Safety preserving control synthesis for sampled data systems

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 this paper we describe an abstract algorithm for approximating the discriminating kernel (also known as the maximal robust control invariant set) for a sampled data system with continuous state space, and then use this operator to construct a switched, set-valued feedback control policy which ensures safety. We show that the approximation is conservative for sampled data systems. We then demonstrate that the key operations–the tensor products of two sets, invariance kernels, and a pair of projections–can be implemented in two formulations: one based on the Hamilton–Jacobi partial differential equation which can handle nonlinear dynamics but which scales poorly with state space dimension, and one based on ellipsoids which scales well with state space dimension but which is restricted to linear dynamics. Each version of the algorithm is demonstrated numerically on a simple example.