Abstract: The boundary characteristic curves, control characteristic curves and switching curves of linear region with second-order discrete time optimal control system are presented by state backstep method. The two-step reachable region is also acquired. If the point is not located in the two step reachable region, a parallel auxiliary line is drawn, which intersects above three curves at three different points. The control variable is acquired according to linear proportion of the three characteristic points about the above three curves, which replace nonlinear boundary transformation, then the time optimal segmental linearized tracking differentiator is constructed, and the control synthetic function is greatly simplified. This algorithm is consistent with nonlinear boundary transform algorithm if the three characteristic points in linear region fall completely in the characteristic curves. Numerical simulation results show that this discrete form of tracking-differentiator can quickly track an input signal without overshooting and chattering, and can produce an excellent differential signal. The sweep-frequency algorithm is employed to compare the above two kinds of tracking differentiator in the field of amplitude-phase frequency characteristic. This tracking-differentiator has the benefits of concision and nonlinearity, and also requires less calculation. It is convenient for engineering applications.