Rational Function Systems and Electrical Networks with Multi-parameters

Preface
Acknowledgments
1 Introduction
References
2 Matrices over Field F(z)of Rational Functions in Multi-parameters
2.1 Polynomials over Field F(z)or Ring F(z)(λ)
2.2 Operations and Determinant of Matrix over F(z)
2.3 Elementary Operations of Matrices over F(z)and Some Conclusions
2.4 Operation and Canonical Form of Matrix over F(z)
2.4.1 Matrix over F(z)and its canonical expression
2.4.2 Characteristic matrix
2.4.3 Two canonical forms of nonderogatory matrix
2.4.4 Rational canonical form and general Jordan canonical form
2.5 Reducibility of Square Matrix over F(z)
2.6 Reducibility Condition of Class of Matrices over F(z)
2.6.1 A class of RFM
2.6.2 Some lemmas and definitions
2.6.3 Reducibility condition
2.6.4 Applications
2.6.5 Summary
2.7 Two Properties
2.7.1 Some lemmas
2.7.2 Type-1 matrix has two properties
2.7.3 Problems
2.8 Independent Parameters and a Class of Irreducible Polynomials over F(z)(λ)
2.9 Conclusions
2.10 New Model and Its Reducibility
2.10.1 The new model
2.10.2 Reducibility condition
References
3 Controllability and Observability of Linear Systems over F(z)
3.1 Controllability and Observability in Time Domain
3.1.1 Preliminaries(Lu,2001)
3.1.2 Controllability criteria(Lu,2001)
3.1.3 The canonical decomposition of controllability and observability of systems
3.1.4 Criterions to linear physical systems
3.1.5 Applications to control systems
3.2 Controllability and Observability in Frequency Domain
3.2.1 General systems(Lu et al.,1991)
3.2.2 SC-SO of composite systems(Lu et al.,1991)
3.2.3 Polynomial matrix(Liu,2008)
References
4 Electrical Networks over F(z)
4.1 Resistor-Source Networks over F(z)
4.1.1 Introduction
4.1.2 General resistor-source networks
4.1.3 Unhinged networks
4.1.4 Effects of single source
4.2 Separability and Reducibility Conditions of RLC Networks over F(z)and Their Applications
4.2.1 Introduction
4.2.2 Preliminaries
4.2.3 Separability condition
4.2.4 Separability and reducibility
4.2.5 Applications
4.3 Controllability and Observability of RLC Networks over F(z)
4.4 Structural Condition of Controllability for RLC Networks over F(z)
4.4.1 Separability conditions
4.4.2 Structural controllability conditions
4.5 Structural Condition of Observability for RLC Networks over F(z)
4.5.1 Node voltage equation and two results
4.5.2 Structural condition of observability over F(z)
4.6 Separability,Reducibility,Controllability and Observability of RLCM Networks over F(z)
4.6.1 Preliminaries
4.6.2 Separability
4.6.3 Reducibility
4.6.4 Controllability and observability
4.6.5 Structural condition of controllability and observability over F(z)
4.7 Existence of State Equations of Linear Active Networks over F(z)
4.7.1 Existence condition of state equation over F(z)
4.7.2 Application
4.8 A Sufficient Condition on Controllability and Observability of Active Networks over F(z)
4.8.1 Preliminaries
4.8.2 Sufficient condition of controllability over F(z)
4.8.3 Applications
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
4.9.1 Preliminaries
4.9.2 Partitioning of y and u2 from u1
4.9.3 Conditions of B11≠0
4.9.4 Reducibility of A and conditions of C≠0
4.9.5 Examples
4.9.6 Applications to controllability and observability over F(z)
4.9.7 Method of designing a structural controllable and observable active network with normal form
4.10 Computer Assistant Analysis Program for Networks over F(z)
4.10.1 Software interface illumination
4.10.2 Structural analysis process description of the software
4.10.3 Software functions
References
5 Further Thought
5.1 Independent Parameters-The Third Type of Variables of Systems
5.2 Physical Realization
5.2.1 Canonical state space description of linear time-invariant systems
5.2.2 Two basic properties
5.3 Some Issues
5.3.1 Is it irreducible when exist interaction?
5.3.2 Dimension of nonzero mode ≤ number of independent parameters
5.3.3 Design of active networks being SC-SO and stable
5.4 Quasi Structural Controllability and Observability Concept of Nonlinear Systems and Its Applications
5.4.1 Preliminaries
5.4.2 Quasi-structural controllability of nonlinear systems
5.4.3 Applications
5.4.4 Conclusions
References
Appendix
Appendix A Some Well-known Results
A.1 Linear systems theory over R
A.1.1 Controllability criterions in time domain
A.1.2 Polynomial matrix theory in frequency domain
A.2 Graph theory
A.3 Linear graph
A.3.1 Linear graph
A.3.2 Relationships in RLC networks
Appendix B Some Relevant Proofs in Theorem 4.11
B.1 The proof of C12≠0
B.2 The proof of Y12≠0
B.3 The proof of G12≠0
B.4 The proof of L12≠0 or Z12≠0
Appendix C Proof of Theorem 4.15
Appendix D Proofs of some conclusions
D.1 The proof of Theorem 4.18
D.2 The proof of Theorem 4.19
D.3 Some results
References
Index