Weighted Matrix Padé-type Approximation And Its Applications In Control Theory

In this paper, theory and method of matrix Padé-type approximation are improved and generalized in the inner product space. The explicit determinant formula with matrix norm elements of matrix Padé-type approximation is introduced. The structure of matrix Padé-type table is given. Weighted matrix Padé-type approximation is constructed. Specially, the method of determining the best weight factors for (m/1)(?)(z) is presented. Moreover the applications of matrix Padé-type approximation to matrix exponent computation and model reduction in control theory are given.This paper consists of three parts:In the first part, the explicit determinant formula with matrix norm elements of matrix Padé-type approximation is presented by exploiting a linear mapping A. It is simple and flexible to compute matrix Padé-type approximation by choosing different matrix norms. It is also adaptive to compute large-scale matrix . The recursive formula is given. H-table and the structure of matrix Padé-type table are obtained.In the second part, weighted matrix Padé-type approximation by introducing a set of scalar weights {ω_k} for the generating polynomial v(z) is presented. Three explicit determinant formulas are given. The method of determining the best weight factors for (m/1)(?)(z) is presented. And higher order approximation and local approximation of weighted matrix Padé-type approximation are studied. A good approximation to e^Az can be obtained by choosing the appropriate weight factors.In the third part, the applications of matrix Padé-type approximation to model reduction in control theory are introduced. Based on the concept of matrix Padé-type approximation, some model reduction methods are given including matrix Padé-type method, Modal-Padé type method, dominent pole-Padé type method, partial fraction-Pade type method and Routh-Pade type method. It is shown that the methods are efficient.