Differential-algebraic systems: Analytical aspects and circuit applications

Contents......Page 12
Preface......Page 8
1. Introduction......Page 16
1.1 Historical remarks: Different origins, different names......Page 17
1.2 DAE analysis......Page 19
1.2.1 Indices......Page 20
1.2.2 Dynamics and singularities......Page 23
1.2.3 Numerical aspects......Page 25
1.3 State vs. semistate modeling......Page 26
1.4 Formulations......Page 27
1.4.1 Input-output descriptions......Page 28
1.4.2 Leading terms......Page 30
1.4.3 Semiexplicit, semilinear and quasilinear DAEs......Page 31
1.4.3.1 Semiexplicit and semilinear DAEs......Page 32
1.4.3.2 Hessenberg DAEs......Page 33
1.4.3.3 Quasilinear DAEs......Page 34
1.5 Contents and structure of the book......Page 35
Analytical aspects of DAEs......Page 38
2. Linear DAEs and projector-based methods......Page 40
2.1 Linear time-invariantDAEs......Page 41
2.1.1 Matrix pencils and the Kronecker canonical form......Page 42
2.1.2 Solving linear time-invariant DAEs via the KCF......Page 43
2.1.3 A glance at projector-based techniques......Page 44
2.1.3.1 Index one characterization via projectors......Page 45
2.1.3.2 Decoupling of linear time-invariant index one DAEs......Page 46
2.1.3.3 Geometrical remarks......Page 47
2.1.3.5 Some auxiliary properties of the projectors Pi and Qi......Page 49
2.2 Properly stated linear time-varying DAEs......Page 50
2.2.1 On standard form index one problems......Page 51
2.2.2 Properly stated leading terms......Page 53
2.2.3.1 Matrix chain......Page 54
2.2.3.2 The tractability index of regular linear DAEs......Page 57
2.2.4 The Π-framework......Page 58
2.2.4.1 Alternative chain construction......Page 59
2.2.4.2 Equivalence of the P- and Π-chains......Page 60
2.2.4.3 Some properties of the projectors Πi and Mi......Page 64
2.2.5 Decoupling......Page 66
2.2.6 A tutorial example......Page 80
2.2.6.1 Index one......Page 82
2.2.6.2 Index two......Page 84
2.2.6.3 Index three......Page 86
2.2.7 Regular points......Page 89
2.3.1 The tractability index of standard form DAEs......Page 90
2.3.2 Decoupling......Page 93
2.3.3 Time-invariant problems revisited......Page 94
2.4 Other approaches for linear DAEs: Reduction techniques......Page 96
3. Nonlinear DAEs and reduction methods......Page 98
3.1 Semiexplicit index one DAEs......Page 100
3.2 Hessenberg systems......Page 103
3.3 Some notions from differential geometry......Page 105
3.4 Quasilinear DAEs: The geometric index......Page 108
3.4.1 The framework of Rabier and Rheinboldt......Page 109
3.4.2 Index zero and index one points......Page 112
3.4.2.2 Index one points......Page 113
3.4.3.1 Index two points......Page 118
3.4.3.2 Index ν points......Page 120
3.4.4 Manifold sequences and locally regular DAEs......Page 123
3.4.4.1 Regular manifold, solution manifold, and locally regular DAEs......Page 125
3.4.5 Local equivalence......Page 126
3.4.5.1 The index: Independence of reduction pairs and invariance......Page 127
3.4.5.2 C-conjugacy of state space descriptions......Page 131
3.4.5.3 On the link between local equivalence and reduction operators......Page 132
3.4.6 Examples......Page 133
3.4.6.1 Semiexplicit index one DAEs......Page 134
3.4.6.2 Hessenberg DAEs......Page 135
3.4.6.3 A locally regular DAE with di.erent indices......Page 136
3.4.7.1 Geometric index and reduction in the nonautonomous context......Page 138
3.4.7.2 Semiexplicit index one DAEs......Page 139
3.4.7.3 Nonautonomous Hessenberg DAEs......Page 140
3.4.7.4 Schur reduction and semiexplicit DAEs......Page 142
3.5 Dynamical aspects......Page 145
3.6 Reduction methods for fully nonlinear DAEs......Page 148
3.7 The differentiation index and derivative arrays......Page 149
4.1 What is a singular DAE?......Page 152
4.2 Singularities of properly stated linear time-varying DAEs......Page 154
4.2.1 Classification of singular points......Page 155
4.2.2.1 Working hypotheses......Page 160
4.2.2.2 Decoupling......Page 164
4.2.2.3 Campbell’s example and harmless singularities......Page 166
4.3.1 Classification......Page 169
4.3.2 Decoupling......Page 171
4.3.3 Analytic problems......Page 172
4.4 Singularities of autonomous quasilinear DAEs......Page 175
4.4.1 Quasilinear ODEs and impasse phenomena......Page 177
4.4.1.2 Impasse points......Page 178
4.4.1.3 Image singularities and singularity crossing phenomena......Page 181
4.4.2.1 0-singular points......Page 183
4.4.2.2 k-singular points......Page 185
4.4.3.1 Working hypotheses......Page 186
4.4.3.2 Local reduction of singular quasilinear DAEs......Page 189
4.4.4 Dynamical aspects......Page 194
4.4.5 Singular semiexplicit index one DAEs......Page 196
4.4.6.1 Impasse points: A simple instance......Page 199
4.4.6.2 Singularity crossing in a singular index one problem......Page 200
4.4.6.3 Singular reduction......Page 201
4.4.6.4 Semi-implicit DAEs......Page 202
4.4.6.5 A DAE with 0- and 1-singularities......Page 204
4.4.6.6 The differentiation-perturbation example of Campbell and Gear Consider .nally the DAE......Page 205
Semistate models of electrical circuits......Page 206
5. Nodal analysis......Page 208
5.1 Background on graphs and electrical circuits......Page 210
5.1.1.1 Graphs, digraphs, subgraphs......Page 211
5.1.1.3 Trees......Page 212
5.1.1.4 Incidence matrix......Page 213
5.1.1.5 Loop matrix......Page 214
5.1.1.6 Cutset matrix......Page 215
5.1.1.7 Relations among digraph matrices......Page 216
5.1.1.8 Fundamental loops and cutsets and their associated matrices......Page 217
5.1.2 Elementary aspects of circuit theory......Page 218
5.1.2.1 Topological aspects......Page 219
5.1.2.2 Circuit elements: dynamic relations and device characteristics......Page 222
5.2 Formulation of nodalmodels......Page 227
5.2.1 Node Tableau Analysis......Page 228
5.2.2 Augmented Nodal Analysis......Page 229
5.2.3 Modified Nodal Analysis......Page 230
5.3.1 Structural form of nodal models......Page 231
5.3.2 On the tractability index of quasilinear DAEs......Page 233
5.4 Index analysis: Passive circuits......Page 234
5.4.1 Tableau equations and Augmented Nodal Analysis......Page 235
5.4.2 Modified Nodal Analysis......Page 247
5.5 Index analysis: Tree methods for non-passive circuits......Page 251
5.5.1 Augmented Nodal Analysis......Page 252
5.5.2 Modified Nodal Analysis......Page 258
6. Branch-oriented methods......Page 270
6.1 Branch-oriented semistate models......Page 272
6.1.1 The basicmodel......Page 273
6.1.2 Tree-based formulations......Page 274
6.1.3 The state formulation problem......Page 277
6.2 Geometric index analysis and reduction of branch models......Page 279
6.2.2 Implicitly described resistors and strict passivity......Page 280
6.2.3.1 Multiport model......Page 286
6.2.3.2 Additional remarks on the invertibility of the matrix M......Page 292
6.2.4 Index characterization.......Page 294
6.2.5 State space reduction......Page 298
6.2.6 Controlled sources......Page 304
6.2.6.2 Index two......Page 306
6.2.6.3 Assumptions on controlling variables for the sources......Page 307
6.3 Qualitative properties......Page 308
6.3.1 Equilibria of DC circuits......Page 310
6.3.2 Nonsingularity......Page 312
6.3.3 Hyperbolicity and exponential stability......Page 315
Bibliography......Page 324
Index......Page 340