Research And Application Of Singular Optimal Control

Singular optimal control problems are common in chemical,pharmaceutical and other production processes.The control variables of the singular part are oscillating,so in order to solve the optimal solution of the problem,the singular problem must be solved.The singular optimal problem is linear with respect to the control variable.Then the second order of the Hamiltonian function on the control variable is zero in the singular optimal problem,which cannot be directly solved by the maximum value principle.It is then proposed to use the regularization method to solve the singular problem into a non-singular problem,that is,to add a variable integral term to the objective function.During the calculation,the variable coefficient is reduced by a certain multiple,but when the variable coefficient is close to 0,the amount of calculation becomes large.This paper proposes a new processing method based on the traditional processing method.1.Using the state variables directly related to the control variables to solve the optimal control curve,it is expected to avoid the calculation of the ill-conditioned Hessian matrix in the singular control problem.At the same time,in order to simplify the calculation steps,the singular optimal control problem is discretized into a nonlinear programming problem by finite element orthogonal configuration,and then solved by the solver.But there are many drawbacks in the way that control variables are not considered.Mainly because the control variables are transferred to the state variable to solve the problem too many constraints,can't solve more types of problems,so need to continue to improve.2.In order to solve more types of problems,the control variables can add additional constraints,and use the orthogonal configuration to discretize the problem.On this basis,the Hamilton function is combined with the regularization method.The variable coefficients of the regularization method are fixed every time.The multiple is reduced,and the Hamiltonian function is used to determine whether the finite element node needs to be inserted at the value of the discrete point until the Hamilton function value of all nodes is constant under the current variable coefficient.While this approach can solve more types of problems,and control variables can add additionalconstraints,this way of adding finite element nodes is computationally intensive.Moreover,the obtained point is not necessarily the point of switching between the arc segment and the arc segment,and has certain contingency.3.In order to reduce the amount of calculation and reduce the unnecessary finite element nodes,the exact inflection point between the arc segment and the arc segment can be obtained.A solution scheme for the singular optimal control problem based on partial moving finite element is proposed.Firstly,the problem is discretized by the orthogonal configuration method,and the fixed point is obtained by calculating the global error,and the approximate contour of the problem control curve needs to be solved.Based on this,the switching function is used to determine whether the control amount is at the upper and lower boundaries,the approximate position of the inflection point between the arc segment and the arc segment is determined,and the finite element node is inserted therein.This method uses a small number of finite element nodes to determine the exact inflection point between the problem arc and the arc,with less computation.