Lattice point sets for state sampling in approximate dynamic programming

This paper investigates the use of lattice point sets as an efficient method to sample uniformly the state space of discrete-time dynamic systems for the solution of finite-horizon optimal control problems using approximate dynamic programming. Lattice point sets are a kind of discretization method, commonly employed for efficient numerical integration, providing a regular and balanced sampling of the state space based on the repetition of elementary unit cells. A convergence analysis of the approximate solution of the control problem to the optimal one is provided, pointing out that such sampling schemes allow one to efficiently exploit possible regularities of the cost-to-go functions. Furthermore, it is shown that a higher accuracy may be obtained through suitable transformations of the state vector of the dynamic system. Another advantage of lattice point sets over other sampling schemes is the possibility of evaluating a priori the goodness of a given set over another through the explicit computation of a specific parameter. Simulation results concerning the opti- mal control of a water reservoirs system are presented to show the effectiveness of the proposed approach.