Stochastic hybrid systems involve the interaction of continuous discrete and probabilistic dynamics, and thus pose considerable conceptual, theoretical, and practical challenges. In this chapter an overview of the modeling issues that arise in the study of stochastic hybrid systems is presented. Based on this discussion, a study of the problem of reachability analysis for stochastic hybrid systems is presented.

Nondeterminism in hybrid systems

Deterministic and nondeterministic models

Much of the work on hybrid systems has focussed on deterministic models that completely characterize the future of the system without allowing any uncertainty. In practice, it is often desirable to introduce uncertainty in the models, to allow, for example, under-modeling of certain parts of the system, external unmodeled disturbances, etc. To address this need, researchers in discrete-event and hybrid systems have introduced what are known as nondeterministic models. Here the evolution is defined in a declarative way (the system specifies what solutions are allowed) as opposed to the imperative way more common in continuous dynamical/control systems (the system specifies what the solution must be).

Nondeterministic hybrid systems allow uncertainty to enter in a number of places: choice of continuous evolution (modeled, for example, by a differential inclusion), choice of discrete transition destination, or choice between continuous evolution and a discrete transition. “Choice” in this setting may reflect disturbances that add uncertainty about the system evolution, but also control inputs that can be used to steer the system evolution.