Reachability Analysis of Nonlinear Systems Using Matrix Measures

Matrix measures, also known as logarithmic norms, have historically been used to provide bounds on the divergence of trajectories of a system of ordinary differential equations. In this technical note we use them to compute guaranteed overapproximations of reachable sets for nonlinear continuous-time systems using numerically simulated trajectories and to bound the accumulation of numerical simulation errors along simulation traces. Our method employs a user-supplied bound on the matrix measure of the system's Jacobian matrix to compute bounds on the behavior of nearby trajectories, leading to efficient computation of reachable sets when such bounds are available. We demonstrate that the proposed technique scales well to systems with a large number of states.