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

Tonametl Sanchez,Fernando A Ortiz-Ricardez,Jaime A Moreno

doi:10.1155/2016/3740834

Tonametl Sanchez, Fernando A Ortiz-Ricardez + Show 1 more

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https://doi.org/10.1155/2016/3740834

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Abstract

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.

Highlights

IntroductionDifferentiators play an important role in (continuous) feedback control systems
Differentiators play an important role in feedback control systems
A sufficient condition for establishing this property is the existence of a robust Lyapunov function for (2), that is, a continuously differentiable and positive definite function V(x), having a negative definite derivative Valong the trajectories of (2) for every |π(t)| ≤ Δ

Summary

Introduction

Differentiators play an important role in (continuous) feedback control systems. It is usually required to differentiate the system’s output in order to construct the feedback controller. From sliding mode control theory, a class of exact differentiators has emerged. We can mention the firstorder robust and exact differentiator (RED) [6] ( known as Super-Twisting algorithm). Such an algorithm was studied through geometric methods, but later the Lyapunov approach provided several interesting results [7,8,9,10,11]

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