Synthesis of Safe Controllers for Nonlinear Systems Using Dynamic Programming Techniques

A dynamic programming based procedure for the synthesis of near-optimal controllers for sampled data systems is proposed. The main features of the proposed procedure are the approximation of nonlinear systems by piecewise affine systems, the synthesis of control strategy in the form of a piecewise constant map, and handling of state constraints and exogenous inputs. In order to increase accuracy, the dynamic programming equations are evaluated for an horizon consisting of several sampling periods instead of the typical single time step approach. The dynamic programming equations are decomposed into set-valued operations and approximations, such that the resulting closed loop system is ensured to be stable and safe, even though the control strategy is computed by numerical methods. Based on that decomposition, two different implementations, of varying complexity and accuracy, are compared. Results are illustrated with the solution of a pursuit-evasion problem.