Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005

Preface......Page 6
Contents......Page 8
1.1 Introduction......Page 16
1.2.1 Preliminaries......Page 17
1.2.2 The Weak Maximum Principl......Page 18
1.2.4 Maximum Principle, Fixed Time......Page 21
1.2.5 Maximum Principle, General Case......Page 27
1.2.7 Introduction to the Micro-analysis of the Extremal Solutions......Page 29
1.3.1 High-order Maximum Principle......Page 30
1.3.2 Intrinsic Second-order Derivative and Conjugate Times......Page 36
1.3.3 Examples......Page 52
1.4.1 Model and Basic Properties......Page 55
1.4.2 Maximum Principle and Extremal Solutions......Page 57
1.4.3 Numerical Resolution......Page 59
1.5 Introduction to Optimal Control with State Constraints......Page 63
1.5.1 The Geometric Framework......Page 65
1.5.3 Junction and Re.ection Conditions......Page 68
1.5.4 Proof of the Necessary Conditions in the Riemannian Case......Page 69
Bibliographical Notes......Page 74
References......Page 75
2.1 Introduction......Page 76
2.2.1 Problem Formulation......Page 78
2.2.2 Conditions for a Solution......Page 80
2.3.1 Observer designs for Linear Structures......Page 88
2.3.2 Observer Designs for Nonlinear Structures......Page 91
2.4.1 Interconnection-based Design......Page 95
2.4.2 Transformation-based Design......Page 99
References......Page 102
3.1 Introduction......Page 105
3.2 Mathematical Preliminaries......Page 109
3.3 Zero-order-hold Equivalent Models......Page 110
3.4 Motivating Counter-examples......Page 112
3.5 Preliminary Results on Stability and Stabilization......Page 115
3.6 Framework for Controller Design......Page 117
3.6.1 Global Exponential Stabilization......Page 118
3.6.2 Semiglobal Practical Stability......Page 122
3.7 Controller Design within the Framework......Page 124
3.7.1 Emulation......Page 125
3.7.2 Continuous-time Controller Redesign......Page 127
3.7.3 Discrete-time Interconnection and Damping Assignment - Passivity-based Control (IDA-PBC)......Page 129
3.7.4 Backstepping via the Euler Model......Page 135
3.8.1 Jet Engine System......Page 140
3.8.2 Inverted Pendulum......Page 141
3.9 Overview of Related Literature......Page 144
3.10 Open Problems......Page 147
References......Page 148
4.1 Introduction......Page 152
4.2.1 Systems of Retarded Type......Page 154
4.2.3 Linear Systems......Page 155
4.3 Stability......Page 156
4.4.1 Delay-independent Stability......Page 158
4.4.2 Delay-dependent Stability Using Model Transformation......Page 161
4.4.3 Implicit Model Transformation......Page 162
4.5.1 Analytical Expression......Page 164
4.5.2 Discretization......Page 166
4.6 A Comparison of Lyapunov-Krasovskii Functionals......Page 168
4.7 Dealing with Time-varying Delays......Page 169
4.8 Razumikhin Theorem......Page 172
4.9.1 Introducation......Page 174
4.9.2 Fundamental Solutions......Page 176
4.9.3 Lyapunov-Krasovskii functional......Page 179
4.9.4 Further Comments......Page 180
4.10 Conclusions......Page 181
References......Page 182
5.1.1 De.nitions......Page 184
5.1.2 The Cauchy Problem......Page 186
5.2.1 A Short Overview on Controllability of Finite-dimensional Linear Control Systems......Page 190
5.2.2 Controllability of Linear Partial Di.erential Equations in Banach Spaces......Page 191
5.3.1 Introduction......Page 200
5.3.2 Uniform Controllability of Semidiscrete Approximations of Parabolic Control Systems......Page 202
References......Page 209
6.1.1 About the Chapter......Page 212
6.1.2 Stability, Generally Speaking......Page 214
6.2 Lagrange’s Stabilit......Page 216
6.2.1 Modern Interpretations of Lagrange-Dirichlet Stability......Page 220
6.3 Lyapunov’s Stability......Page 222
6.3.1 Lyapunov’s Methods to Test for Stability......Page 228
6.4 Asymptotic Stability......Page 232
6.5 Globalisation of Asymptotic Stability......Page 235
6.5.1 Global, i.e. in the Large or in the Whole ?......Page 236
6.5.2 Asymptotic Stability in the Large......Page 237
6.5.3 Asymptotic Stability in the Whole......Page 239
6.5.4 An Illustrative Example......Page 242
6.6.1 Autonomous Systems......Page 245
6.6.2 Time-varying Periodic Systems......Page 247
6.7 Uniformity......Page 249
6.7.1 Uniform Stability......Page 250
6.7.2 Uniform Global Stability......Page 251
6.7.3 Uniform Asymptotic Stability......Page 252
6.7.4 Uniform Asymptotic Stability in the Large......Page 254
6.7.5 Uniform Asymptotic Stability in the Whole......Page 255
6.8 Stability with Respect to Perturbations......Page 258
6.9 Further Bibliographical Remarks......Page 264
6.10 Conclusions......Page 267
References......Page 268
7.2.1 Di.erential Polynomials: A Short Review......Page 272
7.2.3 Formal Power Series......Page 273
7.2.4 A Canonical Cogenerator......Page 274
7.2.5 Matrices over S......Page 275
7.2.6 Formal Laurent Series......Page 277
7.3.1 Transfer Matrix of an Input-Output System......Page 278
7.3.2 Structure at In.nity of a Transfer Matrix......Page 279
7.4.1 Temporal Systems......Page 281
7.4.2 A Key Isomorphism......Page 283
7.4.3 Impulsive Behavior......Page 284
7.4.4 Impulsive System......Page 285
7.4.5 Generalization of the Notion of Temporal System......Page 288
7.5 Poles and Zeros at In.nity......Page 290
7.5.2 System Poles at In.nity......Page 291
7.5.3 Hidden Modes at In.nity......Page 292
7.5.6 Relations between the Various Poles and Zeros at In.nity......Page 294
7.7 errata and addenda for [7]......Page 295
References......Page 296
A.1 Di.erent UGAS De.nitions......Page 298
A.2 A Locally Lipschitz (Uniform-in-time) System that is ULS and UGA but not UGB......Page 299
References......Page 301
Series List......Page 303