Smooth Lyapunov function and gain design for a Second Order Differentiator

We design a smooth Lyapunov function for the Levant´s Second Order Differentiator. The Lyapunov function construction method takes advantage of the structure of the system vector field to choose a candidate function. Both, the vector field and the candidate function belong to a special class of homogeneous functions. The problem of proving the positiveness of the function and the negativeness of its derivative is reduced, by using Pólya´s Theorem, to the problem of solving a system of inequalities. Such inequalities are linear in the coefficients of the candidate function and also linear in the system parameters, but bilinear in both. The gains of the differentiator are designed during the construction process, and through the Lyapunov function, convergence time is estimated.