Researches On Module Structure Of State Space And Controllability Of Linear Systems

There are four major branches for studying linear system theory according to being used mathematic methods and being described system. They are state space method, geometry method, algebra theory and multi-variable frequency-domain method. This dissertation combines state space method with algebra method, presents a module structure in state space for linear system. The state space can be associated with a module over the polynomial ring. By means of the properties of finitely generated module over a principal ideal domain, the decomposition in state space, the necessary and sufficient condition for controllability, controllability canonical forms and pole placement problem for state feedback are discussed. The main results are following.(1) Module structure analysis for state space of linear time-invariant system is presented. A non-zero finite dimensional vector space over a field F can be associated with a module over the ring of polynomials with coefficients in F. Those over the fields of complex numbers and real numbers are especially studied. The definition of the generated generalized eigenvectors of a linear transformation is introduced. The relations between the generators of the cycle modules and the generated generalized eigenvectors are obtained. The decomposition theorems of a non-zero vector space on module structure can be used to discuss the matrices over the field of real numbers and some canonical forms are obtained, which are used to modal structure analysis of multivariable. By means of the properties of finitely generated module over a principal ideal domain, a method of direct decomposition of space is presented. The quasi-diagonal algorithm on system matrix is deduced. A perfect theory and method to predigest system matrix is provided.(2) Controllability problem of linear system is studied and new block diagonal controllability canonical forms are obtained. By use of the transform function (or transform function matrix), the polynomial criterion on the controllability of a kind of time-invariant linear systems is discussed. The properties of the rank of the controllability matrix in time-invariant linear system is discussed and it is inferred that the controllability is not changed when perform elementary column operation on the input-matrix. A fast algorithm and its improvement for judging the controllability in the system are deduced by using the elementary column transformation on the controllability matrix. The most iteration steps judging the controllability is proved and the system is not controllable when the rank of the iteration matrix is not added. A new modified algorithm for getting controllable canonical forms of the multi-input system has been put forward. The annihilator of a vector space is discussed and applied to study the controllability in constant linear systems. A sufficient and necessary condition judging the controllability is easily obtained. Based on the decomposition theorem for vector space with module structure and rational canonical form of matrix, a kind of new block diagonal controllability canonical forms are inferred when the time-invariant linear multivariable system is completely controllable.(3) Some special matrices are studied by using controllability. A kind of circulant matrices, example of Hankel matrices and r-circulant matrices, are viewed as controllability matrices on linear system. The invertible condition and the algorithms for finding inverse matrices are obtained. The properties of the generalized Sylvester matrix are discussed when two polynomial matrices are right coprime. The relations between an R-block circulant matrix and a suitable generalized Sylvester matrix are presented. The necessary and sufficient condition for R-block circulant matrix invertible is obtained. A new method to study the kind of circulant matrices is presented.(4) Some New methods of pole placement are proposed. There are based on No.1 controllability canonical form, the generalized Wonham controllability canonical form, and circulant matrix. The advantage of this algorithm is no need to compute the characteristic polynomical of the system. A special new method of pole placement based on the new block diagonal controllability canonical forms in multi-input system is proposed. This method changes the problem of multi-input system to several problems of single-input. The number is the cycle index of system matrix. Compare with some known results, this method put forward that the least number of unknown entries of the feedback gain matrix is the order of the system matrix. The general expression containing arbitrary parameter is obtained for the feedback gain matrix.