The minimum-time problem for discrete-time linear systems: A non-smooth optimization approach

This paper addresses the problem of driving the state of a linear discrete-time system to zero in minimum time. The inputs are constrained to lie in a bounded and convex set. The solution presented in the paper is based on the observation that the state sequence induced by the minimum-time control sequence is the sparsest possible state sequence over a certain finite horizon. That is, the desired state sequence must contain as many zero vectors as possible, all those zeros corresponding to the highest values of the time index. Hence, by taking advantage of some recent developments in sparse optimization theory, we propose a numerical solution. We show in simulation that the proposed method can effectively solve the minimum-time problem even for multi-inputs linear discrete-time systems.