A Multiple Shooting Algorithm For The Efficient Solution Of Optimal Control For Time-delay Systems

In order to extend the model predictive control(MPC)to the fast dynamic system,the design of optimal control algorithm with high computational efficiency and small resource demand is the key problem to be solved.Parallelization,sensitivity updating and application of reduced order model are all used to speed up the solving process of optimal control.However,most of the existing algorithms ignore time delays in the system,which widely exists in biological systems,chemical reactions,feedback loops,communication networks and other practical systems.Based on this background,a parallelizable and easy-to-implement algorithm for optimal control of time-delay system(TDS)is designed in this paper,and the software implementation has been discussed in detail.This algorithm is expected to provide some key technologies for the extension of nonlinear MPC to fast dynamic TDS.Firstly,a parallelizable multiple shooting algorithm is designed for nonlinear time-varying systems with state and control delays.This algorithm is important for function evaluation and gradient calculation in gradient-based optimization algorithms.By using the implicit continuous Runge-Kutta(CRK)integrator,the restriction that the integration step is less than the minimum time delay is removed,and very dense time grids can also be avoided.At the same time,the prediction-correction strategy is introduced to reduce the steps of Newton iterations and improve the computational efficiency.Based on the implicit function theorem and internal numerical differentiation technique,a gradient calculation method without additional Newton iterations is designed.In the forward simulation,the multiple shooting strategy is introduced to divide the simulation and gradient calculation processes into independent sub-processes,which are implemented by multi-threading programming.Numerical experiments show that the efficiency of optimal control is improved significantly.In the previous algorithm,the calculation of Jacobian matrix is very complex and time-consuming.In order to reduce the computational cost in further,inexact Jacobian matrix is used to reduce the computation related with Newton iterations.By reusing the existing Jacobian matrix and reducing the calculation times of Jacobian matrix,the solving process can be sped up.Numerical experiments show that the strategy of inexact Jacobian matrix is effective to improve the efficiency of the algorithm.In this paper,the proposed multiple shooting algorithms,implemented by using multi-threading programming technology,are combined with the nonlinear programming solver Ipopt to construct a fast solver for the optimal control of nonlinear TDS.Numerical experiments show that the parallel algorithm can significantly improve the computational efficiency compared with the sequential algorithm.