非线性轨迹优化问题的保辛自适应求解方法

Symplectic adaptive algorithm for solving nonlinear trajectory optimization problem

非线性轨迹优化问题一般是一个非线性最优控制问题。将非线性系统的最优控制问题导入到哈密顿体系的辛几何空间当中,基于对偶变量变分原理提出了求解非线性最优控制问题的一种保辛自适应方法。以时间区段两端协态作为独立变量,在时间区段内采用拉格朗日插值近似状态和协态变量,并利用对偶变量变分原理将非线性最优控制问题转化为非线性方程组的求解,保持了哈密顿系统的辛几何结构。并进一步,提出了基于多层次迭代的自适应算法,提高了非线性最优控制问题的求解效率。数值实验验证了该算法在求解非线性轨迹优化问题中的有效性。

In general,nonlinear trajectory optimization problem is also a nonlinear optimal control problem.In this paper,the nonlinear optimal control problem is transformed into the symplectic space of Hamiltionian system and a symplectic adaptive algorithm based on dual variable principle is proposed for solving nonlinear optimal control problem.Costate variables at two ends of time interval are taken as independent variables and the state and costate variables inside the time interval are approximated by Lagrange interpolation.Then,nonlinear optimal control problems are replaced by nonlinear equations through dual variable principle and the symplectic algorithm for solving nonlinear optimal control problems can be obtained at the same time.The computation efficiency of the proposed symplectic algorithm is improved by using adaptive idea,and numerical simulation shows the validity of the proposed algorithm for solving nonlinear trajectory optimization problems.