Research On Optimal Control Problem Based On Numerical Calculation Method

Under certain constraint conditions,the optimal control problem find an admissible control law,which can make the performance index of dynamic system reach the maximum or minimum value.In this paper,Runge-Kutta method,Gauss pseudospectral method and adaptive pseudospectral method were used to study the optimal control problem.Piezoelectric actuator which is a kind of precision positioning device is an ideal nano driving component.However,for its nonlinear hysteresis,it is very difficult to position it.In order to improve the positioning accuracy of piezoelectric actuator,a control method based on Runge-Kutta method is proposed.Based on the ECNLP dynamics model of piezoelectric actuator,and the absolute stability condition of the fourth-order Runge-Kutta method,the step length is automatically updated.The fourth-order Runge-Kutta method is used to solve the ECNLP dynamics Model of piezoelectric actuator to obtain the variables of state variables and control variables.And the minimum value of performance index function and the global optimal solution are proposed.To solve the optimal control problem,an improved Gauss pseud ospectral method is used.Firstly,any definition interval of the optimal control problem can be mapped to [-1,1] by this method.Then the state variables,control variables and performance index functions of optimal control problem are discretized at the Gauss nodes,and the optimal control problem is transformed into a nonlinear programming problem.And the new costate theorem of the method is proved.Finally,using this method to solve the ECNLP dynamics model of piezoelectric actuator,an optimal trajectory tracking control method is proposed.The method has higher positioning displacement accuracy of piezoelectric actuator under the requirement of actual industry tolerance.Aiming at the autonomous parking system,an autonomous parking model based on the time energy optimal control model is established,and a novel adaptive pseudospectral method is proposed to solve the model.The adaptive pseudospectral method combines the global pesudospectral method with the Lagrange interpolation method,and adopts double layer optimization strategy to solve the optimal control.The adaptive pseudospectral method is used to discretize the time energy optimal control model of autonomous parking into a nonlinear programming problem under the Gauss nodes.The necessary conditions that optimal programming exists are given by the Hamiltonian function,and the Karush-Kuhn-Tucker condition is obtained by using the augmented performance index function with Lagrange multiplier vector.Compared with the traditional interior point method,the proposed adaptive pseudospectral method has higher computational efficiency and accuracy for solving automatic car parking systems.Simulation results show that the adaptive pseudospectral method has better efficiency on solving the time energy optimal control model problem.