Approximate predictability of Pseudo-Metric Systems

In this paper, we introduce and characterize the notion of approximate predictability for the general class of pseudo-metric systems, which are a powerful modeling framework to deal with complex heterogeneous systems such as hybrid systems. Approximate predictability corresponds to the possibility of predicting the occurrence of specific states belonging to a particular subset of interest, in advance with respect to their occurrence, on the basis of observations. We establish a relation between approximate predictability of a given pseudo-metric system and approximate predictability of a pseudo-metric system that approximately simulates the given one. This relation allows checking approximate predictability of a system with an infinite number of states, provided that one is able to construct a pseudo-metric system with a finite number of states and inputs, approximating the original one in the sense of approximate simulation. To demonstrate the usefulness of our results, we apply them to the analysis of approximate predictability of Piecewise Affine (PWA) systems, a well studied class of hybrid systems for which, however, to the best of our knowledge, there are no results available in the current literature on predictability. An illustrative example is also included, which demonstrates the applicability of our results.