This work presents a general theory as well as constructive methodology in order to solve "observation problems," namely, those problems that pertain to reconstructing the full information about a dynamical process on the basis of partial observed data. A general methodology to control processes on the basis of the observations is also developed. Illustrative but practical applications in the chemical and petroleum industries are shown.
Author(s): Jean-Paul Gauthier, I. A. K. Kupka
Edition: First edition.
Year: 2001
Language: English
Pages: 240
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
1. Systems under Consideration......Page 13
3. Summary of the Book......Page 14
4. The New Observability Theory Versus the Old Ones......Page 15
5. A Word about Prerequisites......Page 16
6.2.2. About Universal Inputs......Page 17
6.2.3. About the Applications......Page 18
Part I Observability and Observers......Page 19
1. Infinitesimal and Uniform Infinitesimal Observability......Page 21
2. The Canonical Flag of Distributions......Page 23
3. The Phase-Variable Representation......Page 24
4. Differential Observability and Strong Differential Observability......Page 26
5. The Trivial Foliation......Page 27
6. Appendix: Weak Controllability......Page 31
1. Relation Between Observability and Infinitesimal Observability......Page 32
2. Normal Form for a Uniform Canonical Flag......Page 34
3. Characterization of Uniform Infinitesimal Observability......Page 36
4.2. Control Affine Systems......Page 38
4.3. Bilinear Systems (Single Output)......Page 40
5. Proof of Theorem 3.2......Page 41
Step 1. We Claim that the Codimension of Z is at Least 1......Page 43
Step 2. On…......Page 44
Step 4. Proof of the Fact that M has Codimension 1......Page 45
4 The Case…......Page 48
1. Definitions and Notations......Page 49
2. Statement of Our Differential Observability Results......Page 52
3. Proof of the Observability Theorems......Page 54
3.1.2. Estimate of the Codimension of B(k)......Page 55
3.1.3. Proof of the Openness and Density, for Immersivity......Page 60
3.2.2. Estimation of the Codimension of B(k)......Page 61
3.3. Proof of the Observability Theorems 2.1, 2.2, 2.3, and 2.4......Page 62
4.1. Preliminaries......Page 63
4.2. Back to Observability......Page 65
1. Proof of Lemma 4.1.......Page 68
5. The Approximation Theorem......Page 69
6. Complements......Page 70
7.1. Unobservable Linear Systems......Page 71
7.2. Lemmas......Page 72
1. Assumptions and Definitions......Page 80
1.2. Rings of Functions......Page 81
2. The Ascending Chain Property......Page 83
3.1. Finite Germs......Page 85
3.2. The Lemma......Page 86
3.4. Counterexample......Page 87
3.5. Consequences of the Key Lemma......Page 89
4. The ACP(N) in the Controlled Case......Page 90
5.1. Preliminaries......Page 93
5.2. Main Result......Page 94
5.3. Consequences......Page 95
6. The Controllable Case......Page 96
6 Observers: The High-Gain Construction......Page 98
1.1.1. Definitions......Page 99
1.1.2. Comment......Page 100
1.2.1. Definitions......Page 101
1.2.2. Peak Phenomenon......Page 103
1.3. Relations between State Observers and Output Observers......Page 105
2.2. The Luenberger Style Observer......Page 107
2.3. The Case of a Phase-Variable Representation......Page 113
2.4.1. Introduction and Main Result......Page 116
2.4.2. Preparation for the Proof of Theorems 2.6 and 2.8......Page 120
3.1. Continuity of Input-State Mappings......Page 132
Part II Dynamic Output Stabilization and Applications......Page 135
7 Dynamic Output Stabilization......Page 137
1.1. Semi-Global Asymptotic Stabilizability......Page 138
1.2. Stabilization with the Luenberger-Type Observer......Page 139
1.3. Stabilization with the High-Gain EKF......Page 143
2.1.1. Rings of C Functions......Page 144
2.1.2. Assumptions......Page 145
2.1.4. A Crucial Lemma......Page 146
2.2. Output Stabilization Again......Page 149
2.2.1. First Step: Local Asymptotic Stability of (125)......Page 151
2.2.3. Third Step: All Semitrajectories Starting From… Stay in…......Page 152
3.1. Systems with Positively Invariant Compact State Spaces......Page 153
1.1.1. Binary Distillation Columns......Page 155
1.1.2. The Equations of the Column......Page 157
1.1.3. The Problems to Be Solved......Page 158
1.2.1. Invariant Domains......Page 160
1.2.2. Stationary Points......Page 162
1.2.3. Asymptotic Stability......Page 165
1.2.4. Observability......Page 168
1.3. Observers......Page 169
1.4. Feedback Stabilization......Page 172
1.5. Output Stabilization......Page 174
2.1. The Equations of the Polymerization Reactor......Page 175
2.2. Assumptions and Simplifications of the Equations......Page 177
2.3. The Problems to Be Solved......Page 179
2.4.1. The Equilibria......Page 180
2.4.3. Observability......Page 184
2.6. Feedback Stabilization......Page 186
2.7. Dynamic Output Stabilization.......Page 189
1.2. Stability Properties of Subanalytic Sets......Page 191
1.4. Whitney Stratifications......Page 192
2.1. The Concept of Transversality......Page 193
2.2.2. Abstract Transversality Theorem......Page 194
2.2.3. Transversality to Stratified Sets......Page 195
2.3. The Precise Arguments We Use in Chapter 4......Page 196
3.1. The Concept of Stability and Asymptotic Stability......Page 197
3.1.1. Consequences of Stability......Page 198
3.2. Lyapunov’s Functions......Page 200
3.3. Inverse Lyapunov’s Theorems and Construction of Lyapunov’s Functions......Page 201
4.1. Definitions and Center Manifold Theorem......Page 202
4.2. Applications of the Center Manifold Theorem......Page 205
1. Chapter 2......Page 207
2. Chapter 3......Page 209
3. Chapter 4......Page 213
4. Chapter 5......Page 215
5. Chapter 6......Page 218
Bibliography......Page 229
Index of Main Notations......Page 233
Index......Page 236
