This paper considers the optimal control problem of a single train,which is formulated as an optimal control problem of nonlinear systems with switching controller.The switching sequence and the switching time are decision variables to be chosen optimally.Generally speaking,it is very difficult to solve this problem analytically due to its nonlinear nature,the complexity of the controller,and the existence of system state and control input constraints.To obtain the numerical solution,by introducing binary functions for every value of the control input,relaxing the binary functions,and imposing a penalty function on the relaxation,the problem is transformed into a parameter optimization problem,which can be efficiently solved by using any gradient-based numerical approach.Then,the authors propose an adaptive numerical approach to solve this problem.Convergence results indicate that any optimal solution of the parameter optimization problem is also an optimal solution of the original problem.Finally,an optimal control problem of a single train illustrates that the adaptive numerical approach proposed by us is less time-consuming and obtains a better cost function value than the existing approaches.
Journal of Systems Science and Complexity
the Chinese National Natural Science Foundation under Grant Nos.61563011,61473158,61703012,and 61374006
the Ph.D Research Fund of Guizhou Normal University under Grant No.11904–0514170.
Corresponding author,WU Xiang,School of Mathematical Sciences,Guizhou Normal University,Guiyang 550001,China;School of Electrical Engineering,Southeast University,Nanjing 210096,China.Email:email@example.com;ZHANG Kanjian,School of Automation,Southeast University,Nanjing 210096,China;Key Laboratory of Measurement and Control of CSE,Ministry of Education,Southeast University,Nanjing 210096,China.Email:firstname.lastname@example.org;CHENG Ming,School of Electrical Engineering,Southeast University,Nanjing 210096,China.Email:email@example.com.