Analysis and Control Of Nonlinear Systems With Stationary Sets

by Wang, Jinzhi; Duan, Zhisheng; Yang, Ying; Huang, Lin

• ISBN13: 9789812814692
• ISBN10: 9812814698
• Format: Hardcover
• Copyright: 2009-03-12
• Publisher: World Scientific Pub Co Inc

Summary

Nonlinear systems with stationary sets are important because they cover a lot of practical systems in engineering. Previous analysis has been based on the frequency-domain for this class of systems. However, few results on robustness analysis and controller design for these systems are easily available. This book presents the analysis as well as methods based on the global properties of systems with stationary sets in a unified time-domain and frequency-domain framework. The focus is on multi-input and multi-output systems, compared to previous publications which considered only single-input and single-output systems. The control methods presented in this book will be valuable for research on nonlinear systems with stationary sets.

Table of Contents

Preface	p. v
Notation and Symbols	p. xxi
Linear Systems and Linear Matrix Inequalities	p. 1
Controllability and observability of linear systems	p. 1
Controllability and observability	p. 2
Stabilizability and detectability	p. 6
Algebraic Lyapunov equations and Lyapunov inequalities	p. 7
Continuous-time algebraic Lyapunov equations	p. 7
Continuous-time Lyapunov inequalities	p. 10
Discrete-time algebraic Lyapunov equations and inequalities	p. 11
Formulation related to linear matrix inequalities	p. 12
Schur complements	p. 12
Projection lemma	p. 13
The S-procedure	p. 15
The S-procedure for nonstrict inequalities	p. 15
The S-procedure for strict inequalities	p. 15
Kalman-Yakubovic-Popov (KYP) lemma and its generalized forms	p. 16
Notes and references	p. 21
LMI Approach to H∞ Control	p. 23
L∞ norm and H∞ norm of the systems	p. 23
L∞ and H∞ spaces	p. 24
Computing L∞ and H∞ norms	p. 25
Linear fractional transformations	p. 27
Redheffer star product	p. 29
Algebraic Riccati equations	p. 30
Solvability conditions for Riccati equations	p. 31
Discrete Riccati equations	p. 33
Bounded real lemma	p. 34
Small gain theorem	p. 36
LMI approach to H∞ control	p. 37
Continuous-time H∞ control	p. 37
Discrete-time H∞ control	p. 42
Notes and references	p. 43
Analysis and Control of Positive Real Systems	p. 45
Positive real systems	p. 45
Positive real lemma	p. 52
LMI approach to control of SPR	p. 63
Relationship between SPR control and SBR control	p. 66
Multiplier design for SPR	p. 69
Notes and references	p. 73
Absolute Stability and Dichotomy of Lur'e Systems	p. 75
Circle criterion of SISO Lur'e systems	p. 75
Popov criterion of SISO Lur'e systems	p. 80
Aizerman and Kalman conjectures	p. 82
MIMO Lur'e systems	p. 84
Dichotomy of Lur'e systems	p. 89
Bounded derivative conditions	p. 93
Notes and references	p. 97
Pendulum-like Feedback Systems	p. 99
Several examples	p. 99
Pendulum-like feedback systems	p. 102
The first canonical form of pendulum-like feedback system	p. 103
The second canonical form of pendulum-like feed-back system	p. 105
The relationship between the first and the second forms of pendulum-like feedback systems	p. 106
Dichotomy of pendulum-like feedback systems	p. 107
Dichotomy of the second form of autonomous pendulum-like feedback systems	p. 107
Dichotomy of the first form of pendulum-like feedback systems	p. 112
Gradient-like property of pendulum-like feedback systems	p. 114
Gradient-like property of the second form of pendulum-like feedback systems	p. 114
Gradient-like property of the first form of pendulum-like feedback systems	p. 117
Lagrange stability of pendulum-like feedback systems	p. 118
Bakaev stability of pendulum-like feedback systems	p. 124
Notes and references	p. 129
Controller Design for a Class of Pendulum-like Systems	p. 131
Controller design with dichotomy or gradient-like property	p. 131
Controller design with dichotomy	p. 131
Controller design with gradient-like property	p. 137
Controller design with Lagrange stability	p. 139
Notes and references	p. 147
Controller Designs for Systems with Input Nonlinearities	p. 149
Lagrange stabilizing for systems with input nonlinearities	p. 149
Bakaev stabilizing for systems with input nonlinearities	p. 155
Control for systems with input nonlinearities guaranteeing dichotomy	p. 159
Notes and references	p. 162
Analysis and Control for Uncertain Feedback Nonlinear Systems	p. 163
Dichotomy of systems with norm bounded uncertainties	p. 163
Robust analysis for dichotomy	p. 164
Robust control for systems with dichotomy	p. 168
Dichotomy of pendulum-like systems with uncertainties	p. 174
Controller design with dichotomy for uncertain pendulum-like systems	p. 179
Lagrange stability for uncertain pendulum-like systems	p. 184
Gradient-like property for pendulum-like systems with uncertainties	p. 187
Control of uncertain systems guaranteeing gradient-like property	p. 191
Gradient-like property of systems with norm bounded uncertainties	p. 199
Notes and references	p. 205
Control of Periodic Oscillations in Nonlinear Systems	p. 207
Periodic solutions in systems with cylindrical phase space	p. 207
Nonexistence of periodic solutions in Lur'e systems	p. 211
LMI-based conditions for nonexistence of periodic solutions	p. 211
Robustness analysis	p. 213
Robust synthesis	p. 214
Nonexistence of cycles of the second kind in interconnected systems	p. 218
Nonexistence of cycles of the second kind in interconnected systems	p. 220
Nonlinear interconnection design	p. 224
Cycle slipping in phase synchronization systems	p. 228
Notes and references	p. 236
Interconnected Systems	p. 237
Linearly interconnected systems	p. 238
The effect of the unstable subsystem	p. 238
Interconnected feedbacks	p. 241
Decentralized controller design	p. 244
The effect of small gain theorem	p. 246
Interconnected Lur'e systems	p. 250
Lagrange stability of a generalized smooth Chua circuit	p. 252
Input and output coupled nonlinear systems	p. 257
Notes and references	p. 264
Chua's Circuit	p. 267
Chua's circuit	p. 267
Dichotomy: application to chaos control for Chua's circuit system	p. 270
Kalman conjecture: application to the stabilization of Chua's circuit	p. 280
An extended Chua circuit	p. 286
Coupled Chua circuit	p. 288
Notes and references	p. 293
Bibliography	p. 295
Index	p. 309
Table of Contents provided by Ingram. All Rights Reserved.