Approximate Safety Verification and Control of Partially Observable Stochastic Hybrid Systems

Abstract

Assuring safety in stochastic hybrid systems is particularly difficult when only noisy or partial observations of the state are available. We first review a formulation of the probabilistic safety problem under partial hybrid observations as a dynamic program over an equivalent information state. Two methods for approximately solving the dynamic program are presented. The first approximates the hybrid system as a finite state Markov decision process, so that the information state is a probability mass function. The second method approximates an indicator function over the safe region using radial basis functions, to represent the information state as a Gaussian mixture. In both cases we discretize the hybrid observation process, then use point-based value iteration to under-approximate the safety probability and synthesize a safety-preserving control policy. We obtain error bounds and convergence results in both cases, assuming switched affine dynamics and additive Gaussian noise on the continuous states and observations. We compare the performance of the finite state and Gaussian mixture approaches on a simple numerical example.