最优控制问题的Legendre 伪谱法求解及其应用

伪谱法通过全局插值多项式参数化状态和控制变量，将最优控制问题(OCP)转化为非线性规划问题(NLP)进行求解，是一类具有更高求解效率的直接法。总结Legendre伪谱法转化Bolza型最优控制问题的基本框架，推导OCP伴随变量与NLP问题KKT乘子的映射关系，建立基于拟牛顿法的LGL配点数值计算方法，并针对非光滑系统，进一步研究分段伪谱逼近策略。基于上述理论开发通用OCP求解器，并对3个典型最优控制问题进行求解，结果表明了所提出方法和求解器的有效性。

Theory and application of Legendre pseudo-spectral method for solving optimal control problem

The pseudo-spectral method approximates control and state variables through global interpolation polynomials, then discrete the optimal control problem(OCP) to a nonlinear programming problem(NLP) effectively. It’s a kind of direct method with higher solving efficiency. The basic framework of the Legendre pseudo-spectral method converting the Bolza OCP into NLP is summarized, and the mapping between the costates of OCP and the KKT multiplier to NLP is derived. Furthermore, a numerical method is elaborated based on the quasi-Newton method in order to calculate the LGL collocation accurately. The multiphase strategy is also being introduced for non-smooth systems. Finally, a universal optimal control solver POPS(pseudo-spectral optimal control problem solver) is developed based on the Legendre pseudo-spectral method in Matlab. Three typical optimal control problems are solved by using the solver POPS, and the results show the effectiveness of the proposed method and solver POPS.