Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3740834, 12 pages

http://dx.doi.org/10.1155/2016/3740834

## 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.

#### 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.

#### 1. Introduction

Differentiators play an important role in (continuous) feedback control systems. For example, it is usually required to differentiate the system’s output in order to construct the feedback controller. There are several options to approximate the derivatives of a signal, for instance, the very classical linear filters or Luenberger observers, the digital filters approach [1, 2], the high-gain observers [3], and some nonlinear observers [4, 5]. However, from sliding mode control theory, a class of exact differentiators has emerged. We can mention the first-order robust and exact differentiator (RED) [6] (also known as Super-Twisting algorithm). Initially, such an algorithm was studied through geometric methods, but later the Lyapunov approach provided several interesting results [7–11].

Theoretically, the Super-Twisting algorithm can provide exactly the first derivative of a signal in finite time, if the second derivative is uniformly bounded. To obtain higher order derivatives, one could use first-order RED in cascade. However, this configuration produces a significant loss of precision [12]. Hence, for higher order derivatives, a RED of arbitrary order was proposed in [12], and their properties were analyzed by means of geometric methods and homogeneity properties [13]. Unlike those kinds of proofs, a Lyapunov based approach would be very useful to analyze robustness properties, to design the differentiator’s parameters, and to estimate the convergence time.

Lyapunov’s direct method is one of the most important tools in analysis and design of nonlinear control systems [14–17]. It has been used for analysis and design for a wide class of nonlinear systems as, for example, continuous [15], variable structure [18], or hybrid [19] systems, adaptive fuzzy controllers [20], and fuzzy optimal control for chaotic discrete-time systems [21].

For the case of the second-order RED [12], a continuous but not differentiable Lyapunov function was proposed in [22]. Although it can be used to design the parameters of the differentiator, a set of nonlinear inequalities involving the parameters of the function and the gains of the differentiator must be solved. Thus, it is desirable to have a differentiable Lyapunov function and an easier procedure to design the parameters of the differentiator.

The contributions of this paper improve the analysis and design of the second-order RED as stated below:(i)We provide a constructive method to determine a differentiable Lyapunov function for the second-order RED. This is the first time that a Lyapunov function for the second-order RED is provided in the literature.(ii)The Lyapunov function designing process is useful to obtain a procedure to design the gains of the second-order RED.(iii)We also provide some different sets of gains for the second-order RED.

This is achieved by using the Lyapunov function designing method proposed in [23]. Such a method has been applied to second-order systems, and in this paper we apply it for a third-order one: the second-order RED. One of the main characteristics of the method is that it allows us to design the parameters of the system by solving a linear system of inequalities or a linear matrix inequality. A preliminary version of these results was presented in [24].

This paper is organized as follows. In Section 2, a brief description is given of the second-order RED and the Lyapunov function we are proposing. Section 3 is dedicated to the design process of the Lyapunov function, and in Section 4 we finish with some concluding remarks.

#### 2. Lyapunov Analysis and Design for the Second-Order RED

In [12], an arbitrary order RED was proposed, but [12] does not provide a* method* to determine whether for a certain set of gains the RED will converge or how to design gains to achieve convergence of the RED.

In this section, we will first recall Levant’s RED [12] and motivate the necessity of selecting appropriate gains. We then show that it is possible to provide a (smooth) Lyapunov function to prove the convergence of the RED for appropriate gains and how to scale these gains. Although the ideas presented in the paper are valid for an arbitrary order RED, we will restrict ourselves to the second-order differentiator for simplicity and concreteness of the presentation.

##### 2.1. The Differentiator

Consider the class of signals , containing time functions having continuous first-order and second-order derivatives and a third-order derivative which exists almost everywhere and is bounded: that is, , , for some nonnegative constant . The second-order RED given by [12]can provide* exactly* the first- and second-order derivatives of in* finite time*; that is, after a finite time , , , and , for all . In (1), as in the whole paper, the following notation is used: for a real variable and a real number , . Note that since (1) has a discontinuous right-hand side, we should interpret its solutions in the sense of Filippov [25].

Differentiator (1) will work only if the gains are designed properly, as is illustrated in the following example.

*Example 1. *Consider the function given by . The third derivative of is bounded by . We simulate (1) with the gains . In Figure 1, it can be seen that , , and converge in finite time to , , and , respectively. However, with , and converge to and , respectively, but does not converge to ; see Figure 2. Now, with , the differentiator’s trajectories are* bounded*, but they do not converge to , , and ; see Figure 3. Moreover, there are gains that can produce* unstable* differentiator’s trajectories: for instance, ; see Figure 4.