High-order approximations of nonholonomic affine control systems

Abstract

This paper contributes to the theory of approximations of continuous-time control/uncertain systems by discrete-time ones. Dis- crete approximations of higher than ¯rst order accuracy are known for a±ne control systems only in the case of commutative controlled vector fields. The novelty in this paper is that constructive second order discrete approximations are obtained in the case of two non-commutative vector fields. An explicit parameterization of the reachable set of the Brockett non-holonomic integrator is a key auxiliary tool. The approach is not limited to the present deterministic framework and may be extended to stochastic differential equations, where similar di±culties appear in the non-commutative case.