Analysis And Control Of Nonlinear Systems With Stationary Sets: Time-Domain and Frequency-Domain Methods

Contents......Page 16
Preface......Page 6
Notation and Symbols......Page 22
1.1 Controllability and observability of linear systems......Page 24
1.1.1 Controllability and observability......Page 25
1.1.2 Stabilizability and detectability......Page 29
1.2.1 Continuous-time algebraic Lyapunov equations......Page 30
1.2.2 Continuous-time Lyapunov inequalities......Page 33
1.2.3 Discrete-time algebraic Lyapunov equations and inequalities......Page 34
1.3.1 Schur complements......Page 35
1.3.2 Projection lemma......Page 36
1.4.2 The S-procedure for strict inequalities......Page 38
1.5 Kalman-Yakubovi·c-Popov (KYP) lemma and its general- ized forms......Page 39
1.6 Notes and references......Page 44
2.1 L1 norm and H1 norm of the systems......Page 46
2.1.1 L1 and H1 spaces......Page 47
2.1.2 Computing L1 and H1 norms......Page 48
2.2 Linear fractional transformations......Page 50
2.3 Redheffer star product......Page 52
2.4 Algebraic Riccati equations......Page 53
2.4.1 Solvability conditions for Riccati equations......Page 54
2.4.2 Discrete Riccati equation......Page 56
2.5 Bounded real lemma......Page 57
2.6 Small gain theorem......Page 59
2.7.1 Continuous-time H1 control......Page 60
2.7.2 Discrete-time H1 control......Page 65
2.8 Notes and references......Page 66
3.1 Positive real systems......Page 68
3.2 Positive real lemma......Page 75
3.3 LMI approach to control of SPR......Page 86
3.4 Relationship between SPR control and SBR control......Page 89
3.5 Multiplier design for SPR......Page 92
3.6 Notes and references......Page 96
4.1 Circle criterion of SISO Lur'e systems......Page 98
4.2 Popov criterion of SISO Lur'e systems......Page 103
4.3 Aizerman and Kalman conjectures......Page 105
4.4 MIMO Lur'e systems......Page 107
4.5 Dichotomy of Lur'e systems......Page 112
4.6 Bounded derivative conditions......Page 116
4.7 Notes and references......Page 120
5.1 Several examples......Page 122
5.2 Pendulum-like feedback systems......Page 125
5.2.1 The first canonical form of pendulum-like feedback system......Page 126
5.2.2 The second canonical form of pendulum-like feed- back system......Page 128
5.2.3 The relationship between the ¯rst and the second forms of pendulum-like feedback systems......Page 129
5.3.1 Dichotomy of the second form of autonomous pendulum-like feedback systems......Page 130
5.3.2 Dichotomy of the first form of pendulum-like feed- back systems......Page 135
5.4.1 Gradient-like property of the second form of pendulum-like feedback systems......Page 137
5.4.2 Gradient-like property of the first form of pendulum-like feedback systems......Page 140
5.5 Lagrange stability of pendulum-like feedback systems......Page 141
5.6 Bakaev stability of pendulum-like feedback systems......Page 147
5.7 Notes and references......Page 152
6.1.1 Controller design with dichotomy......Page 154
6.1.2 Controller design with gradient-like property......Page 160
6.2 Controller design with Lagrange stability......Page 162
6.3 Notes and references......Page 170
7.1 Lagrange stabilizing for systems with input nonlinearities......Page 172
7.2 Bakaev stabilizing for systems with input nonlinearities......Page 178
7.3 Control for systems with input nonlinearities guaranteeing dichotomy......Page 182
7.4 Notes and references......Page 185
8.1 Dichotomy of systems with norm bounded uncertainties......Page 186
8.1.1 Robust analysis for dichotomy......Page 187
8.1.2 Robust control for systems with dichotomy......Page 191
8.2 Dichotomy of pendulum-like systems with uncertainties......Page 197
8.3 Controller design with dichotomy for uncertain pendulum- like systems......Page 202
8.4 Lagrange stability for uncertain pendulum-like systems......Page 207
8.5 Gradient-like property for pendulum-like systems with un- certainties......Page 210
8.6 Control of uncertain systems guaranteeing gradient-like property......Page 214
8.7 Gradient-like property of systems with norm bounded un- certainties......Page 222
8.8 Notes and references......Page 228
9.1 Periodic solutions in systems with cylindrical phase space......Page 230
9.2.1 LMI-based conditions for nonexistence of periodic solutions......Page 234
9.2.2 Robustness analysis......Page 236
9.2.3 Robust synthesis......Page 237
9.3 Nonexistence of cycles of the second kind in interconnected systems......Page 241
9.3.1 Nonexistence of cycles of the second kind in inter- connected systems......Page 243
9.3.2 Nonlinear interconnection design......Page 247
9.4 Cycle slipping in phase synchronization systems......Page 251
9.5 Notes and references......Page 259
10. Interconnected Systems......Page 260
10.1.1 The effect of the unstable subsystem......Page 261
10.1.2 Interconnected feedbacks......Page 264
10.1.3 Decentralized controller design......Page 267
10.1.4 The effect of small gain theorem......Page 269
10.2 Interconnected Lur'e systems......Page 273
10.3 Lagrange stability of a generalized smooth Chua circuit......Page 275
10.4 Input and output coupled nonlinear systems......Page 280
10.5 Notes and references......Page 287
11.1 Chua's circuit......Page 290
11.2 Dichotomy: application to chaos control for Chua's circuit system......Page 293
11.3 Kalman conjecture: application to the stabilization of Chua's circuit......Page 303
11.4 An extended Chua circuit......Page 309
11.5 Coupled Chua circuit......Page 311
11.6 Notes and references......Page 316
Bibliography......Page 318
Index......Page 332