Subtleties in robust stability of discrete-time piecewise affine systems

In this paper we consider (inherent) robustness ofdiscrete-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 thatarbitrarily 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. Theseresults issue a warning regarding existing stability analysis and synthesis methods for PWA systems that rely on discontinuous Lyapunov functions, as no robustness might be present. In many cases though, the search for Lyapunov functions in discrete-time PWA systems employs discontinuous Lyapunov functions (e.g. piecewise quadratic ones). To establish robustness(or non-robustness) of nominally stable PWA systems in these cases, when a continuous Lyapunov function is not known, robustness tests based on discontinuous Lyapunov functions are needed. Such tests are proposed in this article.