Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method
Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method
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DOI：
10.1007/s10589-009-9291-0
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发表时间：
2011-06-01
影响因子：
2.2
通讯作者：
Rao, Anil V.
中科院分区：
文献类型：
--
作者：
Garg, Divya;Patterson, Michael A.;Rao, Anil V.
提出了一种使用勒让德-高斯-拉道 (LGR) 点全局配置的有限范围和无限范围最优控制问题的直接轨迹优化和共状态估计方法。该方法的一个关键特征是它提供了一种将非线性规划问题的 KKT 乘子映射到最优控制问题的共性的准确方法。更准确地说，表明离散格式的对偶乘法器对应于使用比状态方程小一级多项式的伴随方程的伪谱近似。建立了状态方程和伴随方程的伪谱格式的系数之间的关系。此外，还表明伪谱 LGR 微分矩阵的逆正是与隐式 LGR 积分方案相关的矩阵。因此，本文提出的方法可以被认为是全局隐式积分方法或伪谱方法。数值结果表明，使用本文所述的 LGR 搭配可以为有限和无限范围最优控制问题确定准确的原始解和对偶解。
A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.