Structural Controllability Of The System Over F(z) In Frequency Domain And Its Computer Aided Analysis

Modern control theory was marked with state space methods introduced into the system and control theory systematically by Kalman, and put forward the controllability and observability of system which are two important concepts of representation. The controllability and observability theories over the real number field are efficacious for the analysis of the controllability and observability that are determined by the system structure and the value of the physical parameters together. Nevertheless a pratical system has approximate, even unknown parameter value sometimes because of a limit to experimental conditions or to manufacturing process, the error of observations and the approximate and artificial processing to data. Only with the controllability, we can not know whether it is due to structure or due to inappropriate choice of parameter values when the system does not meet the conditions of complete controllability. Different to the traditional research over the real number field, the research over the multivariate rational function field can have the conclusions that are only about the system structure, but have nothing to do with the values of physical parameters, which reflects alone the structural nature of the system.This paper focuses on structural controoolbility and the frequency domain method, promotes the frequency domain theory of linear invariant systems (real number system) to the multivariate rational function system, and derives a number of analysis results about structural characteristics of the system over F(z) in frequency domain. This paper consists of the following components:The matrix, polynomial and polynomial matrix over F(z) and their operation, irreducibility and canonical matrix; A class of rational function matrices that satisfy two properties of linear systems and structural controllability; The structural properties of the system over F(z) in frequency domain based on the polynomial matrix theory; The structural properties of composite system over F(z) in frequency domain.; The research and design on the structural property analysis software.The main conclusions of this paper are:The degree of irreducible polynomials onλover F(z) can be arbitrarily large. The independence of the coefficients of the polynomial onλover F(z) is related to its irreducible. The only available form of Smith matrix and non-derogatory matrix can be derived by the elementary transformation to the polynomial matrix over F(z)[λ].Type-1 Matrix satisfies two properties:its the characteristic polynomial over the ring of F(z)[λ] has no nonzero constant eigenvalues; and has no non-zero multi roots over the ring of F(z)[λ]. At the same time these properties are applied to the structure analysis of realistic control systems.Based on the definition of unimodular matrix, Lebesgue measure, the definition of coprime, the PBH structural controllability criterion over F(z), the structural controllability criterion of polynomial matrix description of the corresponding to the state space descriptions and the structural controllability criterion of the composite system are related to the coprimeness of the polynomial matrices. The common factor of sub-determinant method which judge the coprimeness of the polynomial matrix over F(z), can simplify the judgement of the structural controllability of the tandem or parallel systems which has two independent subsystems that have two properties.A universal composite system is proposed and the polynomial matrix description of composite systems and structural controllability estimation is derived.The requirements of software analyzing the structural controllability and observability are analyzed and the design of that is descripted in detail in terms of the software engineering. Using the theory over F(z), the analysis software is developed by the model of signal flow graphs and block diagrams. The software can analyze the topology of block diagram and the signal flow graph. The topology results are into MATLAB to make the symbolic computation to analyze the structural controllability and observalibility, and to obtain state-space realization.