Probabilistic bounded reachability for hybrid systems with continuous nondeterministic and probabilistic parameters

TL;DR

This paper presents a novel algorithm for computing the probability of reaching unsafe states in hybrid systems with nonlinear dynamics, combining probabilistic and nondeterministic parameters, using $elta$-complete decision procedures.

Contribution

It introduces a relaxation of the undecidable reachability problem enabling guaranteed probability bounds for complex hybrid systems with continuous and probabilistic parameters.

Findings

Validated the approach on nonlinear hybrid models.

Achieved guaranteed bounds matching Monte Carlo simulations.

Extended the method to systems with nondeterministic and probabilistic parameters.

Abstract

We develop an algorithm for computing bounded reachability probability for hybrid systems, i.e., the probability that the system reaches an unsafe region within a finite number of discrete transitions. In particular, we focus on hybrid systems with continuous dynamics given by solutions of nonlinear ordinary differential equations (with possibly nondeterministic initial conditions and parameters), and probabilistic behaviour given by initial parameters distributed as continuous (with possibly infinite support) and discrete random variables. Our approach is to define an appropriate relaxation of the (undecidable) reachability problem, so that it can be solved by $\delta$-complete decision procedures. In particular, for systems with continuous random parameters only, we develop a validated integration procedure which computes an arbitrarily small interval that is guaranteed to contain the reachability probability. In the more general case of systems with both nondeterministic and probabilistic parameters, our procedure computes a guaranteed enclosure for the range of reachability probabilities. We have applied our approach to a number of nonlinear hybrid models and validated the results by comparison with Monte Carlo simulation.