Du C., Xie L. Modeling and Control of Vibration in Mechanical Systems
Подождите немного. Документ загружается.
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 vibr at ions, their effects on
system stability and perfor mance, 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 t o demonstrate t he modeling and control techni ques.
Vibration exists in a w ide spectra of engineering systems such as hard disk drives,
automotives, aerospace and aeronautic syst ems, manufacturing systems, etc. Vibra-
tion is undesir able in most engineering appli cations, lowering system performance,
wasting energy and creating unwanted noise. Although the problem of vibration
control has been stu died for a lo ng 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 r equired. 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 r ecent
developments in this field.
The book presents the latest r esults in vibration modeling and advanced con trol
design for vibrati on attenuation in mechanical actuation systems to achieve high
precision positioning performance. It focuses on vibration and disturbance rejec-
tions using recently developed contro l techniques for high precision positioning, and
demonstration of the benefits gained from the applications of these techniqu es. The
theoretical developments and principles of control design are elaborated in detail so
that the r eader can apply the techniques developed to obtain so lutions with the help
of MATLAB
r
. Examples are presented thr oughout the book so t hat the subject can
be better understood. A n umber 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 b ackgro und knowledge.
The boo k summarizes a collective research effort which we have had the plea-
sure to contribute to. Many results r epo rted in the book are due to the coll abo ration
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
Arlingto n. The research work contained in this book was mainly performed at DSI
and the School o f Electrical and Electron ic Engineering (EEE) of N TU, Singapo re.
xi
xii Modeling and Control of Vibration in Mechanical Systems
Algorith ms applied in magnetic recording systems were impl emented at DSI and
those in the Stewart platform at the School of EEE, NTU. We would like to express
our sincere appreciation to D SI f or 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 Divi sion of DSI, and EEE, NTU for their
support.
Lihua Xie
Chun ling Du
MATLAB
r
is a registered trademark of The MathWo rks, Inc. For product in-
formation, please contact: The MathWo rks, Inc., 3 Apple Hill Drive, Natick, MA
01760-2098 USA, Tel: 508 6 47 7000, Fax: 508-647-7001, E-mail: info@mathworks.
com, Web: www.mathworks.com.
List of Tables
2.1 σ values o f the modeling error (
ˆ
d
1
−d
1
) 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 Contro l performance comparison . . . . . . . . . . . . . . . . . . . 1 31
7.1 Contro l 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 h ard 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 Spectru m density of broad-band vib ration. . . . . . . . . . . . . . . 9
1.6 Spectru m density of narrow-band vibration. . . . . . . . . . . . . . 1 0
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 f unction. . . . . . . 2 1
2.3 Measured V CM Bode plots (straight lines: pure double integrator
k/s
2
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Multiplicative uncertainty o f a VCM. . . . . . . . . . . . . . . . . . 24
2.5 The operator z
r
versus x. . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Friction f versus actuator displacement x. . . . . . . . . . . . . . . 28
2.7 A PZT actuated suspension. . . . . . . . . . . . . . . . . . . . . . 2 8
2.8 Equivalent spring mass system of PZT microactuator. . . . . . . . . 29
2.9 A t ypical 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 frict ion
measurement via LDV. . . . . . . . . . . . . . . . . . . . . . . . . 33
2.13 Control signal u versus displacement x. . . . . . . . . . . . . . . . 34
2.14 VC M actuator modeling with friction nonlinearity model F (x). . . 34
2.15 Measured and modeled fri ct ion and error. . . . . . . . . . . . . . . 36
2.16 Actuat or frequency response for sinusoi dal reference with amplitude
of 1 and 3 V, respectively. . . . . . . . . . . . . . . . . . . . . . . . 37
2.17 Actuat or frequency response for sinusoi dal reference with amplitude
of 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.18 Preload and two-slope model for friction modeling. . . . . . . . . . 3 8
2.19 Plant input voltage u versus displacement x. . . . . . . . . . . . . . 38
2.20 Closed-lo op control system disturbances d
1
, d
2
and noise n. . . . . 39
2.21 Closed-lo op 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. . . . . . . . . . . . . . . . . . . . 4 4
2.25 Original disturbance d
1
and the modeled
ˆ
d
1
. . . . . . . . . . . . . . 49
2.26 Power spectrum of
ˆ
d
1
. . . . . . . . . . . . . . . . . . . . . . . . . 50
2.27 NR RO power spectrum from measurement and disturbance models,
i.e., e = −P (s) ·
ˆ
d
1
−
ˆ
d
2
+ ˆn. . . . . . . . . . . . . . . . . . . . . 5 1
2.28 Mo deling error (d
1
−
ˆ
d
1
) for different Γ with p = 1. . . . . . . . . 5 2
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 filt er. . . . . . . . . . . . . . . 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 spin dle motor assembly of the spin stand. . . . . . . . . . 67
4.2 Comparison of single-disk and dual-disk axial vibratio ns measured
via LDV at 7 200 RPM. . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Comparison of single-disk and dual-disk RRO power spectrum at
7200 RPM (23% impr ovement 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 7200
RPM (21% reduction of σ value). . . . . . . . . . . . . . . . . . . 72
4.6 Comparison of single-disk and dual-disk axial vibratio ns measured
via LDV at 8 400 RPM. . . . . . . . . . . . . . . . . . . . . . . . . 73
4.7 Comparison of single-disk and dual-disk RRO power spectrum at
8400 RPM (41% impr ovement 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 8400
RPM (38% reduction of σ value). . . . . . . . . . . . . . . . . . . 76
4.10 Comparison of single-disk and dual-disk axial vibrati ons measured
via LDV at 1 0200 RPM. . . . . . . . . . . . . . . . . . . . . . . . 77
4.11 Comparison of single-disk and dual-disk RRO power spectrum at
10200 RPM (th e 3rd and 5th harmonics reduced si gnificantly, 33%
improvement of σ value). . . . . . . . . . . . . . . . . . . . . . . . 78
4.12 Comparison of si ngle-disk and dual-disk NRRO power spectrum at
10200 RPM (27% redu ction of σ value). . . . . . . . . . . . . . . . 79
4.13 Comparison of single-disk and dual-disk PES in time domain at 10200
RPM (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 un certain ty. . . . . . . . . . . . . . . . . 102
List o f Figures xvii
5.3 A clo sed-loop system with additive uncertainty for robust stabili ty
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 03
5.4 A closed-loop system wit h multipl icative uncertainty for robust sta-
bility analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Contro l system structure for Youla parametrization. . . . . . . . . . 107
5.6 Sensitivity function for continuou s- time system. . . . . . . . . . . . 110
5.7 Sensitivity function for di screte-time system. . . . . . . . . . . . . 110
5.8 Sensitivity function in discrete-time domain. . . . . . . . . . . . . . 113
6.1 Mixed H
2
/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 o f the VCM actuator. . . . . . . . . . . . 126
6.4 Frequency response of sensitivity functions. . . . . . . . . . . . . . 129
6.5 Frequency response of sensitivity functions. . . . . . . . . . . . . . 130
7.1 Parallel structu re of a dual-stage actuation system with disturbances
and noise injected. . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4
7.2 Power spectrum of PES nonrepeatable runout in open loop. . . . . . 135
7.3 Decoupled structure of dual-st age actuatio n systems. . . . . . . . . 136
7.4 Structure of H
∞
loop shaping. . . . . . . . . . . . . . . . . . . . . 136
7.5 Frequency responses of S
v
(s) and S
m
(s). . . . . . . . . . . . . . . 139
7.6 Frequency responses of P
v
(s)C
v
(s) (solid line) and P
m
(s)C
m
(s)(dotted
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.7 Frequency response of VCM actuator P (s). . . . . . . . . . . . . . 141
7.8 Frequency response of VCM controller C
v
(s). . . . . . . . . . . . . 142
7.9 Frequency response of microactuator controller C
m
(s). . . . . . . . 143
7.10 Sensitivity function S
m
(s) (sol id) and its w eighting function inverse
(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.11 Open loop frequency response of the dual-stage system. . . . . . . . 144
7.12 Sensitivity and complementary sensitivit y functions. . . . . . . . . 145
7.13 Microactuator frequency response. . . . . . . . . . . . . . . . . . . 146
7.14 Frequency response of microactuator controller C
m
(s). . . . . . . . 147
7.15 Open loop frequency response of the dual-stage system. . . . . . . . 147
7.16 Sensitivity and complementary sensitivit y functions. . . . . . . . . 148
7.17 Experi mental structure. . . . . . . . . . . . . . . . . . . . . . . . . 149
7.18 Sensitivity function of the dual-stage system ( smoo th line: simula-
tion result; rough line: testing result; d otted 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 r esponse of the dual-stage system. . . . . . . . . . . . . . . . 151
7.21 3σ of PES NRRO versus frequencies. . . . . . . . . . . . . . . . . 151
7.22 VC M contro ller C
v
(z). . . . . . . . . . . . . . . . . . . . . . . . . 155
7.23 Microactuator cont roller C
m
(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 P
v
(z)C
v
(z) (solid curve) and P
m
(z)C
m
(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 r esponse of the dual-stage system. . . . . . . . . . . . . . . . 160
7.31 3σ value of PES NRRO versus frequency. . . . . . . . . . . . . . . 160
8.1 Q parameterization fo r control desig n. . . . . . . . . . . . . . . . . 163
8.2 Peak filter F in t he nominal feedback loop. . . . . . . . . . . . . . 167
8.3 Peak filter in the frequency domain. . . . . . . . . . . . . . . . . . 167
8.4 Sensitivity functions befor e and aft er group peak filtering activated. 170
8.5 PZT microactuator attached to VCM actuato r 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 befor e and aft er the K YP lemma-based design:
simulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.9 Open-loop Bode plot befor e and aft er the KY P lemma-based design. 175
8.10 Structure of experimental setup. . . . . . . . . . . . . . . . . . . . 175
8.11 Sensitivity functions before and after the KYP lemma-based design:
experimental r esults. . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.12 σ value of PES versus frequ ency. . . . . . . . . . . . . . . . . . . . 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 vibratio n 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: 1.7
kHz))). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.17 NR RO power sp ectrum with PLPF and KYP (50% reduction before
1 kHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.1 H
2
control scheme with Q parametrization for controller design. . . 184
9.2 Frequency response of a PZT mi croactuator. . . . . . . . . . . . . . 190
9.3 Open-loop frequency responses. . . . . . . . . . . . . . . . . . . . 191
9.4 Designed sensitivity functions. . . . . . . . . . . . . . . . . . . . . 192
9.5 Comparison of sensitivity functions obtained from experi ment. . . . 192
9.6 NRRO power spectrum with KYP Lemma-based controller with and
without H
2
minimization. . . . . . . . . . . . . . . . . . . . . . . 1 93
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: wi th Spec. (i) and (ii)). . . . . . . . . . . . . . . . . . 195
List o f Figures xix
9.10 Resultant sensitivity function with all the three requirements ful-
filled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.11 NR RO power spectrum with rejection of two specific disturbances at
0.65 and 2 k H z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.1 Blending control scheme. . . . . . . . . . . . . . . . . . . . . . . . 1 98
10.2 Control loop with in jected d isturbances 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 8
kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.6 Simulated sensitivity function with disturbance rejection at 4 and 8
kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.7 Measured (solid curve) sensitivity function with disturbance rejec-
tion at 4 and 8 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.8 Open-loop with di sturbance r ej ection at 3, 6. 5, and 10 kHz. . . . . . 210
10.9 Sensitivity function with d isturbance rejection at 3, 6.5, and 1 0 kHz. 211
10.10 Open-loop with disturbance rejection s at 0.65 and 2 kH z. . . . . . . 212
10.11 Sensitivi ty 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 g eneral di sturbance ob-
server. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.3 Frequency response of the designed Q(z). . . . . . . . . . . . . . . 222
11.4 The sensitivit y functions without and with the general disturbance
observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
11.5 The sensitivit y function comparison with the general and the con-
ventional disturbance observers. . . . . . . . . . . . . . . . . . . . 223
11.6 Disturbance d
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.7 Error signal e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.8 Measured sensitivity functio ns without and with the general distur-
bance observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 25
11.9 Comparison of T
EQ−OL
about the general disturbance observer. . . 225
11.10 Comparison of T
EQ−OL
about conventional dist urbance ob server. . 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 dur ing propagation.
(The time sequence and the σ value i ncrease with the track number.) 233
12.4 SSTW servo loop modeling in tw o dimensions. . . . . . . . . . . . 234
12.5 SSTW servo loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
12.6 Frequency response of a VCM actuator. . . . . . . . . . . . . . . . 237
12.7 Frequency respon se of the closed-loop transfer function with the PD
controller, PID control ler, and H
2
controller. . . . . . . . . . . . . 238