Stabilization of linear systems with fixed order controllers

Abstract

This paper deals with the problem of stabilizing a linear time invariant plant with a feedback compensator of fixed dynamic order. This is an important unsolved problem in control theory, because existing solutions to the regulator problem can only generate controllers that are of high enough order that arbitrary pole placement becomes possible. In general the dynamic order of controllers can be much lower if only stabilization is desired. In the transfer function domain a new necessary condition for stabilizability by a fixed order controller is derived. This result, which can be used sequentially to derive a lower bound on the order of a stabilizing controller, is based on a classical result (1873) of linear programming known as Gordan's theorem of the alternative. The stabilization algorithm successively selects Hurwitz vectors to make the largest stability hypersphere intersect a linear subspace determined by the plant parameters and the order of the controller. Occurrence of such an intersection is a sufficient condition for stabilizability, and the intersection also determines a stabilizing compensator. The algorithm is illustrated by practical examples which demonstrate that a large reduction in controller order is achievable by this method.