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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> ) 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm of interval rational functions.