最优控制问题的Legendre 伪谱法求解及其应用
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清华大学汽车安全与节能国家重点实验室,北京100084.

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徐少兵

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TP273.5

基金项目:

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


Theory and application of Legendre pseudo-spectral method for solving optimal control problem
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State Key Laboratory of Automotive Safety and Energy,Tsinghua University,Beijing 100084,China.

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    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.

    Abstract:

    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.

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徐少兵 李升波 成波.最优控制问题的Legendre 伪谱法求解及其应用[J].控制与决策,2014,29(12):2113-2120

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  • 收稿日期:2013-10-18
  • 最后修改日期:2014-01-16
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  • 在线发布日期: 2014-12-20
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