机械多体系统动力学非线性最优控制问题的Noether理论
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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.
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.
引文
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[14]白龙,董志峰,戈新生.基于李群的水下航体动力学建模及最优控制[J].系统仿真学报,2016,28(5):1150-1157.(BAI Long,DONG Zhifeng,GE Xinsheng.Lie group modeling and optimal control of underwater vehicle[J].Journal of System Simulation,2016,28(5):1150-1157.(in Chinese))
[15]彭海军,李飞,高强.多体系统轨迹跟踪的瞬时最优控制保辛方法[J].力学学报,2016,48(4):784-781.(PENG Haijun,LI Fei,GAO Qiang.Symplectic method for instantaneous optimal control of multibody system trajectory tracking[J].Acta Mechanica Sinica,2016,48(4):784-781.(in Chinese))
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[17]HUSSEIN I I,BLOCH A M.Optimal control of underactuated nonholonomic mechanical systems[J].IEEE Translations on Automatic Control,2008,53(3):668-682.
[18]TORRES D F M.On the Noether theorem for optimal control[J].European Journal of Control,2002,8(1):56-63.
[19]TORRES D F M.Carathdory equivalence,Noether theorems,and Tonelli full-regularity in the calculus of variations and optimal control[J].Journal of Mathematical Sciences,2004,120(1):1032-1050.
[20]FABIEN B.Analytical System Dynamics[M].Berlin:Springer,2009:46-3.
[2]YUN C,ZONG G H,ZHANG Q X.Study on active control for a flexible beam under the condition of zero gravity[J].Chinese Journal of Aeronautics,2000,13(1):51-58.
[3]GANG B S,BRIJ N A.Vibration suppression of flexible spacecraft during attitude control[J].Aeta Aetrnnautina,2001,49(2):73-83.
[4]KIM H K,CHOI S B.Compliant control of a two-link flexible manipulator featuring piezoelectric actuators[J].Mechanism and Machine Theory,2001,36(3):411-424.
[5]SUN D,MILLS J K.A PZT actuator control of a single-link flexible manipulator based on linear velocity feedback and actuator placement[J].Mechatronic,2004,14(4):381-401.
[6]徐小明,钟万勰.基于四元数表示的多体动力学系统及其保辛积分算法[J].应用数学和力学,2014,35(10):1071-1080.(XU Xiaoming,ZHONG Wanxie.Symplectic integration for multibody dynamics based on quaternion parameters[J].Applied Mathematics and Mechanics,2014,35(10):1071-1080.(in Chinese))
[7]董雪仰,戈新生.航天器太阳帆板展开过程最优控制的自适应Gauss伪谱法[J].应用数学和力学,2016,37(6):655-664.(DONG Xueyang,GE Xinsheng.The adaptive Gauss pseudospectral method for the optimal control of spacecraft solar array deployment[J].Applied Mathematics and Mechanics,2016,37(6):655-664.(in Chinese))
[8]VASQUES C M A,RODRIGUES J D.Active vibration control of smart piezoelectric beams:comparison of classical and optimal feedback control strategies[J].Computers and Structures,2006,84(22/23):1402-1414.
[9]CAI G P,LIM C W.Active control of a flexible hub-beam system using optimal tracking control method[J].International Journal Mechanical Sciences,2006,48(10):1150-1162.
[10]KOBILAROV M.Discrete geometric motion control of autonomous vehicles[D].PhD Thesis.Los Angeles:University of Southern California,2008:12-18.
[11]张耀欣,丛爽.平面二自由度冗余驱动并联机构的最优运动控制及其仿真[J].系统仿真学报,2005,17(10):2450-2454.(ZHANG Yaoxin,CONG Shuang.Optimal motion control and simulation of redundantly actuated 2-Dof planar parallel manipulator[J].Journal of System Simulation,2005,17(10):2450-2454.(in Chinese))
[12]戈新生,陈立群,刘延柱.一类多体系统的非完整运动规划最优控制[J].工程力学,2006,23(3):63-68.(GE Xinsheng,CHEN Liqun,LIU Yanzhu.Optimal control of a nonholonomic motion planning for mutilbody systems[J].Engineering Mechanics,2006,23(3):63-68.(in Chinese))
[13]LEYENDECKER S,OBER-BLOBAUM S,MARSDEN J E,et al.Discrete mechanics and optimal control for constrained systems[J].Optimal Control Applications and Methods,2010,31(6):505-528.
[14]白龙,董志峰,戈新生.基于李群的水下航体动力学建模及最优控制[J].系统仿真学报,2016,28(5):1150-1157.(BAI Long,DONG Zhifeng,GE Xinsheng.Lie group modeling and optimal control of underwater vehicle[J].Journal of System Simulation,2016,28(5):1150-1157.(in Chinese))
[15]彭海军,李飞,高强.多体系统轨迹跟踪的瞬时最优控制保辛方法[J].力学学报,2016,48(4):784-781.(PENG Haijun,LI Fei,GAO Qiang.Symplectic method for instantaneous optimal control of multibody system trajectory tracking[J].Acta Mechanica Sinica,2016,48(4):784-781.(in Chinese))
[16]FREDERICO G S F,TORRES D F M.Fractional conservation laws in optimal control theory[J].Nonlinear Dynamics,2008,53(3):215-222.
[17]HUSSEIN I I,BLOCH A M.Optimal control of underactuated nonholonomic mechanical systems[J].IEEE Translations on Automatic Control,2008,53(3):668-682.
[18]TORRES D F M.On the Noether theorem for optimal control[J].European Journal of Control,2002,8(1):56-63.
[19]TORRES D F M.Carathdory equivalence,Noether theorems,and Tonelli full-regularity in the calculus of variations and optimal control[J].Journal of Mathematical Sciences,2004,120(1):1032-1050.
[20]FABIEN B.Analytical System Dynamics[M].Berlin:Springer,2009:46-3.