Approximate Abstractions of Stochastic Hybrid Systems

We present a constructive procedure for obtaining a finite approximate abstraction of a discrete-time stochastic hybrid system. The procedure consists of a partition of the state space of the system and depends on a controllable parameter. Given proper continuity assumptions on the model, the approximation errors introduced by the abstraction procedure are explicitly computed and it is shown that they can be tuned through the parameter of the partition. The abstraction is interpreted as a Markov set-Chain. We show that the enforcement of certain ergodic properties on the stochastic hybrid model implies the existence of a finite abstraction with finite error in time over the concrete model, and allows introducing a finite-time algorithm that computes the abstraction.