Robust Stability of Interval Plants: A Review

Abstract

In this paper we present a complete set of results concerning the robust stability analysis of single input single output Interval Plants in continuous time. Robust stability is considered under bounded real perturbations, non linear, sector-bounded perturbations, and unstructured (H¿) feedback perturbations. In each case, a solution to the problem is given based on the generalization of Kharitonov's theorem obtained in [1] and called the Box Theorem. The Box Theorem gives necessary and sufficient conditions for stabilization of an interval plant. This theorem introduced the so-called Kharitonov Segments associated with an interval plant, and the paper shows that these segments play a fundamental role in the robust stability analysis of such systems. Next we analyse the absolute stability of a closed loop system containing an interval plant in the forward path. The resulting theorem gives conditions for robust stability under nonlinear perturbations. This theorem is based on a result concerning the strict positive realness of families of interval rational functions. Finally, robust stability under unstructured (H¿ type) perturbations is considered and we deduce the necessary and sufficient conditions for robust stabilization in the presence of perturbations of this type. This result is again a generalization of a theorem on the H¿ norm of interval rational functions.