Matrix Valued Rational Approximation And Applications In Control Theory

The direct inner product of two matrices is introduced, and then the generalized product of two block matrices is introduced. Based on the generalized product a matrix-type Lanczos algorithm is established to define a matrix sequence that constitute approximations to the given higher order matrix in large-scale multivariable linear systems. The generalized Hadamard product of two block matrices is introduced, and then a new generalized inverse of matrices is defined by virtue of the direct inner product.' The algebraic structures and properties of this generalized inverse are analyzed. To show the usefulness of applying these properties of the generalized Hadamard products, the solution of large-scale linear system problems with sparse matrix is considered. In terms of the direct inner product and the generalized inverse of matrices, the theory and method of generalized inverse matrix Pade approximation are first established. By using this approximation, a new efficient approach is presented to solve " moment matching problem" in control theory. As its an application, the computation of matrix exponential functions is presented. Multivariable generalized inverse Thiele-type matrix Pade approximation is defined for the first time. It is shown that this approximation can be applied to the state-space realization problem of multivariable linear systems in control theory. A generalized matrix-valued linear functional is introduced on the polynomial space, and then matrix Pade-type approximation is first defined by means of the direct inner product of matrices. A matrix Pade-type algorithm and a matrix Pade-type-Routh mixed algorithm are given to solve the reduction problem in control theory. Multivariable matrix Pade-type approximation is defined for the first time and applied to the state-space realization problem of multivariable linear systems in control theory. One-variable and multivariable Lagrange-type matrix rational interpolation are established by using the direct inner product of matrices. In terms of the direct inner product and the generalized inverse of matrices, the theory and method of generalized inverse Thiele-type matrix rational interpolation are first established. It is shown that this interpolation method can be solved the unconstrained tangential rationalinterpolation problem in control theory. Based on the direct inner product and the generalized inverse of matrices, multivariable generalized inverse Thiele-type and Stieltjes-type matrix rational interpolation are constituted.