Generalized (C´, A, B) Pairs and the Linear Servomechanism Problem

The objective of this paper is to solve the Disturbance Rejection Problem [1] in the time domain setting under measurement feedback for systems with feedforward terms (proper as opposed to strictly proper) in the controlled and measured output equations. This is a generalization of the disturbance rejection problem studied by Schumacher [2], Willems and Commault [3] and Imai and Akashi [4] where only the strictly proper case was considered. The solution in the strictly proper case is based on the fundamental concept of (C´, A, B) subspace pairs, an idea first introduced by Schumacher [5]. Our motivation for considering the more general case is that the linear servomechanism problem, where control over both steady state and transient response components of the tracking error is desired, can be formulated as a special case of the disturbance rejection problem only when the feedforward terms are non-zero. In fact most multivariable control problems with input-output response specifications can be treated in this framework. A transfer function version of this problem has been treated by Pernebo [6], where solvability is given in terms of a matrix polynomial equation. Our time domain treatment of the problem involves the definition of (C´, A, B) pairs which are generalized to include the effect of the feedforward terms. The concepts are a nontrivial generalization of the results in references [2-5].