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Mathematical Problems in Engineering
Volume 2016, Article ID 3740834, 12 pages
Research Article

Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator

Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510 Mexico City, DF, Mexico

Received 5 January 2016; Revised 28 February 2016; Accepted 14 March 2016

Academic Editor: Yan-Jun Liu

Copyright © 2016 Tonametl Sanchez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Differentiators play an important role in (continuous) feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used to provide a procedure to design the differentiator’s parameters. Also, some sets of such parameters are provided. The determination of the positive definiteness of the Lyapunov function and negative definiteness of its derivative is converted to the problem of solving a system of inequalities linear in the parameters of the Lyapunov function candidate and also linear in the gains of the differentiator, but bilinear in both.