Nonlinear and Optimal Control Theory: Lectures Given at the C.i.m.e. Summer School Held in Cetraro, Italy, June 19-29, 2004

by Agrachev, Andrei A.; Morse, A. Stephen; Sontag, Eduardo D.; Sussmann, Hector J.; Utkin, Vadim I.

• ISBN13: 9783540776444
• ISBN10: 3540776443
• Edition: 1st
• Format: Paperback
• Copyright: 2008-05-01
• Publisher: Springer Verlag

Summary

The lectures gathered in this volume present some of the different aspects of Mathematical Control Theory. Adopting the point of view of Geometric Control Theory and of Nonlinear Control Theory, the lectures focus on some aspects of the Optimization and Control of nonlinear, not necessarily smooth, dynamical systems. Specifically, three of the five lectures discuss respectively: logic-based switching control, sliding mode control and the input to the state stability paradigm for the control and stability of nonlinear systems. The remaining two lectures are devoted to Optimal Control: one investigates the connections between Optimal Control Theory, Dynamical Systems and Differential Geometry, while the second presents a very general version, in a non-smooth context, of the Pontryagin Maximum Principle. The arguments of the whole volume are self-contained and are directed to everyone working in Control Theory. They offer a sound presentation of the methods employed in the control and optimization of nonlinear dynamical systems.

Table of Contents

Geometry of Optimal Control Problems and Hamiltonian Systems	p. 1
Lagrange Multipliers' Geometry	p. 1
Smooth Optimal Control Problems	p. 1
Lagrange Multipliers	p. 4
Extremals	p. 6
Hamiltonian System	p. 7
Second Order Information	p. 10
Maslov Index	p. 14
Regular Extremals	p. 22
Geometry of Jacobi Curves	p. 25
Jacobi Curves	p. 25
The Cross-Ratio	p. 26
Coordinate Setting	p. 28
Curves in the Grassmannian	p. 29
The Curvature	p. 30
Structural Equations	p. 33
Canonical Connection	p. 35
Coordinate Presentation	p. 38
Affine Foliations	p. 39
Symplectic Setting	p. 41
Monotonicity	p. 44
Comparison Theorem	p. 49
Reduction	p. 51
Hyperbolicity	p. 53
References	p. 58
Lecture Notes on Logically Switched Dynamical Systems	p. 61
The Quintessential Switched Dynamical System Problem	p. 62
Dwell-Time Switching	p. 62
Switching Between Stabilizing Controllers	p. 65
Switching Between Graphs	p. 66
Switching Controls with Memoryless Logics	p. 67
Introduction	p. 67
The Problem	p. 67
The Solution	p. 67
Analysis	p. 68
Collaborations	p. 68
The Curse of the Continuum	p. 69
Process Model Class	p. 69
Controller Covering Problem	p. 73
A Natural Approach	p. 74
A Different Approach	p. 75
Which Metric?	p. 75
Construction of a Control Cover	p. 76
Supervisory Control	p. 76
The System	p. 77
Slow Switching	p. 86
Analysis	p. 87
Analysis of the Dwell Time Switching Logic	p. 102
Flocking	p. 110
Leaderless Coordination	p. 111
Symmetric Neighbor Relations	p. 142
Measurement Delays	p. 148
Asynchronous Flocking	p. 155
Leader Following	p. 158
References	p. 159
Input to State Stability: Basic Concepts and Results	p. 163
Introduction	p. 163
ISS as a Notion of Stability of Nonlinear I/O Systems	p. 163
Desirable Properties	p. 164
Merging Two Different Views of Stability	p. 165
Technical Assumptions	p. 166
Comparison Function Formalism	p. 166
Global Asymptotic Stability	p. 167
0-GAS Does Not Guarantee Good Behavior with Respect to Inputs	p. 168
Gains for Linear Systems	p. 168
Nonlinear Coordinate Changes	p. 169
Input-to-State Stability	p. 171
Linear Case, for Comparison	p. 172
Feedback Redesign	p. 173
A Feedback Redesign Theorem for Actuator Disturbances	p. 174
Equivalences for ISS	p. 176
Nonlinear Superposition Principle	p. 176
Robust Stability	p. 177
Dissipation	p. 178
Using "Energy" Estimates Instead of Amplitudes	p. 180
Cascade Interconnections	p. 180
An Example of Stabilization Using the ISS Cascade Approach	p. 182
Integral Input-to-State Stability	p. 183
Other Mixed Notions	p. 183
Dissipation Characterization of iISS	p. 184
Superposition Principles for iISS	p. 185
Cascades Involving iISS Systems	p. 186
An iISS Example	p. 188
Input to State Stability with Respect to Input Derivatives	p. 190
Cascades Involving the D[superscript k]ISS Property	p. 190
Dissipation Characterization of D[superscript k]ISS	p. 191
Superposition Principle for D[superscript k]ISS	p. 191
A Counter-Example Showing that D[superscript 1]ISS [not equal] ISS	p. 192
Input-to-Output Stability	p. 192
Detectability and Observability Notions	p. 194
Detectability	p. 195
Dualizing ISS to OSS and IOSS	p. 196
Lyapunov-Like Characterization of IOSS	p. 196
Superposition Principles for IOSS	p. 197
Norm-Estimators	p. 197
A Remark on Observers and Incremental IOSS	p. 198
Variations of IOSS	p. 199
Norm-Observability	p. 200
The Fundamental Relationship Among ISS, IOS, and IOSS	p. 201
Systems with Separate Error and Measurement Outputs	p. 202
Input-Measurement-to-Error Stability	p. 202
Review: Viscosity Subdifferentials	p. 203
RES-Lyapunov Functions	p. 204
Output to Input Stability and Minimum-Phase	p. 205
Response to Constant and Periodic Inputs	p. 205
A Remark Concerning ISS and H[subscript infinity] Gains	p. 206
Two Sample Applications	p. 207
Additional Discussion and References	p. 209
References	p. 213
Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints	p. 221
Introduction	p. 221
Preliminaries and Background	p. 222
Review of Some Notational Conventions and Definitions	p. 222
Generalized Jacobians, Derivate Containers, and Michel-Penot Subdifferentials	p. 228
Finitely Additive Measures	p. 229
Cellina Continuously Approximable Maps	p. 230
Definition and Elementary Properties	p. 231
Fixed Point Theorems for CCA Maps	p. 234
GDQs and AGDQs	p. 243
The Basic Definitions	p. 244
Properties of GDQs and AGDQs	p. 246
The Directional Open Mapping and Transversality Properties	p. 255
Variational Generators	p. 267
Linearization Error and Weak GDQs	p. 267
GDQ Variational Generators	p. 269
Examples of Variational Generators	p. 270
Discontinuous Vector Fields	p. 277
Co-Integrably Bounded Integrally Continuous Maps	p. 277
Points of Approximate Continuity	p. 280
The Maximum Principle	p. 281
References	p. 285
Sliding Mode Control: Mathematical Tools, Design and Applications	p. 289
Introduction	p. 289
Examples of Dynamic Systems with Sliding Modes	p. 289
VSS in Canonical Space	p. 296
Control of Free Motion	p. 298
Disturbance Rejection	p. 300
Comments for VSS in Canonical Space	p. 301
Preliminary Mathematical Remark	p. 302
Sliding Modes in Arbitrary State Spaces: Problem Statements	p. 303
Sliding Mode Equations: Equivalent Control Method	p. 305
Problem Statement	p. 305
Regularization	p. 306
Boundary Layer Regularization	p. 311
Sliding Mode Existence Conditions	p. 313
Design Principles	p. 316
Decoupling and Invariance	p. 316
Regular Form	p. 318
Block Control Principle	p. 320
Enforcing Sliding Modes	p. 322
Unit Control	p. 325
The Chattering Problem	p. 327
Discrete-Time Systems	p. 330
Discrete-Time Sliding Mode Concept	p. 331
Linear Discrete-Time Systems with Known Parameters	p. 333
Linear Discrete-Time Systems with Unknown Parameters	p. 335
Infinite-Dimensional Systems	p. 336
Distributed Control of Heat Process	p. 337
Flexible Mechanical System	p. 338
Control of Induction Motor	p. 340
References	p. 344
List of Participants	p. 349
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