Rational function systems and electrical networks with multi-parameters. (Q2913825)

This book is built on the doctoral course ``Rational Function Systems and electrical Networks with Multi-parameters'' of the same author and it is mainly based on his research activity. Finding some inconveniences in exploring structural properties using linear system theory and electrical network theory over the real field \(R\), the author proposed an alternative approach using the matrices over the field \(F(z)\) of rational functions in multi-parameters to describe coefficient matrices of systems and electrical networks and investigates their structural properties based on the description of systems and electrical networks over \(F(z)\).NEWLINENEWLINE The book is divided in five chapters.NEWLINENEWLINEChapter 1, \textit{Introduction}, introduces the background and meaning of systems and electrical networks over \(F(z)\). It should be emphasized that the conclusions over \(F(z)\) are determined by structures of systems and networks only, not the values of \(z\), because parameters are not evaluated with regard to conclusions over \(F(z)\), which eliminates the value effect and only leaves the structure effect.NEWLINENEWLINEIn Chapter 2, \textit{Matrices over Field \(F(z)\) of Rational Functions in Multi-parameters}, the matrix theory over the real field \(R\) is extended to \(F(z)\). The reducibility condition of a class of square matrix and its characteristic polynomial over \(F(z)\) is discussed in detail. The definition of a so called type-1 matrix is given and two basic properties of such a kind of matrices are proved. The variable replacement condition for independent parameters is introduced.NEWLINENEWLINEChapter 3, \textit{Controllability and Observability of Linear Systems over \(F(z)\)}, explores the structural controllability and observability of linear systems over \(F(z)\) and introduces some new conclusions in time domain and frequency domain.NEWLINENEWLINEChapter 4, \textit{Electrical Networks over \(F(z)\)}, provides the structural properties of the electrical networks over \(F(z)\): {\parindent=8mm \begin{itemize}\item[(i)] the separability, reducibility, controllability and observability of RLC networks over \(F(z)\) and structural conditions of controllability and observability over \(F(z)\); \item[(ii)] the separability, reducibility, controllability and observability of RLCM networks over \(F(z)\); \item[(iii)] the state equation existence condition of active networks over \(F(z)\) and controllability and observability conditions over \(F(z)\). NEWLINENEWLINE\end{itemize}} Chapter 5, \textit{Further thought}, describes further thoughts on the field under consideration leading the readers towards more general approach of the domain, as nonlinear systems. Nonlinearity is universal, but it is difficult to directly research properties of nonlinear systems such as controllability and observability.NEWLINENEWLINEThe book end with four Appendices:NEWLINENEWLINEAppendix A, \textit{Some well known results}, which summarizes, for the readers convenience the general theory of linear systems over the real line \({R}\); graph theory and linear graph.NEWLINENEWLINEAppendix B--Appendix D provide detailed proofs of some results from Chapter 4.NEWLINENEWLINEEach chapter ends with a list of adequate References.NEWLINENEWLINEThe results derived in this book are in general clear and intuitive and are useful in analyzing and synthesis of systems and electric networks.NEWLINENEWLINEThe description of systems, electrical networks and matrices over \(F(z)\) in this book, is different from other relevant books, which introduced linear systems, electrical networks and matrices over \(R\). In this book the systems, electrical networks and matrices are explored over the field \(F(z)\).NEWLINENEWLINEThis book can be used by postgraduate students, PhD students, college teachers, researchers and engineers of the field of system theory, electronic and electrical engineering, automatic control and applied mathematics matrix theory.