On discrete time optimal control: a closed-form solution

In this paper, the mathematical derivation of a closed-form discrete optimal control law is presented. Unlike the well-known results for continuous plants, the closed-form time optimal control for discrete time plants was never attained. The recent work of Jingqing Han sheds lights on this problem and is introduced. In particular, a time optimal control law is constructed in the form of state feedback for a discrete time, double-integral plant by using the isochronic region method. This closed-form nonlinear state feedback clearly demonstrates that time optimal control in discrete time is not necessarily bang-bang control, i.e., the control signal does not always take on extreme values. In fact, this characteristic makes the new control law advantageous in engineering applications.