Table of Contents
Preface v
Acknowledgments vii
1 Introduction 1
References 8
2 Matrices over Field F(z) of Rational Functions in Multi-parameters 11
2.1 Polynomials over Field F(z) or Ring F(z)[λ] 11
2.2 Operations and Determinant of Matrix over F(z) 12
2.3 Elementary Operations of Matrices over F(z) and Some Conclusions 12
2.4 Operation and Canonical Form of Matrix over F(z) 14
2.4.1 Matrix over F(z) and its canonical expression 14
2.4.2 Characteristic matrix 23
2.4.3 Two canonical forms of nonderogatory matrix 27
2.4.4 Rational canonical form and general Jordan canonical form 32
2.5 Reducibility of Square Matrix over F(z) 35
2.6 Reducibility Condition of Class of Matrices over F(z) 36
2.6.1 A class of RFM 36
2.6.2 Some lemmas and definitions 36
2.6.3 Reducibility condition 39
2.6.4 Applications 48
2.6.5 Summary 51
2.7 Two Properties 51
2.7.1 Some lemmas 52
2.7.2 Type-1 matrix has two properties 54
2.7.3 Problems 59
2.8 Independent Parameters and a Class of Irreducible Polynomials over F(z)[λ] 59
2.9 Conclusions 64
2.10 New Model and Its Reducibility 68
2.10.1 The new model 68
2.10.2 Reducibility condition 69
References 71
3 Controllability and Observability of Linear Systems over F(z) 73
3.1 Controllability and Observability in Time Domain 73
3.1.1 Preliminaries [Lu, 2001] 73
3.1.2 Controllability criteria [Lu, 2001] 75
3.1.3 The canonical decomposition of controllability and observability of systems 87
3.1.4 Criterions to linear physical systems 89
3.1.5 Applications to control systems 100
3.2 Controllability and Observability in Frequency Domain 102
3.2.1 General systems [Lu et al., 1991] 102
3.2.2 SC-SO of composite systems [Lu et al., 1991] 106
3.2.3 Polynomial matrix [Liu, 2008] 113
References 120
4 Electrical Networks over F(z) 121
4.1 Resistor-Source Networks over F(z) 121
4.1.1 Introduction 121
4.1.2 General resistor-source networks 123
4.1.3 Unhinged networks 131
4.1.4 Effects of single source 135
4.2 Separability and Reducibility Conditions of RLC Networks over F(z) and Their Applications 140
4.2.1 Introduction 140
4.2.2 Preliminaries 141
4.2.3 Separability condition 143
4.2.4 Separability and reducibility 146
4.2.5 Applications 147
4.3 Controllability and Observability of RLC Networks over F(z) 149
4.4 Structural Condition of Controllability for RLC Networks over F(z) 151
4.4.1 Separability conditions 151
4.4.2 Structural controllability conditions 154
4.5 Structural Condition of Observability for RLC Networks over F(z) 155
4.5.1 Node voltage equation and two results 156
4.5.2 Structural condition of observability over F(z) 158
4.6 Separability, Reducibility, Controllability and Observability of RLCM Networks over F(z) 160
4.6.1 Preliminaries 160
4.6.2 Separability 161
4.6.3 Reducibility 166
4.6.4 Controllability and observability 168
4.6.5 Structural condition of controllability and observability over F(z) 173
4.7 Existence of State Equations of Linear Active Networks over F(z) 177
4.7.1 Existence condition of state equation over F(z) 177
4.7.2 Application 181
4.8 A Sufficient Condition on Controllability and Observability of Active Networks over F(z) 185
4.8.1 Preliminaries 186
4.8.2 Sufficient condition of controllability over F(z) 189
4.8.3 Applications 189
4.9 Conditions on B11 ≠ 0 and C ≠ 0 of Active Network over F(z) and Reducibility Condition of A and Their Applications to Controllability and Observability 190
4.9.1 Preliminaries 191
4.9.2 Partitioning of y and u2 from u1 193
4.9.3 Conditions of B11 ≠ 0 198
4.9.4 Reducibility of A and conditions of C ≠ 0 203
4.9.5 Examples 207
4.9.6 Applications to controllability and observability over F(z) 213
4.9.7 Method of designing a structural controllable and observable active network with normal form 223
4.10 Computer Assistant Analysis Program for Networks over F(z) 224
4.10.1 Software interface illumination 224
4.10.2 Structural analysis process description of the software 226
4.10.3 Software functions 232
References 235
5 Further Thought 237
5.1 Independent Parameters - The Third Type of Variables of Systems 237
5.2 Physical Realization 238
5.2.1 Canonical state space description of linear time-invariant systems 238
5.2.2 Two basic properties 239
5.3 Some Issues 240
5.3.1 Is it irreducible when exist interaction? 241
5.3.2 Dimension of nonzero mode ≤ number of independent parameters 242
5.3.3 Design of active networks being SC-SO and stable 242
5.4 Quasi Structural Controllability and Observability Concept of Nonlinear Systems and Its Applications 243
5.4.1 Preliminaries 243
5.4.2 Quasi-structural controllability of nonlinear systems 245
5.4.3 Applications 246
5.4.4 Conclusions 249
References 249
Appendix 251
Appendix A Some Well-known Results 251
A.1 Linear systems theory over R 251
A.1.1 Controllability criterions in time domain 251
A.1.2 Polynomial matrix theory in frequency domain 253
A.2 Graph theory 258
A.3 Linear graph 262
A.3.1 Linear graph 262
A.3.2 Relationships in RLC networks 272
Appendix B Some Relevant Proofs in Theorem 4.11 274
B.1 The proof of C12 ≠ 0 274
B.2 The proof of Y12 ≠ 0 275
B.3 The proof of G12 ≠ 0 277
B.4 The proof of L12 ≠ 0 or Z12 ≠ 0 279
Appendix C Proof of Theorem 4.15 282
Appendix D Proofs of some conclusions 287
D.1 The proof of Theorem 4.18 287
D.2 The proof of Theorem 4.19 293
D.3 Some results 297
References 302
Index 303
