Robust stabilization of singularly perturbed systems

Abstract

We discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models. In [1], it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the ‘true’ system containing both fast and slow modes. This phenomenon was then expanded to include the case where the simplified system is strictly proper in [2]. The objectives of this note are threefold: (i) to show that, given any proper system and any stabilizing controller for it that is proper but not strictly proper, there exists a singular perturbation of the system that is destabilized by that controller, (ii) to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter, and (iii) to provide a characterization, in the same spirit as [3,4], of the set of all strictly proper controllers that stabilize a given proper plant. By combining these results, it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system.