An analysis of the pole-zero cancellations in a class of H∞ optimal control problems

The aim of this paper is to study the pole-zero cancellations which occur in a class of H^∞ control problems which may be embedded in the configuration of Fig. 1. The class is characterized by the assumption that both P₁₂(s) and P₂₁(s) are square but not necessarily of the same size. A general bound on the McMillan degree of all controllers which are stabilizing and lead to a closed loop which satisfies |R(s)| ⩽ p(p need not be optimal in the L^∞-norm sense) is desired if the McMillan degree of P(s) in Fig. 1 is n, we show that in the single-loop (SISO) case the corresponding (unique) H^∞-optimal controller never requires more than n-1 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 exist.