Research On Numerical Solution Of Several Classes Of Nonlinear Optimal Control Problems

Optimal control theory is an important part of modern control theory.It is a discipline that researches on finding the optimal solution from all possible control solutions.It is widely used in aerospace,robotic,precision motion control and other fields.At present,the linear optimal control problem has been well solved,but many high dimensional nonlinear optimal control problems still can not be solved.In order to obtain the approximate optimal solution to the nonlinear optimal control problem,this paper researches the numerical solution to the optimal control problem from the following aspects.A data-driven neural dynamic programming method is proposed for the model-free nonlinear optimal control problem.This method firstly uses two sets of base functions to approximate the Q-function and the control strategy respectively.The residual of the Q-function and the control strategy are operated to be zero with its basis function through the inner product,and the calculation formula of the approximation coefficients are obtained.Then the offline data set and online data are substituted into the calculation formula to update the coefficients.Finally,the required control strategy is obtained and the convergence of this algorithm is proved.In order to solve the optimal control problem of inequality path constraints with control variables and state variables,a new method of quadratic penalty function based on smoothing function is proposed.This method defines a new smoothing method.The inequality path constraint is processed by adjusting the penalty factor.It is proved that under certain conditions the transformation between the optimal control problems is equivalent.A new Galerkin approximation method based on Chebyshev polynomial is proposed for the high dimensional nonlinear optimal control systems.This method uses Chebyshev polynomial as the basis function and it is substituted in the generalized HJB(Hamilton-Jacobi-Bellman)equation.Using that this error is operated to be zero with the basis function through the inner product.Then the update formula is obtained.Finally,the numerical exampleproves the validity of this method.In order to solve the precise motion optimal control problem based on piezoelectric ceramic actuator,a control method of piezoelectric ceramic actuator based on particle swarm optimization is proposed.The control problem is transformed into a nonlinear programming problem by using the Gaussian pseudospectral method.Finally,the particle swarm optimization algorithm is used to solve the nonlinear programming problem.