Structural Controllibility Study Of Electrical Network Over Field F(z) Of Rational Functions In Multi-parameters

Great interests have been put on the security and reliability of system operations in the designs for linear systems at present. The stability of the system is one of the important conditions for system operation safely. Controllability and observability have closely relations with stability: if the system is controllable and observable, it will be stable unquestionably. Now the controllability theory in linear systems over the field R of real numbers has been heavily studied. This theory is efficacious for analyzing the characters which are determined by the value of physical parameters and system structure, nevertheless a practical system has a definite structure and has approximate, even unknown parameter sometimes because of a limit to experiment conditions, or a limit to making technology, the errors of observation. It is inconvenience for a linear system over the field R of real numbers to analyze the structure characters of a physical system, such as structural controllability. If all physical parameters of the system are regarded as mutually independently variable parameters, not as real constants, then it is a system over F(z). The controllability of the system over F(z) is independent of the values of its physical parameters. The controllability and observability of a system over F(z) always means the controllability and observability of the system over the field R of real numbers. In this paper we study the structural controllability of a kind of a typical physical system—the electrical system over F(z). Some analyzing results of structural characters for linear electrical network have been found. The dissertation is mainly composed of four parts.1. The structural controllability and obervability theory of the system over F(z) in frequency domain and in time domain.2. Synthetical applications of electrical structural analyses in cross researches between Graph Theory, Electrical Network Theory/System Theory Over F(z) and Matrix Theory.3. The research and conclusion about the relation between coprime matrix, reducibility of the coefficient matrix (or characteristic equation) of the electrical network and the structural separability and the electric separability of the network.4. The research and conclusion about the structural controllability over F(z) for the passive electrical network and active electrical network.The main contributions of this dissertation are as follows:1. After discussing the structural conditions of the existence of the active network solutions, the state equations in normal form over F(z) of electrical network which are suited for studying the character of network structure are derived. If there is no loop containing only capacitor and independent voltage sources, and if there is no cut-set containing only inductors and independent current-sources, both matrix A and B in (?)=Ax+Be, a state equation in normal form, have alike structure, whether for passive network or active network.2. Taking the passive network as a research task, Network Graph Theory, Matrix Theory and System Theory over F(z) etc are used to analyze the relation between coprime matrix, reducibility of the coefficient matrix of the state equation and the structural separability of the network for the first time, acquiring sufficient and necessary conditions for RLC network and RLCM network that characteristic equation det(sI-A) is reducible in F(z)[s] or coefficient matrix A of the state equation is reducible. Then according to these theoretic results, the structural controllability problems of RLC network and RLCM network over F(z) are studied, acquiring several structural controllability conclusions of the passive network.3. Taking the active network as a research task, the input accessibility, the Coates graph of matrix [A B] and graph transmission etc are used to analyze the relation between reducibility of the coefficient matrix of the state equation and the electric separability of the network or the electric accessibility of the network for the first time, acquiring sufficient and necessary conditions that coefficient matrix [A B] of the state equation is reducible. Network Graph Theory, Matrix Theory and System Theory over F(z) etc are used to derive the structural controllability criterion of the passive network over F(z) for the first time. Then according to these theoretic results, the structural controllability problems of the active network over F(z) are studied, acquiring several structural controllability conclusions of the active network. Meanwhile these conclusions are used to analyze the structural controllability of the practical linear time-invariant control system. It provides an effective way to study the relation between the independent variable parameter and the structure of the linear time-invariant control system from the more common and more systemic point of view.