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

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.


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 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.

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 where x n t x 1 t , . . ., x n t T ∈ R n , f n x n t and g n x n t are smooth functions, u t and y are control input and output variables, respectively.d 1 t , . . ., d n 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.

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 x n t to an output variable f x n t where m is the number of rules.F i1 , F i2 , . . ., F in and G i are the fuzzy sets and x 1 , x 2 , . . ., x n are states of system.Using a singleton function, center average defuzzification, and product inference, the fuzzy systems output is where μ F il x l is the membership of x 1 in the fuzzy set F i1 and y i max y∈R μ G i y i 1.Then, 2.3 can be rewritten as where θ f y 1 , y 2 , . . ., y m T and ϕ f x n t ϕ 11 , . . ., ϕ 1m T .We can define ϕ 1i as 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 x n be continuous functions defined on a compact set U ∈ R n and arbitrary ε > 0, and there exists a fuzzy logic system f x n in the form of 2.4 such that After some simple manipulations in 2.4 and 2.5 , we can obtain g n x n θ T g ϕ g x n as the approximator of g n x n .The nonlinear functions f n x n and g n x n requires successful estimates θ f and θ g 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 as g n x n − θ T g ϕ g .

2.7
Based on fuzzy mechanism, 2.1 can be rewritten as below: where ε f and ε g are internal modeling error variables 2.9 Therefore, the mathematical model includes internal modeling error variables and external disturbance.We will discuss our proposed method in the next section.

Adaptive Backstepping Fuzzy Controller Technique and Fuzzy-Immune Mechanism
f n x n t and g n x n t are the system dynamic functions, these cannot be exactly obtained in general, d 1 t , . . ., d n 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.

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 where r is the command trajectory.The first Lyapunov function is defined as Differentiating 3.2 with respect to time and it is obtained that V1 e 1 ė1 e 1 e 2 a 1 d 1 e 1 e 2 e 1 a 1 d 1 .

3.3
Define the following stabilizing function: where τ 1 and η 1 are positive constants, η 1 represent the attenuation level of disturbances.From substitute 3.3 into 3.4 , then we obtain e 1 e 2 μ 1 , 3.5 where Step 2. The derivative of e The second Lyapunov function is defined as Define the following stabilizing function: where τ 2 and η 2 are positive constants and η 2 represent the attenuation level of disturbances.From substitute 3.9 into 3.8 , then we obtain e 2 e 3 μ 1 μ 2 , 3.10 where

3.11
Step i After some simple manipulations and the same method, the derivative of generalized error is The ith Lyapunov function is defined as

3.15
Define the following stabilizing function:

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 where η n is the given correct factor and τ n is a positive constant.

Fuzzy-Immune Mechanism Design
Assume that the number of the kth-generation antigens is ε k , the output of the helper Tcells, stimulated by antigens, is T H k , and the suppressor T-cells affect B-cells to the amount of T s k 42, 43 where T H k h 1 ε k and T s k h 2 f T e k , ΔT e k ε k .The feedback control rules are defined as

3.21
where h 1 , h 2 , and λ are scaling factors and h 2 /h 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:  where max and min denote the maximum and minimum of , respectively.k max denotes the total number of evolution generations, and k now 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: 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.
The membership functions for u k , Δu k , and f u k , Δu k are shown in Figures 1-3, respectively.
The output of the controller has the following expression:

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   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 Proof.Firstly, consideration of fuzzy approximating error, the Lyapunov functional is set as 3.28 28 where P and Q are positive symmetric matrices.Differentiating 3.28 with respect to time, it is obtained that

3.30
Equation 3.29 can be rewritten as Mathematical Problems in Engineering 11 By using 3.18 and adaptive laws 3.25 -3.26 , 3.31 can be written as τ l e 2 l .

3.32
Integrating 3.32 from t 0 to T yields This completes the proof of the theorem.

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 : where a, b, c and q are constants, ω is the frequency, d 1 t and d 2 t are external disturbance, and u is the control effort.For observing chaotic unpredictable behavior, the open-loop system behavior with u 0, d 1 t d 2 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 y m sin t.Now, choose y x 1 , and there exist external disturbances when the chaotic systems have the form of strict-feedback.It is assumed that the external disturbance d 1 t 0.1 sin t 0.2 cos t and d 2 t 0.1 cos t.By Lemma 2.1, we use using some fuzzy rules for approximation of the function f n x n t and g n x n t .The membership functions of the fuzzy sets are expressed as

4.2
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 4.1 , it was compared to fuzzy adaptive control FAC 40 .Figures 5 and 6 show the simulation curves of the two controls for x 1 and x 2 , for the system described by 4.1 under the initial conditions x 0 2, 2 with external disturbance d 1 t 0.1 sin t 0.2 cos t and d 2 t 0.1 cos t.In order to compare the stabilization and tracking performance, we consider the example introduced in 40 .FAC and the proposed method require 3.5∼4.0seconds 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 x 1 and x 2 for the nonlinear system described by 4.1 under initial states −1.5, −1.5 , 5, −2 , and −5, 2 with external disturbance d 1 t 0.1 sin t 0.2 cos t and d 2 t 0.1 cos t, respectively.Finally, all simulation results are given in Table 1.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 In this simulation, let the sampling time equal 0.01 and the initial system states be  9.By Lemma 2.1, we use some fuzzy rules for approximation of the function f n x n t and g n x n 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 y m sin t .Simulated results are demonstrated in Figures 10, 11 respectively.In order to evaluate the performance of the adaptive backstepping fuzzyimmune 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.   proposed scheme can also suppress disturbance, and its ISE integral square error criterion is lower than RAFC.show the controlled stabilization of x 1 , x 2 , and x 3 for the nonlinear system described by 4.

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.

Figure 1 :
Figure 1: Membership functions for u k .

1 Figure 13 :
Figure 13: Trajectories of state x 1 under initial conditions.

2 Figure 14 :
Figure 14: Trajectories of state x 2 under initial conditions.

3 Figure 15 :
Figure 15: Trajectories of state x 3 under initial conditions.

Table 1 :
Experiments results for FAC and proposed control methods.
2, 1, 3 .Now, let d 1 t , d 2 t and d 3 t are external disturbance.If d 1 t RFAC and the proposed method require 3.0∼3.2seconds and 2.5∼3.0 seconds to track the reference signal, respectively.The 3 under the initial conditions x 0 2, 1, 3 with external disturbance d 1 t 0.1 cos t 0.4 cos t, d 2 t 0.2 sin t and d 3 t 0.1 sin t.
3 under initial states 4, −4, 1 , and −4, 4, −1 with external disturbance d 1 t 0.1 cos t 0.4 cos t, d 2 t 0.2 sin t and d 3 t 0.1 sin t.It seems to be satisfactory.Finally, all simulation results are given in Table 2.

Table 2 :
Experiments results for RAFC and proposed control methods.