We propose an Arnoldi-based numerical method for solving a Sylvester-type equation arising in the construction of the Luenberger observer. Given an N×N matrix A and an N×m matrix G, the method simultaneously constructs an m×m Hessenberg matrix H with a preassigned spectrum and an N×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.
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*A part of the work in this paper was completed when the first-named author was visiting the University of California, San Diego. The author would like to express his sincere thanks to the Mathematics Department for its hospitality. The research of this author was supported by Air Force O%ce of Scientific Research under grant AFOSR 83-0334.