The spectrum of dynamical systems possessing non convex positively invariant sets

Stability of linear systems is equivalent to the existence of ellipsoidal positively invariant or contractive sets. Stable systems are also characterized by the location of the spectrum of system matrix on the complex plane. Such a characterization has also been established for stable systems possessing convex polyhedral positively invariant sets. In this paper the class of unstable systems possessing expansive convex polyhedral and nonconvex polyhedral invariant sets is studied. For this type of systems, encountered in obstacle avoidance control problems, necessary and sufficient conditions for the expansiveness of polyhedral sets are developed. Then the spectral characterization of this class of systems is presented.