Brief A phase-plane approach to time-optimal control of single-DOF mechanical systems with friction

In this paper, we attempt to solve the time-optimal control problem for single-degree-of-freedom (DOF) mechanical systems with friction, while taking into account not only the velocity-dependent control input constraint but also the state constraint. Direct application of the Pontryagin's maximum principle (PMP) leads to a sixth-order nonlinear two-point boundary-value problem (TPBVP) which is very difficult to solve numerically. In this context, we take a phase-plane analysis without resorting to the PMP. Thereby, the exact time-optimal solution can be obtained simply by solving a set of first-order differential equations with continuous right-hand sides. We also present some simulation results to demonstrate the practical use of the time-optimal solution.