Rational Function Systems and Electrical Networks with Multi-Parameters

The following sections are included:

• Preface

• Acknowledgments

• Contents

A matrix A = (a_ij)_n×m is called the matrix over the real field or real number matrix, if each entry a_ij of this matrix is a real number. A system ẋ = AX + BU, Y = CX + DU is called a linear system (including an electrical network, or a network for short) over the real field or a real number system, if its coefficient matrices A, B, C, and D are real number matrices…

The following sections are included:

• Polynomials over Field F(z) or Ring F(z)[⋋]

• Operations and Determinant of Matrix over F(z)

• Elementary Operations of Matrices over F(z) and Some Conclusions

• Operation and Canonical Form of Matrix over F(z)
  • Matrix over F(z) and its canonical expression
  • Characteristic matrix
  • Two canonical forms of nonderogatory matrix
  • Rational canonical form and general Jordan canonical form

• Reducibility of Square Matrix over F(z)

• Reducibility Condition of Class of Matrices over F(z)
  • A class of RFM
  • Some lemmas and definitions
  • Reducibility condition
  • Applications
  • Summary

• Two Properties
  • Some lemmas
  • Type-1 matrix has two properties
  • Problems

• Independent Parameters and a Class of Irreducible Polynomials over F(z)[⋋]

• Conclusions

• New Model and Its Reducibility
  • The new model
  • Reducibility condition

• References

The following sections are included:

• Controllability and Observability in Time Domain
  • Preliminaries [Lu, 2001]
  • Controllability criteria [Lu, 2001]
  • The canonical decomposition of controllability and observability of systems
  • Criterions to linear physical systems
  • Applications to control systems

• Controllability and Observability in Frequency Domain
  • General systems [Lu et al., 1991]
  • SC-SO of composite systems [Lu et al., 1991]
  • Polynomial matrix^➁[Liu, 2008]

• References

The following sections are included:

• Resistor-Source Networks over F(z)
  • Introduction
  • General resistor-source networks
  • Unhinged networks
  • Effects of single source

• Separability and Reducibility Conditions of RLC Networks over F(z) and Their Applications
  • Introduction
  • Preliminaries
  • Separability condition
  • Separability and reducibility
  • Applications

• Controllability and Observability of RLC Networks over F(z)

• Structural Condition of Controllability for RLC Networks over F(z)
  • Separability conditions
  • Structural controllability conditions

• Structural Condition of Observability for RLC Networks over F(z)
  • Node voltage equation and two results
  • Structural condition of observability over F(z)

• Separability, Reducibility, Controllability and Observability of RLCM Networks over F(z)
  • Preliminaries
  • Separability
  • Reducibility
  • Controllability and observability
  • Structural condition of controllability and observability over F(z)

• Existence of State Equations of Linear Active Networks over F(z)
  • Existence condition of state equation over F(z)
  • Application

• A Sufficient Condition on Controllability and Observability of Active Networks over F(z)
  • Preliminaries
  • Sufficient condition of controllability over F(z)
  • Applications

• Conditions on and of Active Network over F(z) and Reducibility Condition of Ā and Their Applications to Controllability and Observability
  • Preliminaries
  • Partitioning of ȳ and u₂ from u₁
  • Conditions of
  • Reducibility of Ā and conditions of
  • Examples
  • Applications to controllability and observability over F(z)
  • Method of designing a structural controllable and observable active network with normal form

• Computer Assistant Analysis Program for Networks over F(z)
  • Software interface illumination
  • Structural analysis process description of the software
  • Software functions

• References

The following sections are included:

• Independent Parameters — The Third Type of Variables of Systems

• Physical Realization
  • Canonical state space description of linear time-invariant systems
  • Two basic properties

• Some Issues
  • Is it irreducible when exist interaction?
  • Dimension of nonzero mode number of independent parameters
  • Design of active networks being SC-SO and stable

• Quasi Structural Controllability and Observability Concept of Nonlinear Systems and Its Applications
  • Preliminaries
  • Quasi-structural controllability of nonlinear systems
  • Applications
  • Conclusions

• References

The following sections are included:

• Appendix A Some Well-known Results

• Appendix B Some Relevant Proofs in Theorem 4.11

• Appendix C Proof of Theorem 4.15

• Appendix D Proofs of Some Conclusions

• Index