On the Time-Discretization of Control Systems
On the Time-Discretization of Control Systems
复制标题
控制系统的时间离散化
DOI：
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发表时间：
1997
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通讯作者：
V. Veliov
中科院分区：
文献类型：
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作者：
V. Veliov
本文提出了一种非线性（仿射）控制系统的离散逼近方法，该方法在离散化步长h下具有高于一阶精度的逼近精度。该方法由两部分组成：首先用一个适当的有限维子集${cal U}_N$代替可测容许控制集${cal U}$，然后用单步离散方法离散来自${cal U}_N$的控制（在合理意义上是“正则的”）所对应的微分方程.主要结果估计的准确性，在第一部分，衡量的规定收集的性能指标。结果可以解释的上下文中的最优控制问题的近似和可达集的近似。在第一种情况下，证明了适当的Runge-Kutta型离散化方法的精度O（h2），而不显式或隐式地要求最优解的任何规律性。在凸可达集的情况下，我们得到O（h2）的逼近的Hausdorff距离和O（h1.5）的精度在非凸的情况下。应用程序的离散时间控制系统的时间聚集。
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.