On the periodicity of symbolic observations of piecewise smooth discrete-time systems

A study is made of 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. Under these assumptions, it is shown 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.