On finite-time stable tracking differentiator without Lipschitz continuity of Lyapunov function

In this paper, the strong and weak convergence of nonlinear finite-time stable tracking differentiators is presented under some easy checkable conditions. A second order nonlinear differentiator is constructed by using homogeneity in geometry. Numerical simulation shows that this tracking differentiator has advantages compared with the existing ones. The application to the stabilization control of one-dimensional wave equation is illustrated numerically. This result relaxes the strong condition required in [IEEE Transactions on Automatic Control, 52(2007), 1731-1737] that the Lyapunov function satisfies global Lipschitz condition, which seems very restrictive in applications.