Scalable techniques for the computation of viable and reachable sets : safety guarantees for high-dimensional linear time-invariant systems

Reachability analysis and viability theory are key in providing guarantees of safety and proving the existence of safety-preserving controllers for constrained dynamical systems. The minimal reachable tube and (by duality) the viability kernel are the only constructs that can be used for this purpose. Unfortunately, current numerical schemes that compute these constructs suffer from a complexity that is exponential in the dimension of the state, rendering them impractical for systems of dimension greater than three or four. In this thesis we propose two separate approaches that improve the scalability of the computation of the minimal reachable tube and the viability kernel for high-dimensional systems. The first approach is based on structure decomposition and aims to facilitate the use of computationally intensive yet versatile and powerful tools for higher-dimensional linear time-invariant (LTI) systems. Within the structure decomposition framework we present two techniques—Schur-based and Riccati-based decompositions—that impose an appropriate structure on the system which is then exploited for the computation of our desired constructs in lower-dimensional subspaces. The second approach is based on set-theoretic methods and draws a new connection between the viability kernel and maximal reachable sets. Existing tools that compute the maximal reachable sets are efficient and scalable with polynomial complexity in time and space. As such, these scalable techniques can now be used to compute our desired constructs and therefore provide guarantees of safety for high-dimensional systems. Based on this new connection between the viability kernel and maximal reachable sets we propose a scalable algorithm using ellipsoidal techniques for reachability. We show that this algorithm can efficiently compute a conservative under-