Evolution Function Based Reach-Avoid Verification for Time-varying Systems with 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 with respect to the uncertain parameter. Afterward, based on the partial sums of this series, over- and under-approximations of the 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. We implement the algorithms and use some benchmarks with comparisons to show that our methodology is both efficient and promising. Finally, we additionally extend our methodology to deal with systems with complex initial sets and time-dependent switchings. The performance of our extended method for these systems is also shown by four examples with comparisons and discussions.