Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 351524, 13 pages

http://dx.doi.org/10.1155/2015/351524

## Improved Adaptive Sliding Mode Control for a Class of Uncertain Nonlinear Systems Subjected to Input Nonlinearity via Fuzzy Neural Networks

^{1}Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan^{2}Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan

Received 8 September 2014; Revised 28 December 2014; Accepted 29 December 2014

Academic Editor: Cheng Shao

Copyright © 2015 Tat-Bao-Thien Nguyen 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 paper presents an improved adaptive sliding mode control method based on fuzzy neural networks for a class of nonlinear systems subjected to input nonlinearity with unknown model dynamics. The control scheme consists of the modified adaptive and the compensation controllers. The modified adaptive controller online approximates the unknown model dynamics and input nonlinearity and then constructs the sliding mode control law, while the compensation controller takes into account the approximation errors and keeps the system robust. Based on Lyapunov stability theorem, the proposed method can guarantee the asymptotic convergence to zero of the tracking error and provide the robust stability for the closed-loop system. In addition, due to the modification in controller design, the singularity problem that usually appears in indirect adaptive control techniques based on fuzzy/neural approximations is completely eliminated. Finally, the simulation results performed on an inverted pendulum system demonstrate the advanced functions and feasibility of the proposed adaptive control approach.

#### 1. Introduction

Due to the wide existence of nonlinear systems in many fields of engineering, the controller design for nonlinear systems still received much attention from many researchers. The early control techniques were developed for nonlinear systems and presented their good performances [1–6]. The fundamental ideas of these control techniques are to transform a nonlinear dynamic system into a linear one through state feedback mechanism and then apply the existing methods developed for linear systems. Although good performances can be obtained with these control techniques, the major deficiency remains. The controller design largely relies on the exact cancellation of nonlinear terms or restricts to conditions in that the unknown parameters of nonlinear systems are assumed to appear linearly. This leads to awful performances of the controllers when the existing uncertainties or nonlinear terms of the nonlinear systems are completely unknown. In addition, all control methods above are carried out with an ideal assumption of linear input. Nevertheless, in practical conditions, there exist nonlinearities in the control input because of physical limitations. The existence of the nonlinear input may lead to degradation or even make the system unstable [7].

Nowadays, fuzzy logic and neural networks are found to be powerful tools for modeling and controlling highly uncertain, nonlinear, and complex systems due to their abilities of universal approximation [8–15]. Although fuzzy logic and neural networks have universal approximation abilities, some differences exist between them. The fuzzy logic has characteristics of linguistic information and logic control, while neural networks possess characteristics of learning, parallelism, and fault-tolerance. The combination of fuzzy logic and neural networks, known as fuzzy neural networks, which incorporate the advantages of fuzzy inference and neurolearning, was developed and has presented advanced functions in modeling and controlling nonlinear systems [15–22]. Based on universal approximation theorem, fuzzy logic and neural networks have been developed and incorporated into adaptive control techniques. In such techniques, Lyapunov approach is used to analyze the system stability and obtain the adaptive laws as well. Conceptually, there are two distinct approaches to design the adaptive controllers: direct and indirect adaptive control methods. In the direct control method, a fuzzy logic system or neural networks are employed to simulate the action of the ideal controller and the parameters are directly adjusted to meet the control objective [15, 19, 23–31]. In contrast, the indirect control method uses a fuzzy logic system or neural networks to approximate the unknown nonlinear terms of model dynamics and then synthesizes control laws based on these approximations [15, 20, 21, 32–40]. In the indirect control method [15, 21, 32–40], the authors considered the single-input single-output (SISO) nonlinear systems in the form, , where is the overall state vector, is the control input, and is the system output. and are unknown nonlinear functions. In order to meet the control objectives, they developed the indirect adaptive controllers which are in the form , where is a new input transforming the nonlinear system into the linear one. and represent the parameterized approximations of the actual nonlinear functions, and , respectively. The weighting vectors, and , which vary according to adaptive laws, are adjusted parameters of the approximations. Since the approximations and are calculated by a fuzzy logic system or neural networks, it is well known that these approximations cannot be guaranteed to be bounded away from zero for all time . In other words, may tend to zero or may be close to zero in some points in time. Such situations lead to very large control signals which may cause the controlled systems to lose their controllability or even damage the whole systems. This problem was known as a singularity problem which usually appears in indirect adaptive control method based on fuzzy/neural approximations.

Since the input nonlinearity and the singularity problem may cause negative effects on controlled systems, it is necessary to develop a new indirect adaptive control method which can bypass the singularity problem even with the existence of input nonlinearity. Therefore, by incorporating both advantages of fuzzy neural networks and sliding mode control technique, we developed a new indirect adaptive control method for a class of uncertain nonlinear systems with input nonlinearity. The proposed controller uses fuzzy neural networks to approximate the unknown nonlinear terms of model dynamics and then synthesizes the sliding mode controller. Due to the novel modifications in design of the proposed controller especially, the denominator in the adaptive control law is guaranteed to be away from zero, and, therefore, the singularity problem is completely avoided. Moreover, in order to treat the undesired effects of approximation errors, a robust compensation is added to the controller to ensure the stability of the controlled system and force the tracking errors to converge to zero as well. In contrast, many previous works dealing with indirect adaptive fuzzy control [15, 21, 32–40] still have a weakness in that the controllers may face to the singularity problem for some cases. When the controlled systems fall into the singularity problem, these controllers may produce very large control signals. These situations may lead the systems to lose their controllability or cause the serious damage to the whole systems. Moreover, these control methods are only proper when the inputs are assumed to be linear ideally. This problem induces the controllers to the restriction of applications because the control inputs may appear nonlinearly due to the physical limitations of some components in many physical systems. For this condition, the linear input may not be suitable to use or show the awful performance. Therefore, in comparison with previous methods, the proposed control method shows the improvements in controller design in that the singularity problem is completely solved. In addition, the proposed controller can show the advanced tracking performance even the controlled system is under the influence of the input nonlinearity. With the proposed controller, the output of the system is forced to follow the desired trajectory successfully and the tracking error converges to zero asymptotically. Finally, the simulations are carried out to illustrate the effectiveness and robustness of the proposed controller.

The rest of this paper is organized as follows. The conventional sliding mode control and problem statement are presented in Section 2. The design of fuzzy neural networks is addressed in Section 3, and the design of the adaptive controller is described in Section 4. In Section 5, simulation results are given to confirm the validity of the proposed method. Finally, the conclusion is given in Section 6.

#### 2. Problem Statement and Sliding Mode Control Design

Considering the th order SISO nonlinear system and assuming that the control input is nonlinearly perturbed due to physical limitations, the dynamic equations can be expressed in normal form as follows:where is the overall state vector of the nonlinear system which is assumed to be available for measurement and , are completely unknown smooth functions. is the scalar control input, while is the scalar system output. is a continuous nonlinear function and inside the sector ; that is,where and are positive constants and . The scalar nonlinear function is illustrated in Figure 1.