On the Time-Discretization of Control Systems

• Authors: Vladimir Veliov
• Venue: SIAM Journal on Control and Optimization
• Year: 1997

This paper develops an approach for obtaining discrete approximations to nonlinear (affine) control systems that are of higher than first order of accuracy with respect to the discretization step h. The approach consists of two parts: first the set ${\cal U}$ of measurable admissible controls is replaced by an appropriate finite-dimensional subset ${\cal U}_N$; then the differential equations corresponding to controls from ${\cal U}_N$ (which are in a reasonable sense "regular") are discretized by single step discretization methods. The main result estimates the accuracy in the first part, measured in terms of a prescribed collection of performance indexes. The result can be interpreted both in the context of approximation of optimal control problems and in the context of approximation of the reachable set. In the first case, accuracy O(h2) is proven for appropriate Runge--Kutta-type discretization methods, without explicitly or implicitly requiring any regularity of the optimal solutions. In the case of a convex reachable set we obtain O(h2) approximation with respect to the Hausdorff distance and O(h1.5) accuracy in the nonconvex case. An application to the time-aggregation of discrete-time control systems is also presented.