Ensuring Safety of Nonlinear Sampled Data Systems through Reachability1

In sampled data systems the controller receives periodically sampled state feedback about the evolution of a continuous time plant, and must choose a constant control signal to apply between these updates; however, unlike purely discrete time models the evolution of the plant between updates is important. In contrast, for systems with nonlinear dynamics existing reachability algorithms–-based on Hamilton-Jacobi equations or viability theory–-assume continuous time state feedback and the ability to instantaneously adjust the input signal. In this paper we describe an algorithm for determining an implicit surface representation of minimal backwards reach tubes for nonlinear sampled data systems, and then construct switched, set-valued feedback controllers which are permissive but ensure safety for such systems. The reachability algorithm is adapted from the Hamilton-Jacobi formulation proposed in Ding and Tomlin [2010]. We show that this formulation is conservative for sampled data systems. We implement the algorithm using approximation schemes from level set methods, and demonstrate it on a modified double integrator.