Trajectory-based reachability analysis of switched nonlinear systems using matrix measures

Matrix measures, or logarithmic norms, have historically been used to provide bounds on the divergence of trajectories of a system of ordinary differential equations (ODEs). In this paper we use them to compute guaranteed overapproximations of reachable sets for switched nonlinear dynamical systems using numerically simulated trajectories, and to bound the accumulation of numerical errors along simulation traces. To improve the tightness of the computed approximations, we connect these classical tools for ODE analysis with modern techniques for optimization and demonstrate that minimizing the volume of the computed reachable set enclosure can be formulated as a convex problem. Using a benchmark problem for the verification of hybrid systems, we show that this technique enables the efficient computation of reachable sets for systems with over 100 continuous state variables.