Calculating switching times for the time-optimal control of single-input, single-output second-order systems

In this paper, we consider the time-optimal control of a single-input, single-output second-order system with bounded input and describe a method for calculating the number of switches and the switching times to drive the system from any initial state to a target state in a particular class. A pair of affine mappings are derived that transform the original system into one where the switching curve becomes a collection of similar sections of a logarithmic spiral. In this coordinate system, the number of switches and the times of those switches are calculated and a feedback control law is synthesized.