Fully Automated Verification of Linear Systems Using Inner and Outer Approximations of Reachable Sets

Reachability analysis is a formal method to guarantee safety of dynamical systems under the influence of uncertainties. A substantial bottleneck of all reachability algorithms is the necessity to adequately tune specific algorithm parameters, such as the time step size, which requires expert knowledge. In this work, we solve this issue with a fully automated reachability algorithm that tunes all algorithm parameters internally such that the reachable set enclosure respects a user-defined approximation error bound in terms of the Hausdorff distance to the exact reachable set. Moreover, this bound can be used to extract an inner approximation of the reachable set from the outer approximation using the Minkowski difference. Finally, we propose a novel verification algorithm that automatically refines the accuracy of the outer and inner approximations until specifications given by time-varying safe and unsafe sets can be verified or falsified. The numerical evaluation demonstrates that our verification algorithm successfully verifies or falsifies benchmarks from different domains without requiring manual tuning.