Robust stability of piecewise linear discrete time systems

This paper presents efficient algorithms for verifying the stability of uncertain discrete time piecewise linear (PL) systems. While PL systems are intuitively simple, they are computationally hard. Two approaches to verifying stability are presented. For each approach, separate necessary and sufficient conditions are given. The first approach requires the solution of a linear matrix inequality. This method is only applicable to a restricted class of PL systems, and is generally very conservative. It is demonstrated that for most PL systems, these conditions yield no information. The second, more general, approach is based upon robust simulation. This method is useful for all PL systems, and will always yield a definitive answer. If a system initially satisfies necessity, but fails sufficiency, these algorithms can be systematically refined and after a finite number of refinements, a definitive answer is guaranteed. The algorithms are illustrated on four examples.