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

ZHENG Mingliang

doi:10.21656/1000-0887.380295

Volume 39 Issue 7

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ZHENG Mingliang. The Noether Theorem for Nonlinear Optimal Control Problems of Mechanical Multibody System Dynamics[J]. Applied Mathematics and Mechanics, 2018, 39(7): 776-784. doi: 10.21656/1000-0887.380295

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

doi: 10.21656/1000-0887.380295

ZHENG Mingliang

School of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, P.R.China

Funds: The National Natural Science Foundation of China（11472247）

Received Date: 2017-11-23
Rev Recd Date: 2018-01-07
Publish Date: 2018-07-15

Abstract

Abstract

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.
- multibody system,
- optimal control,
- Noether symmetry,
- conservation law,
- mechanical arm dynamics

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References

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