Pseudospectral methods for solving infinite-horizon optimal control problems
Pseudospectral methods for solving infinite-horizon optimal control problems
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DOI：
10.1016/j.automatica.2011.01.085
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发表时间：
2011-04-01
期刊：
影响因子：
6.4
通讯作者：
Rao, Anil V.
中科院分区：
文献类型：
--
作者：
Garg, Divya;Hager, William W.;Rao, Anil V.
数值逼近无限时域最优控制问题解的一个重要方面是时域的处理方式。通常，无限时域最优控制问题近似为有限时域问题。在这种情况下，不管近似的有限持续时间，最终时间位于距离t = +无穷大的实际视界的无限持续时间。本文描述了两种新的直接拟谱方法，分别用Legendre-Gauss（LG）和Legendre-Gauss-Radau（LGR）配点法数值求解无限时域最优控制问题。一个光滑的，严格单调的变换被用来映射的无限时域t是一个元素[0，无穷）到一个半开区间τ是一个元素[-1，1）。在有限区间上，利用配置法将问题转化为非线性规划问题。所提出的方法产生近似的状态和costate在整个地平线上，包括近似在t = +无穷大。这些伪谱方法可以等价地写成微分或隐积分形式。在数值实验中，离散解表现出指数收敛作为配置点的数量的函数。结果表明，映射φ：[-1，+1）--> [0，+infinity）可以被调节以提高离散近似的质量. (C)2011爱思唯尔有限公司保留所有权利。
An important aspect of numerically approximating the solution of an infinite-horizon optimal control problem is the manner in which the horizon is treated. Generally, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. In such cases, regardless of the finite duration of the approximation, the final time lies an infinite duration from the actual horizon at t = +infinity. In this paper we describe two new direct pseudospectral methods using Legendre-Gauss (LG) and Legendre-Gauss-Radau (LGR) collocation for solving infinite-horizon optimal control problems numerically. A smooth, strictly monotonic transformation is used to map the infinite time domain t is an element of [0, infinity) onto a half-open interval tau is an element of [-1, 1). The resulting problem on the finite interval is transcribed to a nonlinear programming problem using collocation. The proposed methods yield approximations to the state and the costate on the entire horizon, including approximations at t = +infinity. These pseudospectral methods can be written equivalently in either a differential or an implicit integral form. In numerical experiments, the discrete solution exhibits exponential convergence as a function of the number of collocation points. It is shown that the map phi : [-1, +1) --> [0, +infinity) can be tuned to improve the quality of the discrete approximation. (C) 2011 Elsevier Ltd. All rights reserved.