Optimal control of constrained switched systems and application to electrical vehicle energy management

In this paper, we consider an optimal control problem of switched systems with input and state constraints. It is difficult to numerically integrate the switched system with the initial condition due to the switching times being unknown. In addition, since an input and state constraint is equivalent to an uncountable number of conventional constraints, it is challenging to solve this problem by using conventional optimization approaches. To address the challenge and avoid the shortcoming of existing methods, a numerically tractable method is provided to solve the optimal control problem of constrained switched systems. Firstly, based on introducing binary functions for each subsystem, relaxing the binary functions and including a penalty term on the relaxation, we obtain an equivalent optimal control problem. Then, by using the time-scaling transformation, the smoothing technique, and the idea of l1 penalty function method, the equivalent optimal control problem is transformed into a nonlinear parameter optimization problem, which can be solved by using any gradient-based optimization method. The convergent properties of the nonlinear parameter optimization to the original optimal control problem are discussed. Finally, an optimal energy management problem for an electrical vehicle is used to illustrate our method is less time-consuming with faster convergence speed, and achieves a better objective function value than the existing algorithms.