An Analysis of the Pole-Zero Cancellations in $H^\infty $-Optimal Control Problems of the First Kind

Abstract

The aim of this paper is to study the pole-zero cancellations which occur in a class of H-optimal control problems which may be embedded in the configuration of Fig. 1. H control problems are said to be of the first kind if both P12(s) and P21(s) are square but not necessarily of the same size. It is primarily this class Of problems which will concern us here. A general bound on the McMillan degree of all controllers which are stabilizing and lead.to a closed loop which satisfies (((s)((o -< p (p need not be optimal in the L-norm sense) is derived. As illustrated in Fig. 1, (s) is the transfer function relating yl(s) to Ul(S). If the McMillan degree of P(s) in Fig. is n, we show that in the single-loop (SISO) case the corresponding (unique) H-optimal controller never requires more than n states. In the multivariable case, there is a continuum of optimal controllers whose McMillan degree satisfies this same bound, although other controllers with higher McMillan degree also exist. The derivation of these bounds require several steps, each of which is of independent system theoretic interest.