ARNOLDI METHODS FOR LARGE SYLVESTER-LIKE OBSERVER MATRIX EQUATIONS, AND AN ASSOCIATED ALGORITHM FOR PARTIAL SPECTRUM ASSIGNMENT

We propose an Arnoldi-based numerical method for solving a Sylvester-type equation arising in the construction of the Luenberger observer. Given an N x N matrix A and an N x m matrix G, the method simultaneously constructs an m x m Hessenberg matrix H with a preassigned spectrum and an N x m orthonormal matrix X such that AX - XH = G. We consider the case when A is large and sparse, so that the standard techniques such as the well-known Hessenberg-Schur method for solving a Sylvester equation cannot be easily applied. As a byproduct, we propose an algorithm for the partial pole-assignment problem for large matrices.