Nonsmooth Modeling and Simulation for Switched Circuits (eBook)

Preface 5
Acknowledgements 6
Contents 7
List of Abbreviations 12
List of Algorithms 13
List of Figures 14
List of Tables 18
Theoretical Framework 19
Introduction to Switched Circuits 20
1.1 Simple Examples of Switched Circuits 20
1.2 A Unified Dynamical Framework: Lur'e Dynamical Systems 43
1.3 An Aside on Nonsmooth Mechanics: The Bouncing Ball 44
1.4 Conclusions 46
1.5 Historical Summary 47
Mathematical Background 50
2.1 Basics from Convex and Nonsmooth Analysis 50
2.2 Non Convex Sets 68
2.3 Basics from Complementarity Theory 68
2.4 Mathematical Formalisms 76
2.5 The Dynamics of the Simple Circuits 101
2.6 Time-Discretization Schemes 116
2.7 Conclusions and Recapitulation 121
Dynamics Generation and Numerical Algorithms 123
Conventional Circuit Equation Formulation and Simulation 124
3.1 Circuit Topology and Kirchhoff's Laws 124
3.2 The Sparse Tableau Analysis (STA) 127
3.3 The Modified Nodal Analysis 127
3.4 The Charge/Flux Oriented MNA 130
3.5 Standard DAEs Stemming from the MNA 130
3.6 Semi-Explicit DAE Forms 139
3.7 Basics on Standard Circuit Simulation 146
Nonsmooth Modeling of Electrical Components 151
4.1 General Nonsmooth Electrical Element 151
4.2 Nonsmooth Elements as Inclusions into the Subdifferential of Convex Functions and Variational Inequality ( VI) 152
4.3 Nonsmooth Elements as Inclusions into Normal Cones and Variational Inequalities 154
4.4 Complementarity Problems 155
4.5 The Linear Input/Output Relation Case 156
4.6 Generic Piecewise-Linear Components 157
4.7 Special Instances of Nonsmooth Components 159
Time-Stepping Schemes and One Step Solvers 168
5.1 Summary of the Mathematical Formalisms 168
5.2 Principles of the Numerical Time-Integration Scheme 171
5.3 Time-Discretization of the General Cases 192
5.4 One-Step NonSmooth Problems (OSNSP) Solvers 193
Numerical Simulations 196
The Automatic Circuit Equations Formulation ( ACEF) Module and the SICONOS Software 197
6.1 An Insight into SICONOS 197
6.2 SICONOS Software 202
6.3 The ACEF Module and Algorithms 210
Simple Circuits 227
7.1 Maffezzoni's Example 227
7.2 A First Diode-BridgeWave Rectifier 233
7.3 A Second Diode-Bridge Wave Rectifier 236
7.4 The Cuk Converter 237
7.5 A Circuit Exhibiting Sliding Modes 243
Buck and Delta-Sigma Converters 250
8.1 The Buck Converter with Load Resistor 250
8.2 The Buck Converter Loaded by a Resistor and an Inverter Chain 264
8.3 The Delta-Sigma Converter 265
8.4 Conclusions 276
Some Facts in Real Analysis 277
A.1 Absolutely Continuous Functions and Sets 277
A.2 Lipschitz Continuous Functions and Sets 278
A.3 Functions of Bounded Variations in Time 278
A.4 Multifunctions of Bounded Variation in Time 280
A.5 Differential Measures 280
A.6 Measure Differential Inclusion (MDI) 281
References 283
Index 290