A controller degree bound for H∞-optimal control problems of the second kind

The aim of this paper is to present further results on the cancellation phenomena which occur in H∞-optimal control problems which may be embedded in the generalized regulator configuration illustrated in Fig. 1. This is a continuation of the work in [6]. In this paper we concentrate on problems of the second kind which are characterized by the assumption that one of the off-diagonal blocks P12(s) or P21(s) is non-square. To fix ideas, we assume that P21(s) is square while P12(s) is tall. We also assume that P12(jω) and P21(jω) have full rank for all ω. If the McMillan degree of P(s) in Fig. 1 is n, we show that there is always an H
^∞
-optimal controller requiring no more than 2n-1 states. Space limitations make it impossible to present proofs of the main steps in the argument.