Subtleties in Robust Stability of Discrete-time Piecewise Affine Systems

In this paper we consider (inherent) robustness of discrete-time piecewise affine (PWA) systems. We demonstrate, via examples, that globally exponentially stable discrete-time PWA systems may have no robustness. More precisely, we show that the exponential stability property cannot prevent that arbitrarily small additive disturbances keep the state trajectory far from the origin. Mathematically speaking, this means that the system is not input-to-state stable with respect to arbitrarily small disturbances. The non-robustness property is related to the absence of a continuous Lyapunov function. These results indicate that one should be careful with existing stability analysis and synthesis methods for PWA systems that rely on discontinuous Lyapunov functions, as no robustness might be present. However, as the search for Lyapunov functions for discrete-time PWA systems often employs discontinuous Lyapunov functions (e.g. piecewise quadratic ones), robustness tests based on discontinuous Lyapunov functions are needed. Such tests are proposed in this article.