求解最优控制问题的Chebyshev-Gauss伪谱法

提出了一种求解最优控制问题的Chebyshev-Gauss伪谱法, 配点选择为Chebyshev-Gauss点. 通过比较非线性规划问题的Kaursh-Kuhn-Tucker条件和伪谱离散化的最优性条件, 导出了协态和Lagrange乘子的估计公式. 在状态逼近中, 采用了重心Lagrange插值公式, 并提出了一种简单有效的计算状态伪谱微分矩阵的方法. 该法的独特优势是具有良好的数值稳定性和计算效率. 仿真结果表明, 该法能够高精度地求解带有约束的复杂最优控制问题.

A Chebyshev-Gauss Pseudospectral Method for Solving Optimal Control Problems

A pseudospectral method is presented for direct trajectory optimization of optimal control problems using collocation at Chebyshev-Gauss points, and therefore, it is called Chebyshev-Gauss pseudospectral method. The costate and constraint multiplier estimates for the proposed method are rigorously derived by comparing the discretized optimality conditions of an optimal control problem with the Karush-Kuhn-Tucker conditions of the resulting nonlinear programming problem from collocation. The distinctive advantages of the proposed method over other pseudopsectral methods are the good numerical stability and computational efficiency. In order to achieve this goal, the barycentric Lagrange interpolation is substituted for the classic Lagrange interpolation in the state approximation. Furthermore, a simple yet efficient method is presented to alleviate the numerical errors of state differential matrix using the trigonometric identity especially when the number of Chebyshev-Gauss points is large. The method presented in this paper has been taken to two optimal control problems from the open literature, and the results have indicated its ability to obtain accurate solutions to complex constrained optimal control problems.