Reach-Avoid Verification for Time-varying Systems with Uncertain Disturbances

Abstract

In this work, we investigate the reach-avoid problem of a class of time-varying analytic systems with disturbances described by uncertain parameters. Firstly, by proposing the concepts of maximal and minimal reachable sets, we connect the avoidability and reachability with maximal and minimal reachable sets respectively. Then, for a given disturbance parameter, we introduce the evolution function for exactly describing the reachable set, and find a series representation of this evolution function with its Lie derivatives, which can also be regarded as a series function w.r.t. the uncertain parameter. Afterward, based on the partial sums of this series, over- and under-approximations of evolution function are constructed, which can be realized by interval arithmetics with designated precision. Further, we propose sufficient conditions for avoidability and reachability and design a numerical quantifier elimination based algorithm to verify these conditions; moreover, we improve the algorithm with a time-splitting technique. Finally, we implement the algorithm and use some benchmarks with comparisons to show that our methodology is both efficient and promising.