Normalization,Regularization And Zero Assignment Of Polynomial Matrices

In the research on polynomial matrix descriptions(PMD)of systems,the multivariable frequency domain method has been developed as an entire theory system.It bases on the mathematics theory of rational fraction matrix to analyze and synthesize systems on the condition that the system is regular.But from the point of polynomial matrices,some properties of polynomial matrices such as regularity,zero structure are useful to analyze and synthesize systems.The theory is not yet mature in this field.In view of research situation of polynomial matrix descriptions of systems,this thesis will focus on two aspects of polynomial matrices:The normalization of polynomials matrices is studied in this thesis(the word “normalization” means making the coefficient matrix of the highest degree of a polynomial matrix be nonsingular).We reveal that for an arbitrary nonsingular polynomial matrix,there will be different normalizing polynomial matrices,but all this normalizing polynomial matrices have the same degree.And we show the degree is related to the zero structure at infinity of the original polynomial matrix.An equation describing the relation between the degree of normalizing polynomial matrices and the degree of the original polynomial matrix is given in this thesis,and then two sufficient and necessary conditions about whether a polynomial matrix has infinite zeros are deduced.Three numerical examples are given to illustrate the results.The regularization and zero assignment of a rectangular polynomial matrix are studied in this thesis.For an arbitrary rectangular polynomial matrix with full row rank,the problem that how to compensate it to a square polynomial matrix by adding another rectangular polynomial matrix is investigated,which makes the square polynomial matrix be regular and achieve arbitrary zero(finite and infinite)assignment.In particular,we can make the square polynomial matrix have no infinite zeros and all finite zeros located in the left half complex plane.We investigate rectangular polynomial matrices with one degree and higher degree respectively.We give the method and the sufficient and necessary condition of solving this problem using some related theories of polynomial matrix and descriptor system,and then two numerical examples are given to illustrate the results.