate system
response due to the many simplifying assumptions required to develop these simplified models. This fact
is why large complex (finite element model (FEM)) simulations are required for critical systems when an
accurate response prediction is desired.
If the entire model remains linear, which is not very common, it can be analyzed with a fairly low computation
effort; but, if the system includes some nonlinear elements, the computation cost rises significantly, sometimes
by orders of magnitude. Yet, the nonlinear features of these large systems are oftentimes localized. Local
nonlinear features include uncertainty in the properties of select structural elements (such as strength, stiffness
and damping), nonlinear behavior of some connections or elements, nonlinear nature of the feedback control
11
force.
Brute force methods, such as ordinary differential equation (ODE) solvers, are the typical straightforward
techniques to determine the nonlinear system response with any desirable accuracy. However, applying
these methods (such as theode45 command in MATLAB) for a large FEM with thousands of degrees of
freedom (DOFs) can result in a computationally intensive solution as the semiactive control design parameter
study requires a large number (hundreds or even thousands) of these simulations be performed.
Since the classical methods neglect the localized nature of the nonlinearities and treat the entire system as
nonlinear, researchers have sought alternate methods that can take advantage of the locality of the nonlinear
elements and possibly reduce the computation cost of determining the system response. Some of the
contributing literature that suggests those alternate methods are summarized here.
Linearizing the nonlinear model: Since numerical methods for linear systems have been well defined
for many decades and can provide fast and reliable solutions, generally the first choice for simplifying
the nonlinear analysis is linearizing the nonlinear terms. However, developing a linear model that
represents the critical characteristics of a nonlinear system, if possible, is not easy and straightforward
and often requires expert knowledge. Linearization results in a simpler model and, consequently, a
faster analysis, but at the expense of losing some accuracy.
Model reduction techniques: Model reduction techniques (e.g., Guyan reduction, modal truncation,
and balanced reduction technique) (Friswell et al., 1995, 1996, Segalman, 2007) are of great interest
when the size of the model is significantly large. Although, they look very promising, they can be
very tricky when dealing with nonlinear problems. The performance of the reduced model can vary
dramatically based on the basis functions kept in the reduced system, which is very subjective to the
judicious choice of the designer and requires reconsideration from one application to another.
Combined approximations: Kirsch and Bogomolni (2007) proposed a similar approach, called
combined approximations, which tries to approximate the response of the locally nonlinear system
using a linear set of basis functions. This method has the same difficulty of choosing appropriate basis
functions which is subjective to the judicious choice of the designer (Wojtkiewicz and Johnson, 2014).
12
Convolution integral methods: Another class of methods calculates the system response using the
convolution integral equations. The system response is sum of the response of the nominal linear
system with the response due to the pseudo-force representing the effect of the local nonlinear features.
Some significant contributions in the literature are the studies performed by Clough and Wilson (1979),
Hagedorn and Schramm (1988), Chiang and Noah (1990), Gordis and Radwick (1999), Gordis and
Neta (2001), Iwata et al. (2003). However, in all these, the response of a system is calculated by directly
solving for displacements; which sometimes can result in a large error compared to the response
derived by analyzing the main large-size FEM. Preferably, if a model is generally considered linear
but has a few localized nonlinear features, alternate methods, such as the nonlinear V olterra integral
equation (NVIE) approach introduced by Gaurav et al. (2011), which can take advantage of the locality
of the nonlinearities, can provide a fast and accurate solution. This NVIE approach is also based
on solving the convolution integral; however, it is different from the other convolution algorithms in
the sense that the nonlinear integral equation governs the evolution of the force(s) in the nonlinear
element(s) rather than the displacement directly (Gaurav et al., 2011). This NVIE method is the main
algorithm used in this thesis because of its unique capability of providing accurate and computationally
inexpensive system response. The method is discussed in depth in the next chapter.
1.8 Overview of this Dissertation
This dissertation continues in Chapter 2 by introducing the method, developed by Gaurav et al. (2011), that
uses a convolution integral to determine the system response very computationally efficiently when there is a
high-order linear nominal system with a few local modifications or nonlinearities.
Next, in Chapter 3, the author has adapted this methodology to provide a fast and computationally inexpensive
design of an hysteretic base isolator system (a passive control design) which is one of the systems widely
used to protect structures against severe external forces by limiting the energy transferred into the structure.
In design-level earthquakes, the superstructure is expected to remain elastic, so the system model has a linear
superstructure and a rather nonlinear isolation layer, resulting in a system that is very suitable to apply the
proposed approach and achieve significant computational efficiency.
13
As previously discussed, designing semiactive (“smart”) controllers is an active topic in the civil control
community; yet, it is very challenging because of the computational burden of the design procedure. Therefore,
the objective of the rest of this dissertation is providing a computationally fast and accurate methodology for
semiactive control design.
The development of control strategies for controllable passive dampers, i.e., “semiactive” damping devices, is
complicated by the nonlinear and dissipative nature of the devices and the nonlinear nature of the closed-
loop systems with any feedback control. Chapter 4 demonstrates how the NVIE method can be adapted
for designing a full state feedback semiactive control strategy and significantly decrease its computational
burden.
Next, this study expands the applicability of the proposed method in Chapter 5 by demonstrating that the
approach can also be adapted to accommodate the more realistic cases when, instead of full-state feedback,
only a limited set of noisy response measurements are available to the controller, which requires incorporating
a Kalman filter (KF) estimator into the nominal linear model. The results show that the proposed method can
achieve even higher computational efficiency while retaining an appropriate level of accuracy.
The primary controller is rarely designed using a high-order model because it is impractical due to numerical
difficulties, as well as often unnecessary since high-order models, such as complex finite element structure
models, have high frequency dynamics that remain mostly unexcited by an external disturbance. To bring the
proposed method to full maturity, in Chapter 6, a reduced-order model for control design is incorporated with
the full model to simulate semiactively controlled structural responses using the NVIE approach.
Finally, since uncertainties in key structural elements cannot be neglected in real-world systems and may
intensify the computational burden by demanding a Monte Carlo simulation (MCS) to characterize the effects
of uncertain properties, Chapter 7 explains how the proposed approach can be implemented when uncertainties
are also involved in the system. Numerical results confirm the accuracy, stability, and computational efficiency
of the proposed simulation methodology.
The last chapter of this dissertation (Chapter 8) provides a list of key conclusions and also some of the areas
that need further investigation.
14
Chapter 2
Fast Analysis of Linear Systems with Local
Nonlinearities/Modifications
2.1 Introduction
This thesis investigates the efficient analysis of passive and semiactive damping devices in smart structures.
Although the dynamical model of such systems is nonlinear, the nonlinearity is often localized in the damping
device model. As discussed inx 1.7, many methodologies have been proposed to limit the required memory
and processing capacity required for analyzing these systems. The investigations in this dissertation adapt the
method developed by Gaurav et al. (2011), which is capable of providing a faster and more accurate solution
of large systems that are mostly linear but with some nonlinear, uncertain or modified local elements. This
approach uses a convolution integral to determine how much these local elements change system responses,
resulting in a low-order nonlinear V olterra integral equation (NVIE) in the force of these local elements. This
method has unique characteristics including “(i) the ability to directly compute the response of only a select
subset of the full response, (ii) the ability to handle both smooth and non-smooth nonlinearities, and (iii) good
computational efficiency so that a large number of response samples can be generated cheaply for uncertainty
quantification” (Gaurav et al., 2011). This chapter summarizes the method formulation presented in Gaurav
(2011) and explains the procedure to reduce a large-order system of equations to a low-order NVIE and an
efficient method for solving the NVIE using fast Fourier transforms (FFTs).
15
2.2 Efficient Response Computation
2.2.1 Nominal System
Consider a nominal uncontrolled structure model, without perturbation (e.g., control devices) forces, in state
space form
˙ x(t)= Ax(t)+ Bw(t), x(0)= x
0
(2.1)
where x(t) is the n
x
1 state vector; w(t) is the n
w
1 external excitation vector; A is the n
x
n
x
system
matrix; B is the n
x
n
w
excitation distribution matrix;
˙
() denotes a derivative with respect to time; and x
0
is
the initial condition.
2.2.2 Modified System
When the n
p
1 nonlinear perturbation force vector p(t) (which can be a control force, a nonlinearity or
modification in the model, or combination thereof) is applied, the equation of motion becomes
˙
X(t)= AX(t)+ Bw(t)+ Lp(t), X(0)= x
0
(2.2)
where X(t) is the n
x
1 state vector; L is an n
x
n
p
constant influence matrix that maps the forces to the
appropriate time derivatives of system states; and the initial condition remains the same as for the nominal
system. If the force p(t) were known a priori, (2.2) could be efficiently solved through standard linear
response simulation methods. However, this force is a function, usually nonlinear, of the states or, more
commonly, of some subset or linear combination of states X(t)= GX(t), where G is an n
x
n
x
matrix. The
functional dependence of p(t)= g(X(t)) on X(t) depends on the nature of the nonlinearity.
2.2.3 Response Computation
The response x(t) of the nominal system can be computed in a variety of manners, e.g., using the state
impulse response function H
B
(t)= e
At
B of the nominal system (2.1) in a Duhamel convolution integral to
16
find the nominal response
x(t)= e
At
x
0
+
Z
t
0
H
B
(tt)w(t)dt (2.3)
Using superposition, the modified response can be written in terms of the nominal response.
X(t)= x(t)+
Z
t
0
H
L
(tt)p(t)dt (2.4)
where H
L
(t) denotes the impulse response function of the nominal system to a load applied in the pattern of
the nonlinear force; i.e., ifd(t) is the Dirac delta function, then H
L
(t)= e
At
L is the response of
˙
H
L
(t)=
AH
L
(t)+ Ld(t) with initial condition H
L
(0

)= 0. Defining x(t)= Gx(t) and H
L
(t)= GH
L
(t)= Ge
At
L,
the subset or linear combination of responses can be computed with
X(t)= x(t)+
Z
t
0
H
L
(tt)p(t)dt (2.5)
Substituting (2.5) into p(t)= g(X(t)) results in a system of n
x
NVIEs in nonstandard form
p(t) g

x(t)+
Z
t
0
H
L
(tt)p(t)dt

= 0 (2.6)
that can be solved to obtain the forces p(t). Once p(t) is known, the responses of the modified/controlled
system can be computed using a conventional convolution integral. Note that (2.6) can be solved most
efficiently if n
p
n
x
and n
x
n
x
, i.e., if both the number of local nonlinearities as well as the number of
linear combinations of states required to compute the nonlinear forces are small relative to the system order.
2.2.4 Solution for NVIEs
V olterra integral equations (VIEs) were first introduced by an Italian mathematician and physicist, Vito
V olterra (Schetzen, 2006). Gaurav et al. (2011) has summarized the important studies on fast solutions of
NVIEs. Some of the significant studies were conducted by Hairer et al. (1985), Capobianco et al. (2007), and
Isaacson and Kirby (2011). Hairer et al. (1985) proposed dividing the convolution domain to smaller sections
and using FFT to provide a fast solution. The method proposed by Gaurav et al. (2011) is very similar to
that study with the main difference being that Hairer et al. (1985) assumed a scalar V olterra kernel where as
17
herein the method is generalized for higher-order kernels. Capobianco et al. (2007) introduced Laplace-based
methods to evaluate the V olterra kernel, which provides an efficient Runge-Kutta method to solve the NVIEs
in a general form. Alternatively, Isaacson and Kirby (2011) suggested a collocation-based method for solving
the convolution integral but for a scalar V olterra kernel and is limited to only linear VIEs.
As discussed in Gaurav et al. (2011), most of these methods require having the closed-form of the V olterra
kernel (which is not available because of the dimensionality of the nominal system) and also need to evaluate
it at intermediate time points (which increase the computation time significantly). The interested reader is
encouraged to see Miller (1971), Atkinson (1997), and Brunner (2004) for overviews of multiple methods for
solving (2.6), a system of NVIEs in nonstandard form.
Herein, following Gaurav et al. (2011), a numerical approach consists of a quadrature rule and Newton’s
method is applied to solve (2.6) as follows. First, discretize (2.6) at equally spaced time intervals and evaluate
the convolution integral using a conventional numerical method. For simplicity, the second order trapezoidal
rule (see Figure 2.1) is discussed here; however, the interested reader is directed to Gaurav et al. (2011) for
detailed explanation of 2
nd
, 4
th
, and 6
th
order methods and their accuracy comparison. Using the second order
trapezoidal rule, (2.6) can be written as
p(t
k
) g

x(t
k
)+
1
2
H
L
(t
k
)p(t
0
)Dt+
k1
å
j=1
H
L
(t
k j
)p(t
j
)Dt+
1
2
H
L
(t
0
)p(t
k
)Dt
!
= 0 (2.7)
0
x
1
x
2
x
3
x
4
x
x
y
x
y
0
x
1
x
2
x
)) ( , (
0 0
x f x
)) ( , (
1 1
x f x
Figure 2.1: Trapezoidal rule
18
At step k, the only unknown in (2.7) is p(t
k
); therefore, one can definea
k1
to be the portion of (2.7) that is
dependent, beside unmodified system responses and impulse responses, only on p(t
0
) through p(t
k1
)
a
k1
= x(t
k
)+
1
2
H
L
(t
k
)p(t
0
)Dt+
k1
å
j=1
H
L
(t
k j
)p(t
j
)Dt (2.8)
Substituting (2.8) into (2.7), the NVIE can be written as a nonlinear equation of p(t
k
) at each time k
p(t
k
) g

a
k1
+
1
2
H
L
(t
0
)p(t
k
)Dt
!
= 0 (2.9)
This equation can be solved at each time step using iterative Newton’s method (see Figure 2.2) as follows
˜ p
k,0
= p(t
k1
)
Evaluate g and
¶g
¶X
at X
k,i
=a
k1
+
1
2
H
L
(0)˜ p
k,i
Dt
˜ p
k,i+1
= ˜ p
k,i

I
1
2
¶g
¶X
H
L
(0)Dt

1
(˜ p
k,i
g) (2.10)
p(t
k
) = ˜ p
k,¥
where ˜ p
k,i
and X
k,i
are the updated values of p(t
k
) and X(t
k
), respectively, in the i
th
iteration. It is worth
noting that, for the semiactive control design discussed in the next chapters, the algorithm converges very fast
and typically requires only 2 3 iterations to result in a sufficient accuracy.
0
x
1
x
2
x
3
x
4
x
x
y
x
y
0
x
1
x
2
x
)) ( , (
0 0
x f x
)) ( , (
1 1
x f x
Figure 2.2: Newton’s method
19
2.2.5 Fast Solution of NVIEs using Fast Fourier Transform (FFT)
Most of the computation time and effort spent by the solution algorithm for (2.6) as outlined previously
involves the computation of convolution integrals. The straightforward solution consists of discretizing
the integration space to N
t
discrete points and find the solution using a direct summation such as the 2
nd
trapezoidal rule shown in (2.7). Solving a single convolution integral by this direct summation method
requires O(N
2
t
) operations (multiplications and additions). If both sequences in a convolution are known a
priori, the solution can be calculated much faster using a FFT and the computation order will be reduced to
O(N
t
logN
t
). However, when solving the NVIE equation of (2.6), the second term p(t) is not known a priori
for the entire time domain. But, if the integration domain is divided into two partitions and p is determined in
the first part, this can be used to accelerate the computation of the second part by evaluating some parts of
that convolution using a FFT. This is explained as follows:
Direct computation: Assuming no partitioning (See Figure 2.3a), the convolution integral in (2.6) can
be calculated as:
Z
t
0
H
L
(tt)p(t)dt =Dt
k
å
i=0
H
L,ki
p
i
(2.11)
where k= 0,1, ,N
t
; this requires a computational effort of an order O(n
x
n
p
N
2
t
)) for large N
t
.
Domain divided in two sub-divisions: If the domain is divided to two partitions as shown in Figure 2.3b,
then the convolution integral can be divided in two parts:
1) Evaluate the force p from t= 0 to N
t
/2 1 using the direct method:
Z
t
0
H
L
(tt)p(t)dt =Dt
k
å
i=0
H
L,ki
p
i
(2.12)
where k= 0,1, ,N
t
/2 1, which requires a computational effort of order O(n
x
n
p
N
2
t
/4).
2) Evaluate the force p from t= N
t
/2 to N
t
1:
Z
t
0
H
L
(tt)p(t)dt =Dt
k
å
i=0
H
L,ki
p
i
(2.13)
20
where k= N
t
/2,N
t
/2+1, ,N
t
1. The summation in (2.13) can be divided in two parts as following:
k
å
i=0
H
L,ki
p
i
=
N
t
/21
å
i=0
H
L,ki
p
i
+
k
å
i=N
t
/2
H
L,ki
p
i
(2.14)
Since p is known from t = 0 to N
t
/2 1, both sequences in the first right-hand side term of (2.14)
are known and a FFT can be implemented to evaluate that term efficiently. The second right-hand
side term should be evaluated normally. So, the computational effort for evaluating the force p from
t= N
t
/2 to N
t
is of order of O(n
x
n
p
N
t
/2logN
t
/2+ n
x
n
p
N
2
t
/4).
This means, in case of dividing the integral domain to two partitions, the total computation time is
reduced to O(n
x
n
p
N
2
t
/2+ n
x
n
p
N
t
/2logN
t
/2).
Domain divided in four sub-divisions: Similarly if the domain is divided to four partitions as shown
in Figure 2.3c, then the computation necessary for calculating the convolution integral is of order of
O(n
x
n
p
N
2
t
/4+ n
x
n
p
N
t
/2logN
t
/4+ n
x
n
p
N
t
/2logN
t
/2).
The computational speed of evaluating the convolution integral increases by the number of partitions.
However, the upper limit on the optimal number of partitions is when “the overhead of the function
calls becomes equal to the computation time gained by using the FFT-based convolution” (Gaurav et al.,
2011). For the numerical studies discussed in the next chapters, assuming that N
t
= 2
i
, a preliminary
study showed that the optimal number of partitions is i/2 1.
τ
t
N
t
1
τ
t
N
t
1
2
3
N
t
/2
τ
t
N
t
4
N
t
/2
1
2
3
5
6
7
N
t
/4
(a)
τ
t
N
t
1
τ
t
N
t
1
2
3
N
t
/2
τ
t
N
t
4
N
t
/2
1
2
3
5
6
7
N
t
/4
(b)
τ
t
N
t
1
τ
t
N
t
1
2
3
N
t
/2
τ
t
N
t
4
N
t
/2
1
2
3
5
6
7
N
t
/4
(c)
Figure 2.3: Subdivision of the convolution space: squares by FFT and triangles by Trapezoidal rule
(adapted from Gaurav et al. (2011))
21
The analysis method formulation discussed in this chapter (which is derived by Gaurav et al. (2011)) is the
theoretical basis of this thesis. It provides an efficient solution for the large dynamical systems with local
nonlinearities and/or modifications in the model and is the most effective when many simulations are required.
In the following chapters, this NVIE approach is adapted for efficient design of passive and semiactive
damping systems for complex models of building structures.
2.3 Sensitivity of Responses
This method can also calculate the response sensitivities in a very fast and efficient manner (Wojtkiewicz and
Johnson, 2014). The sensitivity of modification p(t) to some parameterq
i
in g(X;q q q) is given by
s
i
(t)=
¶p(t)
¶q
i
=
¶g
¶q
i
+
¶g
¶X
¶X
¶q
i
(2.15)
The modified states depend onq q q only through p(t) so
¶X(t)
¶q
i
=
Z
t
0
H
L
(tt)s
i
(t)dt (2.16)
¶X
k
¶q
i
=
Z
t
k
0
H
L
(t
k
t)s
i
(t)dt (2.17)
k k k
k1
+
1
2
H
L,0
s
i
k
Dt where k k k
k1
=
1
2
H
L,k
s
i
0
Dt+
k1
å
j=1
H
L,k j
s
i
j
Dt
where a trapezoidal integration is used to approximate the convolution. Substituting (2.17) into (2.15)
evaluated at time t
k
, and solving for sensitivity s
i
k
involves just solving a linear system at each time step.
s
i
k
=

I
1
2
E
k
H
L,0
Dt

1
[d
k
+ E
k
k k k
k1
] (2.18a)
where d
i
k
=

¶g
¶q
i

X
k
,q q q
and E
k
=

¶g
¶X

X
k
,q q q
(2.18b)
The full state sensitivity ¶X
k
/¶q
i
can then be found from convolution in (2.16) using any conventional
method (e.g., quadratures, FFT, etc.). If needed by the optimization method, second-order sensitivities (i.e.,
Hessians) can be found with a similar approach (Wojtkiewicz and Johnson, 2014).
22
Chapter 3
Efficient Optimal Design of Passive
Structural Control with Application to
Isolator Design
1
3.1 Introduction
Passive structural control strategies (Housner et al., 1997, Soong and Dargush, 1997) — such as passive
(linear or nonlinear) dampers, friction elements, yielding metal elements, isolation devices, and tuned-mass
dampers — are some of the most commonly implemented forms of structural control. Typical base isolated
buildings are designed such that the superstructure remains elastic in design-level earthquakes, though the
isolation layer is often quite nonlinear using, for example, hysteretic elements such as lead-rubber bearings
or friction pendulum bearings. Similarly, other types of structural control systems, if they perform well,
keep the structure within the linear range except for the most extreme of excitations. For linear structures
with linear structural control elements, the simulation, design and optimization of such systems is relatively
straightforward as linear response computation, driven by deterministic or stochastic excitation, is well-
understood and can be performed with good efficiency. However, if the passive control element is nonlinear,
or if there are nonlinearities in the structural system model, then response computation, and any design
1
This chapter is based on Kamalzare et al. (2014c) and Johnson et al. (2013)
23
optimization using the responses, becomes more computationally challenging. The usual approach simulates
these systems with a generic nonlinear solver, a general tool that can be used to simulate a wide variety of
dynamical systems, but cannot exploit the localized nature of the passive control elements within the overall
model.
This chapter proposes an approach for the computationally-efficient optimal design of passive isolators by
extending the analysis approach outlined in Chapter 2 for simplifying the response calculation for systems
with local features (linear or nonlinear, deterministic or random), but that are otherwise linear, to a low-order
V olterra integral equation (VIE) that can be integrated in time, as well as analogous VIEs for the sensitivities
of responses to isolator design parameters.
While the approach proposed herein is general, the example application considered herein is an optimal design
problem for the parameters of a hysteretic lead-rubber bearing in a base-isolated building. A brief background
of the optimal design of base isolated systems is given. Next, the proposed formulation for exploiting the
localized nature of the nonlinearities to rapidly perform design optimization problems is summarized. Then,
a numerical example of an 11-story 2-bay superstructure on a one-dimensional isolation layer will be used
to demonstrate the efficacy and computational advantages of the proposed method. A lead-rubber bearing
is modeled as a Bouc-Wen hysteresis in which the yield force, the pre-yield stiffness and the post-yield
stiffness are varied. The responses and their gradients to the three design variables are used to minimize a
mean-square measure of the base drift and roof acceleration subject to physically-meaningful constraints.
Finally, a robustness analysis of the sharpness of the hysteresis is conducted.
3.1.1 Background in Optimal Base Isolation Design
Base isolation (Kelly, 1986, Buckle and Mayes, 1990, Skinner et al., 1993, Naeim and Kelly, 1999) seeks
to separate the superstructure from ground motion, insulating it, insofar as is possible, from the excitation.
However, there are distinct tradeoffs between the displacement across the isolation layer and the motion
of (and within) the superstructure. The isolation design example considered herein is not intended to be
an exhaustive study of the optimal isolation problem, but a demonstration of the computational advantages
of the proposed simulation methodology for design optimization problems. One of the earliest studies of
optimal base isolation design is by Bhatti et al. (1978). A series of papers by Constantinou and Tadjbakhsh
24
examined optimal base isolation systems: with linear stiffness and damping alone using a frequency domain
analysis (Constantinou and Tadjbakhsh, 1983), with rubber bearings and frictional elements (Constantinou
and Tadjbakhsh, 1984), and with hysteretic dampers (using linearization) (Constantinou and Tadjbakhsh,
1985). Park and Otsuka (1999) performed a parameter study with a (small) grid of values for each key isolator
parameter, of the responses of a bridge, to scaled versions of the 1940 El Centro earthquake. Jangid studied
linearization (Jangid, 2000) and time-domain approaches (Jangid, 2005, 2007) to find optimal isolation
parameters, primarily the isolator yield force, for different isolator types/models. Fragiacomo et al. (2003)
used energy measures to search for optimal isolator parameters. Others have used probabilistic/reliability
approaches, such as Taflanidis and Beck (2008a,b, 2009, 2010), Bucher (2009), Jensen and Sepulveda (2012),
and Roy et al. (2012).
Most studies of optimal isolation design make some simplifying assumptions, such as using statistical
linearization of the nonlinear hysteresis, or using very simplified models of the superstructure, such as rigid
or single degree of freedom (SDOF) models. The nonlinear nature of these systems make computationally
challenging the use of a complex superstructure model to perform a full parameter study or true optimization
without simplifications. Conventional model reduction must necessarily make assumptions on the behavior
of the nominal system without regard for whether the added localized nonlinearities may excite dynamics
that are only approximated by the model reduction. In contrast, the method outlined in Chapter 2 has no need
to eliminate dynamics of the nominal system.
3.2 Methodology: Optimization Procedure
Consider a design optimization problem with cost functional J(X(t;q q q)), where X= GX is some linear
combination of the states, and the “optimal” value of the design parameter (vector)q q q is desired. To find the
design point where J is minimized, one may employ non-gradient-based or gradient-based approaches. For
the former, the proposed approach can be used to efficiently compute the response X(t), from which the
cost functional J is determined. To implement a gradient-based optimization algorithm to find the minimum
of a cost functional, the gradient of the cost functional with respect to the design parameters,¶J/¶q q q, may
be approximated by numerical methods such as finite differences, or it can be provided using the response
sensitivities in (2.18), requiring fewer function evaluations and, thus, more computationally efficiently and
25
typically also more accurately. The cost functional derivatives with respect to the design parameters, evaluated
at some candidate design vectorq q q

, can be calculated using
¶J(X(t;q q q))
¶q q q

q q q=q q q

=
¶J(X(t;q q q))
¶X

q q q=q q q

¶X
¶q q q

q q q=q q q

(3.1)
where ¶X(t)/¶q
i

q q q=q q q

=
R
t
0

GH
L
(tt)s
i
(t)

q q q=q q q

dt can be computed using the sensitivities s
i
k
from
(2.18), evaluated atq q q

, in any usual way. (Note that an additional term appears in (3.1) if J also has explicit
dependence onq q q.)
With this computationally-efficient approach for determining the cost functional J and its sensitivities in
(3.1) to the design variables, a design optimization that requires many function evaluations can be performed
in a manner that is much faster than if the response and sensitivity were determined using conventional
nonlinear solvers that cannot exploit the localized nature of the isolators or other structural control devices.
The primary focus of this study is to investigate the use of this VIE approach for design optimization of
(nonlinear) structural control devices.
3.3 Numerical Example: Optimal Design of Isolation Hysteresis for Isolated
Building
This example demonstrates how the proposed method can be utilized for designing an optimal hysteretic
isolator in the isolation layer of the 100 degree of freedom (DOF) frame structure shown in Figure 3.1. In
this example, the proposed method calculates both the responses and sensitivities for use in a gradient-based
optimization algorithm that determines the optimal isolator in a computationally efficient manner.
3.3.1 Model Description
Consider the base isolated building in Figure 3.1. The superstructure is 11 stories tall and 2 bays wide; the
superstructure is treated as linear, with horizontal, vertical and rotational DOFs at each moment-bearing joint.
The superstructure, if it were a fixed-base structure, would have a fundamental period of 1.05 s. As a fixed
26
base structure, the equations of motion would be
M
s
¨ u
s
+ C
s
˙ u
s
+ K
s
u
s
=M
s
r ¨ u
g
(3.2)
where u
s
is a vector of generalized displacements of the structure relative to the ground, for a total of 99
DOFs, three at each of the 33 nodes; the superstructure consistent mass and stiffness matrices are denoted
M
s
and K
s
, respectively; the damping matrix C
s
is computed using the Rayleigh method by assuming
that the 1
st
and 10
th
modal damping ratios are each 3%; ¨ u
g
is the horizontal ground acceleration; and
r=[1 0 0 1 0 0 1 0 0]
T
(i.e., a 1 in each element corresponding to a horizontal displacement in u
s
,
and zeros elsewhere) is the influence vector.
Base
nd
2 floor
th
10 floor
th
11 floor
st
1 floor
Ground
hysteretic
isolation bearings
u
b
u
b
u
3
u
6
u
9
u
12
u
15
u
18
u
84
u
87
u
90
u
1
u
4
u
7
u
10
u
13
u
16
u
82
u
85
u
88
u
91
u
94
u
97
u
2
u
5
u
8
u
11
u
14
u
17
u
83
u
86
u
89
u
92
u
95
u
98
u
93
u
96
u
99
Figure 3.1: 100-DOF base-isolated structure.
27
The structure sits on a base mass which is supported by an isolation layer composed of lead rubber bearings
(LRBs). The base drift is assumed sufficiently moderate that the low-damping rubber acts as a linear stiffness
and viscous damping element. The lead plug is assumed to provide linear stiffness before yielding, and then a
much lower (or zero) stiffness thereafter. Together, the rubber and the lead provide a hysteretic stiffness such
as shown in Figure 3.2. While a bilinear model is commonly used for LRBs, it has been shown to result in
computed accelerations that are larger than those observed (Skinner et al., 1993, Nagarajaiah and Sun, 2000)
as it overstates the sharpness of the lead transition from fully elastic to (partially) plastic; instead here, the
computationally more tractable, smooth Bouc-Wen model (Bouc, 1967, Wen, 1976) is used. The effect of
considering different elastic-to-partially-plastic transition sharpness is discussed in more detail at the end of
this study.

Q
y
k
post
k
pre

Bouc-Wen
Bilinear
force
drift
Figure 3.2: Bilinear and Bouc-Wen models to represent the hysteretic behavior of materials.
The nominal post-yield stiffness k
post
and base mass m
b
are chosen such that the fundamental mode of the
isolated building, if the yield force Q
y
were zero, would have a period of 2.76 s, which is in the typical
expected range (Skinner et al., 1993); the isolation-layer viscous damping coefficient c
b
, provided by the
isolator (and/or supplemental passive viscous dampers), is chosen such that the isolation mode has a damping
ratio of about 5.5%. The building weight (base plus superstructure) is W = 1.28 MN and its height from the
base upwards is h= 44 m. The isolation layer is assumed to be constrained to move only in the horizontal
direction, so multiple identical LRBs can be modeled as a single LRB. The result is a 100-DOF isolated
structure model. The primary ground excitations used herein are the El Centro (N-S Imperial Valley Irrigation
District substation record of the 1940 Imperial Valley earthquake; peak ground acceleration (PGA) 0.348g)
28
and Northridge (N-S Sylmar County Hospital parking lot record of the 1994 Northridge earthquake; PGA
0.843g) ground motions both sampled at 50 Hz (i.e., Dt= 0.02 s).
To model the hysteresis, the Bouc-Wen model introduces an evolutionary variable, z, that is proportional to
the base drift u
b
for small motion, but asymptotically approaches1 for large motion as the lead shears (or a
friction pendulum isolator surface slides). The equations of motion of the base mass are given by
m
b
¨ u
b
+ c
b
˙ u
b
+ k
post
u
b
+az=m
b
¨ u
g
+ r
T
C
s
( ˙ u
s
r ˙ u
b
)+ r
T
K
s
(u
s
ru
b
) (3.3a)
˙ z= A ˙ u
b
b ˙ u
b
jzj
n
gzj ˙ u
b
jjzj
n1
(3.3b)
wherea= Q
y
[1(k
post
/k
pre
)] is the peak of the non-elastic force; A= 2b = 2g= k
pre
/Q
y
(which constrains
z2 [–1,1] and makes the loading and unloading stiffnesses identical); and the exponent n controls the
sharpness of the hysteresis loop. The superstructure equations of motion, with displacements u
s
relative to
the ground, are given by
M
s
¨ u
s
+ C
s
( ˙ u
s
r ˙ u
b
)+ K
s
(u
s
ru
b
)=M
s
r ¨ u
g
(3.4)
Combining the equations of motion in (3.3a) and (3.4) yields
M¨ u+ C ˙ u+ Ku=M˜ r ¨ u
g
l[az+(k
post
k
b
)u
b
] (3.5)
M=
2
6
4
m
b
0
T
0 M
s
3
7
5 , K=
2
6
4
k
b
+ r
T
K
s
r r
T
K
s
K
s
r K
s
3
7
5 , C=
2
6
4
c
b
+ r
T
C
s
r r
T
C
s
C
s
r C
s
3
7
5
where u(t)=[u
b
(t) u
T
s
(t)]
T
is the generalized displacement vector; ˜ r=[1 r
T
]
T
; and l=[1 0
T
]
T
. Note
that since the term k
post
u
b
+az cannot be in the nominal system in (3.3a), because it depends on design
parameters, the term k
b
u
b
is added to both sides of the equation to preserve the stability of the nominal
system (otherwise, the nominal system remains without any stiffness), where k
b
= 750 kN/m is the nominal
post-yield stiffness that results in the 2.76 s isolation period. The equation of motion in (3.5) can be rewritten
in the state space form of (2.2) with
29
X=
8
>
>
>
>
<
>
>
>
>
:
u
˙ u
z
9
>
>
>
>
=
>
>
>
>
;
, B=
8
>
>
>
>
<
>
>
>
>
:
0
˜ r
0
9
>
>
>
>
=
>
>
>
>
;
, L=
2
6
6
6
6
4
0 0
M
1
l 0
0 1
3
7
7
7
7
5
, A=
2
6
6
6
6
4
0 I 0
M
1
K M
1
C 0
0
T
0
T
0
3
7
7
7
7
5
(3.6)
and a modification g that is a 2 1 vector function
g(X;q q q)=
8
>
<
>
:
az+(k
post
k
b
)u
b
A ˙ u
b
b ˙ u
b
jzj
n
gzj ˙ u
b
jjzj
n1
9
>
=
>
;
(3.7)
where X=[u
b
˙ u
b
z]
T
andq q q =[Q
y
k
pre
k
post
]
T
. Note that although design variables Q
y
and k
pre
are not
shown explicitly in (3.7),a, A,b andg are functions of them. Also, note that g
1
is linear in the states, so
it would typically be included in the nominal system (in the A matrix); however, herein it depends on the
design parameters Q
y
, k
pre
and k
post
, therefore it is excluded from the nominal system so that it can be varied
as a modification to the nominal system.
The selection of the hysteretic isolator parameters is used to demonstrate how the proposed approach can
be used for computationally efficient design of structural control elements. For simplicity, two isolation
parameters are assumed fixed: the base mass m
b
and damping c
b
; the three design parameters to be optimized
are: the yield force Q
y
at which the transition occurs from elastic to partially plastic; the pre-yield stiffness
k
pre
; and the post-yield stiffness k
post
. The exponent n is set equal to 1 at first, which results in smooth
hysteresis loops; other possible values of n will be analyzed in a subsequent section of the chapter.
3.3.2 Baseline Lead Rubber Bearing (LRB) Design
To provide a baseline against which to compare performance, the same structure equipped with an LRB system
as suggested by Skinner et al. (1993) is considered. They suggest a yield force of 5% of the building weight
for small to moderate earthquakes such as El Centro and 15% for strong earthquakes such as Northridge, and
pre-yield to post-yield stiffness ratios of 6 and 10 for moderate and strong excitations, respectively (which are
the values most commonly used in literature). Thus, the two baseline designs are: (a) for El Centro, an LRB
with a yield force 0.05W, pre-yield stiffness 6k
b
and post-yield stiffness k
b
; i.e., q q q
0
EC
=[0.05W 6k
b
k
b
]
T
;
30
and (b) for Northridge, an LRB with a yield force 0.15W, pre-yield stiffness 10k
b
and post-yield stiffness k
b
;
i.e., q q q
0
N
=[0.15W 10k
b
k
b
]
T
.
For a baseline LRB system, one may select either a smooth or bilinear hysteresis; a bilinear one (i.e.,
n= 100 in a Bouc-Wen model is used as an approximation) is considered here due to its wide usage in the
literature. This results in root mean square (RMS) base drift and roof acceleration of 1.84 cm and 77.40 cm/s
2
,
respectively, for El Centro and 6.34 cm and 130.22 cm/s
2
, respectively, for Northridge.
3.3.3 Sensitivity Formulation
For this numerical example, the sensitivity to each design parameter is computed analytically. The required
partial derivatives of g with respect to the parametersq q q and to the states X, defined as d
i
k
=[¶g/¶q
i
]
X
k
,q q q
and
E
k
=[¶g/¶X]
X
k
,q q q
in (3.8)-(3.9), respectively, are
d
1
k
=
¶g
¶q
1

k
=
8
>
<
>
:
¶g
1
/¶Q
y
¶g
2
/¶Q
y
9
>
=
>
;
k
=
8
>
<
>
:
z(1 k
post
/k
pre
)
( ˙ u
b

1
2
˙ u
b
jzj
n

1
2
zj ˙ u
b
jjzj
n1
)k
pre
/Q
2
y
9
>
=
>
;
k
(3.8a)
d
2
k
=
¶g
¶q
2

k
=
8
>
<
>
:
¶g
1
/¶k
pre
¶g
2
/¶k
pre
9
>
=
>
;
k
=
8
>
<
>
:
zQ
y
k
post
/k
2
pre
( ˙ u
b

1
2
˙ u
b
jzj
n

1
2
zj ˙ u
b
jjzj
n1
)/Q
y
9
>
=
>
;
k
(3.8b)
d
3
k
=
¶g
¶q
3

k
=
8
>
<
>
:
¶g
1
/¶k
post
¶g
2
/¶k
post
9
>
=
>
;
k
=
8
>
<
>
:
zQ
y
/k
pre
+ u
b
0
9
>
=
>
;
k
(3.8c)
E
k
=
¶g
¶X

k
=
2
6
6
6
6
4
¶g
1
/¶u
b
¶g
2
/¶u
b
¶g
1
/¶ ˙ u
b
¶g
2
/¶ ˙ u
b
¶g
1
/¶z ¶g
2
/¶z
3
7
7
7
7
5
T
k
=
2
6
6
6
6
4
k
post
k
b
0
0 Abjzj
n
gzjzj
n1
sgn( ˙ u
b
)
a bn ˙ u
b
jzj
n1
sgn(z)gnj ˙ u
b
jjzj
n1
3
7
7
7
7
5
T
k
(3.9)
where the subscript k denotes evaluation at time kDt. For example, the sensitivities of the base drift with
respect to Q
y
, k
pre
and k
post
(at the design point discussed subsequently) are shown in Figure 3.3 along with
those obtained by numerical integration of the analytical sensitivity equations.
31
0 5 10
−200
−100
0
100
200
Time (s)
∂u
b
/∂Q
y
(cm/MN)

Exact solution
Proposed method
(a) Sensitivity w.r.t. Q
y
0 5 10
−200
−100
0
100
200
300
Time (s)
∂u
b
/∂k
pre
(cm
2
/MN)

Exact solution
Proposed method
(b) Sensitivity w.r.t. k
pre
0 5 10
−1500
−1000
−500
0
500
1000
Time (s)
∂u
b
/∂k
post
(cm
2
/MN)

Exact solution
Proposed method
(c) Sensitivity w.r.t. k
post
Figure 3.3: Base drift response sensitivity to the design parameters when subjected to the 1940 El Cen-
tro ground motion at the design point.
In Figure 3.3, the “exact” solution is based on the analytical sensitivity equations shown subsequently in
(3.10)-(3.12). The numerical integration of these equations was conducted by a conventional nonlinear solver:
theode45 command in MATLAB. For the “exact” solution, the relative and absolute tolerances ofode45
were both set to 10
–10
.
If the partial derivatives of the states X with respect to an elementq
i
of the parameter vectorq q q is denoted with
S
q
i
=¶X/¶q
i
, then the analytical sensitivity equations can be calculated by finding the partial derivatives of
the system of equations of the model in state space form with the corresponding matrices shown in (3.6) and
32
(3.7) with respect to the parameterq
i
which leads to
˙
S
q
i
= AS
q
i
+ 0+ L
8
>
<
>
:
¶g
1
/¶q
i
¶g
2
/¶q
i
9
>
=
>
;
. This equation can be
simplified forq
i
= Q
y
, k
pre
, and k
post
, respectively, as following
˙
S
Q
y
= AS
Q
y
+ L
1

(1 k
post
/k
pre
)z+a
¶z
¶Q
y
+(k
post
k
b
)
¶u
b
¶Q
y

+ L
2

(k
pre
/Q
2
y
) ˙ u
b
+ A
¶ ˙ u
b
¶Q
y
+(k
pre
/2Q
2
y
) ˙ u
b
jzj
n
b
¶ ˙ u
b
¶Q
y
jzj
n
b ˙ u
b
njzj
n1
¶z
¶Q
y
sgn(z)+(k
pre
/2Q
2
y
)j ˙ u
b
jzjzj
n1
g
¶ ˙ u
b
¶Q
y
sgn( ˙ u
b
)zjzj
n1
gj ˙ u
b
jn
¶z
¶Q
y
jzj
n1

(3.10)
˙
S
k
pre
= AS
k
pre
+ L
1

Q
y
(k
post
/k
2
pre
)z+a
¶z
¶k
pre
+(k
post
k
b
)
¶u
b
¶k
pre

+ L
2

(1/Q
y
) ˙ u
b
+ A
¶ ˙ u
b
¶k
pre
(1/2Q
y
) ˙ u
b
jzj
n
b
¶ ˙ u
b
¶k
pre
jzj
n
b ˙ u
b
njzj
n1
¶z
¶k
pre
sgn(z)(1/2Q
y
)j ˙ u
b
jzjzj
n1
g
¶ ˙ u
b
¶k
pre
sgn( ˙ u
b
)zjzj
n1
gj ˙ u
b
jn
¶z
¶k
pre
jzj
n1

(3.11)
˙
S
k
post
= AS
k
post
+ L
1

(Q
y
/k
pre
)z+a
¶z
¶k
post
+ u
b
+(k
post
k
b
)
¶u
b
¶k
post

+ L
2

A
¶ ˙ u
b
¶k
post
b
¶ ˙ u
b
¶k
post
jzj
n
b ˙ u
b
njzj
n1
¶z
¶k
post
sgn(z)
g
¶ ˙ u
b
¶k
post
sgn( ˙ u
b
)zjzj
n1
gj ˙ u
b
jn
¶z
¶k
post
jzj
n1

(3.12)
where L
1
and L
2
are the first and second columns of L, respectively.
3.3.4 Design of the Optimal Base Isolation
To find the “best” choice of the design parameters, one may use a parameter study over a fine grid of
design variable values or a conventional gradient-based optimization algorithm, with comparisons of some
33
performance metric(s) to those of a baseline design. Studies in the literature have employed objectives such
as minimizing the superstructure drift or absolute acceleration subject to a constraint on the base drift; others
have used reliability measures. Here, a cost functional expressed in terms of mean square (MS) responses is
used, defined as a weighted linear combination of the MS base drifts
2
u
b
and the MS absolute roof acceleration
s
2
¨ u
a
r
, where ¨ u
a
r
= ¨ u
94
+ ¨ u
g
, as follows:
J(q q q)=
s
2
u
b
(q q q)
s
2
u
b
(q q q
0
)
+
s
2
¨ u
a
r
(q q q)
s
2
¨ u
a
r
(q q q
0
)
(3.13)
wheres
2
()
(q q q
0
) is a MS response of a baseline design. A MS response is approximated here as
s
2
u
b
=
1
t
f
Z
t
f
0
u
2
b
(t)dt
1
N
N
å
k=0
u
2
bk
(3.14)
where t
f
= NDt is the simulation duration (and N = N
t
1), and u
b
k
= u
b
(kDt). The optimization here is
performed using a gradient-based active-set algorithm, implemented in thefmincon command in MATLAB,
to determine the optimal design point. The required derivatives of the cost functional with respect to the
design parameters are required, and are given by
¶J
¶q q q
=
2
N
N
å
k=0
"
u
b
k
s
2
u
b
(q q q
0
)
¶u
b
k
¶q q q
+
¨ u
a
rk
s
2
¨ u
a
r
(q q q
0
)
¶ ¨ u
a
rk
¶q q q
#
(3.15)
fmincon is selected because it can exploit gradient information, which is available through (3.15), and can
accommodate the physically-based constraints: (i) the yield force is always strictly positive, Q
y
> 0; (ii) the
pre-yield stiffness is greater than or equal the post-yield stiffness, k
pre
k
post
; and (iii) the post-yield stiffness
is always non-negative, k
post
0.
3.3.5 Preliminary Optimization overQ
y
andk
pre
A preliminary study of this example in which only Q
y
and k
pre
are free design variables is useful as a first
step since the response metrics can be computed over a fine grid of the two design variables to verify that the
method is efficient and accurate. This simpler design optimization problem, discussed in greater detail in
Johnson et al. (2013), uses the same model but fixes exponent n= 1 and post-yield stiffness k
post
= k
b
.
34
Considering the cost functional (3.13), the design point for the El Centro earthquake (PGA 0.348g) is
determined to be (0.0507W , 5.94k
b
), which results in basically the same RMS base drift and roof acceleration
performances as the baseline (0.05W , 6k
b
). A record from the 1994 Northridge earthquake with a PGA similar
to that of the El Centro record was also used for this 2-D optimization: the E-W motion of the Northridge
earthquake recorded at the USC 90003 station at 17645 Saticoy St (PGA 0.368g); using the same baseline
(0.05W , 6k
b
), the optimization converges to a different design point (0.0401W , 8.73k
b
) for Northridge-Saticoy,
which results in reductions of about 1% and 4% in RMS base drift and roof acceleration, respectively, relative
to the baseline. To converge to the design point, these solutions required only 6 iterations, making a total
of 19 function evaluations, for the El Centro earthquake, and 13 iterations (29 function evaluations) for
Northridge-Saticoy; the quick convergence is facilitated by providing the gradient information. The proposed
method was found to perform the optimization about an order of magnitude faster than the conventional solver
ode45. To verify that these optimizations converged to the correct results, a parameter study was performed
over the two-dimensional design space as shown in Figure 3.4, which confirm that the cost functionals are
convex and well-behaved around the design points.
2.03
2.07
2.1
2.2
2.2
2.3
2.3
2.3
2.4
2.4
2.4
2.5
2.5
2.5
3
3
3
4
4
4
5
5
5
9
9
18
18
yield force Q
y
[%W]
pre−yield to post−yield stiffness ratio k
pre
/k
post
2 4 6 8 10
2
4
6
8
10
12
14
(a) 1940 El Centro
1.92
1.94
1.97
1.97
2
2
2.1
2.1
2.2
2.2
2.4
2.4
2.4
2.6
2.6
2.6
3
3
3
4
4
4
5
5
5
8
8
11
11
yield force Q
y
[%W]
pre−yield to post−yield stiffness ratio k
pre
/k
post
2 4 6 8 10
2
4
6
8
10
12
14
(b) 1994 Northridge-Saticoy
Figure 3.4: Cost contours as a function of two design parameters for two historical earthquakes.
3.3.6 Optimization overQ
y
,k
pre
andk
post
The full design space in this example includes the yield force Q
y
, the pre-yield stiffness k
pre
and the post-yield
stiffness k
post
. It is hypothesized that the optimization will converge to different designs, given that k
post
is a
35
design variable as well, that will provide additional improvements in the response metrics. The initial guess
to start the optimization is chosen to be the same as the baseline values: (Q
y
, k
pre
, k
post
)
i
= (0.05W, 6k
b
, k
b
)
and (0.15W, 10k
b
, k
b
) for the El Centro and Northridge earthquakes, respectively.
The optimal design point for the El Centro earthquake is determined to be (0.0641W , 5.75k
b
, 0.484k
b
), which
results in reductions about 6% and 11% in RMS base drift and roof acceleration, respectively, relative to
the baseline (0.05W, 6k
b
, k
b
). When the Northridge (Sylmar) earthquake excitation (recorded at the parking
lot of the Sylmar County Hospital in the N-S direction with a PGA 0.843g) is applied to the system, the
optimization yields a different design point (0.1371W, 12.45k
b
, 0.61k
b
), which results in reductions of about
4% and 8% in RMS base drift and roof acceleration, respectively, relative to the baseline (0.15W, 10k
b
, k
b
).
To converge to the design point, these solutions required only 12 iterations (making a total of 26 function
evaluations) for the El Centro earthquake, and 18 iterations (43 function evaluations) for the Northridge
excitation; the quick convergence is again facilitated by providing the analytical gradient information from
(3.8)-(3.9).
To verify that these optimizations converged to the correct results, a small parameter study was performed
for the two earthquakes. Figure 3.5 shows contour line slices of the cost functional for the El Centro and
Northridge (Sylmar) earthquake excitations. Clearly, each design point found by the optimization is in the
region of the cost functional minimum, around which the cost functional is convex.
5.5
6.0
6.5
7.0
7.5
5.0
5.5
6.0
6.5
7.0
0.4
0.5
0.6

Q
y
k
pre

k
post
1.7 1.75 1.8 1.85 1.9

(a) 1940 El Centro earthquake
12.5
13.0
13.5
14.0
14.5
11.5
12.0
12.5
13.0
13.5
0.4
0.6
0.8

Q
y
k
pre

k
post
1.75 1.8 1.85 1.9 1.95 2

(b) 1994 Northridge (Sylmar) earthquake
Figure 3.5: Contour lines of the cost as a function of the design parameters for two earthquakes. The
vertex of the cutout shows the optimal design location.
36
3.3.7 Timing and Accuracy of the Proposed Approach
In this section, the computational cost of the optimization method described in the previous section is
compared with that employing a conventional nonlinear solver: a fourth-fifth order variable time step Runge-
Kutta solver implemented with MATLAB isode45. The accuracy of bothode45 and the proposed method
can be tuned, the former by setting options for the absolute and relative tolerances, and the latter by choosing
the integration time step. To ensure a fair comparison, preliminary studies showed that the accuracy of
ode45 using the default tolerance parameters (relative tolerance 10
–3
, absolute tolerance 10
–6
) and the
proposed method, using a second-order accurate trapezoidal integration with 2
15
time steps ofDt = 0.92 ms
duration each, both produce relative response accuracy of order 10
–3
. Figure 3.6 compares base drift and
absolute roof acceleration of the structure, respectively, at the design point for the El Centro excitation as
calculated by the proposed method as well as a reference “exact” solution calculated byode45 with the
relative and absolute tolerances both set to 10
–10
.
0 2 4 6 8 10
−6
−4
−2
0
2
4
6
8
10
12
Time (s)
Base Drift (cm)

Exact solution
Proposed method
(a) Base drift response.
0 2 4 6 8 10
−3
−2
−1
0
1
2
3
4
Time (s)
Abs. Roof Accel. (m/s
2
)

Exact solution
Proposed method
(b) Absolute roof acceleration response.
Figure 3.6: Structural responses to the El Centro earthquake using the design point isolation.
The computational cost of the proposed method includes one time calculations and the repeated ones that, in
this example at the design point, take about 2.13 s and 5.81 s, respectively, on a computer with a 3.4 GHz
Intel core i7-2600 processor and 8 GB of RAM, running MATLAB R2013a under Windows 7. The same
calculation takes about 87.50 s if MATLAB’sode45 (with default tolerances) is used as the solver. This
leads to a computation speed-up of 11.0 for a single simulation but 14.8 in a typical optimization where 25
37
function evaluations performed.
Computing the sensitivities of the cost functional with respect to the three design parameters at each step
roughly doubles or triples the computation time of both MATLAB’sode45 and the proposed method relative
to computing the cost functional alone; however, as expected, including gradient information results in much
faster convergence to the “optimal” design point. In this example, for El Centro MATLAB’sfminsearch
(a derivative-free downhill simplex method based on the Nelder-Mead simplex algorithm (Lagarias et al.,
1998)) takes about 53 iterations and a total of 118 function evaluations, which is 4–5 times larger than for the
gradient-basedfmincon, so, at least for this example, there is a clear computational benefit from including
the analytical gradients.
Note that the proposed method here uses a subdivision of the convolution space to compute portions of the
integral in the VIE using fast Fourier transforms (FFTs) as proposed by Gaurav et al. (2011) and discussed
inx 2.2.5; further, the accuracy and computational efficiency would be expected to be even more superior
if the 4
th
-order integration previously discussed by Gaurav et al. (2011) were used instead of the 2
nd
-order
trapezoidal integration used in this study.
3.3.8 Investigation of the Transitional Region of the Bouc-Wen Model
Traditionally, for the sake of simplicity, many researchers have used the bilinear behavior for the hysteretic
loops. However, the Bouc-Wen model introduced in (3.3) provides a formulation that allows modeling a
smooth transition from the pre-yield to post-yield regions. This is mainly controlled by the exponent n, which
is greater than or equal to 1. Using n= 1 results in a very smooth transition and using a very large n models
the strict bilinear loops; the most common values used in literature are n = 1, 2, and¥. Among all parameters
in this Bouc-Wen model (i.e., Q
y
, k
pre
, k
post
, and n), n is the one with the least tangible physical meaning;
thus, this final section investigates the effect of assuming different values for n.
As shown in Figure 3.7, the shape of the hysteresis loops changes significantly for different values of n and,
as expected, will result in a different “optimal” design point. Assuming the same baseline performance, the
optimization is repeated for n = 1, 2, 10, 100; the design points and performance metrics (RMS base drift
and roof acceleration) are shown in Table 3.1 for the El Centro earthquake. It is clear that the “optimal” Q
y
38
-6 -4 -2 0 2
-1.5
-1
-0.5
0
0.5
1
1.5
Relative base drift, u
b
(cm)
Hysteretic variable z

n=1
n=2
n=10
n=100
Figure 3.7: Single hysteresis loop using the optimal base isolator found for different exponentsn when
building is subjected to the El Centro earthquake.
Table 3.1: Design points for different exponentn values for the El Centro earthquake.
n 1 2 10 100
Q
y
/W [%] 6.4142 5.8800 5.0892 5.0613
k
pre
/k
b
5.7506 5.3100 5.3099 5.3527
k
post
/k
b
0.4840 0.5490 1.0508 1.1144
s
u
b
(% change compared tos
0
u
b
) –6.42 –2.54 3.20 2.96
s
¨ u
a
r
(% change compared tos
0
¨ u
a
r
) –10.81 –10.20 –6.44 –4.82
J 1.67 1.76 1.94 1.97
and k
post
are very sensitive to the value of n as they change by 21% and 130%, respectively, in this example
as n changes from 1 to 100. In contrast, k
pre
changes very slightly and appears relatively insensitive to the
exponent n.
It is also important to investigate the effect on response when the isolators are designed using an inaccurate
value of n. The effects of this incorrect assumption are only studied herein for exponent n because the
remaining parameters (Q
y
, k
pre
and k
post
) can be experimentally determined fairly easily for a particular
device whereas the exponent n is often neglected in curve-fitting from laboratory tests. The response metrics
for the system at the design points shown in Table 3.1 are evaluated for isolators with different n values
and the percent change in their performance metrics, compared to the original design point responses, are
calculated and shown in Table 3.2. It is reported in the literature (Skinner et al., 1993, Nagarajaiah and Sun,
39
Table 3.2: RMS base drift and roof acceleration changes (%) when the design points are evaluated at
different exponentn values.
RMS base drift changes (%) RMS roof acceleration changes (%)
Design n
Actual exponent n Actual exponent n
1 2 10 100 1 2 10 100
1 0 –1.57 8.27 10.93 0 9.98 20.62 22.31
2 3.51 0 6.24 9.25 –8.18 0 8.51 9.64
10 3.66 –0.72 0 0.58 –11.19 –5.87 0 0.86
100 3.94 –0.73 –0.47 0 –11.56 –6.48 –0.80 0
2000) that assuming bilinear hysteresis overestimates the roof acceleration; this is confirmed by comparing
the corresponding columns of Table 3.2. This study shows that designing with a larger n always resulted in
overestimating the actual RMS roof acceleration; but, no clear conclusions can be made about the RMS base
drift. Further investigations (not included here for the sake of brevity) showed that the peak base drift and
peak roof acceleration have variation trends similar to the corresponding RMS metrics, though the changes
are less significant and mostly remain less than 5%.
3.3.9 Case Study: Convergence Rate of the Nonlinear Volterra Integral Equation (NVIE)
Approach for a Strictly Bilinear Hysteretic Model
Most of the computation time of the proposed nonlinear V olterra integral equation (NVIE) approach for
solving (2.6) involves the computation of convolution integrals. As discussed before, one way of solving
(2.6) consists of discretizing the integration space and find the solution using a direct summation such as the
2
nd
trapezoidal rule shown in (2.7). This equation can be solved at each time step using an iterative Newton’s
method (2.10). It is important to determine, for different form of nonlinear function g, whether the Newton’s
method converges and, if so, how many iterations are required since it directly affects the computation time
of whole procedure. A preliminary study for the aforementioned Bouc-Wen study inx 3.3.6 showed that the
Newton’s method converges very fast, within one or two iterations. This section investigates the required
number of iterations for convergence of an extreme Bouc-Wen model (with a very large exponent n) where
the smooth transitional region is replaced by a sharp corner and the model represents a bilinear behavior.
40
The Newton’s method is used in the NVIE approach to find the solution of (2.9), which is basically
f(p
k
)= p
k
g(p
k
)= 0 (3.16)
Assuming a bilinear function g(p
k
)
g(p
k
)=
8
>
<
>
:
a
1
p
k
+ b
1
, p
k
< c
a
2
p
k
+ b
2
, p
k
> c
(3.17)
then f(p
k
) is also a bilinear function and can be described with
f(p
k
)= p
k
g(p
k
)=
8
>
<
>
:
(1 a
1
)p
k
b
1
, p
k
< c
(1 a
2
)p
k
b
2
, p
k
> c
(3.18)
Expanding f as a Taylor series, it is clear that the second derivative of f(p
k
) is always zero (except right at the
corner, at which it is undefined); therefore, Newton’s method converges in one iteration if the initial guess is
located on the line passing through the final solution (as shown in Figure 3.8a); otherwise, if the initial point
is on the other section, Figure 3.8b shows that the convergence occurs in two iterations.
iter. 1
p
k
f(p )
k
p
k
0
p
k
1
iter. 1
p
k
f(p )
k
p
k
0
p
k
1
p
k
2
iter. 2
iter. 1
p
k
0
p
k
1 p
k
f(p )
k
(a) Initial guess and the solution on the same branch.
iter. 1
p
k
f(p )
k
p
k
0
p
k
1
iter. 1
p
k
f(p )
k
p
k
0
p
k
1
p
k
2
iter. 2
iter. 1
p
k
0
p
k
1 p
k
f(p )
k
(b) Initial guess and the solution on different
branches.
Figure 3.8: Newton’s method convergence for a bilinear function.
41
Finally, it is worth noting, if g is only a function of displacements (and not velocities), regardless of the
functional form, H
L
(t
0
) in (2.9) is zero and, therefore, p
k
= g(a
k1
) and no iteration is necessary.
3.4 Conclusions
To enable the optimal design of passive isolation systems, this study proposes extending a computationally
efficient approach previously developed by Gaurav et al. (2011) for systems with local features (linear or
nonlinear, deterministic or random) but are otherwise linear. This approach can provide highly efficient
simulation of both responses and their sensitivities to the design parameters in the isolation element models.
The methodology was applied to the optimal design of a base isolation system to find the optimal yield force,
and pre-yield and post-yield stiffnesses of a hysteretic isolation layer for an 11-story 2-bay isolated building.
The isolator was modeled with Bouc-Wen hysteresis and (small) viscous damping. A baseline design, using a
yield force that is 5% or 15% of the building weight and a pre-yield stiffness that is 6 or 10 times that of the
post-yield stiffness, as suggested in the literature for a moderate and strong ground motions, respectively, are
used for comparison. The optimal design for the El Centro earthquake results in reduction of about 6% and
11% in RMS base drift and roof acceleration, respectively, relative to the baseline; and for the Northridge
earthquake the corresponding reductions are 4% and 8%, respectively.
The proposed approach was able to perform the simulations over an order of magnitude faster; the compu-
tation speed-up was 14.8 for this relatively small-sized example, for a typical parameter study or iterative
optimization. The computational efficiency would be expected to be much greater if the proposed method is
utilized for large-size models with thousands of DOFs.
42
Chapter 4
Computationally Efficient Design of
Optimal Strategies for Semiactive Control
Devices
1
4.1 Introduction
Controllable passive energy dissipation devices (i.e., “semiactive” damping devices) — such as semiactive
impact dampers (Karyeaclis and Caughey, 1989, Masri et al., 1989, Nayeri et al., 2007), magnetorheological
(MR) fluid dampers (Jolly et al., 1996, Spencer et al., 1997, Dyke et al., 1998, Ramallo et al., 2002, Xu et al.,
2013), electrorheological (ER) fluid dampers (Stanway et al., 1996, Gavin et al., 1996, Makris et al., 1996,
Makris, 1997), controllable tuned liquid dampers (Fujino et al., 1992, Yalla et al., 2001), controllable friction
dampers (Dowdell and Cherry, 1994, Lu, 2004, Ng and Xu, 2004) and variable orifice dampers (Feng and
Shinozuka, 1990, Jung et al., 2004, Wongprasert and Symans, 2005) (see Housner et al. (1997) and Symans
and Constantinou (1999) for more references) — have been studied by many researchers for mitigating the
responses of structures to various excitations, including earthquakes and strong winds. Extensive numerical
simulations (e.g., Domaneschi (2010), Wang and Dyke (2013)) and experimental tests (e.g., Ou and Li (2010),
Renzi and De Angelis (2010), Zapateiro et al. (2010), Lu et al. (2011)) have verified the capability of these
1
This chapter is based on Kamalzare et al. (2014b) and Kamalzare et al. (2012)
43
devices for improving system performance significantly. Since these devices have properties that can be
adjusted in real time based on measured structure responses, they are sometimes called “smart” damping
devices (e.g., Kareem and Kline (1995), Dyke et al. (1996), Song et al. (2006), Johnson et al. (2007)) in
contrast with conventional passive devices that exert forces based on a fixed function of (local) structural
response. These devices are also called semiactive since they inherit not only the controllable characteristics
of actuators (e.g., hydraulic cylinders) but also the inherent energy dissipation and fail-safe nature of passive
dampers.
A primary challenge of using controllable dampers is developing appropriate control strategies that provide
optimal performance. Several characteristics of these devices make this process challenging. First, control-
lable dampers cannot exert all possible forces; only those forces that resist motion and dissipate energy are
feasible. This dissipativity constraint is, however, a significant advantage in that the device cannot add energy.
Consequently, even if the sensors fail, the control system is poorly designed, power to sensors or actuators or
control system is lost, or the structure acts significantly differently (e.g., nonlinearly) relative to the model
used for the design, the device still dissipates energy and acts, at worst, like a conventional passive damper.
Second, the controllable properties that appear in a structural/mechanical model of such a device are often
nonlinearly related to the command, which is often current or voltage applied to the device electronics. For
example, MR dampers are commonly controlled by adjusting the electric current in an electromagnet coil;
the relationship between current level and damper force in this case is nonlinear (Dyke et al., 1996, Yang
et al., 2002). Third, since the controllable properties usually enter parametrically in the model (multiplied by
some response quantity), any control command computed from system responses will result in a closed-loop
structure/controller/damper system that is inherently nonlinear. While some researchers have addressed
aspects of this problem, there are no well-established analytical design methodologies that provide the optimal
force strategy.
One common approach for determining the forces to be exerted by controllable damping devices is the
clipped-optimal control strategy (Dyke et al., 1996). In the simplest form of clipped-optimal control, an
ideal controllable damper is commanded to exert some desired force if it is dissipative, and zero force (or
as close to zero as possible) if the desired force is not dissipative. For physical controllable dampers, or
models of them, clipped-optimal control is more complicated but still commands the device to be inactive
at certain times. The desired force is computed, assuming all devices are actuators that can exert any
44
desired force, using standard linear control design strategies — usually an optimal control strategy such
as a linear quadratic regulator (LQR), linear quadratic Gaussian (LQG), H
2
or H
¥
designs (Johnson et al.,
1998, Christenson et al., 2006). However, the use of such a linearized model generally results in suboptimal
(and occasionally poor) performance because the linear actuator assumption differs, sometimes significantly,
from the actual implementation with a dissipative damping device. Thus, one must resort to a large-scale
parameter study in which the nonlinear system is simulated repeatedly to determine control strategies that
are truly optimal for the nonlinear controlled closed-loop system. As each simulation can require significant
computational effort, which is further amplified by the large number of control strategies to be explored, this
design problem can be quite computationally expensive, if not prohibitively so, for determining the optimal
control. For example, to design the optimal semiactive control for a single cable of a cable-stayed bridge, the
parameter study simulations to determine the root mean square (RMS) responses of a cable with an attached
optimal controllable damping device required 9 CPU-months in 2005–06 (Johnson et al., 2007). Clearly,
this computational burden renders the design of such systems impractical without advances in simulation
methodologies.
This chapter demonstrates how the approach introduced in Chapter 2, using a set of nonlinear V olterra integral
equations (NVIEs) to determine response of systems with local modifications, can significantly reduce the
computational burden of a complex control design study for controllable dampers. The computationally
efficient approach is first adapted to use a state feedback control with a bang-bang, on-off secondary controller.
Then, the structure and damping device models are detailed. The optimal semiactive control strategy is then
determined through both a parameter study and an optimization algorithm. Numerical examples demonstrate
that the approach is effective and, compared to a conventional simulation approach, reduces computation
time by two orders of magnitude.
4.2 Methodology
4.2.1 Clipped-Optimal Control
While the proposed method is amenable to a wide variety of control strategies, the study herein uses a
clipped-optimal control incorporating a primary controller, using linear state feedback, with an ideal damper
45
model. An “ideal” controllable damper is one that can exert any dissipative force, which is a force that resists
motion, i.e., a force in the direction opposite that of the velocity across the device. Assume some control
design strategy (discussed later) provides a vector p
d
(t)=K
d
X(t) of desired device forces, where K
d
is
some state feedback control gain. The clipping algorithm to determine the force command for the i
th
device
is, then, shown graphically in Figure 4.1 and given by
g
i
(X)= p
SA
i
= p
d
i
H(p
d
i
v
i
)=
8
>
<
>
:
p
d
i
, p
d
i
v
i
< 0 (dissipate energy)
0, p
d
i
v
i
0 (cannot add energy)
(4.1)
dissipative
dissipative
p
i
SA
= p
i
d
p
i
SA
= p
i
d
nondissipative
nondissipative
p
i
SA
= 0
p
i
SA
= 0
v
i
p
i
d
Figure 4.1: Clipping algorithm for “ideal” semiactive damper.
where p
SA
i
(t) is the actual force exerted by the i
th
device, p
d
i
(t) is the i
th
element of desired force vector
p
d
(t), H() is the Heaviside unit step function, and v
i
(t) is the velocity across the i
th
device. If the vector
of device velocities is given by v(t)= VX(t), then the control force function g() is only dependent on
X=[(p
d
)
T
v
T
]
T
= GX where G=[(K
d
)
T
V
T
]
T
is a 2n
p
n
x
matrix, and g
i
(X)= p
d
i
H(p
d
i
v
i
)=
X
i
H(X
i
X
i+n
p
) for i= 1,...,n
p
. Figure 4.2 shows the semiactive control block diagram including the clipped
optimal control. In part of the numerical examples, a saturation of the peak damper force is also enforced, as
shown in Figure 4.2; this saturation could be included in (4.1) as well.
It should be noted that g() for this idealized model of a controllable damper is continuous except at exactly
zero velocity; theoretically, this could cause some difficulty in the Newton-Raphson iteration (2.10) due to
singularity of the Jacobian of g at zero velocity but, in practice, it was not found to present problems for
the simulations herein (more realistic damper models, such as those with continuous transition from zero to
non-zero force, should be studied in the future) and the iteration converged rapidly within a few iterations.
46
Figure 4.2: Semiactive control block diagram.
4.2.2 Semiactive Control Design Parameter Study
For a low order system with very few controllable dampers, it may be possible to perform a parameter study
over a mesh of all elements of gain matrix K
d
. For a system of real size, such a parameter study would be a
monumental task. For example, in the subsequent numerical example, the control gain K
d
i
is a vector of 200
elements; it is impractical and prohibitively computationally expensive to search for an optimum in this 200
dimensional space. An alternate approach is to study a family of control gains generated by parameterizing
an optimal control design. Herein, a LQR approach is used to compute a family of state feedback gains K
d
i
,
each of which minimizes, for a linear actuator, the cost function
J
i
=
Z
T
0
[Z
T
i
(t)Q
i
Z
i
(t)+ p
T
i
(t)R
i
p
i
(t)]dt (4.2)
where Z(t)= C
z
X(t)+ D
z
w(t)+ F
z
p(t) is a vector of controlled responses to be regulated, with corre-
sponding uncontrolled responses z(t)= C
z
x(t)+ D
z
w(t); C
z
is n
z
n
x
, D
z
is n
z
n
w
, and F
z
is n
z
n
p
.
For the optimization to be well-posed, the weighting matrices must satisfy Q
i
0 and R
i
0; i.e., Q
i
must
be positive semidefinite, and R
i
must be positive definite. Parameterizing the weighting matrices with a few
parameters, such as(a
i
,b
i
) in the subsequent numerical example, reduces the dimension of the optimization
space from the high-dimensional space of all control gains to a much lower dimensional parameter space (see
Figure 4.3).
47
Space of all possible K’s
(200 dimensional space)
(2 dimensional space)
LQR K
i
Figure 4.3: Dimension of optimization space.
To compute the regulated responses with all of the candidate control gains, one must:
1. Compute the uncontrolled responses x(t) and z(t).
2. (a) Compute the full impulse response function H
L
(t); and
(b) use matrix multiplication to compute the partial impulse response functions
H
V
(t)= VH
L
(t) and H
C
z
(t)= C
z
H
L
(t) (4.3)
3. For each candidate control gain K
d
i
:
(a) compute H
K
d
i
(t)=K
d
i
H
L
(t);
(b) solve the NVIEs to compute the device force p
i
(t) using candidate control gain K
d
i
;
(c) compute controlled regulated responses
Z
i
(t)= z(t)+ F
z
p
i
(t)+
Z
t
0
H
C
z
(tt)p
i
(t)dt. (4.4)
Since the impulse response functions H
K
d
i
(t) and H
V
(t) in (4.3) are of low dimension and are
pre-computed, the time integration to compute p(t) can be done efficiently. Further, H
C
z
(t) is
computed only once, and the convolution in (4.4) can be computed very efficiently using fast Fourier
transforms (FFTs).
4. Evaluate the resulting setfZ
1
(t),Z
2
(t),...g of regulated responses to determine which control gain
is optimal. For example, define the actual cost J
c
i
of the simulated responses when using candidate
48
control gain K
d
i
to be
J
c
i
=
Z
T
0
[Z
T
i
(t)QZ
i
(t)+ p
T
i
(t)Rp
i
(t)]dt, (4.5)
for a single specific set of response weight Q and constant weight R, where Q 0 and R 0. Then,
choose the candidate control gain K
d
i
that gives minimum J
c
i
. If the weights in (4.2) are parameterized,
such as in the numerical example that follows, then a response surface of J
c
can be formed as a function
of those parameters, and a minimum of the response surface found, yielding an optimal control gain
K
d
.
There are many ways to parameterize the LQR cost function (4.2). One approach, which is used to evaluate
the tradeoffs between control effort and response reduction, is to let force weight R
i
= I and scale the
response weight Q
i
=a
2
i
Q for some base weight Q. Another approach, which is used in the subsequent
numerical example to evaluate the tradeoffs between different components of the regulated responses, is to
again let force weight R
i
= I but scale portions of the response weight differently: Q
i
= I
a
i
QI
a
i
where
I
a
i
= diag[a
i,1
1
1n
1
...a
i,m
1
1n
m
] and n
1
+ n
2
++ n
m
= n
z
(if m= 1, then Q
i
reverts back toa
2
i
Q).
4.3 Numerical Example
4.3.1 Model Description: Base Isolated Frame Structure
While the method proposed herein is applicable to a variety of applications of controllable passive dampers,
a numerical example of a base-isolated building is used to demonstrate the efficacy of the method. Base
isolation systems (Housner et al., 1997, Skinner et al., 1993), among the most commonly-adopted methods
to protect civil structures against severe excitations, are developed based on the fundamental concept of
separating the structure from the ground and preventing the ground excitation energy from being transferred
to the superstructure. Passive base isolation can be augmented with actuators (Kelly et al., 1987, Reinhorn
et al., 1987, Nagarajaiah et al., 1993, Schmitendorf et al., 1994, Reinhorn and Riley, 1994, Yoshida et al.,
1994, Yang et al., 1996) or semiactive devices (Feng and Shinozuka, 1990, Nagarajaiah, 1994, Makris, 1997,
Johnson et al., 1999, Kurata et al., 1999, Niwa et al., 1999, Symans and Constantinou, 1999, Symans and
Kelly, 1999, Yoshida et al., 1999, Ramallo et al., 2002) to provide significant improvements in mitigating
49
structural responses and, thereby, reducing damage to both structural and nonstructural components.
Consider the isolated superstructure introduced inx 3.3.1 with linear stiffness and viscous damping that has a
semiactive control device in the isolator layer as shown in Figure 4.4.
Damper Base
nd
2 floor
th
10 floor
th
11 floor
st
1 floor
Ground
low-damping
isolation bearings
u
b
u
b
u
3
u
6
u
9
u
12
u
15
u
18
u
84
u
87
u
90
u
1
u
4
u
7
u
10
u
13
u
16
u
82
u
85
u
88
u
91
u
94
u
97
u
2
u
5
u
8
u
11
u
14
u
17
u
83
u
86
u
89
u
92
u
95
u
98
u
93
u
96
u
99
Figure 4.4: 11-story base-isolated frame model.
Note that when linear stiffness and viscous damping are considered, if no supplemental controllable damping
devices were to be added, a higher damping ratio would be used. However, a low linear damping ratio is
chosen here so that the controllable damper can add damping when required; if the linear damping ratio
were large, the controllable damper would have difficulty lowering the effective damping, even if that
provided optimal performance, since the controllable damper primarily adds damping. The 100-degree of
freedom (DOF) isolated structure model with the semiactive device has the equation of motion
M¨ u+ C ˙ u+ Ku= B
w
¨ u
g
+ B
p
p (4.6)
where u(t) is the same generalized displacement vector, B
w
=M[1 r
T
]
T
, B
p
=[1 0
T
]
T
, and p(t) is the
force exerted by the controllable damper.
50
Consequently, the equation of motion can be written in state space form (2.2) using w= ¨ u
g
, x
0
= 0, and
X=
2
6
4
u
˙ u
3
7
5 , A=
2
6
4
0 I
M
1
K M
1
C
3
7
5 , B=
2
6
6
6
6
4
0
1
r
3
7
7
7
7
5
, L=
2
6
4
0
M
1
B
p
3
7
5 (4.7)
The ground excitation applied in this study is the N-S El Centro record of the 18 May 1940 Imperial Valley
earthquake sampled at 50 Hz (i.e.,Dt= 0.02 s).
4.3.2 Baseline Performance: lead rubber bearing (LRB) System
An LRB, shown in Figure 4.5, is an isolator that is often used in conventional base isolated buildings. To
provide a basis for performance comparison, an LRB system is used as a baseline (Ramallo et al., 2002). As
discussed inx 3.3.2, an LRB isolator is often modeled with a bilinear hysteretic stiffness, parameterized by a
pre-yield stiffness k
pre-yield
, a post-yield stiffness k
post-yield
, and a yield force Q
y
at which the stiffness transition
occurs (see Figure 3.2). The same recommended values discussed inx 3.3.2 for El Centro earthquake are
used here. However, as the sharp nonlinearity of a bilinear hysteresis is both inconvenient for computational
approaches and results in overestimation of the accelerations (Skinner et al., 1993, Nagarajaiah and Sun,
2000), a smooth Bouc-Wen model (Wen, 1976) of the LRB (i.e., n= 1) is used in this chapter.
Figure 4.5: Schematic view of an LRB device.
The force applied by an LRB system, using a Bouc-Wen model, can be written as
f
LRB
= Q
Pb
z+ k
b
u
b
+ c
b
˙ u
b
(4.8)
51
where Q
Pb
= (1 k
post-yield
/k
pre-yield
) Q
y
is the yield force of the lead plug, and z is an evolutionary variable
that incorporates hysteretic behavior into the model.
The response of the isolated structure with LRB isolators to the 1940 El Centro earthquake is simulated. The
resulting RMS base drift and absolute roof acceleration are 18.5 mm and 0.668 m/s
2
, respectively. These
values are the baseline against which semiactive responses are evaluated.
4.3.3 Control with Ideal Smart Controllable Dampers
For this numerical example, X(t) is a 2 1 vector containing desired device force, p
d
(t) =K
d
X(t),
and the base velocity ˙ u
b
(t). Consequently, G=[(K
d
)
T
V
T
]
T
where V=[0
T
1 0
T
]
T
; and function
g(X(t))= p(t), which is now a scalar function, can be written as g(X(t))= X
1
(t)H(X
1
(t)X
2
(t)). The
outputs Z(t) to be regulated include the base drift (1
st
DOF) and the absolute horizontal roof acceleration
(acceleration at the 95
th
DOF). If A and L are partitioned as[a
1
a
2
]
T
(i.e., a
T
i
is the i
th
row of A) and
[l
1
l
2
]
T
, respectively, then C
z
, D
z
, and F
z
can be defined as:
C
z
=
2
6
4
1 0 0
a
T
195
3
7
5 , D
z
=
2
6
4
0
0
3
7
5 , F
z
=
2
6
4
0
l
195
3
7
5 (4.9)
The cost function is defined as in (4.2) with (now scalar) control weight R
i
= W
2
and diagonal response
weight Q
i
= diag[a
i
/h
2
,b
i
(T
i
1
)
4
/(16p
4
h
2
)], where a
i
and b
i
are dimensionless parameters that will be
tuned to achieve the best semiactive control performance; the scaling by 1/h
2
and (T
i
1
)
4
/(16p
4
h
2
) are
used to nondimensionalize the cost function. The control gain K
d
i
is designed using thelqr command in
MATLAB for each pair (a
i
,b
i
). The commanded force is clipped to 0.15W (i.e., 15% of the building weight)
to accommodate control device practical performance limits, and non-dissipative forces are clipped to zero.
This control gain K
d
i
would minimize the cost function J
i
if neither type of clipping exists; however, the
clipping controller changes the desired force p
d
i
to some other actual force p
i
(see Figure 4.6).
52
Figure 4.6: Modified cost function.
4.3.4 Parameter Study: Design of the Optimal Controller
To find the “best” semiactive design, one can search over a fine grid of (a
i
,b
i
) values and compare the
responses with the baseline to determine the design parameters (a

,b

) that achieve the greatest performance
improvement (see Figure 4.7). The simulation (the oval in Figure 4.7) can be done with traditional solvers
(such asode45) or replaced by the fast response approach discussed in Chapter 2 which displays greater
computational efficiency. A preliminary parameter study is performed overa
i
2[10
4
,10
6
] andb
i
2[10
3
,10
5
]
Figure 4.7: Parameter study procedure.
(a total of 400 sample points); the variation of the differences in responses compared to the baseline LRB
design over the sampled parameter space is shown in Figure 4.8. The thick black dashed line in Figure 4.8
denotes the curve on which, using a smart damper, the absolute acceleration is the same as the LRB design; the
gray region to the left is where the smart damper provides acceleration reductions relative to LRB. The solid
blue contour lines show the percent change in the base drift provided by the controllable damper relative to
the LRB. The point chosen as the “optimal” control design for this example is(a

,b

)=(5 10
5
,2 10
4
),
53
−20% −20% −20%
−20%
−20%
−20% −20% −20% −20%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
−60%
−40%
−20%
0%
Drift Weight α
t h g i e W n o i t a r e l e c c A β
improvements (decrease)
in roof acceleration
10
4
10
5
10
6
10
3
10
4
10
5
Design point results in
28% base drift reduction,
no increase in roof accel.
decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB
decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB
decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB
decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB
decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB decreasing RMS base drift relative to LRB
Figure 4.8: RMS of the semiactive responses relative to those with the LRB.
which provides a smart controller with about a 28% decrease in the base drift without increasing the roof
absolute acceleration relative to the LRB; this point is shown in Figure 4.8 as a red circle.
4.3.5 Timing and Accuracy of the Proposed Approach
In this section, the computational cost of the proposed method is compared with MATLAB’sode45 command,
which is a typical solver used to obtain the response of nonlinear systems. To perform a fair comparison,
theode45 parameters (i.e., relative and absolute error tolerance) and the time-step in the proposed method
should be chosen so that the responses computed by the two methods have the same level of accuracy. For
this example,ode45 with the default parameters (relative tolerance 10
3
, absolute tolerance 10
6
) and the
proposed method, using a 2
nd
-order accurate trapezoidal integration with 2
16
time steps ofDt = 0.46 ms
duration each, both give response relative accuracy of order 10
3
at the design point. The accuracy of the
proposed method can also be seen in Figure 4.9 which illustrates the base drift and absolute roof acceleration
of the structure, using a control strategy at the design point(a

,b

), as calculated by both methods.
54
0 2 4 6 8 10
−6
−4
−2
0
2
4
6
Time [s]
Base Drift [cm]
Traditional solver
Proposed method
0 2 4 6 8 10
−3
−2
−1
0
1
2
3
Time [s]
Abs. Roof Accel.[m/s
2
]
Figure 4.9: Base drift (left) and absolute roof acceleration (right) responses to the 1940 El Centro
earthquake using the design point controller
The computational cost of the proposed method includes one time calculations and the repeated ones that,
in this example, take approximately 8.00 s and 2.29 s, respectively, on a computer with a 3.4 GHz Intel core
i7-2600 processor and 8 GB of RAM, running MATLAB R2011a under Windows 7. Performing the same
analysis at the design point takes 243.73 s when usingode45, which is about 24 times longer (see Table
4.1). In the case of multiple simulations, greater gains in computational efficiency are achieved since the
proposed method can perform each new simulation in just 2.29 s.
Note that the computational cost for each pair(a
i
,b
i
) varies when usingode45, primarily due to variations
in the aggressiveness of the control strategies; however, it remains constant in the proposed method for
constant time integration step. Conducting the parameter study of 400 simulations, each with a different K
d
i
,
with bothode45 and the proposed method, shows that the proposed method is much faster, about 298 times
faster, as shown in Table 4.1. Further, the accuracy and computational efficiency would be expected to be
even more superior if the 4
th
-order integration previously discussed by Gaurav et al. (2011) were used instead
of the 2
nd
-order trapezoidal integration used in this study. The net result is that it is feasible to simulate
systems of much greater complexity (e.g., bridge designs with thousands of DOFs) than is computationally
tractable with conventional nonlinear solvers.
55
Table 4.1: Cost ratio comparison.
ode45 Proposed method
RelTol = 10
3
# of steps = 2
16
Cost ratio
AbsTol = 10
6
Dt= 0.46 ms
at the design point
(1 simulation)
243.73 s 8.00+ 2.29= 10.29 s 24
parameter study 2.75 10
5
s 8.00+ 400 2.29= 924.00 s
(400 simulations) (3.2 days) (15.4 mins)
298
4.3.6 Design an Optimal Semiactive Control Strategy Using an Optimization Algorithm
While the parameter study, discussed inx 4.3.4, can provide a global view of how the controlled base drift
and roof acceleration vary in the design space, it might be too computationally expensive to perform if a high
level of accuracy is desired, which would require a very fine(a,b) mesh. To design an optimal controller
with much fewer function evaluations, one can instead apply an appropriate optimization algorithm to the
objective function along with the necessary equality/inquality constraints.
The same 11-story isolated structure introduced inx 4.3.1 is considered here to investigate the possibility of
utilizing an optimization algorithm in designing a semiactive control device. The main goal is to determine
an optimal design point that results in the maximum reduction in the base drift compared to the baseline
(the LRB passive design) while the absolute roof acceleration remains almost the same. Moreover, since the
demands on the semiactive damper are related to the RMS control force, keeping the RMS device force at, or
below, a specific level is considered as an additional constraint for this optimization.
Assume s
SA
d
(a,b), s
SA
a
(a,b), and s
SA
p
(a,b) represent the RMS base drift, absolute roof acceleration,
and actual semiactive actuator force, respectively, for a particular excitation and a controller designed with
cost function parameters(a,b). In addition, let s
LRB
d
and s
LRB
a
denote the RMS base drift and absolute
roof acceleration corresponding to the baseline LRB design introduced inx 4.3.2. Since the scaling of
the objective function and constraints affect the performance of the optimization, the objective function
is defined as the relative change in the RMS base drift compared to the corresponding baseline design,
s
d
(a,b)=[s
SA
d
(a,b)/s
LRB
d
1]; the constraints are represented as the relative change in the RMS roof
absolute acceleration compared to the corresponding baseline design,s
a
(a,b)=[s
SA
a
(a,b)/s
LRB
a
1],
and the RMS actuator force level is expressed as a fraction of the building weight,s
p
(a,b)=s
SA
p
(a,b)/W .
56
The optimization problem, then, is to minimize the controlled base drift, relative to that of the LRB baseline,
while ensuring that the absolute roof acceleration is the same as, or smaller than, with the LRB baseline, and
the RMS damper forces are constrained to be no larger than a critical values
d
p
(that will be subsequently
varied to see the tradeoffs between force levels and response reduction):
min
a,b
s
d
(a,b) (4.10)
subject tos
a
(a,b) 0 ands
p
(a,b)s
d
p
Since changes in RMS base drift and roof acceleration are opposites in most of the search space, the maximum
reduction in the base drift can be achieved when the roof acceleration is not improved at all. Consequently,
the inequalitys
a
(a,b) 0 is modified and considered as an equality constraint in (4.10).
To solve (4.10), a variety of optimization algorithms could be adopted. Since the equality constraint
s
p
(a,b) = s
d
p
is not smooth, preliminary studies found that gradient-based optimization methods did
not convergence as quickly and consistently as a derivative-free approach. This study uses MATLAB’s
optimization toolbox function optimizerfminsearch, which is a derivative-free downhill simplex method
based on the Nelder-Mead simplex algorithm (Lagarias et al., 1998). Sincefminsearch does not require
gradient evaluation, it is a useful optimizer in finding the minimum of non-smooth objectives or constraints.
The Nelder-Mead algorithm (Nelder and Mead, 1965) evaluates the cost function at the vertices of a simplex
(which is a triangle in this study with a two-dimensional search space); it then iteratively rejects the worse
vertex (with larger cost) and replaces that point by applying reflection, expansion, contraction or shrinking
operations, with the simplex shrinking in size from iteration to iteration and converging to a local minimum.
To implementfminsearch for (4.10), the constraints are reformulated as penalty functions, resulting in
augmented objective function
min
a,b
s
d
(a,b)+ a
0
[s
a
(a,b)]
2
+ b
0
[s
p
(a,b)s
d
p
]
2
(4.11)
where a
0
and b
0
are the weights corresponding to the constraints s
a
and s
p
, respectively; a
0
= 10
2
and
b
0
= 10
4
are considered for this study; the former ensures the optimal point is selected near the s
a
= 0
region, whereas the latter forces the optimization algorithm to stay strictly on the s
p
= s
d
p
level. The
57
initial guess used to start the optimization is (a,b)=(1000,10); the optimization algorithm terminates
when changes in the objective function are less than 0.01. The control design is calculated for different
RMS force levels s
d
p
under El Centro (N-S Imperial Valley Irrigation District substation record of the
1940 Imperial Valley earthquake) and Northridge (N-S Sylmar County Hospital parking lot record of the
1994 Northridge earthquake) excitations, which may be considered commonly-used moderate and strong
earthquakes, respectively. To understand how severely the saturation constraint may be violated by various
designs, the saturation is not enforced in this section.
It is found that base drift reduction can be achieved for weight-normalized RMS control force s
d
p
in
[1.9%,2.6%] and[3.9%,5.0%] intervals for El Centro and Northridge earthquakes, respectively, whiles
a
is forced to be zero (all s
a
’s are on the order of 0.001% except a few where the objective function and
constraints are parallel, causing the optimization algorithm to stop ats
a
= 0.1%). Figures 4.10 and 4.11 show
the candidate design points along with thes
d
ands
d
p
contours for these two excitations. In these figures, the
shaded area represents the region where the absolute roof acceleration is decreased. Note that the curves
which represents
a
= 0 in Figures 4.10 and 4.11 are smoother than the one in Figure 4.8 because no force
saturation is applied in this section. (Note that a full parameter study over a mesh of(a,b) values is used
to generate the contour graphs in Figures 4.10 and 4.11, which are not needed for the optimization, but are
instructive for explaining the design tradeoffs.)
Since very small tolerances are considered for this study, on averagefminsearch takes about 135 function
evaluations to determine the design point for a specifics
d
p
for the El Centro excitation. If the termination
criterion is relaxed to allow a final objective function value up to 1% larger than its true minimum (which
results in less than 1% error in each ofs
d
,s
a
, ands
p
), only about half the number of function evaluations
are required without losing significant accuracy.
The design points and their corresponding objectives
d
and constraints
p
values are summarized in Tables 4.2
and 4.3 for the El Centro and Northridge excitations, respectively. The peak force is another important criteria
in designing semiactive controllers because it has a direct impact on determining the size of the actuators.
Large magnitude forces are unfavorable due to the cost of the device and space limitations where they are
placed in the structure. Therefore, the peak actuator force for each individual design point is also included in
these tables.
58
10
2
10
3
10
4
10
5
10
6
10
0
10
1
10
2
10
3
10
4
10
5
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80% −80% −80% −80% −80% −80% −80% −80% −80%
−80%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0% 0% 0% 0% 0% 0% 0% 0% 0%
0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0% 2.0%
2.0%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2% 2.2%
2.2%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4% 2.4%
2.4%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6% 2.6%
2.6%
Drift Weight α
Acceleration Weight β
roof accel. reduced
base drift change
RMS force
design points
Figure 4.10: Design points at different RMS force levels under the El Centro excitation
Table 4.2: Design points at different RMS force levels under the El Centro excitation.
Design Point RMS Forces
p
Peak Force Base Drifts
d
a

b

[% of building weight] [% change relative to LRB]
1243.04 5.88 1.90 7.69 2.78
1566.68 10.77 2.00 7.98 8.15
2007.99 18.82 2.10 8.23 12.78
2664.67 34.02 2.20 8.50 16.75
3851.38 68.98 2.30 8.76 20.12
7120.54 184.97 2.40 11.68 23.09
32834.31 1177.03 2.50 22.75 26.14
149205.38 5467.54 2.60 54.73 28.20
In Figure 4.12, base drift reduction and peak actuator force are shown as functions of RMS force. As expected,
there is a trade-off between the resulting base drift reduction and the required peak actuator force. The last
few design points under each excitation result in the greatest decrease in the base drift, but at the expense of
requiring dramatically large (and impractical) peak force. Therefore, to have a significant reduction in the base
drift with a feasible peak device force, the suggested design points are[a

,b

]=[7120.54,184.97] (s
p
=
2.4%) and[4804.32,548.79] (s
p
= 4.8%) for the El Centro and Northridge excitations, respectively, which
provide, respectively, about a 23% and 61% base drift reduction, without increasing the roof acceleration; the
59

10
2
10
3
10
4
10
5
10
6
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
=80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80% =80%
−80%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
=60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60% =60%
−60%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
=40% =40% =40% =40% =40% =40% =40% =40% =40%
−40%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9% 3l9%
3.9%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5% 4l5%
4.5%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0% 5l0%
5.0%
Drift Weight α
Acceleration Weight β
roof accel. reduced
base drift change
RMS force
design points
Figure 4.11: Design points at different RMS force levels under the Northridge excitation
Table 4.3: Design points at different RMS force levels under the Northridge excitation.
Design Point RMS Forces
p
Peak Force Base Drifts
d
a

b

[% of building weight] [% change relative to LRB]
540.82 0.17 3.90 22.28 49.81
592.86 0.99 4.00 23.52 51.14
652.22 2.39 4.10 24.80 52.40
720.85 4.68 4.20 26.04 53.61
805.91 8.86 4.30 27.28 54.72
926.31 18.03 4.40 28.51 55.76
1129.13 39.58 4.50 29.65 56.89
1562.95 96.17 4.60 30.58 58.18
2468.49 221.38 4.70 31.28 59.45
4804.32 548.79 4.80 37.20 60.57
17443.23 2326.10 4.90 58.06 61.56
58966.15 8028.90 5.00 89.99 62.14
RMS and peak actuator forces are 2.4% and 12% of the building weight for the El Centro excitation, and
are 4.8% and 37% for the Northridge excitation. (Since force saturation was not used in this section, the
Northridge peak force is somewhat large, but would be significantly reduced by force saturation without
a significant performance penalty. Further, lower peak force design points can be found, using smallers
d
p
60
and relaxing the roof acceleration equality constraint back to the inequality in (4.10), such as [a

,b

]=
[379.31,0.01] (s
p
= 3.5%) that results in Northridge reductions of about 44% in RMS base drift and 5% in
RMS roof acceleration, relative to the baseline, while using a peak force of 18.9% of building weight.)
1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 4.1 4.3 4.5 4.7 4.9
−100
−50
0
RMS Base Drift (relative to LRB baseline) [%]
RMS Force [%W ]

1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 4.1 4.3 4.5 4.7 4.9
0
50
100
Peak Force [%W ]
El Centro base drift
Northridge base drift
El Centro peak force
Northridge peak force
Figure 4.12: Base drift (relative to LRB) and corresponding peak force (relative to building weight)
under El Centro and Northridge excitations for different weight-normalized RMS force levels
4.3.7 Semiactive Performance Compared to a Linear Passive Viscous Damper
Inx 4.3.4 and 4.3.6, the performance baseline was that of the structure with an LRB in the isolation layer.
However, one may be interested in considering a different baseline design, such as an “optimal” linear passive
viscous damper that has characteristic behavior similar to the controllable damper. The goal of this section is
to perform this comparison and investigate whether the same level of improvements, relative to this different
baseline, is still achievable.
61
A typical passive viscous damper consists of a piston forcing a highly viscous oil (Housner et al., 1997);
nonlinear terms can often be neglected (Symans and Constantinou, 1999), so the device force can be expressed
f
d
=c
d
˙ u
b
(4.12)
where c
d
is the constant damping coefficient of the device and ˙ u
b
is the relative velocity across the device
(which, here, is the base velocity). To design the “best” viscous damper, a brute force parameter study is
performed over c
d
2[0,10
6
]; the system RMS response and the RMS device force are depicted in Figure 4.13
as functions of damper coefficient c
d
for the 1940 El Centro earthquake. As shown in this figure, by
10
3
10
4
10
5
10
6
0
20
40
60
80
100
RMS Base Drift [mm], RMS Abs. Roof Accel. [cm/s
2
]
c
d
[N ⋅s/m]
10
3
10
4
10
5
10
6
0
0.6
1.2
1.8
2.4
3.0
RMS Viscous Damper Force [%W ]
RMS Base Drift
RMS Abs. Roof Accel.
RMS Damper Force
design point
Figure 4.13: Responses and device force of the system with a linear viscous damper under El Centro
excitation
increasing the damping coefficient, the RMS base drift strictly decreases and the RMS device force strictly
increases; the RMS absolute roof acceleration first decreases, comes to a minimum, and then increases. (This
qualitative behavior also occurs when considering other ground motions as well.) The point at which the
RMS absolute roof acceleration has the minimum is a reasonable candidate for the design point; for the
El Centro excitation, this point corresponds to c
d
= 104.648 kNs/m, resulting in RMS base drift, absolute
62
roof acceleration and passive damper force of 40.0 mm, 0.407 m/s
2
and 10.982 kN, respectively. Compared
to the LRB performance, the viscous damper results in a larger RMS base drift but smaller RMS absolute
roof acceleration and device force (0.9% of the building weight for the viscous damper against 2.3% for the
LRB system).
Relative to the performance of this new baseline design, the parameter study shown in Figure 4.10 can
be repeated; the updated performance contours (generated using the proposed computationally efficient
approach, with a parameter study over a mesh of weights a and b) are shown in Figure 4.14 along with
the new design points (found with an optimization similar to those performed inx 4.3.6) that are also listed
in Table 4.4. The viscous damper results in RMS base drift that is larger than with the LRB, and a smaller
RMS absolute roof acceleration; so optimization, as expected, results in new design points that provide larger
improvements in RMS base drift and require smaller RMS device forces.
10
2
10
3
10
4
10
5
10
6
10
0
10
1
10
2
10
3
10
4
10
5
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60% −60% −60% −60% −60% −60% −60% −60% −60%
−60%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40% −40% −40% −40% −40% −40% −40% −40% −40%
−40%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20% −20% −20% −20% −20% −20% −20% −20% −20%
−20%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%
0.9%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%
1.2%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%
1.5%
Drift Weight α
t h g i e W n o i t a r e l e c c A β
roof accel. reduced
base drift change
RMS force
design points
Figure 4.14: Design points and response improvements compared to the viscous damper performance
under the El Centro excitation.
To conclude, it is important to emphasize that, independent of the choice of the baseline performance (which
is subjective according to the designer’s judgement), the proposed method is equally applicable and can
63
Table 4.4: Design points at different RMS force levels under the El Centro excitation using the passive
linear viscous damper as a baseline.
Design Point RMS Forces
p
Peak Force Base Drifts
d
a

b

[% of building weight] [% change rel. to viscous damper]
132.34 10.24 0.90 3.80 –4.63
181.62 15.50 1.00 4.27 –13.37
251.31 26.29 1.10 4.72 –21.11
362.74 51.15 1.20 5.14 –28.06
586.57 119.43 1.30 5.45 –33.81
1449.69 438.94 1.40 7.86 –38.84
11763.07 4221.48 1.50 25.62 –42.29
provide a significant reduction in the computational cost of the semiactive design procedure.
4.4 Conclusions
The method, previously discussed in Chapter 2 to find the response of a dynamical system with local
nonlinearities and uncertainties, is adapted specifically to accommodate semiactive control design. The
primary focus of this chapter is to investigate the possible improvements in computational efficiency that
the proposed method can provide for large-scale semiactive control design parameter studies. A parameter
study was performed on a representative frame superstructure with a controllable damping device in the
isolation layer. While the computational efficiency is directly related to the size of the system, the efficiency
is improved by two order of magnitudes for this small-size model. Further, to minimize a cost function
with a few linear/nonlinear constraints, the proposed approach can be implemented to provide function
evaluations for an optimization algorithm to provide a fast and accurate optimal control design method. Thus,
the proposed method promises the capability of performing, in a computationally tractable way, a large
number of simulations required for designing optimal control strategies for controllable damping devices
in complex structural models, in a manner that is much faster than conventional nonlinear solvers without
sacrificing response accuracy.
There are a number of directions of future research. First, unique solutions for the time history of (2.2) put
some modest restrictions of the form of the function g(), regardless of whether the proposed approach, or
some other nonlinear ODE solver, is used to simulate the system; the discontinuous nature of the semiactive
64
device force (4.1) across zero velocity has the potential to be problematic, though this was not seen in the
examples herein, so the proposed approach using continuous — and more realistic — damper models that
have some smooth transition as the velocity changes sign (e.g., a hyperbolic or inverse tangent model) should
be investigated. Second, the computational effectiveness of the proposed approach, as well as the efficacy in
finding optimal control strategies, should be evaluated with real models of controllable dampers (such as MR
dampers (Dyke et al., 1996, Yang et al., 2002, Jiang and Christenson, 2011)). Third, while the control design
herein is focused on two specific historical earthquakes, an approach considering stochastic representations of
the ground motions, or multiobjective approaches that incorporate optimality in a wide suite of earthquakes,
should be developed to ensure that the control design is not overly tuned to the specific design earthquakes
and possibly suboptimal in other ground motions.
65
Chapter 5
Computationally Efficient Design of
Semiactive Structural Control in Presence
of Measurement Noise
1
5.1 Introduction
The previous chapter showed that the proposed method is capable of performing, in a quick and accurate way,
the simulations required for determining the optimal semiactive control design. However, the formulation
of Chapter 4 assumed that full-state feedback is available to the controller. Real-world systems are more
challenging in that (a) full-state feedback is rarely available, so the control must be based on a limited
number of sensor measurements (e.g., accelerometers, strain gauges, LVDTs, force transducers, etc.), and (b)
those measurements are inevitably corrupted by sensor noise. To overcome these difficulties, an appropriate
estimator, capable of minimizing the effects of noise and providing estimates of the the full state response,
must be incorporated into the controller. The Kalman filter (KF) (Kalman, 1960, Kalman and Bucy, 1961,
Kalman, 1963), which is one of the most well known methods to provide the “best” estimates of the states, is
utilized here. The main goal of this chapter, then, is to demonstrate that the proposed approach can provide,
in a computationally efficient manner, multiple simulations of clipped-optimal semiactive control employing
1
This chapter is based on Kamalzare et al. (2014a) and Kamalzare et al. (2013b)
66
a KF to estimate the states from noisy sensor measurements.
The following section explains the theoretical formulation required to combine the KF with the system
equations. The efficiency of the approach is demonstrated inx 5.3 through a numerical study of a 100-degree
of freedom (DOF) frame model excited by one realization of a filtered Gaussian white noise excitation.
Ground and story-level accelerations are measured by the sensors, corrupted by noise. The results show that
the proposed method can achieve significant computational efficiency while retaining a level of accuracy
comparable to traditional nonlinear solvers.
5.2 Methodology
5.2.1 Equations of Motion, Sensor Measurements, and Regulated Output
Consider the same state-space structural model introduced in (2.2), which is augmented in (5.1) with the
measurement and output equations. Assume now that the controller has available, rather than the full state
vector, an n
y
1 noise-corrupted measurement vector Y(t).
˙
X(t) = AX(t)+ Bw(t)+ Lp(t), X(0)= x
0
(5.1a)
Y(t) = C
y
X(t)+ D
y
w(t)+ F
y
p(t)+ n(t) (5.1b)
Z(t) = C
z
X(t)+ D
z
w(t)+ F
z
p(t) (5.1c)
where X(t) is the n
x
1 state vector; p(t) is the n
p
1 control device force vector; w(t) is the n
w
1 ground
excitation vector; A is the n
x
n
x
system matrix; B is the n
x
n
w
excitation distribution matrix; L is an
n
x
n
p
constant influence matrix that maps the control forces to the appropriate system states; x
0
is the initial
state; Y(t) is the n
y
1 measurement vector; n(t) is the n
y
1 measurement noise vector; Z(t) is the n
z
1
output vector; C
y
, D
y
, and F
y
are the measurement influence matrices corresponding to states, process noise,
and control force, respectively; and C
z
, D
z
, and F
z
are similar influence matrices for the output.
67
w(t) and n(t) are assumed to be zero-mean stationary Gaussian processes with statistics
E[w(t)]= 0, E[n(t)]= 0, E[w(t)w
T
(t)]= Q
n
, E[n(t)n
T
(t)]= R
n
, and E[w(t)n
T
(t)]= N
n
= 0
(5.2)
5.2.2 Incorporating Kalman Filter
The algorithm most typically used in the literature and in practice for estimating the states of a linear system
based on a limited set of noisy sensor measurements is the KF. The KF (Kalman, 1960, Kalman and Bucy,
1961, Kalman, 1963), properly called the Kalman-Bucy filter in the continuous-time case, minimizes the
effects of system disturbance and measurement noise on the estimated states by means of the equations
˙
b
X = A
b
X+ Lp+ L
k
(Y
b
Y) (5.3a)
= A
b
X+ Lp+ L
k
(Y C
y
b
X F
y
p)
= (A L
k
C
y
)
b
X+(L L
k
F
y
)p+ L
k
Y
= (A L
k
C
y
)
b
X+(L L
k
F
y
)p+ L
k
(C
y
X+ D
y
w+ F
y
p+ n)
= L
k
C
y
X+(A L
k
C
y
)
b
X+ Lp+ L
k
D
y
w+ L
k
n
b
Y = C
y
b
X+ F
y
p (5.3b)
where
b
X is the optimal estimate of the state vector; L
k
is the Kalman gain matrix
L
k
=(S
n
C
T
y
+ N
n
)R
1
n
(5.4)
where N
n
= B(Q
n
D
T
y
+ N
n
); R
n
= R
n
+ D
y
N
n
+ N
T
n
D
T
y
+ D
y
Q
n
D
T
y
; and S
n
is the error covariance matrix
for the state estimate, containing information about the accuracy of the estimate, determined by solving the
algebraic Riccati equation AS
n
+ S
n
A
T
+ Q
n
(S
n
C
T
y
+ N
n
)R
1
n
(C
y
S
n
+ N
T
n
)= 0 where Q
n
= BQ
n
B
T
(see
Appendix 5.A for the derivation).
Combining (2.2) and (5.3a), the equation of motion can be rewritten
˙
e
X(t)=
e
A
e
X(t)+
e
Lp(t)+
e
Be w(t) (5.5)
68
where
e
X(t)=[X
T
b
X
T
]
T
is the generalized state vector;e w(t)=[w
T
n
T
]
T
is the generalized excitation vector;
and
e
A=
2
6
4
A 0
L
k
C
y
A L
k
C
y
3
7
5 ,
e
L=
2
6
4
L
L
3
7
5 ,
e
B=
2
6
4
B 0
L
k
D
y
L
k
3
7
5 (5.6)
Thus, with the KF equations incorporated with the linear system model, the combined state-space system
(5.5) has a form similar to (2.2) and, therefore, it can be solved using the rapid nonlinear V olterra integral
equation (NVIE) analysis approach of Chapter 2 without extra complication. Since the previous chapter
showed that significant gains in computational efficiency can be achieved, relative to traditional solvers
such as MATLABode45 command, for the optimal design of semiactive control strategies using full state
feedback, it is expected that similar computational advantages are achievable in the output feedback case
as well. Note that the size of (5.5) is twice that of state equation (2.2), resulting in approximately double
the computational cost when using typical nonlinear solvers such as ode45. However, the order of the
nonlinear equations solved by the proposed approach (is the same as in Chapter 4 and) does not change, so
the computational effort required for solving this augmented system is essentially unchanged by adding the
KF.
The proposed approach includes one time calculations to compute the uncontrolled response and the impulse
response functions and also costs repeated for each candidate control law. The inclusion of the KF has a
negligible effect on these repeated costs. For the one time calculations, computing the impulse response
H
L
(t) using (5.6) with X= G
b
X=[0 G]
e
X requires 2–4 times the computation required without the KF.
However, as shown in Appendix 5.B, this impulse response is actually the same as that with state feedback
— i.e., it can be computed using the original system (2.2) instead of the augmented system (5.5) — so the
change in the one time costs due to the incorporation of the KF filter is only a very modest increase. In either
case, as the one-time cost is leveraged over the many repeated solutions for the control force, the overall cost
should increase only very modestly (Kamalzare et al., 2013b); the only significant increase in computational
cost is due to the computation of the uncontrolled response, which will be approximately twice as much as
for the system without the KF.
It is worth noting that the assumption for the KF to be optimal may not hold here. For a linear system, if
the noise is a white Gaussian stochastic process, the KF can be shown to provide an optimal estimate of
69
the original signal. But herein, the semiactive force p is found through a nonlinear transformation of the
responses and is non-Gaussian, and therefore, the system studied here does not have Gaussian responses, and
the KF may not be optimal. However, since finding a suitable replacement is very computationally intensive,
if not impossible, the KF is used here and, as shown in the following section for the example investigated in
this study, it provides an appropriate level of accuracy for state estimates.
5.3 Numerical Examples
The same structural model introduced in Chapter 4 and shown in Figure 4.4 is used here as well.
5.3.1 Ground Excitation
A Kanai-Tajimi filter (Soong and Grigoriu, 1993) is incorporated into the model to generate a ground
excitation with frequency content similar to several historical earthquakes. The filter is a second order
dynamical system characterized by the transfer function
F(s)=
2z
g
w
g
s+w
2
g
s
2
+ 2z
g
w
g
s+w
2
g
, (5.7)
or, equivalently, as the response of the following single DOF system
¨ x
g
(t)+ 2z
g
w
g
˙ x
g
(t)+w
2
g
x
g
(t)=w(t) (5.8a)
¨ u
g
(t)=w
2
g
x
g
(t) 2z
g
w
g
˙ x
g
(t) (5.8b)
where w(t) is a zero-mean Gaussian white noise. The parameter valuesz
g
= 0.3 andw
g
= 17 rad/sec best
approximate (Ramallo et al., 2002), in a least-squares sense, the spectral content of two moderate earthquakes
(1940 El Centro and 1968 Hachinohe) and two severe earthquakes (1994 Northridge and 1995 Kobe), as
shown in Figure 5.1. For the control design, a single 30 s realization of the white noise excitation w(t) and
of the sensor noise n(t), sampled every 0.02 s, is used. Subsequent studies inx 5.3.5 will use Monte Carlo
simulation (MCS) that use multiple realizations of both the excitation and sensor noise.
70
10 1 0.1
10
–4
10
–3
10
–2
10
–1
10
0
Kanai-Tajimi
Magnitude
Frequency [Hz]
Figure 5.1: Frequency content of design earthquakes and Kanai-Tajimi shaping filter (adapted from
Ramallo et al. (2002)).
The Gaussian white noise w(t) is scaled in a way that results in a Kanai-Tajimi output signal ¨ u
g
(t) with
root mean square (RMS) and peak ground acceleration (PGA) similar to the El Centro earthquake. The time
history and frequency content (fast Fourier transform (FFT) of the time history) of this excitation are shown
in Figures 5.2 and 5.3, respectively.
0 5 10 15 20 25 30
-4
-3
-2
-1
0
1
2
3
4
Time (s)
Acceleration (m/s
2
)
Figure 5.2: Time history of the applied excitation
71
0 5 10 15 20 25
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Frequency (Hz)
Magnitude
Figure 5.3: Frequency content of the applied excitation
5.3.2 State-Space Model
By combining the states of the Kanai-Tajimi filter with those of the base and superstructure, the state-space
equation of motion of the system can be written in the form of (2.2) where X(t)=[u
b
u
T
s
˙ u
b
˙ u
T
s
x
g
˙ x
g
]
T
is
the state vector, with base u
b
and superstructure u
s
displacements measured relative to the ground, and x
g
is
the displacement of ground relative to bedrock. A, B, and L can be written as
A=
2
6
6
6
6
6
6
6
6
6
6
6
4
0 I 0
M
1
K M
1
C
2
6
4
1
r
3
7
5

w
2
g
2z
g
w
g

0
2
6
4
0 1
w
2
g
2z
g
w
g
3
7
5
3
7
7
7
7
7
7
7
7
7
7
7
5
, B=
2
6
6
6
6
6
6
6
6
6
6
6
4
0
0
1
3
7
7
7
7
7
7
7
7
7
7
7
5
, L=
2
6
6
6
6
6
6
6
6
6
6
6
4
0
M
1
2
6
4
1
0
3
7
5
0
0
3
7
7
7
7
7
7
7
7
7
7
7
5
(5.9)
Now, assume the structure is instrumented with 13 accelerometers: one on the base, one at each superstructure
story level, and one on the ground to record the ground excitation. It is assumed that all accelerometers are
identical and will add a low-level noise to the signals. The measurement noise is modeled as a Gaussian
pulse process with an RMS of approximately 5% of the average RMS of the story acceleration responses.
72
Based on the ground excitation and its corresponding structural response, the noise levels represented by
Q
n
= 3.26 m
2
/s
4
and R
n
= 0.00390625I
1313
m
2
/s
4
are used to design the KF gain.
Let C
a
y
be a matrix that selects the measured accelerations from among all structure accelerations; i.e., it
has elements[C
a
y
]
i, j
=d
i,1
d
j,1
+d
9i13, j
whered
i, j
is the Kronecker delta (i.e., C
a
y
is a matrix with ones in
elements(1,1),(2,5),(3,14),, (12,95) and zeros elsewhere):
C
a
y
=
2
6
6
6
6
6
6
6
4
1
0
13
1 0
15
0
13
1 0
15
.
.
.
3
7
7
7
7
7
7
7
5
(5.10)
The measurement equation matrices can then be more compactly expressed as
C
y
=
2
6
4
C
a
y
M
1
[K C] 0 0
0 w
2
g
2z
g
w
g
3
7
5 , D
y
=
2
6
4
0
0
3
7
5 , F
y
=
2
6
4
C
a
y
M
1
[1 0
T
]
T
0
3
7
5 (5.11)
If C
a
z
, with elements [C
a
z
]
i, j
=d
95i, j
, selects the roof acceleration from among all accelerations, then the
outputs (shown in (5.1c)) to be regulated, which include the base drift (1
st
DOF) and the absolute horizontal
roof acceleration (at the 95
th
DOF), can be found using output equation matrices
C
z
=
2
6
4
1 0 0 0 0
C
a
z
M
1
[K C] 0 0
3
7
5 , D
z
=
2
6
4
0
0
3
7
5 , F
z
=
2
6
4
0
C
a
z
M
1
[1 0
T
]
T
3
7
5 (5.12)
5.3.3 Augmented System with Kalman Filter
Now, the system of equations can be augmented with KF state estimates and written in the form of (5.5). This
is a system of 404 state equations; however, the nonlinearities are located exclusively in feedback force p, so
the NVIE is still only a single equation. Therefore, it can be solved very efficiently using the NVIE approach
discussed in the previous chapter. Figure 5.4 shows a block diagram representation of the semiactive control
strategy, which includes the linear part (the Kanai-Tajimi filter, the structural model, the KF and the state
feedback gain) and the nonlinear part (clipped optimal controller).
73
5.3.4 Accuracy and Computational Cost of the Semiactive Control Design
In this study, the cost function is defined as in (4.2) with control weight R
i
= W
2
and diagonal response
weight Q
i
= diag

a
i
/h
2
, b
i
(T
i
1
)
4
/(16p
4
h
2
)

, wherea
i
andb
i
are dimensionless parameters that will be
tuned to achieve the best semiactive control performance. The control gain K
i
is designed using thelqr
command in MATLAB for each pair (a
i
,b
i
).
Structure
Structure
Kanai
Tajimi
filter
Kanai
Tajimi
filter
Kalman
filter
Kalman
filter
Control
gain
Control
gain
Clipping
Clipping
White noise Excitation Output
Measurement noise
Measurement
Estimated states
Velocity across the device
Desired force
Actual force
Nominal
system
Nonlinear
part
+
Figure 5.4: Combined filter/structure model for semiactive control design using clipped-optimal con-
troller.
As in the previous chapter, assume s
SA
d
(a,b), s
SA
a
(a,b), and s
SA
p
(a,b) represent the RMS base drift,
absolute roof acceleration, and actual semiactive actuator force, respectively, for a particular excitation and a
controller designed with cost function parameters(a,b). In addition, lets
LRB
d
ands
LRB
a
denote the RMS
base drift and absolute roof acceleration corresponding to a typical (Skinner et al., 1993) baseline lead rubber
bearing (LRB) design, as introduced inx 4.3.2, modeled with a Bouc-Wen hysteresis: the post-yield stiffness
is the same as that of the uncontrolled structure; the pre-yield to post-yield stiffness ratio is six; and the
yield force is 5% of the building weight. Since the scaling of the objective function and constraints affect
the performance of the optimization, the objective function is again defined as the relative change in the
RMS base drift compared to the corresponding baseline design, s
d
(a,b)=[s
SA
d
(a,b)/s
LRB
d
] 1, and
the constraints are represented as the relative change in the RMS roof absolute acceleration compared to
74
the corresponding baseline design,s
a
(a,b)=[s
SA
a
(a,b)/s
LRB
a
] 1, and the RMS actuator force level is
expressed as a fraction of the building weight,s
p
(a,b)=s
SA
p
(a,b)/W.
Similar tox 4.3.6, thefminsearch algorithm in MATLAB’s optimization toolbox is used to find an optimal
semiactive design where the proposed approach is implemented for the function evaluation. The objective is
to minimize the base drifts
d
(a,b) subjected to two constraints: (1) the RMS roof acceleration remains at
(or below) its corresponding value when the isolator is a passive LRB design (baseline) (i.e.,s
a
0), and
(2) the RMS control force is at most 4% of building weight W (i.e.,s
p
4%). The optimization algorithm
is started at (a, b) = (1000, 10); the algorithm terminates when absolute tolerances for both the search
point and the objective function value are within 0.01. The optimization results in the design point of
(a

, b

) = (3931.40, 54.23) which results in about 49.8% base drift reduction, with no increase in roof
acceleration, relative to the LRB baseline performance. Also, the peak and RMS of the semiactive force are
11.9% and 4.0% of the building weight, respectively. Table 5.1 shows the RMS responses (base drift, roof
acceleration, and device force) and peak device force for both LRB and semiactive design. The semiactive
control design reduces the base drift significantly at the expense of applying a larger control force.
Table 5.1: RMS responses comparison.
Damping system
RMS responses
Base drift [cm] Roof acceleration [m/s
2
] Force [kN] Peak force [kN]
LRB 5.67 1.03 40.78 53.23
Semiactive 2.85 1.03 51.02 151.88
To have a better understanding of how the objective function and constraints vary surrounding the design
point, a coarse-grid parameter study is done overa
i
2 [10
3
,10
4
] andb
i
2 [10
1
,10
2
]; the responses compared
to the LRB design are shown in Figure 5.5. In this figure, the shaded area is the region where the semiactive
controller provides acceleration reductions relative to the LRB,s
a
(a, b)< 0; the blue solid contour lines
show the percentage reduction in the base drift provided by the controllable damper relative to the LRB,
s
d
(a, b); and the black dash-dot lines show different RMS force levels as a percentage of the building
weight,s
p
(a, b).
A family of design points can be found using alternate constraints on RMS force metrics
p
(a,b). For this
specific structure, with the design excitation realization discussed inx 5.3.1, assuming the peak force is
75
10
3
10
4
10
1
10
2
Drift Weight α
Acceleration Weight β
−60%
−50%
−40%
3.8%
4.0%
4.2%
roof accel. reduced
base drift change
RMS force
design point
Figure 5.5: RMS semiactive responses relative to those with the LRB.
limited to 15% of the building weight W and ensuring that the RMS absolute roof acceleration is no more
than that with the LRB baseline (i.e., s
a
= 0), base drift reduction can be achieved for force constraints
s
p
2[3.0%,4.4%]. These design points and their corresponding objectives
d
and constraints
p
values, along
with the associated required peak force, are summarized in Table 5.2.
Table 5.2: Design points at different RMS force levels.
Design Point RMS Forces
p
Peak Force Base Drifts
d
a

b

[% of building weight] [% change relative to LRB]
870.97 0.63 3.00 9.65 –28.85
1130.79 2.22 3.20 10.20 –34.66
1459.58 5.11 3.40 10.64 –39.42
1918.98 10.72 3.60 11.02 –43.59
2616.42 22.85 3.80 11.35 –47.01
3931.40 54.23 4.00 11.89 –49.78
8364.08 191.72 4.20 12.70 –51.83
36060.49 1128.74 4.40 15.61 –53.04
To investigate the computational efficiency that can be achieved by implementing the proposed method,
the analysis is performed both using a traditional nonlinear ordinary differential equation (ODE) solver,
MATLAB’s ode45, and by the proposed approach solving the NVIE using FFT (Gaurav et al., 2011)
introduced inx 2.2.5. To perform a fair comparison, the relative and absolute error tolerance ofode45 are set
to be 10
–3
and 10
–6
, respectively, and the proposed method is applied using a 2
nd
-order accurate trapezoidal
76
integration with 2
15
time steps ofDt = 0.92 ms duration each. Both of these have relative accuracy on the
order of 10
–3
.
Figure 5.6 illustrates actual roof acceleration response, measurement noise at the roof sensor, and the
estimated response using the KF at the design point with a RMS force level that is of 4% of the building
weight (s
p
= 4%); the KF is successful in estimating the response very accurately. Figures 5.7 and 5.8 show
estimated base drift and commanded semiactive device force, respectively, calculated by both methods at
the design point. As shown in these figures, the results are almost identical for both methods, which verifies
their comparable level of accuracy. (It may be observed in Figure 5.8, that the semiactive controller may
turn on and off rapidly when the velocity across the device is near zero.) The relative differences between
the responses computed by the proposed method and those byode45 can be quantified by computing the
difference in their RMS values and normalized it by the RMS of the response computed by ode45; the
result is a relative RMS difference of 1.210
–4
for the base drift, 8.610
–4
for the absolute roof acceleration,
and 1.010
–3
for the control force. The relative tolerance requested of theode45 solution is 10
–3
, so the
proposed method is at least as accurate as theode45 solution to which it is compared.
0 2 4 6 8 10
−3
−2
−1
0
1
2
3
Time [s]
Roof Acceleration [m/s
2
]

Actual response
Estimated response
Noise
Figure 5.6: Actual and estimated roof absolute acceleration response and the measurement noise
signal.
77
0 2 4 6 8 10
−8
−6
−4
−2
0
2
4
6
8
Time [s]
Estimated Base Drift [cm]

Proposed approach
Traditional solver
Figure 5.7: Estimated base drift response to the generated excitation.
0 2 4 6 8 10
−100
−50
0
50
100
Time [s]
Commanded Semiactive Force [kN]

Proposed approach
Traditional solver
Figure 5.8: Commanded device force.
The simulations are performed on a computer with a 3.4 GHz Intel core i7-2600 processor and 8 GB of RAM,
running MATLAB R2011a under Windows 7. For the ideal full-state feedback system of 202 states, using
the proposed method, the computational cost of a single simulation includes one time calculations requiring
4.9 s and the repeated ones requiring 1.5 s; for a single function evaluation,ode45 requires 163.8 s. When
the estimated states are incorporated to the system of equations, resulting in a 404-state system, the one
time calculation time of the proposed approach increase to 10.4 s, while the repeated cost remains almost
78
Table 5.3: Cost ratio comparison.
Feedback # states # simulations
ode45 Proposed method Computational
RelTol = 10
–3
# of steps = 2
15
cost
AbsTol = 10
–6
Dt = 9.210
–4
s comparison
State 202
1 simulation 163.8 s 6.4 s 25.6
100 simulations 4.5 hr 2.6 min 105.7
Output
404
1 simulation 286.7 s 11.9 s 24.1
(w/Kalman filter) 100 simulations 8.0 hr 2.7 min 178.7
nearly identical because the size of the nonlinear system of equations does not change. Usingode45, the
computational time increases to 286.7 s with the addition of the filter states. While the number of function
evaluations required for the optimization to converge to the “best” design may vary significantly, depending
on termination tolerances and the initial guess, the optimization here converged with 100–200 function
evaluations. Table 5.3 summarizes the required computational time of a single simulation and 100 simulations
(at the design point) for both methods and shows the achievable cost ratio. Clearly, the proposed method can
again achieve two orders of magnitude reduction in computational cost.
5.3.5 Effects of Sensor Noise Magnitude and Kalman Filter Tuning on Estimated Outputs
In the preceding section, the measurements collected by sensors are assumed to be corrupted by noise with
RMS of about 5% of the mean RMS of the all story accelerations. However, the noise magnitude depends on
many different parameters — e.g., the length of connecting cables, thermal and electrical effects in the sensor
piezoelectric material (Levinzon, 2004), mechanical relaxation of packaging and circuit noise (Barzilai et al.,
1998) — and can vary significantly from one application to another.
Further, tuning the KF parameters affects the estimated outputs; designing a KF requires knowledge of the
process noise (herein, the excitation) covariance Q
n
, and the sensor noise covariance R
n
(the cross correlation
between the excitation and the noise is assumed zero herein, i.e., N
n
= 0). In a scalar sense, the ratio Q
n
/R
n
dictates to the KF whether a change in the measurement should be considered as noise or as an actual change
in response; in other words, this ratio determines how fast the KF reacts to changes in the measurements.
However, the actual measurement noise magnitude could be different in practice from what is assumed at the
design stage.
79
In this section, the effects of (a) noise magnitude and (b) KF design assumptions on the accuracy of the
estimated response are investigated. Noise signals with RMS of 0, 5, 10, 25, and 50% of the mean RMS of
all the story accelerations are applied to the measurements, and the KF is designed for 5, 10, 25, and 50%
noise levels; the relative error between the estimated outputs Z and the actual outputs Z
0
(where no noise
and, therefore, no KF is embedded into the system) is evaluated for each of those cases. The relative error in
the RMS of output Z
i
(t), for a single realization of the excitation and the sensor noise vector, is defined
e
i
=

Z
t
f
0
Z
2
i
(t)dt

Z
t
f
0
fZ
0
i
(t)g
2
dt

1/2
1

(5.13)
where t
f
= 30 s is the duration of the excitation. For each combination of sensor noise level and KF design, a
MCS of 10,000 realizations of excitation and noise signals is performed.
Tables 5.4, 5.5, and 5.6 show comparisons of the mean errors for the various combinations of noise level and
KF design for base drift, absolute roof acceleration and device force, respectively. As expected, the error is
lower when the appropriate KF design is implemented; while an improperly-tuned KF increases the response
error, still this error is negligible compared to the effect of the measurement noise magnitude. The magnitude
in the measurement noise is the primary factor controlling the estimated response accuracy.
Table 5.4: Base drift error mean [%] over 10,000 realizations.
KF is designed for
noise level
0% 5% 10% 25% 50%
5% 0.29 0.51 0.96 2.75 8.05
10% 0.30 0.51 0.95 2.73 7.97
25% 0.53 0.57 0.84 2.26 7.03
50% 4.02 3.95 3.76 2.80 4.20
Table 5.5: Roof absolute acceleration error mean [%] over 10,000 realizations.
KF is designed for
noise level
0% 5% 10% 25% 50%
5% 0.27 0.32 0.49 1.42 4.57
10% 0.26 0.31 0.45 1.24 3.78
25% 0.57 0.55 0.50 0.73 2.58
50% 4.20 4.17 4.10 3.55 1.90
Finally, it is worth noting that the analysis in this section required many realizations of a MCS; implementing
the analysis with the proposed NVIE approach provides a very fast and practical method for this purpose and
80
Table 5.6: Device force error mean [%] over 10,000 realizations.
KF is designed for
noise level
0% 5% 10% 25% 50%
5% 0.04 0.18 0.36 1.07 3.23
10% 0.18 0.22 0.36 0.96 2.84
25% 1.22 1.20 1.13 0.99 1.99
50% 4.51 4.49 4.43 3.97 2.60
significantly reduces the computational cost. The one-time calculations (to construct the necessary matrices,
find the impulse responses to the control force pattern and to the sensor noise and to determine control gain
K
d
) take 56.0 s, and repeated calculations require 3.5 s per realization (to calculate the responses x and z of
the uncontrolled system and to evaluate the nonlinear function g). Sinceode45 takes 450.5 s per realization,
the proposed method provides a computational speedup of 128.5 for the 10,000 realization MCS, requiring
only 9.7 hours instead of 52.1 days.
5.4 Conclusions
A method previously introduced in Chapter 4 is adapted to include a KF estimator to accommodate noisy
output feedback measurements. It is shown that the augmented state space system, which includes both
actual and estimated states, can be written in a form similar to what was implemented for the ideal full-state
feedback control; therefore, the proposed approach can be directly implemented. The proposed method is
tested with a representative frame model where only ground, base and story-level accelerations, polluted by
5% noise signals, are available to measure. It is shown that the proposed approach can compute the structural
responses with an appropriate level of accuracy but with computational cost over two orders of magnitude
smaller. In addition, the effects of sensor noise magnitude and KF tuning on the estimated outputs accuracy
is investigated, with two-orders-of-magnitude computational improvements for the required MCS.
81
Appendices
5.A Kalman Gain and its Corresponding Riccati Equation
Consider the general state-space structural model and measurement equations as follows
˙
X(t) = AX(t)+ Bw(t)+ Lp(t), X(0)= x
0
(5.14a)
Y(t) = C
y
X(t)+ D
y
w(t)+ F
y
p(t)+ n(t) (5.14b)
w(t) and n(t) are assumed to be zero-mean stationary Gaussian white processes with statistics E[w(t)]= 0,
E[n(t)]= 0, E[w(t)w
T
(t)]= Q
n
d(tt), E[n(t)n
T
(t)]= R
n
d(tt), and E[w(t)n
T
(t)]= N
n
d(tt).
The Kalman-Bucy filter minimizes the effect of system disturbance and measurement noise on the estimated
state by means of the system
˙
b
X(t) = A
b
X(t)+ Lp(t)+ L
k
(t)(Y(t)
b
Y(t))
= L
k
(t)C
y
X(t)+(A L
k
(t)C
y
)
b
X(t)+ Lp(t)+ L
k
(t)D
y
w(t)+ L
k
(t)n(t) (5.15a)
b
Y(t) = C
y
b
X(t)+ F
y
p(t) (5.15b)
where
b
X(t) and
b
Y(t) are the optimal estimates of the state and measurement vectors, respectively; and
L
k
(t) is the Kalman gain matrix. If error is defined as e(t)= X(t)
b
X(t), then ˙ e(t) can be calculated by
subtracting (5.15a) from (5.14a),
˙ e(t)=(A L
k
(t)C
y
)e(t)+(B L
k
(t)D
y
)w(t) L
k
(t)n(t) (5.16)
Covariance matrix of error, S
n
(t), and its first derivative can be written as
S
n
(t) = E[e(t)e
T
(t)] (5.17a)
˙
S
n
(t) = E[˙ e(t)e
T
(t)]+ E[e(t)˙ e
T
(t)] (5.17b)
82
Replacing (5.16) into (5.17b) results in
˙
S
n
(t) = (A L
k
(t)C
y
)E[e(t)e
T
(t)]+(B L
k
(t)D
y
)E[w(t)e
T
(t)] L
k
(t)E[n(t)e
T
(t)]
+ E[e(t)e
T
(t)](A L
k
(t)C
y
)
T
+ E[e(t)w
T
(t)](B L
k
(t)D
y
)
T
E[e(t)n
T
(t)]L
T
k
(t)(5.18)
where
E[e(t)e
T
(t)] = S
n
(t) (5.19a)
E[w(t)e
T
(t)] = E
"
w(t)

Z
t
0
e
(AL
k
C
y
)(tt)
h
(B L
k
(t)D
y
)w(t) L
k
(t)n(t)
i
dt

T
#
=
Z
t
0
e
(AL
k
C
y
)(tt)
h
E[w(t)w
T
(t)](B L
k
(t)D
y
)
T
E[w(t)n
T
(t)]L
T
k
(t)
i
dt
=
Z
t
0
e
(AL
k
C
y
)(tt)
h
Q
n
d(tt)(B L
k
(t)D
y
)
T
N
n
d(tt)L
T
k
(t)
i
dt
=
1
2
[Q
n
(B L
k
(t)D
y
)
T
N
n
L
T
k
(t)] (5.19b)
E[n(t)e
T
(t)] = similar calculation =
1
2
[N
T
n
(B L
k
(t)D
y
)
T
R
n
L
T
k
(t)] (5.19c)
The Dirac delta functiond(tt) has an infinitely high and thin spike att = t (symmetric function from
te to t+e wheree is an infinitesimal value), with a total area of one under the spike. Since the upper limit
of the integral is t, the integration is performed up to the middle of the delta function, which results in the
1/2 coefficient in the final result.
Substituting (5.19) into (5.18) results in
˙
S
n
(t) = (A L
k
(t)C
y
)S
n
(t)+ S
n
(t)(A L
k
(t)C
y
)
T
+
1
2
(B L
k
(t)D
y
)[Q
n
(B L
k
(t)D
y
)
T
N
n
L
T
k
(t)]
+
1
2
[(B L
k
(t)D
y
)Q
T
n
L
k
(t)N
T
n
](B L
k
(t)D
y
)
T

1
2
L
k
(t)[N
T
n
(B L
k
(t)D
y
)
T
R
n
L
T
k
(t)]

1
2
[(B L
k
(t)D
y
)N
n
L
k
(t)R
T
n
]L
T
k
(t) (5.20)
83
Since Q
n
and R
n
are symmetric matrices, (5.20) can be simplified to
˙
S
n
(t) = (A L
k
(t)C
y
)S
n
(t)+ S
n
(t)(A L
k
(t)C
y
)
T
+ (B L
k
(t)D
y
)Q
n
(B L
k
(t)D
y
)
T
+ L
k
(t)R
n
L
T
k
(t)
(B L
k
(t)D
y
)N
n
L
T
k
(t) L
k
(t)N
T
n
(B L
k
(t)D
y
)
T
(5.21)
Defining Q
n
= BQ
n
B
T
, R
n
= R
n
+ D
y
N
n
+ N
T
n
D
T
y
+ D
y
Q
n
D
T
y
, and N
n
= B(Q
n
D
T
y
+ N
n
), (5.21) can be
written as
˙
S
n
(t) = AS
n
(t)+ S
n
(t)A
T
+ Q
n
(S
n
(t)C
T
y
+ N
n
)R
1
n
(C
y
S
n
(t)+ N
T
n
)
+ (L
k
(t)(N
n
+ S
n
(t)C
T
y
)R
1
n
)R
n
(L
T
k
(t) R
1
n
(C
y
S
n
(t)+ N
T
n
)) (5.22)
By defining the Kalman gain as in (5.23a), (5.22) can be simplified to the Riccati equation shown in (5.23b):
L
k
(t) = (S
n
(t)C
T
y
+ N
n
)R
1
n
(5.23a)
˙
S
n
(t) = AS
n
(t)+ S
n
(t)A
T
+ Q
n
(S
n
(t)C
T
y
+ N
n
)R
1
n
(C
y
S
n
(t)+ N
T
n
) (5.23b)
Assuming the signals are stationary, (5.23) can be rewritten as
L
k
= (S
n
C
T
y
+ N
n
)R
1
n
(5.24a)
0 = AS
n
+ S
n
A
T
+ Q
n
(S
n
C
T
y
+ N
n
)R
1
n
(C
y
S
n
+ N
T
n
) (5.24b)
84
5.B Impulse Response of the System Augmented by Kalman Filter
With the estimator incorporated, the control force now depends on X= G
b
X=[0 G]
e
X. Then, impulse
response H
L
(t) is the product of G and the inverse Laplace transform of the transfer function T
b
Xp
(s)=
[0 I](sI
e
A)
1
[L
T
L
T
]
T
, which can be simplified, due to the structure of
e
A, by introducing transformation
matrix T and its inverse
T=
2
6
4
I 0
I I
3
7
5 and T
1
=
2
6
4
I 0
I I
3
7
5 (5.25)
Then, transfer function T
b
Xp
(s) can be written as
T
b
Xp
(s)=

[0 I]T

n
T
1
(sI
e
A)
1
T
o

T
1
[L
T
L
T
]
T

=

[0 I]T

(sI T
1
e
AT)
1

T
1
[L
T
L
T
]
T

=
8
>
<
>
:
[0 I]
2
6
4
I 0
I I
3
7
5
9
>
=
>
;
0
B
@sI
2
6
4
I 0
I I
3
7
5
2
6
4
A 0
L
k
C
y
A L
k
C
y
3
7
5
2
6
4
I 0
I I
3
7
5
1
C
A
1
8
>
<
>
:
2
6
4
I 0
I I
3
7
5
2
6
4
L
L
3
7
5
9
>
=
>
;
=[I I]
0
B
@sI
2
6
4
I 0
I I
3
7
5
2
6
4
A 0
A A L
k
C
y
3
7
5
1
C
A
1
2
6
4
L
0
3
7
5
=[I I]
0
B
@sI
2
6
4
A 0
0 A L
k
C
y
3
7
5
1
C
A
1
2
6
4
L
0
3
7
5
=[I I]
2
6
4
sI A 0
0 sI A+ L
k
C
y
3
7
5
1
2
6
4
L
0
3
7
5
=[I I]
2
6
4
(sI A)
1
0
0 (sI A+ L
k
C
y
)
1
3
7
5
2
6
4
L
0
3
7
5
=[I I]
2
6
4
(sI A)
1
L
0
3
7
5
=(sI A)
1
L (5.26)
which is the same as the corresponding transfer function from p to X. As a result impulse response function
H
L
(t) is the same with the KF and X= G
b
X as without the KF and X= GX; this fact can be exploited to
85
reduce the one-time computational cost of producing the impulse responses by using the original system (2.2)
instead of the augmented system (5.5).
86
Chapter 6
Rapid Controllable Damper Design for
Complex Structures with a Hybrid
Reduced-Order Modeling/Simulation
Approach
1
6.1 Introduction
Preceding chapters discussed how controllable passive (semiactive) devices often exhibit inherent nonlin-
earities or do so through a control strategy which must be nonlinear due to the passivity limitations of the
devices; even if the controllable damper is linear and its control is linear, then a feedback strategy to control
its damping coefficient would introduce a multiplicative nonlinearity into the closed-loop dynamics. These
behaviors make difficult the design of controllable passive devices. Yet, the rest of the structure is often
linear, possibly with the exception of some localized nonlinear behaviors. Design optimization and stochastic
response analysis of these smart structures require multiple, usually very many, nonlinear response analyses
that can become very computationally intensive.
1
This chapter is based on Kamalzare et al. (2014d) and Kamalzare et al. (2013c)
87
To remedy this computational challenge, the author and colleagues have adapted to controllable damping
design a computationally-efficient approach that was previously developed (Gaurav et al., 2011) for simulating
the response of systems that are mostly linear and deterministic but with localized nonlinear and/or uncertain
features. This approach reduces the full equations of motion to a low-order nonlinear V olterra integral
equation (NVIE) that can be solved in a very rapid manner.
The author and colleagues used this approach to propose optimal state feedback (Kamalzare et al., 2014b) and
output feedback (Kamalzare et al., 2014a) control strategies (in Chapter 4 and 5, respectively) for mitigating
structural response with controllable dampers. It was shown that the system including a Kalman filter (KF)
to estimate the states from noisy sensor measurements can also be written in the form of a mostly-linear
system but with nonlinearities at the controllable damper locations, and the numerical studies demonstrated
simulations that were 100–300 times faster, for a moderate-order system, than with a conventional nonlinear
ordinary differential equation (ODE) solver.
However, there still exists a major obstacle to be addressed before the method can be implemented for large-
scale real-world structural engineering applications, namely, full-order control strategies are rarely designed
and implemented for very high-order systems. Conversely, finite element models (FEMs) of many civil,
mechanical and aerospace structures often use thousands, or even millions, of degrees of freedom (DOFs)
to accurately reproduce the dynamic behavior of the real system. While some dynamics of such FEMs
(e.g., ultra-high frequency modes) typically have limited impact on important responses in linear analyses,
nonlinearities can give rise to higher harmonics and energy transfers such that it is difficult to determine in
advance which dynamics can be safely neglected; as a result, it is necessary for controllable damping design
strategy simulations to include the full-order system dynamics. However, the implementation of the control
strategy, sensor measurements and controllable damping force commands is accomplished using lower-order
dynamics, partly due to hardware computation limitations. Further, as discussed in the subsequent paragraphs,
the computation time required for the many (possibly thousands of) candidate control strategy designs, each
of which requires solutions of high-order Lyapunov and/or Riccati equations, can be prohibitive and the
memory requirements can be challenging.
For example, a preliminary investigation of the 1623-DOF 20-story structure, discussed further in the
Numerical Examplesx 6.3.2 (and shown in Figure 6.10), shows that the required time to compute a state
88
feedback gain matrix, via solution of an algebraic Riccati equation, for the full-order model is orders of
magnitude larger than for the reduced-order models. Table 6.1 lists the computation time and memory
requirements for each of the various Riccati solutions using the full-order model and 500-, 100-, 50- and
10-state reduced models, as well as the one-time requirements for balanced model reduction to determine
these reduced models. If only one or two feedback gain matrices need to be computed, then the use of the
full-order model is acceptable; however, if many such gain matrices must be computed, the total cost of
the many Riccati solves rapidly becomes a much larger cost than the one-time model reduction that can
be subsequently used to design a state observer and the repeated state feedback gain matrices. Further, the
reduced models are more efficient in memory use as well. This significant improvement in computational
efficiency, along with the hardware limitations of implementing only lower-order dynamic controllers, gives
a clear rationale for investigating whether the authors’ proposed approach can be efficiently implemented
using a reduced model for designing the controllable damper force command strategy (both state observer
and feedback gain design) while still verifying the controller design using the full fidelity model to provide a
robust, accurate and highly computationally efficient approach for controllable damper design.
Table 6.1: Computation (CPU) time and memory usage for solving the primary control gain Riccati
equation.
Model (# states) Computation time (s) Peak memory (MB)
Foreach Riccati equation & control gain
full (3246) 662.52 1569.44
reduced ( 500) 4.87 63.21
reduced ( 100) 0.12 5.77
reduced ( 50) 0.03 2.54
reduced ( 10) 0.002 1.01
One-time calculation of reduced model
any 1870.50 1169.95
Thus, this chapter demonstrates that, indeed, model reduction techniques can generate lower-order control
strategies, which can be incorporated with the full-order structure model into a mostly linear system that can
be simulated in a highly efficient manner for controllable damping design. The simulations here are different
from the typical implementation of reduction techniques commonly used in literature since the system is
nonlinear and the control strategies, in low-order as would be used in a physical implementation, must be
evaluated with the dynamics of the original full-order model to capture all possible relevant dynamics in the
responses. The block diagram shown in Figure 6.1 presents how the original system and its reduced version
89
can be used in the design algorithm.
Desired
force
Velocity across the device
Excitation Output
Response at sensors
Estimated states
of reduced model
Measurement noise
Nominal System Nonlinear Part
White noise
Actual force
Kalman filter
(designed for
reduced model)
Control gain
(designed for
reduced model)
Spectral
filter
Large-scale
structural
model
Clipping
non-dissipative
forces
Figure 6.1: Block diagram of a semiactive design
The efficacy of the proposed method is investigated for two examples. First, a base-isolated 100-DOF
frame model introduced previously inx 4.3.1, with a controllable damper in the isolation layer, excited by a
Kanai-Tajimi filtered Gaussian white noise ground motion with spectral content similar to a few representative
historical earthquakes. In lieu of the usual focus on minimizing statistics of response, system reliability will be
examined. The second example consists of a large-scale 3-D building, modeled with 1623 DOFs, subjected
to wind excitation; the wind-induced response is mitigated by three roof-mounted tuned mass dampers
(TMDs), each with a controllable damping device. The efficacy of the proposed approach is demonstrated by
comparing its results to those of a conventional passive TMD design. The speed of the proposed approach,
with the fast response simulation and reduced-order models for feedback gain computation, is shown to
approach three orders of magnitude faster than using a conventional simulation approach with full-order
models.
6.2 Methodology
6.2.1 Model Reduction Techniques
To capture the “exact” characteristics of a structure, usually a very large FEM with thousands or millions of
DOFs is used; on the other hand, most control techniques can be applied only to low-order models due to
complications in solving for and implementing very high-order controllers. Since it is impractical to design
the primary controller using the full order model, Figure 6.1 shows how a KF and a primary controller, based
90
on a reduced-order model, can be implemented; all other blocks remain the same. Some methods such as
balanced reduction techniques are more rigorous and automatic (i.e., user-free) while others such as Guyan
reduction and modal truncation require more thought and experience by the designer to select the appropriate
number of states to be kept in the reduced-order model. It is important to choose an appropriate order for the
reduced-order model because reduction to a model with very few states may poorly reproduce the response
of the real system, resulting in a controller that performs poorly: the first numerical example demonstrates
that the error increases as the reduced model order decreases.
6.2.1.1 Balanced Reduction
Balanced reduction, one of the most common methods (particularly in the controls and systems communities)
to reduce the order of a linear time invariant dynamic model, removes the (linear combinations of) states that
are less observable and controllable, contribute little to the system response and are less likely to be excited
by the external excitation (Inman, 2006).
Consider a state-space model in the form of (5.1a) with measurement vector Y and outputs Z shown in (5.1b)
and (5.1c), respectively. The degrees of controllability and observability can be measured by the rank and
singular values of the controllability (W
c
) and observability (W
o
) Gramians:
W
2
c
=
Z
¥
0
e
At
[B L][B L]
T
e
A
T
t
dt (6.1a)
W
2
o
=
Z
¥
0
e
A
T
t
[C
T
y
C
T
z
][C
T
y
C
T
z
]
T
e
At
dt (6.1b)
Moore (1981) showed that a transformation X= TX
0
can be chosen such that the system is “balanced” with
equal and diagonal Gramians W
0
c
= T
1
W
c
T and W
0
o
= T
1
W
o
T; this transformation can be computed using
the Cholesky factors of the Gramians (Varga, 1991). Next, the elements of the equal Gramians’ diagonals —
called the system Hankel singular values, which can be computed from the original untransformed Gramians
— are arranged (often sorted) into the larger significant ones and the smaller negligible ones; the states
corresponding to the latter are removed to form a reduced-order model. Balanced reduction is implemented
in MATLAB by:
91
(i) usinghsvd to compute system Hankel singular values,
(ii) selecting a cut-off point to retain the linear combinations of states that significantly contribute to the
system input/output map, and
(iii) using eitherbalred ormodred to treat as infinitely fast the (linear combinations of) states that do
not contribute (or just truncate, which gives a better frequency-domain approximation but a less accurate
representation as in the time domain).
The resulting reduced-order model can be written in state-space form
˙
X
red
= A
red
X
red
+ B
red
w+ L
red
p
Y
red
= C
y,red
X
red
+ D
y,red
w+ F
y,red
p+ n
Z
red
= C
z,red
X
red
+ D
z,red
w+ F
z,red
p,
(6.2)
where X
red
is an m 1 reduced state vector, and Y
red
and Z
red
are measurement and output vectors the same
sizes as Y and Z, respectively.
6.2.1.2 Guyan Reduction
While FEMs of complex systems often have many DOFs, some DOFs (e.g., those on a rigid links, etc.)
may not be necessary to represent the dynamics of the system. Guyan reduction (Guyan, 1965) applies a
transformation based on a partitioning of the stiffness matrix, neglecting the inertial effect of the DOFs that
are removed (Chen and Pan, 1988). The linear time invariant configuration-space system can be partitioned
as (Inman, 2006)
2
6
4
M
11
M
12
M
21
M
22
3
7
5
8
>
<
>
:
¨ u
1
¨ u
2
9
>
=
>
;
+
2
6
4
C
11
C
12
C
21
C
22
3
7
5
8
>
<
>
:
˙ u
1
˙ u
2
9
>
=
>
;
+
2
6
4
K
11
K
12
K
21
K
22
3
7
5
8
>
<
>
:
u
1
u
2
9
>
=
>
;
=
8
>
<
>
:
B
w
1
B
w
2
9
>
=
>
;
w+
8
>
<
>
:
B
p
1
B
p
2
9
>
=
>
;
p (6.3)
To guarantee that u
2
can be truncated, two conditions should be satisfied: (a) u
2
should not carry any
external force, so B
w
2
= 0 and B
p
2
= 0 and (b) a small change in u
2
does not change the system potential
energy, or¶[
1
2
u
T
(t)Ku(t)]/¶u
2
= 0
T
, so u
2
(t)=K
1
22
K
21
u
1
(t). Using the transformation u= Tu
1
where
92
T=[I (K
1
22
K
21
)
T
]
T
, the resulting reduced-order structural matrices are
M
r
= T
T
MT= M
11
K
T
21
K
1
22
M
21
M
12
K
1
22
K
21
+ K
T
21
K
1
22
M
22
K
1
22
K
21
(6.4a)
C
r
= T
T
CT= C
11
K
T
21
K
1
22
C
21
C
12
K
1
22
K
21
+ K
T
21
K
1
22
C
22
K
1
22
K
21
(6.4b)
K
r
= T
T
KT= K
11
K
12
K
1
22
K
21
(6.4c)
B
w
r
= T
T
B
w
and B
p
r
= T
T
B
p
(6.4d)
from which state-space matrices for the form of (6.2) can be constructed.
6.2.1.3 Modal Truncation
The third method discussed here is based on neglecting those eigenvalues (and their corresponding modes)
which are farthest from the origin (Davison, 1966) and therefore will not be excited with the applied force
(and, even if excited, will rapidly decay due to damping). For the state-space model with state equation (5.1a),
measurement equation (5.1b) and output equation (5.1c), using eigensolution AF=FL with eigenvector
matrixF and diagonal eigenvalue matrixL, a transformation X=Fx x x can decouple the individual modes of
the state-space system. Partitioning the modes into two groups gives
8
>
<
>
:
˙
x x x
1
˙
x x x
2
9
>
=
>
;
=
2
6
4
L
1
0
0 L
2
3
7
5
8
>
<
>
:
x x x
1
x x x
2
9
>
=
>
;
+
2
6
4
Y
1
Y
2
3
7
5[Bw+ Lp] (6.5a)
Y= C
y
[
F
1
F
2
][x x x
T
1
x x x
T
2
]
T
+ D
y
w+ F
y
p (6.5b)
Z= C
z
[
F
1
F
2
][x x x
T
1
x x x
T
2
]
T
+ D
z
w+ F
z
p (6.5c)
wherex x x =[x x x
T
1
x x x
T
2
]
T
and[Y
T
1
Y
T
2
]
T
=F
1
=[F
1
F
2
]
1
. As in the balanced reduction, the second set of
modes are taken to be infinitely fast by letting
˙
x x x
2
= 0, solving the second set of equations in (6.5a) forx x x
2
and
substituting back into the measurement and output equations (6.5b)-(6.5c) to find the reduced model: A
red
=
L
1
, B
red
=Y
1
B, L
red
=Y
1
L; C
y,red
= C
y
F
1
, D
y,red
= D
y
C
y
F
2
L
1
2
Y
2
B, F
y,red
= F
y
C
y
F
2
L
1
2
Y2L;
C
z,red
= C
z
F
1
, D
z,red
= D
z
C
z
F
2
L
1
2
Y
2
B, and F
z,red
= F
z
C
z
F
2
L
1
2
Y2L.
93
6.2.2 Incorporating a Kalman Filter to Estimate the Reduced-Order Model States
Full-state feedback is unavailable in real implementations (in the laboratory or in full-scale) where only
a limited set of noisy response measurements are available. The most commonly used observer is the KF
(Kalman, 1960), or continuous-time Kalman-Bucy filter (Kalman and Bucy, 1961), which estimates states
b
X
from noise-corrupted measurements. A KF can be used to estimate either the full states or the reduced-order
model states; the latter is used in this chapter:
˙
b
X
red
= A
red
b
X
red
+ L
red
p+ L
k
(Y
b
Y)
b
Y= C
y,red
b
X
red
+ F
y,red
p
(6.6)
where L
k
is the Kalman gain matrix.
Since the Kalman filter equation is linear, it is easily combined with the original state-space system giving
˙
e
X(t)=
e
A
e
X(t)+
e
Be w(t)+
e
Lp(t) (6.7)
e
A=
2
6
4
A 0
L
k
C
y
A
red
L
k
C
y,red
3
7
5 ,
e
B=
2
6
4
B 0
L
k
D
y
L
k
3
7
5 ,
e
L=
2
6
4
L
L
red
+ L
k
(F
y
F
y,red
)
3
7
5
where
e
X=[X
T
b
X
T
red
]
T
is the generalized state vector (the corresponding nominal response ise x=[x
T
b x
T
red
]
T
);
ande w(t)=[w
T
n
T
]
T
is the generalized excitation vector.
6.3 Numerical Examples
Although the proposed method can be applied to large variety of problems, it is tested herein for two examples:
a moderate size base-isolated frame structure with a controllable damper in the isolation layer and a large
size 3D building with three controllably-damped tuned mass dampers on the roof.
94
6.3.1 100-DOF Base-Isolated Structure Example
6.3.1.1 Model Description and Ground Excitation
The same 11-story 2-bay frame superstructure, modeled as a plane frame with 99 horizontal, vertical and
rotational DOFs, sitting on an isolation layer — which is introduced inx 4.3.1 and shown in Figure 4.4 —
is used herein. Assuming that the isolation layer is rigid in plane and moves only horizontally, the isolated
structure model has 100 DOFs. For brevity, the other characteristics of the model (discussed inx 4.3.1)
are not repeated here. To generate an excitation consistent with the frequency content of some historical
earthquakes (two moderate earthquakes: 1940 El Centro and 1968 Hachinohe; and two severe earthquakes:
1994 Northridge and 1995 Kobe), the same Kanai-Tajimi filter (Soong and Grigoriu, 1993) designed in the
previous chapter inx 5.3.1 is implemented here. The filter is a second order dynamical system characterized
by (5.8).
6.3.1.2 Effectiveness of Controllable Damping Designed using a Reduced Model
As discussed previously, it is impractical or infeasible to design controllers of very high order so the utilization
of reduced-order models for designing estimators and controllers is commonplace. If V=[0 1 0] (a 1 202
row vector with a 1 in the 101
st
entry) selects the base velocity, then to ensure accurate tracking of the
measurements Y, the outputs Z, and the base velocity ˙ u
b
, the system for balanced reduction in a state space
realization is:

A, [B L], [C
T
y
C
T
z
V
T
]
T
,

[D
T
y
D
T
z
0]
T
[F
T
y
F
T
z
0]
T

(6.8)
To determine an appropriate order for the reduced model, the Hankel singular values of the system are
calculated using MATLAB’shsvd command and are shown in Figure 6.2. Since there is a large drop (about 2
orders of magnitude) in the state energy between the 28
th
and 29
th
order models, the reduced model is chosen
to have 28 states (14 modes).
To understand how the choice of reduction technique and the order of the reduced model may affect the system
response fidelity, the closed-loop system responses are computed using the full-order system model coupled
95
0 10 20 30 40 50
10
−8
10
−6
10
−4
10
−2
10
0
10
2
State
State Energy
Figure 6.2: Hankel singular values of the system.
0 20 40 60 80 100
10
−6
10
−4
10
−2
10
0
Number of modes
Relative error

Modal truncation
Balanced reduction
Guyan reduction
Figure 6.3: Relative base drift error for various reduction techniques.
to observers and feedback gains computed (at the design point discussed in the subsequent section) for each
reduction technique and each reduced model order, all compared to the closed-loop responses with observer
96
0 20 40 60 80 100
10
−6
10
−4
10
−2
10
0
Number of modes
Relative error

Modal truncation
Balanced reduction
Guyan reduction
Figure 6.4: Relative roof acceleration error for various reduction techniques.
and feedback gain computed with the full-order model. The relative error in root mean square (RMS) base
drift and roof acceleration are shown in Figures 6.3 and 6.4, respectively, for different reduced model orders
with the three reduction techniques. Interestingly, the relative error for modal truncation technique declines
significantly as the order of the reduced model increases to 16 modes, but shows no further improvement
until at least 70 modes are retained. Guyan reduction removes the DOFs with negligible effect on the system
performance and with no applied external force. In this example, the vertical and rotational DOFs satisfy
these constraints, giving a transformation matrix T
10034
, which produces a reduced model of order 70 (i.e.,
the 34 horizontal displacement states[u
b
u
1
u
4
u
97
]
T
, the 34 corresponding velocity states, and 2
Kanai-Tajimi states). Further, a shear building assumption can be added, lumping a floor’s horizontal external
forces at the center node and condensing out the other horizontal DOFs, using the transformation matrix
T
10012
= T
10034
T
3412
where T
3412
= blkdiag(1,1
31
, ,1
31
). The Guyan-reduced model has error
comparable to modal truncation when both are reduced to 12 modes but the 34-mode Guyan-reduced model is
superior to the 34-mode modal truncation model. Finally, the balanced reduction algorithm is implemented to
find the reduced models as well. As shown in Figure 6.3, for any model order, the reduced model calculated
by the balanced reduction technique results in relative error that is several orders of magnitude smaller than
the other reduction techniques. (Note that, due to numerical errors in zero-pole cancellation, the maximum
97
number of states that can be kept in the balanced reduction model is 70 or, equivalently, 34 modes.)
6.3.1.3 Design Point, Accuracy, and Computation Time
The structural performance objective for the controllable damper design is to minimize the base drift subject
to two constraints: (1) RMS roof acceleration remains at (or below) that using the baseline lead rubber
bearing (LRB) design, a typical isolation implementation, detailed inx 4.3.2; and (2) the RMS control force
is at most 4% of building weight W . Consequently, the cost function defined in (4.2) can be used to determine
the best controllable damping performance by tuning the diagonal response weight Q
i
and the damper force
weight R
i
. The damper force weight is a constant R
i
= W
2
with building weight W = 1.28 MN; using
building height h= 44 m and, uncontrolled isolation period T
i
1
= 2.76 s, the tunable diagonal response
weight Q
i
= diag(a
i
/h
2
,b
i
(T
i
1
)
4
/(16p
4
h
2
)) depends on dimensionless tuning parametersa
i
andb
i
. Prior
numerical studies inx 5.3.4 showed that, for one realization of the Kanai-Tajimi stochastic excitation, the
fminsearch algorithm in MATLAB’s optimization toolbox determined the optimal design point to be
(a

,b

)=(3931.40,54.23), which leads to about a 49.8% base drift reduction without any roof acceleration
increase compared to the baseline performance. Also, the peak and RMS of the controllable damping force
are 11.9% and 4.0% of the building weight, respectively.
The system is simulated at the same design point, when the KF and the primary controller are designed for
the full model and for the reduced-order model. The responses of both systems are calculated by a traditional
nonlinear solver, MATLAB’sode45 with the default relative and absolute error tolerance of 10
–3
and 10
–6
,
respectively. Figures 6.5 and 6.6 show the base drift and absolute roof acceleration of the closed-loop
controlled system, calculated by both design strategies. Clearly, the responses are almost identical (relative
RMS differences are 0.55% and 0.01%, respectively), indicating that using an appropriate reduced-order
model for designing the KF and the primary controller can result in very similar controlled responses.
Finally, the computational efficiency achievable with the proposed approach is compared with that of
traditional nonlinear solver MATLAB’s ode45. To perform a fair comparison, the relative and absolute
error tolerance of ode45 are set to be 10
–3
and 10
–6
, respectively, and the proposed method is tuned to
use a second-order accurate trapezoidal integration rule with 2
15
time steps ofDt= 0.92 ms duration each.
Preliminary studies indicated that both of these have relative accuracy on the order of 10
–3
. Graphs of
98
0 2 4 6 8 10
−8
−6
−4
−2
0
2
4
6
8
Time (s)
Base Drift (cm)

Original system
Reduced system
Figure 6.5: Base drift with the full and reduced-order design models.
0 2 4 6 8 10
−3
−2
−1
0
1
2
3
Time (s)
Absolute roof acceleration (m/s
2
)

Original system
Reduced system
Figure 6.6: Roof absolute acceleration: full and reduced-order designs.
controlled base drift and roof acceleration (Figures 6.7 and 6.8) calculated at the design point using the
proposed approach andode45 are visually identical, which verifies their comparable levels of accuracy.
However, the computational intensity of the methods are quite different: on a computer with a 3.4 GHz
Intel core i7-2600 processor and 8 GB of RAM, running MATLAB R2013a under Windows 7, the proposed
99
0 2 4 6 8 10
−8
−6
−4
−2
0
2
4
6
8
Time (s)
Estimated Base Drift (cm)

Proposed approach
Traditional solver
Figure 6.7: Base drift calculated at the design point using the proposed approach andode45.
method requires one-time calculations taking 6.6 s (all timings reported herein are CPU times measured with
MATLAB’scputime function) and repeated ones each taking 1.4 s; for a single function evaluation,ode45
takes about 266 s. This results in a computational cost ratio of 35 for a single simulation, which is already
significant, and about 190 for the many required simulations as shown in the last row of Table 6.2. This
demonstrates that using the proposed method instead of conventional nonlinear solvers provides accurate
response calculation with much less computational effort. Table 6.2 also shows the computation times with no
model reduction; for this moderately-sized full model, the reduced model is modestly faster (the subsequent
higher-order example will significantly amplify this advantage).
6.3.1.4 Investigation of the System Reliability
This section investigates the reliability of the 100-DOF base isolated building model augmented with a
controllable damper commanded by a controller designed using a reduced-order control-oriented model. The
design point(a

,b

) found in the previous section has been determined based on a single realization of the
excitation. Here, a Monte Carlo simulation with 1000 realizations is performed to investigate the reliability of
the controlled system, with a goal to minimize the likelihood of the base drift exceeding a critical drift level
while ensuring that (a) exceeding a critical absolute roof acceleration is no more likely than with the baseline
100
0 2 4 6 8 10
−3
−2
−1
0
1
2
3
Time (s)
Estimated Roof Acceleration (m/s
2
)

Proposed approach
Traditional solver
Figure 6.8: Roof absolute acceleration calculated at the design point using the proposed approach and
ode45.
LRB system, and (b) the controllable damping force will exceed the median LRB force is very unlikely.
Mathematically:
min
K
d
P

max
tT

x
SA
b
(t)

> x
crit
b

subject to: (6.9)
P

max
tT

Z
SA
2
(t)

> Z
crit
2

P

max
tT

Z
LRB
2
(t)

> Z
crit
2

and P

max
tT

p
SA
(t)

> p
crit

is “small”
where P

max
tT

p
LRB
(t)

> p
crit

= 0.5 defines p
crit
; T is the duration of the excitation; Z
2
, the second ele-
ment of output Z, is the roof absolute acceleration; and the primary control state feedback gain is determined
from(a,b) as described previously. The thresholds for the base drift and absolute roof acceleration depend
on many factors, such as excitation magnitude, human comfort, size of the isolator (or controllable damping)
device. This reliability study uses the same 1000 realizations of the stochastic excitation, scaled so that the
RMS ground acceleration is one-third of 1g, so that the peak ground acceleration (PGA) is about 1g, for each
control design in the parameter study. The critical drift and excitation are chosen to be x
crit
b
= 40 cm and
Z
crit
2
= 10 m/s
2
.
101
Table 6.2: Computational cost comparison for 1000 simulations (averages of 5–10 runs since individ-
ual timings vary roughly up to 5%).
Computation (CPU) time
Full System + Full System +
Analysis Full Controller Reduced Controller
Steps (2202 states) (202+28
*
states)
ode45 (RelTol = 10
–3
, AbsTol = 10
–6
)
reduce model — 1 0.4 s
Kalman gain 1 0.3 s 1 0.1 s
control gain K
d
i
i 1000 0.2 s 1000 0.0 s
ode45 for
e
X; calc. Z + 1000 321 s + 1000 266 s
TOTAL 3.7 days
†
3.1 days
†
(Cost factor
‡
) (228) (189)
Proposed method (2
15
steps,Dt = 0.00092 s)
reduce model — 1 0.4 s
Kalman gain 1 0.3 s 1 0.1 s
one-time:e x, H
L
1 13.5 s 1 6.1 s
control gain K
d
i
i 1000 0.2 s 1000 0.0 s
simulate
{
p; calc. Z + 1000 1.7 s + 1000 1.4 s
TOTAL 31.9 mins
†
23.4 mins
† #
(Cost factor
‡
) (1.4) (1)

reduced model consists of 28 states
†
projected times based on single simulation times
‡
time relative to proposed method w/reduced controller
#
#
method used for cost factor comparisons
{
including negligible time to compute K
d
i
iH
L
and x
To determine the optimal design point, a parameter study over different primary controllers is performed,
each determined via a linear quadratic regulator (LQR) design using cost function (4.2) with weights Q
i
and R
i
parameterized by a and b. Here, a grid of points, over a
i
2[400,40000] and b
j
2[5,500], is
chosen, centered around the point(a

,b

)=(3931.40,54.23) used in the Design Point section. At each
grid point, a Monte Carlo simulation (MCS) of 1000 realizations is used to approximate the probabilities
in (6.9). Figure 6.9 shows the reliability contours of the controlled system along with the optimal design
point at(a,b) = (2917.45,79.50) where: (i) the peak drift is only 0.93% likely to exceed x
crit
b
— a significant
improvement over the 73.8% likelihood for the baseline LRB system; (ii) the peak controllable damping
force exceeds p
crit
only 7.6% of the time, much less than the 50% level for the passive LRB system; and
(iii) the absolute roof acceleration has a small 6% likelihood of exceeding Z
crit
2
, very close to that of the
LRB system. Even for this small 5-by-5 grid of design points, each simulated for 1000 realizations, the
102
conventional simulation withode45 would require 93 days for the control designed with the full model or
77 days for the reduced-order control design; in contrast, the proposed method requires only 13 hrs or 9.7 hrs,
respectively, for the full and reduced-order controllers.
0.01
0.03
0.06
0.10
0.15
0.01
0.02
0.05
0.1
0.3
0.5
0.7
0.9

drift weight α
acceleration weight β

10
3
10
4
10
1
10
2
P(Z
2
SA
,max
> Z
2
crit
)
< P(Z
2
LRB
,max
> Z
2
crit
)
P(p
SA
max
> p
crit
)
P(x
b
SA
,max
> x
b
crit
)
design point
Figure 6.9: Reliability contours of the controlled system
6.3.2 High-Order Structure Example with Three Roof-Mounted Controllably-Damped Tuned
Mass Dampers (TMDs)
6.3.2.1 Model Description
For the second example, consider the complex 3D model in Figure 6.10; relative to the first example, this is
a much higher-order model (1623 DOFs), the excitation is a simulated wind force, and there are multiple
controllable dampers. The building model, adapted from Wojtkiewicz and Johnson (2014), represents a
20-story building structure with 5-bay-by-3-bay, 3-bay-by-2-bay, and 2-bay-by-2-bay plans, respectively, in
stories 1–5, 6–10, and 11–20. The frequencies of the first six modes of the uncontrolled system range from
0.57 Hz to 2.03 Hz. Three TMDs, each with a controllable damper, are installed on the roof of the building,
two in the x-direction and one in the y-direction, to mitigate the response of the structure. More details of the
103
model are reported in Wojtkiewicz and Johnson (2014).
The building equations of motion are similar to the form of (2.2) but with the 3246 1 state vector X
s
(t),
the 3246 3246 state matrix A
s
, the 1 1 wind excitation W(t), the 3 1 force vector p(t) for the con-
trollable dampers in the TMDs, the 3246 1 wind influence matrix B
s
, and the 3246 3 damper force
influence matrix L
s
. The 14 1 output vector Z=[
d
1
d
2
d
3
˙
d
1
˙
d
2
˙
d
3
a
1
a
8
]
T
includes three
relative displacements (strokes) of the controllable dampers, the three corresponding relative velocities,
and eight absolute accelerations of the top-floor corners in both x- and y-directions; and therefore, the
output matrices C
sz
, D
sz
and F
sz
are 14 3246, 14 1, and 14 3, respectively. The measurement vec-
tor is Y=[
d
1
d
2
d
3
a
1
a
8
]
T
+ n and results in similar corresponding C
sy
, D
sy
and F
sy
which are
11 3246, 11 1, and 11 3 matrices, respectively.
−15
−5
5
15
−20
−12
−4
4
12
20
0
8
16
24
32
40
48
56
64
72
80
x [m]
E
30 °
wind
N
y [m]
z [m]
Figure 6.10: 1623-DOF structural model with three controllably-damped roof-mounted TMDs
(adapted from Wojtkiewicz and Johnson (2014))
104
6.3.2.2 Wind Excitation
The wind force, adapted from Wojtkiewicz and Johnson (2014), is one realization of a unidirectional
colored Gaussian noise process, created with a bandpass filter that suppresses energy at frequencies outside
[1/1.2,1.2] times the structure’s fundamental frequency, oriented toward the ENE at a 30

degree angle from
the x-axis, and vertically shaped proportional to the 0.3 power of the vertical location (Holmes, 1996). The
state-space form of the 16th order Butterworth bandpass filter is
˙ x
w
= A
w
x
w
+ B
w
w (6.10)
W= C
w
x
w
+ D
w
w
where A
w
is 16 16, B
w
is 16 1, C
w
is 1 16, and D
w
is 1 1; x
w
(t) are the filter states; w(t) is a
Gaussian white noise; and W(t) is the generated excitation to be applied to the model as a wind force. Wind
filter (6.10) can be combined with the building model into combined state-space equations of motion of the
form (5.1), with 3262 1 state vector X=[X
T
s
x
T
w
]
T
, output and measurement matrices C
z
=[C
sz
D
sz
C
w
],
D
z
= D
sz
D
w
, and F
z
= F
sz
, and similarly, C
y
=[C
sy
D
sy
C
w
], D
y
= D
sy
D
w
, and F
y
= F
sy
, and state matrices
A=
2
6
4
A
s
B
s
C
w
0 A
w
3
7
5 , B=
2
6
4
B
s
D
w
B
w
3
7
5 , L=
2
6
4
L
s
0
3
7
5 . (6.11)
6.3.2.3 Baseline Performance: a Linear Passive Viscous Damper
A baseline design is necessary to evaluate the improvements provided by the controllable dampers. The
baseline here is chosen to be the same structure but with three similar passively-damped TMDs. The TMD
damping is assumed to be linear and viscous (Symans and Constantinou, 1999) where the damping force is
proportional to the velocity across the damping element. The damping coefficient for the one x-direction
TMD is denoted c
x
and the damping coefficients for both of the y-direction TMDs (chosen the same because
of building symmetry) are denoted c
y
. To provide a fair comparison, the “best” passive baseline should
be used. When the TMD damping coefficients increase from zero: the RMS device strokes monotonically
decrease; the RMS device forces monotonically increase; and the RMS roof acceleration decreases at first,
105
coming to a minimum, and then increases. The minimum roof acceleration is a reasonable candidate for the
baseline design point. An optimization is performed over the two parameters, using the derivative-free Nelder-
Mead (Nelder and Mead, 1965) simplex algorithm (Lagarias et al., 1998) in MATLAB’sfminsearch, to
minimize RMS roof acceleration for the realization of the wind excitation. To provide a single optimization
metric, s
a
=[
1
8
å
8
i=1
s
2
a
i
]
1/2
is chosen as it is an RMS measure of the baseline’s eight roof accelerations
a
i
(t), i2f1, ,8g, where s
2
a
i
=
1
T
R
T
0
a
2
i
(t)dt (the overline denotes a baseline response). This resulting
baseline has damper coefficients c
x
= 27.11 kNs/m and c
y
= 22.31 kNs/m, resulting in RMS device stroke
s
d
=[
1
3
å
3
i=1
s
2
d
i
]
1/2
= 11.2cm, RMS top-floor accelerations
a
= 0.652m/s
2
, and RMS passive damper force
s
p
=[
1
3
å
3
i=1
s
2
p
i
]
1/2
= 9.673 kN= 0.029%W, where W = 33.58 MN is the building weight (not including
the TMDs).
6.3.2.4 Reduced-Order Model and Kalman Filter Design
While the method proposed herein is not limited to any specific model reduction technique, balanced reduction
is used for this example since it was clearly the best performing reduction technique in the first example
(Figure 6.3). For this 1623 DOF model, the state energy, as determined by the system Hankel singular values
and shown in Figure 6.11, decrease smoothly and, unlike Figure 6.2 for the previous example, does not
exhibit any sudden drops to pinpoint obvious “good” model order choices. For the sake of convenience, a
small-size model of order 50 is used here for designing the KF and to calculate the desired damping force
gain matrices.
It is assumed that each of the three TMD controllable dampers is equipped with an identical relative
displacement sensor, e.g., a linear variable displacement transducer (LVDT), to measure the damper stroke
d
i
(t), i2f1,2,3g; further, identical bi-directional (x and y) accelerometers are installed at each corner of the
top floor of the structure to measure the accelerations a
i
(t), i2f1, ,8g. These eleven sensor measurements,
are corrupted by low-level additive measurement noises, modeled as independent Gaussian pulse processes
with RMS that is 5% of the baseline RMS of the corresponding sensor classes (i.e., 5% of s
d
and s
a
,
respectively).
106
0 20 40 60 80 100
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
State
State Energy
Figure 6.11: Hankel singular values of the 1623-DOF structural model.
6.3.2.5 Design Point, Accuracy, and Computation Time
The controllable damper is constrained by space (stroke constraint) and physical device limitations (force
constraint). The objective of the controllable damper design here is to minimize the RMS controllable damping
device strokes while the RMS top-floor absolute acceleration remains at or below the corresponding baseline
performance; i.e., minimizes
d
subject tos
a
s
a
, wheres
d
=[
1
3
å
3
i=1
s
2
d
i
]
1/2
ands
a
=[
1
8
å
8
i=1
s
2
a
i
]
1/2
.
Similar to the previous example, the space of possible controllable damping strategies is parameterized
by a family of LQR control gains minimizing the cost function (4.2), with controllable damper force
weight a constant R
i
= 10/W
2
, and a response weight Q
i
= diag(a
i
1
13
/h
2
,b
i
1
18
(T
1
)
4
/(16p
4
h
2
)) where
T
1
= 1.75 s is the building’s fundamental natural period; and the building height h= 80 m. As shown in
Figure 6.12, the optimal design point is (a

,b

)=(87,185), which leads to about a 43.87% reduction
in RMS damper stroke relative to the baseline without any increase in RMS top-floor accelerations; i.e.,
1(s
d
/s
d
)= 0.4387 ands
a
s
a
. Also, the peak and RMS of the controllable damping force are 0.35%W
and s
p
= 0.066%W, respectively. To demonstrate that controllable damping design using the full and
reduced-order models are nearly the same, the corresponding closed loop systems are both simulated at the
107

10
0
10
1
10
2
10
3
10
4
10
1
10
2
10
3
Drift Weight α
Acceleration Weight β
−99%
−75%
−50%
−25%
0%
25%
0.05%
0.06%
0.07%
0.08%
roof accel. reduced
damper stroke change
RMS force/bldg.wt.
design point
Figure 6.12: RMS controllable damping responses relative to those with the passive viscous damper
0 2 4 6 8 10
−30
−20
−10
0
10
20
30
Time (s)
Roof−center TMD stroke (cm)

Full−order controllers
Reduced−order controllers
Figure 6.13: Roof-center TMD (x-direction) stroke with the full and reduced-order models.
design point; the resulting responses, as shown in Figures 6.13 and 6.14, are visually identical (relative RMS
differences are 0.80% and 0.67%, respectively), indicating that using an appropriate reduced-order model for
designing the KF and the primary controller results in very nearly the same controlled responses.
108
0 2 4 6 8 10
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (s)
Top−floor SW−corner x−dir. accel. (m/s
2
)

Full−order controllers
Reduced−order controllers
Figure 6.14: Top-floor acceleration at the southwest corner in the x-direction with the full and reduced-
order models.
Finally, to test this chapter’s primary hypothesis that the proposed method provides greatly increased
computational efficiency, its computation time is compared to that with the traditional solver MATLAB’s
ode45. The proposed method simulations use a second-order accurate trapezoidal integration with 2
17
time steps ofDt= 0.23 ms duration each, andode45’s default tolerances (10
–3
relative and 10
–6
absolute)
are used; both of these have relative error on the order of 10
–3
. However, as shown in the last column of
Table 6.3, the proposed method with a control strategy designed with a reduced model requires one-time
calculations taking 4.04 hrs (0.52 hrs to reduce the model, 1.41 hrs to calculate the unmodified responsee x,
and 2.11 hrs to determine the impulse response H
L
) and repeated ones taking only 45 s for each candidate
design point; in contrast, for each candidate design point evaluation, ode45 takes about 4.07 hrs. Thus,
the computation time reduces very modestly for a single simulation, but reduces by a factor of 215 for
1000 simulations, demonstrating that the proposed approach provides significant computational benefits
for controllable damping design. Further, this shows that a reduced-order observer and control force gain
design, even though evaluated in simulation with a full-order model of the system, provides significant
gains. The proposed computational simulation approach and the reduced-order control design together give
a computational speed-up of 786 relative to ode45 simulation of the full model using a controller also
109
designed from the full order model.
6.4 Conclusions
To design controllable damping strategies for real-world complex dynamical systems, it is impractical,
infeasible and/or unnecessary to design controllers for very high-order systems. This chapter proposes using
a computationally efficient simulation technique, in conjunction with reduced-order models for designing
observers and feedback gain matrices for controllable damping device design. The closed-loop system formed
Table 6.3: Computational cost comparison for 1000 simulations (averages of 5–10 runs since individ-
ual timings vary up to about 5%).
Computation (CPU) time
Full System + Full System +
Analysis Full Controller Reduced Controller
Steps (23262 states) (3262+66
*
states)
ode45 (RelTol = 10
–3
, AbsTol = 10
–6
)
reduce model — 1 0.52 hrs
Kalman gain 1 8.98 mins 1 0.13 s
control gain K
d
i
i 1000 0.18 hrs 1000 0.03 s
ode45 for
e
X 1000 0.53 days 1000 3.55 hrs
calculate Z + 1000 0.09 s + 1000 0.06 s
TOTAL 540.1 days
†
148.1 days
†
(Cost factor
‡
) (786) (215)
Proposed method (2
17
steps,Dt = 0.00023 s)
reduce model — 1 0.52 hrs
Kalman gain 1 8.98 mins 1 0.13 s
nominal resp.e x 1 5.23 hrs 1 1.41 hrs
impulse resp. H
L
1 0.49 days 1 2.11 hrs
control gain K
d
i
i 1000 0.18 hrs 1000 0.03 s
imp. resp. K
d
i
iH
L
1000 2.62 mins 1000 0.73 s
nominal resp. x 1000 0.10 mins 1000 0.08 s
simulate p 1000 0.69 mins 1000 0.69 mins
calculate Z + 1000 3.05 s + 1000 3.05 s
TOTAL 10.8 days
†
16.5 hrs
† #
(Cost factor
‡
) (16) (1)

reduced model consists of 50 structural and 16 wind filter states
†
projected times based on single simulation times
‡
time relative to proposed method w/reduced controller
#
#
method used for cost factor comparisons
110
by the full structure model and the dynamic controllable damping strategies, whether designed using full or
reduced-order models, satisfies the conditions for applying the NVIE computationally-efficient simulation
technique for systems that are mostly linear but with localized nonlinearities (at the controllable damper
locations in this case). The chapter first provided brief explanations of three commonly-used model reduction
techniques: balanced, Guyan and modal reductions. Then, the paradigm for parameterizing the control design
is detailed and the incorporation of a KF observer based on reduced-order models of the structure is given.
The method is evaluated with two numerical examples, a moderate-order frame structure model and a 3D
high-order building model. For each example, the model, excitation, and controllable damping objectives
and constraints are described. For the first example, all three reduction techniques are tested in the proposed
methodology; balanced reduction alone is used for the second example.
The fidelity of the controllable damping strategies based on the reduced-order models is demonstrated by com-
parisons with those using the full models. For the first example, it is demonstrated that the proposed method
is effective in reducing the computational requirements of a reliability-based control strategy optimization
as well. For both examples, the proposed method is shown to provide significant computational advantages
relative to a traditional approach. The efficient simulation technique for mostly linear systems provides
a reduction in computation time by a factor of about 200 for both examples. For the second, high-order
example, the combined effects of simulation efficiency gains and evaluating observer/control strategies based
on reduced-order models provides a computational speed-up approaching three orders of magnitude relative
to the standard approach. Thus, it is demonstrated that controllable damping strategies can be efficiently
designed by making the simulations orders of magnitude faster by exploiting the localized nature of the
nonlinearities in the system, and that reduced-order observers and feedback gains can be incorporated into
this paradigm.
111
Chapter 7
Computationally Efficient Control Design
for a Smart Structure with Uncertainty
1
7.1 Introduction
The previous chapters introduced a methodology to adapt the nonlinear V olterra integral equation (NVIE)
approach (Gaurav et al., 2011) for computationally efficient design of nonlinear passive and semiactive
control devices for large and complex structural systems. However, the systems investigated in the previous
chapters were entirely deterministic. In this chapter, uncertainty is introduced into the system model, using
a Monte Carlo simulation (MCS) to characterize the effects of uncertain elements on the performance of
the semiactive controller. The design of a semiactive control strategy is studied for the same 100-degree of
freedom (DOF) building model (Figure 3.1) that includes a superstructure on a base isolation layer. The
randomness is induced by uncertain isolation layer structural parameters: the rubber bearing stiffness and
damping; the control force is provided by a controllable damper in the isolation layer. Numerical results
confirm the accuracy, stability and computational efficiency of the proposed algorithm.
1
This chapter is based on Kamalzare et al. (2013a)
112
7.2 Methodology
7.2.1 System of Equations along with Uncertainty
Gaurav et al. (2011) introduced an approach for efficient analysis of linear systems with a few local modi-
fications. This approach evaluates the effect of the local modifications (by solving a low-order, generally
nonlinear, V olterra integral equation) and replaces them with the equivalent force. The resulting equation
of motion is linear (with respect to those equivalent forces) and can be solved easily using superposition.
This section adapts this approach for efficient uncertainty analysis in the systems controlled by a semiactive
control device.
Consider a nominal linear deterministic structure model in state space form shown in (2.1). Let u= g
u
(X)
be a n
u
1 vector of control device forces that will be used to mitigate the structural responses. Additionally,
let f= g
f
(X,q q q) be a n
f
1 vector of forces within the structure caused by the uncertainty in parameter vector
q q q. Incorporating these forces, the equations of motion can be written as follows
˙
X(t)= AX(t)+ Bw(t)+ L
u
g
u
(X(t))+ L
f
g
f
(X(t),q q q), X(0)= x
0
(7.1)
where X(t) is the n
x
1 state vector; L
u
is an n
x
n
u
constant influence matrix that maps the (nonlinear)
control force to the appropriate system states; L
f
is an n
x
n
f
constant influence matrix that maps the
uncertainty effect to the appropriate system states; and the initial condition remains the same as that of the
nominal system.
The functional forms of g
u
(X) and g
f
(X,q q q) depend on the nature of the modifications (i.e., the model used
for semiactive control force and the form of the equation describing the uncertainty behavior) in the system
and are not required to be in any specific form. Since these effects are localized, they are direct functions of
only some subset or linear combination of states, X
u
(t)= G
u
X(t) and X
f
(t)= G
f
X(t), respectively, where
G
u
is a n
x,u
n
x
matrix and G
f
is a n
x,f
n
x
matrix.
The two modifications — control force and force from uncertainty — can be combined by merging the state
subsets X
u
and X
f
on which the control and uncertainty depend, respectively, and combining the modification
113
influence matrices L
u
and L
f
of the forces from control and uncertainty, respectively. If the state subsets are
independent, then one may define a corresponding combined n
x
1 vector X=[(X
u
)
T
(X
f
)
T
]
T
= GX, with
n
x
n
x
matrix G=[(G
u
)
T
(G
f
)
T
]
T
, where n
x
= n
x,u
+ n
x,f
; if there are elements of X
u
and X
f
in common,
then the redundant elements of X and redundant rows of G may be removed, leaving n
x
< n
x,u
+ n
x,f
.
Similarly, if the influence matrices L
u
and L
f
have unique columns, then they can be combined into the
n
x
n
p
influence matrix L= [L
u
L
f
], with the resulting combined n
p
1 modification vector g(X) =
[fg
u
(X
u
)g
T
fg
f
(X
f
)g
T
]
T
, where n
p
= n
u
+ n
f
; if columns of the influence matrices are in common, then
redundant columns of L can be eliminated and the corresponding rows of g(X) added together, resulting in
n
p
< n
u
+ n
f
. Therefore, (7.1) can be written in the form of (2.2) and be solved efficiently as discussed in
Chapter 2.
7.3 Numerical Example
7.3.1 Model Description
This example uses the same 100-DOF model of an isolated building, introduced inx 4.3.1 and shown in
Figure 4.4, to demonstrate computationally efficient semiactive control design, except the isolation layer
parameters are uncertain herein. Therefore, the model has equations of motion
2
6
4
m
b
0
T
0 M
s
3
7
5
8
>
<
>
:
¨ x
b
¨ x
s
9
>
=
>
;
+
2
6
4
c
b
+ r
T
C
s
r r
T
C
s
C
s
r C
s
3
7
5
8
>
<
>
:
˙ x
b
˙ x
s
9
>
=
>
;
+
2
6
4
k
b
+ r
T
K
s
r r
T
K
s
K
s
r K
s
3
7
5
8
>
<
>
:
x
b
x
s
9
>
=
>
;
=
2
6
4
m
b
0
T
0 M
s
3
7
5
8
>
<
>
:
1
r
9
>
=
>
;
¨ x
a
g
+
8
>
<
>
:
1
0
9
>
=
>
;
u (7.2)
where x
b
is the base displacement relative to the ground; m
b
is the base mass; c
b
and k
b
are the isolation linear
damping and stiffness, respectively; x
s
is a vector of horizontal and vertical displacements and rotation of the
superstructure nodes relative to the ground; M
s
, C
s
and K
s
are the superstructure mass, damping and stiffness
matrices, respectively; r = [1 0 0 1 0 0 1 0 0]
T
(i.e., ones in elements corresponding to horizontal
displacements in x
s
) is the influence vector; and u(t) is the control force. Ground excitation ¨ x
a
g
is the N-S
El Centro record of the 18 May 1940 Imperial Valley earthquake sampled at 50 Hz.
114
Cost function (4.2) has diagonal response weight Q
i
= diag(a
i
/h
2
,b
i
(T
i
1
)
4
/(16p
4
h
2
)) and control weight
R
i
= W
2
, wherea
i
andb
i
are dimensionless parameters that will be tuned to achieve the best semiactive
control performance. The control gain K
d
i
is designed using thelqr command in MATLAB for each pair
(a
i
,b
i
). The commanded force is clipped to 0.15W to accommodate control device practical performance
limits.
7.3.2 Model Uncertainty
While the proposed approach is capable of dealing with a wide variety of model uncertainties, this study
investigates the uncertainty in k
b
and c
b
, the stiffness and damping of the isolation layer, respectively. To
model the disturbances, replace k
b
and c
b
in (7.2) with k
b
+Dk
b
and c
b
+Dc
b
, respectively. By movingDk
b
andDc
b
to the right-side of the equation, (7.2) can be written
2
6
4
m
b
0
T
0 M
s
3
7
5
8
>
<
>
:
¨ x
b
¨ x
s
9
>
=
>
;
+
2
6
4
c
b
+ r
T
C
s
r r
T
C
s
C
s
r C
s
3
7
5
8
>
<
>
:
˙ x
b
˙ x
s
9
>
=
>
;
+
2
6
4
k
b
+ r
T
K
s
r r
T
K
s
K
s
r K
s
3
7
5
8
>
<
>
:
x
b
x
s
9
>
=
>
;
=
2
6
4
m
b
0
T
0 M
s
3
7
5
8
>
<
>
:
1
r
9
>
=
>
;
¨ x
a
g
+
8
>
<
>
:
1
0
9
>
=
>
;
u+
8
>
<
>
:
1
0
9
>
=
>
;
f (7.3)
where f =(Dk
b
x
b
+Dc
b
˙ x
b
) is a function representing the force caused by parameter uncertainties.
Generally, scalar u and scalar f results in a system of two NVIEs; i.e., p(t)= g(X(t)) where p(t)=[u f]
T
and g(X(t))=[u
d
H(u
d
˙ x
b
) Dk
b
x
b
Dc
b
˙ x
b
]
T
. However, in this specific example, the influence matrices
of both u and f are the same[1 0
T
]
T
and L
u
= L
f
, so these two terms can be combined into a single NVIE
equation with three parameters
p(t)= g(u
d
,x
b
, ˙ x
b
)= u
d
H(u
d
˙ x
b
)(Dk
b
x
b
+Dc
b
˙ x
b
) (7.4)
Applying (7.4), the equation of motion (7.3) can be simplified to
M
8
>
<
>
:
¨ x
b
¨ x
s
9
>
=
>
;
+ C
8
>
<
>
:
˙ x
b
˙ x
s
9
>
=
>
;
+ K
8
>
<
>
:
x
b
x
s
9
>
=
>
;
=M
8
>
<
>
:
1
r
9
>
=
>
;
¨ x
a
g
+
8
>
<
>
:
1
0
9
>
=
>
;
p (7.5)
115
where M, C and K are the mass, damping and stiffness matrices on the left side of (7.3).
Typical values for normalized standard deviation of stiffness and damping of a rubber bearing (Adachi and
Unjoh, 2003, Feng et al., 2004) are adopted here:Dk
b
andDc
b
are assumed to be independent, zero mean
Gaussian random variables with standard deviations of 0.05k
b
and 0.20c
b
, respectively. Note that when the
magnitude of modifications is large, the MCS should be implemented carefully to ensure that the stiffness and
damping of the system remain positive and the system does not become unstable for any realization. In this
example, the standard deviations of the modifications are more than one order of magnitude smaller than the
nominal stiffness and damping of the isolation layer (k
b
+ r
T
K
s
r and c
b
+ r
T
C
s
r, respectively) and therefore,
the probability of generating a realization with negative stiffness and damping is negligible (less than 10
-24
).
Uncertainty analysis often utilizes a MCS to investigate the effect of different parameter values. Herein,
10,000 random samples of stiffness and damping modifications (Dk
b
,Dc
b
) are generated based on their
corresponding distributions. A MCS is used to compute the statistics of root mean square (RMS) and peak
base drift and roof acceleration. As shown in Table 7.1, the coefficients of variation of the RMS and peak
responses are limited to 0.10 to 1.25 percent, which are much smaller than the corresponding values of the
uncertain parameters. This indicates that the control design is effective and the stochastic behavior of the
isolation layer parameters has a very modest effect on the structural responses. Note that the simulations
are performed at the optimal design point of the semiactive controller for a deterministic system (i.e., the
semiactive control strategy is designed for a system withDk
b
=Dc
b
= 0).
Table 7.1: RMS and peak of base drift and roof acceleration of a Monte Carlo simulation
Responses Mean Standard deviation (coeff. of variation)
Base drift [cm]
RMS 1.34 0.00134 (0.10%)
Peak 5.06 0.0297 (0.59%)
Roof acceleration [m/s
2
]
RMS 0.66 0.00162 (0.25%)
Peak 3.31 0.0413 (1.25%)
7.3.3 Computational Cost
One goal of this study is to investigate the magnitude of the reduction in computational effort that can be
achieved when the simulations required by MCS are performed using the proposed method instead of a
116
traditional solver.
Traditional Solver: To perform a MCS for the isolation layer parameters using a traditional solver such as
ode45, one should generate many pair of stiffness and damping variationsq
i
(Dk
i
b
,Dc
i
b
), i= 1, , 10,000,
based on their corresponding distributions, update the structural model for each pair, and analyze the system to
determine the structural responses. This procedure is shown in Figure 7.1. As is demonstrated in Table 7.2, an
ode45 algorithm with the default relative and absolute tolerances will perform one simulation in 243.73 sec
and, consequently, for a 10,000 realization MCS, it will take about a month.
Calculate Unmodified Response
Calculate Unmodified Response
˙ x(t)=Ax(t)+Bw(t)
Solve NVIE to
Find
Solve NVIE to
Find P(t)=g(
̄
X(t))
Calculate Modified Response
Calculate Modified Response
˙
X(t)=AX(t)+Bw(t)+Lp(t)
Control Gain (K)
Control Gain (K)
Excitation
Excitation
Calculate Impulse Response
Calculate Impulse Response
One-Time
Calculations
Repeating
Calculations
Calculate Modified Response
Calculate Modified Response
Excitation
Excitation
˙
X(t)=AX(t)+Bw(t)+Lg( ¯ X(t);θ
i
)
Control Gain (K)
Control Gain (K)
Next Uncertain
Parameter Values
(Monte Carlo Simulation)
Next Uncertain
Parameter Values
(Monte Carlo Simulation)
Calculate Unmodified Response
Calculate Unmodified Response
Solve NVIE to
Find
Solve NVIE to
Find
Calculate Modified Response
Calculate Modified Response
Control Gain (K)
Control Gain (K)
Excitation
Excitation
Calculate Impulse Response
Calculate Impulse Response
One-Time
Calculations
Repeated
Calculations
Next Realization
Next Realization
Next Realization
Next Realization
˙ x(t)=Ax(t)+Bw(t)
P(t)=g(¯ X(t);θ
i
)
˙
X(t)=AX(t)+Bw(t)+Lp(t)
i=i+1
i=i+1
i=1 i=1
i=1 i=1
Figure 7.1: Uncertainty analysis using a traditional solver.
NVIE Approach: Significant increases in computational efficiency can be achieved if the proposed approach
is implemented for system model (7.5) because the nominal system remains unchanged for the MCS;
therefore, the unmodified response and the necessary impulse responses need to be calculated just once. As
shown in Figure 7.2, each new simulation requires solving only a single NVIE equation and finding the
modified response, which results in much smaller computational effort compared to the traditional solver.
For a fair timing comparison, both the traditional solver (i.e.,ode45) and the NVIE method should have the
same level of accuracy. Therefore, a preliminary parameter study was performed to determine the integration
time step in the proposed method which will give an accuracy comparable to that ofode45 with the default
tolerances (i.e., relative tolerance 10
3
and absolute tolerance 10
6
); this preliminary study found that a time
stepDt of about half of a millisecond gives accuracy comparable to the defaultode45.
For this example, one simulation at the control design point takes 12.20 sec (8 sec for one-time calculations
and 4.20 sec for the repeated calculations). However, in the case of multiple simulations, computational
117
Calculate Unmodified Response
Calculate Unmodified Response
˙ x(t)=Ax(t)+Bw(t)
Solve NVIE to
Find
Solve NVIE to
Find P(t)=g(
̄
X(t))
Calculate Modified Response
Calculate Modified Response
˙
X(t)=AX(t)+Bw(t)+Lp(t)
Control Gain (K)
Control Gain (K)
Excitation
Excitation
Calculate Impulse Response
Calculate Impulse Response
One-Time
Calculations
Repeating
Calculations
Calculate Modified Response
Calculate Modified Response
Excitation
Excitation
˙
X(t)=AX(t)+Bw(t)+Lg( ¯ X(t);θ
i
)
Control Gain (K)
Control Gain (K)
Next Uncertain
Parameter Values
(Monte Carlo Simulation)
Next Uncertain
Parameter Values
(Monte Carlo Simulation)
Calculate Unmodified Response
Calculate Unmodified Response
Solve NVIE to
Find
Solve NVIE to
Find
Calculate Modified Response
Calculate Modified Response
Control Gain (K)
Control Gain (K)
Excitation
Excitation
Calculate Impulse Response
Calculate Impulse Response
One-Time
Calculations
Repeated
Calculations
Next Realization
Next Realization
Next Realization
Next Realization
˙ x(t)=Ax(t)+Bw(t)
P(t)=g(¯ X(t);θ
i
)
˙
X(t)=AX(t)+Bw(t)+Lp(t)
i=i+1
i=i+1
i=1 i=1
i=1 i=1 Figure 7.2: Uncertainty analysis using the proposed approach.
efficiency will be increased since each new simulation takes only another 4.20 sec for the repeated calculations.
By implementing the proposed method, the 10,000 realization MCSs can be performed in half of a day;
relative to the traditional solver, this represents a computation time ratio of about 58 (See Table 7.2).
Table 7.2: Cost ratio comparison.
ode45 Proposed method
RelTol = 10
3
# of steps = 2
16
Cost ratio
AbsTol = 10
6
Dt= 0.46 ms
at the design point
243.73 s 8.00+ 4.20= 12.20 s 20
(1 simulation)
Monte Carlo simulation 10
4
243.73 s 8.00+ 10
4
4.20 s
58
(10,000 simulations) = 28.2 days = 0.5 days
7.4 Conclusions
The proposed method previously introduced by Gaurav et al. (2011) for fast computational analysis of linear
systems with local modifications has been adapted to include uncertainty analysis of a system with a smart
control device. It was shown that the equation incorporating the uncertain behavior of model parameters can
be augmented by the nonlinear equation of semiactive control device. Consequently, the system of equations
118
remains in the same form as the deterministic model and the proposed method can be implemented without
any modification. The uncertainty analysis was performed using a MCS for the isolation layer parameters,
the rubber bearing stiffness and damping. It was shown that the stochastic behavior of the isolation layer
parameters has a very modest effect on the structural responses. Further, by implementing the proposed
approach instead of a traditional solver such asode45, the computational cost can be reduced by a factor of
58.
119
Chapter 8
Concluding Remarks and Future Directions
Significant improvements in computer processors have raised the interests of the control community to
investigate large and complex structural dynamic computational models for modeling their complicated
nonlinear behaviors, simulating the responses for various level of excitation and uncertainties, and designing
optimal control strategies to achieve certain objectives. However, while many of these large and complicated
systems contain nonlinearities that are localized, many standard nonlinear solvers ignore the localized nature
of the nonlinearities when computing responses, which can result in a very time-consuming process.
This dissertation starts by introducing an alternate method, introduced recently by Gaurav et al. (2011), for
efficient analysis of complex linear systems with few local nonlinearites, modifications, or uncertainties.
The method first segregates the nonlinear forces from a nominal linear system and evaluates them based
on the response of the nominal linear system. This reduces the high-order system to a much lower-order
system of nonlinear V olterra integral equations (NVIEs), which can be subsequently solved to provide a very
computationally efficient solution. The total response of the system can, then, be easily calculated using
superposition.
This dissertation first adapts the NVIE approach to provide an efficient method for the optimal design of
passive isolation systems. It is shown that the method can provide highly efficient simulation of both responses
and their sensitivities to the design parameters in the isolation element models. The computational efficiency
to perform the simulations is over an order of magnitude faster — the computation speed-up was 14.8 for
120
a typical parameter study or iterative optimization of a relatively small-sized example — compared to a
common traditional nonlinear solver.
The remainder of the dissertation focuses on efficient design of “semiactive” damping strategies, which is one
of the most active research topics in the civil engineering community. The development of control strategies
for controllable passive dampers is complicated by the nonlinear and dissipative nature of the devices and
the nonlinear nature of the closed-loop system with any feedback control. One must generally resort to a
large-scale parameter study (or perform an optimization algorithm) in which the nonlinear system is simulated
many times to determine control strategies that are actually optimal for the nonlinear controlled closed-loop
system. Since, in these types of design problems, the nonlinearity is local (and is often limited to the control
force), it is demonstrated how the NVIE approach can significantly decrease the computational burden of a
complex control design study for controllable dampers, sometimes by over two orders of magnitude.
The methodology for efficient semiactive control design is first developed for a full-state feedback control
and then expanded to the more realistic case when, instead of full-state feedback, only a limited set of noisy
response measurements are available to the controller, which requires incorporating a Kalman filter estimator,
which is linear, into the nominal linear model.
Next, the methodology is improved one step further by providing the capability of incorporating a reduced-
order model for control design with the full fidelity model to simulate semiactively controlled structural
responses. This is an essential step since the primary controller is rarely designed using a high-order model
because it is impractical due to numerical difficulties, as well as often unnecessary since high-order models,
such as complex finite element structure models, have high frequency dynamics that remain mostly unexcited
by an external disturbance. However, evaluation of the reduced-order control with the full structure model is
essential because the system is nonlinear.
Finally, it is discussed how the proposed approach can be implemented when uncertainties are involved in
the system. In presence of uncertainty, a Monte Carlo simulation (MCS) can be used to characterize the
effects of uncertain elements on the system performance. Herein, it is shown that the modification terms
corresponding to the uncertain parameters can be augmented by other nonlinearities/modifications of the
system (e.g., semiactive control force); consequently, the system of equations remains in the form that the
proposed method can be implemented without any modification. Numerical studies demonstrate that the
121
computational cost of uncertainty analysis for a small-size model can be reduced by more than one order of
magnitude using this proposed methodology.
In summary, this dissertation provides a broad, comprehensive, and computationally efficient methodology for
designing control strategies for smart structures. The proposed method promises the capability of performing,
in a computationally tractable way, the large number of simulations required for designing optimal control
strategies for controllable damping devices in complex structural models, in a manner that is much faster than
conventional nonlinear solvers without sacrificing response accuracy. Numerical results confirm the accuracy,
stability, and computational efficiency of the proposed simulation methodology.
It is worth noting that despite all the above-mentioned benefits, the main limitation of the method is: it is
most efficient on problems with very few local nonlinearities, i.e., the computational time efficiency will be
degraded if many semiactive devices are installed or the nonlinearities are distributed throughout the entire
system. For future studies, this work can be extended in various directions. First, more complicated and
realistic damper models should be investigated. The semiactive damper model considered here is an ideal
model that employs a simple Heaviside expression for the semiactive control force and also assumes any
force can be applied by the damper instantly. More realistic models, such as the one introduced by Dyke et al.
(1996), should be utilized to evaluate the accuracy and computational efficiency of the proposed method for
complex damper models. It is unclear whether implementing the NVIE approach for designing more complex
damper models would result in a higher computational time efficiency (because of higher complexity of the
nonlinear term and, possibly, more degrees of freedom (DOFs) required to express the damper behavior) or
a lower computational time efficiency (because of a nonlinear damper model smoother than the very strict
Heaviside nonlinearity assumed in this study). Second, this study shows a significant computational speed
up for repeated simulations, which suggests the proposed NVIE approach as an excellent candidate to be
implemented for real-time control strategies where very limited time is available for determining the optimal
control force at each time step. Third, the ultimate goal would be combining the aforementioned two tasks to
validate the accuracy and robustness of a semiactive control design strategy calculated by the proposed NVIE
approach by performing an experimental test in a lab.
122
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Abstract (if available)

Abstract In recent years, significant improvements in memory capacity and processing speed of computers have provided the ability of modeling and analyzing large and complex dynamical systems. These systems usually consist of many elements, of which some have nonlinear properties. Standard nonlinear solvers ignore the localized nature of the nonlinearities when computing responses, which can result in a very time-consuming process. However, since the nonlinearities are often limited to only a few of the many degrees of freedom (DOFs), an alternate method has been developed in which the nonlinear perturbation dynamics are excluded from the nominal linear system and evaluated based on the response of the nominal system. This reduces the high-order system to a much lower-order system of nonlinear Volterra integral equations (NVIEs), which provides a very computationally efficient solution. The total response of the system can be then easily calculated using superposition. ❧ This study adapts the methodology to provide a fast and computationally inexpensive method for designing control strategies implemented in—but not limited to—smart building structures. The development of control strategies for controllable passive dampers, i.e., “semiactive” damping devices, is complicated by the nonlinear and dissipative nature of the devices and the nonlinear nature of the closed-loop system with any feedback control. Control design for nonlinear systems is often achieved by designing a control for a linearized model since strategies for linear systems are straightforward. One such approach is clipped optimal control in which the desired damper forces are determined from an optimal controller (e.g., linear quadratic regulator (LQR), linear quadratic Gaussian (LQG), H₂, etc.), which is designed assuming that the damping devices are fully linear actuators that can exert any forces (dissipative or non-dissipative), and a secondary bang-bang controller commands the controllable damper to exert forces as close as possible to the desired forces. However, designs using any linearized model generally results in suboptimal (and sometimes lousy) performance because the linear actuator assumption differs from the actual implementation with a dissipative damping device. Thus, one must generally resort to a large-scale parameter study (or performing an optimization algorithm) in which the nonlinear system is simulated many times to determine control strategies that are actually optimal for the nonlinear controlled closed-loop system. Herein, it is demonstrated how the proposed approach can significantly decrease the computational burden of a complex control design study for controllable dampers. ❧ Next, this study expands the applicability of the proposed method by demonstrating that the approach can also be adapted to accommodate the more realistic cases when, instead of full-state feedback, only a limited set of noisy response measurements are available to the controller, which requires incorporating a Kalman filter estimator, which is linear, into the nominal linear model. Furthermore, since the primary controller is rarely designed using a high-order model (because it is impractical due to numerical difficulties, as well as often unnecessary since high-order models, such as complex finite element structure models, have high frequency dynamics that remain mostly unexcited by an external disturbance), to bring the method to full maturity, a reduced-order model for control design is incorporated with the full model to simulate semiactively controlled structural responses using the proposed NVIE approach. Finally, it is explained briefly how the proposed approach can be implemented when uncertainties are involved in the system. ❧ This dissertation provides a broad and comprehensive methodology for designing control strategies for smart structures using the proposed computationally efficient method. Numerical results confirm the accuracy, stability, and computational efficiency of the proposed simulation methodology and specifically show about two orders of magnitude speed up relative to the conventional solvers for the typical semiactive design parameter studies.

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Creator Kamalzare, Mahmoud (author)

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

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering (Environmental Engineering)

Publication Date 11/13/2014

Defense Date 09/29/2014

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

Tag computational efficiency,nonlinear Volterra integral equation,OAI-PMH Harvest,optimal design strategy,passive control design,semiactive control design,smart structure

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Johnson, Erik A. (committee chair), Wojtkiewicz, Steven F. (committee chair), Jonckheere, Edmond A. (committee member), Masri, Sami F. (committee member), Savla, Ketan (committee member)

Creator Email kamalzar@usc.edu,mahmoud.kamalzare@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-516836

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