كتاب Analysis and Control of Nonlinear Systems with Stationary Sets Time

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Analysis and Control of Nonlinear Systems with Stationary Sets Time-Domain and Frequency-Domain Methods
Jinzhi Wang • Zhisheng Duan
Ying Yang • Lin Huang
Peking University, China

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Contents
1Linear Systems and Linear Matrix Inequalities 1
1.1 Controllability and observability of linear systems 1
1.1.1 Controllability and observability 2
1.1.2 Stabilizability and detectability 6
1.2 Algebraic Lyapunov equations and Lyapunov inequalities 7
1.2.1 Continuous-time algebraic Lyapunov equations 7
1.2.2 Continuous-time Lyapunov inequalities 10
1.2.3 Discrete-time algebraic Lyapunov equations and
inequalities 11
1.3 Formulation related to linear matrix inequalities 12
1.3.1 Schur complements 12
1.3.2 Projection lemma 13
1.4 The S-procedure 15
1.4.1 The S-procedure for nonstrict inequalities 15
1.4.2 The S-procedure for strict inequalities 15
1.5 Kalman-Yakubovi·c-Popov (KYP) lemma and its general-
ized forms 16
1.6 Notes and references 21
2LMI Approach to H1 Control 23
2.1 L1 norm and H1 norm of the systems 23
2.1.1 L1 and H1 spaces 24
2.1.2 Computing L1 and H1 norms 25
2.2 Linear fractional transformations 27
2.3 Redhe®er star product 29
2.4 Algebraic Riccati equations 30
2.4.1 Solvability conditions for Riccati equations 31
2.4.2 Discrete Riccati equations 33
2.5 Bounded real lemma 34
2.6 Small gain theorem 36
2.7 LMI approach to H1 control 37
2.7.1 Continuous-time H1 control 37
2.7.2 Discrete-time H1 control 42
2.8 Notes and references 43
3Analysis and Control of Positive Real Systems 45
3.1 Positive real systems 45
3.2 Positive real lemma 52
3.3 LMI approach to control of SPR 63
3.4 Relationship between SPR control and SBR control 66
3.5 Multiplier design for SPR 69
3.6 Notes and references 73
4Absolute Stability and Dichotomy of Lur'e Systems 75
4.1 Circle criterion of SISO Lur'e systems 75
4.2 Popov criterion of SISO Lur'e systems 80
4.3 Aizerman and Kalman conjectures 82
4.4 MIMO Lur'e systems 84
4.5 Dichotomy of Lur'e systems 89
4.6 Bounded derivative conditions 93
4.7 Notes and references 97
5Pendulum-like Feedback Systems 99
5.1 Several examples 99
5.2 Pendulum-like feedback systems 102
5.2.1 The ¯rst canonical form of pendulum-like feedback
system 103
5.2.2 The second canonical form of pendulum-like feed-
back system 105
5.2.3 The relationship between the ¯rst and the second
forms of pendulum-like feedback systems 106
xvi Contents
5.3 Dichotomy of pendulum-like feedback systems 107
5.3.1 Dichotomy of the second form of autonomous
pendulum-like feedback systems 107
5.3.2 Dichotomy of the ¯rst form of pendulum-like feed-
back systems 112
5.4 Gradient-like property of pendulum-like feedback systems 114
5.4.1 Gradient-like property of the second form of
pendulum-like feedback systems 114
5.4.2 Gradient-like property of the ¯rst form of
pendulum-like feedback systems 117
5.5 Lagrange stability of pendulum-like feedback systems 118
5.6 Bakaev stability of pendulum-like feedback systems 124
5.7 Notes and references 129
6Controller Design for a Class of Pendulum-like Systems 131
6.1 Controller design with dichotomy or gradient-like property 131
6.1.1 Controller design with dichotomy 131
6.1.2 Controller design with gradient-like property 137
6.2 Controller design with Lagrange stability 139
6.3 Notes and references 147
7Controller Designs for Systems with Input Nonlinearities 149
7.1 Lagrange stabilizing for systems with input nonlinearities 149
7.2 Bakaev stabilizing for systems with input nonlinearities 155
7.3 Control for systems with input nonlinearities guaranteeing
dichotomy 159
7.4 Notes and references 162
8Analysis and Control for Uncertain Feedback Nonlinear Systems 163
8.1 Dichotomy of systems with norm bounded uncertainties 163
8.1.1 Robust analysis for dichotomy 164
8.1.2 Robust control for systems with dichotomy 168
8.2 Dichotomy of pendulum-like systems with uncertainties 174
8.3 Controller design with dichotomy for uncertain pendulum-
like systems 179
8.4 Lagrange stability for uncertain pendulum-like systems 184
8.5 Gradient-like property for pendulum-like systems with un-
certainties 187
Contents xvii
8.6 Control of uncertain systems guaranteeing gradient-like
property 191
8.7 Gradient-like property of systems with norm bounded un-
certainties 199
8.8 Notes and references 205
9Control of Periodic Oscillations in Nonlinear Systems 207
9.1 Periodic solutions in systems with cylindrical phase space 207
9.2 Nonexistence of periodic solutions in Lur'e systems 211
9.2.1 LMI-based conditions for nonexistence of periodic
solutions 211
9.2.2 Robustness analysis 213
9.2.3 Robust synthesis 214
9.3 Nonexistence of cycles of the second kind in interconnected
systems 218
9.3.1 Nonexistence of cycles of the second kind in inter-
connected systems 220
9.3.2 Nonlinear interconnection design 224
9.4 Cycle slipping in phase synchronization systems 228
9.5 Notes and references 236
10Interconnected Systems 237
10.1 Linearly interconnected systems 238
10.1.1 The e®ect of the unstable subsystem 238
10.1.2 Interconnected feedbacks 241
10.1.3 Decentralized controller design 244
10.1.4 The e®ect of small gain theorem 246
10.2 Interconnected Lur'e systems 250
10.3 Lagrange stability of a generalized smooth Chua circuit 252
10.4 Input and output coupled nonlinear systems 257
10.5 Notes and references 264
11Chua's Circuit 267
11.1 Chua's circuit 267
11.2 Dichotomy: application to chaos control for Chua's circuit
system 270
11.3 Kalman conjecture: application to the stabilization of
Chua's circuit 280
11.4 An extended Chua circuit 286
11.5 Coupled Chua circuit 288
11.6 Notes and references 293
Bibliography 295
Index

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