Symplectic Multi-Level Method for Solving Nonlinear Optimal Control Problem
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摘要: 将非线性系统的最优控制问题导向Hamilton系统,提出了求解非线性最优控制问题的保辛多层次方法.首先,以时间区段两端状态为独立变量并在区段内采用Lagrange插值近似状态和协态变量,通过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.然后,在保辛算法的具体实施过程中提出了多层次求解思想,以2N类算法为基础由低层次到高层次加密离散时间区段,利用Lagrange插值得到网格加密后的初始状态与协态变量作为求解非线性方程组的初值,可提高计算效率.数值算例验证了算法在求解效率与求解精度上的有效性.Abstract: The optimal control problem for nonlinear system was transformed into Hamiltonian system and a symplectic-preserving method was proposed.The state and costate variables were approximated by Lagrange polynomial and state variables at two ends of the time interval were taken as the independent variables, and then based on the dual variable principle, nonlinear optimal control problems were replaced by nonlinear equations.In the implement of symplectic algorithm, based on the 2N algorithm, a multi-level method was proposed.When the time grid was refined from the low level to the high level, the initial state variables and costate variables of nonlinear equations could be obtained from Lagrange interpolation at the low level grid, which could improve the efficiency.Numerical simulations show the precision and efficiency of the proposed algorithm.
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[1] Sage A P, White C C. Optimum Systems Control[M]. New Jersey: Prentice-Hall, 1977. [2] Bryson A E, Ho Y C. Applied Optimal Control[M]. New York: Hemisphere Publishing Corporation, 1975. [3] 钟万勰, 吴志刚, 谭述君. 状态空间控制理论与计算[M]. 北京:科学出版社,2007. [4] Schley C H, Lee I. Optimal control computation by the Newton-Raphson method and the Riccati transformation[J]. IEEE Transactions on Automatic Control, 1967, 12(2):139-144. doi: 10.1109/TAC.1967.1098542 [5] 谭述君,钟万勰. 非线性最优控制系统的保辛摄动近似求解[J]. 自动化学报, 2007, 33(9): 1004-1008. [6] Beeler S C, Tran H T, Banks H T. Feedback control methodologies for nonlinear systems[J]. Journal of Optimization Theory and Applications, 2000, 107(1):1-33. doi: 10.1023/A:1004607114958 [7] Nedeljkovic N. New algorithms for unconstrained nonlinear optimal control problems[J]. IEEE Transactions on Automatic Control, 1981, 26(4): 868-884. doi: 10.1109/TAC.1981.1102732 [8] Benson D A, Huntington G T, Thorvaldsen T P, Rao A V. Direct trajectory optimization and costate estimation via an orthogonal collocation method[J]. Journal of Guidance Control and Dynamics, 2006, 29(6): 1435-1439. doi: 10.2514/1.20478 [9] Badakhshan K P, Kamyad A V. Numerical solution of nonlinear optimal control problems using nonlinear programming[J]. Applied Mathematics and Computation, 2007, 187(2): 1511-1519. doi: 10.1016/j.amc.2006.09.074 [10] Arnold V I. Mathematical Methods of Classical Mechanics[M]. New York: Springer-Verlag, 1989. [11] 高强, 谭述君, 张洪武,钟万勰. 基于对偶变量变分原理和两端动量独立变量的保辛方法[J]. 动力学与控制学报, 2009, 7(2): 97-103. [12] Lew A, Marsden J E, Ortiz M, West M. Variational time integrators[J]. International Journal for Numerical Methods in Engineering, 2004, 60: 153-212. doi: 10.1002/nme.958 [13] de Leon M, de Diego D Martin, Santamaria-Merino A. Discrete variational integrators and optimal control theory[J]. Advances in Computational Mathematics, 2007, 26(1/3): 251-268. doi: 10.1007/s10444-004-4093-5 [14] Srinivas R, Vadali R S. Optimal finite-time feedback controllers for nonlinear systems with terminal constraints[J]. Journal of Guidance Control and Dynamics, 2006, 29(4): 921-928. doi: 10.2514/1.16790
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