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Mathematical Problems in Engineering
Volume 2011, Article ID 439324, 20 pages
http://dx.doi.org/10.1155/2011/439324
Research Article

New Nonlinear Controller for a Class of Chaotic Systems Based on Adaptive Backstepping Fuzzy-Immune Control

1aiRobots Laboratory, Department of Electrical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
2Graduate Institute of Automation Technology, National Taipei University of Technology, Taipei 10608, Taiwan

Received 31 July 2010; Revised 15 March 2011; Accepted 28 March 2011

Academic Editor: E. E. N. Macau

Copyright © 2011 Kai-Shiuan Shih 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

An adaptive backstepping fuzzy-immune controller for a class of chaotic systems is proposed. An adaptive backstepping fuzzy method and adaptive laws are used to approximate nonlinear functions and the unknown upper bounds of uncertainty, respectively. The proposed adaptive backstepping fuzzy-immune controller guarantees the stability of a class of chaotic systems while maintaining good tracking performance. The fuzzy-immune algorithm is used for mathematical calculations. The intelligence algorithm consists of the adaptive backstepping fuzzy method and a novel fuzzy-immune scheme which generates optimal parameters for the control schemes. Finally, two simulation examples are given to illustrate the effectiveness of the proposed approach.

1. Introduction

Adaptive fuzzy logic controllers provide a systematic and efficient framework for incorporating linguistic fuzzy information from human experts. In [1], an adaptive fuzzy logic control theory was derived for a class of uncertain nonlinear single-input single-output (SISO) systems. Moreover, many scientists have since dedicated a lot of effort to solving the adaptive fuzzy control problem of uncertain nonlinear systems [210]. Furthermore, the stability of uncertain nonlinear systems has been addressed by the integration of fuzzy logic control and the adaptive laws. [1120]. Subsequently, several methodologies have been instituted for controlling nonlinear systems [2029]. The primary advantage of adaptive fuzzy control scheme is insensitive to internal uncertainty and external disturbances. Adaptive fuzzy control approaches only can perform desired performance for a simple class of nonlinear systems. If nonlinear systems without satisfying the matching conditions, the adaptive fuzzy control methodologies cannot be implemented.

In the past decade, many adaptive fuzzy control schemes have been developed by combining the backstepping technique [3032]. The primary advantage of adaptive backstepping fuzzy control is that the matching conditions are not needed. Backstepping is based on the nonlinear stabilization technique of adding an integrator. Adaptive backstepping fuzzy control schemes can provide a systematic framework for tracking or regulation strategies [3340].

In the past decade, the research area of controlling chaos has received increasing attention. Chaos is a complex nonlinear dynamical system, and it is commonly difficult to exactly predict the behavior of a chaotic system. Recently, many successful methods for controlling chaos have been developed [3, 8]. In the present study, we propose an adaptive backstepping fuzzy-immune controller for a class of chaotic systems. Based on the backstepping algorithm, the fuzzy methodology augmented by an immune algorithm is proposed as a new evolution algorithm, which maintains the advantages of simplicity and easy handling. The four main contributions are (1) an adaptive backstepping fuzzy-immune tracking controller for a class of chaotic systems is proposed, (2) the controller does not require a priori knowledge of the sign of the control coefficient, (3) a novel fuzzy-immune algorithm is used to find the optimal solution, and (4) a correct term can be used to eliminate disturbance.

The rest of this paper is organized as follows. In Section 2, system statement and description of fuzzy systems for chaotic system are presented. The adaptive backstepping fuzzy controller technique and a novel fuzzy-immune mechanism are discussed in Section 3. The results of simulations for chaotic systems are presented to confirm the validity of the proposed control scheme in Section 4. Finally, the conclusions are given in Section 5.

2. System Statement and Description of Fuzzy Systems for Chaotic System

In this paper, we consider a class of chaotic systems that can be shown in strict-feedback systems with nonlinear functions and disturbanceṡ𝑥1=𝑥2+𝑑1(𝑡),̇𝑥2=𝑥3+𝑑2(𝑡),̇𝑥𝑛=𝑓𝑛𝑥𝑛(𝑡)+𝑔𝑛𝑥𝑛(𝑡)𝑢(𝑡)+𝑑𝑛(𝑡),𝑦=𝑥1,(2.1) where𝑥𝑛(𝑡)=[𝑥1(𝑡),,𝑥𝑛(𝑡)]𝑇𝑅𝑛, 𝑓𝑛(𝑥𝑛(𝑡)) and 𝑔𝑛(𝑥𝑛(𝑡)) are smooth functions, 𝑢(𝑡) and 𝑦 are control input and output variables, respectively. 𝑑1(𝑡),,𝑑𝑛(𝑡) denote external disturbance. However, the bound of external disturbance is difficult to obtain in the practical applications. The control objective is to design a stabilizing controller for the system described by (2.1) so that the tracking error converges to zero asymptotically despite the presence of unknown nonlinearities and disturbances.

2.1. Description of Fuzzy Systems

Fuzzy logic systems have been successfully employed to approximate the mathematical models of nonlinear systems. The fuzzy systems can be divided into four parts: fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzy mechanism is described by IF-THEN rules from an input linguistic vector 𝑥𝑛(𝑡) to an output variable 𝑓(𝑥𝑛(𝑡))𝑅𝑖If𝑥1is𝐹𝑖1,𝑥2is𝐹𝑖2,,𝑥𝑛is𝐹𝑖𝑛,then𝑦is𝐺𝑖,𝑖=1,2,,𝑚,(2.2) where𝑚 is the number of rules. 𝐹𝑖1,𝐹𝑖2,,𝐹𝑖𝑛 and 𝐺𝑖 are the fuzzy sets and 𝑥1,𝑥2,,𝑥𝑛 are states of system. Using a singleton function, center average defuzzification, and product inference, the fuzzy systems output is𝑓𝑥𝑛=𝑚𝑖=1𝑦𝑖𝑛𝑙=1𝜇𝐹𝑖𝑙𝑥𝑙𝑚𝑖=1𝑛𝑙=1𝜇𝐹𝑖𝑙𝑥𝑙,(2.3) where 𝜇𝐹𝑖𝑙(𝑥𝑙) is the membership of 𝑥1 in the fuzzy set 𝐹𝑖1 and 𝑦𝑖=max𝑦𝑅𝜇𝐺𝑖(𝑦𝑖)=1. Then, (2.3) can be rewritten as𝑓𝑥𝑛=𝜃𝑇𝑓𝜑𝑓𝑥𝑛,(2.4) where 𝜃𝑓=[𝑦1,𝑦2,,𝑦𝑚]𝑇 and 𝜑𝑓(𝑥𝑛(𝑡))=[𝜑11,,𝜑1𝑚]𝑇. We can define 𝜑1𝑖 as𝜑1𝑖=𝑛𝑙=1𝜇𝐹𝑖𝑙𝑥𝑙𝑚𝑖=1𝑛𝑙=1𝜇𝐹𝑖𝑙𝑥𝑙,𝑖=1,,𝑚.(2.5)

The logic fuzzy system shown in (2.5) is a universal approximator. It can be proved using the following lemma.

Lemma 2.1 (see [41]). Let 𝑓(𝑥𝑛) be continuous functions defined on a compact set 𝑈𝑅𝑛 and arbitrary 𝜀>0, and there exists a fuzzy logic system 𝑓(𝑥𝑛) in the form of (2.4) such that sup𝑥𝑈|||𝑓𝑥𝑛𝑓𝑛𝑥𝑛|||𝜀.(2.6)

After some simple manipulations in (2.4) and (2.5), we can obtain 𝑔𝑛(𝑥𝑛)=𝜃𝑇𝑔𝜑𝑔(𝑥𝑛) as the approximator of 𝑔𝑛(𝑥𝑛). The nonlinear functions 𝑓𝑛(𝑥𝑛) and 𝑔𝑛(𝑥𝑛) requires successful estimates ̂𝜃𝑓 and ̂𝜃𝑔 in order to perform the performance shown in (2.6).

Typically, there exists optimal parameter estimates 𝜃, and the approximation error is the smallest. The optimal parameter estimate is defined as𝜃𝑓=argmin𝜃𝑓Ω𝑓sup𝑥Ω𝑥||𝑓𝑛𝑥𝑛𝜃𝑇𝑓𝜑𝑓||,𝜃𝑔=argmin𝜃𝑔Ω𝑔sup𝑥Ω𝑥||𝑔𝑛𝑥𝑛𝜃𝑇𝑔𝜑𝑔||.(2.7)

Based on fuzzy mechanism, (2.1) can be rewritten as below:̇𝑥1=𝑥2+𝑑1(𝑡),̇𝑥2=𝑥3+𝑑2(𝑡),̇𝑥𝑛=𝜃𝑇𝑓𝜑𝑓𝑥𝑛+𝜃𝑇𝑔𝜑𝑔𝑥𝑛𝑢(𝑡)+𝑑𝑛(𝑡)+𝜀𝑓+𝜀𝑔,𝑦=𝑥1,(2.8) where 𝜀𝑓 and 𝜀𝑔 are internal modeling error variables𝜀𝑓=𝑓𝑛𝑥𝑛𝜃𝑇𝑓𝜑𝑓𝑥𝑛,𝜀𝑔=𝑔𝑛𝑥𝑛𝜃𝑇𝑔𝜑𝑔𝑥𝑛,(2.9)

Therefore, the mathematical model includes internal modeling error variables and external disturbance. We will discuss our proposed method in the next section.

3. Adaptive Backstepping Fuzzy Controller Technique and Fuzzy-Immune Mechanism

𝑓𝑛(𝑥𝑛(𝑡)) and 𝑔𝑛(𝑥𝑛(𝑡)) are the system dynamic functions, these cannot be exactly obtained in general, 𝑑1(𝑡),,𝑑𝑛(𝑡) are unknown parameters in practical application Thus, in the adaptive backstepping fuzzy controller (ABFC) system, the fuzzy system is designed to estimate the system dynamic functions.

3.1. Backstepping Design Principle

The design of ABFC for the chaotic dynamic system is described step-by-step as follows:

Step 1. Consider the tracking error ̇𝑒1=̇𝑥1̇𝑟=𝑥2̇𝑟+𝑑1=𝑒2+𝑎1+𝑑1,(3.1) where 𝑟 is the command trajectory. The first Lyapunov function is defined as 𝑉1=𝑒212.(3.2) Differentiating (3.2) with respect to time and it is obtained that ̇𝑉1=𝑒1̇𝑒1=𝑒1𝑒2+𝑎1+𝑑1=𝑒1𝑒2+𝑒1𝑎1+𝑑1.(3.3)
Define the following stabilizing function: 𝑎1=𝜏1𝑒11𝜂21𝑒1,(3.4) where 𝜏1 and 𝜂1 are positive constants, 𝜂1 represent the attenuation level of disturbances. From substitute (3.3) into (3.4), then we obtain ̇𝑉1=𝜏1𝑒21+𝑒1𝑒2+𝜇1,(3.5) where 𝜇1=(1/𝜂21)𝑒1+𝑒1𝑑1=((1/𝜂1)𝑒1(1/2)𝜂1𝑑1)2+(1/4)𝜂21𝑑21.

Step 2. The derivative of 𝑒2=𝑥2̇𝑟𝑎1 is ̇𝑒2=̇𝑥3+𝑑2̈𝑟𝜕𝑎1𝑥2+𝑑1𝜕𝑥1𝜕𝑎1̇𝑟𝜕𝑟1=𝑒3+𝑎2𝜕𝑎1𝑥2𝜕𝑥1𝜕𝑎1̇𝑟𝜕𝑟1+𝑑2𝜕𝑎1𝑑1𝜕𝑥1.(3.6) The second Lyapunov function is defined as 𝑉2=𝑉1+𝑒222.(3.7)
Differentiating (3.7) with respect to time, it is obtained that ̇𝑉2=𝑒2𝑒3𝜏1𝑒21+𝑒2𝑒1+𝑎2𝜕𝑎1𝑥2𝜕𝑥1𝜕𝑎1̇𝑟𝜕𝑟1+𝜇1+𝑒2𝑑2𝑒2𝜕𝑎1𝑑1𝜕𝑥1.(3.8) Define the following stabilizing function: 𝑎2=𝑒1𝜏2𝑒21𝜂22𝑒2+𝜕𝑎1𝜕𝑥1𝑥2+𝜕𝑎1𝜕𝑟̇𝑟,(3.9) where 𝜏2 and 𝜂2 are positive constants and 𝜂2 represent the attenuation level of disturbances.
From substitute (3.9) into (3.8), then we obtain ̇𝑉2=𝜏1𝑒21𝜏2𝑒22+𝑒2𝑒3+𝜇1+𝜇2,(3.10) where 𝜇21=𝜂22𝑒22+𝑒2𝑑2𝜕𝑎1𝜕𝑥1𝑑11=𝜂2𝑒212𝜂2𝑑2𝜕𝑎1𝜕𝑥1𝑑12+14𝜂22𝑑2𝜕𝑎1𝜕𝑥1𝑑12.(3.11)

Step i
After some simple manipulations and the same method, the derivative of generalized error iṡ𝑒𝑖=𝑥𝑖+1+𝑑𝑖𝑖1𝑙=1𝜕𝑎𝑖1𝜕𝑥𝑙𝑑𝑙𝑑𝑖𝑟𝑑𝑟𝑖1𝑙=1𝜕𝑎𝑖1𝑥𝑙+1𝜕𝑥𝑙+𝑖1𝑙=1𝜕𝑎𝑖1𝜕𝑟(𝑙1)𝑟𝑙.(3.12) The ith Lyapunov function is defined as 𝑉𝑖=𝑉𝑖1+𝑒2𝑖2.(3.13)
Differentiating (3.13) with respect to time, it is obtained that ̇𝑉𝑖=𝑒𝑖𝑒𝑖+1𝑖𝑙=1𝜏𝑙𝑒2𝑙+𝑖𝑙=1𝜂𝑙,(3.14) where 𝜇𝑙1=𝜂𝑙𝑒𝑙12𝜂𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2+14𝜂2𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2.(3.15)
Define the following stabilizing function: 𝑎𝑖=𝑒𝑖1𝜏𝑖𝑒𝑖1𝜂2𝑖𝑒𝑖+𝑖1𝑙=1𝜕𝑎𝑖1𝜕𝑥𝑙𝑥𝑙+1+𝑖1𝑙=1𝜕𝑎𝑖1𝜕𝑟(𝑙1)𝑟𝑙.(3.16)

Step n
The final control law will be determined in the final step, and the derivative of step generalized error can be described as ̇𝑒𝑛=̇𝑥𝑛𝑑𝑛𝑟𝑑𝑟𝑛1𝑙=1𝜕𝑎𝑛1𝑥𝑙+1𝜕𝑥𝑙+𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑟(𝑙1)𝑟𝑙=𝜃𝑇𝑓𝜑𝑓𝑥𝑛+𝜃𝑇𝑔𝜑𝑔𝑥𝑛𝑑𝑢𝑛𝑟𝑑𝑟𝑛1𝑙=1𝜕𝑎𝑛1𝑥𝑙+1𝜕𝑥𝑙+𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑟(𝑙1)𝑟𝑙+𝑑𝑛𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑥𝑙𝑑𝑙+𝜀𝑓+𝜀𝑔1𝑢,(3.17)𝑢=̂𝜃𝑇𝑔𝜑𝑔𝑎𝑛+𝑑𝑛𝑟,̂𝜃𝑑𝑟𝑇𝑔𝜑𝑔𝑎0,(3.18)𝑛=𝑒𝑛1𝜏𝑛𝑒𝑛1𝜂2𝑛𝑒𝑛̂𝜃𝑇𝑓𝜑𝑓𝑥𝑛+𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑥𝑙𝑥𝑙+1+𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑟𝑙1𝑟𝑙,(3.19) where 𝜂𝑛 is the given correct factor and 𝜏𝑛 is a positive constant.

3.2. Fuzzy-Immune Mechanism Design

Assume that the number of the kth-generation antigens is 𝜀(𝑘), the output of the helper T-cells, stimulated by antigens, is 𝑇𝐻(𝑘), and the suppressor T-cells affect B-cells to the amount of 𝑇𝑠(𝑘) [42, 43]𝑇𝑒(𝑘)=𝑇𝐻(𝑘)𝑇𝑠(𝑘),(3.20) where 𝑇𝐻(𝑘)=1𝜀(𝑘) and 𝑇𝑠(𝑘)=2𝑓(𝑇𝑒(𝑘),Δ𝑇𝑒(𝑘))𝜀(𝑘). The feedback control rules are defined as𝑢(𝑘)=1𝑒(𝑘)2𝑓(𝑢(𝑘),Δ𝑢(𝑘))𝑒(𝑘)=1121𝑓(𝑢(𝑘),Δ𝑢(𝑘))𝑒(𝑘)=1[]𝑒1𝑓(𝑢(𝑘),Δ𝑢(𝑘))(𝑘),(3.21) where 1, 2, and 𝜆 are scaling factors and =2/1 is utilized to control the stabilization effect. Then, we propose a novel correct constriction coefficient to improve the performance of the immune mechanism. In this, the constriction coefficient can be expressed as follows: =maxmaxmin𝑘max×𝑘now,(3.22) where max and min denote the maximum and minimum of , respectively. 𝑘max denotes the total number of evolution generations, and 𝑘now denotes the current number of evolution generations. Let de(𝑘) denote the difference between the 𝑒(𝑘)’s for two consecutive iterations, that is, at iteration 𝑘:de(𝑘)=𝑒(𝑘)𝑒(𝑘1)0.(3.23)

Based on the above inferences, the following nine fuzzy rules are suggested.(1)If 𝑢(𝑘) is big and Δ𝑢(𝑘) is big, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is small.(2)If 𝑢(𝑘) is big and Δ𝑢(𝑘) is small, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is medium.(3)If 𝑢(𝑘) is small and Δ𝑢(𝑘) is big, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is medium.(4)If 𝑢(𝑘) is medium and Δ𝑢(𝑘) is big, then𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is small.(5)If 𝑢(𝑘) is big and Δ𝑢(𝑘) is medium, then𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is small.(6)If 𝑢(𝑘) is medium and Δ𝑢(𝑘) is medium, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is medium.(7)If 𝑢(𝑘) is medium and Δ𝑢(𝑘) is small, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is medium.(8)If 𝑢(𝑘) is small and Δ𝑢(𝑘) is medium, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is big.(9)If 𝑢(𝑘) is small and Δ𝑢(𝑘) is small, then 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) is big.

The membership functions for 𝑢(𝑘), Δ𝑢(𝑘), and 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) are shown in Figures 13, respectively.

439324.fig.001
Figure 1: Membership functions for 𝑢(𝑘).
439324.fig.002
Figure 2: Membership functions for Δ𝑢(𝑘).
439324.fig.003
Figure 3: Membership functions for 𝑓(𝑢(𝑘),Δ𝑢(𝑘)).

The output of the controller has the following expression:1𝑢=̂𝜃𝑇𝑔𝜑𝑔𝑎𝑛+𝑑𝑛𝑟𝑑𝑟1[]𝑒1𝑓(𝑢(𝑘),Δ𝑢(𝑘))𝑛.(3.24)

The proposed method is trying to find more efficient ways of utilizing immune mechanism to correct controller input, and the correct item is effective in keeping out perturbation by external or internal. Then, we have the following theorem to gain the objective.

Theorem 3.1. Consider the nonlinear systems (2.1) and with the controller 𝑢 is given by (3.18). By utilizing parameter adjusting law (3.25) and (3.26) ̇̂𝜃𝑓=𝑃1𝜑𝑓𝑥𝑛𝑒𝑛,𝜃(3.25)̇̂g=Q1𝜑𝑔xnenu,(3.26) then the proposed adaptive backstepping fuzzy-immune control scheme can guarantee the following properties: (1)The asymptotic stability of the system is guaranteed.(2)The tracking error can be described as𝑛𝑙=1𝑇0𝜏𝑙𝑒2𝑙𝑑𝑡𝑉(0)+𝑛𝑙=1𝑇014𝜂2𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2𝑑𝑡.(3.27)

Proof. Firstly, consideration of fuzzy approximating error, the Lyapunov functional is set as (3.28) 𝑉=𝑉𝑛1+12𝑒2𝑛+12̃𝜃𝑇𝑓𝑃̃𝜃𝑓+12̃𝜃𝑇𝑔𝑄̃𝜃𝑔,(3.28) where 𝑃 and 𝑄 are positive symmetric matrices.
Differentiating (3.28) with respect to time, it is obtained that ̇̇𝑉𝑉=𝑛1+𝑒𝑛̇𝑒𝑛+̃𝜃𝑇𝑓𝑃1̇̃𝜃𝑓+̃𝜃𝑇𝑔𝑄1̇̃𝜃𝑔=𝑒𝑛1𝑒𝑛+𝑒𝑛̂𝜃𝑇𝑔𝜑𝑔𝑥𝑛𝑑𝑢𝑛𝑟𝑑𝑟𝑎𝑛+𝑛1𝑙=1𝜇𝑙𝜏𝑙𝑒2𝑙+̃𝜃𝑇𝑓𝑃𝑃1𝜑𝑓𝑥𝑛𝑒𝑛̇̂𝜃𝑓+̃𝜃𝑇𝑔𝑄𝑄1𝜑𝑔𝑥𝑛𝑒𝑛𝜃𝑢̇̂𝑓+𝑒𝑛𝜃𝑇𝑓𝜑𝑓𝑥𝑛+𝑎𝑛𝑛1𝑙=1𝜕𝑎𝑛1𝑥𝑙+1𝜕𝑥𝑙+𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑟(𝑙1)𝑟𝑙+𝑑𝑛𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑥𝑙𝑑𝑙+𝜀𝑓+𝜀𝑔𝑢.(3.29) Define ̇𝑉𝑛=̇𝑉𝑛1+𝑒𝑛𝜃𝑇𝑓𝜑𝑓𝑥𝑛+𝑎𝑛𝑛1𝑙=1𝜕𝑎𝑛1𝑥𝑙+1𝜕𝑥𝑙+𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑟(𝑙1)𝑟𝑙+𝑑𝑛𝑛1𝑙=1𝜕𝑎𝑛1𝜕𝑥𝑙𝑑𝑙+𝜀𝑓+𝜀𝑔𝑢.(3.30)  Equation  (3.29) can be rewritten as ̇̇𝑉𝑉=𝑛+𝑒𝑛̂𝜃𝑇𝑔𝜑𝑔𝑥𝑛𝑑𝑢𝑛𝑟𝑑𝑟𝑎𝑛+̃𝜃𝑇𝑓𝑃𝑃1𝜑𝑓𝑥𝑛𝑒𝑛̇̂𝜃𝑓+̃𝜃𝑇𝑔𝑄𝑄1𝜑𝑔𝑥𝑛𝑒𝑛𝜃𝑢̇̂𝑔,(3.31) Define ̇𝑉=𝑛𝑙=1𝜏𝑙𝑒2𝑙+𝑛𝑙=1𝜇𝑙.
By using (3.18) and adaptive laws (3.25)-(3.26), (3.31) can be written as ̇𝑉=𝑛𝑙=1𝜏𝑙𝑒2𝑙+𝑛𝑙=1𝜇𝑙=𝑛𝑙=11𝜂𝑙𝑒𝑙12𝜂𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2+14𝜂2𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2𝑛𝑙=1𝜏𝑙𝑒2𝑙.(3.32) Integrating (3.32) from 𝑡=0 to 𝑇 yields 𝑉(𝑇)𝑉(0)𝑛𝑙=1𝑇0𝜏𝑙𝑒2𝑙+14𝜂2𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2𝑑𝑡.(3.33) Furthermore, one can derive that 𝑛𝑙=1𝑇0𝜏𝑙𝑒2𝑙𝑑𝑡𝑉(0)+𝑛𝑙=1𝑇014𝜂2𝑙𝑑𝑙𝑙1𝑘=1𝜕𝑎𝑙1𝜕𝑥𝑘𝑑𝑘2𝑑𝑡.(3.34) This completes the proof of the theorem.

4. Illustrative Examples

In this section, two examples are provided to illustrate the usefulness of our method.

Example 4.1. Consider the following Duffing-Holmes chaotic system [40]: ̇𝑥1=𝑥2+𝑑1(𝑡),𝑑1(𝑡)=0.1sin𝑡+0.2cos𝑡,̇𝑥2=𝑎𝑥2𝑏𝑥31𝑐𝑥1+𝑞cos(𝜔𝑡)+1+cos𝑥1𝑢+𝑑2(𝑡),𝑑2(𝑡)=0.1cos𝑡,(4.1) where 𝑎, 𝑏, 𝑐 and 𝑞 are constants, 𝜔 is the frequency, 𝑑1(𝑡) and 𝑑2(𝑡) are external disturbance, and 𝑢 is the control effort. For observing chaotic unpredictable behavior, the open-loop system behavior with 𝑢=0, 𝑑1(𝑡)=𝑑2(𝑡)=0 was simulated with 𝑎=0.1, 𝑏=1.0, 𝑐=1, 𝑞=12, and 𝜔=1.0. The phase plane plots from an initial condition point (0,0) are shown in Figure 4. The proposed controller is designed to force the system output to track the given desired trajectory 𝑦𝑚=sin𝑡. Now, choose 𝑦=𝑥1, and there exist external disturbances when the chaotic systems have the form of strict-feedback. It is assumed that the external disturbance 𝑑1(𝑡)=0.1sin𝑡+0.2cos𝑡 and 𝑑2(𝑡)=0.1cos𝑡. By Lemma 2.1, we use using some fuzzy rules for approximation of the function 𝑓𝑛(𝑥𝑛(𝑡)) and 𝑔𝑛(𝑥𝑛(𝑡)). The membership functions of the fuzzy sets are expressed as 𝜇𝐹1𝑖𝑥𝑖=1𝑥1+exp5×𝑖,𝜇+2𝐹2𝑖𝑥𝑖𝑥=exp𝑖+12,𝜇𝐹3𝑖𝑥𝑖=exp𝑥2i,𝜇𝐹4𝑖𝑥𝑖𝑥=exp𝑖12,𝜇𝐹5𝑖𝑥𝑖=1𝑥1+exp5×𝑖.2(4.2)
The initial membership functions for 𝑢(𝑘), Δ𝑢(𝑘), and 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) are shown in Figures 13, respectively. In order to evaluate the performance of the adaptive backstepping fuzzy-immune control applied to the above system (4.1), it was compared to fuzzy adaptive control (FAC) [40]. Figures 5 and 6 show the simulation curves of the two controls for 𝑥1 and 𝑥2, for the system described by (4.1) under the initial conditions 𝑥(0)=[2,2] with external disturbance 𝑑1(𝑡)=0.1sin𝑡+0.2cos𝑡 and 𝑑2(𝑡)=0.1cos𝑡. In order to compare the stabilization and tracking performance, we consider the example introduced in [40]. FAC and the proposed method require 3.5~4.0 seconds and 2.5~3.0 seconds to track the reference signal, respectively. The proposed scheme can also suppress system uncertainty and disturbance, and its ISE (integral square error criterion) is lower than FAC. Figures 7 and 8 show 𝑥1 and 𝑥2 for the nonlinear system described by (4.1) under initial states [1.5,1.5], [5,2], and [5,2] with external disturbance 𝑑1(𝑡)=0.1sin𝑡+0.2cos𝑡 and 𝑑2(𝑡)=0.1cos𝑡, respectively. Finally, all simulation results are given in Table 1.

tab1
Table 1: Experiments results for FAC and proposed control methods.
439324.fig.004
Figure 4: State trajectories of the Duffing equation.
439324.fig.005
Figure 5: The tracking result for 𝑥1.
439324.fig.006
Figure 6: The tracking result for 𝑥2.
439324.fig.007
Figure 7: Trajectories of state 𝑥1 under initial conditions.
439324.fig.008
Figure 8: Trajectories of state 𝑥2 under initial conditions.

Example 4.2. As a second example, consider the following forced chaotic attractor of the modified Chua circuit. The dynamics of the systems can be described as [44] ̇𝑥1̇𝑥2̇𝑥3=𝑥0100010001𝑥2𝑥3+001𝑥𝑓+1+sin1+𝑑𝑢(𝑡)1𝑑(𝑡)2𝑑(𝑡)3(𝑡).(4.3)
In this simulation, let the sampling time equal 0.01 and the initial system states be [2,1,3]. Now, let 𝑑1(𝑡), 𝑑2(𝑡) and 𝑑3(𝑡) are external disturbance. If 𝑑1(𝑡)=𝑑2(𝑡)=𝑑3(𝑡)=𝑢(𝑡)=0 and 𝑓=(14/1805)𝑥1(168/9025)𝑥2+(1/38)𝑥3(2/45)×((28/321)𝑥1+(7/95)𝑥2+𝑥3)3, then the system (4.3) is chaotic system and the trajectories of the state variables 𝑥1, 𝑥2, and 𝑥3 are shown in Figure 9. By Lemma 2.1, we use some fuzzy rules for approximation of the function 𝑓𝑛(𝑥𝑛(𝑡)) and 𝑔𝑛(𝑥𝑛(𝑡)). In order to control this chaotic system, the proposed control is utilized and the corresponding adaptation laws are also applied such that the system output 𝑦 tracks a desired trajectory. In addition, let the desired output be 𝑦𝑚=sin(𝑡).
Simulated results are demonstrated in Figures 10, 11, 12, 13, 14 and 15. The initial membership functions for 𝑢(𝑘), Δ𝑢(𝑘) and 𝑓(𝑢(𝑘),Δ𝑢(𝑘)) are shown in Figures 13, respectively. In order to evaluate the performance of the adaptive backstepping fuzzy-immune control applied to the above system (4.3), it was compared to robust adaptive fuzzy controller (RAFC) [44]. Figures 1012 show the simulation curves for the system described by (4.3) under the initial conditions 𝑥(0)=[2,1,3] with external disturbance 𝑑1(𝑡)=0.1cos𝑡+0.4cos𝑡, 𝑑2(𝑡)=0.2sin𝑡 and 𝑑3(𝑡)=0.1sin𝑡. RFAC and the proposed method require 3.0~3.2 seconds and 2.5~3.0 seconds to track the reference signal, respectively. The proposed scheme can also suppress disturbance, and its ISE (integral square error criterion) is lower than RAFC. Figures 1315 show the controlled stabilization of 𝑥1, 𝑥2, and 𝑥3 for the nonlinear system described by (4.3) under initial states [4,4,1], and [4,4,1] with external disturbance 𝑑1(𝑡)=0.1cos𝑡+0.4cos𝑡, 𝑑2(𝑡)=0.2sin𝑡 and 𝑑3(𝑡)=0.1sin𝑡. It seems to be satisfactory. Finally, all simulation results are given in Table 2.

tab2
Table 2: Experiments results for RAFC and proposed control methods.
439324.fig.009
Figure 9: Chaotic attractor of the modified Chua circuit.
439324.fig.0010
Figure 10: The tracking result for 𝑥1.
439324.fig.0011
Figure 11: The tracking result for 𝑥2.
439324.fig.0012
Figure 12: The tracking result for 𝑥3.
439324.fig.0013
Figure 13: Trajectories of state 𝑥1 under initial conditions.
439324.fig.0014
Figure 14: Trajectories of state 𝑥2 under initial conditions.
439324.fig.0015
Figure 15: Trajectories of state 𝑥3 under initial conditions.

5. Conclusion

A hybrid optimization algorithm that combines the adaptive backstepping principle and the fuzzy-immune algorithm for a class of chaotic dynamical systems was presented. The four main contributions of this paper are: (1) an adaptive backstepping fuzzy-immune tracking control method for a class of chaotic systems is designed, (2) the controller does not require a priori knowledge of the sign of the control coefficient, (3) fuzzy-immune algorithm is used to self-adjustment controller’s coefficient for the optimal solution, and (4) a correct term is introduced to eliminate internal uncertainty and external disturbance. The proposed hybrid intelligence adaptive backstepping fuzzy-immune controller guarantees closed-loop stability while maitaining the desired tracking performance. Simulation results show that the proposed controller scheme provides better tracking performance than those of two existing methods.

Acknowledgment

This work was supported by the National Science Council of Taiwan, under Grant NSC-99-2221-E-027-101.

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