Laurent Fribourg & Romain Soulat 
Control of Switching Systems by Invariance Analysis [PDF ebook] 
Applcation to Power Electronics

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

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.