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Computing Flowpipe of Nonlinear Hybrid Systems with Numerical Methods

Modern control-command systems often include controllers that perform nonlinear computations to control a physical system, which can typically be described by an hybrid automaton containing high-dimensional systems of nonlinear differential equations. To prove safety of such systems, one must compute all the reachable sets from a given initial position, which might be uncertain (its value is not precisely known). On linear hybrid systems, efficient and precise techniques exist, but they fail to handle nonlinear flows or jump conditions. In this article, we present a new tool name HySon which computes the flowpipes of both linear and nonlinear hybrid systems using guaranteed generalization of classical efficient numerical simulation methods, including with variable integration step-size. In particular, we present an algorithm for detecting discrete events based on guaranteed interpolation polynomials that turns out to be both precise and efficient. Illustrations of the techniques developed in this article are given on representative examples.