Matrix Rational H 2 Approximation: A Gradient Algorithm Based on Schur Analysis

This paper deals with the rational approximation of specified order n to transfer functions which are assumed to be matrix-valued functions in the Hardy space for the complement of the closed unit disk endowed with the L2-norm. An approach is developed leading to a new algorithm, the first one to our knowledge which concerns matrix-transfer functions in L2-norm. This approach generalizes the ideas presented in [L. Baratchart, M. Cardelli, and M. Olivi, Automatica, 27(1991), pp. 413--418] in the scalar case but involves substantial new difficulties. Using the Douglas--Shapiro--Shields factorization of transfer functions, the criterion for the rational approximation problem above is expressed in terms of inner matrix functions of McMillan degree n. These functions, which possess a manifold structure, are represented by means of local coordinate maps obtained in [D. Alpay, L. Baratchart, and A. Gombani, Oper. Theory Adv. Appl., 73(1994), pp. 30--66] from a tangential Schur algorithm and for which the coordinates range over n copies of the unit ball. A gradient algorithm is then employed to solve the approximation problem using the coordinate maps to describe the manifold locally and changing from one coordinate map to another when required. However, while processing the gradient algorithm a boundary point can be reached. It is proved that such a point can be considered as an initial point for searching for a local minimum of lower degree while a local minimum of McMillan degree k < n provides a starting point for searching for a local minimum at degree k+1. The minimization process then pursues through different degrees. The convergence of this algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example.