The boundary characteristic curve of linear regions with the second-order discrete time optimal control is presented by means of the isochronous region. Then the control variable is linearized, which is decided by the switching curve and the boundary curve, and the reachable region and the boundary region are given. The linear synthetic functions are obtained. Based on this function, the discrete form of tracking-differentiator is constructed. Numerical simulation shows that, this discrete form of tracking-differentiator can quickly track an input signal without overshooting and chattering, and can acquire an excellent differential signal. It is simple and practical.