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

Authors: Kai-Shiuan Shih; Tzuu-Hseng S. Li; Shun-Hung Tsai
Journal: Mathematical Problems in Engineering
 (2011)

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## 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.


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## Body

## 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 [2–10]. Furthermore, the stability of uncertain nonlinear systems has been addressed by the integration of fuzzy logic control and the adaptive laws. [11–20]. Subsequently, several methodologies have been instituted for controlling nonlinear systems [20–29]. 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 [30–32]. 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 [33–40].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 Section2, 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 disturbances(2.1)ẋ1=x2+d1(t),ẋ2=x3+d2(t),⋮ẋn=fn(x̅n(t))+gn(x̅n(t))u(t)+dn(t),y=x1,
wherex̅n(t)=[x1(t),…,xn(t)]T∈Rn, fn(x̅n(t)) and gn(x̅n(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 vectorx̅n(t) to an output variable f(x̅n(t))(2.2)Ri:Ifx1isFi1,x2isFi2,…,xnisFin,thenyisGi,i=1,2,…,m,
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 is(2.3)f̅(x̅n)=∑i=1my̅i(∏l=1nμFil(xl))∑i=1m(∏l=1nμFil(xl)),
where μFil(xl) is the membership of x1 in the fuzzy set Fi1 and y̅i=maxy∈RμGi(y̅i)=1. Then, (2.3) can be rewritten as(2.4)f̅(x̅n)=θfTφf(x̅n),
where θf=[y̅1,y̅2,…,y̅m]T and φf(x̅n(t))=[φ11,…,φ1m]T. We can define φ1i as(2.5)φ1i=(∏l=1nμFil(xl))∑i=1m(∏l=1nμFil(xl)),i=1,…,m.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]).
Letf(x̅n) be continuous functions defined on a compact set U∈Rn and arbitrary ε>0, and there exists a fuzzy logic system f̅(x̅n) in the form of (2.4)  such that
(2.6)sup⁡x∈U|f̅(x̅n)-fn(x̅n)|≤ε.After some simple manipulations in (2.4) and (2.5), we can obtain g̅n(x̅n)=θgTφg(x̅n) as the approximator of gn(x̅n). The nonlinear functions fn(x̅n) and gn(x̅n) requires successful estimates θ̂f and θ̂g in order to perform the performance shown in (2.6).Typically, there exists optimal parameter estimatesθ̅, and the approximation error is the smallest. The optimal parameter estimate is defined as(2.7)θ̅f=argmin⁡θf∈Ωf{sup⁡x̅∈Ωx̅|fn(x̅n)-θfTφf|},θ̅g=argmin⁡θg∈Ωg{sup⁡x̅∈Ωx̅|gn(x̅n)-θgTφg|}.Based on fuzzy mechanism, (2.1) can be rewritten as below:(2.8)ẋ1=x2+d1(t),ẋ2=x3+d2(t),⋮ẋn=θ̅fTφf(x̅n)+θ̅gTφg(x̅n)u(t)+dn(t)+εf+εg,y=x1,
where εf and εg are internal modeling error variables(2.9)εf=fn(x̅n)-θ̅fTφf(x̅n),εg=gn(x̅n)-θ̅gTφg(x̅n),Therefore, the mathematical model includes internal modeling error variables and external disturbance. We will discuss our proposed method in the next section.


## 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 vectorx̅n(t) to an output variable f(x̅n(t))(2.2)Ri:Ifx1isFi1,x2isFi2,…,xnisFin,thenyisGi,i=1,2,…,m,
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 is(2.3)f̅(x̅n)=∑i=1my̅i(∏l=1nμFil(xl))∑i=1m(∏l=1nμFil(xl)),
where μFil(xl) is the membership of x1 in the fuzzy set Fi1 and y̅i=maxy∈RμGi(y̅i)=1. Then, (2.3) can be rewritten as(2.4)f̅(x̅n)=θfTφf(x̅n),
where θf=[y̅1,y̅2,…,y̅m]T and φf(x̅n(t))=[φ11,…,φ1m]T. We can define φ1i as(2.5)φ1i=(∏l=1nμFil(xl))∑i=1m(∏l=1nμFil(xl)),i=1,…,m.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]).
Letf(x̅n) be continuous functions defined on a compact set U∈Rn and arbitrary ε>0, and there exists a fuzzy logic system f̅(x̅n) in the form of (2.4)  such that
(2.6)sup⁡x∈U|f̅(x̅n)-fn(x̅n)|≤ε.After some simple manipulations in (2.4) and (2.5), we can obtain g̅n(x̅n)=θgTφg(x̅n) as the approximator of gn(x̅n). The nonlinear functions fn(x̅n) and gn(x̅n) requires successful estimates θ̂f and θ̂g in order to perform the performance shown in (2.6).Typically, there exists optimal parameter estimatesθ̅, and the approximation error is the smallest. The optimal parameter estimate is defined as(2.7)θ̅f=argmin⁡θf∈Ωf{sup⁡x̅∈Ωx̅|fn(x̅n)-θfTφf|},θ̅g=argmin⁡θg∈Ωg{sup⁡x̅∈Ωx̅|gn(x̅n)-θgTφg|}.Based on fuzzy mechanism, (2.1) can be rewritten as below:(2.8)ẋ1=x2+d1(t),ẋ2=x3+d2(t),⋮ẋn=θ̅fTφf(x̅n)+θ̅gTφg(x̅n)u(t)+dn(t)+εf+εg,y=x1,
where εf and εg are internal modeling error variables(2.9)εf=fn(x̅n)-θ̅fTφf(x̅n),εg=gn(x̅n)-θ̅gTφg(x̅n),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(x̅n(t)) and gn(x̅n(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(3.1)ė1=ẋ1-ṙ=x2-ṙ+d1=e2+a1+d1,
where r is the command trajectory. The first Lyapunov function is defined as
(3.2)V1=e122.
Differentiating (3.2) with respect to time and it is obtained that
(3.3)V̇1=e1ė1=e1(e2+a1+d1)=e1e2+e1(a1+d1).
Define the following stabilizing function:(3.4)a1=-τ1e1-1η12e1,
where τ1 and η1 are positive constants, η1 represent the attenuation level of disturbances. From substitute (3.3) into (3.4), then we obtain
(3.5)V̇1=-τ1e12+e1e2+μ1,
where μ1=-(1/η12)e1+e1d1=-((1/η1)e1-(1/2)η1d1)2+(1/4)η12d12.Step 2.
The derivative ofe2=x2-ṙ-a1 is
(3.6)ė2=ẋ3+d2-r̈-∂a1(x2+d1)∂x1-∂a1ṙ∂r1=e3+a2-∂a1x2∂x1-∂a1ṙ∂r1+(d2-∂a1d1∂x1).
The second Lyapunov function is defined as
(3.7)V2=V1+e222.
Differentiating (3.7) with respect to time, it is obtained that
(3.8)V̇2=e2e3-τ1e12+e2(e1+a2-∂a1x2∂x1-∂a1ṙ∂r1)+μ1+e2d2-e2∂a1d1∂x1.
Define the following stabilizing function:
(3.9)a2=-e1-τ2e2-1η22e2+∂a1∂x1x2+∂a1∂rṙ,
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
(3.10)V̇2=-τ1e12-τ2e22+e2e3+μ1+μ2,
where
(3.11)μ2=-1η22e22+e2(d2-∂a1∂x1d1)=-[1η2e2-12η2(d2-∂a1∂x1d1)]2+14η22(d2-∂a1∂x1d1)2.Step i
After some simple manipulations and the same method, the derivative of generalized error is(3.12)ėi=xi+1+[di-∑l=1i-1(∂ai-1∂xl)dl]-dirdr-(∑l=1i-1∂ai-1xl+1∂xl+∑l=1i-1∂ai-1∂r(l-1)rl).
The ith Lyapunov function is defined as
(3.13)Vi=Vi-1+ei22.
Differentiating (3.13) with respect to time, it is obtained that
(3.14)V̇i=eiei+1-∑l=1iτlel2+∑l=1iηl,
where
(3.15)μl=-[1ηlel-12ηl(dl-∑k=1l-1∂al-1∂xkdk)]2+14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2.
Define the following stabilizing function:(3.16)ai=-ei-1-τiei-1ηi2ei+∑l=1i-1∂ai-1∂xlxl+1+∑l=1i-1∂ai-1∂r(l-1)rl.Step n
The final control law will be determined in the final step, and the derivative of step generalized error can be described as(3.17)ėn=ẋn-dnrdr-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl)=θ̅fTφf(x̅n)+θ̅gTφg(x̅n)u-dnrdr-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl)+dn-∑l=1n-1(∂an-1∂xl)dl+εf+εgu,(3.18)u=1θ̂gTφg(an+dnrdr),θ̂gTφg≠0,(3.19)an=-en-1-τnen-1ηn2en-θ̂fTφf(x̅n)+∑l=1n-1∂an-1∂xlxl+1+∑l=1n-1∂an-1∂rl-1rl,
where ηn is the given correct factor and τn is a positive constant.


### 3.2. Fuzzy-Immune Mechanism Design

Assume that the number of thekth-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](3.20)Te(k)=TH(k)-Ts(k),
where TH(k)=h1ε(k) and Ts(k)=h2f(Te(k),ΔTe(k))ε(k). The feedback control rules are defined as(3.21)u(k)=h1e(k)-h2f(u(k),Δu(k))e(k)=h1[1-h2h1f(u(k),Δu(k))]e(k)=h1[1-ℏf(u(k),Δu(k))]e(k),
where h1, h2, and λ are scaling factors and ℏ=h2/h1 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: (3.22)ℏ=ℏmax⁡-ℏmax⁡-ℏmin⁡kmax⁡×know,
where ℏmax and ℏmin denote the maximum and minimum of ℏ, 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:(3.23)de(k)=e(k)-e(k-1)≥0.Based on the above inferences, the following nine fuzzy rules are suggested.(1)
Ifu(k) is big and Δu(k) is big, then f(u(k),Δu(k)) is small.(2)
Ifu(k) is big and Δu(k) is small, then f(u(k),Δu(k)) is medium.(3)
Ifu(k) is small and Δu(k) is big, then f(u(k),Δu(k)) is medium.(4)
Ifu(k) is medium and Δu(k) is big, thenf(u(k),Δu(k)) is small.(5)
Ifu(k) is big and Δu(k) is medium, thenf(u(k),Δu(k)) is small.(6)
Ifu(k) is medium and Δu(k) is medium, then f(u(k),Δu(k)) is medium.(7)
Ifu(k) is medium and Δu(k) is small, then f(u(k),Δu(k)) is medium.(8)
Ifu(k) is small and Δu(k) is medium, then f(u(k),Δu(k)) is big.(9)
Ifu(k) is small and Δu(k) is small, then f(u(k),Δu(k)) is big.The membership functions foru(k), Δu(k), and  f(u(k),Δu(k)) are shown in Figures 1–3, respectively.Figure 1
Membership functions foru(k).Figure 2
Membership functions forΔu(k).Figure 3
Membership functions forf(u(k),Δu(k)).The output of the controller has the following expression:(3.24)u=1θ̂gTφg(an+dnrdr){h1[1-ℏf(u(k),Δu(k))]en}.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)
(3.25)θ̂.f=P-1φf(x̅n)en,(3.26)θ̂.g=Q-1φg(x̅n)enu,
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 as(3.27)∑l=1n∫0T(-τlel2)dt≤V(0)+∑l=1n∫0T(14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2)dt.Proof.
Firstly, consideration of fuzzy approximating error, the Lyapunov functional is set as (3.28) (3.28)V=Vn-1+12en2+12θ̃fTPθ̃f+12θ̃gTQθ̃g,
where P and Q are positive symmetric matrices.
Differentiating (3.28) with respect to time, it is obtained that
(3.29)V̇=V̇n-1+enėn+θ~fTP-1θ~.f+θ~gTQ-1θ~.g=en-1en+en(θ̂gTφg(x̅n)u-dnrdr-an)+∑l=1n-1(μl-τlel2)+θ̃fTP(P-1φf(x̅n)en-θ̂.f)+θ̃gTQ(Q-1φg(x̅n)enu-θ̂.f)+en((θ̅fTφf(x̅n)+an-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl+dn-∑l=1n-1(∂an-1∂xl)dl+εf+εgu))).
Define
(3.30)V̇n=V̇n-1+en{θ̅fTφf(x̅n)+an-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl+dn-∑l=1n-1(∂an-1∂xl)dl+εf+εgu)}.
Equation  (3.29) can be rewritten as
(3.31)V̇=V̇n+en(θ̂gTφg(x̅n)u-dnrdr-an)+θ~fTP(P-1φf(x̅n)en-θ̂.f)+θ~gTQ(Q-1φg(x̅n)enu-θ̂.g),
Define V̇=-∑l=1nτlel2+∑l=1nμl.
By using (3.18) and adaptive laws (3.25)-(3.26), (3.31) can be written as
(3.32)V̇=-∑l=1nτlel2+∑l=1nμl=∑l=1n{-[1ηlel-12ηl(dl-∑k=1l-1∂al-1∂xkdk)]2+14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2}-∑l=1nτlel2.
Integrating (3.32) from t=0 to T yields
(3.33)V(T)-V(0)≤∑l=1n∫0T(-τlel2+14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2)dt.
Furthermore, one can derive that
(3.34)∑l=1n∫0T(-τlel2)dt≤V(0)+∑l=1n∫0T(14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2)dt.
This completes the proof of the theorem.


## 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(3.1)ė1=ẋ1-ṙ=x2-ṙ+d1=e2+a1+d1,
where r is the command trajectory. The first Lyapunov function is defined as
(3.2)V1=e122.
Differentiating (3.2) with respect to time and it is obtained that
(3.3)V̇1=e1ė1=e1(e2+a1+d1)=e1e2+e1(a1+d1).
Define the following stabilizing function:(3.4)a1=-τ1e1-1η12e1,
where τ1 and η1 are positive constants, η1 represent the attenuation level of disturbances. From substitute (3.3) into (3.4), then we obtain
(3.5)V̇1=-τ1e12+e1e2+μ1,
where μ1=-(1/η12)e1+e1d1=-((1/η1)e1-(1/2)η1d1)2+(1/4)η12d12.Step 2.
The derivative ofe2=x2-ṙ-a1 is
(3.6)ė2=ẋ3+d2-r̈-∂a1(x2+d1)∂x1-∂a1ṙ∂r1=e3+a2-∂a1x2∂x1-∂a1ṙ∂r1+(d2-∂a1d1∂x1).
The second Lyapunov function is defined as
(3.7)V2=V1+e222.
Differentiating (3.7) with respect to time, it is obtained that
(3.8)V̇2=e2e3-τ1e12+e2(e1+a2-∂a1x2∂x1-∂a1ṙ∂r1)+μ1+e2d2-e2∂a1d1∂x1.
Define the following stabilizing function:
(3.9)a2=-e1-τ2e2-1η22e2+∂a1∂x1x2+∂a1∂rṙ,
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
(3.10)V̇2=-τ1e12-τ2e22+e2e3+μ1+μ2,
where
(3.11)μ2=-1η22e22+e2(d2-∂a1∂x1d1)=-[1η2e2-12η2(d2-∂a1∂x1d1)]2+14η22(d2-∂a1∂x1d1)2.Step i
After some simple manipulations and the same method, the derivative of generalized error is(3.12)ėi=xi+1+[di-∑l=1i-1(∂ai-1∂xl)dl]-dirdr-(∑l=1i-1∂ai-1xl+1∂xl+∑l=1i-1∂ai-1∂r(l-1)rl).
The ith Lyapunov function is defined as
(3.13)Vi=Vi-1+ei22.
Differentiating (3.13) with respect to time, it is obtained that
(3.14)V̇i=eiei+1-∑l=1iτlel2+∑l=1iηl,
where
(3.15)μl=-[1ηlel-12ηl(dl-∑k=1l-1∂al-1∂xkdk)]2+14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2.
Define the following stabilizing function:(3.16)ai=-ei-1-τiei-1ηi2ei+∑l=1i-1∂ai-1∂xlxl+1+∑l=1i-1∂ai-1∂r(l-1)rl.Step n
The final control law will be determined in the final step, and the derivative of step generalized error can be described as(3.17)ėn=ẋn-dnrdr-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl)=θ̅fTφf(x̅n)+θ̅gTφg(x̅n)u-dnrdr-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl)+dn-∑l=1n-1(∂an-1∂xl)dl+εf+εgu,(3.18)u=1θ̂gTφg(an+dnrdr),θ̂gTφg≠0,(3.19)an=-en-1-τnen-1ηn2en-θ̂fTφf(x̅n)+∑l=1n-1∂an-1∂xlxl+1+∑l=1n-1∂an-1∂rl-1rl,
where ηn is the given correct factor and τn is a positive constant.


## 3.2. Fuzzy-Immune Mechanism Design

Assume that the number of thekth-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](3.20)Te(k)=TH(k)-Ts(k),
where TH(k)=h1ε(k) and Ts(k)=h2f(Te(k),ΔTe(k))ε(k). The feedback control rules are defined as(3.21)u(k)=h1e(k)-h2f(u(k),Δu(k))e(k)=h1[1-h2h1f(u(k),Δu(k))]e(k)=h1[1-ℏf(u(k),Δu(k))]e(k),
where h1, h2, and λ are scaling factors and ℏ=h2/h1 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: (3.22)ℏ=ℏmax⁡-ℏmax⁡-ℏmin⁡kmax⁡×know,
where ℏmax and ℏmin denote the maximum and minimum of ℏ, 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:(3.23)de(k)=e(k)-e(k-1)≥0.Based on the above inferences, the following nine fuzzy rules are suggested.(1)
Ifu(k) is big and Δu(k) is big, then f(u(k),Δu(k)) is small.(2)
Ifu(k) is big and Δu(k) is small, then f(u(k),Δu(k)) is medium.(3)
Ifu(k) is small and Δu(k) is big, then f(u(k),Δu(k)) is medium.(4)
Ifu(k) is medium and Δu(k) is big, thenf(u(k),Δu(k)) is small.(5)
Ifu(k) is big and Δu(k) is medium, thenf(u(k),Δu(k)) is small.(6)
Ifu(k) is medium and Δu(k) is medium, then f(u(k),Δu(k)) is medium.(7)
Ifu(k) is medium and Δu(k) is small, then f(u(k),Δu(k)) is medium.(8)
Ifu(k) is small and Δu(k) is medium, then f(u(k),Δu(k)) is big.(9)
Ifu(k) is small and Δu(k) is small, then f(u(k),Δu(k)) is big.The membership functions foru(k), Δu(k), and  f(u(k),Δu(k)) are shown in Figures 1–3, respectively.Figure 1
Membership functions foru(k).Figure 2
Membership functions forΔu(k).Figure 3
Membership functions forf(u(k),Δu(k)).The output of the controller has the following expression:(3.24)u=1θ̂gTφg(an+dnrdr){h1[1-ℏf(u(k),Δu(k))]en}.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)
(3.25)θ̂.f=P-1φf(x̅n)en,(3.26)θ̂.g=Q-1φg(x̅n)enu,
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 as(3.27)∑l=1n∫0T(-τlel2)dt≤V(0)+∑l=1n∫0T(14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2)dt.Proof.
Firstly, consideration of fuzzy approximating error, the Lyapunov functional is set as (3.28) (3.28)V=Vn-1+12en2+12θ̃fTPθ̃f+12θ̃gTQθ̃g,
where P and Q are positive symmetric matrices.
Differentiating (3.28) with respect to time, it is obtained that
(3.29)V̇=V̇n-1+enėn+θ~fTP-1θ~.f+θ~gTQ-1θ~.g=en-1en+en(θ̂gTφg(x̅n)u-dnrdr-an)+∑l=1n-1(μl-τlel2)+θ̃fTP(P-1φf(x̅n)en-θ̂.f)+θ̃gTQ(Q-1φg(x̅n)enu-θ̂.f)+en((θ̅fTφf(x̅n)+an-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl+dn-∑l=1n-1(∂an-1∂xl)dl+εf+εgu))).
Define
(3.30)V̇n=V̇n-1+en{θ̅fTφf(x̅n)+an-(∑l=1n-1∂an-1xl+1∂xl+∑l=1n-1∂an-1∂r(l-1)rl+dn-∑l=1n-1(∂an-1∂xl)dl+εf+εgu)}.
Equation  (3.29) can be rewritten as
(3.31)V̇=V̇n+en(θ̂gTφg(x̅n)u-dnrdr-an)+θ~fTP(P-1φf(x̅n)en-θ̂.f)+θ~gTQ(Q-1φg(x̅n)enu-θ̂.g),
Define V̇=-∑l=1nτlel2+∑l=1nμl.
By using (3.18) and adaptive laws (3.25)-(3.26), (3.31) can be written as
(3.32)V̇=-∑l=1nτlel2+∑l=1nμl=∑l=1n{-[1ηlel-12ηl(dl-∑k=1l-1∂al-1∂xkdk)]2+14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2}-∑l=1nτlel2.
Integrating (3.32) from t=0 to T yields
(3.33)V(T)-V(0)≤∑l=1n∫0T(-τlel2+14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2)dt.
Furthermore, one can derive that
(3.34)∑l=1n∫0T(-τlel2)dt≤V(0)+∑l=1n∫0T(14ηl2(dl-∑k=1l-1∂al-1∂xkdk)2)dt.
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]:
(4.1)ẋ1=x2+d1(t),d1(t)=0.1sin⁡t+0.2cos⁡⁡t,ẋ2=-ax2-bx13-cx1+qcos⁡⁡(ωt)+(1+cos⁡⁡x1)u+d2(t),d2(t)=0.1cos⁡⁡t,
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(x̅n(t)) and gn(x̅n(t)). The membership functions of the fuzzy sets are expressed as
(4.2)μFi1(xi)=11+exp⁡⁡(5×(xi+2)),μFi2(xi)=exp⁡⁡[-(xi+1)2],μFi3(xi)=exp⁡⁡[-xi2],μFi4(xi)=exp⁡⁡[-(xi-1)2],μFi5(xi)=11+exp⁡⁡(5×(xi-2)).
The initial membership functions foru(k), Δu(k), and f(u(k),Δu(k)) are shown in Figures 1–3, 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.Table 1
Experiments results for FAC and proposed control methods.
Time (sec)ISEPerformanceFAC [40]3.5~4.00.30Affected by noiseProposed2.5~3.00.05Rejects noiseFigure 4
State trajectories of the Duffing equation.Figure 5
The tracking result forx1.Figure 6
The tracking result forx2.Figure 7
Trajectories of statex1 under initial conditions.Figure 8
Trajectories of statex2 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]
(4.3)[ẋ1ẋ2ẋ3]=[010001000][x1x2x3]+[001][f+(1+sin(x1)u(t))]+[d1(t)d2(t)d3(t)].
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(x̅n(t)) and gn(x̅n(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 Figures10, 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 1–3, 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 10–12 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 13–15 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.Table 2
Experiments results for RAFC and proposed control methods.
Time (sec)ISEPerformanceRAFC [44]3.0~3.20.2Affected by noiseProposed2.5~3.00.05Rejects noiseFigure 9
Chaotic attractor of the modified Chua circuit.Figure 10
The tracking result forx1.Figure 11
The tracking result forx2.Figure 12
The tracking result forx3.Figure 13
Trajectories of statex1 under initial conditions.Figure 14
Trajectories of statex2 under initial conditions.Figure 15
Trajectories of statex3 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.



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