基于Radau伪谱法的非线性最优控制问题的收敛性

Convergence of nonlinear optimal control problem using Radau pseudospectral method

中文关键词  Radau伪谱法  收敛性  最优解  存在性

英文关键词  Radau PS method  convergence  optimal solution  feasibility

中文摘要

      在过去的10年里, 伪谱方法(如Legendre伪谱法、Gauss伪谱法、Radau伪谱法)逐步成为求解不同领域中非线 性最优控制问题的一种高效、灵活的数值解法. 本文从最优控制问题解的存在性、收敛性以及解的可行性3个方面 对采用Radau伪谱法求解一般非线性最优控制问题解的收敛性进行研究. 证明了原最优控制问题的离散解存在、存 在收敛到原最优控制问题解上的离散解和离散形式的收敛解是原最优控制问题的最优解. 在此基础上, 证明了 Radau伪谱法的收敛性. 本文结论与现有文献相比, 去掉了一些必要条件, 更适合一般的非线性时不变系统.

英文摘要

      In the last decade, pseudospectral (PS) methods, such as Legendre PS method, Gauss PS method and Radau PS method, have emerged as effective and flexible approaches to solve nonlinear optimal control in a variety of areas in applications. We investigate the Radau PS method in solving nonlinear optimal control problems, from the aspects of solution existence, solution convergence and solution feasibility. The investigation results show that the discrete form of the original optimal control problem has solutions, the solutions of the discrete form converge to the solutions of the original problems, and the convergent solution of the discrete form is the optimal solution of the original problem. Thus, the convergence of the Radau PS method is proved. Compared with existing results in the literature, the conclusions of this paper remove some necessary conditions, making this method more applicable to general nonlinear time-invariant systems.