Control vector parameterization approach with variable time nodes

To solve optimal control problem,the control vector parameterization(CVP)approach is often used and the time grid nodes are usually fixed.In each time intervals,the controls are approximated by a function of time,which is determined by parameters.The partition of time grid will take effects on the accuracy and performance of the numerical optimization algorithm used to solve the original optimal control problem.A CVP method with variable time nodes is presented to optimize both of the control parameters and time nodes in the grid partition.The time node parameter is introduced by sigmoid function which approximate the switch procedure in the piecewise constant parameterization method.The derivatives of the performance to time nodes are derived and the time nodes constraints are handled by a given algorithm.An example of optimal control problem with two controls is solved by the proposed method.The different time grid partitions of two controls are obtained which show a higher approximation of the optimal control trajectories.