Existence and Generic Properties of L2 Approximants for Linear Systems

Abstract

The problem examined in this paper is the best approximation of a (possibly non-rational) given stable discrete system by a finite-dimensional one, where a bound on the dimension of the approximant is imposed. The criterion is the L ² norm of the transfer function. In this paper, we are interested only in generic qualitative properties of the set of approximants. We first show eixstence. We then prove, for instance, that best approximants are almost always cyclic, and finite in number.