Minimal Pade-Sense Matrix Approximations Around s = 0 and s = ∞

Abstract

An algorithm is presented for the solution of the minimal Padé-sense matrix approximation problem around s = 0 and s = ∞ or what is called in linear system theory, the minimal partial realization problem with Markov parameters and time moments. It gives a solution in the form of a polynomial matrix fraction. The algorithm is a generalization of the Berlekamp-Massey algorithm for the scalar case [1,2,7]. Our approach is mainly based on the work of Dickinson, Morf and Kailath who treated the problem with Markov parameters only [6]. Y. Bistritz [3,4,5] gives a solution of the above problem in a state space setting and refers to [6] for a polynomial approach. However an extra condition should be imposed on the solution as will be shown in this paper. Our main tool is the construction of a nested basis for the null spaces of block Hankel matrices. The proof of the algorithm and other aspects of the minimal partial realization can be found in [8]. We only give the main result.