The Symplectic Algorithm For Optimal Control Of Constrained Dynamical Systems

Optimal control is the core of modern control theory.It mainly studies the method to optimize the objective function of the systems.Optimal control plays an important role in increasing economic efficiency,reducing energy consumption and improving production efficiency in industrial manufacturing.Optimal control theory is widely used in aerospace,complex multi-body systems,robots and vehicle manufacturing.For complex engineering problems,the state variables or control variables of the systems contain constraints,so constraints need to be introduced in the process of dynamic modeling.Therefore,the numerical algorithm research on the optimal control problem of constrained dynamic systems has attracted the attention of a large number of researchers.The traditional numerical algorithm mainly focuses on the accuracy of the solution,while the physical properties of the optimal control problem are rarely considered.Compared with traditional algorithms,the symplectic algorithm applies the theory of computational structural mechanics to the solution of optimal control problem,so that not only the accuracy and efficiency of the numerical solutions can be guaranteed,but also the inherent physical properties of the original system can be maintained.Therefore,the research on the symplectic algorithm for the optimal control problem of constrained systems is of great significance.In this paper,the symplectic algorithm for optimal control of holonomic constrained dynamical systems is studied.The main research contents include:(1)Based on dual variables,the variational principle of optimal control problem for constrained dynamic systems is established,and based on the variational principle,a high-order symplectic algorithm is constructed for the optimal control problem of constrained dynamic systems.Based on dual variables and performance index,the dynamic equations and constraints are introduced into performance index by using Lagrange multipliers,then the extended performance index is obtained and the Hamiltonian canonical equation is derived by the least action principle.Based on the generating function theory and the canonical transformation,and by using the independent variables of the costate variables at initial time and the state variables at terminal time,a high-precision symplectic algorithm is constructed to solve the optimal control problem of constrained dynamic systems.In the time domain,the state variables,costate variables and Lagrange multipliers are approximated by Lagrange polynomials,then the optimal control problem of constrained dynamic systems is transformed into solving a set of nonlinear equations.Finally,the conservation of Hamiltonian function and the symplectic property of the algorithm are proved.(2)A lot of numerical examples of fixed terminal state and free terminal state are analyzed by the symplectic algorithm constructed in this paper.The effects of different time steps and interpolation parameters on the accuracy,order and convergence of the proposed algorithm are studied.The results show that increasing interpolation parameters or reducing the time step can improve the precision of the symplectic algorithm.The relation between the order s of algorithm precision and interpolation parameter m satisfies s=m+1.The performance index converges with the decrease of time step and the increase of interpolation parameters.The algorithm has good convergence and can satisfy the holonomic constrains precisely and it is proved to preserve the symplectic structure of the systems.