In this paper, we present a method and associated theory for solving the multi-input Sylvester-Observer equation arising in the construction of the Luenberger observer in control theory. The proposed method is a particular generalization of the algorithm described by Datta and Saad in 1991 to the multi-output. We give some theoretical results and present some numerical experiments to show the accuracy of the proposed algorithm. Mathematical subject classification: 65F10, 65F30.

In the present paper, we propose a numerical method for solving the coupled Lyapunov matrix equations A P + P A T + B B T = 0 and A T Q + Q A + C T C = 0 where A is an n ×n real matrix and B, C T are n × s real matrices with rank(B) = rank(C) = s and s << n . Such equations appear in control problems. The proposed method is a Krylov subspace method based on the nonsymmetric block Lanczos process. We use this process to produce low rank approximate solutions to the coupled Lyapunov matrix equations. We give some theoretical results such as an upper bound for the residual norms and perturbation results. By approximating the matrix transfer function F(z) = C (z In - A)-1 B of a Linear Time Invariant (LTI) system of order n by another one Fm(z) = Cm (z Im - Am)-1 Bm of order m, where m is much smaller than n , we will construct a reduced order model of the original LTI system. We conclude this work by reporting some numerical experiments to show the numerical behavior of the proposed method.