Differential-algebraic Systems: Analytical Aspects And Circuit Applications

by Ricardo Riaza

Overview

Differential-algebraic equations (DAEs) provide an essential tool for system modeling and analysis within different fields of applied sciences and engineering. This book addresses modeling issues and analytical properties of DAEs, together with some applications in electrical circuit theory.Beginning with elementary aspects, the author succeeds in providing a self-contained and comprehensive presentation of several advanced topics in DAE theory, such as the full characterization of linear time-varying equations via projector methods or the geometric reduction of nonlinear systems. Recent results on singularities are extensively discussed. The book also addresses in detail differential-algebraic models of electrical and electronic circuits, including index characterizations and qualitative aspects of circuit dynamics. In particular, the reader will find a thorough discussion of the state/semistate dichotomy in circuit modeling. The state formulation problem, which has attracted much attention in the engineering literature, is cleverly tackled here as a reduction problem on semistate models.

Product Details

ISBN-13:	9789812791801
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/30/2008
Pages:	344
Product dimensions:	9.20(w) x 6.00(h) x 1.10(d)

Table of Contents

Preface vii

1 Introduction 1

1.1 Historical remarks: Different origins, different names 2

1.2 DAE analysis 4

1.2.1 Indices 5

1.2.2 Dynamics and singularities 8

1.2.3 Numerical aspects 10

1.3 State vs. semistate modeling 11

1.4 Formulations 12

1.4.1 Input-output descriptions 13

1.4.2 Leading terms 15

1.4.3 Semiexplicit, semilinear and quasilinear DAEs 16

1.5 Contents and structure of the book 20

Analytical aspects of DAEs 23

2 Linear DAEs and projector-based methods 25

2.1 Linear time-invariant DAEs 26

2.1.1 Matrix pencils and the Kronecker canonical form 27

2.1.2 Solving linear time-invariant DAEs via the KCF 28

2.1.3 A glance at projector-based techniques 29

2.2 Properly stated linear time-varying DAEs 35

2.2.1 On standard form index one problems 36

2.2.2 Properly stated leading terms 38

2.2.3 P-projectors: Matrix chain and the tractability index 39

2.2.4 The [Pi]-framework 43

2.2.5 Decoupling 51

2.2.6 A tutorial example 65

2.2.7 Regular points 74

2.3 Standard form linear DAEs 75

2.3.1 The tractability index of standard form DAEs 75

2.3.2 Decoupling 78

2.3.3 Time-invariant problems revisited 79

2.4 Other approaches for linear DAEs: Reduction techniques 81

3 Nonlinear DAEs and reduction methods 83

3.1 Semiexplicit index one DAEs 85

3.2 Hessenberg systems 88

3.3 Some notions from differential geometry 90

3.4 Quasilinear DAEs: The geometric index 93

3.4.1 The framework of Rabier and Rheinboldt 94

3.4.2 Index zero and index one points 97

3.4.3 Higher index points 103

3.4.4 Manifold sequences and locally regular DAEs 108

3.4.5 Local equivalence 111

3.4.6 Examples 118

3.4.7 Nonautonomousproblems 123

3.5 Dynamical aspects 130

3.6 Reduction methods for fully nonlinear DAEs 133

3.7 The differentiation index and derivative arrays 134

4 Singularities 137

4.1 What is a singular DAE? 137

4.2 Singularities of properly stated linear time-varying DAEs 139

4.2.1 Classification of singular points 140

4.2.2 Decoupling 145

4.3 Singularities of standard form linear time-varying DAEs 154

4.3.1 Classification 154

4.3.2 Decoupling 156

4.3.3 Analytic problems 157

4.4 Singularities of autonomous quasilinear DAEs 160

4.4.1 Quasilinear ODEs and impasse phenomena 162

4.4.2 Singular points of quasilinear DAEs 168

4.4.3 A reduction framework for singular problems 171

4.4.4 Dynamical aspects 179

4.4.5 Singular semiexplicit index one DAEs 181

4.4.6 Examples 184

Semistate models of electrical circuits 191

5 Nodal analysis 193

5.1 Background on graphs and electrical circuits 195

5.1.1 Graphs and digraphs 196

5.1.2 Elementary aspects of circuit theory 203

5.2 Formulation of nodal models 212

5.2.1 Node Tableau Analysis 213

5.2.2 Augmented Nodal Analysis 214

5.2.3 Modified Nodal Analysis 215

5.3 Index analysis: Fundamentals 216

5.3.1 Structural form of nodal models 216

5.3.2 On the tractability index of quasilinear DAEs 218

5.4 Index analysis: Passive circuits 219

5.4.1 Tableau equations and Augmented Nodal Analysis 220

5.4.2 Modified Nodal Analysis 232

5.5 Index analysis: Tree methods for non-passive circuits 236

5.5.1 Augmented Nodal Analysis 237

5.5.2 Modified Nodal Analysis 243

6 Branch-oriented methods 255

6.1 Branch-oriented semistate models 257

6.1.1 The basic model 258

6.1.2 Tree-based formulations 259

6.1.3 The state formulation problem 262

6.2 Geometric index analysis and reduction of branch models 264

6.2.1 Operating points 265

6.2.2 Implicitly described resistors and strict passivity 265

6.2.3 Multiport reduction 271

6.2.4 Index characterization 279

6.2.5 State space reduction 283

6.2.6 Controlled sources 289

6.3 Qualitative properties 293

6.3.1 Equilibria of DC circuits 295

6.3.2 Nonsingularity 297

6.3.3 Hyperbolicity and exponential stability 300

Bibliography 309

Index 325