Recent and New Results in Rational L2-Approximation

Abstract

The problem under consideration in this paper is the best approximation of a stable linear constant dynamical system by one whose Mac-Millan degree does not exceed a prescribed n. This problem can be formulated in many ways, according to the criterion which is used. In particular, striking results have been obtained concerning the Hankel norm, and a bibliographical account can be found in (KL, 81) and (Gl, 84). The criterion here will be the l 2 norm of the discrete transfer function, or equivalently the L2 \( (\frac{{d\omega }}{{1 + {\omega ^{2}}}}) \) norm of the continuous time transfer function. This problem has already been examined in the scalar case in (Ro, 78) (Ruc, 78) (De, 80) (Du, 73) and in the multivariable case in (Ba, 82) (Bs, 85) (Ba, *). In fact, the present paper can be considered as a sequel to (Ba, *), to which we refer on several occasions. Thereby, it does not really need a new introduction. Let us mention, however, that while studying qualitative generic properties as in (Ba, *), we focuse mainly on uniqueness, and give an account of the case of finite sequences. Though we do not develop here effective applications of the differential theory, we hope our results will help clarifying the situation.