The Study Of Structural Controllability Over F(z) Based On Algebra, Graph And Matroid Theoretic Approach

Great interests have been put on the security and reliability of system operations in the designs for linear systems. The stability of the system in one of the important conditions for system operation safely. A system can work normally it must firstly be stable. Controllability and observability have closely relations with stability:if the system is controllable and observable, the compensation system will be stable unquestionably by feedback compensation even if it has an unstable mode. The contrary, if the system has an unstable mode and is not controllable, it can not be stable by feedback. So long as the system is controllable, it can be able to stabilize. So it is very important to analysis controllability of systems. 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. From the point of view of complete controllable of Kalman, 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. 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.Structural controllability is the prerequisite for real number system can be controlled. If all physical parameters of the system are regarded as mutually independently variable parametes, 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. A system is controllable and observable over F(z) that is structural controllable and observable, then the system is controllable and observable over the field R of real number in the parameter space Rq. That always means the controllability and observability of the system over the field R of real number. The study of structural controllability is the main content in this paper. The structural controllability of system in time and frequency domain over F(z) is researched. The graph theoretic and matroid theoretic methods in the study of structural controllability are also researched. Some algebraic, graph and matroid criteria of structural controllability of systems over F(z) are obtained. This paper consists of the following components:(1) The matrix over F(z), polynomial and polynomial matrix over ring F(z)[λ] and their operation, irreducibility of a class of RFM and canonical matrix.(2) Algebraic criteria of structural controllability of systems over F(z)—based on the methods in time and frequency domain.(3) Graph criteria of structural controllability of systems over F(z).(4) Matroid criteria of structural controllability of systems over F(z).The main contributions of this dissertation are as follows:(1) The previous conclusions of structural controllability of systems in time domain over F(z) is summarized based on the method described by state space. The structural controllability conditions of system (A,B) are given in the case of coefficient matrix of system A is reducible and irreducible. Particularly, a class of coefficient matrix as form A=(T+V)-1U over F(z) can describe most of the physical system. Therefore, it is more practical significance to study of the structure controllability of the system with this standard form. The conclusion is that system (A,B) is structural controllable if and only if (A,B) is irreducible under permutation transform where B≠0. The system described in frequency domain over F(z) is established on the basis of transfer function matrix and polynomial matrix description. The controllability and observability of system is relative to whether det(sI-A) and Q(s) are coprime, while the polynomial matrix description is a general description of the linear system. The PBH and Smith normal form PBH structural controllability criterion are acquired according to the polynomial matrix theory.(2) According to the reducible characteristics of a class of rational function matrix in multi-parameters the relationship between the reducibility and the corresponding signal-flow graph connectivity of which as form A=(C+V)-1U or G=C+D is studied. So as arrive that the graph criterion which the system of a physical parameter canonical form is structural controllable if G(A) is strongly connected. For such system, whether it is structural controllable will simply known by observation of the signal-flow graph connectivity of G(A). Then a necessary and sufficient condition that system is structural controllable is derived by analyzes about the relationship between the reducibility of (A,B) and the accessibility of G(A,B). Finally, a necessary and sufficient condition on the structural controllability of a general system is gained by the concept of1-factors connected.(3) The vector matroid is defined which is combine the system over F(z) with matroid theory from the basic concepts of matroid. The basic conditions for structural controllability of linear systems over F(z) are given by using the definitions of base and rank function of matroid. Then full rank conditions of [sI-A|B](s∈(?),(?) denotes complex field) are resealrched by use of the concepts of matroid union and the matroid criterion that the linear system is structural controllable is also given. At last, parallel composite system is studied and the sufficient conditions for parallel structural controllability of composite systems are attained.