High-Order Approximations to Nonholonomic Affine Control Systems
This paper contributes to the theory of approximations of continuous-time control/uncertain systems by discrete-time ones. Discrete approximations of higher than first order accuracy are known for affine 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 difficulties appear in the non-commutative case.
Unable to display preview. Download preview PDF.
- 2.Brockett, R.W.: Asymptotic stability and feedback stabilization. Differential geometric control theory. In: Proc. Conf., Mich. Technol. Univ., Prog. Math., vol. 27, pp. 181–191 (1982)Google Scholar
- 6.Kawski, M., Sussmann, H.J.: Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. In: Helmke, U., Prätzel-Wolters, D., Zerz, E. (eds.) Operators, Systems, and Linear Algebra, pp. 111–128. Teubner (1997)Google Scholar
- 8.Kloeden, E., Platen, E.: Numerical Solutions to Stochastic Differential Equations. Springer, Heidelberg (1992) (third revised printing, 1999)Google Scholar
- 10.Sussmann, H.: A product expansion of the Chen series. In: Byrnes, C.I., Lindquist, A. (eds.) Theory and Applications of Nonlinear Control Systems, pp. 323–335. Elsevier, North-Holland, Amsterdam (1986)Google Scholar
- 13.Veliov, V.M.: Best Approximations of Control/Uncertain Differential Systems by Means of Discrete-Time Systems. WP–91–45, International Institute for Applied Systems Analysis, Laxenburg, Austria (1991)Google Scholar