On the Periodicity of Symbolic Observations of Piecewise Smooth Discrete-Time Systems

We study the behavior of discrete-time systems composed of a set of smooth transition maps coupled by a “quantized” feedback function. The feedback function partitions the state space into disjoint regions, and assigns a smooth transition function to each region. The main result is that under a constraint on the norm of the derivative of the transition maps, a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic. Indeed, under these assumptions, we show that eventually the feedback function is determined by a finite state automaton. A similar result is proved in the case of finite state dynamic feedback.