计算最优控制辛数值方法

Computational symplectic numerical methods for optimal control

针对最优控制问题(OCP)的辛数值方法研究及应用进行综述。主要涉及内容包括,动力学系统为常微分方程描述的一般无约束、含不等式约束和状态时滞的最优控制问题,微分代数方程描述的一般无约束、含不等式约束和含切换系统的最优控制问题,以及闭环最优控制问题。从间接法和直接法两个求解框架出发,重点介绍本课题组在保辛算法方面的研究工作。在间接法框架下,首先基于生成函数和变分原理,将OCP保辛离散为非线性方程组,再数值求解方程组。在直接法框架下,将OCP保辛离散为有限维的非线性规划问题(NLP),再数值求解。针对闭环最优控制问题,提出了保辛模型预测控制、滚动时域估计和瞬时最优控制算法。研究表明,保辛算法具有高精度和高效率的特点,在航空航天和机器人等领域有着广泛应用前景和价值。

This review focuses on symplectic numerical methods for different types of nonlinear optimal control problems(OCP).It covers:OCPs where dynamic systems are described by ordinary differential equations(ODE)with unconstrained,inequality constraints and time delays,OCPs where dynamic systems are described by differential algebraic equations(DAE)with unconstrained,inequality constraints and switching systems,together with the closed-loop optimal control problems.Symplectic algorithms can be constructed in both the direct and indirect frameworks.In indirect methods,OCPs are transformed into nonlinear equations by generating functions and the variational principle.In direct methods,dynamic systems are discretized in a symplectic manner,then the OCPs are transformed into nonlinear programming(NLP)problems.For closed-loop OCPs,symplectic model predictive control,rolling horizon estimation,and instantaneous optimal control algorithms are introduced.The results reveal that symplectic algorithms have high precision and efficiency,which find applications in aeronautical and aerospace engineering,robotics and other fields.