On the discrete linear quadratic minimum-time problem

E. I. Verriest and F. L. Lewis have presented in (1) a new method to approximate the minimum-time control of linear continuous-time systems avoiding the Bang-Bang control. Their method relied on the optimization of a cost including time energy and precision terms. Our purpose in this article is to extend their work to discrete-time linear systems. Indeed, we consider a linear discrete-time system and a quadratic cost including time and energy term, and we look for the minimum-time and the control which minimizes the cost function and leads the system from a given initial state to a fixed final one. By selecting the magnitude of the energy term, one may balance off the requirement for minimum-time versus the one for keeping the state and inputs small over the considered interval of time. Finally we give a method to compute the minimum-time and the optimal control.