Control of Switching Systems by Invariance Analysis (eBook)

Laurent Fribourg is head of the LSV (Laboratoire Spécification et Vérification) and Scientific Coordinator of the Institut Farman, Institut Fédératif de Recherche CNRS, which brings together the expertise of five laboratories from ENS Cachan, in France, in the fields of modeling, simulation and validation of complex systems. He has published over 70 articles in international journals and reviewed proceedings of international conferences, in the domain of the theory of formal methods and their applications. Romain Soulat is in the third year of his doctorate at the LSV at ENS Cachan in France, under the supervision of Laurent Fribourg. He is working on the modeling and verification of hybrid systems. In particular, his interests concern robustness in scheduling problems - especially as part of a collaborative project with EADS Astrium on the verification of a component in the launcher for the future Ariane 6 rocket. He has published 5 articles in reviewed proceedings of international conferences.

PREFACE ix

ACKNOWLEDGMENTS xi

INTRODUCTION xiii

CHAPTER 1. CONTROL THEORY: BASIC CONCEPTS 1

1.1. Model of control systems 1

1.2. Digital control systems 3

1.2.1. Digitization 3

1.2.2. Quantization 6

1.2.3. Switching 6

1.3. Control of switched systems using invariant sets 8

1.3.1. Controlled invariants 9

1.3.2. Safety control problem 9

1.3.3. Stability control problem 10

1.3.4. Other controllers 11

1.4. Notes 11

CHAPTER 2. SAMPLED SWITCHED SYSTEMS 13

2.1. Model 13

2.2. Illustrative examples 18

2.3. Zonotopes 21

2.4. Notes 23

CHAPTER 3. SAFETY CONTROLLERS 25

3.1. Backward fixed point computation (direct
approach) 26

3.2. Approximate bisimulation (indirect approach) 29

3.3. Application to a three-cell Boost DC-DC
converter 35

3.3.1. Model 35

3.3.2. Direct method 37

3.3.3. Indirect method 37

3.4. Notes 40

CHAPTER 4. STABILITY CONTROLLERS 41

4.1. Motivation 42

4.2. Preliminaries 42

4.2.1. Control induced by the decomposition 45

4.3. Decomposition function 46

4.3.1. Basic procedure 46

4.3.2. Enhancement for safety 48

4.4. Limit cycles 52

4.4.1. Discussion of the assumptions H1 and H2 53

4.4.2. Illustrative examples 54

4.5. Implementation 58

4.6. Notes 59

CHAPTER 5. APPLICATION TO MULTILEVEL CONVERTERS
61

5.1. Multilevel converters 62

5.2. Application of the decomposition procedure 62

5.2.1. Five-level converter 63

5.2.2. Seven-level converter 67

5.3. Physical experimentations 70

5.4. Notes 73

CHAPTER 6. OTHER ISSUES: REACHABILITY, SENSITIVITY,
ROBUSTNESS AND NONLINEARITY 75

6.1. Reachability control 75

6.2. Sensitivity 78

6.3. Robust safety control 79

6.4. Nonlinearity 82

6.5. Notes 87

CONCLUSIONS AND PERSPECTIVES 89

APPENDIX 1. SUFFICIENT CONDITION OF DECOMPOSITION 93

APPENDIX 2. APPLICATIONS OF THE ENHANCED DECOMPOSITION PROCEDURE
97

APPENDIX 3. PROOF OF THEOREM 4.3 103

APPENDIX 4. EXAMPLE WITH |R*Delta| = infinity 107

APPENDIX 5. CODE 109

BIBLIOGRAPHY 121

INDEX 127