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 |  موضوع: كتاب Linear Feedback Control - Analysis and Design with MATLAB الجمعة 08 أكتوبر 2021, 11:33 pm | |
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أخواني في الله
أحضرت لكم كتاب
Linear Feedback Control - Analysis and Design with MATLAB
Dingyü Xue
Northeastern University
Shenyang, People’s Republic of China
YangQuan Chen
Utah State University
Logan, Utah, USA
Derek P. Atherton
University of Sussex
Brighton, United Kingdom
و المحتوى كما يلي :
Contents
Preface xi
1 Introduction to Feedback Control 1
1.1 Introduction 1
1.2 Historical Background 3
1.3 Structure of the Book . 4
1.4 A Survival Guide to MATLAB 6
1.4.1 A Brief Overview of MATLAB 6
1.4.2 Standard MATLAB Statements and Functions 6
1.4.3 Graphics Facilities in MATLAB . 7
1.4.4 On-Line Help Facilities in MATLAB . 7
1.4.5 MATLAB Toolboxes . 8
Problems 9
2 Mathematical Models of Feedback Control Systems 11
2.1 A Physical Modeling Example 11
2.2 The Laplace Transformation . 12
2.3 Transfer Function Models . 14
2.3.1 Transfer Functions of Control Systems 14
2.3.2 MATLAB Representations of Transfer Functions . 14
2.3.3 Transfer Function Matrices for Multivariable Systems 16
2.3.4 Transfer Functions of Discrete-Time Systems 16
2.4 Other Mathematical Model Representations . 17
2.4.1 State Space Modeling . 17
2.4.2 Zero-Pole-Gain Description 19
2.5 Modeling of Interconnected Block Diagrams . 20
2.5.1 Series Connection . 20
2.5.2 Parallel Connection 20
2.5.3 Feedback Connection . 21
2.5.4 More Complicated Connections 22
2.6 Conversion Between Different Model Objects 24
2.6.1 Conversion to Transfer Functions . 25
2.6.2 Conversion to Zero-Pole-Gain Models 26
2.6.3 State Space Realizations 27
v2007/1
page v
vi Contents
2.6.4 Conversion Between Continuous and Discrete-Time Models . 34
2.7 An Introduction to System Identification . 35
2.7.1 Identification of Discrete-Time Systems . 35
2.7.2 Order Selection 40
2.7.3 Generation of Identification Signals 41
2.7.4 Identification of Multivariable Systems 44
Problems 45
3 Analysis of Linear Control Systems 51
3.1 Properties of Linear Control Systems . 52
3.1.1 Stability Analysis . 52
3.1.2 Controllability and Observability Analysis 55
3.1.3 Kalman Decomposition of Linear Systems 59
3.1.4 Time Moments and Markov Parameters 62
3.1.5 Norm Measures of Signals and Systems . 64
3.2 Time Domain Analysis of Linear Systems 66
3.2.1 Analytical Solutions to Continuous Time Responses . 66
3.2.2 Analytical Solutions to Discrete-Time Responses . 69
3.3 Numerical Simulation of Linear Systems . 70
3.3.1 Step Responses of Linear Systems 70
3.3.2 Impulse Responses of Linear Systems 75
3.3.3 Time Responses to Arbitrary Inputs 76
3.4 Root Locus of Linear Systems 78
3.5 Frequency Domain Analysis of Linear Systems . 84
3.5.1 Frequency Domain Graphs with MATLAB 84
3.5.2 Stability Analysis Using Frequency Domain Methods 87
3.5.3 Gain and Phase Margins of a System . 88
3.5.4 Variations of Conventional Nyquist Plots . 90
3.6 Introduction to Model Reduction Techniques . 92
3.6.1 Padé Approximations and Routh Approximations 92
3.6.2 Padé Approximations to Delay Terms . 96
3.6.3 Suboptimal Reduction Techniques for Systems with Delays . 98
3.6.4 State Space Model Reduction . 101
Problems 104
4 Simulation Analysis of Nonlinear Systems 111
4.1 An Introduction to Simulink . 111
4.1.1 Commonly Used Simulink Blocks 112
4.1.2 Simulink Modeling 115
4.1.3 Simulation Algorithms and Control Parameters 116
4.2 Modeling of Nonlinear Systems by Examples 118
4.3 Nonlinear Elements Modeling 126
4.3.1 Modeling of Piecewise Linear Nonlinearities . 126
4.3.2 Limit Cycles of Nonlinear Systems 129
4.4 Linearization of Nonlinear Models 131
Problems 1352007/1
page v
Contents vii
5 Model-Based Controller Design 139
5.1 Cascade Lead-Lag Compensator Design . 140
5.1.1 Introduction to Lead-Lag Synthesis 140
5.1.2 Lead-Lag Synthesis by Phase Margin Assignment 146
5.2 Linear Quadratic Optimal Control 151
5.2.1 Linear Quadratic Optimal Control Strategies . 151
5.2.2 Linear Quadratic Regulator Problems . 152
5.2.3 Linear Quadratic Control for Discrete-Time Systems . 155
5.2.4 Selection of Weighting Matrices . 156
5.2.5 Observers and Observer Design 159
5.2.6 State Feedback and Observer-Based Controllers . 162
5.3 Pole Placement Design 165
5.3.1 The Bass–Gura Algorithm . 166
5.3.2 Ackermann’s Algorithm 166
5.3.3 Numerically Robust Pole Placement Algorithm 167
5.3.4 Observer Design Using the Pole Placement Technique 169
5.3.5 Observer-Based Controller Design Using the Pole Placement
Technique . 169
5.4 Decoupling Control of Multivariable Systems 171
5.4.1 Decoupling Control with State Feedback . 171
5.4.2 Pole Placement of Decoupling Systems with State Feedback . 172
5.5 SISOTool: An Interactive Controller Design Tool 175
Problems 177
6 PID Controller Design 181
6.1 Introduction 182
6.1.1 The PID Actions 182
6.1.2 PID Control with Derivative in the Feedback Loop 184
6.2 Ziegler–Nichols Tuning Formula . 185
6.2.1 Empirical Ziegler–Nichols Tuning Formula 185
6.2.2 Derivative Action in the Feedback Path 189
6.2.3 Methods for First-Order Plus Dead Time Model Fitting . 191
6.2.4 A Modified Ziegler–Nichols Formula . 194
6.3 Other PID Controller Tuning Formulae 197
6.3.1 Chien–Hrones–Reswick PID Tuning Algorithm . 197
6.3.2 Cohen–Coon Tuning Algorithm 198
6.3.3 Refined Ziegler–Nichols Tuning . 200
6.3.4 The Wang–Juang–Chan Tuning Formula . 203
6.3.5 Optimum PID Controller Design . 203
6.4 PID Controller Tuning Algorithms for Other Types of Plants 210
6.4.1 PD and PID Parameter Setting for IPDT Models . 210
6.4.2 PD and PID Parameters for FOIPDT Models . 211
6.4.3 PID Parameter Settings for Unstable FOPDT Models 213
6.5 PID_Tuner: A PID Controller Design Program for FOPDT Models . 213
6.6 Optimal Controller Design 216
6.6.1 Solutions to Optimization Problems with MATLAB . 2162007/1
page v
viii Contents
6.6.2 Optimal Controller Design 218
6.6.3 A MATLAB/Simulink-Based Optimal Controller Designer and Its
Applications 221
6.7 More Topics on PID Control . 225
6.7.1 Integral Windup and Anti-Windup PID Controllers 225
6.7.2 Automatic Tuning of PID Controllers . 227
6.7.3 Control Strategy Selection . 230
Problems 231
7 Robust Control Systems Design 235
7.1 Linear Quadratic Gaussian Control 236
7.1.1 LQG Problem . 236
7.1.2 LQG Problem Solutions Using MATLAB 236
7.1.3 LQG Control with Loop Transfer Recovery 241
7.2 General Descriptions of the Robust Control Problems 247
7.2.1 Small Gain Theorem 247
7.2.2 Unstructured Uncertainties 248
7.2.3 Robust Control Problems . 249
7.2.4 Model Representation Under MATLAB . 250
7.2.5 Dealing with Poles on the Imaginary Axis 251
7.3 H∞ Controller Design 253
7.3.1 Augmentations of the Model with Weighting Functions . 253
7.3.2 Model Augmentation with Weighting Function Under MATLAB 255
7.3.3 Weighted Sensitivity Problems: A Simple Case 256
7.3.4 H∞ Controller Design: The General Case 261
7.3.5 Optimal H∞ Controller Design 267
7.4 Optimal H2 Controller Design 271
7.5 The Effects of Weighting Functions in H∞ Control . 273
Problems 281
8 Fractional-Order Controller: An Introduction 283
8.1 Fractional-Order Calculus and Its Computations . 284
8.1.1 Definitions of Fractional-Order Calculus . 285
8.1.2 Properties of Fractional-Order Differentiations 286
8.2 Frequency and Time Domain Analysis of Fractional-Order Linear Systems . 287
8.2.1 Fractional-Order Transfer Function Modeling 287
8.2.2 Interconnections of Fractional-Order Blocks . 288
8.2.3 Frequency Domain Analysis of Linear Fractional-Order Systems 289
8.2.4 Time Domain Analysis of Fractional-Order Systems . 290
8.3 Filter Approximation to Fractional-Order Differentiations 292
8.3.1 Oustaloup’s Recursive Filter . 292
8.3.2 A Refined Oustaloup Filter 294
8.3.3 Simulink-Based Fractional-Order Nonlinear Differential Equation
Solutions 296
8.4 Model Reduction Techniques for Fractional-Order Systems . 298
8.5 Controller Design Studies for Fractional-Order Systems . 3002007/1
page ix
Contents ix
Problems 304
Appendix 307
CtrlLAB: A Feedback Control System Analysis and Design Tool 307
A.1 Introduction 307
A.1.1 What Is CtrlLAB? . 307
A.1.2 Installation and Requirements . 308
A.1.3 Execution of CtrlLAB . 308
A.2 Model Entry and Model Conversion . 309
A.2.1 Transfer Function Entry 309
A.2.2 Entering Other Model Representations 309
A.2.3 A More Complicated Model Entry 310
A.3 Model Transformation and Reduction 311
A.3.1 Model Display . 311
A.3.2 State Space Realizations 314
A.3.3 Model Reduction . 314
A.4 Feedback Control System Analysis 316
A.4.1 Frequency Domain Analysis 316
A.4.2 Time Domain Analysis 318
A.4.3 System Properties Analysis 321
A.5 Controller Design Examples . 322
A.5.1 Model-Based Controller Designs . 322
A.5.2 Design of PID Controllers . 322
A.5.3 Robust Controller Design . 325
A.6 Graphical Interface-Based Tools . 327
A.6.1 A Matrix Processor 327
A.6.2 A Graphical Curve Processor . 331
Problems 334
Bibliography 337
Index of MATLAB Functions 345
Index
Index
Ackermann’s algorithm, 166
actuator saturation, 220, 226, 302
additive uncertainty, 248
AIC, 40, 41
Akaike’s information criterion, 337
algebraic Riccati equation (ARE), 152,
158, 237, 238, 262
analytical solution, 66–70, 135, 160, 291,
321
anti-windup, 5, 226
ARE (algebraic Riccati equation), 152,
158, 237, 238, 262
automatic tuning, 207, 208, 227–228
relay, 5, 128, 207, 228, 229
Tsypkin’s method, 228–229
autonomous system, 67
balanced realization, 31–32, 58, 59,
101–103, 314
Schur’s, 102
Bass–Gura algorithm, 166
Bezout equation, 259, 260
bilinear transform, 251, 252, 266
block diagram, 1, 4, 20–24, 60, 111, 163,
201, 248, 309
Bode diagram, 7, 85–88, 317, 322
magnitude, 259, 262, 275, 279, 282,
300
bounded input–bounded output, 52
canonical form, 56, 57, 59, 62
controllable, 29
Jordanian, 29–31, 314
observable, 29
Caputo’s definition, 284, 286
cascade PI controller, 223
Cauchy’s definition, 284, 285
Chien–Hrones–Reswick formula, 181,
197–198
class, 287, 288
Cohen–Coon formula, 181, 198–200
complementary sensitivity function, 108,
243, 255
complex plane, 194, 251
connection
feedback, 21–22, 288
parallel, 20–21, 32, 288
series, 11, 20, 22, 288
constrained optimization, 131, 216, 217
control strategy, 2, 3, 157, 158, 162,
182–184, 230
Control Systems Toolbox, 2, 6, 8
controllability, 51, 55–60, 168
Gramian, 51, 58, 59, 179
staircase form, 56, 57
controllable canonical form, 29
controller
H∞, 236, 249, 262, 263, 266, 270, 325
H2, 272, 273, 325
fractional-order, 283, 284, 300
PD, 200, 210–212, 223, 300
PI, 123, 183, 186, 188, 189, 194–196,
198, 200, 203, 205–207, 222, 226,
300, 324
PID, 181–233
coprime factorization, 259–261
crossover frequency, 142, 146–149, 186,
189, 192, 207, 228, 297, 322
CtrlLAB, 5–7, 9, 307
damping ratio, 78, 81
iso-, 78, 81, 82
DC (direct-current) gain, 42, 192, 193
3492007/1
page 3
350 Index
decoupling, 5, 139, 171–174, 270
dynamic, 172, 174
with state feedback, 171–174
default discretization, 34
delayed system, 79, 120
describing function, 126, 228–229
descriptor system, 250
difference equation, 44
differential equation, 12, 14, 17, 283
fractional-order, 283, 290, 291
differential Riccati equation, 152, 158
differentiation, 14, 284
fractional-order, 285, 286, 292
direct-current (DC) gain, 42, 192, 193
discrete-time Riccati equation, 156
discretization, 34
disturbance, 53, 198, 203, 205, 235, 241,
248
rejection, 197, 198, 205–207
dominant poles, 81
dual, 29, 58, 169
dynamic decoupling, 172, 174
feedback connection, 21–22, 288
filter
Kalman, 236–239, 241–243, 245, 272
low-pass, 184, 254, 297
Oustaloup’s, 292–293, 298, 299
refined Oustaloup’s, 294–299
first-order lag and integrator plus dead
time (FOIPDT), 211, 212, 222
first-order plus dead time (FOPDT), 181,
186, 188, 193, 198, 209, 324
fixed step, 117
FOIPDT(first-order lag and integrator plus
dead time), 211, 212, 222
FOPDT (first-order plus dead time), 181,
186, 188, 193, 198, 209, 324
Fourier series expansion, 41, 229
fractional transformation representation,
249, 254
fractional-order, 283–305
calculus, 284, 286
controller, 283, 284, 300
differential equation, 283, 290, 291
differentiation, 285, 286, 292
Caputo’s definition, 284, 286
Cauchy’s definition, 284, 285
Grünwald–Letnikov definition,
284–286, 290, 292
Riemann–Liouville definition,
284–286
transfer function, 287–289, 298, 299
frequencyresponses, 5, 43, 64, 65, 84–92,
186, 191–192, 194, 317
gain margin, 88–89, 141, 144, 189, 244
general mixed sensitivity problem, 254
genetic algorithm (GA), 224
GeneticAlgorithm Optimization Toolbox
(GAOT), 9, 224
Grünwald–Letnikov definition, 284–286,
290, 292
H-norm, 65
H2-norm, 65–66, 98, 99, 236, 249
H∞-norm, 236, 249, 259, 261
H2 controller, 272, 273, 325
H∞ controller, 236, 249, 262, 263, 266,
270, 325
optimal, 267, 270, 274, 276, 280, 302,
325
standard, 249
Hankel matrix, 166
Hankel norm, 103
Hardy space, 3, 5, 65
identification
system, 4, 11, 35–45, 139, 194
impulse response, 51, 62, 63, 70, 75–77,
125, 250, 315, 319
impulse signal, 65, 76, 77, 98, 125, 320,
321
integral of absolute error (IAE), 98, 173,
203, 218, 223, 278, 301
integral of squared error (ISE), 98–100,
203–206
integrator plus dead time(IPDT), 181, 210
internal stability, 51–55
internal structure, 4, 17, 35, 57, 226
inverse system, 83
inverse Z transform, 692007/1
page 3
Index 351
IPDT(integrator plus dead time), 181, 210
ISE (integral of squared error) criterion,
98–100, 203–206
iso-damping, 78, 81, 82
iso-frequency, 78
ITAE (integral of absolute error) criterion, 98, 173, 203, 218, 223, 278, 301
Jordanian canonical form, 29–31, 314
Kalman decomposition, 51, 59–61
Kalman filter, 236–239, 241–243, 245,
272
L-norm, 65
L1-norm, 65
L2-norm, 65
L∞-norm, 65
L
p-norm, 64
Laplace transform, 11–14, 25, 62, 64,
68–69, 77, 98, 99, 286, 287, 290
inverse, 13, 69
lead-lag compensator, 139–151, 218, 308,
322
Lebesgue space, 65
limit cycle, 111, 126, 129, 131, 228, 229
linear quadratic Gaussian control (LQG),
3, 235–247
linear quadratic regulator (LQR), 3, 152,
156, 180, 216
linear system
fractional-order, 283–305
state space, 3, 4, 11, 17–19, 24–33, 51,
55–57, 59, 62, 64, 101–103, 281
transfer function, 4, 7, 11, 14–17,
19–22, 24–28, 44, 288, 295
linear time invariant (LTI), 14, 18, 131,
133, 134, 138, 151
logarithmicNyquist plot, see Nyquist plot,
logarithmic
loop transfer recovery (LTR), 3, 236, 243,
245, 247
low-pass filter, 184, 254, 297
LQG (linear quadratic Gaussian control),
3, 235–247
LQR (linear quadratic regulator), 3, 152,
156, 180, 216
LTI (linear time invariant), 14, 18, 131,
133, 134, 138, 151
LTR (loop transfer recovery), 3, 236, 243,
245, 247
Lyapunov equation, 10, 58
Maclaurin series, 62, 96, 97
magnitude Bode diagram, 259, 262, 275,
279, 282, 300
Markov parameters, 51, 63–64
MATLAB toolbox
CtrlLAB, 5–7, 9, 307
Genetic Algorithm Optimization Toolbox (GAOT), 9, 224
Optimal Controller Designer (OCD),
216, 221–225, 303
PID_ Tuner, 213–216
Robust Control, 9, 235, 250–252, 255
Simulink, 111–135, 296–298
Symbolic, 9, 13, 14, 68–70
System Identification, 9, 36, 39
measurement noise, 53, 239
minimum
phase, 164, 257–259, 261
realization, 21, 32–33, 44, 61, 62
sensitivity problem, 257, 258
Mittag–Leffler function, 291, 292
mixed stability, 262
model conversion, 4, 11, 25, 26, 38, 43,
44, 67
model mismatch, 235
model reduction, 4, 51, 58, 59, 92–103,
194, 271, 293, 314–316
optimal Hankel norm approximation,
103, 314
Padé approximation, 92, 94, 96, 97, 99,
120, 133, 298, 314
Routh approximation, 94, 95, 314
Schur’s balanced realization, 102
suboptimal reduction, 191, 215, 298,
299, 314
multiple input–multiple output, 7, 16
multiplicative uncertainty, 248
multivariable system, 16, 44–45, 120,
171–1742007/1
page 3
352 Index
natural frequency, 174, 180, 282, 325
Nichols chart, 85, 148–151, 289
nominal value, 262, 301
nonminimum phase model, 246, 259,
261–267
nonlinear system, 5, 17, 111, 112, 116,
126, 129, 131–134, 136, 313, 319, 321
nonlinearity, 111, 112, 127, 128, 228, 310
double-valued, 111, 126–128
piecewise linear, 111, 126
relay, 128, 228, 229
saturation, 112, 123, 224
single-valued, 111, 126–128
static, 126, 128, 228
Nyquist plot, 42, 51, 84, 85, 87–90
atan, 90
logarithmic, 90–92
Nyquist Theorem, 87, 88
observability, 51, 57–60
Gramian, 58, 59
staircase form, 58
observable canonical form, 29
observer, 3, 139, 159–162, 164, 165, 169,
236, 262
observer-based
controller, 139, 322
regulator, 165, 169
OCD(Optimal ControllerDesigner), 216,
221–225, 303
operating point, 131, 132
optimal control, 181, 216, 218–225
Optimal ControllerDesigner(OCD), 216,
221–225, 303
optimalHankel norm approximation, 103,
314
optimization, 99, 181, 216–219, 221, 223,
224, 239
constrained, 131, 216, 217
Genetic Algorithm Toolbox, 9, 224
unconstrained, 216–217
optimum PID controller, 181, 209, 324
ordinary differential equations(ODE), 12,
14, 17, 283
Oustaloup recursive approximation,
292–293, 298, 299
refined, 294–299
overshoot, 71, 72, 74, 196–198
Padé approximation, 92, 94, 96, 97, 99,
120, 133, 298, 314
parallel connection, 20–21, 32, 288
PD controller, 200, 210–212, 223, 300
phase margin, 88–89, 141, 144, 146–151,
175, 240, 243, 244, 281, 321, 322
assignment, 207
PI controller, 183, 186, 188, 189, 194–196
PIλDµ controller, 300
PID controller, 181–233
anti-windup, 5, 226
Chien–Hrones–Reswick, 181, 197–198
Cohen–Coon, 181, 198–200
for FOIPDT plant, 211, 212, 222
for IPDT plant, 181, 210
fractional-order, 300
modified Ziegler–Nichols, 181, 202
optimum setting, 181, 209, 324
phase margin assignment, 207
refinedZiegler–Nichols, 181, 200–202,
323
Wang–Juang–Chan, 181, 203, 300
Ziegler–Nichols, 181, 185–198, 200–202,
209, 323
PID_ Tuner, 213–216
plant augmentation, 247, 249, 255
plant model, 2, 53, 82
FOIPDT, 211, 212, 222
FOPDT, 181, 186, 188, 193, 198, 209,
324
IPDT, 181, 210
minimum phase, 164, 257–259, 261
nonminimum phase, 246, 259, 261–267
unstable FOPDT, 213
pole placement, 139, 165–170, 173, 260
Ackermann’s algorithm, 166
Bass–Gura’s algorithm, 166
robust algorithm, 167–169
prefilter, 2
pseudorandom binary sequence (PRBS),
42–442007/1
page 3
Index 353
ramp response, 77
realization, 58, 59, 61, 62, 101, 102, 163,
307, 314
balanced, 31–32, 58, 59, 101–103, 314
minimum, 21, 32–33, 44, 61, 62
reduced-order model, 59, 92–95, 98, 298,
299, 315
refined Oustaloup recursive approximation, 294–299
refinedZiegler–Nichols tuning, 181, 200–202,
323
relay, 128, 228, 229
autotuning, 5, 207, 228
Riccati equation, 155, 156, 237, 241, 262
algebraic, 152, 158, 237, 238, 262
differential, 152, 158
discrete-time, 156
Riemann–Liouville definition, 284–286
rise time, 72, 73
Robust Control Toolbox, 235, 250–252,
255, 278
robust pole placement algorithm, 167–169
root locus, 3, 51, 78–83, 316, 317
Routh approximation, 94, 95, 314
sampling interval, 15, 17, 19, 39, 74, 87,
122, 123
saturation, 112, 123, 224
actuator, 220, 226, 302
Schur decomposition, 329
Schur’s balanced realization, 102
sensitivity function, 243, 255, 256, 259,
275, 278
sensitivity problem, 254, 256, 265, 325
general mixed, 262
minimum, 257, 258
series connection, 11, 20, 22, 288
settling time, 72, 74
similarity transformation, 28, 59–62
Simulink, 111–135, 296–298
single input–single output, 7, 16
SISOTool, 175–177
small gain theorem, 247–248
stability, 3, 51–55, 84, 86–88, 90, 94, 95
assessment, 51–53
internal, 51–55
stability margins, 3, 241
stabilizing controller, 249, 257, 260, 271
standard transfer function, 11, 173, 174,
278
state augmentation, 67, 68, 254
state feedback, 152, 153, 155, 156,
163–167, 171–174, 236, 239, 243, 272
decoupling with, 171–174
state space, 3, 4, 11, 17–19, 24–33, 51,
55–57, 59, 62, 64, 101–103
steady-state, 42
error, 183, 189, 210, 211, 322
response, 62, 64, 231
value, 71, 72, 152, 192, 266
step response, 70, 73–75, 121, 291, 299,
301–303
suboptimal reduction, 191, 215, 298, 299,
314
Symbolic Toolbox, 9, 13, 14, 68–70
System Identification Toolbox, 4, 9, 11,
35–45, 139, 194
Taylor series expansion, 62–64, 92, 294
time domain response, 77, 87, 290
impulseresponse, 51, 62, 63, 70, 75–77,
125, 250, 315, 319
ramp response, 77
step response, 70, 73–75, 121, 291,
299, 301–303
time moment, 62–63, 96
time varying system, 111, 118, 123–125,
152
transfer function, 4, 7, 11, 14–17, 19–22,
24–28, 44, 288, 295
discrete-time, 16, 35, 39, 42, 43, 69,
79, 134
fractional-order, 287–289, 298, 299
matrix, 16, 24, 25, 28, 38, 44, 45, 120,
172
standard, 11, 173, 174, 278
transmission zero, 27, 243
tree variable, 250–252, 255, 262, 268
Tsypkin’s method, 228–229
Tustin transform, 252
bilinear, 251, 252, 266
two degrees-of-freedom control, 22007/1
page 3
354 Index
two-port state-space, 250, 253, 255, 256,
261–263, 268, 270, 272
uncertainty, 64, 159, 235, 247, 248, 262, 269
additive, 248
multiplicative, 248
unstructured uncertainty, 248–249
unconstrained optimization, 216–217
undershoot, 266
unity negative feedback, 53, 78, 87, 88,
163, 289
unstable FOPDT(first-order plus dead time),
213
variable step, 117
Wang–Juang–Chanformula, 181, 203, 300
weightingfunction, 99, 236, 243, 253–256,
258, 262, 273–281, 302, 325
weighting matrix, 152, 154, 157, 158,
164, 180
well-posedness, 53–54, 248
Youla parameterization, 256, 257
Z transform, 16
inverse, 69
zero initial conditions, 13, 14, 25, 106
zero-order-hold (ZOH), 34, 121, 123
zero-pole-gain model, 19, 25–27, 32, 94,
112
Ziegler–Nichols formula, 181, 185–198,
200–202, 209, 323
modified algorithm, 181, 202
refined, 181, 200–202, 323
ZOH (zero-order-hold), 34, 121, 123
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