机械多体系统动力学非线性最优控制问题的Noether理论  

The Noether Theorem for Nonlinear Optimal Control Problems of Mechanical Multibody System Dynamics

作　　者：郑明亮[1] ZHENG Mingliang(School of Mechanical Engineering and Automation,Zhejiang Sci-Tech University,Hangzhou 310018,P.R.China)

机构地区：[1]浙江理工大学机械与自动控制学院,杭州310018

出　　处：《应用数学和力学》2018年第7期776-784,共9页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11472247)~~

摘　　要：基于群不变性原理求解了机械多体动力学系统非线性最优控制问题的Noether型守恒定律.该文主要研究一类理想完整约束下的受控机械多刚体系统,通过增广向量法将动力学Euler-Lagrange方程以状态空间形式表示,利用变分法得到最优控制问题最优解的状态方程、伴随方程和控制方程,对系统性能指标泛函进行包含时间、状态变量、协态变量和控制变量的Noether对称无限小变换,进而得到最优解方程组的守恒量,使最优解关系以一组代数方程形式表达,为最优解的积分方法以及各种数值算法都奠定了坚实基础.最后,以基础振动下机械臂非线性动力学的能量最优控制实例分析,说明了该文对称性方法的正确性.A Noether-type conservation law for the nonlinear optimal control problems of mechanical multibody system dynamics was proposed based on the group invariance principle. The controlled mechanical multi-rigid-body systems under ideal holonomic constraints were studied,and the dynamic Euler-Lagrange equations were expressed in the form of the state space with the augmented vector method. The state equations,adjoint equations and governing equations for the optimal solution to the optimal control problem were obtained with the variational method. The Noether symmetric infinitesimal transformation with time,state variables,covariate variables and control variables was applied to the system performance index functional,then the conservation laws of the optimal solution equations were obtained,and the optimal solution relation was expressed in the form of a set of algebraic equations,which lays a solid foundation for the integral method and various numerical algorithms of the optimal solution. Finally,an example about the optimal energy control of the nonlinear dynamics of the mechanical arm under the basic vibration was given to illustrate the correctness of the proposed symmetry method.

关 键 词：多体系统 最优控制 NOETHER对称性 守恒律 机械臂动力学 

分 类 号：TH122[机械工程—机械设计及理论] O316[理学—一般力学与力学基础]