Mathematical Problems in Engineering

Volume 2019, Article ID 4509674, 9 pages

https://doi.org/10.1155/2019/4509674

## Adaptive Chattering-Free Sliding Mode Control of Chaotic Systems with Unknown Input Nonlinearity via Smooth Hyperbolic Tangent Function

Correspondence should be addressed to Jason Sheng-Hong Tsai; wt.ude.ukcn.liam@iasths and Jun-Juh Yan; wt.ude.uts@nayjj

Received 17 May 2019; Revised 17 July 2019; Accepted 1 August 2019; Published 7 October 2019

Academic Editor: Alessandro Lo Schiavo

Copyright © 2019 Jiunn-Shiou Fang 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

The design of adaptive chattering-free sliding mode controller (SMC) for chaotic systems with unknown input nonlinearities is studied in this paper. A smooth hyperbolic tangent function is utilized to replace the discontinuous sign function; therefore, the proposed adaptive SMC ensures that not only the chaos phenomenon can be suppressed effectively but also the chattering often appearing in the traditional discontinuous SMC with sign function is eliminated, even when the unknown input nonlinearity is present. A sufficient condition for stability of closed-loop system is acquired by Lyapunov theory. The numerical simulation results are illustrated to verify the proposed adaptive sliding mode control method.

#### 1. Introduction

In recent decades, sliding mode control is one of the popular control theories due to its robustness and insensitivity to parameter uncertainty and external disturbance [1, 2]. The trajectories of SMC systems can be driven onto a specified sliding surface and slide toward desired stable condition, which is called sliding motion. There are many methodologies used to select sliding surfaces, while the different control law is obtained, and it is guaranteed that the controlled system is stable [2]. In general, the SMC and switched system control strategies are usually adopted to the practical application. For example, in [3, 4], the new SMC-based fuzzy controller and robust hybrid controller are applied to the 7-degree-of-freedom upper-limb exoskeleton robot such that the undesired external disturbance can be suppressed. Therefore, the SMC can be used to overcome undesired external disturbance successfully; in [5], a class of linear switched system is discussed for the cooperative stabilization problem and a novel class of switching signals is proposed. Based on above descriptions, the SMC is adopted to cope with input nonlinearity considered in this paper due to its robustness.

In traditional SMC, the control force is difficult to achieve, which has high frequency chattering phenomenon due to the discontinuous sign function. Therefore, many quasisliding mode control techniques have been proposed [2, 6–8], and the chattering phenomenon is eliminated due to the controller designed by using a continuous function. On the other hand, chaotic systems have been proven the existence which is extensively and become an interesting topic in physical systems [9–11]. Also, the chaotic systems have special nonlinear dynamic behavior [7, 12]. In general, there are two sides to such physical systems [11]; the first side is the beneficial feature, such as chemical reactions and encrypted communication systems. On the other side, the chaos phenomenon in some engineering systems is highly unexpected for its applications, such as electronic systems, power converters, and high-precision mechanical systems. Therefore, many different methods have been proposed to solve the problems of chaos control and suppression for chaotic systems, such as adaptive feedback control [13–16], sliding mode control [11, 17, 18], PID control [19–21], optimal control [22–26], and robust control [27–29] among many others [30, 31].

As described above, many modified methodologies have been developed to overcome the chattering phenomenon in traditional SMC, but they have not been well discussed when the system structure is subjected to input nonlinearity and even the input nonlinearity is unknown. It is well known that in practical systems, unavoidable external nonlinear perturbations do exist in control input, which not only affect system stability but also decrease the input performance and response. Furthermore, since the feature of chaotic system is very sensitive to parameter uncertainty and external perturbation [11], the stability analysis of controlled system with input nonlinearity is an important issue for chaotic systems. Recently, some papers [11, 32, 33] have researched SMC design for the chaotic systems with nonlinear input, but they only discussed the control design when input nonlinearity is well known. Hu et al. [34] proposed a SMC-based adaptive controller to deal with nonlinear input, but the adaptive law is high order. However, we present a low-order adaptive law to omit the harmful effects of nonlinear input. To our knowledge, the chattering-free SMC design with unknown input nonlinearity is still not well discussed.

In this paper, the adaptive continuous SMC design for chaotic systems with unknown input nonlinearity is studied. The main contribution is to introduce a new technique of adaptive sliding mode control with a smooth hyperbolic tangent function to avoid chattering. In addition, the undesired chaotic behavior of the considered chaotic systems can be fully suppressed even with unknown input nonlinearities. Finally, we present numerical simulation results to illustrate the effectiveness of the proposed adaptive continuous SMC scheme.

##### 1.1. Notations

In this paper, represents the *n*-dimensional Euclidean space, denotes the set of all real *n* by *m* matrices, the superscript “+” denotes the matrix generalized inverse, and stands for *m* by *m* identity matrices. denotes the Euclidean norm, when *W* is a vector, and the induced norm, when *W* is a matrix. denotes the maximum eigenvalue of matrix *W*. represents the absolute value of , and is the sign function of *s*; if *s* > 0, ; if *s* = 0, ; and if *s* < 0, .

#### 2. System Description and Problem Formulation

Consider a general class of chaotic systems given below:where is the system matrix and is the system state vector. denotes the nonlinear vector of systems. Without loss of generality, we make the following assumption about system (1).

*Assumption 1. *The dynamic system (1) can be written aswhere and . The pair is controllable.

*Remark 1. *Assumption 1 is not restrictive. Many nonlinear chaotic systems described by (2) can be found. For example, the Matsumoto–Chua–Kobayashi circuit, system, modified Chua’s circuit, Lorenz system, Duffing–Holmes system, system, and Chen chaotic dynamical system.

To suppress the chaos oscillation behavior, we introduce a control vector subjected to unknown nonlinearity described aswhere is the control vector with nonlinearity. The continuous nonlinear function satisfies , where with the law , satisfyingwhere are unknown positive nonzero constants but bounded. Figure 1 shows a nonlinear function inside the sectors .

In SMC, the traditional controller is difficult to implement because there is an important adverse problem of high-frequency chattering phenomenon. In order to improve this issue, the following lemma with smooth hyperbolic tangent function is introduced.