Using Redundant Constraints for Refinement

This paper is concerned with a method for computing reachable sets of linear continuous systems with uncertain input. Such a method is required for verification of hybrid systems and more generally embedded systems with mixed continuous-discrete dynamics. In general, the reachable sets of such systems (except for some linear systems with special eigenstructures) are hard to compute exactly and are thus often over-approximated. The approximation accuracy is important especially when the computed over-approximations do not allow proving a property. In this paper we address the problem of refining the reachable set approximation by adding redundant constraints which allow bounding the reachable sets in some critical directions. We introduce the notion of directional distance which is appropriate for measuring approximation effectiveness with respect to verifying a safety property. We also describe an implementation of the reachability algorithm which favors the constraint-based representation over the vertex-based one and avoids expensive conversions between them. This implementation allowed us to treat systems of much higher dimensions. We finally report some experimental results showing the performance of the refinement algorithm.