Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1743861, 12 pages

http://dx.doi.org/10.1155/2016/1743861

## A New Fast Nonsingular Terminal Sliding Mode Control for a Class of Second-Order Uncertain Systems

School of Automation, Beijing Institute of Technology, Zhongguancun Street, Haidian District, Beijing, China

Received 12 July 2016; Revised 30 October 2016; Accepted 15 November 2016

Academic Editor: Asier Ibeas

Copyright © 2016 Linjie Xin 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

This paper considers the robust and adaptive nonsingular terminal sliding mode (NTSM) control for a class of second-order uncertain systems. First, a new fast NTSM was proposed which had global fast convergence rate in the sliding phase. Then, a new form of robust NTSM controller was designed to handle a wider class of second-order uncertain systems. Moreover, an exponential-decline switching gain was introduced for chattering suppression. After that, a double sliding surfaces control scheme was constructed to combine the NTSM control with the adaptive technique. The benefit is that a strict demonstration can be given for the stagnation problem in the stability analysis of NTSM. Finally, a case study for tracking control of a variable-length pendulum was performed to verify the proposed controllers.

#### 1. Introduction

It is well known that the sliding mode control (SMC) has invariant property to the matched uncertainties. It has become a popular method to control nonlinear uncertain systems [1, 2]. In the earlier studies of SMC, the research was focused on the linear sliding mode (LSM). In [3, 4], the concept of terminal attractor was proposed and used to design the sliding mode which was known as the terminal sliding mode (TSM). The TSM had advantage of finite time convergence, and it has been widely used in many applications, such as robots, spacecraft, and DC-DC buck converters [5–7].

Compared to the LSM, the TSM has lower convergence speed when the states are far from the origin. Considering this problem, Yu et al. designed a fast terminal sliding mode (FTSM) which had global fast convergence in the sliding phase [8]. The FTSM technique has been developed for controlling nonlinear second-order systems with uncertain terms in [9–11]. Moreover, the concept of terminal attractor was also applied to the reaching law by Yu and Man [12]. As the negative fractional power exists in the TSM control signals, it may result in singularity of the control input. This is hard to accept in real implementations. So, the control input was switched to a general sliding mode control in order to avoid the singularity when the states converged into a small vicinity of origin in [13]. However, this switching method is a suboptimal solution which lost the advantage of TSM. In [14, 15], a new form of TSM known as nonsingular TSM (NTSM) was proposed and applied to robotic manipulators and piezoelectric actuators. It also has a lower convergence speed when the states are far from the origin. Although the NTSM control has solved the singularity problem, it needs an extra demonstration to show that the motion of sliding surface will not stagnate at some nonzero points in the reaching phase. According to the existing literatures, most of the TSM control methods were designed for the second-order systems. In [16–18], the recursive TSM control was developed for higher order nonlinear systems.

For the robust TSM controllers, the boundary information of system uncertainties is usually required to be known in advance. However, it is difficult to obtain in many practical implementation processes. Therefore, the adaptive technique was employed to cooperate with the TSM and FTSM control in [19–21]. Moreover, the adaptive NTSM control method was developed for uncertain systems with input nonlinearity in [22, 23]. Note that a strict proof for the stagnation problem of NTSM can be given in the robust NTSM control, for example, [14, 15]. However, it was difficult to obtain for the case of adaptive NTSM control in [22, 23].

Previous studies have developed the TSM control and thus have great significance. On the other hand, there are still some points valuable for further research. Most of the NTSM controllers were designed for a class of uncertain systems in which the coefficient of control input was a known function (or with an uncertain item). In the case that this coefficient is totally unknown except its boundary, few NTSM controllers were reported. So, the purpose of this paper is to develop the NTSM control for the latter case.

The main contribution of this paper can be summarized as follows.(1)A fast NTSM (FNTSM) is designed to combine the advantages of LSM and NTSM together. Thus, the convergence rate of the NTSM is enhanced when the initial states are far away from the origin.(2)A new robust NTSM controller is proposed for the case that the coefficient of control input is unknown. It is also applicable to the case that the coefficient is known. Moreover, an exponential-decline switching gain is designed to attenuate the chattering phenomenon.(3)An adaptive NTSM control scheme with double sliding surfaces is proposed to give a strict proof for the stagnation problem of NTSM. The system uncertainty is firstly compensated on an integral sliding surface; after that the system trajectory is forced to the designed fast NTSM.

#### 2. Robust NTSM Controller Design

Consider a second-order nonlinear system aswhere and denote bounded uncertain functions, .

*Assumption 1. *The uncertain functions and satisfywhere , , , , and are positive constants. and are assumed to be known. Robust and adaptive controllers are designed for the two cases that is known or not, respectively. In the rest of the paper, and are simply denoted as and .

##### 2.1. FNTSM Design

In the study of [14], a kind of NTSM was designed as

It is equivalent to the following form which can be converted to a conventional TSM:

As illustrated in [8], the TSM has slower convergence rate than the LSM when the initial states are far away from the origin. It can be explained by its eigenvalue. For TSM , it is easy to obtain the eigenvalue as

As , the eigenvalue tends to be negative infinity at the origin. This fact implies that the convergence rate is infinitely large. This is the major merit of TSM. However, when the initial state is far away from the origin, the convergence rate is smaller than the case , that is, a LSM.

In order to enhance the convergence rate of NTSM (3), a new FNTSM is designed as

The FNTSM (6) is a switching sliding surface which transfers the system dynamic from a LSM () to a NTSM () (Figure 1; e.g., ). Thus, it has global fast convergence rate and reserves the advantage of NTSM.