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Modeling and Controlof Vibration in
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Modeling and Control of Vibration in Mechanical Systems, Chungling Du and Lihua Xie
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CRC Press is an imprint of theTaylor & Francis Group, an informa business
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Modeling and Controlof Vibration in
Mechanical Systems
Chunling DuData Storage Institute
Singapore
Lihua XieNanyang Technological University
Singapore
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Contents
Preface xi
List of Tables xiii
List of Figures xv
Symbols and Acronyms xxiii
1 Mechanical Systems and Vibration 1
1.1 Magnetic recording system . . . . . . . . . . . . . . . . . . . . . 1
1.2 Stewart platform . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Vibration sources and descriptions . . . . . . . . . . . . . . . . . 4
1.4 Types of vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Free and forced vibration . . . . . . . . . . . . . . . . . . 6
1.4.2 Damped and undamped vibration . . . . . . . . . . . . . 6
1.4.3 Linear and nonlinear vibration . . . . . . . . . . . . . . . 6
1.4.4 Deterministic and random vibration . . . . . . . . . . . . 6
1.4.5 Periodic and nonperiodic vibration . . . . . . . . . . . . . 7
1.4.6 Broad-band and narrow-band vibration . . . . . . . . . . 8
1.5 Random vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Random process . . . . . . . . . . . . . . . . . . . . . . 11
1.5.2 Stationary random process . . . . . . . . . . . . . . . . . 12
1.5.3 Gaussian random process . . . . . . . . . . . . . . . . . . 12
1.6 Vibration analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.1 Fourier transform and spectrum analysis . . . . . . . . . . 13
1.6.2 Relationship between the Fourier and Laplace transforms . 14
1.6.3 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . 14
2 Modeling of Disk Drive System and Its Vibration 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 System description . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 System modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Modeling of a VCM actuator . . . . . . . . . . . . . . . . 19
2.3.2 Modeling of friction . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Modeling of a PZT microactuator . . . . . . . . . . . . . 29
2.3.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Vibration modeling . . . . . . . . . . . . . . . . . . . . . . . . . 39
v
vi Modeling and Control of Vibration in Mechanical Systems
2.4.1 Spectrum-based vibration modeling . . . . . . . . . . . . 39
2.4.2 Adaptive modeling of disturbance . . . . . . . . . . . . . 43
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Modeling of Stewart Platform 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 System description and governing equations . . . . . . . . . . . . 53
3.3 Modeling using adaptive filtering approach . . . . . . . . . . . . . 55
3.3.1 Adaptive filtering theory . . . . . . . . . . . . . . . . . . 55
3.3.2 Modeling of a Stewart platform . . . . . . . . . . . . . . 58
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Classical Vibration Control 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Passive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Isolators . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.3 Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.4 Suspension . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.5 An application example − Disk vibration reduction via
stacked disks . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Self-adapting systems . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Active vibration control . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.1 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.2 Active systems . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.3 Control strategy . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Introduction to Optimal and Robust Control 89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 H2 and H∞ norms . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.1 H2 norm . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 H∞ norm . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 H2 optimal control . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1 Continuous-time case . . . . . . . . . . . . . . . . . . . . 92
5.3.2 Discrete-time case . . . . . . . . . . . . . . . . . . . . . 94
5.4 H∞ control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4.1 Continuous-time case . . . . . . . . . . . . . . . . . . . . 96
5.4.2 Discrete-time case . . . . . . . . . . . . . . . . . . . . . 99
5.5 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.6 Controller parametrization . . . . . . . . . . . . . . . . . . . . . 104
5.7 Performance limitation . . . . . . . . . . . . . . . . . . . . . . . 108
5.7.1 Bode integral constraint . . . . . . . . . . . . . . . . . . 108
5.7.2 Relationship between system gain and phase . . . . . . . 111
5.7.3 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Table of Contents vii
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Mixed H2/H∞ Control Design for Vibration Rejection 115
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Mixed H2/H∞ control problem . . . . . . . . . . . . . . . . . . 115
6.3 Method 1: slack variable approach . . . . . . . . . . . . . . . . . 116
6.4 Method 2: an improved slack variable approach . . . . . . . . . . 117
6.5 Application in servo loop design for hard disk drives . . . . . . . . 123
6.5.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 123
6.5.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . 128
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7 Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 133
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3 Design in continuous-time domain . . . . . . . . . . . . . . . . . 137
7.3.1 H∞ loop shaping for low-hump sensitivity functions . . . 137
7.3.2 Application examples . . . . . . . . . . . . . . . . . . . . 141
7.3.3 Implementation on a hard disk drive . . . . . . . . . . . . 148
7.4 Design in discrete-time domain . . . . . . . . . . . . . . . . . . . 152
7.4.1 Synthesis method for low-hump sensitivity function . . . . 152
7.4.2 An application example . . . . . . . . . . . . . . . . . . . 153
7.4.3 Implementation on a hard disk drive . . . . . . . . . . . . 158
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 Generalized KYP Lemma-Based Loop Shaping Control Design 161
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3 Generalized KYP lemma-based control design method . . . . . . 163
8.4 Peak filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.4.1 Conventional peak filter . . . . . . . . . . . . . . . . . . 166
8.4.2 Phase lead peak filter . . . . . . . . . . . . . . . . . . . . 168
8.4.3 Group peak filter . . . . . . . . . . . . . . . . . . . . . . 169
8.5 Application in high frequency vibration rejection . . . . . . . . . 169
8.6 Application in mid-frequency vibration rejection . . . . . . . . . . 177
8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9 Combined H2 and KYP Lemma-Based Control Design 183
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 184
9.3 Controller design for specific disturbance rejection and overall error
minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.3.1 Q parametrization to meet specific specifications . . . . . 185
9.3.2 Q parametrization to minimize H2 performance . . . . . 187
9.3.3 Design steps . . . . . . . . . . . . . . . . . . . . . . . . 188
viii Modeling and Control of Vibration in Mechanical Systems
9.4 Simulation and implementation results . . . . . . . . . . . . . . . 189
9.4.1 System models . . . . . . . . . . . . . . . . . . . . . . . 189
9.4.2 Rejection of specific disturbance and H2 performance min-
imization . . . . . . . . . . . . . . . . . . . . . . . . . . 190
9.4.3 Rejection of two disturbances with H2 performance mini-
mization . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10 Blending Control for Multi-Frequency Disturbance Rejection 197
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.2 Control blending . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.2.1 State feedback control blending . . . . . . . . . . . . . . 199
10.2.2 Output feedback control blending . . . . . . . . . . . . . 200
10.3 Control blending application in multi-frequency disturbance rejec-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 203
10.3.2 Controller design via the control blending technique . . . 205
10.4 Simulation and experimental results . . . . . . . . . . . . . . . . 207
10.4.1 Rejecting high-frequency disturbances . . . . . . . . . . . 207
10.4.2 Rejecting a combined mid and high frequency disturbance 211
10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11 H∞-Based Design for Disturbance Observer 215
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.2 Conventional disturbance observer . . . . . . . . . . . . . . . . . 216
11.3 A general form of disturbance observer . . . . . . . . . . . . . . . 217
11.4 Application results . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12 Two-Dimensional H2 Control for Error Minimization 227
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
12.2 2-D stabilization control . . . . . . . . . . . . . . . . . . . . . . . 228
12.3 2-D H2 control . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12.4 SSTW process and modeling . . . . . . . . . . . . . . . . . . . . 231
12.4.1 SSTW servo loop . . . . . . . . . . . . . . . . . . . . . . 232
12.4.2 Two-dimensional model . . . . . . . . . . . . . . . . . . 233
12.5 Feedforward compensation method . . . . . . . . . . . . . . . . . 235
12.6 2-D control formulation for SSTW . . . . . . . . . . . . . . . . . 243
12.7 2-D stabilization control for error propagation containment . . . . 244
12.7.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . 244
12.8 2-D H2 control for error minimization . . . . . . . . . . . . . . . 245
12.8.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . 245
12.8.2 Experimental results . . . . . . . . . . . . . . . . . . . . 247
12.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Table of Contents ix
13 Nonlinearity Compensation and Nonlinear Control 251
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
13.2 Nonlinearity compensation . . . . . . . . . . . . . . . . . . . . . 251
13.3 Nonlinear control . . . . . . . . . . . . . . . . . . . . . . . . . . 252
13.3.1 Design of a composite control law . . . . . . . . . . . . . 256
13.3.2 Experimental results in hard disk drives . . . . . . . . . . 257
13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
14 Quantization Effect on Vibration Rejection and Its Compensation 261
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
14.2 Description of control system with quantizer . . . . . . . . . . . . 261
14.3 Quantization effect on error rejection . . . . . . . . . . . . . . . . 266
14.3.1 Quantizer frequency response measurement . . . . . . . . 266
14.3.2 Quantization effect on error rejection . . . . . . . . . . . 266
14.4 Compensation of quantization effect on error rejection . . . . . . . 269
14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
15 Adaptive Filtering Algorithms for Active Vibration Control 275
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
15.2 Adaptive feedforward algorithm . . . . . . . . . . . . . . . . . . 275
15.3 Adaptive feedback algorithm . . . . . . . . . . . . . . . . . . . . 277
15.4 Comparison between feedforward and feedback controls . . . . . 280
15.5 Application in Stewart platform . . . . . . . . . . . . . . . . . . . 280
15.5.1 Multi-channel adaptive feedback AVC system . . . . . . . 280
15.5.2 Multi-channel adaptive feedback algorithm for hexapod plat-
form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
15.5.3 Simulation and implementation . . . . . . . . . . . . . . 284
15.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
References 293
Index 305
Preface
This book is primarily intended for researchers and engineering practitioners in sys-
tems and control, especially those engaged in the area of modeling and control of
vibrations in mechanical structures and systems. The book aims at empowering read-
ers with a clear understanding of characteristics of various vibrations, their effects on
system stability and performance, and techniques for rejecting vibrations of different
frequency ranges and their limitations. Special attention is given to recently devel-
oped vibration modeling and control techniques in high precision systems. Many
real-world examples are given to demonstrate the modeling and control techniques.
Vibration exists in a wide spectra of engineering systems such as hard disk drives,
automotives, aerospace and aeronautic systems, manufacturing systems, etc. Vibra-
tion is undesirable in most engineering applications, lowering system performance,
wasting energy and creating unwanted noise. Although the problem of vibration
control has been studied for a long time, it remains and indeed becomes more chal-
lenging in many applications such as precision engineering and hard disk drives,
where an extremely high positioning accuracy is required. Therefore, vibration con-
trol has drawn more intensive efforts from researchers and engineering practitioners
in recent years. It is our intention in this book to present to readers some of the recent
developments in this field.
The book presents the latest results in vibration modeling and advanced control
design for vibration attenuation in mechanical actuation systems to achieve high
precision positioning performance. It focuses on vibration and disturbance rejec-
tions using recently developed control techniques for high precision positioning, and
demonstration of the benefits gained from the applications of these techniques. The
theoretical developments and principles of control design are elaborated in detail so
that the reader can apply the techniques developed to obtain solutions with the help
of MATLABr. Examples are presented throughout the book so that the subject can
be better understood. A number of simulation and experimental results with compre-
hensive evaluations are provided in each chapter, except Chapters 1, 4, and 5, which
are dedicated to the review of related background knowledge.
The book summarizes a collective research effort which we have had the plea-
sure to contribute to. Many results reported in the book are due to the collaboration
with Guoxiao Guo from Western Digital Corporation, Jianliang Zhang and Jul Nee
Teoh from Data Storage Institute (DSI) of Singapore, Youyi Wang from Nanyang
Technological University (NTU), and Frank Lewis from the University of Texas at
Arlington. The research work contained in this book was mainly performed at DSI
and the School of Electrical and Electronic Engineering (EEE) of NTU, Singapore.
xi
xii Modeling and Control of Vibration in Mechanical Systems
Algorithms applied in magnetic recording systems were implemented at DSI and
those in the Stewart platform at the School of EEE, NTU. We would like to express
our sincere appreciation to DSI for its supportive environment and vibrant research
atmosphere. We are also sincerely grateful to Dr. Ong Eng Hong and the colleagues
in Mechatronics and Recording Channel Division of DSI, and EEE, NTU for their
support.
Lihua Xie
Chunling Du
MATLABr is a registered trademark of The MathWorks, Inc. For product in-
formation, please contact: The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA
01760-2098 USA, Tel: 508 647 7000, Fax: 508-647-7001, E-mail: info@mathworks.
com, Web: www.mathworks.com.
List of Tables
2.1 σ values of the modeling error (d1 − d1) for different p and Γ . . . 49
4.1 % reduction of σ values of PES, RRO and NRRO and disk vibration
amplitude with stacked disks compared with single disk . . . . . . . 81
6.1 Control performance comparison . . . . . . . . . . . . . . . . . . . 131
7.1 Control performance comparison. . . . . . . . . . . . . . . . . . . 146
9.1 Comparison of performance specifications . . . . . . . . . . . . . . 191
14.1 Quantization and friction effect . . . . . . . . . . . . . . . . . . . 272
xiii
List of Figures
1.1 The servo control loop of a hard disk drive. . . . . . . . . . . . . . 2
1.2 Hexapod from Micromega Dynamics. . . . . . . . . . . . . . . . . 3
1.3 Zoomed-in view of the hexapod. . . . . . . . . . . . . . . . . . . . 4
1.4 An example of random vibration. . . . . . . . . . . . . . . . . . . . 7
1.5 Spectrum density of broad-band vibration. . . . . . . . . . . . . . . 9
1.6 Spectrum density of narrow-band vibration. . . . . . . . . . . . . . 10
1.7 Histogram of signal x. . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 A read back signal of embedded servo. . . . . . . . . . . . . . . . . 20
2.2 Frequency responses of a second order transfer function. . . . . . . 21
2.3 Measured VCM Bode plots (straight lines: pure double integrator
k/s2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Multiplicative uncertainty of a VCM. . . . . . . . . . . . . . . . . . 24
2.5 The operator zr versus x. . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Friction f versus actuator displacement x. . . . . . . . . . . . . . . 28
2.7 A PZT actuated suspension. . . . . . . . . . . . . . . . . . . . . . 28
2.8 Equivalent spring mass system of PZT microactuator. . . . . . . . . 29
2.9 A typical frequency response of the PZT microactuator. . . . . . . . 30
2.10 An opened 1.8-inch hard disk drive. . . . . . . . . . . . . . . . . . 31
2.11 Measured and modeled frequency responses of the VCM actuation
system (LDV range 0.5 µm/V). . . . . . . . . . . . . . . . . . . . . 32
2.12 Closed control loop of a disk drive with a VCM actuator for friction
measurement via LDV. . . . . . . . . . . . . . . . . . . . . . . . . 33
2.13 Control signal u versus displacement x. . . . . . . . . . . . . . . . 34
2.14 VCM actuator modeling with friction nonlinearity model F (x). . . 34
2.15 Measured and modeled friction and error. . . . . . . . . . . . . . . 36
2.16 Actuator frequency response for sinusoidal reference with amplitude
of 1 and 3 V, respectively. . . . . . . . . . . . . . . . . . . . . . . . 37
2.17 Actuator frequency response for sinusoidal reference with amplitude
of 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.18 Preload and two-slope model for friction modeling. . . . . . . . . . 38
2.19 Plant input voltage u versus displacement x. . . . . . . . . . . . . . 38
2.20 Closed-loop control system disturbances d1, d2 and noise n. . . . . 39
2.21 Closed-loop control system with disturbance and noise models. . . . 40
2.22 Sensitivity function S(z). . . . . . . . . . . . . . . . . . . . . . . . 41
2.23 PES NRRO spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . 42
xv
xvi Modeling and Control of Vibration in Mechanical Systems
2.24 Three-layer RBF neural network. . . . . . . . . . . . . . . . . . . . 44
2.25 Original disturbance d1 and the modeled d1. . . . . . . . . . . . . . 49
2.26 Power spectrum of d1. . . . . . . . . . . . . . . . . . . . . . . . . 50
2.27 NRRO power spectrum from measurement and disturbance models,
i.e., e = −P (s) · d1 − d2 + n. . . . . . . . . . . . . . . . . . . . . 51
2.28 Modeling error (d1 − d1) for different Γ with p = 1. . . . . . . . . 52
3.1 Single-axis system using piezoelectric stiff actuator. . . . . . . . . . 54
3.2 Linear discrete time adaptive filter. . . . . . . . . . . . . . . . . . . 55
3.3 Block diagram of the LMS adaptive filter. . . . . . . . . . . . . . . 57
3.4 System identification using LMS adaptive filter. . . . . . . . . . . . 59
3.5 Frequency responses of a PZT actuator. . . . . . . . . . . . . . . . 60
3.6 Estimated and experimental frequency responses. . . . . . . . . . . 61
4.1 Disk and spindle motor assembly of the spin stand. . . . . . . . . . 67
4.2 Comparison of single-disk and dual-disk axial vibrations measured
via LDV at 7200 RPM. . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Comparison of single-disk and dual-disk RRO power spectrum at
7200 RPM (23% improvement of σ value). . . . . . . . . . . . . . 70
4.4 Comparison of single-disk and dual-disk NRRO power spectrum at
7200 RPM (18% reduction of σ value). . . . . . . . . . . . . . . . 71
4.5 Comparison of single-disk and dual-disk PES in time domain at 7200RPM (21% reduction of σ value). . . . . . . . . . . . . . . . . . . 72
4.6 Comparison of single-disk and dual-disk axial vibrations measured
via LDV at 8400 RPM. . . . . . . . . . . . . . . . . . . . . . . . . 73
4.7 Comparison of single-disk and dual-disk RRO power spectrum at
8400 RPM (41% improvement of σ value). . . . . . . . . . . . . . 74
4.8 Comparison of single-disk and dual-disk NRRO power spectrum at
8400 RPM (28% reduction of σ value). . . . . . . . . . . . . . . . 75
4.9 Comparison of single-disk and dual-disk PES in time domain at 8400RPM (38% reduction of σ value). . . . . . . . . . . . . . . . . . . 76
4.10 Comparison of single-disk and dual-disk axial vibrations measured
via LDV at 10200 RPM. . . . . . . . . . . . . . . . . . . . . . . . 77
4.11 Comparison of single-disk and dual-disk RRO power spectrum at
10200 RPM (the 3rd and 5th harmonics reduced significantly, 33%improvement of σ value). . . . . . . . . . . . . . . . . . . . . . . . 78
4.12 Comparison of single-disk and dual-disk NRRO power spectrum at
10200 RPM (27% reduction of σ value). . . . . . . . . . . . . . . . 79
4.13 Comparison of single-disk and dual-disk PES in time domain at 10200RPM (32% reduction of σ value). . . . . . . . . . . . . . . . . . . 80
4.14 Generic feedback control system. . . . . . . . . . . . . . . . . . . . 86
5.1 Configuration of standard optimal control. . . . . . . . . . . . . . . 94
5.2 A closed-loop system with uncertainty. . . . . . . . . . . . . . . . . 102
List of Figures xvii
5.3 A closed-loop system with additive uncertainty for robust stability
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 A closed-loop system with multiplicative uncertainty for robust sta-
bility analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Control system structure for Youla parametrization. . . . . . . . . . 107
5.6 Sensitivity function for continuous-time system. . . . . . . . . . . . 110
5.7 Sensitivity function for discrete-time system. . . . . . . . . . . . . 110
5.8 Sensitivity function in discrete-time domain. . . . . . . . . . . . . . 113
6.1 Mixed H2/H∞ control scheme for HDD servo loop with distur-
bance models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Frequency responses of the VCM actuator. . . . . . . . . . . . . . . 125
6.3 Multiplicative uncertainty of the VCM actuator. . . . . . . . . . . . 126
6.4 Frequency response of sensitivity functions. . . . . . . . . . . . . . 129
6.5 Frequency response of sensitivity functions. . . . . . . . . . . . . . 130
7.1 Parallel structure of a dual-stage actuation system with disturbances
and noise injected. . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2 Power spectrum of PES nonrepeatable runout in open loop. . . . . . 135
7.3 Decoupled structure of dual-stage actuation systems. . . . . . . . . 136
7.4 Structure of H∞ loop shaping. . . . . . . . . . . . . . . . . . . . . 136
7.5 Frequency responses of Sv(s) and Sm(s). . . . . . . . . . . . . . . 139
7.6 Frequency responses of Pv(s)Cv(s) (solid line) and Pm(s)Cm(s)(dotted
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.7 Frequency response of VCM actuator P (s). . . . . . . . . . . . . . 141
7.8 Frequency response of VCM controller Cv(s). . . . . . . . . . . . . 142
7.9 Frequency response of microactuator controller Cm(s). . . . . . . . 143
7.10 Sensitivity function Sm(s) (solid) and its weighting function inverse
(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.11 Open loop frequency response of the dual-stage system. . . . . . . . 144
7.12 Sensitivity and complementary sensitivity functions. . . . . . . . . 145
7.13 Microactuator frequency response. . . . . . . . . . . . . . . . . . . 146
7.14 Frequency response of microactuator controller Cm(s). . . . . . . . 147
7.15 Open loop frequency response of the dual-stage system. . . . . . . . 147
7.16 Sensitivity and complementary sensitivity functions. . . . . . . . . 148
7.17 Experimental structure. . . . . . . . . . . . . . . . . . . . . . . . . 149
7.18 Sensitivity function of the dual-stage system (smooth line: simula-
tion result; rough line: testing result; dotted line: PID design). . . . 150
7.19 Open loop frequency response of the dual-stage system (smooth line:
simulation result; rough line: testing result.) . . . . . . . . . . . . . 150
7.20 Step response of the dual-stage system. . . . . . . . . . . . . . . . 151
7.21 3σ of PES NRRO versus frequencies. . . . . . . . . . . . . . . . . 151
7.22 VCM controller Cv(z). . . . . . . . . . . . . . . . . . . . . . . . . 155
7.23 Microactuator controller Cm(z). . . . . . . . . . . . . . . . . . . . 155
7.24 Sensitivity function of the dual-stage system. . . . . . . . . . . . . 156
xviii Modeling and Control of Vibration in Mechanical Systems
7.25 Sensitivity function. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.26 Sensitivity function of the dual-stage system. . . . . . . . . . . . . 157
7.27 Frequency responses of Pv(z)Cv(z) (solid curve) and Pm(z)Cm(z)(dashed curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.28 Sensitivity function of the dual-stage system. . . . . . . . . . . . . 159
7.29 Open loop frequency responses of the dual-stage system. . . . . . . 159
7.30 Step response of the dual-stage system. . . . . . . . . . . . . . . . 160
7.31 3σ value of PES NRRO versus frequency. . . . . . . . . . . . . . . 160
8.1 Q parameterization for control design. . . . . . . . . . . . . . . . . 163
8.2 Peak filter F in the nominal feedback loop. . . . . . . . . . . . . . 167
8.3 Peak filter in the frequency domain. . . . . . . . . . . . . . . . . . 167
8.4 Sensitivity functions before and after group peak filtering activated. 170
8.5 PZT microactuator attached to VCM actuator arm. . . . . . . . . . 170
8.6 Power spectrum of the position error before servo control. . . . . . 171
8.7 PZT micro actuator frequency response. . . . . . . . . . . . . . . . 172
8.8 Sensitivity functions before and after the KYP lemma-based design:
simulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.9 Open-loop Bode plot before and after the KYP lemma-based design. 175
8.10 Structure of experimental setup. . . . . . . . . . . . . . . . . . . . 175
8.11 Sensitivity functions before and after the KYP lemma-based design:
experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.12 σ value of PES versus frequency. . . . . . . . . . . . . . . . . . . . 176
8.13 Frequency response of the PZT microactuator. . . . . . . . . . . . . 179
8.14 PES NRRO power spectrum calculated from measured PES signal
without servo control, reflecting the vibration distribution of the sys-
tem (3σ = 21 nm including the noise 3σ = 15.2 nm). . . . . . . . . 180
8.15 Comparison of sensitivity functions. . . . . . . . . . . . . . . . . . 181
8.16 Open loop frequency responses (PLPF (GM: 6 dB, PM: 50 deg.,
Bandwidth 1.4kHz)); KYP(GM: 6 dB, PM: 34 deg., Bandwidth: 1.7kHz))). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.17 NRRO power spectrum with PLPF and KYP (50% reduction before
1 kHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.1 H2 control scheme with Q parametrization for controller design. . . 184
9.2 Frequency response of a PZT microactuator. . . . . . . . . . . . . . 190
9.3 Open-loop frequency responses. . . . . . . . . . . . . . . . . . . . 191
9.4 Designed sensitivity functions. . . . . . . . . . . . . . . . . . . . . 192
9.5 Comparison of sensitivity functions obtained from experiment. . . . 192
9.6 NRRO power spectrum with KYP Lemma-based controller with and
without H2 minimization. . . . . . . . . . . . . . . . . . . . . . . 193
9.7 PES NRRO spectrum without servo control. . . . . . . . . . . . . . 194
9.8 Open-loop frequency response. . . . . . . . . . . . . . . . . . . . . 195
9.9 Resultant sensitivity function (Solid line: with Spec. (i), (ii) and (iii);
Dashed line: with Spec. (i) and (ii)). . . . . . . . . . . . . . . . . . 195
List of Figures xix
9.10 Resultant sensitivity function with all the three requirements ful-
filled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.11 NRRO power spectrum with rejection of two specific disturbances at
0.65 and 2 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.1 Blending control scheme. . . . . . . . . . . . . . . . . . . . . . . . 198
10.2 Control loop with injected disturbances at different frequencies. . . 204
10.3 Control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.4 Frequency response of the VCM actuator. . . . . . . . . . . . . . . 207
10.5 Open-loop frequency response with disturbance rejection at 4 and 8kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.6 Simulated sensitivity function with disturbance rejection at 4 and 8kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.7 Measured (solid curve) sensitivity function with disturbance rejec-
tion at 4 and 8 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.8 Open-loop with disturbance rejection at 3, 6.5, and 10 kHz. . . . . . 210
10.9 Sensitivity function with disturbance rejection at 3, 6.5, and 10 kHz. 211
10.10 Open-loop with disturbance rejections at 0.65 and 2 kHz. . . . . . . 212
10.11 Sensitivity function with disturbance rejections at 0.65 and 2 kHz. . 213
11.1 Block diagram of the control loop with a conventional disturbance
observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.2 Block diagram of the control loop with a general disturbance ob-
server. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.3 Frequency response of the designed Q(z). . . . . . . . . . . . . . . 222
11.4 The sensitivity functions without and with the general disturbance
observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
11.5 The sensitivity function comparison with the general and the con-
ventional disturbance observers. . . . . . . . . . . . . . . . . . . . 223
11.6 Disturbance d1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.7 Error signal e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.8 Measured sensitivity functions without and with the general distur-
bance observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
11.9 Comparison of TEQ−OL about the general disturbance observer. . . 225
11.10 Comparison of TEQ−OL about conventional disturbance observer. . 226
12.1 SSTW process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.2 SSTW servo loop with disturbances and noise models. . . . . . . . 232
12.3 PES NRRO and its σ values versus track number during propagation.
(The time sequence and the σ value increase with the track number.) 233
12.4 SSTW servo loop modeling in two dimensions. . . . . . . . . . . . 234
12.5 SSTW servo loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
12.6 Frequency response of a VCM actuator. . . . . . . . . . . . . . . . 237
12.7 Frequency response of the closed-loop transfer function with the PD
controller, PID controller, and H2 controller. . . . . . . . . . . . . 238
xx Modeling and Control of Vibration in Mechanical Systems
12.8 |Φ| versus frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.9 3σ of PES NRRO. . . . . . . . . . . . . . . . . . . . . . . . . . . 240
12.10 Frequency response of the H2 controller. . . . . . . . . . . . . . . 241
12.11 Frequency response of the open-loop system with the H2 controller. 241
12.12 Comparison of sensitivity functions. . . . . . . . . . . . . . . . . . 242
12.13 σ value of PES NRRO versus track number. . . . . . . . . . . . . . 243
12.14 2-D controller for SSTW servo loop. . . . . . . . . . . . . . . . . 245
12.15 σ of PES NRRO versus track number. . . . . . . . . . . . . . . . . 246
12.16 Open-loop frequency response with stabilization controller. . . . . 246
12.17 Sensitivity function with stabilization controller (Ac2, Bc2, Cc, Dc). 247
12.18 Frequency response of controller (Ac2, Bc2, Cc, Dc). . . . . . . . . 248
12.19 Open-loop frequency response with controller (Ac2, Bc2, Cc, Dc). . 249
12.20 Sensitivity function with controller (Ac2, Bc2, Cc, Dc). . . . . . . 249
12.21 Step response (Channel 1/2/3: Reference/Output/Control signal). . 250
13.1 Friction compensation for the actuation system. . . . . . . . . . . . 252
13.2 Input u versus displacement x with and without compensation. . . . 253
13.3 Actuator frequency responses with and without friction compensa-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
13.4 Actuator frequency responses with friction compensation for differ-
ent displacements in voltage with 0.5µm/V (Straight smooth lines:
the pure double integrator). . . . . . . . . . . . . . . . . . . . . . . 254
13.5 Actuator frequency responses with friction compensation for differ-
ent displacements in voltage with 0.5 µm/V (Straight smooth lines:
the pure double integrator). . . . . . . . . . . . . . . . . . . . . . . 254
13.6 Sensitivity functions with and without friction compensation. . . . . 255
13.7 Control structure of a plant P (s) with Youla parametrization ap-
proach and adaptive nonlinear compensation. . . . . . . . . . . . . 256
13.8 Comparison of error rejection frequency response without and with
uN of different p and Γ. . . . . . . . . . . . . . . . . . . . . . . . . 258
13.9 NRRO power spectrum with KYP lemma-based linear control and
nonlinear compensation (80% reduction before 400 Hz). . . . . . . 259
14.1 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . 263
14.2 Frequency response of the VCM actuator measured by injecting a
swept sine wave of 5mV amplitude. . . . . . . . . . . . . . . . . . 263
14.3 The servo loop in experiment. . . . . . . . . . . . . . . . . . . . . 264
14.4 Frequency response of the controller C(z). . . . . . . . . . . . . . 264
14.5 Frequency response of the sensitivity function S(z) with different
reference levels (i.e., actuator moving ranges are different). . . . . . 265
14.6 Frequency response of the quantizer before and after compensation
(bit number n = 6). . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14.7 Frequency response of the quantizer with compensation (bit number
n = 8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
List of Figures xxi
14.8 Frequency response of the quantizer with compensation (bit number
n = 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
14.9 Measured sensitivity function SQ(z) with different bits n. . . . . . . 268
14.10 Sensitivity function |SQ(f)| with f = 10 Hz versus bit n. . . . . . 269
14.11 Compensation scheme of quantization effect. . . . . . . . . . . . . 270
14.12 Choosing the threshold δ and the scaling factor a. . . . . . . . . . . 270
14.13 Sensitivity function with quantization compensation. . . . . . . . . 271
15.1 Block diagram of FXLMS algorithm. . . . . . . . . . . . . . . . . 276
15.2 Filtered-X LMS adaptive feedback algorithm. . . . . . . . . . . . . 279
15.3 Adaptive inverse control scheme. . . . . . . . . . . . . . . . . . . 279
15.4 Block diagram of 2 × 2 adaptive feedback algorithm. . . . . . . . . 282
15.5 Block diagram of 6 × 1 FXLMS adaptive feedback control system. 283
15.6 General layout of the experimental setup. . . . . . . . . . . . . . . 286
15.7 60 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 287
15.8 210 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 287
15.9 240 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 288
15.10 270 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 288
15.11 Simulink diagram of automatic gain control. . . . . . . . . . . . . 291
15.12 180 Hz error signal without automatic gain control. . . . . . . . . . 292
15.13 180 Hz error signal with automatic gain control. . . . . . . . . . . 292
Symbols and Acronyms
Rn: n-dimensional real Euclidean space
Rn×m: set of n × m real matrices
In: n × n identity matrix
(A, B, C, D): state-space representation of a system
A B1 B2
C1 D11 D12
C2 D21 D22
: compact representation of system:
x(k + 1) = Ax(k) + B1w(k) + B2u(k)z(k) = C1x(k) + D11w(k) + D12u(k)y(k) = C2x(k) + D21w(k) + D22u(k)
diagA1, A2, · · · , An: block diagonal matrix with Aj ( not necessarilysquare), j = 1, 2, · · · , n, on the diagonal
XT : transpose of matrix X
X∗: complex conjugate transpose of matrix X
P ≥ 0: symmetric positive semidefinite matrix P ∈ Rn×n
P > 0: symmetric positive definite matrix P ∈ Rn×n
P ≥ Q: P − Q ≥ 0 for symmetric P, Q ∈ Rn×n
P > Q: P − Q > 0 for symmetric P, Q ∈ Rn×n
σ(X): largest singular value of X
Trace(X): trace of X
‖ · ‖: Euclidean vector norm
‖w‖2 : ℓ2-norm of a signal w(k), i.e.,
√
∞∑
k=0
‖w(k))‖2.
xxiii
xxiv Modeling and Control of Vibration in Mechanical Systems
ℓ2[0,∞): space of square summable sequences on [0,∞).The signal w(k) is said to be from ℓ2[0,∞) or simply ℓ2if ‖w‖2 < ∞.
‖G‖2: H2 norm of transfer function G
‖G‖∞: H∞ norm of transfer function G
Re( ): the real part of a complex number
Im( ): the imaginary part of a complex number
ρ( ): spectral radius
AGC : automatic gain control
AV C : active vibration control
deg: degree
det: determinant
DSA: Dynamic Signal Analyzer
FFT : fast Fourier transform
FXLMS: filtered-X LMS
HDD: hard disk drive
LDV : Laser Doppler Vibrometer
LFT : Linear fractional transformation
LMI: linear matrix inequality
LMS: least mean square
LQG: linear quadratic Gaussian
LTR: loop transfer recovery
MEMS: micro electro-mechanical system
MSE: mean square error
NRRO: nonrepeatable runout
PES: position error signal
Symbols and Acronyms xxv
PID : proportional-integral-derivative
PLPF : phase lead peak filter
PZT : lead zirconate titanate/piezoelectric
RBF : radial basis function
RMS: root mean square
RPM : rotations per minute
RRO: repeatable runout
SSTW : self-servo track writing
STW : servo track writing
TMR: track misregistration
V CM : voice coil motor
1
Mechanical Systems and Vibration
When studying mechanical systems, we have to include the subject of dynamics and
vibration. Dynamics is a branch of mechanics that deals with motion and its effect on
a body. Unlike statics, which deals with bodies at rest, dynamics takes into account
the effect of velocities and accelerations on the forces acting on bodies. Vibration is
regarded as a branch of dynamics, since forces and masses are taken into account in
vibration analysis. Thus it is natural that both dynamics and vibrations of mechanical
systems are studied in this book.
A vibration may be a signal, force, or temperature variation that affects the re-
sponse of a system in an unacceptable manner. If our analysis of system response to
a vibration shows that it regularly affects the system performance in an unacceptable
manner, we need to alter or control the vibration response and bring the response
within acceptable levels by adding appropriate forces, called control forces which
are functions of system response such as displacement. This may require a design
by using the value of the system response to generate additional forces according to
certain rules or laws such that the modified response behaves according to desired
performance and within certain bounds. This results in a closed-loop system that
incorporates feedback controls. The purpose of designing a system with feedback
force is to minimize unwanted behavior. Examples include magnetic recording sys-
tems, Stewart platforms, positioning stages [27], the atomic force microscope (AFM)
[28], industry robots [29], as well as some automotive systems [34]. The magnetic
recording system and Stewart platform are examples to be presented next to help
in fixing these ideas more firmly. With such a grounding, more advanced problems
become accessible.
1.1 Magnetic recording system
Figure 1.1 shows a servo control loop of a hard disk drive (HDD) with a voice coil
motor (VCM) and a piezoelectric (PZT) actuated servo system. It consists of a stack
of flat rotating disks with positioning information or servo information embedded in
their surfaces. The servo information is used to position the magnetic heads on the
disk surfaces. Position measurement of the magnetic heads is achieved by means of
analyzing the position error signal (PES) calculated from the read back signal. To
1
2 Modeling and Control of Vibration in Mechanical Systems
have a disk drive with a high storage capacity, the head positioning error with re-
spect to the target track center needs to be as small as possible. The error is mainly
due to, (1) torque disturbances from spindle motor, (2) actuator pivot friction, (3)
airflow-induced non-repeatable disk, suspension and slider vibrations, (4) mechani-
cal resonance vibration, and (5) head sensing and electronic noises, media noise, and
quantization noises. Hence how to deal with the variety of the disturbances is crit-
ical to the head positioning accuracy, and subsequently the track density for a high
capacity disk drive.
FIGURE 1.1
The servo control loop of a hard disk drive.
1.2 Stewart platform
The six-leg parallel linkage mechanism known as the “Stewart platform” was dis-
covered as early as in 1965 [30]. It is designed according to a cubic configuration,
consisting of two triangular parallel plates connected to each other by six active legs
orthogonal to each other. Each leg is equipped with a voice coil actuator, a force
sensor and two flexible joints. The closed kinematical linkage structure of a Stewart
platform has major advantages over any serial link robots: great rigidity, high force
to weight ratio, six degrees of freedom (DOF), etc. [31] The Stewart platform is
Mechanical Systems and Vibration 3
widely used as space born structures, as well as a high precision pointing device and
vibration isolator.
Stewart platforms can be divided into two main classes according to the stiffness
of the legs: stiff type and soft type. For the soft design, each leg essentially acts as
an axial springing parallel with a voice coil actuator, while the stiff design involves
piezoelectric or magneto restrictive legs whose extensions can be controlled [32].
The Stewart platform has been widely used in active vibration control. It has an
important property for vibration control application: forces transmitted between the
mobile plate and the base plate are totally axial forces of actuators. This implies that
if the axial forces can be measured and eliminated, the vibration created by these
forces can thus be eliminated. Thus the Stewart platform has become one of the
most popular approaches for 6-DOF active vibration control in precision systems
due to its attractive properties.
FIGURE 1.2
Hexapod from Micromega Dynamics.
The Stewart platform (Hexapod) from Micromega-Dynamics used as a vibration
isolation device is shown in Figure 1.2. The hexapod has a cubic architecture and
consists of two parallel plates connected to each other by six active legs. The plates
are made of aluminum with a thickness of 20 mm and diameter of 250 mm, with
the weight of the mobile plate at 1 kg. Each leg of the active interface consists of
a linear piezoelectric actuator, a collocated force sensor, and flexible tips to connect
the two end plates. Flexible tips are used in order to avoid the problem of friction and
backlash, which comes with the use of spherical joints. The hexapod can be used to
4 Modeling and Control of Vibration in Mechanical Systems
FIGURE 1.3
Zoomed-in view of the hexapod.
actively increase the structural damping of flexible systems attached to it.
Figure 1.3 shows the zoomed-in view of the inside of the hexapod, revealing the
arrangement of the six collocated sensor-actuator legs. The wires shown in Figure
1.2 are the outputs of the force sensors (each for one sensor) and inputs to the piezo-
electric actuators (each for one actuator), respectively.
Each of the legs in the Stewart platform consists of a PZT force sensor and an
amplified PZT actuator. They form a collocated sensor-actuator pair configuration.
If an actuator and a sensor are collocated, the associated signals for the actuator
and sensor are power conjugated, i.e., the product of the actuated velocity and the
measured force represents the power that is extracted from the mechanical structure.
A collocated actuator-sensor pair thus enables the control of power that is supplied
to the mechanical structure. Collocated actuator-sensor pairs are suitable in active
vibration control applications in the sense that they guarantee damping and stability
robustness if designed properly [33].
In this book, the modeling and vibration controls for the magnetic recording sys-
tem and the Stewart platform will be detailed.
1.3 Vibration sources and descriptions
Any oscillatory motion of bodies that repeatedly appears is called vibration or os-
cillation. There are usually forces associated with vibrations. They can be induced
by various types of excitation. Some of them are: fluid flow; rotating unbalanced
machinery; structure flexible modes; electrical torque; reciprocating machinery; mo-
tion induced in vehicles traveling over uneven surfaces; and ground motion caused
by earthquakes.
Flow induced vibration is generated by the forces exerted on an object by fluid
motion. Such situations can be complicated by the fact that the motion of the vibrat-
Mechanical Systems and Vibration 5
ing object can alter the fluid flow conditions, thus changing the fluid forces. Another
complicating factor is the mass of the fluid, which increases the effective mass of the
system. Examples of vibration caused by fluid motion include: wave action on struc-
ture; vortex-induced vibration such as vibration of transmission cables, underwater
cables used for towing and structural support, and cooling towers and chimneys; vi-
bration caused by internal flows, such as air flow in hard disk drives, flow through
pipes and hoses having bends; structural vibration caused by fluctuating aerodynamic
forces such as turbulence [1][56]. In certain situations, the steady-state excitation
force due to fluid motion is sinusoidal with an amplitude proportional to the square
of the forcing frequency. A model of the system undergoing such excitation is
mx + cx + kx = F0ω2sinωt, (1.1)
where m is the system mass, x is the system response, ω is the forcing frequency, kand c are respectively stiffness and damping, and F0 is the force coefficient. Further
analysis of system response to flow fluid motion is quite complicated and requires
detailed consideration of fluid mechanics of the system.
Some oscillatory systems have simple harmonic motion of the form
y(t) = Bsin(ωt) + Ccos(ωt), (1.2)
where y(t) is the displacement of a mass, and is equivalent to
y(t) = Asin(ωt + φ), (1.3)
A =√
B2 + C2, (1.4)
cos(φ) =B
A, sin(φ) =
C
A. (1.5)
There occur some oscillations having exponential amplitude as follows:
y(t) = Aertsin(ωt + φ), (1.6)
where oscillation amplitude decays exponentially when r < 0, and grows indefi-
nitely when r > 0.
Many types of motion cannot be easily represented by simple functions because
they are essentially random. Examples include air turbulence to arm and suspension
in hard disk drives, ground motion due to an earthquake, and base motion that occurs
when a vehicle travels over an uneven surface. It is possible, however, to characterize
them by means of statistical averages and spectrum plots, in which Fourier analysis
is used to identify the major frequency components in the vibration and they will be
described in more detail in the later part of the chapter.
1.4 Types of vibration
There are several ways to categorize vibrations. Basically, they can be calssified as
follows.
6 Modeling and Control of Vibration in Mechanical Systems
1.4.1 Free and forced vibration
If a system vibrates on its own after an initial disturbance and no external force acts
on it, the ensuing vibration is known as free vibration. A direct example of free
vibration is the oscillation of a simple pendulum.
If a system vibration is due to an external force, the arising vibration is known as
forced vibration. The oscillation in machines such as diesel engines that results from
an external force is an example of forced vibration. If the frequency of the external
force coincides with one of the natural frequencies of the system, the phenomenon
known as resonance occurs, and the system undergoes oscillation. The occurrence of
that resonance causing large oscillation may lead to failures of some structures such
as buildings, bridges, turbines, and airplane wings.
1.4.2 Damped and undamped vibration
If during oscillation there is no energy lost or dissipated in friction or other resistance,
the vibration is known as undamped vibration. On the other hand, if there is energy
lost during oscillation, it is called damped vibration. When analyzing vibration near
resonance in physical systems, consideration of damping becomes extremely impor-
tant.
1.4.3 Linear and nonlinear vibration
If all the basic components in a vibratory system such as spring, mass and damper
behave linearly, the resulting vibration is classified as linear vibration. On the other
hand, if any of the basic components behaves nonlinearly, the vibration is categorized
as nonlinear vibration. Linear and nonlinear differential equations are used to govern
the behaviors of linear and nonlinear vibratory systems, respectively. If a vibration is
linear, the principle of linear systems such as superposition holds, and there are well
developed mathematical tools for analysis. As for nonlinear vibration, the superposi-
tion principle is not valid, and techniques of analysis are more complicated and less
well known. Since all vibratory systems tend to behave nonlinearly with respect to
amplitude level of oscillation, some knowledge of nonlinear vibration is desirable in
dealing with practical vibratory systems. As is known, a describing function is one
approximation method used to analyze nonlinear vibratory systems.
1.4.4 Deterministic and random vibration
A vibration is known as deterministic vibration if it results from an excitation with
value or amplitude known at any given time. In some cases, the excitation acting on
a vibratory system is nondeterministic or random, and the value of the excitation at
a given time cannot be predicted. In these cases, a large amount of excitation data
collected may exhibit some statistical regularity. Statistical methods can be used for
analysis, as it is possible to estimate averages such as the mean and variance values
of the random excitation. Examples of random excitations are air flow inside hard
Mechanical Systems and Vibration 7
disk drives, road roughness and ground motion during earthquakes. If the excitation
is random, the induced vibration is called random vibration, such as that shown in
Figure 1.4. It can be described in terms of statistical quantities.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time(sec)
Vib
ratio
n s
ign
al
FIGURE 1.4
An example of random vibration.
1.4.5 Periodic and nonperiodic vibration
Periodic vibration can be represented by Fourier series as superposition of harmonic
components of various frequencies. That is, if x(t) is a periodic function with period
τ , its Fourier series representation is given by
x(t) =a0
2+ a1cos(ωt) + a2cos(2ωt) + · · ·+ b1sin(ωt) + b2sin(2ωt) + ...
=a0
2+
∞∑
n=1
(ancos(nωt) + bnsin(nωt)), (1.7)
where ω = 2π/τ is the fundamental frequency, and a0, an, bn are constant coeffi-
cients given by
a0 =ω
π
∫ 2π/ω
0
x(t)dt =2
τ
∫ τ
0
x(t)dt, (1.8)
8 Modeling and Control of Vibration in Mechanical Systems
an =ω
π
∫ 2π/ω
0
x(t)cos(nωt)dt =2
τ
∫ τ
0
x(t)cos(nωt)dt, (1.9)
bn =ω
π
∫ 2π/ω
0
x(t)sin(nωt)dt =2
τ
∫ τ
0
x(t)sin(nωt)dt. (1.10)
Let
c0 =a0
2, (1.11)
cn = (a2n + b2
n)1/2. (1.12)
The root mean square (RMS) value of the periodic function x(t) can be determined
as
RMS(x(t)) =
√
√
√
√
∞∑
n=0
|cn|2. (1.13)
Thus the mean square value of x(t) is given by the sum of the squares of the absolute
values of the Fourier coefficients. Equation (1.13) is known as Parseval’s formula for
periodic functions.
Although the series in (1.7) is an infinite sum, most periodic functions can be
approximated by only a few harmonic functions.
A nonperiodic vibration x(t) can be represented by the integral Fourier transform
pair:
x(t) =1
2π
∫ ∞
−∞X(ω)eiωtdω (1.14)
and
X(ω) =
∫ ∞
−∞x(t)e−iωtdt. (1.15)
The RMS value of the nonperiodic function x(t) can be determined as
RMS(x(t)) =
√
∫ ∞
−∞
|X(ω)|22πτ
dω. (1.16)
Equation (1.16) is known as Parseval’s formula for nonperiodic functions.
1.4.6 Broad-band and narrow-band vibration
A broad-band vibration is a stationary random process whose spectral density func-
tion has significant values over a range or band of frequencies which is approximately
of the same order of magnitude as the center frequency of the band. The density in
Mechanical Systems and Vibration 9
Figure 1.5 describes a broad-band vibration, which is composed of components con-
taining frequencies over a wide or broad frequency range. A narrow-band random
vibration is a stationary vibration whose spectral density function has significant val-
ues only in a range of frequency whose width is small compared to the magnitude
of the center frequency. Figure 1.6 shows a vibration containing frequencies over a
narrow band.
101
102
103
104
0
2
4
6
8
10
12
14
16
Frequency(Hz)
Ma
gn
itu
de
(dB
)
FIGURE 1.5
Spectrum density of broad-band vibration.
A random process whose power spectral density is constant over a frequency range
is called white noise. It is called ideal white noise if the band of frequencies is
infinitely wide. An ideal white noise is physically unrealizable, since the variance of
such a random process would be infinite because the area under the spectrum would
be infinite. It is called band-limited white noise if the band of frequencies has finite
cut-off frequencies ω1 and ω2. The variance of a band-limited white noise is given
by the total area under the spectrum, namely, 2S0(ω2 − ω1), where S0 denotes the
constant value of the spectral density.
10 Modeling and Control of Vibration in Mechanical Systems
102
103
0
1
2
3
4
5
6
7
Frequency(Hz)
Ma
gn
itu
de
(dB
)
FIGURE 1.6
Spectrum density of narrow-band vibration.
Mechanical Systems and Vibration 11
1.5 Random vibration
1.5.1 Random process
In contrast to deterministic excitations, in many applications the inputs are not well
known and can be described only in terms of statistical measures such as mean value,
variance and standard deviation.
Mean value
The average is called the mean or expected value, often represented by µ. For dis-
crete values, such as those produced by digital data acquisition, the mean is defined
as
E(x) =1
n
n∑
j=1
xj. (1.17)
For a continuous function x(t), the mean is
E[x(Td)] =1
Td
∫ Td
0
x(t)dt (1.18)
where the time duration of the data sample is Td.
Variance
Two signals may have the same mean value but one may fluctuate with greater
amplitude about the mean. So we also need a measure of the range of fluctuation.
Simply specifying the minimum and maximum values is insufficient because a single
large fluctuation (above or below the mean) can be misleading. A measure that
indicates the spread about the mean is the variance, which is defined as the average
value of the square of the difference between the signal and its mean. The variance
is calculated for a discrete signal as follows.
var(x) = σ2 =1
n
n∑
j=1
(xj − µ)2, (1.19)
where σ2 is the variance and σ is called the standard deviation.
For a signal continuous in time, the variance is calculated from
var(x, Td) =1
Td
∫ Td
0
[x(t) − µ]2dt. (1.20)
The mean-square value of x is the expected value of x2 and is denoted E(x2). The
RMS value of x is√
E(x2). The relation between the variance, the mean square,
and the mean is as follows.
σ2 = E(x2) − [E(x)]2. (1.21)
12 Modeling and Control of Vibration in Mechanical Systems
If the mean of x is zero, the standard deviation σx of x is calculated from
σx =√
E(x2), (1.22)
and is thus the same as the RMS value.
A random input will generate a random response, and the output signal given by
its sensor will appear to be random. Random signals, like the example shown in
Figure 1.4, have no apparent pattern and never repeat. Another characteristic of a
random signal is that it is impossible to predict what the signal will be in the future,
even if we have the past values of the signal.
1.5.2 Stationary random process
A special case of a random process is one that is stationary, which means that its
statistical properties (such as its mean and variance) are time-independent.
If x(t) is stationary, its mean and covariance will be independent of t:
E[x(t)] = E[x(t + τ )] = µ, (1.23)
and
E[x(t)x(t + τ )] = σxx(τ ). (1.24)
1.5.3 Gaussian random process
A Gaussian or normal random process has a number of remarkable properties that
permit the computation of random vibration characteristics in a simple manner. The
probability density function of a Gaussian process x(t) is given by
p(x) =1√
2πσx
e−12 ( x−x
σx)2 , (1.25)
where x and σx denote the mean value and standard deviation of x. The mean x and
standard deviation σx of x(t) vary with t for a nonstationary process but are constants
for a stationary process. A very important property of a Gaussian process is that the
form of its probability distribution is invariant with respect to linear operations. This
means that if the excitation of a linear system is a Gaussian process, the steady-state
response is generally a different random process, but still a normal one. The only
changes are that the magnitude of the mean and standard deviation of the response
are different from those of the excitation.
The graph of a Gaussian probability density function has a bell-shaped envelope
as seen in Figure 1.7, and is symmetric about the mean value; its spread is governed
by the value of the standard deviation.
Mechanical Systems and Vibration 13
−4 −3 −2 −1 0 1 2 3 40
20
40
60
80
100
120
140
160
180
Signal x
Pro
ba
bili
ty d
en
sity
FIGURE 1.7
Histogram of signal x.
1.6 Vibration analysis
1.6.1 Fourier transform and spectrum analysis
We know that a periodic signal can be expressed as a Fourier series of harmonic
functions. The Fourier series coefficients, when plotted versus frequency, gives a
plot of the spectrum of the signal. The spectrum graphically displays the frequency
content of the signal.
A nonperiodic function is expressed with the following Fourier transform pair:
x(t) =
∫ ∞
−∞a(ω)cos(ωt)dω +
∫ ∞
−∞b(ω)sin(ωt)dω, (1.26)
where
a(ω) =1
2π
∫ ∞
−∞x(t)cosωtdt, (1.27)
b(ω) =1
2π
∫ ∞
−∞x(t)sinωtdt. (1.28)
14 Modeling and Control of Vibration in Mechanical Systems
The Fourier transform of x(t) is
X(ω) = a(ω) − ib(ω). (1.29)
An equivalent form of the Fourier transform is given by
x(t) =
∫ ∞
−∞X(ω)eiωtdω, (1.30)
and
X(ω) =1
2π
∫ ∞
−∞x(t)e−iωtdt. (1.31)
The spectrum of a nonperiodic signal is the magnitude of its Fourier transform,
that is, X(ω). Equation (1.31) implies that the transform is symmetric about ω = 0,
that is
X(−ω) = X(ω). (1.32)
The transform of a time-shifted signal x(t − d) is X(ω)e−iωd and its spectrum is
the same as the spectrum of x(t), while the time shift d affects only the phase angle
of the transform.
1.6.2 Relationship between the Fourier and Laplace transforms
The Fourier transform X(ω) of x(t) is related to the Laplace transform X(s) as
follows.
X(ω) =1
2πX(s)|s=iω. (1.33)
1.6.3 Spectral analysis
The spectral density, or power spectral density, Sxx(ω), is one of the most useful
functions in vibration testing. It is defined as
Sxx(ω) =1
2π
∫ ∞
−∞Rxx(τ )e−iωτdτ, (1.34)
where
Rxx(τ ) = limT→∞
1
T
∫ T/2
−T/2
x(t)x(t + τ )dt (1.35)
is the autocorrelation function and T is the time duration.
Mechanical Systems and Vibration 15
Rxx(τ ) can be written as the inverse Fourier transform of Sxx(ω):
Rxx(τ ) =
∫ ∞
−∞Sxx(ω)eiωτ dω. (1.36)
The cross-spectral density is written as
Sxy(ω) =1
2π
∫ ∞
−∞Rxy(τ )e−iωτdτ, (1.37)
where
Rxy(τ ) = limT→∞
1
T
∫ T/2
−T/2
x(t)y(t + τ )dt (1.38)
is the cross-correlation function.
A very useful relation is that
Rxx(0) =
∫ ∞
−∞Sxx(ω)dω = E(x2), (1.39)
which means that the mean-square value can be computed from the spectral density.
2
Modeling of Disk Drive System and ItsVibration
2.1 Introduction
Modeling plays an important role in any control system analysis and design. A
physical system has to be modeled in order to design a control system. Physical
systems are more or less nonlinear and may vary with time. Researchers have made
many attempts to deal with a physical system: approximating it with a linear model
at an operating point, finding a control strategy that is robust and adaptable to the
changes in the physical system.
The system modeling in this chapter is discussed from the first principle of actu-
ators. System dynamics measurement in the frequency domain is used to determine
the parameters of a model in the form of transfer function. To have a more realistic
model, uncertainties, especially due to high frequency unmodeled dynamics, have to
be involved inherently. Moreover, nonlinearity induced by friction is measured under
the condition that the actuator is controlled to avoid unsteady signal measurement.
The hysteresis of friction versus actuator position is then obtained from the measure-
ment in closed-loop. An operator based modeling approach is adopted to model the
hysteresis, and an optimal model is obtained by minimizing the energy gain between
the position and the modeling error.
In this chapter, vibration modeling is based primarily on the spectral decomposi-
tion of the error signal measured in a closed-loop system. A decoupling procedure
is proposed which leads to approximate disturbance and noise models. Particularly,
low-frequency disturbances are modeled as the output of an adaptive nonlinear mech-
anism with the error signal as the input.
2.2 System description
A hard disk drive as shown in Figure 1.1 includes five major parts: baseplate and
cover, spindle and motor assembly, actuator assembly, disk, head/suspension assem-
bly, and electronics card.
17
18 Modeling and Control of Vibration in Mechanical Systems
The spindle and motor assembly includes disk clamps to clamp disks. The actuator
assembly contains an actuator driven by voice-coil motor (VCM) and mounted via
ball-bearing at each end of a pivot shaft, flex cable carrying head and VCM leads, and
arms to support suspension/head extension between the disks. In the head/suspension
assembly, an airbearing surface is created on surface next to the rotating disk, the
slider carrying heads flies on top of the disk surface, and a gimbal attaches the slider
to the suspension. The electronics card involves drivers for the spindle motor and
VCM, read/write (R/W) electronics, a servo demodulator, and micro processors for
servo control and control of interface to host computer.
The actuator servo channel consists of a demodulator producing position informa-
tion from the servo burst read from the disk during seeking and following; a servo
controller to control the position of the R/W head during reading, writing and seek-
ing; spindle control to keep the spindle rotating at a specific speed with a minimum
speed fluctuation and a power driver to drive the spindle motor and the VCM actu-
ator. The servo or position information is used to position the magnetic head on the
disk surfaces. Position measurement of the magnetic head is achieved by means of
analyzing the position error signal (PES) calculated from the read back signal.
The head-positioning servomechanism is a control system that positions the R/W
head from one track to another in minimum time, and repositions the R/W head over
a desired track with minimum statistical deviation from the track center. A settling
controller is used in between the above seeking and following modes.
The seek time is a measure of how fast the disk drive actuators can move the R/W
head to a desired location. The seek time is limited by the actuator behavior, acceler-
ation current level and the control algorithm. The major requirement in the seeking
process is fast and smooth seeking with small or even no overshoot. Dual-stage ac-
tuation with a VCM as primary actuator and a microactuator as secondary actuator
works as one way to achieve fast seeking and settling due to higher bandwidth.
Once the actuator is regulating the position of the R/W head at the desired track,
the smaller the head position deviates from the desired track center, the closer the
tracks can be put together and the higher the track density becomes. In this stage,
the servo performance is limited by mechanical factors in the actuator, disk platter,
spindle motor, etc. An improved mechanical design is supposed to present less dis-
turbance, causing less off-track. On the other hand, a good closed-loop servo system
is expected to reject the disturbances. The error transfer function must be well de-
signed to yield a sufficiently small closed-loop non-repeatable runout. This typically
requires a satisfactory servo loop based on the disturbance spectrum. Generally, it
demands a high servo bandwidth, a high 0-dB crossover frequency and a low hump
of error rejection transfer function. A secondary microactuator activated together
with the VCM primary actuator is generally used to produce a higher bandwidth
closed-loop system.
In a disk drive, the positioning information or servo information (“servo bursts”)
is embedded in each disk surface. Servo bursts are conventionally written by costly
dedicated servo writing equipment external to the disk drive, which uses a laser-
guided push-pin mechanism to position the write head on the disk surface until the
servo burst information is written on the disk completely [53] [54]. The defects such
Modeling of Disk Drive System and Its Vibration 19
as non-circularity caused by the spindle motor vibration in the servo bursts will make
servo tracks more difficult to follow in disk drives. The demand of big storage capac-
ity in hard disk drives requires high track density. More accurate placement of servo
bursts is thus required accordingly. As technologies such as servo mechanism and
head and media technology advance, the capability of writing and reading narrower
tracks is improved, resulting in an increased track density.
Conventional servo writers require a clean room environment because the disk
and head will be exposed to the environment to allow access to the external head and
actuator. A self-servo track writer regenerates timing and radial information from
previously written tracks using the existing R/W head [135]. The external equip-
ment is no longer needed in servo pattern writing and thus no open access is required
for the head disk assembly and servo track writing does not have to be carried out
in a clean room environment. However, the self-servo track writing creates the ra-
dial error propagation problem, which will hinder the whole process of servo track
writing if not properly solved [140].
In an embedded servo, data and servo information are written on all tracks in an
interleaved manner. All tracks are divided into a fixed number of radial sectors. Each
sector starts with a servo pattern followed by user data information. The pattern is
repeated for each sector. The fields in an embedded servo pattern include Pream-
ble, AGC field, sector or index mark, track address in gray code and servo bursts.
The position feedback information for the disk drive servo mechanism consists of
two components: gray code and position error signal. The gray code track num-
ber provides coarse position information. It determines the absolute position of the
read/write head during seeking and tracking. Position error signal, or PES, is the
relative displacement of the R/W head from the track center. When PES and gray
code are combined together, the position of the read/write head is obtained. Figure
2.1 shows one read back signal of the embedded servo. The A, B, C, D bursts give
fine position error quantifying the amount of track misregistration, i.e., the deviation
of the R/W head from the center of the track.
2.3 System modeling
2.3.1 Modeling of a VCM actuator
A linear VCM actuator moves in and out along a disk radius in one direction. It
contains a coil which is rigidly attached to the structure to be moved and suspended
in a magnetic field created by permanent magnets. When a current passes through the
coil, a force is produced which accelerates the actuator radially inward or outward,
depending on the direction of the current. The produced force is a function of the
current ic. Approximately,
fm = ktic, (2.1)
20 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.1
A read back signal of embedded servo.
where kt is a linearized nominal value called torque constant.
The resonance of the actuator is mainly due to the flexibility of the pivot bearing,
arm, suspension, etc. When the bandwidth of a control loop is very low and the
resonance may not be a limiting factor to the control design, the actuator model
can be considered as the simplified and rigid one which is a double integrator with
transfer function k/s2, i.e.,
y =k
s2u, (2.2)
or
y = kyv, (2.3)
v = kvu, (2.4)
where u is the input to the actuator, y and v are the displacement and the velocity of
the read/write head, ky is the position measurement gain, and kv = kt/m with the
actuator mass m.
With higher bandwidth, the actuator resonances have to be considered in the con-
trol design, since the flexible resonance modes will reduce the system stability and
affect control performance if ignored. Then the actuator model becomes
y =kvky
s2Pr(s)u, (2.5)
which includes the resonance model Pr(s). Let ωn = 2πfn correspond to a single
resonance frequency fn, and ξn be the associated damping coefficient. A second
order transfer function can be used to represent the resonance, i.e.,
Pr(s) =ω2
n
s2 + 2ξnωns + ω2n
. (2.6)
Modeling of Disk Drive System and Its Vibration 21
Different ξn gives different frequency responses, shown in Figure 2.2. The peak
of magnitude is higher when ξn decreases.M
agnitude(d
B)
Frequency(Hz)
Phase(d
eg)
0
0
FIGURE 2.2
Frequency responses of a second order transfer function.
Other forms of Pr(s) include
Pr(s) =b1ωns + b0ω
2n
s2 + 2ξnωns + ω2n
(2.7)
and
Pr(s) =b2s
2 + b1ωns + b0ω2n
s2 + 2ξnωns + ω2n
(2.8)
with zeros included to facilitate a phase lift which is usually associated with reso-
nance modes.
For lightly damped resonance, 0.005 ≤ ξn ≤ 0.05 is typical.
22 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.3
Measured VCM Bode plots (straight lines: pure double integrator k/s2).
Modeling of Disk Drive System and Its Vibration 23
Figure 2.3 shows the measured frequency response of a VCM actuator system. By
curve fitting each resonant mode, one can obtain the parameters of the transfer func-
tion. The deviation from the double integrator model in low frequencies in Figure
2.3 is due to the pivot friction and other nonlinearities. The nonlinearity modeling
will be discussed in detail in the next section.
Consider the resonance model in (2.7) ((2.6) and (2.8) can be handled similarly).
y = kyv, (2.9)
v = kvb1ωns + b0ω
2n
s2 + 2ξnωns + ω2n
u. (2.10)
Define
x1 =ω2
n
s2 + 2ξnωns + ω2n
u, x2 =sωn
s2 + 2ξnωns + ω2n
u, (2.11)
i.e.,
x1 = ωnx2. (2.12)
Then, we have a state-space description
yvx1
x2
=
0 ky 0 00 0 kvb0 kvb1
0 0 0 ωn
0 0 −ωn −2ξnωn
yvx1
x2
+
000ωn
u. (2.13)
More resonance modes can be modeled similarly.
Assume that nonlinearities such as friction and bias force and resonances are prop-
erly compensated, the actuator model is a double integrator model. It is written as[
yv
]
=
[
0 ky
0 0
][
yv
]
+
[
0kv
]
u. (2.14)
On the other hand, to make the mathematical models more realistic, we need to
consider uncertainties inherent in the plant model. Note that it is difficult to involve
all high-frequency dynamics in a model-based servo control design. In this case, an
uncertainty is introduced in the plant model. For instance, a model with multiplica-
tive uncertainty can be given by P = (1+∆)Pn. Figure 2.4 shows the multiplicative
uncertainty ∆ of a VCM. To meet the robustness requirement against this unmodeled
high frequency dynamics, we need to properly handle the uncertain system so that it
can perform under stricter margins.
2.3.2 Modeling of friction
In this section, we focus on friction modeling. There are basically two kinds of
methodologies for friction control in the literature: Model-based friction compen-
sation and non-model based friction control. Friction models for the former can be
24 Modeling and Control of Vibration in Mechanical Systems
102
103
104
−60
−50
−40
−30
−20
−10
0
10
20
Frequency(Hz)
Ma
gn
itu
de
(dB
)
FIGURE 2.4
Multiplicative uncertainty of a VCM.
Modeling of Disk Drive System and Its Vibration 25
roughly classified into two categories: static models and dynamic models. There
are various static models for friction, for example, the Coulomb model, the viscous
model, and friction models with the Stribeck effect, etc. However, a static friction
model cannot capture observed friction phenomena like hysteresis, position depen-
dence, and variations in breakaway forces. Therefore, a friction model involving
dynamics is necessary to describe friction phenomena accurately. A relatively new
dynamic friction model proposed in [43] combines the Dahl stiction behavior with
arbitrary steady state friction characteristics, which is able to include the Stribeck
effect. A nonlinear friction observer is then required for position control because
the involved interim state is not measurable and has to be observed in order to es-
timate the friction force. Later, to overcome the limitations of the above model, an
integrated model is proposed in [44], which is used in [41] for VCM pivot friction
modeling in HDDs. The resultant friction model needs to be iteratively improved and
verified using the measured and the simulated responses. Another dynamic model
used in VCM pivot friction modeling is the preload and two-slope model, which is
detailed in [40, 42] respectively in the frequency domain and the time domain. How-
ever, although the time-domain approach provides a good match between the time
domain response of the model and the data collected, it cannot guarantee a good
match in the frequency domain, and vice versa.
The non-model based approaches include the neural network method [46][47] and
the disturbance observer method [45]. The neural network method does not require
full knowledge of the nonlinearity model, but its implementation in real disk drives
seems difficult because of slow convergence [48]. In [45], a novel method for the
cancelation of pivot nonlinearities is proposed and it consists of an accelerometer
and a disturbance observer. The accelerometer is employed to linearize the dynam-
ics from the desired input signal to carriage angular acceleration, and the observer
estimates the nonlinear disturbances due to pivot friction for disturbance cancelation.
In this section, a mathematical model will be developed to closely describe the
friction hysteresis behavior. Among existing hysteresis models in the literature, the
Prandtl model [50] is less complex and more attractive in real-time applications. The
elementary operator in the Prandtl hysteresis model is a rate-independent backlash
or linear play operator, defined by pr(π0, x(t)), where x(t) is the actuator response
and π0 ∈ R is usually initialized to 0. Hysteresis nonlinearity can be modeled by a
linearly weighted superposition of many backlash operators with different threshold
r > 0 and weight values wb, i.e.,
Fh(x(t)) =
∫ ∞
0
wb(r)pr[π0, x(t)]dr, (2.15)
where the weight wb defines the ratio of the backlash operator, as seen in Figure 2.5.
In order to have an accurate mathematical model for the hysteresis, the creep model
proposed in [49] is also incorporated. Hence we consider the operator model given
by
F (x(t)) = ax(t) +
∫ ∞
0
wb(r)pr[π0, x(t)]dr +
∫ ∞
0
wc(λ)lλ[ξ0, x(t)]dλ,
26 Modeling and Control of Vibration in Mechanical Systems
(2.16)
where t ∈ [0, T ], a, wb(r) and wc(λ) are parameters to be determined, pr and lλ are
the elementary hysteresis and linear creep operators, and are defined as follows.
The elementary hysteresis operator pr with threshold r is defined as the solution
operator pr[π0, x(t)] = zr(t) of the rate independent hybrid differential equation
zr(t) =
x(t), if x(t) = zr(t) − r0, if zr(t) − r < x(t) < zr(t) + rx(t), if x(t) = zr(t) + r
with the initial value
zr(0) = maxx(0) − r, minx(0) + r, π0(r). (2.17)
FIGURE 2.5
The operator zr versus x.
Define the linear creep operator lλ with λ > 0 as the solution operator lλ[ξ0, x(t)] =zλ(t) of the differential equation
1
λzλ(t) + zλ(t) = x(t) (2.18)
with the initial value equation
zλ(0) = ξ0(λ).
Modeling of Disk Drive System and Its Vibration 27
The explicit integral formula for the linear creep operator lλ is as follows.
lλ[ξ0, x(t)] = e−λtξ0(λ) + λ
∫ t
0
eλ(τ−t)x(τ )dτ. (2.19)
For numerical implementation of the operator-based modeling, a discrete-time
model F (x(k)) of the operator F (x(t)) in (2.16) is developed as follows.
F (x(k)) = ax(k) +
n∑
i=1
wbipri[π0, x(k)] +
m∑
j=1
wcj lλj[ξ0, x(k)], (2.20)
where
1) the output sequence of the discrete hysteresis operator is calculated by
pri[π0, x(k)] = zri
(k), (2.21)
zri(k) =
x(k) + ri, if zri(k − 1) − ri ≥ x(k)
zri(k − 1), if zri
(k − 1) − ri < x(k) < zri(k − 1) + ri
x(k) − ri, if zri(k − 1) + ri ≤ x(k)
with the initial value zri(0) = maxx(0) − ri, minx(0) + ri, π0(ri);
2) the discrete counterpart to the continuous elementary creep operator is given by
lλj[ξ0, x(k)] = zλj
(k) (2.22)
with
zλj(k + 1) = e−λjTs · zλj
(k) + (1 − e−λjTs) · x(k) (2.23)
and the initial value zλj(0) = ξ0(λj).
Consider the hysteresis curve of the friction f versus the displacement x in Figure
2.6. Let fe = F (x(k)) be the approximated friction, then the approximation error
e = f − fe. We define the energy gain between the actuator position and the error as
‖Tex‖∞ =
√
√
√
√
∑Lk=1 eT (k)e(k)
∑Lk=1 xT (k)x(k)
, (2.24)
where L is the number of data points.
Denote wb = (wb1, wb2, · · · , wbn), wc = (wc1, wc2, · · · , wcm), and Λ = (λ(1),λ(2), · · ·, λ(m)). We aim to find optimal parameters a, wb, wc, and Λ in (2.20) so
that (2.24) is minimized, and thus a model (2.20) can be obtained to approximate the
friction f with the displacement x as the input.
Note that ‖Tex‖∞ is a function of a, wb, wc, and Λ, and is denoted as
‖Tex‖∞ = ℓ (a, wb, wc, Λ) . (2.25)
The MATLAB function fminsearch can be used to minimize ℓ (a, wb, wc, Λ) with
respect to (a, wb, wc, Λ).
28 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.6
Friction f versus actuator displacement x.
FIGURE 2.7
A PZT actuated suspension.
Modeling of Disk Drive System and Its Vibration 29
2.3.3 Modeling of a PZT microactuator
A piezoelectric-based microactuator located on the suspension as shown in Figure
2.7 is considered in this section. The mechanical operation of the microactuator
can be understood via an equivalent spring-mass system. The compliance of the
base plate is simplified as a single spring Kb, and the compliance of the flex hinge
elements is simplified as a single rotational spring Kr.
FIGURE 2.8
Equivalent spring mass system of PZT microactuator.
An important point for PZT microactuator modeling is that the PZT element acts in
series with the base plate springs. Thus the displacement of the PZT element results
in displacements of the springs. The PZT and the base plate with spring constants
Km and Kb can be equivalent to a single spring with constant
KT =2
1Km
+ 1Kb
. (2.26)
The model is derived by applying forces at the interface of the piezo element and
the base plate spring and by summing moments about the pivot point. The free
expansion of the piezo element is expressed as
θf =LmdexpV
cl1, (2.27)
30 Modeling and Control of Vibration in Mechanical Systems
where Lm is the piezo length, dexp is the piezo expansion coefficient, V is the volt-
age, c is the piezo thickness, and l1 is the length as indicated in Figure 2.8.
The following second order differential equation can be derived to capture the
dynamic behavior of the microactutor:
Kd2θ
dt+ C
dθ
dt+ (Kr + KT l21)θ =
KT Lmdexpl1c
V, (2.28)
where K is the torsional inertia, Kr is the torsional spring rate. A typical frequency
response of the PZT microactuator from voltage input to position output is shown in
Figure 2.9.
101
102
103
104
−30
−20
−10
0
Magnitude(d
B)
101
102
103
104
−200
−160
−120
−80
−40
0
Phase(d
eg)
Frequency(Hz)
FIGURE 2.9
A typical frequency response of the PZT microactuator.
2.3.4 An example
2.3.4.1 Dynamics modeling
The hard disk drive under consideration is shown in Figure 2.10. It includes the VCM
actuator mounted with the arm and the suspension/head. The actuator is driven by a
driver which converts voltage differences into current differences linearly.
The actuator dynamics measurement is taken using a Laser Doppler Vibrometer
(LDV) and a Dynamic Signal Analyzer (DSA). The displacement y is measured via
Modeling of Disk Drive System and Its Vibration 31
FIGURE 2.10
An opened 1.8-inch hard disk drive.
the LDV, and the frequency response is measured by using the DSA to generate a
swept sine signal to excite the actuator. The measured frequency response is shown
in Figure 2.11, where the modeled one is plotted from the following transfer function
P (s) obtained by curve fitting to the measured frequency responses.
P (s) = KPf(s)Pres(s), K = 5.3290× 1017,
Pf (s) =1
s2 + 2× 0.25× 2π120s + (2π120)2,
Pres(s) =s2 + 1081s + 7.3× 108
(s2 + 1056s + 6.964× 108)(s2 + 6032s + 2.527× 109). (2.29)
The resonance of Pf(s) at 120 Hz is due to the nonlinearity of actuator pivot
friction. When the friction nonlinearity is neglected, it is replaced with the pure
double integrators model, i.e.,
P (s) =K
s2Pres(s), K = 4.8209× 1017, (2.30)
which is plotted as the dotted curves in Figure 2.11. However, the friction in the
actuator pivot [39] [40] is known to limit the low frequency gain of the control loop.
Translated to the error rejection function or sensitivity function, it lifts the magnitude
32 Modeling and Control of Vibration in Mechanical Systems
101
102
103
104
−40
−20
0
20
40
60
80
100
Magnitude(d
B)
101
102
103
104
−600
−400
−200
0
200
Frequency(Hz)
Phase(d
eg)
MeasuredModeled
FIGURE 2.11
Measured and modeled frequency responses of the VCM actuation system (LDV
range 0.5 µm/V).
Modeling of Disk Drive System and Its Vibration 33
of the sensitivity function at low frequencies, and thus reduces the ability of the
control loop to reject low-frequency vibrations and affects the positioning accuracy.
Therefore, it is necessary to compensate the friction impact.
2.3.4.2 Friction measurement and modeling
Due to the fluctuation of the head when the disk is rotating (rotational speed is 4200
RPM), it is difficult to have a steady displacement signal of the head. Thus the
friction measurement is carried out under the closed loop control as shown in Figure
2.12. The controller C(z) is a PID controller combined with a notch filter and is
expressed in (2.31). The sampling time Ts is 83.3 ms. With the controller, the open-
loop 0 dB crossover frequency is 945 Hz, the gain margin is 8.7 dB, and the phase
margin is 49 deg.
C(z) = kc(kp + kdz − 1
Tsz+ ki
Ts
z − 1) × 0.9023z2 + 0.9467z + 0.7242
z2 + 0.929z + 0.6442,
kc = 0.0625, kp = 0.8, kd = 400× 10−6, ki = 400. (2.31)
FIGURE 2.12
Closed control loop of a disk drive with a VCM actuator for friction measurement
via LDV.
A 10 Hz sinusoidal signal with increasing amplitude of 0.5, 1, and 3 V is re-
spectively used as the reference signal in Figure 2.12. The control signal u and
displacement x are measured, and shown in Figure 2.13.
The VCM actuator model with consideration of nonlinearity F (x) is shown in
Figure 2.14, which includes two pure integrators, the resonance modes Pres(s) and
the gain K given in (2.30).
With the measured u and x, ua can be obtained from x, and thus f = u−ua. The
relation between x and f can be obtained and shown as hysteresis curves in Figure
2.6.
34 Modeling and Control of Vibration in Mechanical Systems
−4 −3 −2 −1 0 1 2 3 4−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
Displacement (V, 0.5 µm/V)
Co
ntr
ol sig
na
l u
(V
)
FIGURE 2.13
Control signal u versus displacement x.
FIGURE 2.14
VCM actuator modeling with friction nonlinearity model F (x).
Modeling of Disk Drive System and Its Vibration 35
In what follows, we shall find F (x) to model the relationship between f and x.
The above mentioned operator based method to approximate hysteresis will be ap-
plied to model the hysteresis of f and x as shown in Figure 2.6.
ri in (2.21) can be chosen as the amplitude of x. Since with the chosen peak-to-
peak values of x = 0.5, 1 and 3V, the ri are respectively
r1 = 0.25; r2 = 0.5; r3 = 1.5. (2.32)
With m = 3, n = 3 and the initial values π0 = 0, ξ0 = 0, after 2000 iterations, a
minimum error e is achieved and the optimal parameters are obtained as
a = 9.0024,
wb1 = −1.5783, wb2 = 0.1667, wb3 = −0.0655,
wc1 = −0.1431, wc2 = −7.3341, wc3 = 0.4403,
which gives the minimal ‖Tex‖∞ = 0.08.
With these parameters, fe = F (x(k)) can be calculated from (2.20). The time
traces of fe and f are compared in Figure 2.15. It is observed that the time trace
from the model (2.20) can closely track f , and the error f−fe is small. The modeled
hysteresis from x to fe is drawn and compared with the measured one in Figure 2.6.
It is seen that the modeled hysteresis and the measured one are close to each other.
If the creep term in the model (2.20) is removed, i.e., wcj = 0, j = 1, · · · , m, the
modeling accuracy decreases. Indeed, in this case, the minimal ‖Tex‖∞ = 0.1293 >0.08.
Figures 2.16 and 2.17 show that the quality of the agreement in the frequency
domain between 70 Hz and 150 Hz decreases with lower excitation amplitude espe-
cially for the 0.5V case.
Note that the operator model fe = F (x(k)) describes the hysteretic characteristics
in Figure 2.6 as a mapping between the actuator position x and the friction force
f . It turns out that the model makes it possible to approximate hysteretic transfer
characteristics without modeling the underlying physics. This is different from the
friction models such as the preload and two-slope model in [40][42]. Compared to
the model in [40] [42], an advantage of our operator based model is that the frequency
response of the actuator can also fit well to the measured one, as shown in Figures
2.16 and 2.17 .
For comparison, we also apply the preload and two slope model [40] for the hys-
teresis, as shown in Figure 2.18. The preload model for velocity v and the two-slope
model for position x are given by
fv = kvv + kssgn(v), (2.33)
and
fx =
kax, |x| ≤ sx
kbx + (ka − kb)sx, |x| > sx.
36 Modeling and Control of Vibration in Mechanical Systems
Using the data with the amplitude of x(t) of 0.25V, we obtain that
ks = 1.85e− 4, kv = 0.48, ka = 0.0145, kb = 0.0014, sx = 0.009.
Using this model, a comparison with the measured data in the time domain is
shown in Figure 2.19 for the amplitude of x(t) of 0.5 and 1 V, respectively. It can
be observed that the plant input u versus position x fits reasonably well to the mea-
sured results in the time domain. However, in the frequency domain, the magnitude
response for the case of the amplitude of x(t) of 0.5V, plotted as the dotted curve in
Figure 2.17 deviates significantly from the measurement results.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5x 10
−3
Time(sec)
Friction a
nd e
rror
From measurement (f)From modeling(f
e)
Error (f−fe)
FIGURE 2.15
Measured and modeled friction and error.
Modeling of Disk Drive System and Its Vibration 37
FIGURE 2.16
Actuator frequency response for sinusoidal reference with amplitude of 1 and 3 V,
respectively.
FIGURE 2.17
Actuator frequency response for sinusoidal reference with amplitude of 0.5 V.
38 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.18
Preload and two-slope model for friction modeling.
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1x 10
−3
Displacement (µm)
Inp
ut
vo
lta
ge
u
Measured
Modeled
1V
0.5V
FIGURE 2.19
Plant input voltage u versus displacement x.
Modeling of Disk Drive System and Its Vibration 39
2.4 Vibration modeling
Vibration in disk drives causes the deviation of the R/W head positioning from the de-
sired track center. It is the combination of the repeatable runout which is synchronous
with the spindle revolution and the nonrepeatable runout. Figure 2.20 shows a sim-
plified block-diagram of disk drive servo loop. y is the position of the R/W head and
e is the position error signal. The signal d1 represents all the torque disturbances
to the system. Such disturbances include any torque due to air-turbulence force to
the actuator, the suspension and the slider. The effects of the torque disturbances are
dominant at frequencies that are relatively low when compared to the servo band-
width. The signal d2 represents disturbances that are due to non-repeatable motions
of the disk and motor, suspension and slider vibrations, which directly add to the
relative position of the R/W head and the servo track. The noise signal n includes
media and head sensing noises and also represents the effects of the PES demodu-
lation noise which includes actual electrical noise and A/D quantization noise. The
noise signal n is thus reasonably modeled as a broad-band white noise.
2.4.1 Spectrum-based vibration modeling
FIGURE 2.20
Closed-loop control system disturbances d1, d2 and noise n.
Since a controlled closed-loop can provide steady signals, the vibration source
analysis is based on the signal that is collected from the closed-loop system. From
Figure 2.20,
e(k) = −P (z)S(z)d1(k) − S(z)d2(k) + S(z)n(k), (2.34)
40 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.21
Closed-loop control system with disturbance and noise models.
where P (z) is the transfer function of the discretized plant model P (s) and the sen-
sitivity function or error rejection function is given by
S(z) =1
1 + P (z)C(z). (2.35)
Figure 2.22 shows the sensitivity function S(z) of a closed-loop control system.
Assume that d1, d2, and n are uncorrelated. The power spectrum denoted by Se of
the error signal e is given by
Se = |P (z)S(z)|2|d1(k)|2 + |S(z)|2|d2(k)|2 + |S(z)|2|n(k)|2. (2.36)
Figure 2.23 shows the spectrum of NRRO component in the error signal e. Two
humps are obviously observed in the baseline curve. One is in the frequency range
lower than 100 Hz, the other one is after 500 Hz. Considering (2.36) and Figure
2.22, the second hump is caused by S(z) through |S(z)|2|n(k)|2, and the first hump
is due to d1 through P (z)S(z) with a hump in a lower frequency range. Hence the
disturbance and noise modeling can be carried out as follows.
In Figure 2.21, models D1(s), D2(s) and N(s) are used to describe d1(s), d2(s)and n(s) respectively, and wi (i = 1, 2, 3) are independent white noises with
variance 1. Eq. (2.36) becomes
Se = |P (z)S(z)|2|D1(z)|2 + |S(z)|2|D2(z)|2 + |S(z)|2|N(z)|2, (2.37)
where D1(z), D2(z) and N(z) are the discrete forms of D1(s), D2(s) and N(s),respectively. D1(z) and N(z) can be determined by fitting weighted versions of
P (z)S(z) and S(z) to the baseline curve of the spectrum and the spikes are consid-
ered as the effect of D2(z). Hence the steps to obtain D1, N and D2 are
Modeling of Disk Drive System and Its Vibration 41
FIGURE 2.22
Sensitivity function S(z).
42 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.23
PES NRRO spectrum.
Step 1). Find Sb(j),
Sb(j) =jq
mini=1+(j−1)q
Se(i), j = 1, 2, · · · , L/q, (2.38)
where L is the length of Se and q is as small as possible.
Step 2). Compute D1(z),
|D1(z)|2 = WL(z)Sb/|P (z)S(z)|2, (2.39)
where WL is a low-pass weighting function used to select Sb in low frequency range.
Step 3). Compute N(z),
|N(z)|2 = WHSb/|S(z)|2, (2.40)
where WH is the high-pass weighting function to select Sb in high frequency range.
Step 4). Till now, the baseline curve Sb can be fit well by the identified D1 and N .
The remaining part of the spectrum is then regarded as D2. Thus,
|D(z)|2 = Se − [|P (z)S(z)|2|D1(z)|2 + |S(z)|2|N(z)|2]/|S(z)|2. (2.41)
Modeling of Disk Drive System and Its Vibration 43
D1(s), D2(s) and N(s) are then obtained as follows from D1(z), D2(z) and
N(z) using the bilinear approximation method [81].
D1(s) =1.3916× 10−5(s + 575.8)(s + 575.6)(s2 + 0.04389s + 161.6)
(s2 + 315.5s + 8.178× 104)(s2 + 315.4s + 8.178× 104), (2.42)
D2(s) =0.701588(s + 1.271× 104)2(s2 + 6.125× 10−5s + 4.373× 108)
(s + 708.4)2(s2 + 0.0001317s + 4.376× 108),
(2.43)
N(s) =1.1695(s + 1.431× 104)(s + 766.2)(s2 + 8609s + 4.672× 107)
(s + 4603)(s + 1538)(s2 + 4451s + 1.507× 107). (2.44)
With the disturbance and noise models, the feedback controller C(z) can be opti-
mized to minimize the error due to w = [w1 w2 w3]T by using the H2 optimal control
method or other advanced control methods.
2.4.2 Adaptive modeling of disturbance
2.4.2.1 Neural network approximation
A neural network usually consists of a large number of simple processing elements
called nodes. The nodes are interconnected by weighted links with weight param-
eters adjustable. The different arrangement of the nodes and the interconnections
defines various architectures of neural networks [73][74], which are suitable to dif-
ferent kinds of applications. In control engineering, a multi-layer neural network
is usually used to generate the mapping from input to output since it can approxi-
mate any function under mild assumption with any desired accuracy. The function
approximation is defined as follows.
Definition 2.1 [75] If f(x) : Rn → Rm is a continuous vector function defined
on a compact set Ω, and any y(W, x) : Rt × Rn → Rm is an approximating
function that depends continuously on W and x, then, the approximation problem is
to determine the optimal W denoted by W⋆, for some index d such that
d(y(W⋆, x), f(x)) ≤ ε, (2.45)
for an acceptable small ε.
There are a number of neural networks studied for function approximation such as
multi-layer perceptron networks, radial basis function (RBF) networks, and higher
order neural networks. The RBF network is suitable for online nonlinear adaptive
modeling and control, because it is a linearly parameterized network, has spatially
localized learning capability and thus has better memory during learning, and ex-
hibits a fast initial rate of learning convergence.
The three-layer neural network shown in Figure 2.24 is a RBF network, where
x ∈ Rn, y ∈ Rm, and s ∈ Rp are respectively the input, the output, and the
activation function vectors, and wij is the second to the third layer interconnection
44 Modeling and Control of Vibration in Mechanical Systems
FIGURE 2.24
Three-layer RBF neural network.
Modeling of Disk Drive System and Its Vibration 45
weights. The output yi is given by
yi =
p∑
j=1
wijsj(‖x − cj‖2), i = 1, 2, · · · , m, (2.46)
or equivalently in a matrix form,
y(x) = WTs(x), (2.47)
where p is the number of nodes, x = [x1 x2 · · · xn]T ∈ Rn is the network
input vector, ‖ • ‖ denotes the Euclidean norm, cj ∈ Rn is the center vector, and
WT = [wij]T .
The approximation of a general function f(x) : Rn → Rm can then be expressed
as
f(x) = WTs(x) + ε(x) (2.48)
where ε(x) ∈ Rm is the reconstruction error vector.
Several functions such as Gaussian, Hardy’s multiquadric and inverse Hardy’s
multiquadric functions have been used as activation functions. They are separately
written as
sj(x) = exp[−(x− cj)
T (x − cj)
σ2j
], (2.49)
sj(x) =√
σ2j + (x − cj)T (x − cj), (2.50)
and
sj(x) =1
√
σ2j + (x − cj)T (x − cj)
. (2.51)
For a mechanical system with dynamics given as a function of position, velocity
and acceleration, a dynamic neural network is needed in order to fully emulate the
dynamics. In a dynamic neural network, dynamic variables such as velocity and
acceleration are involved in the input x. In discrete-time neural networks, past infor-
mation such as x(k − 1) and x(k − 2) is used as input.
2.4.2.2 Disturbance modeling
In some mechanical motion systems, disturbances in frequency range lower than a
few hundred Hz are quite dominant and they may be due to torque disturbances,
nonlinear or unknown vibration sources. Here we use d1 to represent these dis-
turbances. This section aims to model the low-frequency disturbance d1 with an
46 Modeling and Control of Vibration in Mechanical Systems
adaptive nonlinear strategy based on the measured error signal e. In the open-loop
without controller C(z),
e = −P (s) · d1 − d2 + n. (2.52)
The disturbance d1 is modeled as
d1 = f(Φ(e(k))), (2.53)
where f(·) is an unknown function of the bounded vector-valued function Φ(e(k))with
e(k) = [e(k) e(k − 1) · · · e(k − l)]T , (2.54)
where l is to be determined. Thus (2.53) is dependent on some time history of the
measured signal e.
For brevity, f(Φ(e(k))) is denoted as f(Φk). As stated in the previous section,
the unknown function f can be approximated using neural networks. Here we se-
lect a Gaussian RBF based neural network. The Gaussian RBF network has some
attractive properties and thus has been widely used in nonlinear control and signal
approximation. The properties are, (1) it is bounded and strictly positive, and (2)
it possesses a localized response. Based on the Gaussian RBF network, the low
frequency disturbances can be approximated as
d1 = f(Φk) = WT (k)s(Φk) + ε, (2.55)
where ε is the modeling error, s(Φk) ∈ Rp is a basis vector function with a suitable
number of nodes p. The radial basis function s(Φk) is written as
si(Φk) = e− (e(k)−cei
)2
σ2ei · e
−(e(k)−e(k−1)−c∆ei
)2
σ2∆yi ,
s(Φk) = [s1(Φk) s2(Φk) ... sp(Φk)]T , (2.56)
where σeiand cei
(σ∆eiand c∆ei
) are the variance and center position of the mea-
surement e (velocity e).
The weight vector W (k) in (2.55) can be calculated iteratively according to the
following weight update law:
W (k + 1) = (1 − δ)W (k) − Γs(Φk)e(k) (2.57)
with the adaptation gain matrix Γ ∈ Rp×p being diagonal and satisfying Γ > 0 and
the forgetting factor 1 > δ > 0. The forgetting factor δ is to ensure the bounded-
ness of W (k) when the system is subjected to bounded disturbances. The speed of
learning rate is related to the chosen Γ matrix.
Next, the power spectrum in Figure 2.27 of a position error signal e will be used to
verify the adaptive modeling scheme. It is noted that the contribution of disturbances
and noise to e is described as in (2.52).
Modeling of Disk Drive System and Its Vibration 47
A low frequency signal with the spectrum as in the low frequency range in Figure
2.27 is generated and injected as disturbance d1 in Figure 2.21. The disturbance d1
affects the error signal or the output e through the plant P (s), thus the nonlinear
model from e to d1 aims to generate a cancelation signal of the disturbance d1. Since
the low frequency disturbances due to torque and bias are generally nonlinear, the
model from e to d1 is chosen to be nonlinear.
The verification is implemented with the sampling period Ts = 1/45000sec. For
p = 1 with zero center positions cy1 and c∆y1 , after some trials, it was found that
when σ2y1
= σ2∆y1
= 10, δ = 0.5, and
Γ = Ts · 106 · 220, (2.58)
p = 1, l = 1, (2.59)
the time trace d1 calculated from (2.55) gives the best approximation of the distur-
bance d1. The effect of different p and Γ on the modeling accuracy will be evaluated
later. Figure 2.25 shows the simulated time trace comparison of d1 and d1 from the
nonlinear model (2.55). It is observed that the time trace from the model (2.55) can
give a close tracking of the original one. The spectrum of d1 is seen in Figure 2.26.
Moreover, from the spectrum in Figure 2.27 with the component of disturbance
P (s) · d1 removed and considering that noise n is a white noise, the disturbance d2
and the noise n are represented approximately by
d2 = D2(s)w2, (2.60)
D2(s) =
0.0019(s2 + 3329s + 1.695× 107)(s2 + 3340s + 5.61× 108)
(s2 + 245s + 1.668× 107)(s2 + 477.5s + 5.701× 108),
and
n = N(s)w3 = 0.005 w3. (2.61)
The NRRO spectrum obtained by combining the nonlinear and linear models is
compared with the measured one in Figure 2.27. It is found that the adaptive nonlin-
ear modeling method can indeed be used to model the disturbance d1 that is dominant
in low frequency range.
Let the modeling error de = d1 − d1. To evaluate the effects of different values of
p on the modeling error, two more cases with p = 5, 9 and the following parameters
for Γ are investigated.
Γ = Ts · 106 · diag33, 66, 220, 66, 33, p = 5; (2.62)
Γ = Ts · 106 · diag38.5, 88, 88, 38.5, 220, 38.5, 88, 88, 38.5,p = 9. (2.63)
The σ value of the modeling error e can be seen in Table 2.1. With p = 1, 5, 9,
σ increases from 4.43 to 4.60, which means higher p may not give a better result
48 Modeling and Control of Vibration in Mechanical Systems
in this practical application. Compared with p, Γ influences the modeling accuracy
more significantly. For p = 1, with Γ being changed to 10% of the value in (2.58),
the σ value of the error e increases from 4.43 to 11. The time sequence comparison
of d1 − d1 is shown in Figure 2.28, where the difference is clearly seen. The other
two cases are similar.
2.5 Conclusion
In this chapter the models of a VCM actuator and a PZT microactuator have been
derived on the basis of their physical operations. The models have been verified
with the measured frequency responses from voltage input to displacement output.
To have a complete model of an actuation system, this chapter has also addressed
the modeling of uncertainties in high frequency and nonlinearities such as actuator
pivot friction. A Prandtl operator based model has been used to model the friction
nonlinearity, and the optimal model parameters have been obtained by minimizing
the energy gain between the actuator position and the modeling error. It turns out that
the derived model matches well the measured model not only in the time domain, but
also in the frequency domain.
The developed vibration and noise modeling approaches are based on the error
signal power spectrum of the closed-loop system. In particular, for low frequency
disturbance modeling, an adaptive nonlinear scheme with system measurement as
input has been applied to approximate the original disturbance for real-time com-
pensation.
Modeling of Disk Drive System and Its Vibration 49
TABLE 2.1
σ values of the modeling error (d1 − d1) for different p and Γ
p 1 5 9
σ values of (d1 − d1) for Γ and 0.1Γ.(×10−4) 4.43/11 4.55/11 4.60/10
0 0.05 0.1 0.15 0.2 0.25−4
−3
−2
−1
0
1
2
3
4
5x 10
−3
Time(sec)
Origin
al dis
turb
ance a
nd the m
odele
d (
µ m
)
Original disturbance d
1
d1 from nonlinear modeling
FIGURE 2.25
Original disturbance d1 and the modeled d1.
50 Modeling and Control of Vibration in Mechanical Systems
0 1000 2000 3000 4000 5000 60000
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3
Frequency(Hz)
NR
RO
magnitude(µ
m)
FIGURE 2.26
Power spectrum of d1.
Modeling of Disk Drive System and Its Vibration 51
0 1000 2000 3000 4000 5000 60000
0.5
1
1.5
2
2.5
3
3.5
4
x 10−3
Frequency(Hz)
NR
RO
ma
gn
itu
de
(µm
)
ModeledMeasured
FIGURE 2.27
NRRO power spectrum from measurement and disturbance models, i.e., e = −P (s)·d1 − d2 + n.
52 Modeling and Control of Vibration in Mechanical Systems
0 0.05 0.1 0.15 0.2 0.25−4
−3
−2
−1
0
1
2
3
4x 10
−3
Time(sec)
Modelin
g e
rror
(µm
)
Γ
0.1Γ
FIGURE 2.28
Modeling error (d1 − d1) for different Γ with p = 1.
3
Modeling of Stewart Platform
3.1 Introduction
An adaptive control system is able to adapt to the changes of a physical system. In
order to obtain more accurate models for physical systems, adaptive identification
algorithms can be used. Indeed, adaptive filtering is often adopted in control and
signal processing fields, for the modeling of an unknown system.
In this chapter, the governing motion equation of a piezoelectric Stewart platform
will be obtained first, followed by the derivation of a transfer function from the actua-
tor force to the sensor output based on frequency response data. An adaptive filtering
approach will then be introduced and subsequently used to model the platform and
verify the transfer function obtained.
3.2 System description and governing equations
The study of active damping of flexible structures has been induced by the require-
ment of structural stability. Most solutions to the active damping problem rely on
the integration of smart actuators and sensors in the structure itself. Referring to the
discussion in Chapter 1, each leg of the hexapod Stewart platform consists of an am-
plified piezoelectric actuator, a force sensor and two flexible joints. This piezoelec-
tric Stewart platform can be used as a precision pointing device, a vibration isolation
mount, or an active damping mount.
Shown in Figure 1.2 and Figure 1.3, the hexapod consists of two parallel plates
connected to each other by six active legs. The legs are mounted in such a way
to achieve the geometry of cubic configuration. Each active leg consists of a force
sensor, an amplified piezoelectric actuator, and two flexible joints.
Figure 3.1 shows an equivalent system of each leg connecting two rigid bodies:
the disturbance source m and the sensitive payload M that are connected by a force
sensor and a piezoelectric actuator represented by its elongation δ and spring stiffness
k. The Laplace form of the governing equation of the motion in this system is
53
54 Modeling and Control of Vibration in Mechanical Systems
FIGURE 3.1
Single-axis system using piezoelectric stiff actuator.
Ms2xp = −ms2xd = k(xd − xa) = y (3.1)
and
δ = xp − xa. (3.2)
The transfer function between the extension δ of the piezo stack and the sensor
output y is
y
δ= k
Mms2
Mms2 + k(M + m). (3.3)
Consider the piezoelectric hexapod integrated in a structure. Let M and K be the
mass and stiffness matrices of the global passive system, i.e., structure and hexapod.
The dynamic equation governing the system is written as
Ms2x + Kx = Bf, (3.4)
where B is the force Jacobian matrix, x represents payload frame displacement, and
f = kδ represents the equivalent piezoelectric force in the leg. A force sensor is lo-
cated in each leg of the hexapod and collocated with the actuator. The corresponding
sensor output is
y = k(q − δ), (3.5)
where y is the force sensor output and q is the leg extension from the equilibrium
position. Taking into account the relationship between the leg extension and the
payload frame displacement
q = BT x, (3.6)
Modeling of Stewart Platform 55
the sensor output equation becomes
y = k(BT x − δ). (3.7)
The sensor output y is to be used as the input to the controller.
Next an adaptive filtering approach will be used to model the hexapod Stewart
platform and verify the dynamics equation (3.3).
3.3 Modeling using adaptive filtering approach
3.3.1 Adaptive filtering theory
LMS algorithm
FIGURE 3.2
Linear discrete time adaptive filter.
In Figure 3.2, the statistically stationary time sequence of input signal, x(0), x(1),· · ·, is applied to the linear discrete time filter whose coefficients are W0, W1, · · ·.The filter output at time k, y(k), is to be as close as possible to the desired response,
d(k). The difference between y(k) and d(k) is defined as the estimation error, e(k)that is then applied back to the filter to adjust the filter weights so as to minimize
the estimation error in the statistical sense. y(k) = WT (k)X(k), where WT (k) =[W0 W1 W2 · · ·], and XT (k) = [x(k) x(k − 1) x(k − 2) · · ·].
In most practical instances the adaptive process aims to minimize the mean-square
value, or average power of the error signal [37]. Optimization under this criterion is
56 Modeling and Control of Vibration in Mechanical Systems
most effective in statistical sense. So the performance index of the adaptive filter is
determined by the mean-squared error (MSE) criterion in which the cost function is
defined as follows.
ξ(k) = E[e2(k)], (3.8)
where E[ ] denotes the expected value.
e(k) = d(k) − y(k)
= d(k) − WT (k)X(k). (3.9)
ξ(k) = E[(d(k)− WT (k)X(k))2 ]
= E[d2(k)] − 2WT (k)E[d(k)X(k)] + WT (k)E[X(k)XT (k)]W (k)
= Rdd − 2WT (k)Rdx + WT (k)RxxW (k), (3.10)
where Rdx is the cross correlation function between d(k) and x(k), Rdd and Rxx are
respectively the autocorrelation functions of d(k) and x(k).
In many practical applications, the statistics of d(k) and x(k) are unknown; the
exact knowledge of gradient vector is not available. Thus the method of steepest de-
scent [35] cannot be implemented in practical situations. Therefore another method
called the Least Mean Square (LMS) algorithm was introduced by Bernard Widrow
[37]. In the LMS algorithm, instantaneous squared error is used as an estimation
of the mean squared error. The mathematical derivation of the LMS algorithm is as
follows.
ξ(k) = e2(k),
∇ξ(k) = ∇e2(k) = 2[∇e(k)]e(k),
e(k) = d(k) − WT (k)X(k),
∇e(k) =∂e(k)
∂W (k)= −X(k),
∇ξ(k) = −2X(k)e(k). (3.11)
The updating equation of the adaptive filter coefficient:
W (k + 1) = W (k) − µ
2∇ξ(k),
becomes
W (k + 1) = W (k) + µX(k)e(k). (3.12)
Modeling of Stewart Platform 57
FIGURE 3.3
Block diagram of the LMS adaptive filter.
Equation (3.12) is the well-known LMS algorithm that is suitable for practical
signal processing applications because of its simplicity and the availability of the in-
stantaneous error in real time. The block diagram of the LMS algorithm is illustrated
in Figure 3.3.
Normalized LMS algorithm
The LMS adaptation process is very much dependant on the step size µ and the
reference signal power. The step size determines the system convergence rate and
stability [36][37]. The maximum stable step size is inversely proportional to the filter
order L and the power of the reference signal x(k). A technique used to optimize
the convergence speed, independent of the reference signal power, is known as the
normalized LMS algorithm (NLMS).
The NLMS algorithm is given as follows.
W (k + 1) = W (k) + µ(k)X(k)e(k), (3.13)
where each variable is identical to the one for the LMS algorithm except for µ(k)that is an adaptive step size computed as
µ(k) =α
LPx(k), (3.14)
where α is a normalized step size that satisfies 0 < α < 2, L is the filter length, and
Px(k) is the estimated power of the reference signal x(k).The estimation of Px(k) can be done using the mean square value of the reference
signal as follows.
Px(k) =XT (k)X(k)
L. (3.15)
58 Modeling and Control of Vibration in Mechanical Systems
Then Equation (3.14) reduces to
µ(k) =α
XT (k)X(k). (3.16)
This step size is the most widely used for the NLMS algorithm. A variant of the
NLMS algorithm uses a small constant ε as follows.
µ(k) =α
ε + XT (k)X(k). (3.17)
This constant ensures that the step size does not tend to infinity in the case of a
zero input signal.
The NLMS algorithm guarantees an attenuation level of γ ≤ 1, where γ is the
induced-norm from noise or signal inputs to the filtering error. Therefore, a salient
feature of this algorithm is that it lowers the influence of the input signal on the noise
amplification effect, especially when the input signal x(k) is large.
3.3.2 Modeling of a Stewart platform
Adaptive identification is a technique that uses an adaptive filter to model an un-
known system. An adaptive identification method can be applied either online along
with the vibration control or offline prior to the vibration control. In an online identi-
fication system, the number of adaptive filters required for an adaptive control system
will be increased. Furthermore, convergence of the adaptive filter used in the identi-
fication section of the system can be affected by a large amount of primary vibration
signal [36]. Therefore an offline identification technique will be applied for the sys-
tem modeling.
The block diagram of the adaptive identification is illustrated in Figure 3.4. P (z)is the transfer function of the system to be identified. W (z) is a digital filter used
to model P (z) based on the LMS error minimization algorithm. Both systems P (z)and W (z) are excited by a band limited white noise. The difference between the
two outputs d(k) and y(k) is fed back into the LMS algorithm as error signal e(k).The LMS algorithm will adaptively adjust the coefficients of filter W (z) to minimize
e(k) based on the least MSE criterion. When the error signal reaches the minimum
level, the filter W (z) represents a model of P (z).The LMS adaptive filter approach is applied for the modeling of the Stewart plat-
form. A Simulink program is developed for offline identification of the platform.
A 16th order LMS adaptive filter with adaptation step size of 0.01 is chosen. A
band-limited white noise is used as the training signal. Sampling frequency of the
white noise generator is set at 1 kHz so that the response of the PZT actuator will be
confined to the bandwidth of 500 Hz (half of sampling frequency). But the sampling
frequency of the adaptive filter is set to 10 kHz to give enough time for adaptive filter
to compute the filter weights within each sample of the training signal.
The Simulink program for the identification is downloaded into a dSPACE real
time interface board DS1104. Identification process for each PZT actuator is per-
formed alternatively. Filter tap values are recorded through the dSPACE manager
Modeling of Stewart Platform 59
FIGURE 3.4
System identification using LMS adaptive filter.
software during the identification process. From the results of the identification of
six PZT actuators of the Stewart platform in Figure 1.2, one of the resonance fre-
quencies of the smart structure is found to be around 240 Hz. Figure 3.5 is the phase
and magnitude responses of one of the six PZT actuators. The result shows that there
is a resonance peak at about 240 Hz. The phase response between 60 Hz and 200Hz is approximately linear. Experimental results of the active vibration control sys-
tem also indicate that the frequency region that can attenuate the vibration signal is
between 60 Hz and 220 Hz.
By applying a frequency domain modeling approach, an approximate model in
terms of a transfer function for each leg can be obtained. Figure 3.6 shows the fre-
quency responses of an PZT actuator, including the estimated and the experimental
ones. It is clearly shown that the estimated transfer function is well fitted to the real
one. It is also noted that the shape of the frequency responses in Figure 3.6 agrees
with that of the model in (3.3).
The transfer function of the PZT actuator is obtained as:
P (s) =
−190.9s4 − 1.499× 104s3 − 5.703× 108s2 − 2.507× 1010s− 5.624× 1011
s5 + 3232s4 + 5.378× 106s3 + 1.621× 1010s2 + 6.82× 1012s + 1.943× 1016.
(3.18)
60 Modeling and Control of Vibration in Mechanical Systems
FIGURE 3.5
Frequency responses of a PZT actuator.
Modeling of Stewart Platform 61
FIGURE 3.6
Estimated and experimental frequency responses.
62 Modeling and Control of Vibration in Mechanical Systems
3.4 Conclusion
This chapter has studied the modeling of an active piezoelectric Stewart platform.
The LMS adaptive filtering approach has been adopted for the system identification.
The model obtained via the adaptive identification or the experimental testing has
shown consistency with the governing motion equation.
4
Classical Vibration Control
4.1 Introduction
The presence of vibration often leads to undesirable effects such as structural or
mechanical failure, frequent and costly maintenance of machines, worsening posi-
tioning performance, and human pain and discomfort. Vibrations can sometimes be
eliminated theoretically. However, due to the high manufacturing cost that may be
involved in eliminating vibration, reduction of vibration is preferred so as to achieve
a compromise between an acceptable amount of vibration and a reasonable manu-
facturing cost. Various classical techniques of vibration control for the purpose of
vibration reduction have been presented, such as balancing of rotating and recipro-
cating machine, control of natural frequency, damping and stiffness modification,
isolators, and absorbers. Some of the techniques will be introduced briefly in this
chapter.
An active vibration control is required for a system if it needs an external power
to perform its function. Examples of some active vibration control include, (1) using
hybrid mass dampers to apply a control force to a movable mass so as to reduce
building sway caused by wind and seismic waves, (2) reducing aircraft cabin noise
by attenuating the vibration of the large panels of thin metal that form the cabin
walls, (3) damping out vibrations using piezoelectric devices installed on the trailing
edge of helicopter blades, etc.
4.2 Passive control
4.2.1 Isolators
Vibration isolation methods are used to reduce the undesired effects of vibration. It
involves the insertion of a resilient member called an isolator between the vibrating
mass and the source of vibration so that a reduction in the dynamic response of
the system is achieved under specified conditions of vibration excitation [4]. An
isolation system is said to be active or passive depending on whether or not external
power is required for the isolator to perform its function. A passive isolator consists
63
64 Modeling and Control of Vibration in Mechanical Systems
of a resilient member, i.e., stiffness, and an energy dissipator, i.e., damping. Typical
examples of passive isolators include metal springs, cork, pneumatic springs, and
rubber springs.
4.2.2 Absorbers
If a system is acted upon by a force whose excitation frequency nearly coincides with
its natural frequency, the vibration of the system can be reduced by using a dynamic
vibration absorber, which is simply a spring-mass system. We consider an auxiliary
mass m2 attached to a machine of mass m1 through a spring of stiffness k2. The
motion equations of the masses m1 and m2 are written as
m1x1 + k1x1 + k2(x1 − x2) = F0sinωt, (4.1)
m2x2 + k2(x2 − x1) = 0, (4.2)
where x1 and x2 represent displacements of m1 and m2, respectively. The steady-
state amplitudes of the mass m1 and m2 are given by
X1 =(k2 − m2ω
2)F0
(k1 + k2 − m1ω2)(k2 − m2ω2) − k22
, (4.3)
X2 =k2F0
(k1 + k2 − m1ω2)(k2 − m2ω2) − k22
. (4.4)
Equation (4.3) implies that if
ω2 =k2
m2,
the amplitude X1 of the machine m1 will be zero. Consider the machine operat-
ing near its resonance ω2 ≃ k1/m1 before the addition of the dynamic vibration
absorber. If the absorber is designed such that
ω2 =k2
m2=
k1
m1, (4.5)
the vibration amplitude of the machine, while operating at its original resonant fre-
quency, will be zero.
The above mentioned dynamic vibration absorber removes the original resonance
peak in the response of the machine, but introduces two new resonance peaks. If
it is necessary to reduce the amplitude of vibration of the machine over a range of
frequencies, a damped dynamic vibration absorber can be used [5].
4.2.3 Resonators
Resonators are used in certain cases where it is easier and more efficient to locate
anti-vibration systems near the vibration source. The main idea is to create another
source of vibrations which will cancel out the original vibration if it is correctly
tuned. The principle is to use the kinetic energy in the resonant mass system, where
Classical Vibration Control 65
the mass is coupled to the vibrating source either by kinematic coupling or via a very
flexible link [8].
For example, in a rotating structure the stator is often excited by the rotor via
inertial effects such as imbalance or other effects such as the aerodynamic asymmetry
of the blades or flaps. Positioning a resonator on the rotor is to generate loads that
oppose the excitation loads. In the case of a helicopter, the resonant mass is placed
on the axis of the rotor hub. It is supported by three springs which allow it to vibrate
in a plane perpendicular to the axis of rotation. The motion of the counterweight is
mainly in-plane motion so that the system will generate in-plane loads to eliminate
the in-plane vibration.
4.2.4 Suspension
Suspension acts as the link between two structures that is designed in order to isolate
one of the structures from the other. This is a concept for structure isolation that is
achieved by canceling the variable loads transmitted to the support structure. The
characteristics of the link are determined by analyzing the dynamic behavior and vi-
brations without modifying its main function in static characteristics. In practice, the
link characteristics can be modified by varying its stiffness or appropriate positioning
of the system natural frequencies and its damping in terms of energy transfer. An ad-
ditional technique consists of introducing a flapping mass whose inertial effects will
neutralize the excitation inputs. Next we will briefly introduce stiffness modification
and damping modification.
Stiffness modification
The dynamic behavior of structures results from the exchange and dissipation of
energy. Dynamic forces transfer their energy to the structure, which then responds
via several mechanisms, such as bending or extension. Dynamic behavior can be
modeled in several ways. The best known is Newton’s second law of motion. If
the external force is static or quasi-static, structural stiffness forces develop to create
an equilibrium. External dynamic forces are balanced in a more complex way with
inertial and damping forces.
Stiffness, often schematically and conceptually represented by a spring, denotes
the capacity of a system to store strain energy. The stiffness force follows Hooke’s
law, Fs = ks∆x, where the stiffness constant ks is expressed in the unit of force per
unit length. This is a linear model where the spring displacement is measured from
the rest length. More complicated laws exist, for example, the nonlinear relation
F = k(x)∆xa, where the parameter a would depend on the particular material being
modeled, and the stiffness parameter k(x) is a function of how much the spring has
been elongated.
Damping modification
Damping defines the ability of a structure to dissipate energy. For an oscillatory
system, damping is a measure of how much energy is dissipated by the system during
an oscillation cycle. For example, structural connections between components add
damping to a structure.
66 Modeling and Control of Vibration in Mechanical Systems
Most systems possess damping to some extent, which is helpful in vibration con-
trol. If a system undergoes a forced vibration, its response or amplitude of vibration
near resonance tends to become large if there is no damping. The presence of damp-
ing always limits the amplitude of vibration. Damping can be modified in the system
to control its response, by the use of structural materials having high internal damp-
ing, such as laminated or sandwiched materials. An example is the use of viscoelastic
materials.
We know that the response amplitude of a system at resonance ω = ωn under
harmonic excitation F (t) = F0eiωt is given by
F0
kη=
F0
αEη, (4.6)
where η is the energy loss factor, and the stiffness k is proportional to the Young’s
modulus, i.e., k = αE with a constant α.
When viscoelastic materials are used for vibration control, they are subjected to
shear or direct strains. A simplest arrangement is that a layer of the viscoelastic
material is attached to an elastic one. Another arrangement is that a viscoelastic
layer is sandwiched between the elastic layers. The material with the largest loss
factor will be subjected to the smallest stress, while the stress is proportional to
the displacement. Hence viscoelastic materials having large loss factor are used to
provide internal damping for vibration control. Damping tapes, consisting of thin
metal foil covered with a viscoelastic adhesive, are used on vibrating structures.
Another application example is presented as follows with detailed discussion and
evaluation [14].
4.2.5 An application example − Disk vibration reduction via stackeddisks
To support higher track per inch (TPI) density hard disk drive, the positioning accu-
racy of test equipment such as spin stand and servo track writer has to be increased.
The Spin stand is commonly-used equipment for magnetic recording component test-
ing [10]. A servo track writer (STW) [25] is used to pattern the recording media with
servo information for the HDD servo system. In both cases, spindle motor and disk
vibrations [23] affect the positioning accuracy and limit how close adjacent tracks
can be placed together, and thus restrict the TPI number that can be achievable. In
[11], damped laminated disks were used to reduce the amplitude of rocking modes
by sandwiching a viscoelastic layer in between two aluminum layers to increase disk
damping. Reference [24] investigated the effect of suppressing resonance amplitude
of disk vibrations by applying a squeeze film damping. In addition to the laminated
disks and the squeezed air bearing plate methods, in this section we discuss an ap-
proach with minimal mechanical alternation. That is to stack more than one normal
recording media together for reading and writing on the disk surfaces.
The Guzik spin stand shown in Figure 4.1 is used to evaluate the effectiveness of
dual-disk stack in position accuracy improvement. The disk vibrations in the axial
Classical Vibration Control 67
direction are also measured at the outer diameter (OD) region of the disk via LDV.
In the experiment, the spindle motor is spinning at 7200, 8400 and 10200 RPM
respectively, and the disk is an aluminum disk of 3.5-inch in diameter and 0.8 mm in
thickness. Two such disks are stacked together and mounted on the spindle motor of
the spin stand. The experimental process is described as follows.
Step 1: Stack two disks together and mount on the spin stand;
Step 2: Spin the motor and write servo information on the top disk surface;
Step 3: Read back the written information and obtain the PES.
To write servo information on the disk surface in between the two disks, the spin-
dle motor is stopped and the written surface of the disk is flipped over, following
which, steps 1, 2 and 3 are repeated.
FIGURE 4.1
Disk and spindle motor assembly of the spin stand.
The disk vibration in the axial direction of the single disk and the dual-disk stack
are measured via LDV. The position error signals are also collected for both the
single disk and the dual-disk stack cases. To evaluate the improvement of position
accuracy, 20 traces of the position error signal are collected and denoted by PESi
68 Modeling and Control of Vibration in Mechanical Systems
(i = 1, · · · , N). N = 20. The repeatable runout (RRO) and the nonrepeatable
runout (NRRO) are given by
RRO =1
N
N∑
i=1
PESi, NRROi = PESi − RRO. (4.7)
A. Positioning accuracy improvement at 7200 RPM
Tested disk vibration results of the dual-disk and the single disk are shown in Fig-
ure 4.2. Besides the harmonics with respect to the spindle speed, the disk vibration
modes can be seen clearly as denoted by T1-T9. It is found that the frequencies of the
disk vibration modes T1-T9 are shifted and their amplitudes are apparently reduced.
The vibration modes are all shifted by 40 Hz to higher frequencies. The power spec-
tra of the RRO and NRRO, computed from the collected PES data, are shown in
Figure 4.3 and Figure 4.4. As seen in Figure 4.3, except for the 1st harmonic, other
dominant harmonics before the 11th harmonic are all reduced. In Figure 4.4, we
can observe that almost all the reduced disk vibration modes T1-T9 in Figure 4.2
are reflected in NRRO. The other peaks indicated by S1-S4 can be identified to be
caused by slider vibrations. They are both lowered significantly, which may be due
to better slider-disk interaction. It is noted that the peak S2 for the single disk case is
shifted to S3 in the dual-disk case. This randomly happens and the writing process
at a different time may cause such a shifting. This phenomenon can be seen in other
testing results which will be shown later. The standard deviations (or σ values) of
RRO, NRRO and PES are obtained. It turns out that the σ value of RRO is reduced
by 23%, NRRO by 18%, and the PES σ value is reduced by 21%. Figure 4.5 shows
the time sequences of PES in both the cases and the amplitude reduction is seen
apparently.
B. Positioning accuracy improvement at 8400 RPM
The disk or the dual-disk stack is rotating at 8400 RPM, and the disk vibrations
in the axial direction are shown in Figure 4.6. It can be seen that the amplitudes of
the vibration modes are all reduced with shifts in frequencies, as indicated by T1-T6.
The power spectra of the RRO and NRRO are calculated from the tested PES and
shown in Figure 4.7 and Figure 4.8. Figure 4.7 shows that almost all the first seven
harmonics are decreased and as a result, the σ value of RRO is reduced by 41%.
In Figure 4.8, the corresponding T1-T6 in Figure 4.6 can be found and they are all
suppressed in the case of the dual-disk stack. Notice that S1 is reduced significantly,
but can not be traced in the tested disk vibrations in Figure 4.6. This further verifies
that S1 is caused by the disk-slider interaction. The corresponding S4 cannot be
found in this case, while T1 appears very close to S4. The resultant σ value of
NRRO is improved by 28%, and PES by 38%. Figure 4.9 shows the comparison of
single-disk and dual-disk PES in time domain, and the amplitude of PES in the case
of the dual-disk stack approach is much lower as compared to conventional single
disk approach.
Classical Vibration Control 69
0 500 1000 1500 2000 2500 3000−9
−8.5
−8
−7.5
−7
−6.5
−6
−5.5
−5
−4.5
Frequency(Hz)
Dspla
cem
ent am
plit
ude(log10(m
))
2−disk1−disk
T5 T6 T7
T8 T9
T4 T3
T2 T1
FIGURE 4.2
Comparison of single-disk and dual-disk axial vibrations measured via LDV at 7200RPM.
70 Modeling and Control of Vibration in Mechanical Systems
0 500 1000 1500 2000 2500 3000 3500 40000
0.05
0.1
Frequency(Hz)
FF
T(R
RO
)(V
rms)
1−disk2−disk
FIGURE 4.3
Comparison of single-disk and dual-disk RRO power spectrum at 7200 RPM (23%improvement of σ value).
Classical Vibration Control 71
0 500 1000 1500 2000 2500 3000 3500 40000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Frequency(Hz)
FF
T(N
RR
O)(
Vrm
s)
2−disk1−disk
T8 T9
T7 T6
T4
T3
T2 T1
S1
S2
S3
S4
FIGURE 4.4
Comparison of single-disk and dual-disk NRRO power spectrum at 7200 RPM (18%reduction of σ value).
72 Modeling and Control of Vibration in Mechanical Systems
FIGURE 4.5
Comparison of single-disk and dual-disk PES in time domain at 7200 RPM (21%reduction of σ value).
Classical Vibration Control 73
0 500 1000 1500 2000 2500 3000−9
−8.5
−8
−7.5
−7
−6.5
−6
−5.5
−5
Frequency(Hz)
Dspla
cem
ent am
plit
ude(log10(m
))
2−disk1−disk
T1
T2 T3
T4
T5 T6
FIGURE 4.6
Comparison of single-disk and dual-disk axial vibrations measured via LDV at 8400RPM.
74 Modeling and Control of Vibration in Mechanical Systems
0 500 1000 1500 2000 2500 3000 3500 40000
0.05
0.1
Frequency(Hz)
FF
T(R
RO
)(V
rms)
1−disk2−disk
FIGURE 4.7
Comparison of single-disk and dual-disk RRO power spectrum at 8400 RPM (41%improvement of σ value).
Classical Vibration Control 75
0 500 1000 1500 2000 2500 3000 3500 40000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Frequency(Hz)
FF
T(N
RR
O)(
Vrm
s)
1−disk2−disk
T6
T4 T3
T2
T1
T5
S1
S2(S3)
FIGURE 4.8
Comparison of single-disk and dual-disk NRRO power spectrum at 8400 RPM (28%reduction of σ value).
76 Modeling and Control of Vibration in Mechanical Systems
FIGURE 4.9
Comparison of single-disk and dual-disk PES in time domain at 8400 RPM (38%reduction of σ value).
Classical Vibration Control 77
C. Positioning accuracy improvement at 10200 RPM
A higher rotational speed of 10200 RPM is performed on both the dual-disk stack
and single disk cases. Figure 4.10 shows the obtained disk axial vibration by LDV.
The modes T1-T6 are quite obvious with the amplitude reduced and the frequencies
shifted higher. Figure 4.11 and Figure 4.12 show the power spectrum of RRO and
NRRO. The 3rd and 5th harmonics are significantly reduced as seen in Figure 4.11.
Figure 4.12 reflects the corresponding disk vibration modes T1-T6 in Figure 4.10.
As for the slider related vibrations, S1 is reduced significantly again and S2 and S3
are shifted, while S4 cannot be seen at all in Figure 4.12. As a result, the σ value of
RRO is improved by 33%, NRRO by 27%, and PES by 32%, compared with that of
the single disk approach. Figure 4.13 shows the comparison of PES in time domain,
and it is observed that the amplitude of PES in the case of the dual-disk stack is
decreased.
0 500 1000 1500 2000 2500 3000−9
−8.5
−8
−7.5
−7
−6.5
−6
−5.5
−5
−4.5
Frequency(Hz)
Dspla
cem
ent am
plit
ude(log10(m
))
2−disk1−disk
T3
T5
T4
T6
T2
T1
FIGURE 4.10
Comparison of single-disk and dual-disk axial vibrations measured via LDV at
10200 RPM.
The reduction of disk vibration amplitude is evaluated from Figures 4.2, 4.6 and
4.10 and tabulated in Table 4.1 for different rotational speeds. The improvements of
78 Modeling and Control of Vibration in Mechanical Systems
0 500 1000 1500 2000 2500 3000 3500 40000
0.05
0.1
0.15
0.2
Frequency(Hz)
FF
T(R
RO
)(V
rms)
1−disk2−disk
FIGURE 4.11
Comparison of single-disk and dual-disk RRO power spectrum at 10200 RPM (the
3rd and 5th harmonics reduced significantly, 33% improvement of σ value).
Classical Vibration Control 79
0 500 1000 1500 2000 2500 3000 3500 40000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Frequency(Hz)
FF
T(N
RR
O)(
Vrm
s)
1−disk2−disk
T1 T2 T3 T4
T6 T5
S1
S2 S3
FIGURE 4.12
Comparison of single-disk and dual-disk NRRO power spectrum at 10200 RPM
(27% reduction of σ value).
80 Modeling and Control of Vibration in Mechanical Systems
FIGURE 4.13
Comparison of single-disk and dual-disk PES in time domain at 10200 RPM (32%reduction of σ value).
Classical Vibration Control 81
TABLE 4.1
% reduction of σ values of PES, RRO and NRRO and disk
vibration amplitude with stacked disks compared with single disk
Speed(RPM) RRO NRRO PES Disk vibration amplitude
7200 23 18 21 24
8400 41 28 38 45
10200 33 27 32 35
the σ values of RRO, NRRO and PES are also summarized in Table 4.1. Remarkable
improvement is achieved for every rotational speed. It can be seen clearly from the
table that the best improvement is obtained when operating at 8400 RPM. The second
best result obtained is performed at 10200 RPM, where the dual-disk stack approach
leads to an improvement of 32% in positioning accuracy.
For all the cases studied, we have observed a remarkably reduced amplitude of the
disk vibration and its reflection in positioning accuracy improvement via the dual-
disk stack. This indicates that the stacked disk is a simple way of reducing the disk
vibrations in spin stand tests, particularly when a single disk surface is accessed.
In Figures 4.2, 4.6, and 4.10, we have also observed the frequency shifting of disk
vibration modes.
Theoretically the maximum amplitude of disk vibration is given by [22]
V ∝ ρR4ω2
t3Ed, (4.8)
where R is the disk outer radius, ρ is the gas density, ω is the angular speed, t is
the disk thickness, E is the Young’s modulus of the disk substrate, and d is the
disk substrate damping. Equation (4.8) indicates that the disk vibration is inversely
proportional to t3. In the dual-disk case, the disk stack thickness t = 2t. Hence
assuming that the change of E is negligible, the amplitude of disk vibration
V ∝ ρR4ω2
8t3Ed, (4.9)
which means that V < V when the damping d > 18d. The experimental results in
Figures 4.2, 4.6 and 4.10 verified the amplitude reduction of the disk vibration.
On the other hand, it is known that the natural frequency of disk vibration mode is
defined by [23]
fmn =λ2
mn
2πR2
[
Et3
12γ(1 − ν2)
]1/2
, (4.10)
where fmn is the natural frequency of the (m, n) mode, γ is the mass per unit area of
the disk, ν is the Poisson’s ratio, and λmn is the dimensionless frequency parameter
which is generally a function of the boundary conditions on the plate, the ratio of the
82 Modeling and Control of Vibration in Mechanical Systems
inner to outer diameter, and Poisson’s ratio. Denoting Ω as the rotational speed of
the disk, the frequency of the disk vibration mode is given by [23]
f = fmn ± nΩ. (4.11)
Assuming that the change of ν is also negligible, and considering that t = 2t,
fmn = 2√
2λ2
mn
2πR2
[
Et3
12γ(1 − ν2)
]1/2
, (4.12)
which implies some shifting of the disk vibration modes, as can be seen in the ex-
perimental results in Figures 4.2, 4.6 and 4.10.
In addition to the doubled thickness, the mechanical properties such as d and λmn
[22][23] are certain to change correspondingly after stacking two disks. Further
research for details on how the stacked disks change these properties would be sig-
nificant.
To summarize, the proposed process is indeed an approach to reduce the disk
vibration on both disk surfaces for reading and writing, such as multi-disk servo
track writers [25]. Thus such a disk vibration reduction approach can be used as an
alternative or complementary technique to the air shroud design [24] and the active
control approaches to further improve positioning accuracy.
4.3 Self-adapting systems
In certain applications, system dynamic parameters vary with time. For example,
the mass of a system like a car, airplane or helicopter decreases as it consumes fuel.
Distinct flight conditions such as flight level, maneuvers, or landing produce different
types of excitation. When the initial characteristics of the system no longer meet the
requirements of the system’s working conditions, the system characteristics therefore
need to adapt to the parameter variations of the excitation and the system itself. Two
ways are possible: one is to change the stiffness of the structure; the other is to add
moving masses so that their inertial effects counteract the effects of the excitation
variation. Some self-adapting systems in use are briefly introduced as follows [8].
Self-tuning suspension
Self-tuning suspension systems are equipped suitably for slow variation of the sys-
tem characteristics. One method, for example, used in self-tuning systems involves
analyzing the system’s vibration frequency and its timewise variation. The tuning
is usually performed by an actuator. After the system status is measured, the tuning
system will modify the feature of the system, such as stiffness, damping and position
of a mass.
The self-tuning system is tuned by a designed control algorithm. Simulations
using the control algorithm are required to identify the setting parameters of the
Classical Vibration Control 83
algorithm and to verify the reliability of the system. Subsequently, the algorithm
must be validated by experimental tests on the real structure. The car suspension
is an example of self-tuning suspensions. It is able to adapt to various operating
configurations depending on the type of driving, the driving conditions, or the desired
comfort. One method to achieve this is to modify the suspension stiffness or its
damping using various techniques.
Self-adjusting absorbers
The mechanical resonator, which is basically a mass-spring system, is used to con-
trol certain vibration sources. It is placed where the vibrations are to be reduced. The
principle is that the resonance frequency of the mass-spring system as a resonator is
adjusted with the excitation frequency in operation, and as the mass vibrates, the
vibratory level is decreased.
In certain structures, the excitation frequency evolves with time gradually. These
changes take place over a long period, compared to the excitation frequency. As a
result, it is necessary to create a system whose resonance frequency would adjust
automatically to the variations of the excitation frequency. For a 20 Hz excitation
frequency, this variation usually takes place between 1 and 3 seconds. This kind of
system actually works as the case of self-adjusting absorbers.
Self-adapting resonator
Self-adapting resonators are capable of adjusting to changes in the excitation fre-
quency. The hub resonator used in a helicopter is an example. It consists of a flapping
mass that vibrates in a plane perpendicular to the rotation axis of the rotor. The mo-
bile flapping mass is supported by three flexible elements indexed at 120, and it
slides along the rotor axis. The stiffness of the three elements is designed so that
the resulting anti-resonance corresponds to the excitation frequency. The required
position of the mobile mass is determined by control algorithms through an actuator
and the mobile mass is moved by the actuator. The z position of the mass makes the
variation of the inertia of the assembly and thus the anti-resonance frequency varies
accordingly.
4.4 Active vibration control
The self-adapting systems presented previously will not be sufficient in the cases
where the source characteristics vary too fast for the involved algorithms or the re-
quired level of performance is too high. Active methods should be used to decrease
the vibrations.
A vibration control system is called active if it uses external power to perform
its function. It is comprised of a servomechanism with an actuator, a sensor, and a
microprocessor-based system. The actuator applies a force to the mass whose vi-
bration is to be reduced. The sensor measures the motion of the mass in terms of
displacement, velocity, or acceleration, depending on the application. The micropro-
84 Modeling and Control of Vibration in Mechanical Systems
cessor based system consists of analog-to-digital converters to process sensor inputs
and digital-to-analog converters to convert the microprocessor’s output command
into an input signal to the actuator. The control logic, called the control algorithm,
programmed in the computer uses sensor measurement to decide how much force
the actuator should apply.
4.4.1 Actuators
Piezoelectric materials can convert electrical current into motion, and vice versa.
They change shape when an electric current passes through them, and they generate
an electric signal when they flex. Thus they can be used as actuators to create force
or motion, and as sensors to sense motion.
The materials used for high-precision actuation include electrostrictive and mag-
netostrictive materials, which are similar to piezoelectric materials. These are ferro-
magnetic materials that expand or contract when subjected to an electric or a mag-
netic field.
The previously mentioned voice coil motor (VCM) actuator in hard disk drives is
a linear actuator moving in one direction. Because of its similarity to a loudspeaker,
it is referred to as a voice coil motor. A VCM actuator has a coil of wire rigidly
attached to the structure and suspended in a permanent magnetic field. It is driven
when a force is produced to accelerate it radially as a current is passed through the
coil.
Some actuators cannot provide enough force for larger applications such as vehi-
cle suspension and control of building motions. In building application, hydraulic
cylinders are usually used. With a working hydraulic pressure of commonly 2000
psi, a cylinder containing a piston whose area is only 1 square inch will generate 1
ton of force. Active vehicle suspensions use hydraulic devices, electric motors, and
magneto-rheological fluid dampers.
4.4.2 Active systems
An active system is applied to decrease vibrations by introducing dynamic loads
locally in the structure [8]. The dynamic loads are controlled by a processor in order
to minimize the vibratory level. The technology of generating loads is fundamental
in control strategy.
Active suspension
Several active principles are used for suspensions to isolate one structure from an-
other. One of them is internal load control to modify the distribution of internal loads
in the structure. Its principle is to inject a set of dynamic loads into the structure in
order to minimize its vibratory response. The loads depend on the vibratory condi-
tion of the structure, which is sensed with the help of a set of accelerometers or strain
gauges. Hydraulic or electro-dynamic actuators suitably located on the structure are
used to inject the loads. The role of the actuators is to modify the distribution of
vibratory energy for different modes to minimize the structure vibrations, instead of
dispersing the vibratory energy of the structure.
Classical Vibration Control 85
Active resonator
An active resonator works by displacing a mass with an actuator. It uses the dy-
namic amplification of the mass to generate high loads using minimal energy. Differ-
ent types of actuator technologies such as hydraulic, electromagnetic, or piezoelec-
tric actuators, have been used to match the fields of application in terms of forces and
frequencies. We take an electromagnetic actuator as an example. The type of single
stage resonator uses the principle of single stage mechanical systems with displace-
ment operated by an electromagnetic force. The motion equation of the mass M of
a permanent magnet for small motion around its static position is given by
Mx1 + c1x1 + k1x1 = FV (t). (4.13)
With FV (t) = F0cos(ωt), the Laplace transform of the displacement produced by
the mass is
X1(s) =1
Ms2 + c1s + k1F0, (4.14)
where k1 and c1 are respectively the stiffness and damping of the mass.
The load transmitted to the structure equals to
FT (t) = c1x1 − k1x1 − FV (t). (4.15)
The mechanical parameters such as mass and stiffness are designed so that the
magnetic loads remain small and compatible with an acceptable energy consump-
tion. The natural frequency of the mass-spring system must be designed as close as
possible to the frequency of the vibration that needs to be controlled.
A two-stage electromagnetic resonator uses two masses called stages [8]. The
control force FV is introduced between the two masses by an electromagnetic load.
It is produced from a voltage V that is fed to a power amplifier to generate a magnetic
force via the current in the coil. The role of the amplifier is to ensure the law between
the generated force FV and the control voltage V is linear. The function of control
consists of tuning the system parameters such as mass and stiffness so that at the
control frequency, the force generated on the structure is amplified with respect to
the force FV , while maintaining a small relative displacement of the two stages.
The motion equations of the two masses are written as
M1x1 + c1(x1 − x2) + k1(x1 − x2) = FV (t), (4.16)
M2x2 + c1(x2 − x1) + c2x2 + k1(x2 − x1) + k2x2 = −FV (t), (4.17)
with the primary stage M2 and the secondary stage M1.
The load transmitted to the structure is
FT = c2x2 + k2x2. (4.18)
86 Modeling and Control of Vibration in Mechanical Systems
4.4.3 Control strategy
When the system response is not acceptable, the value of the response is used to
generate additional forces according to rules or laws such that the modified response
behaves according to the design and within certain bounds. This results in a closed-
loop system that incorporates feedback control, where the response is evaluated by a
sensor and is fed back to an actuator that generates a force or motion. The purpose of
designing a system with feedback force is to minimize unwanted behaviors and elim-
inate the effect of vibration on system performance in a desired manner. Feedback
control provides a mechanism for tailoring system behavior to specific standards and
needs. The block diagram depicting feedback control is in Figure 4.14. One can see
that the signal from a block goes concurrently to other blocks or a summing point.
The summing point indicates that the input and output signals are compared. The
difference is an error that generates a control action. The disturbance or forcing is
also shown being input directly to the plant or being injected to the output of the
plant.
FIGURE 4.14
Generic feedback control system.
In order to properly design a feedback control system, performance must be de-
fined in terms of system specifications. Standard performance measures are usually
defined in terms of the step response. The general design objectives are the speed of
response, stability, and accuracy or allowable error. The first objective implies that it
is desirable for a control system to respond to a reference command rapidly. Stability
is the prerequisite of any active control system design. The allowable error represents
how close to the desired response the control force must bring the structure and how
much the control system can reject vibrations.
The controller produces control action or signal by using the comparison signal
of plant output with the desired reference value. There are a number of control
Classical Vibration Control 87
methods used by people to design control actions. Classical control is represented
by the well-known proportional-integral-derivative (PID) control. Advanced control
techniques include robust and optimal controls, which will be introduced briefly in
the next chapter.
An adaptive control system is one that can change or adapt either its gain values
or even its control algorithm to accommodate changing conditions. In mathematics
how the control gains or the control algorithm should adapt to changing conditions
is difficult. There are thus fewer practical applications of adaptive controls.
4.5 Conclusion
Various classical techniques of vibration control were discussed in this chapter and
some typical techniques such as using isolators, absorbers, a resonator, and suspen-
sion were introduced individually. Specifically, disk vibration reduction via stacked
disks as an application example was presented with detailed discussion and evalu-
ation. Subsequently, active vibration control was introduced with actuators, active
systems, and control strategies.
5
Introduction to Optimal and Robust Control
5.1 Introduction
This chapter intends to briefly review the optimal and robust control. H2 and H∞norms and their calculation are first introduced since they are two commonly used
measures of performance specifications of a control system. What follow are the
H2 and H∞ control problems and controller design via a linear matrix inequality
approach. Robust control of systems with multiplicative uncertainty and additive un-
certainty, and the parametrization of all stable controllers, i.e., the so-called Youla
parametrization, will be investigated. Moreover, performance limitation of a feed-
back control system is discussed, which is necessary to help understand the vibration
control schemes in the later chapters.
5.2 H2 and H∞ norms
5.2.1 H2 norm
The H2 norm of a matrix transfer function G(s) analytic in Re(s) > 0 (open right-
half plane) is defined as
‖G‖2 :=
√
supσ>0
1
2π
∫ +∞
−∞Trace[G∗(σ + jω)G(σ + jω)]dω, (5.1)
or equivalently [2]
‖G‖2 =
√
1
2π
∫ +∞
−∞Trace[G∗(jω)G(jω)]dω. (5.2)
Although ‖G‖2 can be computed from its definition, there are some simple alter-
natives taking advantage of a state-space representation of G(s).
LEMMA 5.1
[6] Consider a system G(s) with a state-space representation (A, B, C, D). If
89
90 Modeling and Control of Vibration in Mechanical Systems
A is stable and D = 0, then we have
‖G‖22 = Trace(BT Y2B) = Trace(CX2C
T ) (5.3)
where X2 and Y2 are the controllability and observability Gramians that canbe obtained from the following Lyapunov equations:
AX2 + X2AT + BBT = 0, (5.4)
AT Y2 + Y2A + CT C = 0. (5.5)
We also consider a discrete-time linear time-invariant system G(z) with the fol-
lowing state-space representation:
x(k + 1) = Ax(k) + Bw(k), (5.6)
z(k) = Cx(k) + Dw(k), (5.7)
where x ∈ Rnx is the state, z ∈ Rnz is the controlled output, w ∈ Rnw is the
disturbance input. Let Tzw denote the transfer function from the input w to the
output z. Then the H2 norm is defined as
‖Tzw‖2 =
√
1
2πTrace[
∫ π
−π
T ∗zw(ejω)Tzw(ejω)dω]. (5.8)
By Parseval’s theorem, ‖Tzw‖2 can equivalently be defined as
‖Tzw‖2 =
√
√
√
√Trace[
∞∑
k=0
g(k)gT (k)], (5.9)
where g(k) is the impulse response of Tzw .
Let the input w to the system be a wide-band stationary stochastic process. The
H2 norm of Tzw can also be interpreted as the RMS value of the output z(k) when
the system is subject to a white noise having zero mean and unit variance. That is
‖Tzw‖2 =√
E[zTz]. (5.10)
The H2 norm for the discrete-time system Tzw can be computed as
‖Tzw‖2 =√
Trace(DT D + BT Y2B) =√
Trace(DDT + CX2CT ), (5.11)
where Y2 and X2 are the reachability and observability Gramians that can be ob-
tained from the following Lyapunov equations:
AY2AT − Y2 + BBT = 0, (5.12)
AT X2A − X2 + CT C = 0. (5.13)
Introduction to Optimal and Robust Control 91
The following theorem presents an alternative LMI condition for bounding the H2
norm of the discrete-time system Tzw.
LEMMA 5.2
Consider a discrete-time transfer function Tzw of realization (A, B, C, D).Given a scalar µ > 0, ‖Tzw‖2
2 < µ if and only if there exist X2 = XT2 and
Y2 = Y T2 such that Trace(Π) < µ and
Π CX2 DX2C
T X2 0DT 0 I
> 0, (5.14)
Y2 AY2 BY2A
T Y2 0BT 0 I
> 0. (5.15)
Observe that (5.14) and (5.15) are linear in X2 and Y2, and hence can be solved
by employing the LMI Tool [69] in MATLAB. The H2 norm of the system can
be computed by minimizing µ using the function mincx in MATLAB Optimization
Toolbox.
5.2.2 H∞ norm
The H∞ norm of a matrix transfer function G(s) that is analytic and bounded in the
open right-half plane is defined as [3]
‖G‖∞ := supRe(s)>0
σ[G(s)] = supω∈R
σ[G(jω)] = supw 6=0
‖z‖2
‖w‖2, (5.16)
where w and z are respectively the input and output of G(s). A control engineering
interpretation of the infinity norm of a scalar transfer function G(s) is the distance in
the complex plane from the origin to the farthest point on the Nyquist plot of G, and
it also appears as the peak value on the magnitude plot of |G(jω)|. Hence the H∞norm of a transfer function can, in principle, be obtained graphically.
In general, the H∞ norm of a stable matrix transfer function can be read directly
from its singular value plots.
The H∞ norm can also be computed in state-space.
LEMMA 5.3
[6] Let γ > 0 and G(s) : (A, B, C, D) with A stable. Then ‖G(s)‖∞ < γ ifand only if σ(D) < γ and the Hamiltonian matrix H has no eigenvalues onthe imaginary axis, where
H :=
[
A + BR−1DT C BR−1BT
−CT (I + DR−1DT )C − (A + BR−1DT C)T
]
, (5.17)
92 Modeling and Control of Vibration in Mechanical Systems
and R = γ2I − DT D.
The following so-called Bounded Real Lemma gives a matrix inequality condition
for the system G(s) to have a pre-specified level of H∞ norm.
LEMMA 5.4
Continuous-time Bounded Real Lemma Consider a continuous-time transferfunction G(s) of realization G(s) = C(sI − A)−1B + D, where A is a stablematrix. Given a scalar λ > 0, the H∞ norm ‖G‖∞ < λ if and only if thereexists X = XT > 0 such that
AT X + XA XB CT
BT X − λI DT
C D − λI
< 0. (5.18)
In the discrete-time case, the H∞ norm of a stable transfer function matrix Tzw(z)with realization (A, B, C, D) is defined as
‖Tzw‖∞ = supω∈[0,2π]
σ[Tzw(ejω)] = supw 6=0
‖z‖2
‖w‖2, (5.19)
where w and z are respectively the input and output of Tzw.
The following discrete-time Bounded Real Lemma provides a linear matrix in-
equality condition for the system Tzw to have ‖Tzw‖∞ < γ.
LEMMA 5.5
Discrete-time Bounded Real Lemma Consider a discrete-time and a stabletransfer function matrix Tzw(z) with a state-space realization (A, B, C, D).Then, for some given λ > 0, ‖Tzw‖∞ < λ if and only if there exists X =XT > 0 such that
AT XA − X AT XB CT
BT XA BT XB − λ2I DT
C D − I
< 0. (5.20)
5.3 H2 optimal control
5.3.1 Continuous-time case
We consider the closed-loop system described by the block diagram in Figure 5.1.
The continuous-time linear time-invariant plant P (s) is described by the state-space
equations:
x(t) = Ax(t) + B1w(t) + B2u(t), (5.21)
Introduction to Optimal and Robust Control 93
z(t) = C1x(t) + D11w(t) + D12u(t), (5.22)
y(t) = C2x(t) + D21w(t) + D22u(t), (5.23)
where x ∈ Rnx is the state, y ∈ Rny is the measurement output, z ∈ Rnz is the
controlled output, w ∈ Rnw is the disturbance input, u ∈ Rnu is the control input,
and A, B1, B2, C1, D11, D12, C2, and D21 are of appropriate dimensions. We
assume D22 = 0 without loss of generality [89].
Introduce the following dynamic output feedback controller C(s):
xc(t) = Acxc(t) + Bcy(t), (5.24)
u(t) = Ccxc(t) + Dcy(t). (5.25)
Denote ξ = [xT xTc ]T . From (5.21)−(5.23) and (5.24)−(5.25), the closed-loop
system is given by
ξ(t) = Aξ(t) + Bw(t), (5.26)
z(t) = Cξ(t) + Dw(t), (5.27)
where
A =
[
A + B2DcC2 B2Cc
BcC2 Ac
]
, B =
[
B2DcD21 + B1
BcD21
]
, (5.28)
C = [C1 + D12DcC2 D12Cc] , D = D12DcD21 + D11. (5.29)
The continuous-timeH2 control problem is to find a proper, real rational controller
C(s) that stabilizes P internally and minimizes the H2 norm of the closed-loop
transfer function matrix Tzw from w to z in (5.26)−(5.27).
Assume that the system (5.21)−(5.23) satisfies the following conditions:
Assumption 5.1
(1). D12 is of full column rank;
(2). The subsystem (A, B2, C1, D12) has no invariant zeros on the imaginary axis;
(3). D21 is of full row rank;
(4). The subsystem (A, B1, C2, D21) has no invariant zeros on the imaginary
axis.
Let X2 ≥ 0 and Y2 ≥ 0 be the solutions of the following Riccati equations:
AT X2 + X2A − (X2B2 + CT1 D12)(D
T12D12)
−1(X2B2 + CT1 D12)
T
+CT1 C1 = 0, (5.30)
Y2AT + AY2 − (Y2C
T2 + B1D
T21)(D21D
T21)
−1(Y2CT2 + B1D
T21)
T
+B1BT1 = 0. (5.31)
According to the H2 optimal control theory, an H2 optimal controller can be ob-
tained as [7]
Ac = A + B2F + KC2, Bc = −K, Cc = F, Dc = 0, (5.32)
94 Modeling and Control of Vibration in Mechanical Systems
FIGURE 5.1
Configuration of standard optimal control.
where
F = −(DT12D12)
−1(DT12C1 + BT
2 X2),
K = −(Y2CT2 + B1D
T21)(D21D
T21)
−1. (5.33)
The minimal H2 norm of the transfer function Tzw is given by
‖Tzw‖2 =√
Trace(BT1 X2B1) + Trace[(AT X2 + X2A + CT
1 C1)Y2].(5.34)
If the conditions (1)−(4) in Assumption 5.1 are not satisfied, the so-called pertur-
bation method is applied [7] so that the above design method to find an appropriate
controller is still applicable.
5.3.2 Discrete-time case
Consider the discrete-time linear time-invariant system P (z) with the following state-
space representation:
x(k + 1) = Ax(k) + B1w(k) + B2u(k), (5.35)
z(k) = C1x(k) + D11w(k) + D12u(k), (5.36)
y(k) = C2x(k) + D21w(k) + D22u(k), (5.37)
where x ∈ Rnx is the state, y ∈ Rny is the measurement output, z ∈ Rnz is the
controlled output, w ∈ Rnw is the disturbance input, u ∈ Rnu is the control input,
and A, B1, B2, C1, D11, D12, C2, and D21 are of appropriate dimensions. D22 = 0is also assumed.
Introduction to Optimal and Robust Control 95
Introduce the following dynamic output feedback controller C(z):
xc(k + 1) = Acxc(k) + Bcy(k), (5.38)
u(k) = Ccxc(k) + Dcy(k). (5.39)
Denote ξ = [xT xTc ]T . From (5.35)−(5.37) and (5.38)−(5.39), the closed-loop
system is given by
ξ(k + 1) = Aξ(k) + Bw(k), (5.40)
z(k) = Cξ(k) + Dw(k), (5.41)
where
A =
[
A + B2DcC2 B2Cc
BcC2 Ac
]
, B =
[
B2DcD21 + B1
BcD21
]
, (5.42)
C = [C1 + D12DcC2 D12Cc] , D = D12DcD21 + D11. (5.43)
The discrete-time H2 control problem is to find a proper, real rational controller
C(z) that stabilizes P (z) internally and minimizes the H2 norm of the transfer func-
tion matrix Tzw(z) from w to z of the closed-loop system (5.40)−(5.41).
The counterpart of the Riccati equations (5.30)−(5.31) for discrete-time systems
is as follows.
AT X2A − (AT X2B2 + CT1 D12)(D
T12D12 + BT
2 X2B2)−1(AT X2B2 + CT
1 D12)T
+CT1 C1 = 0, (5.44)
AY2AT − (AY2C
T2 + B1D
T21)(D21D
T21 + C2Y2C
T2 )−1(AY2C
T2 + B1D
T21)
T
+B1BT1 = 0. (5.45)
A discrete-time H2 optimal controller can then be obtained as (5.32). And the
minimal H2 norm of the transfer function Tzw is given by
‖Tzw‖2 =√
Trace(BT1 X2B1) + Trace[(AT X2A + CT
1 C1)Y2]. (5.46)
A parametrization of all H2 controllers is developed in terms of LMIs as in the
following theorem which linearizes the H2 norm conditions (5.14)−(5.15) for syn-
thesis.
THEOREM 5.1
[90] Consider system (5.35)−(5.37). There exists a controller (5.38)−(5.39)such that ‖Tzw‖2
2 < µ if and only if the following linear matrix inequalitiesand equality admit a solution:
Trace(Π) < µ, (5.47)
96 Modeling and Control of Vibration in Mechanical Systems
Π C1X + D12E C1 + D12DcC2
∗ X + XT − P2 I + ZT − J∗ ∗ Y + Y T − H
> 0, (5.48)
P2 J AX + B2E A + B2DcC2 B1 + B2DcD21
∗ H U Y A + WC2 Y B1 + WD21
∗ ∗ X + X′ − P2 I + Z′ − J 0∗ ∗ ∗ Y + Y T − H 0∗ ∗ ∗ ∗ I
> 0, (5.49)
and
D11 + D12DcD21 = 0, (5.50)
where ∗ denotes an entry that can be deduced from the symmetry of the matrix,the matrices X, E, Y , W , U , Dc, Z, J , and the symmetric matrices P2, Hand Π are the variables. A feasible H2 controller is obtained by choosing N1
and M1 nonsingular such that N1M1 = Z − Y X and calculating
Cc = (E − DcC2X)M−11 , Dc = Dc, (5.51)
Bc = N−11 (W − Y B2Dc), (5.52)
Ac = N−11 [U − Y (A + B2DcC2)X − N1BcC2X − Y B2CcM1]M
−11 . (5.53)
5.4 H∞ control
5.4.1 Continuous-time case
We consider the continuous-time system P described by (5.21)−(5.23), and the class
of causal, linear, time-invariant and finite-dimensional controllers that internally sta-
bilize P , or namely, all admissible controllers for P . Our aim is to find an admissible
controller C such that the closed-loop system Tzw satisfies
‖Tzw‖∞ < γ. (5.54)
Assumption 5.2
(1). The pair (A, B2) is stabilizable and the pair (A, C2) is detectable.
(2). The matrices D12 and D21 satisfy DT12D12 = Inu
and D21DT21 = Iny
.
(3).
rank
[
A − jωI B2
C1 D12
]
= nx + nu, for all real ω. (5.55)
(4).
rank
[
A − jωI B1
C2 D21
]
= nx + ny, for all real ω. (5.56)
Introduction to Optimal and Robust Control 97
The assumption that (A, B2, C2) is stabilizable and detectable is necessary and
sufficient for the existence of admissible controllers. The full rank assumptions (3)
and (4) are necessary for the existence of stabilizing solutions to the Riccati equations
that are used to obtain the solution to the H∞ control problem.
Define the matrices A, B, A and C as
A = A − B1DT21C2, BBT = B1(I − DT
21D21)BT1 , (5.57)
A = A − B2DT12C1, CT C = CT
1 (I − D12DT12)C1. (5.58)
Suppose that Uy maps y to u and has a minimal realization (Au, Bu, Cu, Du)satisfying det(I − D22Du) 6= 0 for a well-posed closed loop. Then a realization in
terms of LFT is given by
LFT (P, Uy) =
A + B2DuMC2 B2(I + DuMD22)Cu B1 + B2DuMD21
BuMC2 Au + BuMD22Cu BuMD21
C1 + D12DuMC2 D12(I + DuMD22)Cu D11 + D12DuMD21
, (5.59)
with M = (I − D22Du)−1.
THEOREM 5.2
[71] Consider the system (5.21)−(5.23) satisfying Assumption 5.2. Thereexists an admissible C such that the closed-loop system (5.26)−(5.27) satisfies(5.54) if and only if
1. There is a solution X∞ ≥ 0 to the algebraic Riccati equation
X∞A + AT X∞ − X∞(B2BT2 − γ−2B1B
T1 )X∞ + CT C = 0, (5.60)
such that A − (B2BT2 − γ−2B1B
T1 )X∞ is asymptotically stable.
2. There is a solution Y∞ ≥ 0 to the algebraic Riccati equation
AY∞ + Y∞AT − Y∞(CT2 C2 − γ−2CT
1 C1)Y∞ + BBT = 0, (5.61)
such that A − Y∞(CT2 C2 − γ−2CT
1 C1) is asymptotically stable.3. ρ(X∞Y∞) < γ2.In the case when these conditions hold, C is an admissible controller satis-
fying (5.54) if and only if C is given by the LFT
C = LFT (Ca, Uy), ‖Uy‖∞ < γ, (5.62)
where Uy is a stable transfer function. The generator Ca is given by
Ca =
Ak Bk1 Bk2
Ck1 0 ICk2 I 0
, (5.63)
98 Modeling and Control of Vibration in Mechanical Systems
where
Ak = A + γ−2B1BT1 X∞ − B2F∞ − Bk1C2z, (5.64)
[
Bk1 Bk2
]
=[
B1DT21 + Z∞CT
2z B2 + γ−2Z∞F T∞
]
, (5.65)[
Ck1
Ck2
]
=
[
−F∞−C2z
]
, (5.66)
C2z = C2 + γ−2D21BT1 X∞, F∞ = DT
12C1 + BT2 X∞, (5.67)
Z∞ = Y∞(I − γ−2X∞Y∞)−1 = (I − γ−2Y∞X∞)−1Y∞. (5.68)
A point given by Theorem 5.2 is that a solution to the H∞ generalized regulator
problem exists if and only if there exist stabilizing, nonnegative definite solutions
X∞ and Y∞ to the algebraic Riccati equations associated with the full information
H∞ control problem and the H∞ estimation of C1x such that the coupling condition
ρ(X∞Y∞) < γ2 is satisfied.
The optimal H∞ control problem is to find an internally stabilizing controller
C(s) such that ‖Tzw‖∞ of the closed-loop system (5.26)−(5.27) is minimized.
However, in practice it is often not necessary to design an optimal controller, and
it is usually appropriate to obtain a controller that gives rise to an H∞ norm of the
closed-loop system less than a prescribed value. More specifically, a suboptimal H∞control problem is that given γ > 0, find an admissible controller C , if there is any,
such that ‖Tzw‖∞ < γ.
The following theorem gives a design method for a suboptimal H∞ output feed-
back controller.
THEOREM 5.3
[90] Consider system (5.21)−(5.23). Given a scalar γ > 0, there exists anoutput feedback controller (5.24)−(5.25) such that ‖Tzw‖2
∞ < γ if the followingLMI admits a solution (E, W, U, Dc, X, Y ):
AX + XAT + B2E + (B2E)T UT + A + B2DcC2
∗ AT Y + Y A + WC2 + (WC2)T
∗ ∗∗ ∗
,
B1 + B2WDcD21 (C1X + D12E)T
Y B1 + WD21 (C1 + D12DcC2)T
−γI (D11 + D12DcD21)T
∗ −γI
< 0, (5.69)
[
X II Y
]
> 0. (5.70)
In this case, a feasible H∞ controller is obtained from (5.51)−(5.53), whereN1M1 = I − Y X.
Introduction to Optimal and Robust Control 99
5.4.2 Discrete-time case
Assume that the time-invariant discrete-time system (5.35)−(5.37) satisfies:
Assumption 5.3
(1). (A, B2, C2) is stabilizable and detectable.
(2). DT12D12 > 0 and D21D
T21 > 0.
(3).
rank
[
A − ejθI B2
C1 D12
]
= nx + nu, for all θ ∈ (−π, π]. (5.71)
(4).
rank
[
A − ejθI B1
C2 D21
]
= nx + ny, for all θ ∈ (−π, π]. (5.72)
We seek a causal, linear, time-invariant and finite-dimensional controller C(z)such that the closed-loop system (5.40)−(5.41) is stable and
‖Tzw‖∞ < γ, (5.73)
or equivalently, under zero initial conditions,
‖z‖22 − γ2‖w‖2
2 ≤ −ε‖w‖22, (5.74)
for all w ∈ ℓ2[0,∞) and some ε > 0.
Let
B =[
B1 B2
]
, Cd =
[
C1
0
]
, (5.75)
Dd =
[
D11 D12
Inw0
]
, (5.76)
Js =
[
Inz0
0 −γ2Inw
]
, (5.77)
Jt =
[
Inw0
0 −γ2Inu
]
. (5.78)
With the assumptions (1)−(4), a causal, linear, finite-dimensional stabilizing con-
troller that leads to ‖Tzw‖∞ < γ exists if and only if the following two conditions
hold [71].
1. There exists a solution to the Riccati equation
X∞ = CTd JsCd + AT X∞A − WT R−1W, (5.79)
with
R =
[
R1 RT2
R2 R3
]
= DTd JsDd + BT X∞B, (5.80)
W =
[
W11
W21
]
= DTd JsCd + BT X∞A, W21 ∈ Rnu×nx , (5.81)
100 Modeling and Control of Vibration in Mechanical Systems
such that A − BR−1W is asymptotically stable and
X∞ ≥ 0, (5.82)
R1 − RT2 R−1
3 R2 < 0, R1 ∈ Rnw×nw , R3 ∈ Rnu×nu. (5.83)
Denote
∇ = R1 − RT2 R−1
3 R2, W∇ = W11 − RT2 R−1
3 W21, (5.84)
and let E1 be an nu × nu matrix such that
ET1 E1 = R3, (5.85)
and E2 be an nw × nw matrix such that
ET2 E2 = −γ−2(R1 − RT
2 R−13 R2) = −γ−2∇. (5.86)
Define the system
[
At Bt
Ct Dt
]
=
A − B1∇−1W∇ B1E−12 0
E1R−13 (W − R2∇−1W∇) E1R
−13 R2E
−12 I
C2 − D21∇−1W∇ D21E−12 0
. (5.87)
2. There exists a solution to the Riccati equation
Y∞ = BtJtBTt + AtY∞AT
t − MtS−1t MT
t , (5.88)
where
St = DtJtDTt + CtY∞CT
t =
[
S1 S2
ST2 S3
]
, (5.89)
Mt = BtJtDTt + AtY∞CT
t =[
Mt1 Mt2
]
, (5.90)
such that At − MtS−1t Ct is asymptotically stable and
Y∞ ≥ 0, (5.91)
S1 − S2S−13 ST
2 < 0. (5.92)
Note that Ct in (5.87) is partitioned as
Ct =
[
Ct1
Ct2
]
=
[
E1R−13 (W − R2∇−1W∇)C2 − D21∇−1W∇
]
. (5.93)
A controller that achieves the objective (5.73) is given by
xc(k + 1) = Atxc(k) + B2u(k) + Mt2S−13 (y(k) − Ct2xc(k)), (5.94)
E1u(k) = −Ct1xc(k)1 − S2S−13 (y(k) − Ct2xc(k)). (5.95)
Introduction to Optimal and Robust Control 101
All controllers that achieve the objective (5.73) are generated by the LFT C =LFT (Ca, Uc), where Uc is a linear causal system such that ‖Uc‖∞ < γ , and the
generator Ca is given by
I − B2 − Mt2(XT2 )−1
0 E1 S2(XT2 )−1
0 0 X2
xc(k + 1)u(k)η(k)
=
At 0 (Mt1 − Mt2S−13 S2)(γ
2XT1 )−1
−Ct1 0 X1
−Ct2 I 0
xc(k)y(k)φ(k)
, (5.96)
with
X2XT2 = S3, (5.97)
X1XT1 = −γ−2(S1 − S2S
−13 ST
2 ). (5.98)
The following theorem gives one parametrization approach of all suboptimal discrete-
time H∞ output feedback controllers.
THEOREM 5.4
[90] Consider system (5.35)−(5.37). Given a scalar γ > 0, there exists anoutput feedback controller (5.38)−(5.39) such that ‖Tzw‖2
∞ < γ if the followingLMI admits a solution:
P∞ J AX + B2E A + B2DcC2 B1 + B2DcD21 0∗ H U Y A + V C2 Y B1 + V D21 0∗ ∗ X + X′ − P∞ I + Z′ − J 0 XT CT
1 + ET DT12
∗ ∗ ∗ Y + Y T − H 0 CT1 + CT
2 DTc DT
12
∗ ∗ ∗ ∗ I DT11 + DT
21DTc DT
12
∗ ∗ ∗ ∗ ∗ γI
> 0, (5.99)
where the matrices X, E, Y , V , U , Dc, Z, J , and the symmetric matrices P∞and H are the variables. A feasible H∞ controller is obtained from (5.51)-(5.53).
5.5 Robust control
The H∞ norm is used to test robust stability of a nominally stable system under
unstructured perturbations. The following so-called small gain theorem is the basis
for robust stability analysis.
102 Modeling and Control of Vibration in Mechanical Systems
THEOREM 5.5
Small Gain Theorem Consider a proper and stable transfer function matrixT (s). Suppose that a stable ∆(s) is connected from the output of T (s) to theinput of T (s) as shown in Figure 5.2. Then the closed-loop system given inFigure 5.2 is internally stable if
σ[∆(jω)]σ[T (jω)] < 1, ∀ω ∈ R⋃
∞. (5.100)
FIGURE 5.2
A closed-loop system with uncertainty.
The small gain condition is sufficient to guarantee internal stability of the closed-
loop system even if ∆ is nonlinear and time-varying. The small gain theorem tells us
that an H∞ norm bound on T implies closed-loop stability in the presence of certain
H∞ norm bounded system uncertainties. The H∞ norm bound implies a certain
stability robustness. Note from (5.100) that the size of tolerable uncertainty varies
inversely proportional to the H∞ norm bound of T , which means that the robustness
increases as the H∞ norm bound decreases.
In what follows, the small gain theorem will be used to test robust stability under
model uncertainties. The modeling error ∆ is assumed to be stable and suitably
scaled with weighting functions W1 and W2, i.e., the uncertainty can be represented
as W1∆W2.
Additive uncertainty
We assume that the model uncertainty can be represented by an additive perturba-
tion:
Π = P + W1∆W2, (5.101)
Introduction to Optimal and Robust Control 103
which is shown in Figure 5.3.
FIGURE 5.3
A closed-loop system with additive uncertainty for robust stability analysis.
THEOREM 5.6
[70] Let Π = P + W1∆W2, and C be a stabilizing controller for the nominalplant P . Then the closed-loop system is well-posed and internally stable forall ‖∆‖∞ < 1 if and only if ‖W2CSW1‖∞ ≤ 1, where
S =1
1 + PC. (5.102)
Similarly, the closed-loop system is stable for all stable ∆ with ‖∆‖∞ ≤ 1 if and
only if ‖W2CSW1‖∞ < 1.
Multiplicative uncertainty
The system model is described by the following multiplicative perturbation:
Π = (I + W1∆W2)P (5.103)
where W1, W2 and ∆ are stable. Consider the feedback system shown in Figure 5.4.
THEOREM 5.7
[70] Let Π = (I + W1∆W2)P , C be a stabilizing controller for the nominalplant P , and T = 1 − S. Then
(i) the closed-loop system is well-posed and internally stable for all stable∆ with ‖∆‖∞ < 1 if and only if ‖W2TW1‖∞ ≤ 1.
(ii) the closed-loop system is well-posed and internally stable for all stable∆ with ‖∆‖∞ ≤ 1 if ‖W2TW1‖∞ < 1.
(iii) the robust stability of the closed-loop system for all stable ∆ with‖∆‖∞ ≤ 1 does not necessarily imply ‖W2TW1‖∞ < 1.
104 Modeling and Control of Vibration in Mechanical Systems
FIGURE 5.4
A closed-loop system with multiplicative uncertainty for robust stability analysis.
(iv) the closed-loop system is well-posed and internally stable for all stable∆ with ‖∆‖∞ ≤ 1 only if ‖W2TW1‖∞ ≤ 1.
(v) In addition, assume that neither P nor C has poles on the imaginaryaxis. Then the closed-loop system is well-posed and internally stable for allstable ∆ with ‖∆‖∞ ≤ 1 if and only if ‖W2TW1‖∞ < 1.
5.6 Controller parametrization
Consider the standard system block diagram in Figure 5.1 with
P (s) =
Ap B1 B2
C1 D11 D12
C2 D21 D22
. (5.104)
Suppose (Ap, B2) is stabilizable, (C2, Ap) is detectable, and D22 = 0. The problem
discussed here is that given a plant P , parameterize all controllers C that internally
stabilize P .
The parametrization for all stabilizing controllers is known as Youla parametriza-
tion [68], as shown in Figure 5.5. The Youla parametrization starts with a nominal
controller that is an estimated-state feedback. The estimated state feedback controller
is given by
u = −Kx, (5.105)
where the state feedback gain K is some appropriate matrix and x is an estimate of
the component of x, governed by the observer equation
˙x = Apx + B2u + L(y − C2x), (5.106)
Introduction to Optimal and Robust Control 105
where L is the estimator gain. The transfer function of the estimated-state feedback
controller is thus
Kn(s) = −K(sI − Ap + B2K + LC2)−1L. (5.107)
The nominal controller Kn(s) will stabilize P provided K and L are chosen such
that Ap − B2K and Ap − LC2 are stable.
To augment the estimated state feedback controller, we inject v into u as shown in
Figure 5.5, meaning that (5.105) is replaced by
u = −Kx + v, (5.108)
and therefore the signal v does not induce any observer error. For the signal e we
take the output prediction error:
e = y − C2x. (5.109)
In Figure 5.5, the observer based controller applies output prediction error processed
through a stable transfer function Q and added to the output of Kn. This augmen-
tation is able to yield every controller that stabilizes the plant, which means every
stabilizing controller can be realized as an observer based controller with some sta-
ble transfer function Q. Thus in the sequel we can form simple state-space equations
for the parametrization of all controllers that stabilize the plant, and all closed-loop
transfer matrices achieved by controllers that stabilize the plant.
The state-space equations of the augmented controller are given by
˙x = (Ap − B2K − LC2)x + Ly + B2v, (5.110)
u = −Kx + v, (5.111)
e = y − C2x. (5.112)
If Q has a state-space realization
xq = Aqxq + Bqe, (5.113)
v = Cqxq + Dqe, (5.114)
then a state-space realization of the observer based controller by eliminating e and
v from the augmented controller equations (5.110)−(5.112) and the Q realization
(5.113)−(5.114) is obtained as:
˙x = (Ap − B2K − LC2 − B2DqC2)x + B2Cqxq + (L + B2Dq)y, (5.115)
xq = −BqC2x + Aqxq + Bqy, (5.116)
u = −(K + DqC2)x + Cqxq + Dqy, (5.117)
or equivalently,
C(s) = Cc(sI − Ac)−1Bc + Dc, (5.118)
106 Modeling and Control of Vibration in Mechanical Systems
where
Ac =
[
Ap − B2K − LC2 − B2DqC2 B2Cq
−BqC2 Aq
]
, (5.119)
Bc =
[
L + B2Dq
Bq
]
, (5.120)
Cc =[
−K − DqC2 Cq
]
, (5.121)
Dc = Dq. (5.122)
On the other hand, the state-space equations for the closed-loop system with only
the augmented controller (5.110)−(5.112) are found as follows by eliminating u and
y from (5.110)−(5.112) and the plant equations in (5.104).
x = Apx − B2Kx + B1w + B2v, (5.123)
˙x = LC2x + (Ap − B2K − LC2)x + LD1w + B2v, (5.124)
z = C1x − D12Kx + D11w + D12v, (5.125)
e = C2x − C2x + D21w, (5.126)
which are equivalently written as
[
T11(s) T12(s)T21(s) 0
]
= CT (sI − AT )−1BT + DT , (5.127)
where
AT =
[
Ap −B2LLC2 Ap − B2K − LC2
]
, (5.128)
BT =
[
B1 B2
LD1 B2
]
, (5.129)
CT =
[
C1 − D12KC2 − C2
]
, (5.130)
DT =
[
D11 D12
D21 0
]
. (5.131)
It has been verified that by augmenting the stable transfer function Q, the closed-
loop transfer function from w to z is simply an affine function of Q and equals
T11 + T12QT21.
Introduction to Optimal and Robust Control 107
FIGURE 5.5
Control system structure for Youla parametrization.
108 Modeling and Control of Vibration in Mechanical Systems
5.7 Performance limitation
5.7.1 Bode integral constraint
The block diagram in Figure 2.20 shows a typical closed-loop control system. The
closed-loop transfer function from r to y is given by
T =PC
1 + PC. (5.132)
The sensitivity function is also known as the disturbance rejection function or error
rejection function, and given by
S =1
1 + PC. (5.133)
Note that
S + T ≡ 1, (5.134)
hence T is also commonly called the complementary sensitivity function.
Denote Tyd2 the transfer function from d2 to y. In Figure 2.20 , note that
S = Tyd2 = −Ted2 = Ten. (5.135)
The sensitivity function is thus important, because it explains how disturbance d2
goes through the closed-loop system and shows up at the output y, or the error signal
e. It is also important to understand how noise n will be filtered through the closed-
loop system.
The Bode plot of sensitivity function in continuous-time domain is shown in Fig-
ure 5.6. It can be explained by the following Bode integral theorem.
THEOREM 5.8
[67] Bode’s Integral Theorem for Continuous-time Systems For a stable, ra-tional P and C with P (s)C(s) having at least 2-pole roll off,
∫ ∞
0
log|S|dω = 0. (5.136)
An implication of Theorem 5.8 is that if the system is made less sensitive to dis-
turbance at some frequencies, it will be more sensitive at other frequencies. If the
plant P or compensator is not stable, i.e., if P (s) and /or C(s) have a finite number
of unstable poles pk, then (5.136) becomes
∫ ∞
0
log|S|dω = 2π
K∑
k=1
Re(pk) > 0, (5.137)
Introduction to Optimal and Robust Control 109
where K is the number of unstable poles. Equation (5.137) implies that any unstable
poles in the system only make it worse, in that more of the disturbance would have
to be amplified.
Figure 5.7 means that for a discrete-time system, the main difference is the Nyquist
frequency limits the frequency range we have to work with. In both cases, if we
want to attenuate disturbances at one frequency, we must amplify some disturbance
at another frequency.
THEOREM 5.9
[108] Bode’s Integral Theorem for Discrete-time Systems Given a stable closed-loop discrete-time feedback system, its sensitivity function has to satisfy thefollowing integral constraint:
1
π
∫ π
0
log|S(ejφ)|dφ =
K∑
k=1
ln|βk|, (5.138)
where βk are the open-loop unstable poles of the system, K is the total numberof unstable poles, and φ = Tsω with the sampling time Ts and the frequencyω in radians/sec.
Note that Ts is the sampling period, and the upper limit of the frequency spectrum
is π/Ts, the Nyquist frequency. Typical digital control systems assume PC is small
and |S| ≈ 1 at or above the Nyquist frequency, which is in general not practical for
a physical system. Further research beyond Nyquist frequency is needed to address
vibrations at frequencies above the Nyquist frequency, which strives to overcome the
limitation due to the integral theorem.
Theorem 5.9 implies that if for some frequency |S| < 1, then at some other fre-
quency |S| > 1. Unlike the continuous time result, there is no infinite bandwidth to
spread this over. Thus all |S| > 1 happen below the Nyquist frequency, and there-
fore in a finite frequency range. Since the theorem is limited by frequencies up to the
Nyquist frequency, if the closed-loop bandwidth is pushed up, better performance at
low frequency may result in worse performance at high frequency. In a word, with
the linear feedback control whenever we improve the disturbance rejection at one
frequency we pay for it at another. Nevertheless, if we have sufficient knowledge
of system disturbance and place the disturbance amplification at places where the
disturbance is negligible, then we succeed. Otherwise, most of the disturbances may
be amplified.
110 Modeling and Control of Vibration in Mechanical Systems
Frequency(Hz)
0
Ma
gn
itu
de
in
dB
(2
0lo
g10|S
|)
Area of vibration rejection
Area of vibration amplification
FIGURE 5.6
Sensitivity function for continuous-time system.
Frequency(Hz)
Nyquist frequency Area of vibrationrejection
Area of vibration amplification
0
Magnitude in d
B (
20lo
g1
0|S
|)
FIGURE 5.7
Sensitivity function for discrete-time system.
Introduction to Optimal and Robust Control 111
5.7.2 Relationship between system gain and phase
In the classical feedback theory, the Bode’s gain-phase integral relation has been
used as an important tool to express design constraints in feedback systems. Let
L = PC denote the open-loop system. It is noted that ∠L(jω0) will be large if
the gain L attenuates slowly near ω0 and small if it attenuates rapidly near ω0. The
behavior of ∠L(jω) is particularly important near the crossover frequency ωc, where
|L(jωc)| = 1, and π +∠L(jωc) is the phase margin of the feedback system. Further
|1 + L(jωc)| = |1 + L−1(jωc)| = 2|sinπ + ∠L(jωc)
2| (5.139)
must not be too small for good stability robustness. If π + ∠L(jωc) is forced to be
very small by rapid gain attenuation, the feedback system will amplify disturbances
and exhibit little uncertainty tolerance at and near ωc. A non-minimum phase zero
contributes an additional phase lag and imposes limitations upon the roll off rate of
the open-loop gain. Thus the conflict between attenuation rate and loop quality near
crossover is clearly evident.
In the classical feedback control theory, it has been common to express design
goals in terms of the shape of the open-loop transfer function. A typical design
requires that the open-loop transfer function has a high gain at low frequencies and
a low gain at high frequencies while the transition should be well behaved [70].
5.7.3 Sampling
Mathematical relations and operations can be handled by digital microprocessor only
when they are expressed as a finite set of numbers rather than as functions having
an infinite number of possible values. Thus any measured continuous signal must
be converted to a set of pulses by sampling, which is the process used to measure a
continuous-time variable at separated instants of time. The infinite set of numbers
represented by the smooth curve is replaced by a finite set of numbers. Each pulse
amplitude is then rounded off to one of a finite number of levels depending on the
characteristics of the converter. The process is called quantization. Thus a digital
device is one in which signals are quantized in both time and amplitude. In an analog
device, signals are analog; that is, they are continuous in time and are not quantized
in amplitude. The device that performs the sampling, quantization, and converting
to binary form is an analog to digital (A/D) converter.
The number of binary digits or bits generated by the device is its word length,
which is an important characteristic related to the resolution of the converter. The
resolution measures the smallest change in the input signal that will produce a change
in the output signal. An example is that if an A/D converter has a word length of 10bits or more, an input signal can be resolved to 1 in 210 or 1024. If the input signal
has a range of 10 V, the resolution is 10/1024, or approximately 0.01 V. Thus in
order to produce a change in the output the input must change by at least 0.01 V.
A discrete-time signal is extracted by sampling from a continuous-time signal. If
the sampling frequency is not selected properly, the resulting sampled sequence will
112 Modeling and Control of Vibration in Mechanical Systems
not accurately represent the original continuous signal. A proper sampling frequency
is readily determined in many cases by means of the following sampling theorem.
THEOREM 5.10
Sampling Theorem A continuous-time signal y(t) can be reconstructed fromits uniformly sampled values y(KTs) if the sampling period Ts satisfies
Ts ≤ π
ω(5.140)
where ω is the highest frequency contained in the signal, that is, |Y (ω)| = 0for ω > ω.
If a system involves a sampling operation of continuous-time signals to gener-
ate discrete-time signals, a time delay may be induced. Any time delay added into
a closed-loop control system will decrease the stability of the system and in some
cases may even cause system instability. The Nyquist rate shown in Figure 5.7 signi-
fies that the freedom to spread the amplification area around is limited by the Nyquist
frequency, which is half of the sampling rate. Figure 5.7 implies that a certain rejec-
tion amount |S| < 1 must be accompanied by a certain area of |S| > 1, which has
to occur before the Nyquist frequency. With a higher sampling rate and closed-loop
system bandwidth being kept constant, the amplification |S| > 1 will essentially
spread over a broader frequency band and the height of amplification hump shrinks,
as shown in Figure 5.8. If we push the closed-loop system bandwidth from f1 to f2,
as seen in Figure 5.8, better performance in low frequency range may result in worse
performance in high frequency range.
5.8 Conclusion
Before presentating a series of advanced vibration control methodologies, this chap-
ter has been used to recall some standard advanced control techniques, which helps
understand problems to be addressed in the later chapters and possible solutions. It
has reviewed H2 and H∞ performances, H2 and H∞ controls, robust control, con-
troller parametrization, as well as performance limitation of linear feedback control
systems.
Introduction to Optimal and Robust Control 113
Frequency(Hz)
fN1
0
Ma
gn
itu
de
in
dB
(2
0lo
g10|S
|)
2fN1
f2
f1
FIGURE 5.8
Sensitivity function in discrete-time domain.
6
Mixed H2/H∞ Control Design for VibrationRejection
6.1 Introduction
The mixed H2/H∞ control problem is concerned with the design of a controller that
minimizes the H2 norm of a certain closed-loop transfer function while satisfying a
H∞ norm constraint on the same or another closed-loop transfer function. One of the
important applications of this problem is to address the optimal nominal performance
subject to a robust stability constraint.
In this chapter, we employ two methods: the one described in Chapter 5 and an
improved method. The two methods are applied to the control design in disk drives.
In order to meet the robustness requirement against unmodeled high frequency dy-
namics of the VCM actuator in disk drives, an H∞ constraint is to be satisfied when
minimizing the H2 performance of the nominal system. Hence, a mixed H2/H∞control can be formulated for disk drive control. Both the simulation and experiment
results demonstrate that the mixed H2/H∞ control design gives a significant perfor-
mance improvement of TMR over the conventional method of PID control combined
with notch filters.
6.2 Mixed H2/H∞ control problem
This section aims to derive an improved design method for the mixed H2/H∞ control
which will give rise to an equal or better performance than the method in Chapter 5.
We consider the state-space representation for linear time-invariant systems:
x(k + 1) = Ax(k) + B1w(k) + B2u(k), (6.1)
y(k) = C2x(k) + D21w(k), (6.2)
z1(k) = Cz1x(k) + Dz11w(k) + Dz12u(k), (6.3)
z2(k) = Cz2x(k) + Dz21w(k) + Dz22u(k), (6.4)
where x(k) ∈ Rnx is the state, y(k) ∈ Rny is the measurement output, zi(k) ∈
115
116 Modeling and Control of Vibration in Mechanical Systems
Rnz(i = 1, 2) are the controlled outputs, w(k) ∈ Rnw is the disturbance input,
u(k) ∈ Rnu is the control input, and A, B1, B2, C2, D21, Cz1, Cz2, Dz11, Dz12,
Dz21, Dz22 are of appropriate dimensions.
Let a controller of the same dimension as that of the system (6.1)−(6.4) be of the
form:
xc(k + 1) = Acxc(k) + Bcy(k), (6.5)
u(k) = Ccxc(k) + Dcy(k), (6.6)
where the matrices (Ac, Bc, Cc, Dc) are to be determined.
First, denote ξ = [xT xTc ]T . It follows from (6.1)−(6.4) and (6.5)−(6.6) that
ξ(k + 1) = Aξ(k) + Bw(k), (6.7)
z1(k) = C1ξ(k) + D1w(k), (6.8)
z2(k) = C2ξ(k) + D2w(k), (6.9)
where
A =
[
A + B2DcC2 B2Cc
BcC2 Ac
]
, B =
[
B2DcD21 + B1
BcD21
]
, (6.10)
C1 = [Cz1 + Dz12DcC2 Dz12Cc] , D1 = Dz12DcD21 + Dz11, (6.11)
C2 = [Cz2 + Dz22DcC2 H22Cc] , D2 = Dz22DcD21 + Dz21. (6.12)
Denote Tz1w the transfer function matrix from w to z1, and Tz2w the transfer
function matrix from w to z2. The mixed H2/H∞ control problem is then stated as:
Given a positive scalar γ, design a controller of the form (6.5)−(6.6) such that the
H2 norm, ‖Tz1w‖2 is minimized subject to the constraint ‖Tz2w‖∞ < γ.
6.3 Method 1: slack variable approach
Given the solutions of H2 control and H∞ control in Chapter 5, the following solu-
tion to the mixed H2/H∞ control follows directly.
THEOREM 6.1
Consider the system (6.1)−(6.4). Given scalars µ > 0 and γ > 0, a controllerof the form (6.5)−(6.6) that solves the mixed H2/H∞ control problem existsif the following conditions are satisfied.
Trace(Π) < µ, (6.13)
Π Cz1X + Dz12E Cz1 + Dz12DcC2
∗ X + XT − P2 I + ZT − J∗ ∗ Y + Y T − H
> 0, (6.14)
Mixed H2/H∞ Control Design for Vibration Rejection 117
P2 J AX + B2E A + B2DcC2 B1 + B2DcD21
∗ H U Y A + WC2 Y B1 + WD21
∗ ∗ X + XT − P2 I + ZT − J 0∗ ∗ ∗ Y + Y T − H 0∗ ∗ ∗ ∗ I
> 0, (6.15)
Dz11 + Dz12DcD21 = 0, (6.16)
and
P∞ J AX + B2E A + B2DcC2 B1 + B2DcD21 0
∗ H U Y A + WC2 Y B1 + WD21 0
∗ ∗ X + XT − P∞ I + ZT − J 0 XT CTz2 + ET DT
z22
∗ ∗ ∗ Y + Y T − H 0 CTz2 + CT
2 DTc DT
z22
∗ ∗ ∗ ∗ I DTz21 + DT
21DTc DT
z22
∗ ∗ ∗ ∗ ∗ γI
> 0, (6.17)
where the matrices X, E, Y , W , U , Dc, Z, J , J and the symmetric matricesP2, P∞, H, H and Π are the variables. A feasible mixed H2/H∞ controlleris obtained by choosing N1 and M1 nonsingular such that N1M1 = Z − Y Xand calculating
Cc = (E − DcC2X)M−11 , Dc = Dc, (6.18)
Bc = N−11 (W − Y B2Dc), (6.19)
Ac = N−11 (U − Y (A + B2DcC2)X − N1BcC2X − Y B2CcM1]M
−11 . (6.20)
In the early development of the mixed H2/H∞ control, to solve the problem in
terms of LMIs, a single Lyapunov matrix is adopted for both the H2 and H∞ perfor-
mances, which is very conservative in general. A significant improvement was made
in [90] where a slack variable technique is introduced which separates the Lyapunov
matrices from the controller parameters and hence allows them to be different for
the H2 and H∞ performances. This is observed in the above theorem where the
Lyapunov matrices P2, J and H are used for H2 performance and different matrices
P∞, J and H are used for H∞ performance.
6.4 Method 2: an improved slack variable approach
Recall from Chapter 5 that for a given controller that stabilizes the system (6.7)−(6.9),
the H2 norm square, ‖Tz1w‖22, can be computed by the following minimization [95]:
min(Q=QT ,Π=ΠT )
Trace(Π), (6.21)
118 Modeling and Control of Vibration in Mechanical Systems
subject to
AT QA − Q + CT1 C1 < 0, (6.22)
BT QB + DT1 D1 < Π. (6.23)
The following lemma leads to an alternative approach for computing the H2 norm
of the system (6.7)−(6.8).
LEMMA 6.1
The minimization of the H2 norm square in (6.21) is equivalent to the fol-lowing minimization:
min(Q=QT ,Π=ΠT ,Σ)
Trace(Π), (6.24)
subject to
AT diagQ, IA−[
Q ΣΣT Π
]
< 0, (6.25)
where
A =
[
A BC1 D1
]
.
Proof First of all, (6.25) can be rewritten as
[
AT QA − Q + CT1 C1 AT QB + CT
1 D1 − ΣBT QA + DT
1 C1 − ΣT BT QB + DT1 D1 − Π
]
< 0. (6.26)
It is then clear that if there exists a solution (Q, Π, Σ) to (6.25), the same Q and Πalso satisfy (6.22) and (6.23), respectively. On the other hand, if there exist Q and
Π satisfying (6.22) and (6.23), then (6.25) is also met by the same Q and Π and
Σ = AT QB + CT1 D1.
REMARK 6.1 The characterization of the H2 norm in the above lemmahas the advantage that it provides a unified treatment of H2 and H∞ designsvia an LMI approach, as seen later. Furthermore, the additional parameterΣ in (6.25) will offer an additional freedom in optimization of performance
when a mixed H2 and H∞ control design is concerned.
Without loss of generality, we shall assume nw = nz , i.e. the disturbance input
and the signal to be controlled have the same dimension. Note that, if this is not the
case, some simple modification can render the requirement satisfied. For example,
if nw < nz , the matrices B1, D21, Dz11 and Dz21 can be augmented as B1 =[B1 0nx×(nz−nw)], D21 = [D21 0nx×(nz−nw)], Dz11 = [Dz11 0nz×(nz−nw)] and
Dz21 = [Dz21 0nz×(nz−nw)].
Mixed H2/H∞ Control Design for Vibration Rejection 119
LEMMA 6.2
There exists a solution (Σ, Q, Π) with Q = QT to (6.25) if and only if thereexist matrices (Σ, Π, Q, F, G) with Q = QT and Π = ΠT such that
−[
Q ΣΣT Π
]
+ AT F + F T A −F T + AT G
−F + GT A diagQ, I − (G + GT )
< 0.
(6.27)
Proof First, if (6.25) holds for some Q > 0, by applying the Schur complement,
it is easy to know that (6.27) is satisfied with F = 0 and GT = G = diagQ, I.
On the other hand, if (6.27) holds for some (Σ, Q, F, G), multiplying (6.27) from the
left and from the right by ΓT and Γ, respectively, where
Γ =
[
I
A
]
,
(6.25) follows.
REMARK 6.2 It should be pointed out that (6.21)−(6.23) and Lemma6.2 give equivalent computations of the H2 norm of the system. For systemswithout uncertainty, it is well known that (6.21)−(6.23) can be applied toderive the optimal H2 controller [95]. Hence, there is no advantage of usingLemma 6.2. However, as will be seen later, when additional performances suchas the H∞ performance are to be met in addition to the H2 performance, thelatter will result in a less or equally conservative design due to the additionalvariables F and G. We observe that when F = 0 and Σ = 0, Lemma 6.2reduces to the result in Theorem 6.1.
While Lemma 6.2 can be applied to compute the H2 norm of the system (6.7)−(6.8)
when a controller (6.5)−(6.6) is given, it may not be directly applicable to the H2
control design problem due to the presence of the products of F with A and G with
A. To overcome this difficulty, we specialize the matrices F and G as follows.
F =
[
λ1Φ 00 λ2I
]
, G =
[
Φ 00 λ3I
]
, (6.28)
where Φ ∈ R2n×2n and λi, i = 1, 2, 3 are real scaling parameters. While this
specialization of F and G generally introduces some conservatism, it contains three
additional variables λi, i = 1, 2, 3 as compared to the result of Theorem 6.1 which
help reduce the design conservatism in Theorem 6.1.
Substituting (6.28) into (6.27) leads to
−Q + λ1AT Φ + λ1Φ
T A λ2C1T
+ λ1ΦT B − Σ
λ2C1 + λ1BT Φ − ΣT − Π + λ2
(
DT1 + D1
)
−λ1Φ + ΦT A ΦT Bλ3C1 −λ2I + λ3D1
120 Modeling and Control of Vibration in Mechanical Systems
−λ1ΦT + AT Φ λ3C
T1
BT Φ − λ2I + λ3DT1
Q − (Φ + ΦT ) 00 (1 − 2λ3)I
< 0. (6.29)
A similar characterization for the H∞ performance can be derived. Indeed, recall
that when the system (6.7) and (6.9) is known, it is stable with its H∞ norm less than
γ if and only if there exists a matrix P = P T > 0 such that
AT diagP∞, IA − diagP∞, γ2I < 0, (6.30)
where
A =
[
A BC2 D2
]
.
Observe that (6.30) turns out to be a special case of (6.25) with Π = γ2I and Σ = 0.
Thus, following a similar procedure for deriving (6.29), it can be shown that the
system (6.7) and (6.9) has an H∞ performance γ if and only if for some real scalars
εi, i = 1, 2, 3, there exist matrices P∞ = P T∞ > 0 and Φ such that
−P∞ + ε1(AT Φ + ΦT A) ε2C2
T+ ε1Φ
T Bε2C2 + ε1B
T Φ −γ2I + ε2
(
DT2 + D2
)
−ε1Φ + ΦT A ΦT Bε3C2 −ε2I + ε3D2
−ε1ΦT + AT Φ ε3C
T2
BT Φ −ε2I + ε3DT2
P − (Φ + ΦT ) 00 (1 − 2ε3)I
< 0. (6.31)
REMARK 6.3 When setting ε1 = ε2 = 0 (i.e. setting F = 0) and ε3 = 1and by some row-column exchanges, the above inequality reduces to
P∞ − Φ − ΦT ΦT A ΦT B 0AT Φ −P∞ 0 CT
2
BT Φ 0 −γ2I DT2
0 C2 D2 −I
< 0
which is the result in Chapter 5. Therefore, εi, i = 1, 2, 3 are additionalvariables which can be exploited to alleviate the conservatism in the mixedH2/H∞ design of Theorem 6.1.
Denote
Ω (Π, X, Y, E, U, W, Z, Dc, Q11 = QT11, Q12, Q22 = QT
22, Σ1,
Σ2, λi, i = 1, 2, 3) =
Mixed H2/H∞ Control Design for Vibration Rejection 121
−Q11 + λ1(AY + Y T AT + ET BT
2 + B2E)−QT
12 + λ1(AT + CT
2 DcT BT
2 ) + λ1UT
λ2(Cz1Y + Dz21E) + λ1(BT
1 + DT
21DcT BT
2 ) − ΣT
1
−λ1YT + AY + B2E
−λ1I + U
λ3(Cz1Y + Dz21E)
∗
−Q22 + λ1(AT X + XT A) + λ1(C
T
2 WT + WC2)
λ2(Cz1 + Dz21DcC2) + λ1BT X + λ1D
T
21WT − ΣT
2
−λ1Z + A + B2DcC2
−λ1X + XT A + WC2
λ3(Cz1 + Dz21DcC2)
∗ ∗∗ ∗
−Π + λ2(DT21Dc
T DTz21 + Dz21DcD21 + DT
z11 + Dz11) ∗B1 + B2DcD21 Q11 − (Y + Y T )XT B1 + WD21 QT
12 − (I + ZT )−λ2I + λ3(Dz11 + Dz21DcD21) 0
∗ ∗∗ ∗∗ ∗∗ ∗
Q22 − (X + XT ) ∗0 (1 − 2λ3)I
. (6.32)
We have the following solution to the mixed H2/H∞ control.
THEOREM 6.2
Consider the system (6.1)−(6.4). A controller of the form (6.5)−(6.6) thatsolves the mixed H2/H∞ control problem exists if for some λi, εi, i = 1, 2, 3,there exists a solution (P11, P12, P22, Q11, Q12, Q22, Σ1, Σ2, X, Y, Π, U, E, Z, W ,Ψ, Dc) with Q = [Qlj] = QT > 0 and P = [Plj] = P T > 0 to the followingoptimization:
minTrace(Π)
subject to
Ω (Π, X, Y, E, U, W, Z, Dc, Q11, Q12, Q22, Σ1, Σ2, λi,
i = 1, 2, 3) < 0, (6.33)
Ω (γ2I, X, Y, E, U, W, Z, Dc, P11, P12, P22, Σ1, Σ2, εi,
i = 1, 2, 3)|Σ1=Σ2=0 < 0. (6.34)
122 Modeling and Control of Vibration in Mechanical Systems
In this situation, a mixed H2/H∞ controller is given by
Ac = M−T
1 (U − XT (A + B2DcC2)Y − X
TB2CcN1
−MT
1 BcC2Y )N−1
1 , (6.35)
Bc = M−T
1 (W − XTB2Dc), (6.36)
Cc = (E − DcC2Y )N−1
1 , Dc = Dc, (6.37)
where M1, N1 satisfy NT
1 M1 = Z − Y T X .
Proof First, observe from (6.29) that Φ is invertible since Φ + ΦT > Q > 0.
Denote
Φ =
[
X MM1 U
]
, Φ−1 =
[
Y HN1 E
]
,
and
J =
[
Y In
N1 0
]
, J1 = diagJ, I, J, I.
Further, denote
E = DcC2Y + CcN1, (6.38)
U = XT (A + B2DcC2)Y +
XT B2CcN1 + MT1 BcC2Y + MT
1 AcN1, (6.39)
W = XT B2Dc + MT1 Bc, (6.40)
Z = Y T X + NT1 M1. (6.41)
Multiplying from the left and the right of (6.29) by JT1 and J1 respectively and
applying (6.10), we obtain (6.33) where Q = [Qlj ] = JT QJ and [ΣT1 ΣT
2 ] = ΣT J .
By applying a similar procedure to (6.31), (6.34) can be obtained.
If there exists a solution for the LMIs (6.33) and (6.34), it is easy to see that
[
Y + Y T I + ZI + ZT X + XT
]
> Q > 0.
Multiplying the above from the left by [Y −T −I] and from the right by [Y −T −I]T ,
we obtain that
(X − Y −T Z) + (X − Y −T Z)T > 0.
It is then clear that Z −Y T X is invertible. Hence, there exist invertible matrices M1
and N1 such that Z − Y T X = NT1 M1. Thus, it follows from (6.38)−(6.41) that the
controller parameters of (6.5)−(6.6) can be obtained as in (6.35)−(6.37).
REMARK 6.4 It is worth noting that when setting λ1 = λ2 = ε1 =ε2 = 0, λ3 = ε3 = 1 and Σ1 = Σ2 = 0, Theorem 6.2 reduces to Theorem6.1. Hence, by exploring the freedoms offered by these parameters, a less orequally conservative mixed H2/H∞ design can be achieved. Certainly, the
Mixed H2/H∞ Control Design for Vibration Rejection 123
controller from Theorem 6.1 is derived by a convex optimization and can befurther refined by an iterative procedure. In the disk drive application tobe presented later, we demonstrate that Method 2 can give a much betterperformance than Method 1 with Theorem 6.1 together with an iterativerefinement in the latter design.
REMARK 6.5 Observe that for given λi, εi, i = 1, 2, 3, (6.33) and (6.34)are linear in (P11, P12, P22, Q11, Q12, Q22, Σ1, Σ2, X, Y, Π, U, E, Z, W, Dc), andhence can be solved by employing the LMI Tool [69]. The problem of searchingfor the optimal scaling parameters λi, εi, i = 1, 2, 3 in general may be nu-merically costly although the function fminsearch in MATLAB OptimizationToolbox may be applied.
6.5 Application in servo loop design for hard disk drives
6.5.1 Problem formulation
A block-diagram representation of a typical HDD servo loop is shown in Figure
6.1 with disturbances injected. P (z) and C(z) represent transfer functions of the
plant and controller, respectively. v represents all torque disturbances. d represents
disturbances that are due to non-repeatable disk and suspension/slider motions. ndenotes the PES demodulation and measurement noise. z1 is the true position error,
and e is the measured position error or measurement output y in (6.2). D1, D2, and
N are the disturbance and noise models, and w1, w2 and w3 are white noises of zero
mean and unit variance.
FIGURE 6.1
Mixed H2/H∞ control scheme for HDD servo loop with disturbance models.
124 Modeling and Control of Vibration in Mechanical Systems
Through experiments, the frequency responses of the actual VCM is obtained and
is shown in Figure 6.2. A 5th order model is used to approximate the actual frequency
responses of the VCM actuator and is given by
P (s) =5.172× 1012s2 + 1.82× 1017s + 3.267× 1021
s5 + 2.117× 104s4 + 1.032× 109s3 + 1.906× 1013s2
+8.587× 1015s + 7.345× 1018
. (6.42)
Figure 6.2 shows the comparison between the frequency responses of the actual data
and those of P (s). It is clear that their difference is more significant for the frequency
range of over 4 kHz. To capture the unmodeled dynamics in high frequencies, dozens
of frequency response measurements are carried out and Figure 6.3 shows the multi-
plicative uncertainty of the VCM actuator defined by
∆(ω) =Nmeamaxi=1
Pi(jω) − P (jω)
P (jω)
, (6.43)
where Nmea is the number of measurements and Pi(jω) is the actual frequency
response of the plant in the i-th measurement, and P (jω) is the frequency response
of the model in (6.42). An approximate bounding function Wu(s), i.e., the smooth
line in Figure 6.3, is obtained as
Wu(s) =3s2 + 2.903× 104s + 1.433× 108
s2 + 3.016× 104s + 1.421× 109. (6.44)
From Figure 6.3, it is clear that the uncertainty at frequency over 5 kHz is the major
concern. We observe that the actual uncertainties at some frequencies below 5 kHz
exceed the bounding function, which, however, will not cause any major problem.
In fact, we verify that the robust stability of our designed system is guaranteed even
with the worst case of uncertainty.
By discretization using the zero-order hold, the corresponding z-domain models
of the VCM and the bounding function, i.e., P (z) and Wu(z) can be obtained.
The disturbance and noise models D1(s), D2(s) and N(s) are given by (2.42)−(2.44),
and D1(z), D2(z) and N(z) are their discrete-time forms.
As mentioned, one of the most important performance measures for HDDs is the
track misregistration or TMR, the total amount of random fluctuation about the de-
sired track location. TMR is used to judge the required accuracy of positioning and
thus to scale the disk capacity. To achieve a high capacity disk drive, one way in
servo control is to minimize TMR, which is given in terms of the standard deviation
of the true PES, i.e.,
3σz1 = 3
√
√
√
√
1
q − 1
q∑
i=1
z1(i)2, (6.45)
where q is the number of true PES samples.
Mixed H2/H∞ Control Design for Vibration Rejection 125
102
103
104
−40
−20
0
20
40
60
Magnitude(d
B)
measuredmodeled
102
103
104
−1200
−1000
−800
−600
−400
−200
0
Phase(d
eg)
Frequency(Hz)
FIGURE 6.2
Frequency responses of the VCM actuator.
126 Modeling and Control of Vibration in Mechanical Systems
102
103
104
−60
−50
−40
−30
−20
−10
0
10
20
Frequency(Hz)
Magnitude(d
B)
VCM uncertaintybounding curve
FIGURE 6.3
Multiplicative uncertainty of the VCM actuator.
Mixed H2/H∞ Control Design for Vibration Rejection 127
Let w = [w1 w2 w3]T and Tz1w denote the transfer function matrix from w to z1.
When q is large enough, the H2 norm of Tz1w is given by [88]
‖Tz1w‖2 ≈
√
√
√
√
1
q − 1
q∑
i=1
z1(i)2. (6.46)
Thus, the control design problem to minimize TMR can be treated as an H2 optimal
control problem.
On the other hand, we need to ensure the system stability against the unmodeled
high frequency dynamics of the VCM actuator, i.e., the constraint ‖TWu‖∞ < 1is to be met, where T is the closed-loop transfer function and Wu is the bounding
function of the unmodeled dynamics which was derived earlier. Therefore, we have
the mixed H2/H∞ control scheme as shown in Figure 6.1, where w4 ∈ ℓ2[0,∞), a
disturbance input or a reference. Clearly, the transfer function from w4 to z2 is TWu.
We now derive the state-space representation (6.1)−(6.4) for the system in Figure
6.1 with
x =[
xTp xT
d1xT
d2xT
n xTu
]T, (6.47)
w =[
w1 w2 w3 w4
]T, (6.48)
A =
Ap BpCd1 0 0 00 Ad1 0 0 00 0 Ad2 0 00 0 0 An 0
BuCp 0 BuCd2 0 Au
, (6.49)
B1 =
BpDd1 0 0 0Bd1 0 0 00 Bd2 0 00 0 Bn 00 BuDd2 0 0
, B2 =
Bp
0000
, (6.50)
C2 =[
−Cp 0 −Cd2 Cn 0]
, (6.51)
D21 =[
0 −Dd2 Dn 1]
, Cz1 =[
−Cp 0 −Cd2 0 0]
,
Dz11 =[
0 −Dd2 0 0]
, Dz12 = 0, (6.52)
Cz2 =[
DuCp 0 DuCd2 0 Cu
]
, (6.53)
Dz21 =[
0 DuDd2 0 0]
, Dz22 = 0, (6.54)
and xp, xd1 , xd2 , xn, and xu are respectively the state variables of the VCM actua-
tor P (z), the input disturbance model D1(z), the output disturbance model D2(z),the measurement noise model N(z), and the uncertainty Wu(z). (Ap, Bp, Cp, Dp),(Ad1 , Bd1 , Cd1 , Dd1), (Ad2 , Bd2 , Cd2, Dd2 ), (An, Bn, Cn, Dn) and (Au, Bu, Cu, Du)are respectively the state-space models of P (z), D1(z), D2(z), N(z), and Wu(z).Note that for the servo control shown in Figure 6.1, nz = ny = 1.
128 Modeling and Control of Vibration in Mechanical Systems
6.5.2 Design results
In this section, we will apply the mixed H2/H∞ control to hard disk drive servo
formulated previously. The sampling frequency being used is 20 kHz. By applying
Theorem 6.2 and searching for the optimal scaling parameters, we obtain λ1 = 0.3,
λ2 = 0.31, λ3 = 0.9, ε1 = 0.3, ε2 = 0.28, ε3 = 1.1 and the minimum H2 norm,
i.e., the σ value of the true PES z1, of 0.00748 µm.
For the purpose of comparison, we also design a mixed H2/H∞ controller for the
disk drive using the approach in Theorem 6.1. The minimum H2 norm of 0.01013µm is obtained. Starting from this controller, we carried out a further iterative pro-
cedure between controller variables and Lyapunov parameters:
Step 1: obtain the closed-loop system (A, B, C, D) with the controller parameters
(Ac, Bc, Cc, Dc), and let A = A, B = B, C = C, and D = D.
Step 2: solve LMIs (5.14)−(5.15), (5.20) for P and X, and minimize Trace(Π).If Trace(Π) does not differ from the previous value, stop. Otherwise, go to step 3.
Step 3: With the obtained P and X, solve LMIs (5.14)−(5.15), and (5.20) for
(Ac, Bc, Cc, Dc).
Step 4: go to Step 1.
The iterative procedure gives a controller that produces a slight improvement of
the H2 norm, i.e., 0.01002 µm. Hence, the improved approach represents about
25.3% more improvement on TMR than the design method in Theorem 6.1 together
with an iterative refinement.
Figure 6.4 shows the comparison of sensitivity functions, where we can see that
the sensitivity function designed based on the improved design method is better than
that from Theorem 6.1, except that its hump is slightly higher. The comparison of
control performances obtained by the improved method and that of Theorem 6.1
is given as in Table 6.1, where the bandwidth from the improved method is much
higher. Although the H∞ norm of TWu of the improved design is slightly higher
than that of the design using Method 1, it is below one as required, implying the
designed controller makes the closed-loop system robustly stable in the presence of
uncertainty bounded by Wu. Further, from Figure 2.23, the disturbance mainly con-
centrates on frequencies below 1 kHz, hence, the slight higher peak does not degrade
the disturbance rejection performance much. Figure 6.5 shows the testing result of
sensitivity functions, which is consistent with the simulation results in Figure 6.4.
REMARK 6.6 From Table 6.1, one may argue that a reduced H2 normfor the method of Theorem 6.1 may be obtained by a γ value of greaterthan one as the actual H∞ norm is 0.78, lower than that of the improvedmethod. However, based on our simulations, no obvious improvement on theH2 performance has been observed. For example, with γ = 1.5, a slightlyreduced H2 norm of 10.11 nm is obtained whereas the actual H∞ norm is0.79. With γ = 5, the H2 norm is reduced to 10.08nm with an unchangedH∞ norm of 0.78. This means that with a larger γ, the improvement on theH2 norm for the method of Theorem 6.1 is negligible.
Mixed H2/H∞ Control Design for Vibration Rejection 129
The improved method is also compared with conventional PID design. As shown
in Figure 6.4 and Figure 6.5, the sensitivity function by PID control has a lower
bandwidth and almost the same peak value, which gives a higher TMR as listed in
Table 6.1.
The previous simulation is carried out with Σ1 and Σ2 as solutions to the LMI
(6.32). When (3,1) and (3,2) blocks in (6.32) are set to zeros, it is found that the
designed controller gives a 9% lower bandwidth for the closed-loop system, leading
to a worse TMR. This demonstrates that the parameter Σ in Lemma 6.1 is useful in
achieving a better performance.
103
104
−30
−25
−20
−15
−10
−5
0
5
10
Frequency(Hz)
Magnitude(d
B)
Method 1
Method 2
PID
FIGURE 6.4
Frequency response of sensitivity functions.
Next, we shall calculate the H2 norm using the measured plant frequency response
as in Figure 6.3 and sensitivity function as in Figure 6.5. The spectrum of z1 is given
by
|z1(fk)|2 = |P (fk)S(fk)|2|D1(fk)|2 + |S(fk)|2|D2(fk)|2
+|1 − S(fk)|2|N(fk)|2 (6.55)
130 Modeling and Control of Vibration in Mechanical Systems
103
104
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
5
10
Magnitude(d
B)
Frequency(Hz)
Method 1
Method 2
PID
FIGURE 6.5
Frequency response of sensitivity functions.
Mixed H2/H∞ Control Design for Vibration Rejection 131
TABLE 6.1
Control performance comparison
Method Method 2 Method 1 PID
Open-loop crossover frequency (Hz) 1.4k 969 976
H2 norm (σ (nm)) 7.48 10.13 12.11
‖TWu‖∞ 0.89 0.78 0.27
where fk (k = 1, 2, 3..., K) are frequency points, P (fk) represents the measured
frequency response of the plant, and S(fk) is the frequency response of the sensi-
tivity function. The resultant σ value of z1 is 0.01155, better than 0.02714 with the
controller designed by the method in Theorem 6.1.
6.6 Conclusion
This chapter has presented two design methods for the mixed H2 and H∞ control.
One is a slack variable approach, and another one is a less or equally conservative
design in terms of LMIs that contain more free variables than the conventional ap-
proaches. Those variables offer additional freedoms in optimization, resulting in a
less or equally conservative control design. The improved H2/H∞ control design
has been applied to hard disk drives to minimize the track misregistration while guar-
anteeing the system robustness in the presence of actuator uncertainties. Compared
with the slack variable method, the improved design result for hard disk drives has
indicated a marked improvement of 25.3% in the H2 norm or the TMR performance
while guaranteeing the system robustness by satisfying the required H∞ constraint.
The experimental result validates the advantage of the improved design.
7
Low-Hump Sensitivity Control Design forHard Disk Drive Systems
7.1 Introduction
In feedback control systems, sensitivity functions are critical to the determination of
their ability in disturbance and noise rejections. However, Bode has shown the limi-
tation of using a feedback structure in terms of an integral constraint on the sensitivity
function, as discussed in Chapter 5. Briefly speaking, the Bode integral theorem im-
plies that we cannot have a sensitivity function less than unity at all frequencies using
output feedback with a finite-bandwidth controller. Such a sensitivity function must
amplify the disturbances existing in frequencies higher than the system bandwidth.
In view of this, we shall employ the special structure of a secondary actuator system
and design appropriate controllers for primary and secondary actuators such that the
hump of the sensitivity function comes as low as possible without the cost of low-
frequency performance. This optimized sensitivity function is expected to minimize
the amplification of high-frequency disturbances while attenuating low-frequency
and mid-frequency disturbances. With this low-hump sensitivity, the dual-stage con-
trol system is able to reduce the contribution from all existing disturbances to the
error. Two types of microactuator models are considered in this chapter: a MEMS
actuator [102] and a PZT actuator [103]. The purpose is to design controls for the
primary and the secondary actuators such that a low hump of the sensitivity func-
tion can be achieved with the help of secondary actuators. A comparison will be
made to evaluate the effectiveness of the proposed method for the two microactua-
tors. Besides simulations, an implementation with a PZT microactuator verifies that
the HDD servo loop design method leads to a low-hump sensitivity function.
7.2 Problem statement
Figure 7.1 shows a dual-stage actuation system with one primary actuator Pv(s) and
one secondary actuator Pm(s), and two parallel controllers Cv(s) and Cm(s). With
disturbances and noise injected, the error is contributed by the disturbances and noise
133
134 Modeling and Control of Vibration in Mechanical Systems
in terms of
Se = |Pv(fk)S(fk)|2|d1(fk)|2 + |S(fk)|2|d2(fk)|2 + |S(fk)|2|n(fk)|2, (7.1)
where Se is the power spectrum of the error e, S is the sensitivity function, and
fk(k = 1, 2, · · · , N) are frequency points. Figure 7.2 shows an example of main
disturbances existing in disk drive systems, where the low frequency components are
represented by d1, the higher frequency portions such as disk vibration and windage
are lumped as d2, and the base line of the spectrum stands for the noise n.
FIGURE 7.1
Parallel structure of a dual-stage actuation system with disturbances and noise in-
jected.
Equation (7.1) implies that the sensitivity function S is important in determining
the disturbance rejection of the dual-stage closed loop control system. In a conven-
tional single-stage system using one primary actuator it is difficult to have a low-
hump sensitivity function. Thus the dual-stage structure is applied to increase the
system bandwidth and lower the sensitivity function peak such that
sup |S(jω)| ≤ 1 + τ, ω = 0, · · · , ∞ (7.2)
where 0 < τ << 1 is a sufficiently small tolerance. The sensitivity function satisfy-
ing (7.2) is called a nearly “flat” sensitivity function.
A dual-stage actuation system uses a microactuator to increase the system band-
width. Among many control schemes for the dual-stage control loop, Figure 7.1 is
one of the most popular ones [106]. The overall sensitivity function S(s) is
S(s) =1
1 + Pv(s)Cv(s) + Pm(s)Cm(s). (7.3)
Another popular control scheme is the decoupled structure as shown in Figure 7.3
[101], where the overall sensitivity function of the closed-loop system is given by
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 135
cascading each sensitivity function, i.e.,
S(s) = Sm(s)Sv(s),
Sv(s) =1
1 + Pv(s)Cv(s),
Sm(s) =1
1 + Pm(s)Cm(s). (7.4)
In this chapter we are mainly concerned with the parallel structure as in Figure 7.1
since it is a basic structure for dual-stage control systems and can be converted to the
decoupled master-slave structure.
FIGURE 7.2
Power spectrum of PES nonrepeatable runout in open loop.
136 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.3
Decoupled structure of dual-stage actuation systems.
FIGURE 7.4
Structure of H∞ loop shaping.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 137
7.3 Design in continuous-time domain
7.3.1 H∞ loop shaping for low-hump sensitivity functions
The H∞ loop shaping method is used to design controllers for the primary actuator
and the microactuator to achieve a low-hump sensitivity function for the dual-stage
actuator system. The structure of the H∞ loop shaping method is depicted in Figure
7.4, where W (s) is a weighting function of the desired sensitivity function. For
a plant model P (s), a controller C(s) is to be designed such that the closed-loop
system is stable and
‖Tzw‖∞ < 1 (7.5)
is satisfied, where Tzw is the transfer function from w to z, i.e., S(s)W (s). Clearly,
(7.5) means that the sensitivity function S(s) can be shaped similarly to the inverse
of the chosen weighing function W (s). A simple form of W (s) is
W (s) =
1M
s2 + 2ωζ 1√M
s + ω2
s2 + 2ω√
εs + ω2ε, (7.6)
where ω is valued by the desired bandwidth, ε is to determine the low-frequency
level of the desired sensitivity function, and ζ is the damping ratio.
Associated with the weighting function, Figure 7.4 can be formulated as follows.
x(t) = Ax(t) + B1w(t) + B2u(t), (7.7)
z(t) = C1x(t) + D11w(t) + D12u(t), (7.8)
y(t) = C2x(t) + D21w(t) + D22u(t), (7.9)
where
A =
[
Ap 0BwCp Aw
]
, B1 =
[
0Bw
]
, B2 =
[
Bp
BwDp
]
,
C1 =[
DwCp Cw
]
, D11 = Dw , D12 = DwDp,
C2 =[
Cp 0]
, D21 = 1, D22 = Dp,
(Ap, Bp, Cp, Dp) and (Aw , Bw, Cw, Dw) are respectively the state-space realiza-
tions of plant P (s) and weighting function W (s). Let (Ac, Bc, Cc, Dc) be the state
space description of C(s). Then (Ac, Bc, Cc, Dc) is to be designed such that (7.5)
is satisfied. An LMI approach stated in Theorem 5.4 is used to design the controller.
It is known that MATLAB functions, say “hinfsyn.m,” are available to design
the controller. However, numerical errors will occur due to the large gain of VCM
actuator and will be the hindrance for running the function. Thus, we would rather
use the LMI approach in our application to the VCM actuator. There always exists
a minimum level γ that makes the LMI (5.99) solvable, which gives a sensitivity
138 Modeling and Control of Vibration in Mechanical Systems
function closer to the inverse of its weighting function than that given by a larger γ.
Certainly, the solvability is also related to the chosen weighting W (s), which must
be realistic due to the Bode limitation.
Bode’s integral theorem allows the possibility of a “flat” sensitivity function up to
a frequency of our concern in the continuous time domain, since the integral con-
straint in (5.136) is defined on the frequency range from 0 to infinity. Equation (7.2)
can be achieved by choosing appropriate weighting functions for Sv and Sm sep-
arately. One way is to put the peak of Sv within the reduction band of Sm, as in
Figure 7.5, and lower the high-frequency hump of Sm. This would need to decrease
the bandwidth of the primary actuator loop and increase the bandwidth of the mi-
croactuator loop, which can be realized by adjusting the weighting functions Wv(s)for the primary actuator and Wm(s) for the microactuator. Here, the primary actuator
loop and the microactuator loop are designed separately for the dual-stage parallel
structure. An additional reason for this is to meet the goals of ensuring the stability
of the separate primary and microactuator loops and as in Figure 7.6, letting the pri-
mary actuator open loop have a higher gain at low frequencies and the microactuator
open loop have a higher gain at high frequencies. The dual-stage parallel control
scheme could be formulated as an MIMO problem, however, to satisfy these specific
requirements, the controller design would be more complicated.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 139
FIGURE 7.5
Frequency responses of Sv(s) and Sm(s).
140 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.6
Frequency responses of Pv(s)Cv(s) (solid line) and Pm(s)Cm(s)(dotted line).
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 141
7.3.2 Application examples
This section will apply the proposed control design method to a dual-stage actuation
system in hard disk drives with the structure as in Figure 7.1 such that the designed
controllers can produce a low-hump sensitivity function. The dual-stage HDD uses
a microactuator as a fine positioner to increase the positioning accuracy. The mi-
croactuator piggyback on a VCM actuator is driven jointly with the VCM actuator
through suspension, slider or head [101]. Due to various microactuator designs, two
different microactuator models will be studied and they are coupled separately with
one VCM actuator.
Figure 7.7 shows the frequency response of the VCM actuator and the poles, zeros
and gain of its model are listed in (7.10)−(7.11). With ω = 2π350, ε = 10−3.2,
ζ = 0.4 in (7.6), VCM controller Cv is designed as in Figure 7.8 using the H∞ loop
shaping method described in the previous section. The sensitivity function |Sv(s)| is
shown in Figure 7.5.
FIGURE 7.7
Frequency response of VCM actuator P (s).
poles = [−2312.2122± 57759.0421j, −2670.3538± 53340.2745j,
−779.1150± 38947.9570j, −942.4778± 31401.7862j,
−219.9115± 588.5772j, −18849.5560]; (7.10)
142 Modeling and Control of Vibration in Mechanical Systems
zeros = −2356.1945± 47064.9481j,
Gain = 2.127723028788067× 1040. (7.11)
FIGURE 7.8
Frequency response of VCM controller Cv(s).
Various microactuators have been proposed for HDD dual-stage servo systems,
including the MEMS actuator and PZT actuator, which are designed by different
mechanism and possess different dynamics. They can generally be characterized by
a pade delay and a 2-pole roll-off model:
PadeDelay × O(s2) (7.12)
According to their physical behavior and the locations of the poles, microactuators
are categorized as two cases. The low-hump design method in Section 7.3.1 will be
applied to the two cases to investigate their effect on the ultimate dual-stage sensi-
tivity function.
Case 1: Poles are located at low frequency, such as a MEMS actuator [101]
Pm(s) =2 × 106
s2 + 282.7s + 8.883× 107. (7.13)
Equation (7.13) is a special case of (7.12) without delay. With ε = 10−0.6, ζ = 1and ω = 2π2391 in (7.6), the designed controller Cm(s) is shown in Figure 7.9.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 143
Figure 7.10 implies that the resultant sensitivity function Sm(s) follows its weighting
function closely. The dual-stage servo system has a gain margin of 17 dB and a phase
margin of 92, and its open loop frequency response is shown in Figure 7.11. A
“flat” sensitivity function is achieved as shown in Figure 7.12 even though the Bode
integral limitation (5.136) has to be fulfilled in this case.
FIGURE 7.9
Frequency response of microactuator controller Cm(s).
144 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.10
Sensitivity function Sm(s) (solid) and its weighting function inverse (dashed).
FIGURE 7.11
Open loop frequency response of the dual-stage system.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 145
FIGURE 7.12
Sensitivity and complementary sensitivity functions.
Case 2: Poles are located around 10 kHz, such as the active piezoelectric suspen-
sion modeled by
Pm(s) =
−1.438572836× 109(s − 6.157× 105)(s2 + 923.7s + 1.934× 109)
(s + 6.157× 105)(s2 + 791.7s + 1.517× 109)(s2 + 5089s + 7.195× 109)
(7.14)
with the frequency response in Figure 7.13.
For the control design of (7.14), ε is kept unchanged to have the same low-
frequency level of the sensitivity function as in Case 1, and ω is adjusted to be
2π2107 to have a similar open-loop bandwidth. The resultant Cm(s) is shown in
Figure 7.14. The designed Sv(s) and Sm(s) are matched as in Figure 7.5, which
subsequently leads to the overall sensitivity as in Figure 7.16. We can observe that
a “flat” sensitivity function has been achieved. The frequency response of the open
loop dual-stage system is shown in Figure 7.15, where the gain margin is 31 dB and
the phase margin is 88. Note that a non-minimum phase zero is included in the
microactuator model and a nearly “flat” sensitivity function is still achieved.
REMARK 7.1 The control design is based on the relevant sensitivityfunction only and does not consider robustness to plant pole/zero variations.This will, however, not hamper the practical usability of the resulting con-trollers due to the large gain and phase margins.
146 Modeling and Control of Vibration in Mechanical Systems
TABLE 7.1
Control performance comparison.
Microactuator model type Case 1 Case 2
Gain Margin (dB) 17 31Phase Margin (deg) 90 88Sensitivity function Flat Flat
Control performances are summarized in Table 7.1 for the two dual-stage systems
consisting of different microactuators. It can be seen that they both achieve a “flat”
sensitivity function with bandwidth of 2 kHz. Case 2, the PZT actuated suspension,
looks better than Case 1, the MEMS actuator, in the sense that its gain margin is much
higher and phase margin is slightly lower, whereas its behavior is to be improved to
possess a characteristic as simple as a pade-delay which relies on its manufacturing
process. Also, from a manufacturing point of view, the MEMS actuator benefits from
its mass production and low cost.
FIGURE 7.13
Microactuator frequency response.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 147
FIGURE 7.14
Frequency response of microactuator controller Cm(s).
FIGURE 7.15
Open loop frequency response of the dual-stage system.
148 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.16
Sensitivity and complementary sensitivity functions.
7.3.3 Implementation on a hard disk drive
The implementation is carried out for Case 2 in the previous section, i.e., the dual-
stage system consisting of the VCM in Figure 7.7 and the PZT microactuator in
Figure 7.13. The VCM and PZT controllers are obtained by discretizing the designed
controllers with frequency responses shown in Figure 7.8 and 7.14 and sampling rate
of 40 kHz.
The controllers are implemented with dSpace 1103 on TMS320C240 DSP board.
An LDV is used to measure the position of the dual-stage actuator as the feedback
signal. The displacement range used is 2 µm/V. A DSA is used to measure the
frequency response of the sensitivity function by injecting a swept sine signal at
point A in Figure 7.17. The frequency response of points B over A is the sensitivity
function.
The resultant sensitivity functions are shown in Figure 7.18, where the rough line
is the tested result and the smooth line is the simulation result. We can observe that
the hump of the sensitivity function is lower than 3 dB, which is better than that with
the following PID-like controllers designed by the lead-lag method:
Cv(z) =0.9z2 − 1.778z + 0.8788
z2 − 1.532z + 0.5326, (7.15)
Cm(z) =0.4081z + 0.2627
z2 − 1.209z + 0.2631. (7.16)
The tested and simulation results of the dual-stage open loop are shown in Figure
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 149
7.19. The step response of the dual-stage system is shown in Figure 7.20, where
Channel 1 is the output corresponding to the reference Channel 4, Channel 2 is the
control signal of VCM and Channel 3 is the control signal of PZT.
Note that the hump of the sensitivity function S(z) is tested to be 3 dB, instead of
0 dB as in Figure 7.16. This is mainly due to the locations of the microactuator poles
which are around 10 kHz and not sufficiently far away from the bandwidth of 2 kHz.
We know that a pole at 20 kHz may not affect 2 kHz bandwidth that much and thus
it is not needed to include a pole far beyond 20 kHz in the actuator modeling for 2kHz servo bandwidth. Moreover, an actuator with high-frequency poles is helpful to
the low-hump sensitivity function design. It is expected that the microactuator could
be modeled as a pade-delay within the frequency range of our interest. In the next
section we discuss the possibility of achieving low-hump sensitivity functions in the
discrete time domain.
According to (7.1), the sensitivity function as in Figure 7.18 must amplify the cor-
responding high-frequency disturbances due to the hump above 0 dB, and certainly
the one with the lower hump will result in less amplification.
The position error is evaluated according to (7.1) with the sensitivity function in
Figure 7.18 associated with the PES spectrum in Figure 7.2. The Parseval’s formula
stated in Chapter 1 addresses how to calculate σ value in the frequency domain. As
such, Figure 7.21 shows the 3σ value of position error versus frequencies. The low-
hump design outperforms the PID design for disturbance rejection with almost the
same performance at frequencies lower than 1.5 kHz, which is consistent with Figure
7.18.
FIGURE 7.17
Experimental structure.
150 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.18
Sensitivity function of the dual-stage system (smooth line: simulation result; rough
line: testing result; dotted line: PID design).
FIGURE 7.19
Open loop frequency response of the dual-stage system (smooth line: simulation
result; rough line: testing result.)
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 151
FIGURE 7.20
Step response of the dual-stage system.
FIGURE 7.21
3σ of PES NRRO versus frequencies.
152 Modeling and Control of Vibration in Mechanical Systems
7.4 Design in discrete-time domain
This part is concerned with servo loop control design in the discrete-time domain
that aims to achieve a low-hump sensitivity function for dual-stage actuator systems.
Before the control design, sensitivity functions for a discrete-time dual-stage system
will be analyzed on the basis of the Bode integral theorem.
7.4.1 Synthesis method for low-hump sensitivity function
We recall a simplified Bode’s theorem which is helpful to analyze the sensitivity
function of the dual-stage servo system.
THEOREM 7.1
Discrete Bode’s Theorem [104] Consider a discrete-time SISO LTI systemwhose open-loop transfer function G(z) does not have unstable poles. Thesensitivity function of the system is given by S(z) = 1/(1 + G(z)). If theclosed-loop system is stable and kg = limz→∞G(z), then
∫ π
−π
ln |S(ejω)|dω = 2π(−ln|kg + 1|). (7.17)
Based on the discrete Bode’s theorem, we have the following statements.
a). When G(z) is strictly proper, then limz→∞G(z) = 0 and
∫ π
−π
ln |S(ejω)|dω = 0.
b). When the orders of the denominator and numerator of G(z) are the same and
kg < −2 or kg > 0, then
∫ π
−π
ln |S(ejω)|dω < 0.
c). When the orders of the denominator and numerator of G(z) are the same,
−2 ≤ kg < 0 and kg 6= −1, then
∫ π
−π
ln |S(ejω)|dω ≥ 0.
Thus, we can conclude that only if G(z) is non-strictly proper and kg < −2 or
kg > 0, then
|S(ejω)| ≤ 1, ω ∈ [0, π],
i.e., the sensitivity function is bounded by 0 dB.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 153
The open-loop transfer function of the dual-stage parallel system is the sum of
each path, i.e.,
G(z) = Gv(z) + Gm(z), (7.18)
Gv(z) = Pv(z)Cv(z), (7.19)
Gm(z) = Pm(z)Cm(z). (7.20)
Note that a VCM actuator can be approximately represented by a double integrator
(i.e., k/s2) combined with some resonance modes and is often described by a strictly
proper model. When Gm(z) is strictly proper, G(z) is also strictly proper and when
Gm(z) is non-strictly proper, G(z) is non-strictly proper. It can be concluded that
the Bode’s integral of the overall dual-stage loop is determined by the microactuator
loop. Only a non-strictly proper model of microactuator could possibly produce a
“flat” sensitivity function for dual-stage servo systems. An additional condition for
the “flat” sensitivity is kg < −2 or kg > 0. These are necessary conditions and could
be used as a criterion to examine the closed-loop design.
In what follows, the discrete H∞ loop shaping method will be applied to the con-
trol designs of the VCM loop and the microactuator loop such that the sensitivity
functions of the two loops are coupled and achieve a low-hump overall sensitivity.
The structure of the H∞ loop shaping method is the same as in Figure 7.4. In the
discrete time case, Figure 7.4 is formulated as follows.
x(k + 1) = Ax(k) + B1w(k) + B2u(k), (7.21)
z(k) = C1x(k) + D11w(k) + D12u(k), (7.22)
y(k) = C2x(k) + D21w(k) + D22u(k), (7.23)
where
A =
[
Ap 0BwCp Aw
]
, B1 =
[
0Bw
]
, B2 =
[
Bp
BwDp
]
C1 =[
DwCp Cw
]
, D11 = Dw, D12 = DwDp,
C2 =[
Cp 0]
, D21 = 1, D22 = Dp,
(Ap, Bp, Cp, Dp) and (Aw, Bw , Cw, Dw) are respectively the state-space real-
izations of plant P (z) and weighting function W (z). Let (Ac, Bc, Cc, Dc) be a
state space description of C(z). An LMI approach stated in Chapter 5 will be used
to design the controller C(z) : (Ac, Bc, Cc, Dc).
7.4.2 An application example
The VCM actuator and the microactuator are the same as those in Case 2 in Section
7.3.2. Notice that the PZT microactuator is represented using a Pade-delay with two
2nd order resonance terms. The form (7.14) can be regarded as a general model of
PZT actuated suspensions. It is strictly proper and thus according to the analysis in
154 Modeling and Control of Vibration in Mechanical Systems
Section 7.4.1, a “flat” sensitivity function is impossible. The used sampling rate for
the controller design is 40 kHz. The discrete-time actuator models are obtained using
“zero-order-hold” method to ensure that the designed controllers are implementable.
With ω = 2π350, ε = 10−3.2, ζ = 0.4 in (7.6), a VCM controller Cv(z) is de-
signed as in Figure 7.22 using the H∞ loop shaping method in the previous section.
Also applying the LMI approach, the controller Cm(z) for the microactuator is de-
signed as in Figure 7.23 with ω = 2π3700, ε = 10−1.38, ζ = 1, and M = 0.071/2
in (7.6). The sensitivity function of the dual-stage system is shown in Figure 7.24,
where we can observe that the hump is below 3 dB. The open-loop system has the
gain margin of 8.4 dB, the phase margin of 64 and the bandwidth of 2.49 kHz.
REMARK 7.2 If manufacturing processes could produce an ideal actu-ator that can be modeled by a Pade-delay only, which is non-strictly proper,it is possible to have a “flat” sensitivity like that in Figure 7.25 with kg > 0.
An actual microactuator, which resembles the ideal case, is the PZT microactuator
in [107]. It can be described by a 1-pole roll-off model
Pm(s) =1257
s + 1.257× 105, (7.24)
which is strictly proper and does not satisfy the necessary conditions in Section 7.3.1.
The sensitivity function with the hump lower than 2 dB, as shown as in Figure 7.26,
can be obtained with the same weighting function as that for (7.14).
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 155
FIGURE 7.22
VCM controller Cv(z).
FIGURE 7.23
Microactuator controller Cm(z).
156 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.24
Sensitivity function of the dual-stage system.
FIGURE 7.25
Sensitivity function.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 157
FIGURE 7.26
Sensitivity function of the dual-stage system.
FIGURE 7.27
Frequency responses of Pv(z)Cv(z) (solid curve) and Pm(z)Cm(z) (dashed curve).
158 Modeling and Control of Vibration in Mechanical Systems
7.4.3 Implementation on a hard disk drive
The experimetal setup is the same as in Figure 7.17. An LDV with a range of 2µm/V is used to measure the position of the dual-stage actuator. Controllers are
implemented with dSpace 1103 on a TMS320C240 DSP board. When the dual-
stage loop is closed and stabilized with the designed controllers, a swept sine signal
is injected at point A. A DSA is then used to measure the frequency response of
points B over A and obtain the sensitivity function.
The resultant sensitivity function is shown in Figure 7.28, where the rough line is
the testing result and the smooth line is the simulation result. We can observe that the
hump of the sensitivity function is lower than 3 dB, which is better than that by a PID
design as shown by the dotted line in Figure 7.28. The testing and simulation results
of the dual-stage open loop system are shown in Figure 7.29. The step response in
Figure 7.30 shows that the system is stabilized and working in real time. Channel 2
in Figure 7.30 is the control signal of the VCM actuator and Channel 3 is the control
signal of the PZT microactuator.
The sensitivity function as in Figure 7.28 will amplify the corresponding high-
frequency disturbances shown in Figure 7.2 due to the hump above 0 dB. The po-
sition error is evaluated from (7.1) with the designed sensitivity functions. The 3σvalue of the position error versus frequencies is shown in Figure 7.31, where we can
see that the low-hump design outperforms the PID design for disturbance rejection
after 2.4 kHz, which is consistent with Figure 7.28.
The proposed control design is based on the sensitivity weighting function only
and does not consider robustness to plant parameter variations. This, however, will
not hamper the practical application of the resulting controllers due to the large gain
and phase margins. The system is verified to maintain stability in spite of the vari-
ation of resonance frequency, e.g., ±5% shift of PZT resonance frequency around
13.5 kHz. The performance change with the presence of the system parameter un-
certainty is illustrated in Figure 7.31, where we can see that the performance varies
slightly.
7.5 Conclusion
An H∞ method has been proposed in both continuous and discrete time domains
to achieve a low-hump sensitivity function for dual-stage HDD systems using an
LMI approach. Two different microactuator models have been studied, which are
represented by a MEMS actuator and a PZT actuated suspension. With the proposed
selection of sensitivity weighting functions, the sensitivity function with a hump
below 3 dB has been achieved in both simulations and experiments. Such a design
process can generate a robust servo controller with high disturbance rejection in a
low frequency range, and less vibration amplification in a high frequency range, and
thus is effective in achieving higher positioning accuracy.
Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 159
FIGURE 7.28
Sensitivity function of the dual-stage system.
FIGURE 7.29
Open loop frequency responses of the dual-stage system.
160 Modeling and Control of Vibration in Mechanical Systems
FIGURE 7.30
Step response of the dual-stage system.
FIGURE 7.31
3σ value of PES NRRO versus frequency.
8
Generalized KYP Lemma-Based LoopShaping Control Design
8.1 Introduction
To shape frequency responses of closed-loop transfer functions such as sensitiv-
ity/complementary sensitivity functions, H∞ optimization together with frequency
weighting is a commonly used method. The additional weight functions, however,
increase the system and controller complexity since the weighting functions usually
have to be of high orders in order to capture the desired specifications accurately.
This is especially so when a controller is to be designed, aiming at achieving a wider
bandwidth while simultaneously suppressing disturbances of particular frequencies
within or beyond the servo bandwidth. Further, the process of choosing appropriate
weights is tedious and time-consuming.
The KYP Lemma [61], being one of the most fundamental results in systems the-
ory and control, establishes the equivalence between a frequency domain inequality
(FDI) for a transfer function and a linear matrix inequality associated with its state
space realization. It allows us to characterize various properties of dynamic systems
in the frequency domain in terms of linear matrix inequalities. The standard KYP
Lemma is only applicable for the infinite frequency range, while the generalized
KYP Lemma [62] establishes the equivalence between a frequency domain prop-
erty and a linear matrix inequality over a finite frequency range, allowing designers
to impose performance requirements over chosen finite or infinite frequency ranges.
Hence, it is very suitable for analysis and synthesis problems in practical applications
where different specifications over different frequency ranges are usually required.
In this chapter, the generalized KYP Lemma is applied to design a feedback con-
trol such that the specifications of the sensitivity function, required to suppress some
specific frequency disturbances, are satisfied. Unlike the standard KYP Lemma, the
matrix inequality in the generalized KYP Lemma involves a matrix variable which is
not necessarily positive definite and thus the Schur complement cannot be applied to
convexify the controller design. To overcome this difficulty, the Youla parametriza-
tion approach is used to parameterize the closed-loop transfer function. The search
for the coefficients of the parameter Q(z) is then converted to a linear matrix in-
equality problem within the generalized KYP Lemma framework. An application of
the method in the rejection of narrowband high-frequency and mid-frequency distur-
161
162 Modeling and Control of Vibration in Mechanical Systems
bances is presented to demonstrate the simplicity of the design and the improvement
of the positioning accuracy by the resultant controller.
8.2 Problem description
It is known that the power spectrum of the error e in Figure 2.20 is given by
Se = |P (z)S(z)|2|d1|2 + |S(z)|2|d2|2 + |S(z)|2|n|2 (8.1)
which implies that the sensitivity function S(z) is important in determining the dis-
turbance rejection of a control loop. Thus, the design of a controller that gives
rise to a sensitivity function which can reject specific disturbances with known fre-
quencies becomes rather significant. The purpose here is then stated as: to de-
sign a dynamic feedback controller C(z) for plant P (z) such that the closed-loop
system is stable and for some prescribed positive scalars ri and frequency ranges
(fi1, fi2), i = 1, ..., N ,
|S(f)| < ri, fi1 ≤ f ≤ fi2 (8.2)
where S(f) = 1/(1 + C(f)P (f)). Smaller ri means the less contribution of the
disturbance in frequency range (fi1, fi2) to the error.
In view of the constraint stated in the Bode integral theorem, it is impossible to
achieve disturbance rejection at all frequencies higher than the bandwidth of the con-
trol loop for actuators or microactuators used in mechanical motion systems such as
hard disk drives. The specification (8.2) is considered for a specific frequency range,
which is however possible to be achieved through shaping the sensitivity function.
The above design problem may be approached by selecting a proper frequency
weighting function and carrying out an H∞ optimization. However, the problem
of how to select a proper frequency weighting function that can give an accurate
shaping of the sensitivity function is generally difficult and time-consuming. Further,
the resultant controller order will depend on the order of the weighting function and
the plant. Note that a more accurate frequency shaping usually requires a higher
order weighting function.
The generalized KYP Lemma [62] gives a necessary and sufficient condition for a
given transfer function to satisfy a required frequency domain property over a finite
frequency range in terms of a matrix inequality condition. Thus, it may be applied to
address the above design problem. In what follows, we employ the generalized KYP
Lemma to design a feedback control such that the sensitivity function satisfies the
required specifications so as to reject disturbances at specific frequencies. In order
to convexify the matrix inequality, the Youla parametrization approach as shown in
Figure 8.1 is used to design a controller with the generalized KYP Lemma.
Generalized KYP Lemma-Based Loop Shaping Control Design 163
FIGURE 8.1
Q parameterization for control design.
8.3 Generalized KYP lemma-based control design method
The previous analysis indicates that the sensitivity function plays a key role in distur-
bance rejection. To reject disturbance of frequency within a certain frequency range,
a proper shaping of the sensitivity function can be carried out. In this section, we
shall present a frequency shaping method based on the generalized KYP Lemma.
First, it is easy to see that the sensitivity function S(z) is equal to the transfer
function from w to z in Figure 8.1. The state-space model of the plant under consid-
eration is denoted as (Ap, Bp, Cp, Dp). Then, a state-space representation of the
system in Figure 8.1 is given by
x(k + 1) = Apx(k) + Bpu(k), (8.3)
z(k) = −Cpx(k) + w(k) − Dpu(k), (8.4)
where x ∈ Rnx is the state.
Let a state-space representation of the controller C(z) be given by (Ac, Bc, Cc,
Dc). Assuming that D = 1 + DcDp is invertible, then a state-space representation
of the sensitivity function can be given by (A, B, C, D), where
A =
[
Ap − BpD−1DcCp BpD−1Cc
−BcCp + BcDpD−1DcDp Ac − BcDpD−1Cc
]
, (8.5)
B =
[
BpD−1Dc
Bc − BcDpD−1Dc
]
,
C =[
−Cp + DpD−1DcCp − DpD−1Cc
]
, D = 1 − DpD−1Dc. (8.6)
A special case of the generalized KYP Lemma that relates the bounded realness of
the sensitivity function over finite frequency ranges to its state space representation
is given below.
164 Modeling and Control of Vibration in Mechanical Systems
LEMMA 8.1
[62] Consider the sensitivity function S(z) = C(zI − A)−1B + D with A beingstable. Then, for a given scalar r > 0, |S(ejθ)| ≤ r over a finite frequencyrange if and only if there exist Hermitian matrices U and V ≥ 0 such that
[
A BI 0
]∗
Σ
[
A BI 0
]
+
[
0 00 −r2
]
[
C D]∗
[
C D]
−I
≤ 0, (8.7)
where
(i) for low frequency range |θ| ≤ θl,
Σ =
[
−U VV U − (2cosθl)V
]
; (8.8)
(ii) for middle frequency range θ1 ≤ θ ≤ θ2,
Σ =
[
−U ejθcVe−jθcV U − (2cosθd)V
]
,
θc = (θ1 + θ2)/2, θd = (θ2 − θ1)/2; (8.9)
(iii) for high frequency range |θ| ≥ θh,
Σ =
[
−U −V−V U + (2cosθh)V
]
. (8.10)
For a given controller C(z), the above gives a necessary and sufficient condition
for evaluating if |S(z)| ≤ r over some given frequency range in terms of LMI.
However, (8.7) is no longer an LMI when the controller C(z) = (Ac, Bc, Cc, Dc)is to be designed since A and B involve the unknown parameters Ac, Bc, Cc and
Dc. Further, it is not possible to be converted to an LMI by Schur complement since
the matrix Σ is not definite. To overcome this difficulty, in the following we apply a
Youla parametrization approach where the controller C(z) is now of the structure as
shown in Figure 8.1.
Let K(z) be a given observer based controller that can be designed using ap-
proaches such as the LQG control:
x(k + 1) = Apx(k) + Bpu(k) + L(z(k) + Cpx(k)), (8.11)
u(k) = −Mx(k). (8.12)
Then, as stated in Chapter 5, the set of sensitivity functions can be parameterized as
S(z) = T11(z) + T12(z)Q(z)T21(z) (8.13)
Generalized KYP Lemma-Based Loop Shaping Control Design 165
where Q(z) is a stable transfer function to be designed and
[
T11(z) T12(z)T21(z) 0
]
= CT (zI − AT )−1BT + DT ,
[
AT BT
CT DT
]
=
Ap −BpM 0 Bp
−LCp Ap − BpM + LCp L Bp
−Cp DpM 1 −Dp
−Cp Cp 1 0
. (8.14)
If Q(z) has the state-space realization (Aq , Bq, Cq, Dq), then the designed con-
troller C(z) is given by
Ac =
[
Ap − BpM + LCp + BpDqCp BpCq
BqCp Aq
]
, (8.15)
Bc =
[
L + BpDq
Bq
]
, Cc =[
−M + DqCp Cq
]
,
Dc = Dq . (8.16)
Denote the state space representations of T11(z) and T12(z)T21(z) by (At11, Bt11,
Ct11, Dt11) and (At, Bt, Ct, Dt), respectively, then from (8.13) a state space model
of S(z) can be written as
A =
At11 0 00 At 00 BqCt Aq
, B =
Bt11
Bt
BqDt
, (8.17)
C =[
Ct11 DqCt Cq
]
, D = Dt11 + DqDt. (8.18)
Let Q(z) be an FIR filter:
Q(z) = q0 + q1z−1 + q2z
−2 + ... + qτz−τ , (8.19)
q = [q0 q1 q2, ... , qτ ] (8.20)
which is to be designed so that the required bounded realness of the sensitivity func-
tion is satisfied. It is known that a state space realization for Q(z) can be given
by
Aq =
[
0 Iτ−1
0 0
]
, Bq =
[
0(τ−1)×1
1
]
,
Cq = [qτ qτ−1 · · · q1] , Dq = q0,
where Iτ−1 is the identity matrix of dimension (τ −1)× (τ −1) and 0(τ−1)×1 is the
zero matrix of dimension (τ − 1)× 1. Note that the filter parameter q to be designed
only appears in Cq and Dq. Therefore, from (8.17)−(8.18), we know that q exists
in C and D only. In this case, (8.7) defines an LMI in terms of the variables U ,
V , and q. Hence, U , V , and the design parameter q can be computed via a convex
optimization.
166 Modeling and Control of Vibration in Mechanical Systems
The KYP Lemma-based control design can be carried out following the steps:
Procedure 8.1
Step 1. Compute M and L using MATLAB commands M = dlqr(Ap, Bp, CTp Cp,
R) and L = Ap · dlqe(Ap, Bp, − Cp, Wd, Wv), where R is the weighting for the
control input in the cost function
J =∑
(xT CTp Cpx + uT Ru)
for linear quadratic regulator design, and Wd and Wv are the variance matrices of
process noise and measurement noise for the Kalman estimator design. Here in
the KYP Lemma-based control design, R, Wd, and Wv can be chosen as identity
matrices.
Step 2. Compute T11(z), T12(z) and T21(z) from (8.14).
Step 3. Obtain the state space model (A, B, C, D) in (8.17)−(8.18).
Step 4. Based on disturbance spectrum, specify the positive scalars ri and fre-
quency points fi(i = 1, ..., N) for the sensitivity function
|S(fi)| < ri, fi1 ≤ fi ≤ fi2 (8.21)
where fi1 and fi2 define the frequency range. For each specification on the resultant
sensitivity function in the frequency range, construct the LMI (8.7) in terms of the
variables U , V , and q with r = ri.
Step 5. Obtain q, U and V by solving these LMIs using MATLAB LMI toolbox.
If the LMIs are not solvable, the specifications given in Step 4 are to be adjusted.
Step 6. Obtain the controller parameters from (8.15)−(8.16).
8.4 Peak filter
Many other control methods are available to reject narrowband disturbance, such as
linear time invariant feedback methods using classical design and modern frequency
shaping and filter shaping, and adaptive feedforward methods using higher harmonic
control and LMS algorithm [57]. It has been shown that most of these methods would
result in a compensator with lightly damped complex poles at the center frequency
of the disturbance. Such a compensator has high gains at specific frequency. It is
named as peak filter according to its shape of frequency response. The peak filter
F (s) works in the control loop as in Figure 8.2 with its discretized form F (z).
8.4.1 Conventional peak filter
Conventionally, the peak filter is described as the following model
F (s) =s2 + 2ξ1ωps + ω2
p
s2 + 2ξ2ωps + ω2p
, (8.22)
Generalized KYP Lemma-Based Loop Shaping Control Design 167
FIGURE 8.2
Peak filter F in the nominal feedback loop.
FIGURE 8.3
Peak filter in the frequency domain.
168 Modeling and Control of Vibration in Mechanical Systems
where ωp = 2πfp is the center frequency in radian/sec and ξ1 and ξ2 are the damping
ratios with ξ1 > ξ2. The peak filter F (s) can be designed according to its Bode plot,
as illustrated in Figure 8.3. M and N denote the peak height, ∆ stands for the peak
width corresponding to N . Let
m = 10M/20, n = 10N/20. (8.23)
When M ≫ N , ξ1 and ξ2 are determined approximately by
ξ1 =∆2 + 2∆
2∆ + 2
√n − 1, (8.24)
ξ2 =ξ1
m. (8.25)
The maximum phase loss θ caused by the pair of the lightly damped complex poles
is given by
θ = arctanm− 1
2√
m. (8.26)
It is noticed that the conventional version of the peak filter induces additional
phase loss. The phase loss negatively impacts the phase margin and distorts the
gain of the sensitivity function around the disturbance frequency, particularly for
the disturbance near the 0 dB crossover frequency. To overcome the drawback, a
phase-lead peak filter [118] was proposed.
8.4.2 Phase lead peak filter
The phase filter is improved by adding a differentiator to provide additional phase
lead such that the phase margin is preserved and the sensitivity function curve is
smoothly shaped.
Let
T0(s) =P (s)C(s)
1 + P (s)C(s). (8.27)
The peak filter adopts the following form
F (s) = K−sin(φ)s2 + ω0cos(φ)s
s2 + 2ξω0s + ω20
, (8.28)
where ω0 is the disturbance frequency at which high disturbance rejection is required,
ξ ∈ (0, 1) is the damping ratio, φ is the phase angle determined by
φ = arg[T0(jω0)] ∈ [−π, π], (8.29)
and 0 < K < γ is the positive filter gain. Let
G(ω, K) = 1 + T0(jω)F (jω), (8.30)
Generalized KYP Lemma-Based Loop Shaping Control Design 169
and γ is the minimal positive real solution of the following two equations:
Re[G(ω, γ)] = 0, (8.31)
Im[G(ω, γ)] = 0. (8.32)
An estimate of ξ and K in the filter (8.28) is given by
ξ =∆d(ω0 + 0.5∆d)
4ω20
, (8.33)
K = (10M/20 − 1)2ξ
T0(jω0), (8.34)
where the disturbance bandwidth ∆d is defined as the frequency difference between
the two points whose magnitudes are 1/√
2 times of the peak value, and M in unit
dB is the desired reduction ratio of the narrowband disturbance.
The disturbance filter in (8.28) is a general high-gain controller to reject narrow-
band disturbances in a specific frequency range because the filter zero location can
be automatically shifted according to the disturbance frequency associated with the
baseline servo system with C(s) and P (s).
8.4.3 Group peak filter
The group filtering scheme is used to compensate multiple frequency narrowband
disturbance. A parallel structure of the group peak filter is described as
F (s) =
n∑
i=1
F i(s), (8.35)
where the sub-filter F i(s) is given by (8.28) with the corresponding filter parameters
(ωi, φi, ξi, Ki).An illustration example is shown in Figure 8.4 with the sensitivity functions plot-
ted before and after activating the group filter with two sub-filters at ω1 = 2π700and ω2 = 2π2000. The filters associated with solid and dashed curves have different
parameters.
8.5 Application in high frequency vibration rejection
In this section, we shall provide an example to demonstrate the design and effective-
ness of the proposed KYP based control to reject narrowband disturbances at high
frequencies. The previously introduced peak filtering method finds it difficult to deal
with disturbances in high frequency range, especially near the actuator resonance
modes.
170 Modeling and Control of Vibration in Mechanical Systems
102
103
104
−35
−30
−25
−20
−15
−10
−5
0
5
10
Frequency (Hz)
Magnitude (
dB
)
with filter
with filter
without filter
FIGURE 8.4
Sensitivity functions before and after group peak filtering activated.
FIGURE 8.5
PZT microactuator attached to VCM actuator arm.
Generalized KYP Lemma-Based Loop Shaping Control Design 171
0 2000 4000 6000 8000 10000 120000
0.5
1
1.5
2
2.5
3x 10
−3
Frequency(Hz)
PE
S N
RR
O p
ow
er
se
pc
tru
m(µ
m)
8 kHz 10 kHz
FIGURE 8.6
Power spectrum of the position error before servo control.
A. System models
Figure 8.5 shows one kind of microactuators for HDDs, which is a PZT actuatedsuspension attached to a VCM actuator arm. Figure 8.6 shows an example of themain disturbances that exist in the hard disk drive servo system. Besides the distur-bances of low frequencies, the system suffers from the high frequency disturbancesaround 8 kHz and 10 kHz, which are induced by air turbulence to suspensions orsliders in HDDs. For higher rotational speed HDDs, these disturbances appear moreprominent and create a significant impact on the positioning accuracy of the R/Whead. Here we employ the generalized KYP Lemma to design a feedback controlC(z) for the microactuator such that the sensitivity function S(z) satisfies the re-quired specifications so as to reject disturbances around 8 kHz and 10 kHz. Thedesired specifications for the sensitivity function S(z) are set as
Spec.(a) |S(f)| < 0 dB, f ≤ 2 kHz,
Spec.(b) |S(f)| < −0.35 dB, 9950 Hz ≤ f ≤ 10050 Hz,
Spec.(c) |S(f)| < −1.41 dB, 7950 Hz ≤ f ≤ 8050 Hz.
Spec. (a) means to guarantee a 2 kHz bandwidth at least.
The transfer function of the PZT microactuator is given by
Pzt(s) =
180828605599509(s2 + 3079s + 1.934× 109)
(s + 1.257× 105)(s2 + 791.7s + 1.567× 109)(s2 + 5089s + 7.195× 109),
(8.36)
172 Modeling and Control of Vibration in Mechanical Systems
which is obtained by curve-fitting to its frequency response measured via DSA. A
comparison between the measured and the modeled frequency responses is shown in
Figure 8.7, where it is noticed that there are two dominant resonance modes at 6.3kHz and 13.5 kHz. Pzt(s) is discretized using the zero-order-hold method to obtain
Pzt(z). The sampling frequency being used is 40 kHz.
103
104
−40
−30
−20
−10
0
10
20
Magnitude(d
B)
measuredmodeled
103
104
−300
−200
−100
0
100
200
Phase(d
eg)
Frequency(Hz)
FIGURE 8.7
PZT micro actuator frequency response.
B. Controller design and LDV based experiment results
The plant Pzt(s) is pre-compensated by the integrator:
Int(z) =6.3z
z − 1, (8.37)
and two notch filters for suppressing the two main resonances at 6.3 kHz and 13.5kHz before the KYP Lemma approach is applied to design a controller so that the
specifications (a), (b) and (c) can be satisfied. The notch filter is designed as the
following form:
Notch =ω2
2
ω21
· s2 + 2ξ1ω1s + ω21
s2 + 2ξ2ω2s + ω22
, (8.38)
where ω1 is the frequency of the resonance to be suppressed , ξ1 < ξ2, and ω2 and
ω1 should be chosen to be close to each other so that the resultant notch filter will
Generalized KYP Lemma-Based Loop Shaping Control Design 173
not influence the system stability. Here for the model (8.36) the notch filters after
discretization are as follows.
Notch1(z) =1.177z2 − 1.279z + 1.154
z2 − 0.8751z + 0.9259,
Notch2(z) =0.4011z2 + 0.3972z + 0.3532
z2 + 0.1391z + 0.007922.
(Ap, Bp, Cp, Dp) in (8.3)−(8.4) is a state space description of the combined
system P (z) = Int(z) · Notch1(z) · Notch2(z) · Pzt(z). Q(z) is chosen as the
1st-order FIR filter (8.19), and q0 and q1 are to be solved via the KYP Lemma.
As mentioned previously in Step 1 of the control design procedure, M and L are
obtained using MATLAB commands, i.e. M = dlqr(A, B2, CT1 C1, R) and L =
A · dlqe(A, B2, C1, Wd, Wv).
Next we use the KYP Lemma to search for the coefficients of Q(z). Three LMIs of
the form (8.7) with Σ in (8.8) and (8.9) need to be solved in order to achieve Spec. (a),
(b) and (c), respectively. The obtained q0 and q1 are q0 = −0.4117, q1 = 0.6371.
As such, C(z) can be obtained via (8.15)−(8.16). The resulting controller for the
PZT microactuator is C(z) · Int(z) · Notch1(z) · Notch2(z), and the sensitivity
function is shown in Figure 8.8. It can be observed in Figure 8.8 that with Q(z), the
sensitivity function is less than −4 dB and −2 dB at 8 kHz and 10 kHz respectively,
which means that specifications (a)−(c) have been satisfied by the searched Q(z).The sensitivity function before the KYP Lemma based design is also shown in Figure
8.8 where it can be seen that the required specifications are not met. Moreover,
as seen from Figure 8.9, the gain margin and the phase margin are 12 dB and 67deg, higher than 7.7 dB and 63 deg before the KYP design. Observed from Figures
8.8 and 8.9, the KYP design increases the loop gain around 9 kHz, which results
in the reduction of the sensitivity function from 8 to 10 kHz. The price for this
compensation is a bit lower loop gain at lower frequencies, which is consistent with
the Bode Integral constraint for sensitivity function.
In the experiment, the dSpace 1103 on TMS320C240 DSP board was used to im-
plement the controller, and an LDV was used to measure the actuator displacement,
as shown in Figure 8.10. Channel 2 over Channel 1 of DSA is the measured fre-
quency response of the sensitivity function with a swept sine signal as the reference.
Figure 8.11 shows the experimental sensitivity functions, which agree with the sim-
ulation results in Figure 8.8. From (8.1), the σ values of the position errors before
and after the KYP Lemma-based control versus frequencies are obtained and shown
in Figure 8.12. It can be seen that the performance with the KYP Lemma-based con-
trol is much better from 5 kHz onwards, and slightly worse before 5 kHz, which is
consistent with the sensitivity functions in Figure 8.8.
174 Modeling and Control of Vibration in Mechanical Systems
101
102
103
104
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Ma
gn
itu
de
(dB
)
Before KYP designAfter KYP design
FIGURE 8.8
Sensitivity functions before and after the KYP lemma-based design: simulation re-
sult.
Generalized KYP Lemma-Based Loop Shaping Control Design 175
101
102
103
104
−20
−10
0
10
20
30
40
50
Ma
gn
itu
de
(dB
)
After KYP optimizationBefore KYP optimization
101
102
103
104
−800
−600
−400
−200
0
Ph
ase
(de
g)
Frequency(Hz)
FIGURE 8.9
Open-loop Bode plot before and after the KYP lemma-based design.
FIGURE 8.10
Structure of experimental setup.
176 Modeling and Control of Vibration in Mechanical Systems
102
103
104
−25
−20
−15
−10
−5
0
5
10
Frequency(Hz)
Magnitude(d
B)
Before KYP designAfter KYP design
< 0dB
FIGURE 8.11
Sensitivity functions before and after the KYP lemma-based design: experimental
results.
0 2000 4000 6000 8000 10000 120000
0.002
0.004
0.006
0.008
0.01
0.012
Frequency(Hz)
PE
S σ
(µm
)
Before KYP design After KYP design
FIGURE 8.12
σ value of PES versus frequency.
Generalized KYP Lemma-Based Loop Shaping Control Design 177
8.6 Application in mid-frequency vibration rejection
The frequency responses of the microactuator are shown in Figure 8.13. Six reso-
nance modes at 3.7, 4.9, 6.9, 9, 12.7 and 15 kHz are included in the model.
The disturbance distribution is reflected in the non-repeatable runout power spec-
trum of the measured PES in Figure 8.14. It is noticed that there is a vibration mode
at 650 Hz due to disk vibration. The objective here is to use the above KYP method
to design a linear dynamic output feedback controller C(z) for the microactuator in
Figure 8.13 such that its closed-loop system is stable and the disturbance centering at
650 Hz is suppressed sufficiently. 45 kHz sampling rate is used in the servo control
design. The control algorithm is implemented with the digital position error signal
generated from DSP TMS320C6711. Currently, due to the limitation by the DSP
speed, with this sampling rate the platform can support up to 10th order controller.
Because 650 Hz is at a relative low frequency range, we just involve the static
part of the microactuator represented by a pade delay in the control design with the
KYP Lemma. After that, notch filters for the resonance modes at 3.7, 9 and 15kHz will be used to compensate the dynamic part, which will not change a lot the
obtained performance of the low frequency part. The 4.9 and 6.9 kHz resonance
modes, seen in Figure 8.13, have relatively small magnitudes and can be ignored as
long as they are not excited in the control loop. The resonance mode at 12.7 kHz is
not considered in the control design as it is not excited easily and does not affect the
whole loop stability when the 15 kHz mode is compensated.
The pade delay model is given by
Ppade−delay = −5.6234s− 2 · π · 17000
s + 2 · π · 17000, (8.39)
which is pre-compensated by the proportional-integral (PI) controller:
Int(z) = 0.027(− z
z − 0.999+ 0.5). (8.40)
Due to the first order pade-delay model used in the computation of LMIs, the
computation of controller can be very efficient.The desired specifications for the sensitivity function S(z) are set as:
Spec.(a) |S(f)| < 0 dB, f ≤ 500 Hz,
Spec.(b) |S(f)| < −10 dB, 610 Hz ≤ f ≤ 670 Hz,
Spec.(c) |S(f)| < 9.54 dB, f ≥ 19 kHz.
Spec. (b) means to attenuate the disturbances centering at 650 Hz by 10 dB at least.
The parameters of Q(z) in (8.19) with τ = 1 are attained by solving three LMIs of
the form (8.7) corresponding to Spec. (a), (b) and (c). The resultant C(z) is a 10th
order controller.
For the sake of comparison, the phase-lead peak filter (PLPF) of the form in (8.28)
with values K = 0.4, φ = −0.584, ω0 = 2π650, and ξ = 0.0632, is also applied to
178 Modeling and Control of Vibration in Mechanical Systems
suppress the low frequency disturbances around 650 Hz, and the sensitivity function
comparison is shown in Figure 8.15. It can be seen that the KYP method achieves
better disturbance rejection from 60 Hz to 1 kHz, although they have almost the
same rejection capability in the very narrow band around 650 Hz. However, the
KYP method gives a poorer disturbance rejection performance for frequency below
60 Hz than the PLPF method.
In the open loop comparison in Figure 8.16, the phase margin (PM) with the PLPF
method is much higher, while the bandwidth is lower and the gain margin (GM) is
comparable with the KYP Lemma method. Consistent with the sensitivity functions
in Figure 8.15, the PES NRRO power spectrum comparison is shown in Figure 8.17
which clearly shows that the KYP based design gives a better disturbance rejection
around 650 Hz than the PLPF although at 650 Hz they offer a similar performance.
From Figure 2.20, it is known that the spectrum of the true PES y is given by
Sy
= |P (z)S(z)|2 × |d1|2 + |S(z)|2|d2|2 + |T (z)|2 × |n|2 (8.41)
= Se − |S(z)|2 × |n|2 + |T (z)|2 × |n|2, (8.42)
where Se is in (8.1), and T (z) = 1− S(z) is the closed loop transfer function. Thus
the 3σ value of the true PES can be assessed from the power spectrum Se in Figure
8.17 with the known level of noise n. As a result it is improved from 6.4 nm with
the PLPF method to 6 nm with the KYP Lemma method.
In the above application, only the first order Q(z) is used. A higher order Q(z)offers more design freedom and has the potential of achieving better results. How-
ever, whatever Q(z) is used, the resultant sensitivity function has to comply with the
Bode integral theorem, meaning that it is not possible to achieve disturbance rejec-
tion across the entire frequency range. To further improve the disturbance rejection
at low frequency for the KYP Lemma-based design, we shall incorporate a nonlinear
compensation in Chapter 13.
8.7 Conclusion
This chapter has applied the generalized KYP Lemma in the microactuator closed-
loop design to suppress the narrow band disturbances. The system design problem
with multiple specifications on the gain properties of the sensitivity function over
several frequency ranges has been solved by the LMI optimization based on the
KYP Lemma. The Youla parametrization approach has been used in the feedback
controller design. Practical applications have been demonstrated for narrowband
high frequency and mid-frequency disturbance rejection. The resultant controller
verifies that the desired specifications to reject the disturbances have been satisfied
via the search for the coefficients of Q(z) in the Youla parametrization approach.
Generalized KYP Lemma-Based Loop Shaping Control Design 179
102
103
104
0
10
20
30
40
Ma
gn
itu
de
(dB
)
102
103
104
−600
−500
−400
−300
−200
−100
0
100
Ph
ase
(de
g)
Frequency(Hz)
MeasuredModeled
FIGURE 8.13
Frequency response of the PZT microactuator.
180 Modeling and Control of Vibration in Mechanical Systems
0 1000 2000 3000 4000 5000 60000
0.5
1
1.5
2
2.5
3
3.5
4
x 10−3
Frequency(Hz)
NR
RO
ma
gn
itu
de
(µm
)
FIGURE 8.14
PES NRRO power spectrum calculated from measured PES signal without servo
control, reflecting the vibration distribution of the system (3σ = 21 nm including the
noise 3σ = 15.2 nm).
Generalized KYP Lemma-Based Loop Shaping Control Design 181
101
102
103
104
−60
−50
−40
−30
−20
−10
0
10
Mag
nit
ud
e(d
B)
Frequency(Hz)
KYPPLPF
FIGURE 8.15
Comparison of sensitivity functions.
FIGURE 8.16
Open loop frequency responses (PLPF (GM: 6 dB, PM: 50 deg., Bandwidth
1.4kHz)); KYP(GM: 6 dB, PM: 34 deg., Bandwidth: 1.7 kHz))).
182 Modeling and Control of Vibration in Mechanical Systems
0 1000 2000 3000 4000 5000 60000
0.2
0.4
0.6
0.8
1
x 10−3
Frequency(Hz)
NR
RO
magnitude(µ
m)
KYPPLPF
650Hz
FIGURE 8.17
NRRO power spectrum with PLPF and KYP (50% reduction before 1 kHz).
9
Combined H2 and KYP Lemma-BasedControl Design
9.1 Introduction
As a closed-loop shaping method, the KYP Lemma-based approach allows designers
to impose performance requirements over selected finite frequency ranges so as to
have the desired sensitivity function that is able to reject the disturbances in these
specific frequency ranges. Subsequently, the positioning accuracy can be improved
to some extent. However, the KYP Lemma-based loop shaping method does not
count for overall positioning error minimization which can be translated into the
H2 optimal control problem by taking into consideration the disturbance and noise
models. On the other hand, the H2 control design which incorporates all disturbance
and noise models can result in an average performance across the entire frequency
range and a high order controller. Thus it usually does not have the flexibility to
specifically reject disturbances at certain frequency ranges, which however may be
dominant factors that influence the overall performance. Therefore there is a need to
suppress disturbances of specific frequencies when minimizing the positioning error.
This motivates us to incorporate the KYP Lemma-based method with the H2 control
method in this chapter. With the selected specific disturbances handled by the KYP
Lemma-based design, the H2 control is formulated with a lower order disturbance
model, excluding the disturbances covered in the above design. This will not only
release the computation burden in the H2 control design but also result in a lower
order controller.
In this chapter, we will apply the combined control design method to a PZT mi-
croactuator such that a disturbance at 650 Hz is rejected with the KYP Lemma-based
design and at the same time overall positioning error is minimized via the H2 control
design. Then one more disturbance at 2 kHz near the servo bandwidth 1kHz is also
considered as a specific disturbance to be rejected via the the KYP Lemma-based
design. The design procedure will be illustrated and the resultant controller will be
verified via an experiment. A series of simulation and experimental results will show
the effectiveness of the control design method in terms of enhancing positioning ac-
curacy.
183
184 Modeling and Control of Vibration in Mechanical Systems
9.2 Problem formulation
FIGURE 9.1
H2 control scheme with Q parametrization for controller design.
In the previous chapter, specifications on sensitivity function S(z) are described
as
|S(fi)| < ri, fi1 < fi < fi2, i = 1, 2, · · · , m (9.1)
where ri < 1 is a positive scalar, and fi1 and fi2 define the frequency range.
Such an upper-bound specification as in (9.1) will lead to a problem when the fre-
quency fi is larger than and especially near the desired bandwidth or 0-dB crossover
frequency of S(z). The 0-dB crossover frequency of S(z) will be pushed away to-
wards a higher frequency, as seen in Figure 9.9, which tends to damage the system
stability and deteriorate the system high-frequency performance. In view of this, a
lower-bound specification, i.e.,
|S(fi)| ≥ 1, fi1 ≤ fi ≤ fi2 (9.2)
is required. This specification helps to fix the bandwidth or 0-dB crossover frequency
of S(z), which will be seen later in the application results.
The problem of the specific disturbance rejection can be solved by imposing such
performance specifications in (9.1) and then using the KYP Lemma-based control
design method in Chapter 8. However, as shown in Figure 9.1 associated with Fig-
ure 8.14, the servo mechanical system suffers from various kinds of disturbances
and sensing noise. The KYP Lemma-based control design cannot include all dis-
turbances and noises which contribute to the position error. In view of this, we also
Combined H2 and KYP Lemma-Based Control Design 185
take into account the overall performance of the servo control system, which is repre-
sented as the so-called track misregistration (TMR) induced by w =[
w1 w2 w3
]T
passing through D1(s), D2(s), and N(s). It is expressed by the standard deviation
σz of z, and
‖Tzw‖2 = σz, (9.3)
when w is a white noise with zero mean and identity covariance matrix, where Tzwis the transfer function from w to z.
In the next section, we proceed to the controller design to achieve the specifica-
tions in (9.1) and (9.2), and meanwhile to optimize (9.3).
9.3 Controller design for specific disturbance rejection and over-
all error minimization
The generalized KYP Lemma-based design method in Chapter 8 is used to design a
controller for specific disturbance attenuation.
Let (Ap, Bp, Cp, Dp) and (Ac, Bc, Cc, Dc) respectively be the state-space model
of plant P (z) and controller C(z). In order to convexify matrix inequalities, the
Youla parametrization approach with the Q(z) in a FIR filter form is applied and the
controller structure is shown in Figure 9.1. K(z) is an observer based controller that
can be designed using the LQG method as in (8.11)−(8.12).
For the presentation of the KYP Lemma, we denote
σ(S, Π) :=
[
SI
]∗Π
[
SI
]
(9.4)
where S(z) = S(ejθ), I stands for an identity matrix and Π a Hermitian matrix of
the form
Π =
[
Π11 Π12
Π∗12 Π22
]
, (9.5)
which specifies the frequency domain property to be investigated.
9.3.1 Q parametrization to meet specific specifications
A. Specification (9.1)
Recall from (8.13)−(8.14) that a set of sensitivity functions S(z): (A, B, C, D)can be Q-parameterized.
According to the denotation (9.4)−(9.5), the specification |S(z)| ≤ r is written as
σ(S, Π) ≤ 0 with
Π =
[
Π11 Π12
Π∗12 Π22
]
=
[
1 00 − r2
]
. (9.6)
186 Modeling and Control of Vibration in Mechanical Systems
Thus based on the KYP Lemma in Chapter 8, achieving∣
∣S(
ejθ)∣
∣ ≤ r for the
frequency range θ1 ≤ θ ≤ θ2 can be obtained by solving the following matrix
inequality
[
A BI 0
]∗Σ
[
A BI 0
]
+
[
C D0 I
]∗Π
[
C D0 I
]
≤ 0, (9.7)
which is, since Π11 > 0, equivalent to
[
A BI 0
]∗
Σ
[
A BI 0
]
+
[
0 00 −r2
]
[
C D]∗
Π11
Π11
[
C D]
−Π11
≤ 0, (9.8)
where
Σ =
[
−U ejθcVe−jθcV U − (2 cos θd)V
]
, (9.9)
θc = (θ1 + θ2)/2, θd = (θ2 − θ1)/2, (9.10)
U and V are Hermitian matrices and V ≥ 0.
To convexify the matrix inequality (9.8), we shall give a state space realization
of S(z) = T11 (z) + T12 (z)Q (z)T21 (z). Denote the state-space representation
of T11(z) and T12(z)T21(z) by (At11, Bt11, Ct11, Dt11) and (At, Bt, Ct, Dt),respectively. Then a state-space model of S(z) can be written as (8.17)−(8.18).
B. Specification (9.2)
Again, according to the denotation (9.4)−(9.5) the specification |S(z)| ≥ r is
equivalent to σ(S, Π) ≤ 0 with
Π =
[
Π11 Π12
Π∗12 Π22
]
=
[
−1 00 r2
]
. (9.11)
However, because Π11 < 0, (9.7) can not be converted equivalently to (9.8), which
means (9.7) is not possibly convexified according to the method in Section 9.3.1.
Hence we resort to the following specification
σ (S, Π) = aR (S) + bI (S) + c, Π :=
[
0 a + jba − jb 2c
]
(9.12)
where R and I denote the real and the imaginary parts of S(ejθ). When a, b and
c are properly selected, |S(z)| ≥ r can be achieved. A simple selection is a = 0,
b = −1, c = r, and
σ (S, Π) = −I (S) + r. (9.13)
Thus σ(S, Π) ≤ 0 means I (S) ≥ r, and subsequently |S(z)| ≥ r. In this situation,
Π =
[
Π11 Π12
Π∗12 Π22
]
=
[
0 −jj 2r
]
, (9.14)
Combined H2 and KYP Lemma-Based Control Design 187
where Π11 = 0 and (9.7) is equivalent to
[
A BI 0
]∗
Σ
[
A BI 0
]
+
[
0 C∗Π12
Π∗12C D∗Π12 + Π∗
12D + Π22
]
≤ 0, (9.15)
which is a linear matrix inequality with unknown variables in C and D only, and can
be solved using the same method as in Section 9.3.1A.
It should be mentioned that R (S) ≥ r can also be used to achieve |S(z)| ≥ r, if
it is suitable for a specific application. In this case,
Π =
[
Π11 Π12
Π∗12 Π22
]
=
[
0 −1−1 2r
]
, (9.16)
and the linear matrix inequality (9.15) remains applicable.
9.3.2 Q parametrization to minimize H2 performance
Next we focus on the design of Q(z) to minimize the H2 norm ‖Tzw‖2. From Figure
9.1 we have
−z = N (z) w3 + S (z) [P (z) D1 (z) w1 + D2 (z) w2 − N (z) w3] . (9.17)
Denote a state-space realization of P (z)D1(z), D2(z) and N(z) by (A1, B1, C1, D1),(A2, B2, C2, D2), and (A3, B3, C3, D3), respectively. It follows from (8.13) and
(8.17)−(8.18) that
x (k + 1) = Ax (k) + Bw(k), (9.18)
−z (k) = Cx (k) + Dw(k), (9.19)
where,
A =
A1 0 0 00 A2 0 00 0 A3 0
BC1 BC2 −BC3 A
, B =
B1 0 00 B2 00 0 B3
BD1 BD2 −BD3
, (9.20)
C =[
DC1 DC2 −DC3 + C3 C]
, D =[
DD1 DD2 −DD3 + D3
]
.
The H2 norm ‖Tzw‖2 can be minimized as follows:
min(Ξ=ΞT >0, Ω=ΩT >0)
Trace (Ω) (9.21)
subject to
AT ΞA − Ξ + CT C < 0 (9.22)
BT ΞB + DT D < Ω (9.23)
188 Modeling and Control of Vibration in Mechanical Systems
or equivalently,
[
AT ΞA − Ξ CT
C −I
]
< 0 (9.24)
[
−Ω + BT ΞB DT
D −I
]
< 0 (9.25)
where
A =
A1 0 0 0 0 00 A2 0 0 0 00 0 A3 0 0 0
Bt11C1 Bt11C2 −Bt11C3 At11 0 0BtC1 BtC2 −BtC3 0 At 0
BqDtC1 BqDtC −BqDtC3 0 BqCt Aq
,
B =
B1 0 00 B2 00 0 B3
Bt11
D1 Bt11
D2 −Bt11
D3
BtD1 BtD2 −BtD3
BqDtD1 BqDtD2 −BqDtD3
,
C =[
(Dt11 + DqDt)C1 (Dt11 + DqDt)C2
(Dt11 + DqDt)C3 Ct11 DqCt Cq
]
,
D =[
(Dt11 + DqDt)D1 (Dt11 + DqDt) D2
− (Dt11 + DqDt)D3 + D3] . (9.26)
Note that the Q(z) coefficients qi(i = 0, 1, . . . , τ) only appear in Cq and Dq .
Therefore, from (8.17)−(8.18) and (9.26), we know that qi exists only in C, D, Cand D. In this case, (9.8), (9.15) and (9.24)−(9.25) define LMIs in terms of the
variables U , V , Ξ, Ω and qi. Hence, the Q(z) coefficients qi can be computed via a
convex optimization.
With the solved Q(z): (Aq , Bq, Cq, Dq), the controller C(z) is then given by
Ac =
[
Ap − BpM + LCp + BpDqCp BpCq
BqCp Aq
]
,
Bc =
[
L + BpDq
Bq
]
,
Cc =[
−M + DqCp Cq
]
,Dc = Dq .
(9.27)
9.3.3 Design steps
To summarize, a design procedure for controller C(z) is given as follows.
Combined H2 and KYP Lemma-Based Control Design 189
Step 1. Design K(z) from (8.11)−(8.12).
Step 2. Compute T11(z), T12(z) and T21(z) from (8.14), and obtain the state space
model (A, B, C, D) in (8.17)−(8.18).
Step 3. Based on disturbance spectrum and bandwidth requirement, specify the
positive scalars ri and rj , and the frequency points fi (i = 1, . . . , m) and fj (j = 1,
. . . , n) for the sensitivity function S(z), i.e.,
|S (fi)| < ri, fi1 ≤ fi ≤ fi2, (9.28)
and
|S (fj)| > rj, fj1 ≤ fj ≤ fj2. (9.29)
For each specification, construct the LMIs (9.8) and (9.15) in terms of the variables
U , V , Cq and Dq.
Step 4. Construct the LMIs (9.24)−(9.25) in terms of the variables Ξ, Ω, Cq and
Dq .
Step 5. Obtain Q(z) : (Aq, Bq , Cq, Dq) by solving the above LMIs using the
MATLAB LMI toolbox.
Step 6. Obtain the controller C(z) from (9.27).
9.4 Simulation and implementation results
This section will apply the control design method in Section 9.3 for a PZT microac-
tuator to separately reject one or two specific disturbances and meanwhile minimize
the H2 norm of the PES.
9.4.1 System models
The frequency response of the PZT microactuator, shown in Figure 9.2, was obtained
using a LDV and a DSA. The main resonance modes of the plant are at frequencies
6.5 kHz, 9.5 kHz, 11.3 kHz, and 20 kHz. The identified plant model of the micro-
actuator P (s) has the following parameters:
Zeros = 105 × [−0.0296± 0.4927j, − 0.0110± 1.0964j,− 0.7116± 0.6275j, − 0.0093± 0.6198j, 0.8168],
P oles = 105 × [−0.0255± 1.2733j, − 0.0050± 0.7100j,− 0.0048± 0.5981j, − 0.0245± 0.4071j, − 0.8168],
Gain = −0.4819.
The frequency response of the plant model is plotted against the measured data in
Figure 9.2 for comparison and it is subsequently discretized in MATLAB using the
“zoh” method with a sampling rate of 40 kHz.
190 Modeling and Control of Vibration in Mechanical Systems
102
103
104
−40
−20
0
20
40
Frequency (Hz)
Magnitude (
dB
)
102
103
104
−250
−200
−150
−100
−50
0
50
Frequency (Hz)
Phase (
deg)
MeasuredModeled
FIGURE 9.2
Frequency response of a PZT microactuator.
9.4.2 Rejection of specific disturbance and H2 performance minimiza-tion
Consider the disturbances in Figure 8.14; the disturbance around 650 Hz is due to
the disk vibration. A suitable feedback controller, C(z) has to be designed for the
system so that the overall system is stable and the disturbance around 650 Hz is
suppressed sufficiently, while ensuring that the H2 norm of the position error signal
is minimized. Hence, the desired specification of the sensitivity function S(z) is
|S (f)| < −10 dB for 610 Hz ≤ f ≤ 670 Hz and at the same time the position error
signal is to be minimized.
The parameters of a first-order FIR Q(z) are obtained by solving the three LMIs
(9.8), (9.24) and (9.25). For comparison, another controller is designed without H2
minimization and just to suppress the vibration around 650 Hz. With each of the
two designed controllers, the frequency response of the open-loop C(z)× P (z) and
the sensitivity function S(z) are depicted in Figure 9.3 and Figure 9.4. It is seen
that the hump of S(z) is reduced to about 3 dB with the controller designed by
the combined method, i.e., the combined H2 optimization and specific disturbance
rejection method. On the other hand, the performance specifications listed in Table
9.1 show that the proposed method offers better stability margins although the open-
loop crossover frequency is a bit lower.
Experiments are carried out for the KYP Lemma-based controller and the KYP+H2
controller to verify the simulation results. The controllers are implemented using
dSPACE 1103 on a TMS320C240 DSP board and the structure of the experimental
setup is the same as in Figure 8.10. The sensitivity function of the system is obtained
by the ratio of the measurements in Channel 2 and Channel 1 of the DSA, where a
swept sine signal is the reference input. The sensitivity function obtained from the
Combined H2 and KYP Lemma-Based Control Design 191
TABLE 9.1
Comparison of performance specifications
Method KYP KYP+H2
Crossover frequency (kHz) 1.85 1.78
Gain margin (dB) 7.9 12.6
Phase margin (deg) 35.4 45.9
experiment, as shown in Figure 9.5, demonstrates the effectiveness of the proposed
method. As a result, as seen in Figure 9.6, the position error is reduced and its σvalue has a 4% improvement.
101
102
103
104
−50
0
50
Ma
gn
itu
de
(dB
)
101
102
103
104
−250
−200
−150
−100
−50
Ph
ase
(de
g)
Frequency(Hz)
KYP+H2
KYP
Frequency(Hz)
FIGURE 9.3
Open-loop frequency responses.
192 Modeling and Control of Vibration in Mechanical Systems
101
102
103
104
−50
−40
−30
−20
−10
0
10
Ma
gn
itu
de
(dB
)
Frequency(Hz)
KYP+H2
KYP
FIGURE 9.4
Designed sensitivity functions.
102
103
104
−35
−30
−25
−20
−15
−10
−5
0
5
10
Frequency (Hz)
Magnitude (
dB
)
KYP+H2
KYP
FIGURE 9.5
Comparison of sensitivity functions obtained from experiment.
Combined H2 and KYP Lemma-Based Control Design 193
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Frequency (Hz)
NR
RO
magnitude (
nm
)
KYP+H2
KYP
FIGURE 9.6
NRRO power spectrum with KYP Lemma-based controller with and without H2
minimization.
9.4.3 Rejection of two disturbances with H2 performance minimiza-tion
In what follows, we include one more specific disturbance rejection at 2 kHz (see
Figure 9.7), which is caused by air flow and is near the desired servo bandwidth 1kHz. In this case, the specifications on S(z) are set as:
i. |S (f)| < −10 dB for 610 Hz ≤ f ≤ 670 Hz;
ii. |S (f)| < −5 dB for 1950 Hz ≤ f ≤ 2050 Hz;
iii. |S (f)| > 0 dB for 990 Hz ≤ f ≤ 1010 Hz,
where in addition to the rejection of disturbance at 650 Hz, the disturbance around
2 kHz is also to be suppressed. Note that the third specification aims to fix the
bandwidth.
By solving the LMIs (9.8), (9.15) and (9.24)−(9.25) in terms of the variables
P , V , Ξ, Ω, Cq and Dq , a controller C(z) which leads to the S(z) satisfying the
specifications can be obtained with a fourth-order FIR Q(z). With the resultant
controller, the open-loop frequency response and the sensitivity function are shown
in Figures 9.8 and 9.9, where the specifications (i), (ii) and (iii) have been achieved
and the closed-loop system remains stable. The open-loop gain and phase margins
are 14.6 dB and 49.5 degrees, respectively.
We have also carried out experiments to verify the designed controller. The ex-
perimental results of the open-loop frequency response and sensitivity function are
194 Modeling and Control of Vibration in Mechanical Systems
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.5
1
1.5
2
2.5
3x 10
−3
Frequency (Hz)
NR
RO
magnitude (
µm
)
FIGURE 9.7
PES NRRO spectrum without servo control.
plotted in Figure 9.8 and Figure 9.10, respectively. The NRRO spectrum of the sys-
tem with the designed controller is shown in Figure 9.11. Based on Figure 9.11, the
attenuation of disturbance centering at 2 kHz has improved by 35%.
9.5 Conclusion
This chapter has addressed a combined control design method that incorporates the
generalized KYP Lemma-based design and the H2 optimization. With the incorpo-
rated control design method, specific narrowband disturbances have been attenuated
and simultaneously the positioning error of the control system has been minimized.
The method has been applied to design a controller for a PZT microactuator to at-
tenuate disturbances at 650 Hz and 2 kHz where disk vibrations are dominant and
minimize the 3σ value of the overall PES. Simulation and experimental results have
demonstrated the effectiveness of the proposed method and verified that the position-
ing accuracy has been improved.
Combined H2 and KYP Lemma-Based Control Design 195
102
103
104
−60
−40
−20
0
20
40
Magnitude (
dB
)
102
103
104
−800
−600
−400
−200
0
200
Frequency (Hz)
Phase (
deg)
SimulationExperiment
FIGURE 9.8
Open-loop frequency response.
101
102
103
104
−50
−40
−30
−20
−10
0
10
Ma
gn
itu
de
(d
B)
Frequency (Hz)
Bandwidth is increased
FIGURE 9.9
Resultant sensitivity function (Solid line: with Spec. (i), (ii) and (iii); Dashed line:
with Spec. (i) and (ii)).
196 Modeling and Control of Vibration in Mechanical Systems
102
103
104
−30
−25
−20
−15
−10
−5
0
5
10
Frequency (Hz)
Magnitude (
dB
)
SimulationExperiment
FIGURE 9.10
Resultant sensitivity function with all the three requirements fulfilled.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
−3
Frequency (Hz)
NR
RO
magnitude (
µm
)
Without controlClosed loop
FIGURE 9.11
NRRO power spectrum with rejection of two specific disturbances at 0.65 and 2 kHz.
10
Blending Control for Multi-FrequencyDisturbance Rejection
10.1 Introduction
The blending control technique [120] aims to design a controller capable of simul-
taneously coping with different optimality criteria defined for various input-output
channels. If several targeted transfer functions are achieved respectively by their
designed controllers, it is shown that under some mild assumptions there exists a
unique controller capable of replicating these transfer functions and thus simultane-
ously achieving the performance attained by each individual controller. Each indi-
vidual controller can be a static or a dynamic feedback controller derived by means
of standard optimal and robust control methods such as the LQG/LTR control, H2 or
H∞ control. The blended controller can be easily computed by a procedure based on
simple linear algebra. In this chapter, we shall apply the blending control idea to deal
with the problem of rejecting several narrowband disturbances. The multi-frequency
disturbance rejection is formulated as a control blending problem. Control design for
each disturbance rejection is carried out by using the H2 optimal control method. The
ultimate controller is obtained by blending all these H2 controllers and is expected
to be able to reject all these disturbances simultaneously. In this chapter, the con-
trol blending is respectively applied in the control design for a 1.8-inch HDD VCM
actuator in three cases, (1) reject two disturbances of frequencies higher than band-
width, (2) reject three disturbances with frequencies higher than bandwidth, and, (3)
reject one disturbance with frequency lower than bandwidth and another one higher
than bandwidth. Note that this chapter presents a systematic approach using dis-
turbance models only, which leads to the achievement of a satisfactory rejection of
multi-frequency narrow band disturbances.
10.2 Control blending
Control blending accounts for the problem of simultaneous performance achieve-
ments. It involves designing an individual controller for each performance specifica-
197
198 Modeling and Control of Vibration in Mechanical Systems
tion and blending all the controllers to obtain an ultimate controller.
Consider the system in state space:
x(t) = Ax(t) + Bu(t) +
r∑
i=1
Eiwi(t), (10.1)
y1(t) = C1x(t) + D1u(t), (10.2)
y2(t) = C2x(t) + D2u(t), (10.3)
· · ·yr(t) = Crx(t) + Dru(t), (10.4)
e(k) = Cex(k) + Deu(k) + v(k), (10.5)
where x(t) ∈ Rnx , u(t) ∈ Rnu , yi(t) ∈ Rpi , wi(t) ∈ Rqi , e(t) ∈ R, and A, B, Ci,
Di, Ei, Ce, and De are matrices of appropriate dimensions.
The following two assumptions are necessary in the control blending scheme.
Assumption 10.1: (A, B) is stabilizable.
Assumption 10.2: The nx × s matrix,
E =[
E1 E2 · · · Er
]
, (10.6)
where s =∑r
i=1 qi, has full column rank.
FIGURE 10.1
Blending control scheme.
Blending Control for Multi-Frequency Disturbance Rejection 199
10.2.1 State feedback control blending
The feedback control scheme of the above system is shown in Figure 10.1, where the
controller is expressed as
xs(t) = Asxs(t) + Bsx(t) (10.7)
u(t) = Csxs(t) + Dsx(t). (10.8)
The closed-loop system from wi to yi, i = 1, 2, · · ·, r is then described by
[
Acl Ecl
Ccl 0
]
=
A + BDs BCs E1 E2 · · · Er
Bs As 0 0 · · · 0
C1 + D1Ds D1Cs 0 0 · · · 0C2 + D2Ds D2Cs 0 0 · · · 0
......
...... · · ·
...
Cr + DrDs DrCs 0 0 · · · 0
. (10.9)
For the system (10.1)−(10.5), we consider the static state feedback control
u(t) = Kix(t), i = 1, 2, · · · , r. (10.10)
The individual closed-loop transfer function is given by
Ti(s) = Ci(sI − Ai)−1Ei, (10.11)
with
Ai = A + BKi, Ci = Ci + DiKi. (10.12)
The result in [120] reads that there exists a single dynamic compensator of the
form (10.7)−(10.8) of order µ = nx(r − 1), such that 1). the closed-loop system is
stable with spectrum
σ(Acl) = ∪iσ(Ai); (10.13)
and 2). for each i, the transfer function Ti(s) from the ith input to the ith output
coincides with the one given by the ith gain Ki, i.e.,
Ti(s) = Ti(s). (10.14)
The significance of the result lies in the fact that if Ki, i = 1, 2, · · · , r, are the
static compensators each of which optimizes some performance criteria, then there
exists a single dynamic compensator which simultaneously achieves the same per-
formances. The single dynamic compensator of the form in (10.7)−(10.8) can be
found following the procedure below.
Procedure 10.1
200 Modeling and Control of Vibration in Mechanical Systems
• Let Ki, i = 1, 2, · · · , r be the static state feedback controllers each of which
asymptotically stabilizes the plant, i.e., Ai = A + BKi is asymptotically
stable.
• For each i, choose a matrix Ei of dimension nx × (nx − qi) such that [Ei Ei]is square and invertible. This is possible under Assumption 10.2.
• Choose matrices Zi of dimension µ× (nx − qi) (µ = nx(r− 1)) such that the
matrix[
E1 E2 · · · Er E1 E2 · · · Er
0 0 · · · 0 Z1 Z2 · · · Zr
]
(10.15)
is invertible.
• Define
Zi = [0 Zi][Ei Ei]−1, (10.16)
and
Vi = ZiAi. (10.17)
• Form the square matrix Ξ as
Ξ =
[
I I · · · IZ1 Z2 · · · Zr
]
. (10.18)
• Compute the controller matrices As, Bs, Cs, Ds as
[
Ds Cs
Bs As
]
=
[
K1 K2 · · · Kr
V1 V2 · · · Vr
]
Ξ−1. (10.19)
The order of the blended controller is nx(r− 1) which is very high. Theoretically
to match r different transfer functions Ti, nx(r − 1) + nx = nxr different poles
have to be allocated and a full order controller is required. In practical application,
however, reducing the order is needed as long as it is possible.
10.2.2 Output feedback control blending
When the full state variables of the plant are not available, the feedback control has
to rely on some measurements given by
yp(t) = Cyx(t), (10.20)
where yp ∈ Rp. The output feedback controller takes the form
xc(t) = Acxc(t) + Bcy(t) (10.21)
u(t) = Ccxc(t) + Dcy(t). (10.22)
Blending Control for Multi-Frequency Disturbance Rejection 201
The closed-loop system from wi to yi, i = 1, 2, · · · , r is then described by
[
Acl Bcl
Ccl 0
]
=
A + BDcCy BCc E1 E2 · · · Er
BcCy Ac 0 0 · · · 0
C1 + D1DcCy D1Cc 0 0 · · · 0C2 + D2DcCy D2Cc 0 0 · · · 0
......
...... · · ·
...
Cr + DrDcCy DrCc 0 0 · · · 0
. (10.23)
Assumption 10.3: The composed p × s matrix,
E =[
CyE1 CyE2 · · · CyEr
]
, (10.24)
where s =∑r
i=1 qi, has full column rank.
Consider the static state feedback control as stabilizing compensators
u(t) = Kix(t), i = 1, 2, · · · , r. (10.25)
The corresponding closed-loop transfer function from wi to yi is given by
Ti(s) = Ci(sI − Ai)−1Ei, (10.26)
with
Ai = A + BKi, Ci = Ci + DiKi. (10.27)
The theorem on output blending control in [120] says that there exists a single
dynamic compensator of the form (10.21)−(10.22) of order µ = nxr − p, such that
1). the spectrum of the closed-loop system satisfies
∪iσ(Ai) ⊆ σ(Acl); (10.28)
and 2). for each i, the transfer function Ti(s) from the ith input wi to the ith output
yi coincides with the one given by the ith gain Ki, i.e.,
Ti(s) = Ti(s). (10.29)
The single dynamic output controller of the form in (10.21)−(10.22) can be found
following the procedure below.
Procedure 10.2
• Let Ki, i = 1, 2, · · · , r be the static state feedback controllers each of which
asymptotically stabilizes the plant, i.e., the matrix Ai = A + BKi is asymp-
totically stable.
• For each i, choose a matrix Ei of dimension n× (n− qi) such that [Ei Ei] is
square and invertible. This is possible under Assumption 10.3.
202 Modeling and Control of Vibration in Mechanical Systems
• Choose matrices Zi of dimension µ × (nx − qi) (µ = nxr − p) such that the
matrix
[
CyE1 CyE2 · · · CyEr CyE1 CyE2 · · · CyEr
0 0 · · · 0 Z1 Z2 · · · Zr
]
(10.30)
is invertible.
• Define
Zi = [0 Zi][Ei Ei]−1, (10.31)
and
Vi = ZiAi. (10.32)
• Form the square matrix Ξ as
Ξ =
[
Cy Cy · · · Cy
Z1 Z2 · · · Zr
]
. (10.33)
• Compute the controller matrices Ac, Bc , Cc, Dc as
[
Dc Cc
Bc Ac
]
=
[
K1 K2 · · · Kr
V1 V2 · · · Vr
]
Ξ−1. (10.34)
The procedure is quite simple to implement, and it is proved in [120] that the
matrix Ξ is indeed an invertible matrix.
Consider the system
[
A E
N 0
]
=
A1 0 · · · 0 E1 0 · · · 00 A2 · · · 0 0 E2 · · · 0...
... · · ·...
...... · · ·
...
0 0 · · · Ar 0 0 · · · Er
Cy Cy · · · Cy 0 0 · · · 0
. (10.35)
The following Proposition advises on the stability of the controller matrix Ac.
Proposition 10.1 [120]
i). The controller matrix Ac is not stable if system (10.35) has unstable invariant
zeros.
ii). The controller matrix can be generically chosen as a stability matrix if system
(10.35) is of minimum phase.
iii). If E is square then the controller matrix is stable if and only if system (10.35)
is of minimum phase.
Blending Control for Multi-Frequency Disturbance Rejection 203
10.3 Control blending application in multi-frequency disturbance
rejection
In this section, we first formulate the problem of multi-frequency disturbance rejec-
tion into the above control blending framework. Then, Procedure 10.1 is used to
design a blended controller associated with the static state feedback controller de-
signed via the H2 optimal control method. To facilitate the single dynamic state
feedback controller, a full state observer is required.
10.3.1 Problem formulation
The closed control loop of a mechanical actuation system is shown in Figure 10.2,
where P (s) is the actuator model, C(z) is the feedback controller, e is the measured
error signal. Di(s), i = 1, 2, · · · , r are disturbance models and wi are white noises
with unity variances. v is the measurement noise with σv as its standard deviation.
Here, we focus on the rejection of output disturbance Di(s), considering that an
input disturbance can be converted to an output disturbance.
From Figure 10.2, with Ref = 0 we have
y = S(z)(D1(z)w1 + D2(z)w2 + · · ·+ Dr(z)wr) + (1 − S(z))v. (10.36)
The multi-frequency disturbance rejection problem is stated as follows: For nar-
rowband disturbances di, i = 1, 2, · · · , r, design a feedback controller C(z) such
that the closed-loop system is stable, and the disturbances di in different frequency
ranges [fiL, fiH ] can be suppressed simultaneously.
In Figure 10.2, denote (Ap, Bp, Cp, Dp) as the state-space realizations of P (z)with state vector xp∈Rnp, and (Adi
, Bdi, Cdi
, Ddi) as the state-space realizations
of Di(z) with state vector xdi∈Rndi . Di(s) is proposed to have the form:
Di(s) =kdi(s + 2ζiωi)
s2 + 2ξiωis + ω2i
, i = 1, 2, · · · , r, (10.37)
with frequency ωi, damping ratio ξi, ζi used to adjust the damping of Di, and gain
kdi. Clearly, Ddi= 0.
To ensure necessary rejection of low-frequency disturbances, a pre-compensation
integrator Int(z) is introduced which is shown in Figure 10.3 and given by
u(k + 1) = ke(k) + u(k). (10.38)
Denote the output due to di by yi. Then the combined system is given as follows.
x(k + 1) = Ax(k) + Bu(k) +
r∑
i=1
Eiwi(k), (10.39)
y1(k) = C1x(k) + D1u(k), (10.40)
204 Modeling and Control of Vibration in Mechanical Systems
FIGURE 10.2
Control loop with injected disturbances at different frequencies.
FIGURE 10.3
Control structure.
Blending Control for Multi-Frequency Disturbance Rejection 205
y2(k) = C2x(k) + D2u(k), (10.41)
· · ·yr(k) = Crx(k) + Dru(k), (10.42)
e(k) = Cex(k) + Deu(k) + v(k), (10.43)
where x =[
xTp uT xT
d1xT
d2· · · xT
dr
]T ∈ Rnx , u(k) ∈ Rnu , yi(k) ∈ Rpi , wi(k) ∈Rqi , nx = np + 1 +
r∑
i=1ndi
, nu = 1, pi = 1, qi = 1,
A =
Ap Bp 0 0 · · · 0
−kCp 1 − kDp kCd1 kCd2 · · · kCdr
0 0 Ad1 0 · · · 00 0 0 Ad2 · · · 0...
......
......
...
0 0 0 0 · · · Adr
, B =
Bp
−kDp
00...
0
, (10.44)
E1 =
0
−kDd1
Bd1
0...
0
, E2 =
0
−kDd2
0Bd2
...
0
, · · · , Er =
0
−kDdr
00...
Bdr
, (10.45)
C1 =[
−Cp −Dp −Cd1 0 · · · 0]
, D1 = −Dp, (10.46)
C2 =[
−Cp −Dp 0 −Cd2 · · · 0]
, D2 = −Dp, (10.47)
Cr =[
−Cp −Dp 0 0 · · · −Cdr
]
, Dr = −Dp, (10.48)
Ce =[
−Cp −Dp −Cd1 −Cd2 · · · −Cdr
]
, De = −Dp. (10.49)
Note that the measurement error signal e in (10.43) includes all disturbance out-
puts yi, i = 1, 2, · · · , r.
10.3.2 Controller design via the control blending technique
In this section, we present how to design the controller C(z) by using the control
blending technique. Before the blending, the H2 optimal control method will be
used to design a static state feedback controller to minimize the error caused by each
disturbance. Subsequently, the blending technique will be utilized to yield one single
dynamic state feedback controller that is able to reject all disturbances. Next a state
observer will be designed to facilitate the state feedback controller with the mea-
surement signal. Finally, the SISO (single input and single output) output feedback
controller Cb(z) will be given based on the dynamic state feedback controller and
the state observer.
Denote the dynamic state feedback controller by
Σ(z) : xs(k + 1) = Asxs(k) + Bsx(k), (10.50)
206 Modeling and Control of Vibration in Mechanical Systems
u(k) = Csxs(k) + Dsx(k). (10.51)
This dynamic state feedback controller is designed following Procedure 10.1,
where Ki in step 1 is designed based on the H2 optimal control method, which
is described as follows.
Let Ki, i = 1, 2, · · · , r be the static state feedback controllers each of which
asymptotically stabilizes the plant and minimizes the H2 norm of the transfer func-
tion from wi to yi. Ki can therefore be obtained by the following optimization:
min Trace(Si)
subject to the LMIs
−Pi ∗ ∗PiA
T + WTi BT −Pi ∗
0 CiP + DiWi −1
< 0, (10.52)
[
−Si ∗Ei −Pi
]
< 0. (10.53)
With the solved Wi and Pi, Ki is obtained as
Ki = WiP−1i . (10.54)
Next, a dynamic state feedback controller of the form (10.50)−(10.51) is designed
based on Procedure 10.1. Since only the measurement is available for feedback, a
state observer, as in Figure 10.3, is needed to facilitate the controller Σ(z). The state
observer O(z) is designed as follows.
x(k + 1) = Aox(k) + [Bo1 Bo2]
[
u(k)e(k)
]
, (10.55)
yo(k) = Cox(k) + Do
[
u(k)e(k)
]
, (10.56)
where
Ao = A − ALCe, Bo1 = B, Bo2 = AL, (10.57)
Co = I, Do = 0, (10.58)
L = dlqe(A, E, Ce, Q, R), E is as in (10.6), Q = I, and R = σ2v.
The overall controller C(z) is then given by
Ac =
[
As BsCo
Bo1Cs Ao + Bo1DsCo
]
, Bc =
[
0Bo2
]
, (10.59)
Cc =[
Cs DsCo
]
, Dc = 0. (10.60)
Blending Control for Multi-Frequency Disturbance Rejection 207
10.4 Simulation and experimental results
The disk drive under consideration is a 1.8-inch small hard disk drive with a spindle
motor rotational speed of 4200 RPM. Figure 10.4 shows the frequency response of
the VCM actuator. The VCM actuator model P (s) is described by
P (s) =8.326628× 1017(s2 + 1081s + 7.3× 108)(s2 + 6635s + 2.852× 109)
(s2 + 552.9s + 4.777× 105)(s2 + 1056s + 6.964× 108)
× 1
(s2 + 2815s + 2.527× 109)(s2 + 1.131× 104s + 3.948× 109). (10.61)
101
102
103
104
−60
−40
−20
0
20
40
60
Ga
in [
dB
]
MeasuredModeled
101
102
103
104
−600
−500
−400
−300
−200
−100
0
Frequency [Hz]
Ph
ase
[d
eg
ree
]
FIGURE 10.4
Frequency response of the VCM actuator.
10.4.1 Rejecting high-frequency disturbances
In this section, controller C(z) will be designed to reject a few disturbances with
frequencies higher than bandwidth. First, two disturbances will be rejected, and
next, three disturbances are to be attenuated simultaneously.
208 Modeling and Control of Vibration in Mechanical Systems
We consider two disturbances at frequencies 4 kHz and 8 kHz which are the res-
onance frequencies of the plant as shown in Figure 10.4. By fitting the model in
(10.37), we have ω1 = 2π4000, ξ1 = 0.085, ζ1 = 0.6, kd1 = 20 for D1(s), and
ω2 = 2π8000, ξ2 = 0.01, ζ2 = 0.6, kd2 = 43 for D2(s).
Take ki = 100 · Ts. The parameters in (10.19) of Σ(z) are obtained by the design
procedure outlined in the last section. With Σ(z), the observer O(z) (10.55)−(10.56),
we have controller Cb(z) by (10.59)−(10.60). Incorporated with an integrator, a 27th
order C(z) is obtained. After simple order reduction by canceling close zeros and
poles, C(z) can be reduced to 14th order. The final controller C(z) leads to the fre-
quency response of the open-loop P (z)C(z) as shown in Figure 10.5. The open-loop
bandwidth is 1 kHz. The simulated sensitivity function S(z) = 1/(1 + P (z)C(z))is shown in Figure 10.6, where there are two obvious rejection (i.e., |S(z)| < 1)
frequency ranges around 4 kHz and 8 kHz.
Rejection capability in terms of |S(z)| for Di (i = 1, 2) can be changed by
adjusting kdi and ξi. More simulations show that the change of |S(z)| for D1 at 4kHz leads to slight variation of |S(z)| for D2 at 8 kHz, while the change of |S(z)|for D2 at 8 kHz affects more on |S(z)| around 4 kHz. Moreover, changing kd2 will
affect rejection at 4 kHz more than changing damping ξ2.
The controller is implemented via dSpace 1103 with the utilization of LDV to
measure the displacement of the actuator. The sensitivity function S(z) of the closed
control loop is measured via DSA and plotted in Figure 10.7, which agrees well with
the simulated one (the dashed curve).
101
102
103
104
−20
0
20
40
60
80
Magnitude(d
B)
101
102
103
104
−200
−100
0
100
200
Phase(d
eg)
Frequency(Hz)
FIGURE 10.5
Open-loop frequency response with disturbance rejection at 4 and 8 kHz.
Blending Control for Multi-Frequency Disturbance Rejection 209
101
102
103
104
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Ma
gn
itu
de
(dB
)
4 kHz 8 kHz
FIGURE 10.6
Simulated sensitivity function with disturbance rejection at 4 and 8 kHz.
Next, assume that disturbances at three frequencies are to be rejected simultane-
ously, and the frequencies are not the plant resonance frequencies. The disturbances
are modeled as (10.37) with ω1 = 2π3000, ξ1 = 0.02, ζ1 = 0.7, kd1 = 40 for D1(s),ω2 = 2π6500, ξ2 = 0.085, ζ2 = 0.61, kd2 = 50 for D2(s), and ω3 = 2π10000,
ξ3 = 0.3, ζ3 = 0.6, kd3 = 52 for D3(s).
The order of the designed C(z) is 46. After order reduction, a 31st order C(z) is
obtained. With the lower order C(z), the compensated open-loop frequency response
is shown in Figure 10.8. The simulated and the measured sensitivity functions are
shown in Figure 10.9 where three rejection frequency bands around 3 kHz, 6.5 kHz,
and 10 kHz are clearly seen.
The rejection capabilities for these three disturbances are further examined via
more simulations. It is shown that changing rejection capability for 3 kHz distur-
bance by adjusting kd1 and/or ξ1 will change that for 6.5 kHz, while will not affect
much on that for 10 kHz. The rejection change for 6.5 kHz disturbance by chang-
ing kd2, ξ2 or ζ2 will not lead to much change to other two disturbance rejections.
Changing parameters ξ3, ζ3 and kd3 of D3 for 10 kHz disturbance will affect the
rejection for the other two disturbances more than itself.
210 Modeling and Control of Vibration in Mechanical Systems
101
102
103
104
−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency(Hz)
Magnitude(d
B)
FIGURE 10.7
Measured (solid curve) sensitivity function with disturbance rejection at 4 and 8 kHz.
101
102
103
104
−20
0
20
40
60
80
Magnitude(d
B)
101
102
103
104
−200
−100
0
100
200
Phase(d
eg)
Frequency(Hz)
FIGURE 10.8
Open-loop with disturbance rejection at 3, 6.5, and 10 kHz.
Blending Control for Multi-Frequency Disturbance Rejection 211
101
102
103
104
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
FIGURE 10.9
Sensitivity function with disturbance rejection at 3, 6.5, and 10 kHz.
10.4.2 Rejecting a combined mid and high frequency disturbance
Previously, the frequencies of the disturbances that need to be rejected are higher than
the bandwidth. In this section, we consider two disturbances near the bandwidth:
one at 2 kHz, higher than the bandwidth, and the other at 650 Hz, lower than the
bandwidth. The disturbance models are as in (10.37) where ω1 = 2π650, ξ1 = 0.05,
ζ1 = 0.8, kd1 = 77 for D1(s) at 650 Hz, and ω2 = 2π2000, ξ2 = 0.0046, ζ2 = 0.85,
kd2 = 0.1 for D2(s) at 2 kHz.
The resultant controller C(z) is of 27th order. By close zero-pole cancelations, a
21st order controller can be obtained, and is able to accomplish the task of rejecting
D1 and D2, which is illustrated in Figures 10.10 and 10.11. In Figure 10.10, a
significant point is that corresponding to the peaks at 650 Hz and 2 kHz, the phases
are both lifted, which is desired in order not to lose the phase margin.
As observed in Figure 10.11, |S(z)| < 0 at 2 kHz. The experimentally measured
|S(z)| is plotted; see the rough curve in Figure 10.11. Additional simulations show
that decreasing damping ξ1 will not only lower |S(z)| at 650 Hz, but also lower it
at 2 kHz. Increasing kd1 will lower |S(z)| at 650 Hz, but will increase the hump of
S(z) and cause slight increase of |S(z)| at 2 kHz. On the other hand, increasing kd2
in D2(s) for 2 kHz disturbance will not affect much on |S(z)| at 650 Hz. Changing
ζ2 will easily cause phase loss at 2 kHz. It is thus suggested to change kd2 and ξ2 to
have a satisfactory sensitivity function |S(z)| at 2 kHz without causing phase loss.
Using the method of phase lead peak filter in Chapter 8, the rejection capability
212 Modeling and Control of Vibration in Mechanical Systems
for vibrations at 650 Hz and 2 kHz can be achieved with two peak filters, say pf1
and pf2, which are individually designed to deal with the two vibrations, and then
the two filters are connected in parallel, i.e., 1 + pf1 + pf2. In this design method,
a baseline controller needs to be designed first, then the two peak filters are obtained
based on the pre-designed baseline controller. Thus the rejection performance of the
system for vibration in other frequency ranges depends on the pre-designed base-
line controller. Moreover, the phase lead peak filter method is difficult to deal with
multi-frequency vibrations in higher frequency range and at frequencies near plant
resonance modes, that however have been treated in this chapter.
102
103
104
−20
0
20
40
60
Magnitude(d
B)
102
103
104
−200
−100
0
100
200
Phase(d
eg)
Frequency(Hz)
FIGURE 10.10
Open-loop with disturbance rejections at 0.65 and 2 kHz.
REMARK 10.1 From the above application results and analysis, it isnoted that to deal with lower (say ≤ 2 kHz) frequency disturbances, the pro-posed control design method based on the blending technique works like thephase lead peak filtering method. For higher (say > 2 kHz) frequency distur-bances, it works like phase-stabilized control [86]. This feature is beneficial to
the stability of the closed-loop system.
Blending Control for Multi-Frequency Disturbance Rejection 213
101
102
103
104
−70
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
FIGURE 10.11
Sensitivity function with disturbance rejections at 0.65 and 2 kHz.
10.5 Conclusion
The rejection problem of several disturbances around different frequencies has been
formulated as a control blending problem. A controller for each disturbance rejec-
tion has been designed individually by using the H2 optimal control method. The
ultimate controller has been obtained by blending all these H2 controllers so that all
these disturbances can be rejected simultaneously. The control blending method has
been respectively applied to design a controller for a 1.8-inch HDD VCM actuator
in three cases, (1) rejecting two disturbances of frequencies higher than bandwidth
and close to actuator resonance frequencies, (2) rejecting three disturbances with fre-
quencies higher than bandwidth and different from resonance frequencies, and, (3)
rejecting one disturbance with frequency lower than bandwidth and another higher
than bandwidth. Simulation and experimental results have shown that the control
blending technique results in a simultaneous attenuation for these disturbances. In
addition, it is worth noting that the method is able to prevent phase loss when it is
used to deal with disturbances near the bandwidth.
11
H∞-Based Design for Disturbance Observer
11.1 Introduction
The idea of observing disturbance to improve the performance of a servomechanism
was first introduced in [123]. It was suggested that if the disturbances were suppos-
edly generated by a linear dynamic system and the model of the system was known,
they could be estimated from the system output measurement by an asymptotic esti-
mator (Luenberger observer) and the effect of the disturbances could be neutralized
by feeding the disturbance estimates back into the system [124]. Over the years, the
method has been modified and applied. However, it is not always easy to identify the
disturbance model. Further, it is not always true that the disturbance model is linear
time-invariant. Subsequently, a new type of disturbance observer (DOB) has been
introduced [128]. This new method does not require control designers to have the
full information of the disturbance model and does not need the assumption that the
disturbance model is linear time-invariant. However, it requires the model of the con-
trolled plant to be accurately known and invertible, at least within the bandwidth of
interest [129]. Recently, it has been proven that under certain assumptions imposed
on the plant and disturbance models, the two different methods are equivalent in that
the original disturbance observer introduced in [123] is actually a generalization of
the latter method [126]. In this chapter, we study the latter method where a general
form of disturbance observers, which does not need to solve the plant model inverse,
will be introduced.
If majority of the disturbances are of relatively lower frequency, when a DOB
is added on to attenuate the effect of disturbances, the standard and conventional
way of designing a Q-filter is to design it to be a low-pass filter with unity DC gain
[128]. In this chapter, we introduce the conventional disturbance observer first, and
then present a general form of disturbance observer. An H∞ control based method is
applied to the design of a Q-filter. The designed disturbance observer is applied to an
HDD servo system and its effectiveness in disturbance attenuation is demonstrated
by simulations and experiments.
215
216 Modeling and Control of Vibration in Mechanical Systems
11.2 Conventional disturbance observer
Figure 11.1 shows the block diagram of the conventional disturbance observer (DOB)
structure, where C is the feedback controller, Pn is the nominal model of the plant
P , d1 is the input disturbance, n is the noise, d1 is an estimate of d1, and P−1n is the
inverse of Pn. τ represents a delay in the plant P . Theoretically, P (z) = z−τPn(z).Ignoring the nominal feedback loop with C , the transfer functions from the distur-
bance d1 and noise n to the output y are given by
Tyd1 =PPn(1 − Qz−τ)
Pn + Q(P − Pnz−τ ), (11.1)
Tyn =−PQ
Pn + Q(P − Pnz−τ ). (11.2)
To reject the disturbance d1, Q can be set as unity because Tyd1 ≈ 0 when the
delay is negligible. However, when Q = 1, Tyn ≈ −1 which means the measure-
ment noise n is not attenuated. Thus, to eliminate the noise effect, it is known from
(11.2) that an ideal solution of Q is zero, but this will mean Tyd1 ≈ P and hence the
disturbance will be amplified.
FIGURE 11.1
Block diagram of the control loop with a conventional disturbance observer.
H∞-Based Design for Disturbance Observer 217
Considering the overall system in Figure 11.1, the sensitivity function is given by
S(z) =1 − Qz−τ
1 − Qz−τ + PC + PQP−1n
. (11.3)
Theoretically, the plant model is P = Pnz−τ , thus (11.3) becomes
S(z) =1 − Qz−τ
1 + PC, (11.4)
which is stable as long as Q is stable since the loop is stable before the disturbance
observer is added on. Moreover, it is deduced from (11.3) that when Qz−τ = 1 with
zero phase around the disturbance frequency, the disturbance can be rejected because
S(z) → 0. Hence Q is designed such that the phase of Qz−τ is almost zero degree
and the magnitude is close to one in the frequency range where the disturbance d1
dominates. Note that in Figure 11.1, it is needed to solve the inverse P−1n of the
nominal plant model. In what follows, a general form of the disturbance observer is
proposed and the plant model inverse is not needed.
11.3 A general form of disturbance observer
Let
Pn(z) = B(z)/A(z), (11.5)
B(z) = bmzm + bm−1zm−1 + · · ·+ b0,
A(z) = zn + an−1zn−1 + · · ·+ a0
be the nominal model of the plant P . With the same notations as in Figure 11.1,
Figure 11.2 displays a general form of disturbance observer, where
M(z) =B(z)
zdm, N(z) =
A(z)
zdn, (11.6)
and d2 is the output disturbance.
In the conventional disturbance observer in Figure 11.1,
M(z) = z−τ , N(z) = P−1n (z). (11.7)
Denote the state-space descriptions P (z) : (Ap, Bp, Cp, Dp); C(z) : (Ac , Bc ,
Cc, Dc); M(z) : (AM , BM , CM , DM ); N(z) : (AN , BN , CN , DN ). From
Figure 11.2,
x(k + 1) = Ax(k) + B1w(k) + B2uq(k), (11.8)
yq(k) = Cyx(k) + Dyww(k) + Dyuuq(k), (11.9)
y(k) = Czx(k), (11.10)
218 Modeling and Control of Vibration in Mechanical Systems
FIGURE 11.2
Block diagram of the control loop with a general disturbance observer.
where
A =
Ap − BpDcCp BpCc 0 0−BcCp Ac 0 0
−BMDcCp BM Cc AM 0−BNCp 0 0 AN
, (11.11)
B1 =
Bp −BpDc BpDc
0 −Bc Bc
0 −BM Dc BMDc
0 −BN BN
, B2 =
−Bp
0−BM
0
,
Cy =[
−DNCp − DMDcCp DMCc CM CN
]
, (11.12)
Dyw =[
0 −(DN + DmDc) DN + DMDc
]
, Dyu = −DM ,
Cz =[
−Cp 0 0 0]
, Dzw = [0 − 1 0], Dzu = 0, (11.13)
and xT (k) = [xTp (k) xT
c (k) xTM(k) xT
N(k)] is the augmented state of P (z), C(z),
M(z), and N(z), and wT (k) = [d1(k) d2(k) n(k)].Denote the transfer function from w to y as Tyw = [Tyd1 Tyd2 Tyn]. The H∞
optimization method will be applied to design Q(z) to minimize the H∞ norm∥
∥
∥
∥
∥
∥
Wd1Tyd1
Wd2Tyd2
WnTyn
∥
∥
∥
∥
∥
∥
∞
. (11.14)
Wd1 , Wd2 and Wn are weightings. Here the models for disturbances d1, d2 and
noise n are not needed. However, in order to have a desired suppression of the
H∞-Based Design for Disturbance Observer 219
disturbances d1, d2 and the noise n, appropriately chosen weightings Wd1 , Wd2 and
Wn are required which relies on our knowledge of the disturbances.
Different from the conventional disturbance observer, the general disturbance ob-
server does not need the inverse of the nominal plant model. As shown later, it is
able to suppress the disturbances in a low frequency range without much perfor-
mance degradation to higher frequency disturbances and noise.
The objective of the general disturbance observer design is then stated as: Given
a positive scalar γ and appropriate weightings Wd1 , Wd2 and Wn, design a stable
Q(z) : (AQ, BQ, CQ, DQ) such that
‖[Wd1Tyd1 Wd2Tyd2 WnTyn]T ‖∞ < γ. (11.15)
The H∞ control design problem can be solved via Theorem 5.4.
REMARK 11.1 The sensitivity function of the control system in Figure11.2 with the general disturbance observer is given by
S(z) =1 + QM
1 + PC + QM − PQN. (11.16)
With the conventional disturbance observer in (11.7), (11.16) is equal to
S(z) =1 + Qz−d
1 + PC + Qz−d − PQP−1n
. (11.17)
Assume that in (11.16)
Q(z) = Qn(z) × zdn
B(z), P = Pn, (11.18)
then
S(z) =1 + Qnzdn/zdm
1 + PC + Qnzdn/zdm − Qn, (11.19)
which recovers the form in (11.17) when dm ≥ dn, meaning that the con-ventional disturbance observer is a special case of the general disturbanceobserver.
REMARK 11.2
In Figure 11.2, if the loop is cut at u, the transfer function from u to e ory is considered as a new plant, and derived as
Pequ =y
u=
P
1 − (M − PN)Q. (11.20)
220 Modeling and Control of Vibration in Mechanical Systems
An equivalent open loop is denoted by TEQ−OL(z), and
TEQ−OL(z) = PequC =PC
1 − (M − PN)Q. (11.21)
Assume ideally that Pn = P and dm = dn. Equation (11.20) becomesPequ = P , which implies that the proposed general disturbance observer doesnot influence the characteristics of open-loop P (z)C(z) greatly and thus thestability and performance achieved by the nominal control loop with controllerC(z) are maintained. This is similar to the conventional disturbance observer.
11.4 Application results
It has been shown that a disturbance observer is capable of estimating disturbances
and modeling error [128]. Hence, disturbance observers can be used to increase the
R/W head-positioning accuracy in hard disk drives (HDDs) by using its estimation
results to cancel the effect of the disturbances and modeling error. Further, due to its
cost effectiveness and easy “add-on” implementation with minimal change required
to the existing feedback controller, the disturbance observer without using additional
sensors is frequently used to enhance the tracking performance of a hard disk drive
servo system, such as the attenuation of disturbances [127], and the compensation of
VCM pivot friction [130].
Consider the VCM plant with model P (s) given in Chapter 10. The discretized
model with the sampling time Ts = 1/30000 sec is given by (11.5) with
B(z) = 0.02399z5 + 0.1868z4 − 0.03483z3 − 0.02165z2 + 0.1728z + 0.02025,
(11.22)
A(z) = z6 − 3.052z5 + 4.657z4 − 4.979z3 + 3.997z2 − 2.399z + 0.7765.
(11.23)
The controller C(z) is the combination of a PID controller and two notch filters, and
given by
C(z) =
1.474× 10−5z6 − 4.416× 10−5z5 + 6.517× 10−5z4 − 6.621× 10−5z3
+5.015× 10−5z2 − 2.872× 10−5z + 9.053× 10−6
3.333× 10−5z6 − 7.59× 10−5z5 + 8.081× 10−5z4 − 4.846× 10−5z3
+1.545× 10−5z2 − 5.234× 10−6z
.
(11.24)
Let
M(z) =B(z)
z6, N(z) =
A(z)
z6, (11.25)
H∞-Based Design for Disturbance Observer 221
Wd1 = 0.3, Wd2 = 1 and Wn = 1.5. A stable Q(z) is then obtained via the H∞optimization in (11.14) and its frequency response is shown in Figure 11.3. Note
that the plant model inverse is not required in the general disturbance observer. This
benefit is of great significance, especially for nonminimum phase plant.
The sensitivity function |S(z)| is plotted in Figure 11.4, from which it can be
seen that the designed disturbance observer is able to suppress disturbance with fre-
quency lower than 1 kHz without causing much degradation for rejection of higher
frequency disturbance. The servo performance, such as bandwidth, will change with
different weightings Wd1 , Wd2 and Wn. Thus by adjusting the weightings accord-
ing to the weights of d1, d2, and noise n in the position error signal, the designed
disturbance observer will result in a desired reduction rate of the error. To demon-
strate the effectiveness of the disturbance estimation in the time domain, we assume
that the disturbances d1 and d2 and the noise n are generated by d1 = D1(s)w1,
d2 = D2(s)w2 and n = Nn(s)w3, where
D1(s) =0.0004(s2 − 83.39s + 9.741× 105)(s2 + 1616s + 9.626× 106)
(s2 + 125.7s + 3.948× 105)(s2 + 10.05s + 1.011× 106),
D2(s) and Nn(s) are in (2.60) and (2.61), and wi(i = 1, 2, 3) are independent white
noises with unity variance.
With the designed general disturbance observer, the estimate d1 of d1 is shown in
Figure 11.6. It follows the original d1 approximately. As a result, the error signal is
shown in Figure 11.7. 50% reduction is achieved. The general disturbance observer
is more effective to compensate for the input disturbance d1 than d2 and n. Wd2 and
Wn are selected as 1 and 1.5 is to keep the attenuation to d2 and n achieved by the
nominal feedback controller C(z). With lower Wd2 and Wn and higher Wd1 , the
attenuation to d2 and n will be degraded, although more suppression to d1 will be
attained by using the disturbance observer.
Moreover, the conventional disturbance observer is designed for comparison. M(z)and N(z) are given by
M(z) = z−1, (11.26)
N(z) = P−1n (z)
=5.2494(z2 − 1.983z + 0.9834)(z2 − 1.253z + 0.9654)(z2 + 0.1836z + 0.8179)
z(z + 0.9501)(z + 0.1259)(z + 0.116)(z2 − 1.22z + 0.9646).
(11.27)
A stable Q(z) for the conventional disturbance observer is designed with the H∞control method. Figure 11.5 shows the resultant sensitivity function, which is similar
to the one from the general disturbance observer. However, the plant model inverse
needs to be calculated.
Experiment has been done with a LDV and a dSpace 1103. The measured sensi-
tivity functions are shown in Figure 11.8, which agree with the simulation results in
Figure 11.4. To evaluate the effect of the disturbance observer on the stability and
performance achieved by the nominal controller C(z), TEQ−OL is measured with the
222 Modeling and Control of Vibration in Mechanical Systems
general disturbance observer and the conventional disturbance observer, and shown
in Figure 11.9 and Figure 11.10. As expected, performance measures such as gain
margin and phase margin are not affected.
−10
−5
0
5
10
15
20
25
30
35
40
Ma
gn
itu
de
(d
B)
101
102
103
104
45
90
135
180
225
270
315
Ph
ase
(d
eg
)
Bode Diagram
Frequency (Hz)
FIGURE 11.3
Frequency response of the designed Q(z).
11.5 Conclusion
A general form of disturbance observer has been presented and designed based on
the H∞ control method to achieve desired disturbance/noise rejection. The distur-
bance observer does not need to solve the plant model inverse, and thus its design
is simplified and has great advantages over the conventional disturbance observer,
especially for nonminimum phase plant. The simulation and implementation results
show that the general disturbance observer designed using the method employed in
this chapter is able to effectively improve the attenuation of disturbance in low fre-
quency, and will not sacrifice the stability and performance of the nominal feedback
control loop.
H∞-Based Design for Disturbance Observer 223
102
103
104
−50
−40
−30
−20
−10
0
10
Magnitude(d
B)
Frequency(Hz)
No DOBWith DOB
FIGURE 11.4
The sensitivity functions without and with the general disturbance observer.
101
102
103
104
−50
−40
−30
−20
−10
0
10
Magnitude(d
B)
Frequency(Hz)
Conventional DOB
General DOB
FIGURE 11.5
The sensitivity function comparison with the general and the conventional distur-
bance observers.
224 Modeling and Control of Vibration in Mechanical Systems
0 0.05 0.1 0.15 0.2 0.25 0.3−8
−6
−4
−2
0
2
4
6
8x 10
−3
Time(sec)
Am
plit
ude o
f dis
turb
ance d
1 a
nd its
estim
ate
(µ
m)
d1
estimate of d1
FIGURE 11.6
Disturbance d1.
0 0.05 0.1 0.15 0.2 0.25 0.3−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Time(sec)
Am
plit
ude o
f err
or
e (
µm
)
No DOB
With DOB
FIGURE 11.7
Error signal e.
H∞-Based Design for Disturbance Observer 225
102
103
104
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
No DOB
With general DOB
FIGURE 11.8
Measured sensitivity functions without and with the general disturbance observer.
101
102
103
104
−40
−20
0
20
40
60
Magnitude(d
B)
101
102
103
104
−500
−400
−300
−200
−100
0
Phase(d
eg)
Freuqency(Hz)
Nominal
with the general DOB
FIGURE 11.9
Comparison of TEQ−OL about the general disturbance observer.
226 Modeling and Control of Vibration in Mechanical Systems
101
102
103
104
−40
−20
0
20
40
60
Magnitude(d
B)
101
102
103
104
−500
−400
−300
−200
−100
0
Phase(d
eg)
Freuqency(Hz)
Nominal
with the conventional DOB
FIGURE 11.10
Comparison of TEQ−OL about conventional disturbance observer.
12
Two-Dimensional H2 Control for ErrorMinimization
12.1 Introduction
The H2 optimal control for 1-D systems is a classical problem in linear systems
theory. Its objective is to minimize the error energy of the system when the system is
subject to a unit impulse input or, equivalently, a white noise input of unit variance.
Because of this analytically and practically meaningful specification, the H2 problem
and solution has been well studied and applied for several decades. Recently, the
H2 control problem has been studied for 2-D systems and a sufficient condition for
the evaluation of 2-D system H2 performance in terms of LMIs is derived [139].
Using the condition, a systematic method for the design of the H2 controller for 2-
D systems in terms of LMIs has been developed. The developed 2-D H2 control
design method is of great importance to those systems that have 2-D behavior and
can be modeled using 2-D linear system models. Self-servo track writer (SSTW)
for data storage devices is one of these systems [17]. In the self-servo track writing,
due to vibrations and noise the servo controller causes the actuator to follow the
resulting non-circular trajectory in the next burst writing step, so that the new bursts
are written at locations reflecting the errors present in the preceding track via the
closed-loop response of the servo loop, as well as in the present track. Consequently,
each step in the process carries a “memory” of all preceding track shape errors.
This “memory” depends on the particular closed-loop response of the servo loop.
Because of the interdependency of propagation tracks, track shape errors may be
amplified from one track to the next through the closed-loop response when writing
the propagation tracks. Thus self-servo writing systems must provide a means of
accurately writing servo-patterns while controlling the propagation of track shape
errors. Therefore, error propagation containment is critically important. The target
of preventing error propagation is to reject the track shape error due to track non-
circularity recorded in propagated tracks so that the circular concentric tracks are
achieved in every propagation trace.
In this chapter we describe the SSTW process with a two-dimensional (2-D) model.
Then the error propagation containment problem of the SSTW process is formulated
as a 2-D stabilization problem. Instead of the conventional feedforward control, the
2-D stabilizing control is able to prevent the error propagation. Furthermore, the
227
228 Modeling and Control of Vibration in Mechanical Systems
TMR minimization problem of the SSTW process is formulated as a 2-D H2 con-
trol problem. A 2-D H2 controller is designed which is able to prevent the error
propagation and minimize the TMR.
12.2 2-D stabilization control
We consider the following 2-D system model [138]:[
xh(i + 1, j)xv(i, j + 1)
]
= A
[
xh(i, j)xv(i, j)
]
+ B1w(i, j) + B2u(i, j), (12.1)
y(i, j) = C1
[
xh(i, j)xv(i, j)
]
+ D11w(i, j) + D12u(i, j), (12.2)
e(i, j) = C2
[
xh(i, j)xv(i, j)
]
+ D21w(i, j) + D22u(i, j), (12.3)
where xh ∈ Rn1 , xv ∈ Rn2 , w(i, j) ∈ Rq , u(i, j) ∈ Rm, y(i, j) ∈ Rp and
e(i, j) ∈ Rl are, respectively, the horizontal state, the vertical state, the disturbance
input, the control input, the controlled output, and the measurement of the plant.
Let
x(i, j) =
[
xh(i, j)xv(i, j)
]
, (12.4)
the above system is equivalent to
x(i + 1, j + 1) = A1x(i, j + 1) + A2x(i + 1, j)
+B11w(i, j + 1) + B12w(i + 1, j)
+B21u(i, j + 1) + B22u(i + 1, j), (12.5)
y(i, j) = C1x(i, j) + D11w(i, j) + D12u(i, j), (12.6)
e(i, j) = C2x(i, j) + D22u(i, j) (12.7)
where
A1 =
[
In1 00 0
]
A, A2 =
[
0 00 In2
]
A,
Bk1 =
[
In1 00 0
]
Bk, Bk2 =
[
0 00 In2
]
Bk, k = 1, 2. (12.8)
Introduce the following 2-D output feedback controller C(z1, z2):
xc(i + 1, j + 1) = Ac1xc(i, j + 1) + Ac2xc(i + 1, j)
+Bc1e(i, j + 1) + Bc2e(i + 1, j), (12.9)
u(i, j) = Ccxc(i, j) + Dce(i, j). (12.10)
Two-Dimensional H2 Control for Error Minimization 229
Associated with the 2-D controller (12.9)−(12.10), the 2-D stabilization prob-
lem is stated as follows: for the 2-D system (12.1)−(12.3) or (12.5)−(12.7) with
w(i, j) = 0, design a dynamic output feedback controller of the form in (12.9)−(12.10)
such that the resulting closed-loop SSTW servo system is asymptotically stable.
Define
Z = DcCR + CcΞT , Vk = SB2kDc + ΛBck, (12.11)
Uk = S(Ak + B2kDcC)R + SB2kCcΞT + ΛBckCR + ΛAckΞT , k = 1, 2,
(12.12)
where R > 0, S > 0 and Ξ and Λ are invertible matrices satisfying ΞΛT = I −RS.
THEOREM 12.1
[136] Consider the 2-D system (12.5)−(12.7). Then, there exists a full orderoutput feedback controller of the form in (12.9)−(12.10) that asymptoticallystabilizes the system (12.5)−(12.7) if there exist matrices R > 0, S > 0,ΩX > 0, Dc, Uk, Vk(k = 1, 2) and Z such that the following LMI holds:
−ΩX 0 ΩTA1
0 −(ΩF − ΩX) ΩTA2
ΩA1 ΩA2 −ΩF
< 0 (12.13)
where
ΩF =
[
S II R
]
, ΩA1 =
[
SA1 + V1C U1
A1 + B21DcC A1R + B21Z
]
,
ΩA2 =
[
SA2 + V2C U2
A2 + B22DcC A2R + B22Z
]
. (12.14)
In this situation, the controller parameters of (12.9)−(12.10) can be given by
Cc = (Z − DcCR)Ξ−T , Bck = Λ−1(Vk − SB2kDc), (12.15)
Ack = Λ−1[Uk − S(Ak + B2kDcC2)R − SB2kCcΞT − ΛBckC2R]Ξ−T , k = 1, 2.
(12.16)
12.3 2-D H2 control
Let Tyw : w → y denote the closed-loop system subject to the white noise w. The
H2 norm of Tyw is approximately given by
‖Tyw‖2 =
√
√
√
√
1
L
1
K − 1
L,K−1∑
i, j=1
y(i, j)2 (12.17)
230 Modeling and Control of Vibration in Mechanical Systems
where L and K are large enough. The control design problem to minimize the 2-
D H2 norm is stated as follows: find a 2-D output feedback controller of the form
in (12.9)−(12.10) for the 2-D system (12.1)−(12.3) or (12.5)−(12.7) such that the
closed-loop system is stable and the H2 performance ‖Tyw‖2 is minimized.
An LMI approach will be given as follows to design a 2-D H2 controller for the
2-D system (12.1)−(12.3) such that the closed-loop system is stable and the error is
minimized.
THEOREM 12.2
[139] The 2-D H2 control problem for the plant (12.1)−(12.3) is solvable ifthere exist matrices S > 0, Θ, Λ, Γ, Dc and block-diagonal matrices X =diagXh, Xv, Y = diagY h, Y v, M = diagMh, Mv, H11 = diagHh
11,Hv
11 > 0, H22 = diagHh22, H
v22 > 0, and H12 = diagHh
12, Hv12, of appro-
priate dimensions, such that
H11 ∗ ∗ ∗ ∗HT
12 H22 ∗ ∗ ∗XA + ΓC2 Θ X + XT − H11 ∗ ∗
A + B2DcC2 AY T + B2Λ I + MT − HT12 Y + Y T − H22 ∗
C1 + D12DcC2 C1YT + D12Λ 0 0 I
> 0,
(12.18)
S ∗ ∗ ∗XB1 + ΓD21 X + XT − H11 ∗ ∗
B1 + B2DcD21 I + MT − HT12 Y + Y T − H22 ∗
D11 + D12DcD21 0 0 I
> 0, (12.19)
Trace(S) < λ2, (12.20)
are satisfied. If the above stated conditions are satisfied, a feasible H2 con-troller is given by
[
xhc (i + 1, j)
xvc(i, j + 1)
]
= Ac
[
xhc (i, j)
xvc(i, j)
]
+ Bce(i, j), xhc ∈ Rnh , xv
c ∈ Rnv , (12.21)
u(i, j) = Cc
[
xhc (i, j)
xvc (i, j)
]
+ Dce(i, j), (12.22)
with
Ac = U−1[Θ − X(A + B2DcC2)YT − XB2CcV
T − UBcC2YT ]V −T ,(12.23)
Cc = (Λ − DcC2YT )V −T , Bc = U−1(Γ − XB2Dc), (12.24)
where U = diagUh, Uv and V = diagV h, V v satisfy XY T +UV T = M .
REMARK 12.1 The solution of the 2-D controller in the above theoremis in terms of LMIs which can be efficiently solved by convex optimization just
Two-Dimensional H2 Control for Error Minimization 231
like other 1-D control problems involving LMIs. In [139], it is proved that ifthere exist solutions X, Y , M , Dc, S > 0, Θ, Λ, Γ, H11, H22, and H12 forthe LMIs (12.18)-(12.20), M − XY T is invertible. From UV T = M − XY T ,nonsingular matrices U and V can be computed. Then, the computation ofthe controller parameters Cc, Ac, Bc can be carried out by solving (12.23)
and (12.24).
12.4 SSTW process and modeling
FIGURE 12.1
SSTW process.
The process of self-servo writing is shown in Figure 12.1 and is generally known
to involve the following distinct steps [135]: writing some tracks or at least one track
called seed tracks; reader reads back the seed track and writer writes actual product
servopattern for the next track based on the readback signal; writing servopattern for
232 Modeling and Control of Vibration in Mechanical Systems
the next track based on the readback signal from the previous written track till the
whole process is completed. During the process, SSTW generates radial informa-
tion progressively to deploy servopattern. The radial error in track N is inevitably
compounded to the following tracks: tracks N + 1, N + 2, · · ·. This leads to error
propagation in SSTW.
In self-servo track writing, track shape errors such as non-circularity are intro-
duced by mechanical disturbances, spindle motor vibration and other factors when
writing the propagation tracks. The servo controller causes the actuator to follow
the resulting non-circular trajectory in the next burst writing step, so that the new
bursts are written at locations reflecting the errors present in the preceding step via
the closed-loop response of the servo loop, as well as in the present step. Conse-
quently, each step in the process carries a “memory” of all preceding track shape
errors. This “memory” depends on the particular closed-loop response of the servo
loop. Self-servo writing systems must provide a means of accurately writing ser-
vopatterns while controlling the propagation of track shape errors.
12.4.1 SSTW servo loop
FIGURE 12.2
SSTW servo loop with disturbances and noise models.
Figure 12.2 shows the SSTW servo loop with disturbances and noises. D1(s),D2(s), and N(s) are respectively the models of disturbances d1, d2 and noise n.
y(k) is the position of the write head with respect to a perfectly circular track on the
disk and PES(k) is the position error signal. Let Tp be the rotational period of the
disk, and Ts be the sampling rate of the position error signal. Then the sector number
K = Tp/Ts. y(k − K) represents the track profile of the previous track. Similarly,
PES(k−K) represents the position error when writing the previous track. The read
head follows on the track y(k −K) which is the reference input for the SSTW servo
system, i.e., one revolution of y(k) becomes the reference of the next written track
Two-Dimensional H2 Control for Error Minimization 233
or a few subsequent tracks due to the action of self-servo writing.
It follows from Figure 12.2 that
y(k) = T (z)y(k − K) + P (z)S(z)d1(k) + S(z)d2(k) + T (z)n(k), (12.25)
where S(z) = 1/(1 + P (z)C(z)) is the sensitivity function, T (z) = 1−S(z) is the
closed-loop transfer function. The typical closed-loop transfer function will amplify
the error at the frequency where its magnitude is more than 0 dB. In other words,
these frequency components in disturbances or noise will be amplified during prop-
agation, while others will compound to following tracks and decay gradually. With
the same disturbance models D1(s) and D2(s) and noise model N(s) as in Chapter
11, Figure 12.3 shows PES NRRO in the time domain and its σ value versus track
number when a 1-D feedback controller C(z) is used in the closed loop. The error
propagation mentioned previously is clearly observed. Hence the error propagation
problem must be addressed in the servo control design for SSTW.
0 1 2 3 4 5 6 7 8 9 10−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Track number
PE
S a
nd
σ o
f P
ES
Position error signalSigma value
FIGURE 12.3
PES NRRO and its σ values versus track number during propagation. (The time
sequence and the σ value increase with the track number.)
12.4.2 Two-dimensional model
For the 2-D modeling, the following notations are used:
234 Modeling and Control of Vibration in Mechanical Systems
FIGURE 12.4
SSTW servo loop modeling in two dimensions.
i = 0, 1, 2, · · ·, L: the ith track.j · Ts: time with j = 0, 1, 2, · · · , K − 1.
As shown in Figure 12.4, y(i, j) is the position of the write head at track i in the
radial dimension and time j · Ts in the axial dimension and PES(i, j) the position
error. y(i − 1, j) (i.e., y(j − K)) represents the track profile of the (i − 1)-th track.
Similarly, PES(i − 1, j) represents the position error of the (i − 1)-th track. The
read head follows the track y(i − 1, j) which is the reference input for the SSTW
servo system, i.e., one revolution of y(i, j) becomes the reference of the next written
track due to the action of self-servo track writing.
Based on Figure 12.4, we have
xp(i, j + 1) = Apxp(i, j) + Bp(u(i, j) + d1(i, j)), (12.26)
y(i, j) = Cpxp(i, j) + Dp(u(i, j) + d1(i, j)) + d2(i, j), (12.27)
PES(i, j) = y(i − 1, j) − y(i, j) + n(i, j). (12.28)
Denote xp, xd1 , xd2 and xn the corresponding state vectors of P (z), D1(z), D2(z)and N(z), respectively. Let
xh(i + 1, j) = y(i, j), xh ∈ Rnh , xv ∈ Rnv ,
xv(i, j + 1)T = [xp(i, j + 1)T xd1 (i, j + 1)T xd2(i, j + 1)T xn(i, j + 1)T ],
e(i, j) = PES(i, j), w(i, j)T = [w1(i, j)T w2(i, j)
T w3(i, j)T ], (12.29)
it follows from (12.26)−(12.28) that
[
xh(i + 1, j)xv(i, j + 1)
]
= A
[
xh(i, j)xv(i, j)
]
+ B1w(i, j) + B2u(i, j), (12.30)
y(i, j) = C1
[
xh(i, j)xv(i, j)
]
+ D11w(i, j) + D12u(i, j), (12.31)
Two-Dimensional H2 Control for Error Minimization 235
e(i, j) = C2
[
xh(i, j)xv(i, j)
]
+ D21w(i, j) + D22u(i, j), (12.32)
where
A =
0 Cp DpCd1 Cd2 00 Ap BpCd1 0 00 0 Ad1 0 00 0 0 Ad2 00 0 0 0 An
, B1 =
DpDd1 Dd2 0BpDd1 0 0Bd1 0 00 Bd2 00 0 Bn
, (12.33)
B2 =
Dp
Bp
000
, C1 =[
0 Cp DpCd1 Cd2 0]
, (12.34)
D11 =[
DpDd1 Dd2 0]
, D12 = Dp, (12.35)
C2 =[
1 − Cp − DpCd1 − Cd2 Cn
]
, (12.36)
D21 =[
−DpDd1 − Dd2 Dn
]
, D22 = −Dp. (12.37)
As such, the SSTW servo loop is modeled as the 2-D Roesser model (12.30)−(12.37)
[138] where disturbance and noise models are taken into consideration. The SSTW
error propagation problem can then be simplified as the stabilization problem of a
2-D system. Unlike 1-D feedback control plus feedforward compensation, it does
not need an additional feedforward controller to prevent the error propagation. In the
next section, feedforward compensation on the basis of 1-D feedback control will be
presented, followed by 2-D control in a later section.
12.5 Feedforward compensation method
To contain the error propagation, it is easy to come up with the idea of injecting a
correction signal f(k) to the PES by using the available signal PES(k−K) through
a feedforward compensator F (z). Thus a feedforward (FF) compensation method is
used as shown in Figure 12.5.
During the servo writing process, by tracking the previously written adjacent
tracks, adjusting servo reference and closing the existing VCM loop, the head can
be offset to a known radial position with reference to the adjacent tracks and gen-
erates radial information progressively to deploy servo pattern. The radial error in
one track is inevitably compounded to the following tracks and propagates accord-
ing to the closed-loop transfer function from y(k − K) to y(k), as seen in (12.25).
The target of error propagation containment is thus to reject the written-in error due
to track non-circularity recorded in propagated tracks so that the circular concentric
236 Modeling and Control of Vibration in Mechanical Systems
FIGURE 12.5
SSTW servo loop.
tracks are achieved in every propagation trace. Obviously, when there is no hump in
the closed-loop transfer function T (z), the error propagation will be contained.
Next, we use three control schemes for the self-servo track writer: (1). PD con-
trol since it can produce a flat closed-loop response such that the error propagation
is contained even without a feedforward compensation; (2). PID control plus feed-
forward compensation; (3). To minimize TMR or equivalently the H2 norm of the
transfer function from w = [w1 w2 w3]T to track profile y, the H2 control tech-
nique is employed to design the feedback controller C(z) and then a feedforward
compensator F (z) is designed to contain the error propagation.
The frequency response of the VCM plant under consideration is shown in Figure
12.6, and its transfer function P (s) is described by the following zeros, poles and
gain:
zeros = 104 × [3.4558, − 0.6158± 8.7749j, 0.7540± 4.9697j,
−0.1960± 3.2614j,−0.9425± 1.6324j], (12.38)
poles = 105 × [−0.2513, − 0.3456, − 0.0726± 1.0342j, − 0.0090± 0.5968j,
−0.0377± 0.3751j, − 0.0283± 0.2813j,
−0.0011± 0.0036j], (12.39)
gain = 5.8987× 1012. (12.40)
(1). PD feedback control
A typical form of PID controllers is given by
C(z) = Kp + Kiz
z − 1+ Kd
z − 1
z. (12.41)
Let Ki = 0, a PD controller leading to a flat closed-loop transfer function can
be obtained by adjusting Kp and Kd. Denote the closed-loop transfer function as
Two-Dimensional H2 Control for Error Minimization 237
102
103
104
−60
−40
−20
0
20
40
60
Frequency(Hz)
Ma
gn
itu
de
(dB
)
MeasuredModeled
102
103
104
−1000
−800
−600
−400
−200
0
Ph
ase
(de
g)
FIGURE 12.6
Frequency response of a VCM actuator.
238 Modeling and Control of Vibration in Mechanical Systems
FIGURE 12.7
Frequency response of the closed-loop transfer function with the PD controller, PID
controller, and H2 controller.
TPD(z). Figure 12.7 shows a flat magnitude response of TPD(z) resulting from the
PD controller given below. The sampling rate is 12.64 kHz.
C(z) =4.79× 10−5 − 9.573× 10−5 + 4.783× 10−5
z2 − z. (12.42)
Theoretically the error propagation would disappear as ‖TPD‖∞ ≤ 1. However,
the PD control is not able to deal with bias force due to lack of integrator. Simulation
shows that the propagation stops after several tracks when a small bias force of 0.001is added. Therefore, PD control is not practically applicable.
(2). PID feedback control plus feedforward compensation
The closed-loop transfer function with the PID controller
C(z) =2.435× 10−5z2 − 4.406× 10−5z + 1.999× 10−5
z2 − 1.209 + 0.24(12.43)
has a region greater than 0 dB as seen in Figure 12.7. As mentioned earlier, this will
cause the error propagation problem. Thus a feedforward compensator F (z) will be
designed to contain the error propagation. According to Figure 12.5,
y(k) =
(
PC
1 + PC+
F
1 + PC
)
y(k − K) +P
1 + PCd1(k) +
1
1 + PCd2(k)
Two-Dimensional H2 Control for Error Minimization 239
+PC
1 + PCn(k) − PF
1 + PCd1(k − K) − F
1 + PCd2(k − K), (12.44)
which means that if the feedforward compensator F (z) is designed such that the
magnitude of the transfer function
Φ =PC
1 + PC+
F
1 + PC(12.45)
is less than one, i.e., ‖Φ‖∞ < 1, the error propagation can be contained. It is
straightforward from (12.45) that
F (z) = Φ(z)(1 + P (z)C(z)) − P (z)C(z). (12.46)
When Φ is selected as the closed-loop transfer function with the previous PD con-
trol, a 7th order F (z) is obtained from (12.46) after order reduction. The designed
F (z) can prevent the error propagation when the PID feedback control is applied in
the servo loop since ‖Φ‖∞ < 1 with the designed F (z), as observed in Figure 12.8.
Figure 12.9 shows the 3σ value of the PES versus frequency with error propagation
containment.
101
102
103
104
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Frequency(Hz)
Ma
gn
itu
de
(dB
)
FF with H2 FF with PID
FIGURE 12.8
|Φ| versus frequency.
240 Modeling and Control of Vibration in Mechanical Systems
0 1000 2000 3000 4000 5000 6000 7000
2
4
6
8
10
12
14
Frequency(Hz)
3σ
(p
erc
en
t o
f tr
ack)
PID
H2
FIGURE 12.9
3σ of PES NRRO.
(3). H2 feedback control plus feedforward compensation
With the identified disturbance models in Chapter 2, the augmented system for the
optimal H2 control design is described as follows.
x(k + 1) = Ax(k) + B1w(k) + B2u(k), (12.47)
PES(k) = C1x(k) + D11w(k), (12.48)
z(k) = C2x(k) + D21w(k) + D22u(k), (12.49)
where
A =
Ap BpCd1 0 00 Ad1 0 00 0 Ad2 00 0 0 An
, B1 =
BpDd1 0 0Bd1 0 00 Bd2 00 0 Bn
, B2 =
Bp
000
,(12.50)
C1 =[
Cp 0 Cd2 Cn
]
, D11 =[
0 Dd2 Dn
]
, (12.51)
C2 =[
Cp 0 Cd2 0]
, D21 = 0, D22 = 0, (12.52)
x is the combined state variables from the VCM actuator model P (z), the input dis-
turbance model D1(z), the output disturbance model D2(z), and the measurement
noise model N(z). (Ap, Bp, Cp, Dp), (Ad1 , Bd1 , Cd1 , Dd1 ), (Ad2 , Bd2 , Cd2 , Dd2),and (An, Bn, Cn, Dn) are respectively the state-space models of P (z), D1(z),D2(z) and N(z). PES is the measured position error signal and z stands for y in
Figure 12.5. The H2 control problem can be solved using the method in Chapter 5.
Two-Dimensional H2 Control for Error Minimization 241
FIGURE 12.10
Frequency response of the H2 controller.
FIGURE 12.11
Frequency response of the open-loop system with the H2 controller.
242 Modeling and Control of Vibration in Mechanical Systems
FIGURE 12.12
Comparison of sensitivity functions.
Figures 12.10 and 12.11 show the frequency responses of the designed H2 feed-
back controller and the compensated open-loop system with bandwidth of 1.07 kHz,
gain margin 7.3 dB, and phase margin 45.9 deg. Figure 12.12 shows the comparison
of error rejection functions with the PID control and the H2 control, where we can
see that the H2 control outperforms the PID control in error rejection.
The frequency response of the closed-loop transfer function with the H2 control
is shown in Figure 12.7, which implies that a feedforward compensator is needed to
contain the error propagation when the servo loop uses the H2 feedback controller.
For the design of the feedforward compensator F (z), the selected Φ depends on the
feedback controller and it is found that lower Φ may not give a better error propaga-
tion containment. When Φ = 0.9TPD, a satisfactory error propagation containment
is guaranteed when the H2 feedback control is applied in the SSTW servo loop. A
6th order F (z) is thus obtained after model reduction. The resultant |Φ| with the
designed F (z) is shown in Figure 12.8, and it is seen that |Φ| < 1. Figure 12.13
shows the σ value comparison of PES at different tracks with error propagation con-
tainment. It can be observed that PES σ is up and down as track propagation is going
on. Compared with that of the PID control, the σ of PES versus track number is
improved by around 27% when the H2 control is employed in the SSTW servo loop.
Two-Dimensional H2 Control for Error Minimization 243
FIGURE 12.13
σ value of PES NRRO versus track number.
12.6 2-D control formulation for SSTW
With the 2-D model (12.30)−(12.37), the SSTW error propagation problem is sim-
plified as the stabilization problem of a 2-D system. That is to design a dynamic out-
put feedback controller of the form in (12.9)−(12.10) for the model (12.30)−(12.32)
with w(i, j) = 0 such that the resulting closed-loop system is asymptotically sta-
ble. Unlike the design of 1-D feedback plus feedforward compensation previously, it
does not need an additional feedforward controller to prevent the error propagation.
It is not difficult to see from (12.9)−(12.10) that (Ac2, Bc2, Cc, Dc) is acting
along the time direction only, and thus actually it works like the 1-D feedback con-
troller that we are concerned with conventionally.
The 2-D controller (12.21)−(12.22) can be written equivalently to the form (12.9)-
(12.10). Let
Ac1 =
[
Inh0
0 0
]
Ac, Ac2 =
[
0 00 Inv
]
Ac, (12.53)
Bc1 =
[
Inh0
0 0
]
Bc, Bc2 =
[
0 00 Inv
]
Bc. (12.54)
The performance of (Ac2, Bc2, Cc, Dc) can thus be evaluated as a normal 1-D
244 Modeling and Control of Vibration in Mechanical Systems
control in time dimension.
As is known that one of the most important performance measures for SSTW is
the track misregistration or TMR, the total amount of random fluctuation about the
desired track location. TMR is used to judge the required accuracy of positioning. To
achieve a high positioning accuracy, one way in servo control is to minimize TMR,
which is expressed as the standard deviation of the true PES, i.e.
σy(i,j) =
√
√
√
√
1
L
1
K − 1
L,K−1∑
i,j=0
y(i, j)2 . (12.55)
Let Tyw : w → y denote the closed-loop system subject to a white noise w. When
L and K are large enough, the H2 norm of Tyw can be approximately given by
‖Tyw‖2 =
√
√
√
√
1
L
1
K − 1
L,K−1∑
i, j=1
y(i, j)2 . (12.56)
Thus, the control design problem to minimize TMR can be treated as a 2-D H2
optimal control problem, which is stated as follows: find a 2-D output feedback
controller of the form in (12.9)−(12.10) for the SSTW plant P (s) such that the
closed-loop system is stable and the H2 performance ‖Tyw‖2 is minimized.
The problem of minimizing the TMR of the SSTW process is thus formulated as
the 2-D H2 control problem. A 2-D H2 controller will subsequently be designed to
minimize the track mis-registration. Note that since the 2-D controller stabilizes the
system, it also contains the error propagation. Thus, the 2-D H2 control approach
simultaneously addresses the error propagation and TMR minimization problems,
which is different from the previous 1-D method where the problems are addressed
separately by feedback and feedforward controls.
12.7 2-D stabilization control for error propagation containment
12.7.1 Simulation results
The simulation block diagram on 2-D control is shown in Figure 12.14. The sim-
ulation is carried out in MATLAB/Simulink. In the simulation, the sector number
K = 270, the spindle rotational speed is 7200 RPM, and thus the sampling fre-
quency is 270× (7200/60) = 32400 Hz.
To show the capability of the designed 2-D controller to prevent the error propa-
gation, 100 tracks are propagated in the simulation. The results show that the SSTW
process is stabilized and the error propagation is contained. The σ value of PES
NRRO is plotted versus the track number in Figure 12.15, which implies that the
error amplitude is oscillating steadily.
Two-Dimensional H2 Control for Error Minimization 245
FIGURE 12.14
2-D controller for SSTW servo loop.
From (12.9)−(12.10) and Figure 12.14, it is not difficult to see that (Ac2, Bc2,
Cc, Dc) is acting along the time direction only, and thus actually it works like the
1-D controller we usually take into account. Subsequently, the open-loop and the
sensitivity function frequency responses are obtained and shown in Figures 12.16
and 12.17. The open-loop crossover frequency is 1.6 kHz, the gain margin is 6 dB,
and the phase margin is 50 degrees. A bad high frequency part of the sensitivity
function has appeared, because here only the stabilization problem is considered and
no performance optimization is involved. Thus the 2-D H2 control will be presented
which gives a better performance than the stabilizing only controller.
12.8 2-D H2 control for error minimization
12.8.1 Simulation results
The σ value of PES NRRO is plotted versus the track number in Figure 12.15, where
the error amplitude is oscillating steadily, which implies that error propagation is
contained. Additionally, the position error has been minimized in the H2 norm sense.
The σ values of PES NRRO with the stabilization control and the H2 control are
compared in Figure 12.15 and it is seen that the error is reduced by an average of
60% via the 2-D H2 control.
The conventional 1-D H2 feedback control has also been designed to minimize
the error signal, and based on the feedback control a feedforward control is further
designed to contain the error propagation. This method is also compared with the
above two methods in Figure 12.15. It is seen that the proposed 2-D control method
is comparable to the previous 1-D control method.
246 Modeling and Control of Vibration in Mechanical Systems
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9x 10
−3
Track number
σ o
f p
es N
RR
O (
µm
)
2−D H2 control2−D Stabiliation1−D H2 feedback+feedforward control
FIGURE 12.15
σ of PES NRRO versus track number.
102
103
104
−20
−10
0
10
20
30
Ma
gn
itu
de
(dB
)
102
103
104
−250
−200
−150
−100
−50
Ph
ase
(de
g)
Frequency(Hz)
FIGURE 12.16
Open-loop frequency response with stabilization controller.
Two-Dimensional H2 Control for Error Minimization 247
102
103
104
−30
−25
−20
−15
−10
−5
0
5
10
15
20
Ma
gn
itu
de
(dB
)
Frequency(Hz)
FIGURE 12.17
Sensitivity function with stabilization controller (Ac2, Bc2, Cc, Dc).
12.8.2 Experimental results
It is noted that in the 2-D controller (12.9)−(12.10), (Ac2, Bc2, Cc, Dc) acts in
the time dimension, and runs on the same track as a 1-D controller C(z) we have
usually considered. Thus in this section we particularly take into account the 1-D
controller C(z) : (Ac2, Bc2, Cc, Dc) for the plant P (s) (12.38)−(12.40) in the
time dimension.
Straightforwardly from the designed 2-D controller C(z1, z2) : (Ac1, Ac2, Bc1,Bc2, Cc, Dc), the 1-D controller C(z) : (Ac2, Bc2, Cc, Dc) can be obtained
and its frequency response is shown in Figure 12.18, where the notches around 4, 6and 9 kHz are to suppress the resonances as observed in Figure 12.6. With the 1-D
controller C(z), the simulated frequency responses of the open loop C(z)P (z) and
the sensitivity function S(z) = 1/(1 + C(z)P (z)) are drawn in Figures 12.19 and
12.20.
To practically verify the 1-D controller C(z), experiment is carried out with dSpace
1103 on TMS320C240 DSP board and the LDV used to measure the displacement of
the actuator. The measured open-loop frequency response compared with the simu-
lated one is shown in Figure 12.19, and the crossover frequency is 1.2 kHz, the gain
margin is 9 dB, and the phase margin is 54 degrees. Figure 12.20 also shows the
measured sensitivity function with comparison to the simulated one. Additionally,
the step response and the control signal of the closed loop are taken from the oscillo-
248 Modeling and Control of Vibration in Mechanical Systems
scope as a bmp picture shown in Figure 12.21, which testifies that the 1-D controller
drawn from the designed 2-D controller works well.
The results in this section show that the 1-D controller C(z) extracted from the 2-
D controller can stabilize the plant P (s) in the time dimension. However, as shown
in Figure 12.3, one 1-D feedback controller C(z) is not capable enough to contain
error propagation and stabilize the SSTW process modeled in (12.30)−(12.37) in
two dimensions. On the basis of the 2-D model, the designed 2-D controller can
stabilize the SSTW process and minimize the position error simultaneously.
Bode Diagram
Frequency (Hz)
Ph
ase
(d
eg
)M
ag
nitu
de
(d
B)
−30
−20
−10
0
10
20
30
100
101
102
103
104
−90
−45
0
45
90
135
FIGURE 12.18
Frequency response of controller (Ac2, Bc2, Cc, Dc).
12.9 Conclusion
This chapter has employed 2-D controllers in stabilization and error minimization
of systems that have 2-D behavior and can be modeled as a 2-D model. The ap-
plication in a self-servo track writing process described by a 2-D model has been
addressed in detail. By applying the two-dimensional model, the error propagation
Two-Dimensional H2 Control for Error Minimization 249
102
103
104
−60
−40
−20
0
20
40
Ma
gn
itu
de
(dB
)
MeasuredSimulated
102
103
104
−600
−500
−400
−300
−200
−100
0
100
Ph
ase
(de
g)
Frequency(Hz)
FIGURE 12.19
Open-loop frequency response with controller (Ac2, Bc2, Cc, Dc).
102
103
104
−40
−35
−30
−25
−20
−15
−10
−5
0
5
10
Frequency(Hz)
Magnitude(d
B)
MeasuredSimulated
FIGURE 12.20
Sensitivity function with controller (Ac2, Bc2, Cc, Dc).
250 Modeling and Control of Vibration in Mechanical Systems
FIGURE 12.21
Step response (Channel 1/2/3: Reference/Output/Control signal).
containment problem and the TMR minimization problem of the self-servo track
writing process has been formulated as a 2-D H2 control problem. With the stored
error information of the preceding track, the adopted 2-D controller is applicable.
The 2-D approach provides a systematic way of achieving both the error propagation
containment and the TMR minimization simultaneously. The simulation results have
demonstrated that the error propagation is prevented by the 2-D control scheme, and
good positioning accuracy is achieved with the 2-D H2 control scheme. Also, the
time dimensional portion, as a 1-D controller, of the designed 2-D controller has
been implemented via LDV and dSpace and the implementation results have been
shown to verify that the 1-D controller performs well.
13
Nonlinearity Compensation and NonlinearControl
13.1 Introduction
Nonlinearities such as friction in the actuator pivot are known to limit the low fre-
quency gain of a control loop. Translated to the error rejection function or sensitivity
function, it lifts the magnitude of the sensitivity function at low frequencies, and
thus reduces the ability of the control loop to reject vibrations at low frequencies and
affects the system performance. Based on an identified friction model, the friction
can be compensated for by injecting an estimated friction force into the actuator. A
friction compensation method based on a nonlinear hysteresis model is thus studied.
Moreover, on the basis of a linear feedback control, to further improve the re-
jection of low-frequency disturbances such as nonlinear disturbances arising from
friction torque or bias or other unknown disturbances, an adaptive nonlinear com-
pensation scheme will be adopted in this chapter to cancel their effects through a
proper estimation of the disturbances.
13.2 Nonlinearity compensation
As stated in Chapter 2, the nonlinear friction model fe = F (x(k)) is identified as
in the form of (2.20) using the operator based method. With the model, the friction
can be compensated for by injecting the friction force fe into the plant, as seen in
Figure 13.1. A sinusoidal signal of 50 Hz and 1 V amplitude is injected as the
reference. The input u versus the actuator displacement x is compared for the cases
of with and without the nonlinear compensation in Figure 13.2. It is evident that
with the nonlinear compensation the relationship between u and x is linearized very
well. In the frequency domain, the actuator frequency responses before and after the
compensation are measured with a swept sine wave via a DSA and shown in Figure
13.3. The compensated magnitude and phase responses approach those of the pure
double integrator much more closely than those before compensation.
With the friction compensation, the VCM actuator frequency responses are mea-
251
252 Modeling and Control of Vibration in Mechanical Systems
sured with different sinusoidal reference amplitudes and plotted in Figures 13.4 and
13.5, where the straight smooth lines are from the pure double integrator. It is seen
that the linearization effect becomes better when the reference amplitude is higher
than 1 V (0.5 µm/V), and it is not so satisfactory for 0.25 V. Although the friction
model is obtained on the basis of the measurement for 0.5, 1, and 3 V displacement
amplitudes, the compensation based on the obtained model is able to achieve good
linearization effect for any displacement ranging from 0.5 V and above.
FIGURE 13.1
Friction compensation for the actuation system.
Figure 13.6 shows the corresponding simulated and measured sensitivity func-
tions. With the compensation the magnitude at 10 Hz is reduced by around 20 dB
due to the increased open-loop gain at low frequencies as seen in Figures 13.4 and
13.5. The slightly lower magnitude from 60 to 100 Hz before compensation, corre-
sponding to the higher magnitude of the original actuator model from 60 to 100 Hz
in Figure 13.3, is caused by the nonlinearity of the original actuator.
In the above, a model based nonlinearity compensation has improved the ability of
the closed control loop to reject vibrations in low frequency range. In the rest of the
chapter, a non-model based compensation, which is an adaptive and a more flexible
scheme, will be applied to compensate the nonlinear or unknown vibrations in low
frequency range.
13.3 Nonlinear control
While a KYP Lemma-based linear control can achieve disturbance rejection over
some chosen frequency ranges, it cannot run away from performance limitation,
Nonlinearity Compensation and Nonlinear Control 253
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5x 10
−3
Displacement (V, 0.5 µm/V)
Inp
ut
u (
V)
No compensation
With compensation
FIGURE 13.2
Input u versus displacement x with and without compensation.
101
102
40
50
60
70
80
90
100After comp.(measured)After comp.(simulated)
1/s2
Before comp.
101
102
−400
−300
−200
−100
0
Frequency(Hz)
Ph
ase
(d
eg
)M
ag
nitu
de
(dB
)
FIGURE 13.3
Actuator frequency responses with and without friction compensation.
254 Modeling and Control of Vibration in Mechanical Systems
101
102
40
60
80
100
101
102
−300
−200
−100
0
Frequency(Hz)
Phase (
deg)
Magnitude(d
B)
0.25V
0.5V
FIGURE 13.4
Actuator frequency responses with friction compensation for different displacements
in voltage with 0.5µm/V (Straight smooth lines: the pure double integrator).
101
102
40
60
80
100
101
102
−300
−200
−100
0
Frequency(Hz)
Phase (
deg)
Magnitude(d
B)
1V
3V
FIGURE 13.5
Actuator frequency responses with friction compensation for different displacements
in voltage with 0.5 µm/V (Straight smooth lines: the pure double integrator).
Nonlinearity Compensation and Nonlinear Control 255
101
102
103
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
10After compensationAfter compensationBefore compensation
Magnitude(d
B)
Frequency(Hz)
FIGURE 13.6
Sensitivity functions with and without friction compensation.
as observed from the well known Bode integral constraint. For example, the KYP
Lemma-based control design in Chapter 8 results in an excellent attenuation of dis-
turbance around 650 Hz. But its ability in rejecting low frequency disturbance is
not very desirable. On the other hand, in Chapter 2, the low-frequency disturbance
is modeled as the output of an adaptive nonlinear scheme with the error signal as
the input. Here it is used to compensate for the low-frequency disturbance, i.e., a
nonlinear controller is augmented with the KYP Lemma-based linear feedback con-
troller, which remarkably improves the system disturbance rejection capability in
low frequency range without sacrificing performance at other frequencies.
Considering Figure 13.7 with plant P (z): (Ap, Bp, Cp, Dp), we have the fol-
lowing discrete time state-space realization:
x(k + 1) = Apx(k) + Bpu(k) + Bpd1(k), (13.1)
z(k) = −Cpx(k) + w(k) − Dpu(k) − Dpd1(k) − d2(k), (13.2)
e(k) = −Cpx(k) + w(k) − Dpu(k) − Dpd1(k) − d2(k) + n(k). (13.3)
The design of control law u = uL + uN includes two parts:
1. The first is to design a linear dynamic output feedback controller uL = C(z)efor the plant P (z) such that the closed-loop system is stable and satisfies the
performance in (8.2).
256 Modeling and Control of Vibration in Mechanical Systems
FIGURE 13.7
Control structure of a plant P (s) with Youla parametrization approach and adaptive
nonlinear compensation.
2. The second is to design a nonlinear control law uN such that the contribution
of the low-frequency disturbance in d1 to the error can be compensated.
13.3.1 Design of a composite control law
The linear control law design based on the KYP Lemma in Chapter 8 can give good
rejection of disturbance of particular frequencies such as that around 650 Hz. How-
ever, the performance at low frequency needs to be improved. Thus we shall design a
nonlinear compensation based on the nonlinear modeling discussed in Section 2.4.2.
Note that the modeling in Section 2.4.2 is based on the time history of the mea-
surement e. We consider the modeled disturbance d1 given by
d1 = ωT (k)s(Φk) (13.4)
with the update law
ω(k + 1) = (1 − δ)ω(k) − Γs(Φk)e(k), (13.5)
where s(Φk) is given in (2.56).
As demonstrated later, by injecting
uN(k) = −d1 (13.6)
to the plant combined with uL as shown in Figure 13.7, we are able to compensate
the low-frequency disturbance in d1, which is modeled as in (2.55).
Nonlinearity Compensation and Nonlinear Control 257
13.3.2 Experimental results in hard disk drives
The plant under consideration and the linear controller are the same as in Section
8.6 of Chapter 8. The nonlinear control signal uN in Figure 13.7 is calculated from
(13.5)−(13.6) and (2.56) in Section 2.4.2 of Chapter 2. The center positions ceiand
c∆eifor the measurement e and velocity e are chosen as zero. The variances are
σ2ei
= σ2∆ei
= 10, i = 1, ..., p. The forgetting factor δ = 0.5. Γ affects the learning
speed and should be selected to be as large as possible.
To evaluate the disturbance rejection performance of the combined linear control
C(z) designed in Section 8.6 in Chapter 8 and the nonlinear control (13.6), a sinu-
soidal signal with the logarithmically spaced frequency from 10 Hz to 22.5 kHz is
respectively injected as w in Figure 13.7. For each frequency sinusoidal input, the
error signal e will involve multiple frequency components due to the nonlinear con-
trol. In this situation, it is reasonable that the error rejection capability is directly
measured as the amplitude ratio e/w in time domain. At each frequency point, the
error rejection e/w is then plotted and shown in Figure 13.8.
The error rejection capability is evaluated for each value of p = 1, 5, 9. As in
Figure 13.8, the two cases with p = 1 and p = 5 give similar results, and both are
better than that given by p = 9. This implies that a higher p may not necessarily
lead to a better result. This phenomenon is consistent with the observation in mod-
eling. Overall, from the simulation result, the nonlinear control produces a better
rejection of disturbances of low frequencies, while not affecting the high frequency
disturbance rejection performance.
Figure 13.8 also shows the effect of Γ on the error rejection. 10% of the Γ value
in (2.58) is used in the calculation. It is observed that the larger Γ yields a better
accuracy. This also agrees with the modeling result in Section 2.4.2 in Chapter 2.
Consequently, corresponding to the error rejection in Figure 13.8 with p = 1, the
power spectrum of PES NRRO is shown in Fig 13.9. It is observed that the error
is much lowered by 80% before 400 Hz, and no cost in higher frequency range is
paid. Overall it is evaluated from calculation that the 3σ of the true PES NRRO
is improved from 6.0 nm with the KYP Lemma method to 5.5 nm with the KYP
Lemma-based linear control augmented with the nonlinear control.
REMARK 13.1 It should be mentioned that the nonlinear compensationscheme can be combined with any linear control to improve low frequencyvibration rejection without sacrificing disturbance rejection capability in otherfrequency ranges.
258 Modeling and Control of Vibration in Mechanical Systems
101
102
103
104
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
10
p=1p=5p=9No u
N
Ma
gn
itu
de
(dB
)
Frequency(Hz)
0.1Γ
FIGURE 13.8
Comparison of error rejection frequency response without and with uN of different
p and Γ.
Nonlinearity Compensation and Nonlinear Control 259
0 1000 2000 3000 4000 5000 60000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−3
KYPKYP+nonlinear control
NR
RO
magnitude(µ
m)
Frequency(Hz)
400 Hz
FIGURE 13.9
NRRO power spectrum with KYP lemma-based linear control and nonlinear com-
pensation (80% reduction before 400 Hz).
13.4 Conclusion
The nonlinear friction has been compensated for by injecting the modeled friction
force in Chapter 2 into the actuator. With the model based compensation, the lin-
earization effect for the VCM actuator has been verified via the measurement of the
hysteresis in time domain and the frequency response in frequency domain. The
measured error rejection function showed an increased error rejection capability in
low frequency range, as a result of the compensation.
To further improve the disturbance rejection in low frequency range, the linear
control is combined with an adaptive nonlinear compensation. The simulation results
have demonstrated that the proposed controller can effectively reject disturbances at
low frequencies, resulting in a marked improvement for the 3σ value of the PES
NRRO in the data storage system.
14
Quantization Effect on Vibration Rejectionand Its Compensation
14.1 Introduction
A/D and D/A converters are inevitable in digital control systems. Chapter 5 men-
tioned quantization performed by the A/D converter. This chapter investigates the
quantization effect on vibration rejection capability of the closed-loop control sys-
tem. With a proposed quantizer model, the influence of different quantizer bits on
the error rejection ability in low frequency range is analyzed and evaluated based
on frequency response measurement. On the other hand, it is known that nonlinear
behaviors such as actuator pivot friction limit the low frequency gain of the open
loop. Translated to the error rejection function or sensitivity function, it lifts the
magnitude at low frequencies and thus reduces the ability of the closed-loop system
to reject vibrations in the low frequency range. Therefore, quantization and friction
induced problems should be treated differently for more effective control.
A simple and low cost scaling scheme is developed in this chapter to compen-
sate for the effect of the quantizer. It is demonstrated that the compensation scheme
is effective in improving the sensitivity function in the low frequency range with-
out deteriorating performances at other frequencies. With the compensation for the
quantization effect, the impact on the rejection ability is mostly due to the friction.
As such, the effects from quantization and friction on the error rejection function can
be differentiated, and the quantization and friction induced problems can be tack-
led separately in the control loop. Additionally, through the proposed quantization
model and measurement methodology, suitable bit resolution for the quantizer can
be identified with and without the compensation.
14.2 Description of control system with quantizer
The system under investigation is the VCM actuator in a commercial 1.8−inch disk
drive with a spindle motor rotational speed of 4200 RPM. The experiment setup is
shown in Figure 14.1, where the spindle driver is used to spin the spindle motor, the
261
262 Modeling and Control of Vibration in Mechanical Systems
LDV is to measure the position of the read/write head, and a VCM driver to drive
the VCM actuator. The LDV displacement range is set as 2 µm/V. The frequency
response of the VCM actuator is measured via DSA by injecting a swept sine wave
of 5 mV amplitude. Due to experimental limitations, measurements in this chapter
are taken when the head is positioned in the middle-diameter region of the disk. The
measured frequency response of the VCM actuator is shown in Figure 14.2, and its
model P (s) is obtained as
zeros = 105 × [1.6965;−0.0077± 0.2575j];
poles = 105 × [−1.6965;−0.0251± 0.5020j;
−0.0101± 0.2511j;−0.0019± 0.0073j];
gain = −5.3167× 1017.
A PID controller combined with notch filters is designed for the VCM actuator and
shown in Figure 14.4. The closed servo loop of the VCM actuator is seen in Figure
14.3, where dSpace DS1103 with TMS320C240 DSP on board are used to imple-
ment the controller with the sampling rate of 30 kHz. The quantizer model Q(·) is
given by
Q(e) =max(e)
2n× floor
(
e × 2n
max(e)+ 0.5
)
, (14.1)
where n is the quantizer bit, max(e) means the maximum amplitude of the error
signal e, and floor(e) means rounding e to the nearest integer towards minus infinity.
The sensitivity function of the control loop without Q(e) is given by
S(z) =1
1 + P (z)C(z). (14.2)
A swept sine wave with 50 mV amplitude is injected as the reference signal to the
closed control loop and the sensitivity function S(z) is measured and plotted in Fig-
ure 14.5. It is known that as the excitation level to the VCM actuator decreases, the
VCM actuator gain in the low frequency range is lowered due to friction nonlinearity
effect [40], which correspondingly leads to the increased |S(z)|. This is illustrated
also in Figure 14.5, where S(z) changes for different reference amplitudes. In this
chapter, we focus on the effect of quantizer (14.1) on the sensitivity function and
seek to differentiate the quantization and friction nonlinearity effect.
With the quantizer Q(e), the sensitivity function is given by
SQ(z) =e
reference=
1
1 + P (z)Qe(z)C(z)(14.3)
where
Qe(z) =e3(z)
e(z)(14.4)
stands for an approximation of the quantizer (14.1).
Quantization Effect on Vibration Rejection and Its Compensation 263
FIGURE 14.1
Experimental setup.
101
102
103
104
−40
−20
0
20
40
60
Magnitude(d
B)
MeasuredModeled
101
102
103
104
−600
−400
−200
0
200
Frequency(Hz)
Phase(d
eg)
FIGURE 14.2
Frequency response of the VCM actuator measured by injecting a swept sine wave
of 5mV amplitude.
264 Modeling and Control of Vibration in Mechanical Systems
FIGURE 14.3
The servo loop in experiment.
101
102
103
104
−20
−15
−10
−5
0
5
10
15
Ma
gn
itu
de
(dB
)
101
102
103
104
−100
−50
0
50
100
Ph
ase
(de
g)
Frequency(Hz)
FIGURE 14.4
Frequency response of the controller C(z).
Quantization Effect on Vibration Rejection and Its Compensation 265
101
102
103
104
−80
−70
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
0.05v0.1v0.5v1v
FIGURE 14.5
Frequency response of the sensitivity function S(z) with different reference levels
(i.e., actuator moving ranges are different).
266 Modeling and Control of Vibration in Mechanical Systems
14.3 Quantization effect on error rejection
In this section, we investigate how the quantizer (14.1) with different bits affects the
error rejection function, and show that the lower low-frequency disturbance rejection
reflected in the sensitivity function may also be caused by quantization in addition to
friction.
14.3.1 Quantizer frequency response measurement
Before we proceed to investigate the quantization effect on error rejection, we exam-
ine the transfer function (14.4) for the quantizer (14.1) with different bits n through
the measurement of its frequency response.
The frequency response of e3 over e in Figure 14.3 was measured via DSA by
injecting the swept sinusoidal signal as the reference signal. The quantizer model
of the form in (14.1) with max(e) = 0.2 V and different resolution bits of n =6, 8, 10 are investigated respectively. The dashed curves in Figures 14.6−14.8
are the corresponding measured frequency responses of Q(e). It is observed that the
magnitude difference at frequencies less than 100 Hz between bits 6 and 8 are almost
10 dB and the phase difference is about 250 deg. From 200 Hz upwards, the quantizer
gains in the three cases are exactly unity. Moreover, as the bit number increases, the
quantizer approaches unity gain. When bit n = 10, it can be approximated as 1. The
frequency response when n = 12 is almost the same as that when n = 10, and thus
is omitted here.
14.3.2 Quantization effect on error rejection
Figures 14.6−14.8 show that the bit number n affects Q(e) mainly in low frequency
range. In addition, we shall see that Q(e) with lower bit number n will deteriorate
the error rejection capability of the servo system in low frequency range.
The measured sensitivity functions with different quantizer bits are shown in Fig-
ure 14.9, where the effect of the bit n on the low-frequency part can be seen. The
averaged difference of |S(z)| at the low-frequency range when the bit changes from
10 to 6 is about 10 dB. The trend is that the lower number of bits leads to a lower
effective magnitude of the compensated open loop, and thus higher |S(z)|, which
means poorer error rejection ability.
Note that a lower bit number means that the known part due to the quantization
is less. When the error signal is too low for the A/D converter to differentiate, a
high level of error rejection can not be reflected in the sensitivity transfer function.
In Figure 14.5, the lowest sensitivity function level at low frequencies is below −60dB. This implies that a gain of at least 1000 requires more than 10 bit resolution.
Hence, as shown in Figure 14.9, for the two cases of n = 10 and n = 12, no obvious
impact on the sensitivity functions is seen.
Quantization Effect on Vibration Rejection and Its Compensation 267
101
102
103
−50
−40
−30
−20
−10
0
10
20
Ma
gn
itu
de
(dB
)
After compensationBefore compensation
101
102
103
−400
−300
−200
−100
0
100
200
Frequency(Hz)
Ph
ase
(de
g)
FIGURE 14.6
Frequency response of the quantizer before and after compensation (bit number n =6).
101
102
103
−20
−10
0
10
20
Ma
gn
itu
de
(dB
)
After compensationBefore compensation
101
102
103
−150
−100
−50
0
50
100
150
Frequency(Hz)
Ph
ase
(de
g)
FIGURE 14.7
Frequency response of the quantizer with compensation (bit number n = 8).
268 Modeling and Control of Vibration in Mechanical Systems
101
102
103
−10
−5
0
5
10
Ma
gn
itu
de
(dB
)
After compensationBefore compensation
101
102
103
−200
−100
0
100
200
Frequency(Hz)
Ph
ase
(de
g)
FIGURE 14.8
Frequency response of the quantizer with compensation (bit number n = 10).
101
102
103
104
−80
−70
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
n=12n=10n=8n=6
FIGURE 14.9
Measured sensitivity function SQ(z) with different bits n.
Quantization Effect on Vibration Rejection and Its Compensation 269
We take |SQ(f)| at f = 10 Hz as the representative value of |SQ(z)| at low
frequencies. Figure 14.10 shows the relation of |SQ(f)| versus bit number n at
f = 10 Hz, which is decreasing dramatically until n = 10. This means that bit
number n = 10 is necessary for satisfactory performance. Approximately |SQ(z)| =0.4604n3−12.1375n2+99.2833n−303.2000. Similar to Figure 14.5, |SQ(z)| with
a fixed bit number n also changes with plant excitation levels. The trend of |SQ(z)|with the bit number n is almost the same for each excitation level.
6 7 8 9 10 11 12−64
−62
−60
−58
−56
−54
−52
−50
−48
−46
−44
Bit n
|S(f
)| f=
10
FIGURE 14.10
Sensitivity function |SQ(f)| with f = 10 Hz versus bit n.
14.4 Compensation of quantization effect on error rejection
In this section, a scaling method is used to compensate for the effect of the quantizer
on the sensitivity function at low frequencies. The compensation scheme is shown in
Figure 14.11. When the error signal amplitude is less than a threshold δ, it is scaled
up by a factor a > 1, and then scaled down by the factor a−1 after undergoing quan-
tization. The method of choosing the threshold δ and the scaling factor a is illustrated
in Figure 14.12. M is the value at the beginning point where the quantization effect
270 Modeling and Control of Vibration in Mechanical Systems
FIGURE 14.11
Compensation scheme of quantization effect.
can be seen obviously. D is the biggest difference between SQ and S. δ and a can
be generally chosen as follows.
δ = 10M/20 · reference, a = 10D/20. (14.5)
101
102
103
104
−80
−70
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
Target SMeasured S
Q
M
D
FIGURE 14.12
Choosing the threshold δ and the scaling factor a.
The compensation is carried out for three cases with n = 6, 8, and 10 respectively,
and the improvement of the error rejection after compensation will be discussed.
A. n = 6
Quantization Effect on Vibration Rejection and Its Compensation 271
In this case, from the sensitivity function in Figure 14.9, δ and a are obtained as
δ = 10−40/20 · 50 mV = 0.5 mV, a = 1018/20 = 8.
Figure 14.6 shows the improved frequency response of the quantizer after com-
pensation. The sensitivity function improvement after compensation in the low fre-
quency range is shown in Figures 14.9 and 14.13. There is no further improvement
when a is increased. When a = 5, the result is as good as when a = 8, which means
that the optimal value of a is roughly between 5 and 8.
101
102
103
104
−80
−70
−60
−50
−40
−30
−20
−10
0
10
Frequency(Hz)
Magnitude(d
B)
n=10n=8n=6
FIGURE 14.13
Sensitivity function with quantization compensation.
B. n = 8When n = 8,
δ = 10−48/20 · 50 mv = 0.2 mv, a = 1015/20 = 5.6.
Figures 14.7 and 14.13 show the obvious improvement of the quantizer and the
sensitivity function after compensation. When a is increased to 10 and above, no
further improvement is observed. Thus the best value of a can be chosen around 5.6.
C. n = 10In the case with bit n = 10,
δ = 10−48/20 · 50 mV = 0.2 mV, a = 1010/20 = 3.
272 Modeling and Control of Vibration in Mechanical Systems
TABLE 14.1
Quantization and friction effect
Quantizer bit n 6 8 10
Actually measured |S|f=10Hz (dB) −45 −50 −65Compensated quantization effect (dB) 13 18 4
Estimated |S|f=10Hz with friction effect (dB) −58 −68 −69
No obvious difference can be seen in Figure 14.8. When compared with Figure
14.9, Figure 14.13 shows the obvious improvement of the sensitivity function after
compensation. When a = 6, the result is almost the same. As a is increased to 10,
the result becomes worse. Thus in this case a = 3 can be chosen as one of the best
scaling factors.
It can be seen from Figure 14.13 that after compensation, |SQ(z)| is much closer to
|S(z)| in the dotted line in Figure 14.5, which is measured without the quantizer and
thus considered as subjected to nonlinear friction effect only, e.g., |S(z)|f=10Hz=
−69 dB with friction effect alone. From Figure 14.13, we can also see that instead
of bit number n = 10 previously shown in Section 14.3.2 without compensation, bit
number n = 8 is adequate with the compensation for a similar performance.
It has been shown that the low frequency range of the sensitivity function is also
affected by quantization in addition to actuator pivot friction. When the quantiza-
tion effect is compensated for sufficiently, the low frequency portion of the sensi-
tivity function can be regarded as the estimated effect from friction alone. Thus, by
comparing the compensated and the uncompensated |SQ(z)| with Figure 14.13, the
quantization and the friction effect on SQ(z) can then be differentiated and shown in
Table 14.1. We can see that the estimated friction effect for n = 6 is not as accurate
as the other two cases. This is because the compensation for n = 6 is insufficient,
as seen in Figure 14.13. In this situation, a multi-stage scaling, i.e., a = ai for
δi1 < |e| < δi2, i = 1, 2, ..., may be necessary.
To this end, the scaling compensation scheme is conducted for quantization com-
pensation in the disk drive. The series of results demonstrates that the compensation
scheme is effective in improving the sensitivity function in the low frequency range
without influencing other frequencies. Additionally, it is noted that the implementa-
tion of the scheme is simple and therefore incurs low cost.
14.5 Conclusion
The quantization effect on the closed-loop system performance has been investi-
gated. The frequency responses of the quantizers with different bits have been mea-
sured and analyzed. Its effect on the error rejection function or sensitivity function
at low frequencies has shown that the rejection ability of the servo loop for low
Quantization Effect on Vibration Rejection and Its Compensation 273
frequency disturbances is relevant to the quantizer in addition to the pivot friction.
Moreover, a simple and low cost scaling scheme has been used to compensate for the
effect of the quantizer. With sufficient compensation, the effects due to quantization
and friction can be differentiated, and thus the friction and the quantization impact on
the system performance such as disturbance rejection capability can be treated sepa-
rately. Through the proposed quantization model and measurement methodology, a
suitable bit resolution for the quantizer can be easily identified while the pivot fric-
tion nonlinearity effect can be decoupled and tackled for more effective closed-loop
system control.
15
Adaptive Filtering Algorithms for ActiveVibration Control
15.1 Introduction
Many advanced control techniques have been studied for active noise and vibration
control, including optimal and robust control and adaptive control strategies. The ad-
vance of computer technologies has made digital signal processing techniques much
more useful in modern control systems. Adaptive filters have been widely used in
the implementation of adaptive algorithms for active vibration control (AVC). Sam-
pling time constraint is the major drawback of a digital control system that requires
very high processing speed for real time control. Adaptive filtering offers significant
advantages over passive silencers at low frequencies where lower sampling rates are
adequate.
There are two main approaches to adaptive filtering in AVC, feedforward and feed-
back adaptive filtering. The feedforward algorithm requires the source information
in order to attenuate the vibration. In many applications, the source of vibration is
impractical or expensive to measure, and a feedforward algorithm becomes impos-
sible or difficult to implement in those applications. A feedback algorithm has an
advantage of utilizing only the vibration signal to be controlled.
15.2 Adaptive feedforward algorithm
Figure 15.1 shows the so-called filtered-X LMS (FXLMS) algorithm which was first
introduced by Bernard Widrow in 1981 [37].
The filtered-X LMS algorithm is derived as follows with the notation defined in
Chapter 3. The error signal e(k) is given by
e(k) = d(k) + y(k) = d(k) + P (z)u(k)
= d(k) + P (z)(WT (z)x(k)). (15.1)
275
276 Modeling and Control of Vibration in Mechanical Systems
FIGURE 15.1
Block diagram of FXLMS algorithm.
Considering that
∇e(k) =∂e(k)
∂W (k)= p(k) ∗ x(k), (15.2)
the gradient estimate of the mean-squared error will be
∇ξ(k) = ∇e2(k) = 2[∇e(k)]e(k)
= 2[p(k) ∗ x(k)]e(k), (15.3)
where p(k) is the impulse response of P (z). In a practical situation, an exact model
P (z) is not available, and therefore its estimated model P (z) is used to represent
P (z) in the algorithm. Either an FIR or IIR filter can be used to model the AVC
system. Then the reference signal for the LMS algorithm will be
x(k) = P (z)x(k). (15.4)
x(k) is called the filtered reference signal because the reference input signal is passed
through the estimated model of the AVC system. Equation (15.3) becomes
∇ξ(k) = 2[p(k) ∗ x(k)]e(k) = 2x(k)e(k). (15.5)
The coefficients of the weight vector will be updated by the following equation.
Adaptive Filtering Algorithms for Active Vibration Control 277
W (k + 1) = W (k) − µ
2∇ξ(k)
= W (k) − µx(k)e(k). (15.6)
The resulting adaptive algorithm is known as the Filtered-X LMS algorithm.
In the normalized case, the algorithm is known as the Filtered-X NLMS (FXNLMS)
algorithm whereby the weight vector will be updated by
W (k + 1) = W (k) − µ(k)x(k)e(k), (15.7)
µ(k) =α
ε + xT (k)x(k). (15.8)
So far, the feedforward adaptive algorithm has been discussed. The feedforward
algorithm requires the reference signal that cannot be available in many applications.
To overcome this problem an adaptive feedback algorithm is considered below.
15.3 Adaptive feedback algorithm
There is another algorithm that combines the traditional feedback control and adap-
tive filtering approach, and is therefore referred to as the Adaptive Feedback Algo-
rithm. The algorithm utilizes only the feedback error signal to cancel the disturbance
vibration. Since there is no direct reference information available for the vibration
control, the disturbance signal is regenerated (extracted) from the error signal and the
approximated input reference signal is then fed back to the adaptive filter, as shown
in Figure 15.2.
An estimate of the reference signal x(k) is obtained by subtracting the estimated
cancellation signal y(k) of y(k) from the error signal e(k).
x(k) = d(k) = e(k) − y(k). (15.9)
The weight updating algorithm is the same as the feedforward control scheme,
FXLMS or FXNLMS. The feedback loop W (z)P (z) will introduce poles to the
system. The characteristic equation of the system will be:
α(z) = 1 + W (z)P (z). (15.10)
Adaptive feedback control can be seen as adaptive inverse control where an adap-
tive filter is used to track the inverse model of P (z). The analogy is illustrated in
Figure 15.3.
In this control system, a compensator C(z) will adapt to track the inverse model
P−1(z) [38]. The system is analyzed as follows.
X′
(z) = R(z) − X(z) = R(z) − E(z) + P (z)C(z)X′
(z), (15.11)
278 Modeling and Control of Vibration in Mechanical Systems
i.e.,
X′
(z) =R(z) − E(z)
1 − P (z)C(z). (15.12)
E(z) = D(z) +P (z)C(z)R(z) − P (z)C(z)E(z)
1 − P (z)C(z), (15.13)
from which,
E(z) =D(z)(1 − P (z)C(z))
1 − P (z)C(z) + P (z)C(z)+
R(z)
1 − P (z)C(z) + P (z)C(z).(15.14)
With the reference signal R(z) set to zero, the transfer function from D(z) to E(z)will be:
E(z)
D(z)=
1 − P (z)C(z)
1 − P (z)C(z) + P (z)C(z). (15.15)
If the estimated model P (z) is exactly the same as P (z) , (15.15) will be reduced to:
E(z)
D(z)= 1 − P (z)C(z). (15.16)
An adaptive filter will be applied in the place of C(z) and the filter will adapt
its weight vector to approach the inverse model of P (z), P−1(z) [38]. When C(z)becomes the inverse model of P (z), the effect of D(z) over E(z) will be totally
eliminated.
Therefore correct modeling of P (z) is crucial to the adaptive feedback control
system. In a practical situation, it is difficult to obtain the exact model of the system.
Modeling error may lead the system to an unstable situation.
Adaptive Filtering Algorithms for Active Vibration Control 279
FIGURE 15.2
Filtered-X LMS adaptive feedback algorithm.
FIGURE 15.3
Adaptive inverse control scheme.
280 Modeling and Control of Vibration in Mechanical Systems
15.4 Comparison between feedforward and feedback controls
In spite of the advantage of not requiring the information over feedforward control,
adaptive feedback control can only be applicable for narrow band disturbance rejec-
tion. The traditional feedback controller cannot react to the disturbance before the
control error has already occurred. But the adaptive filter has the ability to capture
the statistics of the disturbance signal. For periodic disturbance signals, it is possible
for the adaptive feedback control to track the frequency of the signal and progres-
sively attenuate the disturbance signal. Therefore adaptive feedback control is still
efficient for pure sinusoid signals.
On the other hand, feedforward control has an ability to reject the wide band dis-
turbances, because the controller receives the disturbance signal before it reaches the
point to be controlled and takes a control action in advance to eliminate the distur-
bance impact.
An approximated model of the system appears in the feedback loop of the adaptive
feedback system, which introduces poles to the system. Therefore robustness of sta-
bility of the adaptive feedback system can be improved by minimizing the modeling
error of the system.
15.5 Application in Stewart platform
We have chosen the adaptive feedback algorithm for hexapod smart structure intro-
duced in Chapter 3. Since the structure has six actuators, a multiple-channel adaptive
feedback control system is studied.
15.5.1 Multi-channel adaptive feedback AVC system
In a multiple-channel AVC system, let the number of the secondary sources be Kand the number of error sensors be M . The reference signal synthesizer uses Ksecondary signals, M error signals, and K×M secondary path estimates to generate
M reference signals for K × M adaptive filters. The synthesized reference signals
are expressed as:
xm(k) = em(k) +
K∑
n=1
pmn(k) ∗ un(k), m = 1, 2, · · · , M, (15.17)
Adaptive Filtering Algorithms for Active Vibration Control 281
where pmn(k) is the impulse response of the secondary-path estimate Pmn(z) and
un(k) is the nth secondary signal expressed as
un(k) =
M∑
m=1
wnm(k) ∗ xn(k), n = 1, 2, · · · , K, (15.18)
where wnm(k) is the impulse response of the adaptive filter Wnm(z) .
A 2 × 2 adaptive feedback AVC system shown in Figure 15.4 is presented as an
example for multiple channel AVC systems. Two secondary signals, u1(k) and u2(k)are generated as
u1(k) = w11(k) ∗ x1(k) + w12(k) ∗ x2(k), (15.19)
u2(k) = w21(k) ∗ x1(k) + w22(k) ∗ x2(k). (15.20)
The reference signals are synthesized as:
x1(k) = e1(k) + p11(k) ∗ u1(k) + p12(k) ∗ u2(k), (15.21)
x2(k) = e2(k) + p21(k) ∗ u1(k) + p22(k) ∗ u2(k). (15.22)
The coefficients of the four adaptive filters are adjusted using the FXLMS algo-
rithm expressed as:
W11(k + 1) = W11(k) + µ[p11(k) ∗ X1(k)]e1(k) + [p21(k) ∗ X1(k)]e2(k),W21(k + 1) = W21(k) + µ[p12(k) ∗ X1(k)]e1(k) + [p22(k) ∗ X1(k)]e2(k),W12(k + 1) = W12(k) + µ[p11(k) ∗ X2(k)]e1(k) + [p21(k) ∗ X2(k)]e2(k),W22(k + 1) = W22(k) + µ[p12(k) ∗ X2(k)]e1(k) + [p22(k) ∗ X2(k)]e2(k).
15.5.2 Multi-channel adaptive feedback algorithm for hexapod plat-form
The hexapod smart structure has six secondary sources. If we use six error sensors,
there will be 72 adaptive filters including 36 filters for reference signal synthesizer
and 36 filters for Filtered-X purpose. There will be a very high computational burden
in real time implementation. Therefore a (6 × 1) adaptive control scheme, as shown
in Figure 15.5, has been chosen for the smart structure with only one sensor placed
at the center of the upper plate surface.
In this system, there are six secondary path actuators and one error sensor. The
error signal will be:
e(k) = d(k) +
6∑
n=1
pn(k) ∗ un(k), (15.23)
where pn(k), n = 1, 2, · · · , 6 is the impulse response of the secondary path Pn(z),and un(k), n = 1, 2, · · · , 6 is the secondary signal of the adaptive filter Wn(z).
282 Modeling and Control of Vibration in Mechanical Systems
FIGURE 15.4
Block diagram of 2 × 2 adaptive feedback algorithm.
Adaptive Filtering Algorithms for Active Vibration Control 283
The reference signal x(k) is synthesized as an estimate of the primary disturbance.
x(k) = d(k) = e(k) −6
∑
n=1
pn(k) ∗ un(k), (15.24)
where pn(k) is the impulse response of Pn(z).The FXLMS algorithm is used to minimize the error signal e(k) by adjusting the
weight vector for each adaptive filter Wn(z) according to:
Wn(k + 1) = Wn(k) + µX′
n(k)e(k), n = 1, 2, · · · , 6, (15.25)
where X′
n(k) = pn(k)∗X(k) is the reference signal vector filtered by the secondary
path estimate, Pn(z).
FIGURE 15.5
Block diagram of 6 × 1 FXLMS adaptive feedback control system.
284 Modeling and Control of Vibration in Mechanical Systems
15.5.3 Simulation and implementation
Identification of the actuators in the platform can be found in Chapter 3. The 6 × 1adaptive feedback controller is developed using the MATLAB Simulink platform.
Improvement and modification on the controller structure was performed during an
experiment. There are six Filtered-X LMS adaptive filters which form the main parts
of the controller. The outputs of the adaptive filters are fed into the six DA converters
of the dSPACE real time interface board (DS1104). Another major portion is the
primary disturbance synthesizer with the six estimated filters Pn(z) of the secondary
paths Pn(z). The secondary signals from the adaptive filters are also inputed into
these filters in order to regenerate the primary disturbance signal.
Normally a feedback controller can get into oscillation due to internally generated
noise. To prevent this unstable situation, small nonlinear dead zones are placed at the
output of the primary signal synthesizer and the receiving point of the error signal.
The dead zone levels are small enough to ensure that only the noise signal is prohib-
ited from passing through. An automatic gain control, which will be introduced in
Section 15.5.3.2, is placed at the reference signal input to the six adaptive filters to
improve the stability of the controller.
Sampling frequency is set at 1 kHz. Therefore, the entire control process is carried
out within 1 ms for each sampled signal and the secondary signals are sent out with
1 kHz sampling rate. We may obtain better performance if the sampling frequency is
increased. But the computing demand of the six adaptive filters limits the sampling
frequency. The adaptation step size µ is set between 0.0005 and 0.002 depending on
the vibration frequency. During the experiment, the controller is able to achieve 20dB to 30 dB attenuation for vibration of frequency from 60 Hz to 220 Hz. The result
can be further improved by minimizing the modeling error of the secondary path.
Figure 15.6 shows the experiment setup and the connections between various de-
vices. The Stewart platform is mounted on a shaker (Labworks ET139) through
a custom−made mounting interface. The shaker is powered by a power amplifier,
which in turn is driven by a signal generator. A sine wave is used as the reference
signal for the shaker. The piezoelectric accelerometer (357B21 from PCB Piezoelec-
tronics) is placed at the center of the top plate to detect the error signal of the system
as a whole. A charge amplifier (Sinocera YE5852) will amplify the detected error
signal from the accelerometer and filter the high frequency noise before the signal
is sent to the controller via the ADC unit. An interface unit of DS1104 DSP board,
which includes peripheral outlets for the DS1104 card, serves as a junction point
among the DSP board, charge amplifier and DMU (Drive and Monitoring Unit).
Adaptive Filtering Algorithms for Active Vibration Control 285
15.5.3.1 Experimental results
Experiments are conducted for different frequencies of vibration. Depending on the
frequency of the disturbance (vibration) signal, an adaptation step size µ is chosen.
Recall that the step size is inversely proportional to the reference input power; a
smaller step size µ is chosen for higher frequency since the signal power increases
as the frequency is higher.
The PZT actuator has a maximum stroke length of 50 µm. In order not to exceed
the maximum stroke length of the actuator, the vibration signal is chosen with 200mV peak-to-peak at the ADC input. Error signal at the input of the algorithm will be
20 mV peak-to-peak (−40 dB) due to the scaling factor of the A/D converter.
During the experiments, the controller is observed to be able to attenuate distur-
bances with frequency up to 220 Hz. The actual resonance frequency of the system
is approximately at 230 Hz. When the vibration frequency is around 230 Hz, the
system runs into an unstable state. This may be due to a large modeling error around
the resonance frequency. Theoretically, there is a phase shift of close to 180 around
the resonance frequency. The phase response of the secondary path also shows that
there is a major phase shift around the resonance frequency. But the phase response
in the identification result may not have sufficient changes to represent the resonance
region.
Therefore, the controller is modified to compensate the phase error. Inverted gain
(−1) is inserted in the error signal path and the controller is tested with frequency
starting from 240 Hz. With this modification, disturbance frequency up to 280 Hz is
observed to be attenuated.
Figures (15.7)−(15.10) are the experimental results captured by dSPACE control
desk with frequency 60 Hz, 210 Hz, 240 Hz, and 270 Hz, respectively. The straight
line is the point where AVC is switched on. The error signal is inverted when the
disturbance frequency is set at 240 Hz and 270 Hz.
286 Modeling and Control of Vibration in Mechanical Systems
FIGURE 15.6
General layout of the experimental setup.
Adaptive Filtering Algorithms for Active Vibration Control 287
FIGURE 15.7
60 Hz error signal in dB unit.
FIGURE 15.8
210 Hz error signal in dB unit.
288 Modeling and Control of Vibration in Mechanical Systems
FIGURE 15.9
240 Hz error signal in dB unit.
FIGURE 15.10
270 Hz error signal in dB unit.
Adaptive Filtering Algorithms for Active Vibration Control 289
15.5.3.2 Controller modification and discussion
Observe from Figure 15.7−Figure 15.10 that the level of vibration attenuation varies
with frequency. It may be because of the error of magnitude in the frequency re-
sponse of the secondary path. An accurate modeling is required not only at the
fundamental frequency but also at the harmonics of the disturbance signal. Because
once the controller starts to suppress the disturbance signal, harmonics will be intro-
duced in the reference signal path of the controller.
Therefore, modeling is a big challenge. Nonlinearity of the secondary path ac-
tuator introduces modeling error and instability. Modeling errors include magnitude
error and phase error. Phase error is a crucial factor since the feedback control system
must be able to predict the disturbance before the control signal is sent out. Phase
error in the secondary path model of the synthesizer section will introduce the wrong
phase in the reference signal. Then the phase error in the reference signal makes the
adaptive controller output the wrong phase of cancellation signal to the PZT actuator
as well as to the synthesizer again.
The magnitude response of the secondary path model is altered by ± 6 dB, which
means that the FIR models of the secondary path are multiplied by 2 or divided by
2, to introduce the magnitude error. Then the experiments are conducted again. The
stability of the controller is not affected and there is only a slight decrease in vibration
attenuation level. Furthermore, there is a magnitude peak at 240 Hz in the frequency
response of the secondary path model. Based on the experiments, the actual peak
response is estimated to be at 230 Hz. But the controller is able to attenuate the
disturbance frequency of 240 Hz with the phase compensating gain (−1) in the error
signal path. Therefore, we can say that the adaptive feedback controller is able to
tolerate some magnitude error in the secondary path model.
During the experiment, the controller is observed to have a faster convergence
rate for a higher frequency vibration. It verifies that a larger signal power at higher
frequency can drive the adaptive filter to converge at a faster rate. But the faster
convergence rate due to the higher signal power can lead the system to instability if
the adaptation step size “µ” is not sufficiently small. Minimizing the adaptation step
size can improve the stability of the controller but a slow convergence rate has to be
borne.
Another strategy used to compromise between the stability and convergence rate is
to use the normalized LMS algorithm. The NFXLMS algorithm is able to work well
in the feedforward algorithm. But in the practical implementation of adaptive feed-
back system, surged convergence of the normalized adaptive filter is again observed
to cause some instability to the feedback system.
Therefore, instead of applying the normalized algorithm, another strategy is devel-
oped and applied to the system to ensure the stability and to improve the convergence
of the adaptive algorithm. That is an automatic gain control (AGC), introduced in
the reference signal path as shown in Figure 15.11.
When the adaptive controllers start to adapt their filter weights, it is observed that
the reference signal becomes larger than the normal level before the adaptation is
triggered. It may be because of the error of magnitude of the secondary path model.
290 Modeling and Control of Vibration in Mechanical Systems
In order to prevent the actuator from being overdriven and the control system from
being driven into an unstable situation, an automatic gain control, AGC is inserted
in the reference signal path. AGC is a simple nonlinear attenuator. The reference
signal generated by the synthesizer will be transformed into a 16-order vector form
by a delay line. “Max” block will filter out the maximum value of the vector signal
and this signal will be transformed back to a scalar value by “To Sample” block.
This scalar value is amplified with “Slider Gain” which can be adjusted during the
experiment to obtain the best performance. Constant value 1 is added to avoid the
division by zero. The summed value will be reciprocated by the math function “1/µ”.
This results in a variable attenuating factor for the reference signal by inputting it to
the “Cross Product” block together with the reference signal. Therefore, when the
reference signal becomes larger, AGC will attenuate more and maintain the reference
signal within a safe level. The mathematical description of the automatic gain control
is
Xo(k + 1) =Xi(k + 1)
1 + gXo(k), (15.26)
where g is the AGC gain (slider gain).
The different results between with and without automatic gain control are shown
Figures 15.12 and 15.13. It can be seen that automatic gain control can greatly
improve the performance of the adaptive feedback controller. Another way of min-
imizing the overgrowing of reference signal is to restrict the adaptive filter weights
by putting in a leakage factor. The leakage factor of the adaptive filter prevents the
weight vector from growing without bound after the convergence of the error signal.
A leakage factor of about 0.9999 is chosen so that 1 − 0.9999 = 0.0001 is still less
than the smallest adaptation size 0.0005 in the experiments. But experiments with
the leakage factor show that the attenuation level is decreased from 30 dB to 20 dB.
Although the leakage factor can improve the stability, it can degrade the performance
of the AVC.
15.6 Conclusion
In this chapter, with the adaptive filtering algorithm being conceptualized, a multiple-
reference adaptive feedback controller has been developed. A 6×1 adaptive feedback
system has been implemented using Simulink and Control Desk. Adaptive identifi-
cation has been used to model the six PZT actuators in the secondary path.
Experiments have been performed with different adaptation step µ. Lower step
size has to be chosen for a high frequency vibration signal in order to take into
account its high signal power. Improperly chosen µ has been observed to affect the
stability and performance of the control system.
The regenerated reference signal has been observed to become larger than the nor-
mal level before the adaptive filter has started to adapt. The leaky FXLMS algorithm
Adaptive Filtering Algorithms for Active Vibration Control 291
has been applied to restrict the overflow of the weight vector of the adaptive filter. But
the controller performance is degraded with introducing the leakage factor. Finally
the automatic gain controller has been considered and developed in the reference sig-
nal path. The experimental results are greatly improved by fine-tuning the slider gain
of the automatic gain controller. Actually, the nonlinear property of the automatic
gain controller partially compensates for the modeling error so that the performance
and stability of the adaptive feedback controller are comparatively better than those
without using the automatic gain controller. For vibrations with frequencies from 60to 220 Hz, an attenuation level of up to 30 dB has been achieved. Therefore, the
adaptive feedback controller can provide satisfactory performance among various
constraints.
FIGURE 15.11
Simulink diagram of automatic gain control.
292 Modeling and Control of Vibration in Mechanical Systems
FIGURE 15.12
180 Hz error signal without automatic gain control.
FIGURE 15.13
180 Hz error signal with automatic gain control.
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