求解含复杂约束非线性最优控制问题的改进Gauss伪谱法

针对含有复杂约束条件的非线性最优控制问题,提出了一种改进的Gauss伪谱法 (Improved Gauss pseudospectral method, IGPM). 这类问题难以得到解析解,特别是有些问题不存在解析的模型, 一些参数只能通过查表得到,使得传统方法难以求解. 在传统的Gauss伪谱法的基础上,将非线性的终端状态积分约束等价地转化为线性形式,提出了IGPM, 通过协态映射定理可以计算出协态变量,检验最优性,使得IGPM具有间接法一样的精度. 并且给出了初始时刻协态变量和端点时刻控制变量的计算方法. 为了提高解的精度,基于IGPM提出了迭代算法, 最后将该算法应用于求解高超声速飞行器上升段轨迹优化问题,结果表明最优轨迹基本满足路径约束条件和最优性条件.

Improved Gauss Pseudospectral Method for Solving Nonlinear Optimal Control Problem with Complex Constraints

An improved Gauss pseudospectral method (IGPM) is proposed to solve the nonlinear optimal control problem with complex constraints, which is difficult to get the analytical solution. Traditional methods can hardly solve these problems by getting some parameter values from the look-up table. Based on the traditional Gauss pseudospectral method, the IGPM is put forward by transforming the nonlinear final state integral constraints into linear type. The costate obtained by the costate mapping theorem is used to check the optimality, which makes the IGPM have the same accuracy with the indirect method. Besides, the computation methods of the initial costate and the boundaries controls are proposed. In order to increase the accuracy, an iterative algorithm is given based on the IGPM. Finally, the ascent trajectory optimization problem of the hypersonic vehicle is solved by the proposed algorithm. The results show that the optimal trajectory almost satisfies the path constraints and the optimality.