In this paper we introduce and characterize the notion of approximate predictability for the general class of 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 corrupted by measurement errors. We establish a relation between approximate predictability of a given metric system and approximate predictability of a 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 metric system with a finite number of states and inputs, approximating the original one in the sense of approximate simulation. The analysis of approximate predictability of Piecewise Affine (PWA) systems is carried out as an application of the proposed approach.
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