最优控制问题的Legendre伪谱法求解及其应用  

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

作　　者：徐少兵[1] 李升波[1] 成波[1] 

机构地区：[1]清华大学汽车安全与节能国家重点实验室,北京100084

出　　处：《控制与决策》2014年第12期2113-2120,共8页Control and Decision

基　　金：国家自然科学基金项目(51205228);清华大学自主科研计划项目(2012THZ0)

摘　　要：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.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.

关 键 词：最优控制 伪谱法 非线性规划 数值求解 

分 类 号：TP273.5[自动化与计算机技术—检测技术与自动化装置]