Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation

Given a switched system with multiple operating modes, a fundamental problem is to determine the optimal times at which the system should switch from one mode to another. This paper investigates two computational approaches for solving this problem-the direct optimization approach and the time-scaling approach. The direct optimization approach involves optimizing the mode switching times directly using gradient-based optimization methods such as sequential quadratic programming. The time-scaling approach involves transforming the switched system with variable switching times into an equivalent switched system with fixed switching times, where the decision parameters in the new switched system represent the mode durations in the original system. The optimal values for these new decision parameters-which can be obtained using conventional dynamic optimization techniques-then yield the optimal switching times for the original system. It is widely claimed in the literature that the time-scaling approach is superior to the direct optimization approach. However, the reasons given for its superiority are often vague, and sometimes incorrect. In this paper, we rigorously explicate the major advantages of the time-scaling transformation. We also compare the time-scaling and direct optimization approaches by solving a trajectory optimization problem involving the classical Dubins vehicle model