# Adaptive Fuzzy Command Filtered Control for Chua’s Chaotic System

Authors: Lin Wang; Hui Wei; Chunzhi Yang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, we propose the command filtered adaptive fuzzy backstepping control (AFBC) approach for Chua’s chaotic system with external disturbance. Based on two proposed first-order command filters, the convergence of tracking errors as well as the problem of “explosion of complexity” in traditional backstepping design procedure is solved. In the command filtered AFBC design, we do not need to calculate the complicated partial derivatives of the virtual control inputs. Fuzzy logic systems (FLSs) are used to identify the system uncertainties in real time. Based on Lyapunov stability criterion, the proposed controller can guarantee that all signals in the closed-loop system keep bounded, and the tracking errors converge to a small region eventually. Finally, simulation studies have been provided to verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

It is well known that adaptive backstepping control (ABC) is an effective technique for controlling nonlinear systems in parameter strict-feedback form [1]. For ABC of strict-feedback nonlinear systems without system uncertainties and external disturbances, this issue has been studied by using many control approaches [2–4]. Based on the sliding mode filters, [5, 6] estimated the command derivatives in the design of ABC. Linear filters for derivative generation were considered in [7]. Then, Farrell et al. in [8] introduced a command filtered backstepping control (CFBC) method, in which some new approaches were given to indicate that the virtual tracking errors between the signals of the command filtered and standard ABC methods were of ∘1/W, where W represented the frequency of the command filter. Up to now, many command filter control methods have been reported [9–11]. The above literature only addressed the nonadaptive case for nonlinear feedback systems. Design of ABC with complicated situations was given in, for instance, [12–20]. It should be mentioned that the dimension of the input variables of the estimated system must be extended to include the reference trajectory and its first derivatives. However, the aforementioned works studied the approximation problem of the command derivatives, but the resulting implementation does not achieve the theoretical guarantees of the ABC design. That is to say, new approaches are expected to solve this problem.It has been shown that modeling of plant systems is badly affected by system uncertainties, i.e., parameter uncertainties, modeling errors, external disturbances, etc. This strongly motivates the study to design a robust, flexible, and effective controller, which can suppress complexities that demean the exhibition of the plants [21–37]. For backstepping control of nonlinear systems subject to system uncertainties, some control methods have been proposed, for example, in [38–43]. On the other hand, to tackle system uncertainties, scientists and researchers have proposed a lot of intelligent methods such as fuzzy logic systems (FLSs), neural networks, and neurofuzzy systems. In these methods, FLSs have been shown to be most successful and popular [44–48]. Following later advancement in intelligent control techniques, adaptive fuzzy controllers were developed such as fuzzy gain scheduled PID controller, fuzzy model reference adaptive controller, and self-organizing fuzzy controller. Adaptive fuzzy backstepping control (AFBC) methods also have been reported recently, for example, in [2, 4, 38, 45, 49, 50]. In [4], AFBC has been established for fractional-order strict-feedback systems. In [45], AFBC has been given for uncertain nonlinear systems with input saturation. Wang et al. introduced a command filtered AFBC approach for uncertain nonlinear systems, where the “explosion of complexity” problem in backstepping design and chartering phenomenon were solved [49]. However, in their work, external disturbance was not considered, and a complicated second-order filter was used in the controller design.Motivated by above discussion, this paper will investigate the control uncertain Chua’s chaotic system with external disturbance by means of command filtered AFBC. Combining the ABC method and command filter, a robust command filter AFBC is established. The proposed method can guarantee that all signals in the closed-loop system remain bounded, and the tracking errors converge to a small neighborhood of the origin eventually. The main contributions of this paper can be summarized as follows.(1) The proposed command filter AFBC method works well even in the presence of full unknown system structure and external disturbance. A simple first-order filter has been introduced. Compared with the filter introduced in [49], our method is simpler and easier to be established. The proposed command filter guarantees that the commanded tracking error as well as its first derivative satisfies our control objective.(2) By using the proposed command filter, the conventional “explosion of complexity” problem can be avoided. That is to say, the complicated calculation of partial derivative of the virtual control input is unnecessary. Our controller and adaptation laws are more concise compared with dynamic surface control approach, for example, in [51, 52].The structure of this paper is arranged as follows. Section2 gives the description of FLSs. The description of the problem and the controller design as well as the stability analysis are included in Section 3. The simulation results are indicated in Section 4. Finally, Section 5 gives a brief conclusion of this paper.


## 2. Description of FLS

A FLS contains four parts, i.e., the knowledge base, fuzzifier, fuzzy inference engine basing on the fuzzy rules, and defuzzifier. Thej-th fuzzy rule is written asR(j): if x1 is E1j, x2 is E2j,…,xn is Enj, then f^(x(t)) is Cj(j=1,2,…,N)wherex(t)=[x1(t),x2(t),…,xn(t)]T∈Rn and f^(x(t))∈R are, respectively, the input and the output of fuzzy logic systems. Eij and Cj(i=1,2,…,n) are fuzzy sets belonging to R. The output of fuzzy logic systems can be expressed by (1)f^xt=∑j=1Nθjt∏i=1nμEijxit∑j=1N∏i=1nμEijxit,where θj(t) is a value where fuzzy membership function μCj is maximum. Generally, we can consider that μCj(θj(t))=1, and fuzzy basic function is φj(x(t))=∏i=1nμEij(xi(t))/∑j=1N∏i=1nμEij(xi(t)). Let φ(x(t))=[φ1(x(t)),φ2(x(t)),…,φN(x(t))]T, θ(t)=[θ1(t),θ2(t),…,θN(t)]T, and then output of fuzzy logic systems can be written as (2)f^xt=θTtφxt.Lemma 1.
Suppose thath(x) is a continuous function defined on compact set Ω; for any constants ε>0, there exists a fuzzy logic system approximating function f^(x) forming (2) such that (3)supΩhxt-θ^Ttφxt≤ε,where θ^(t) is an estimator of optimal vector θ∗.


## 3. Main Results

### 3.1. Problem Description

The controlled Chua’s system is described as(4)x˙1t=αx2t-αx1t-fx1t+g1x1t,x˙2t=x3t+x1t-x2t+g2x-2t,x˙3t=βx2t-γx3t+g3xt+dt+ut,yt=x1t,with (5)fx1t=ax1t+bx1t+1-x1t-1where α,β,γ,a, and b are system parameters, y∈R represents the system output, g1(x1(t)), g2(x-2(t)), and g(x(t)) are the system uncertainties with x-2(t)=[x1(t),x2(t)]T∈R2, x(t)=[x1(t),x2(t),x3(t)]T∈R3, d(t)∈R is an unknown external disturbance, and u(t)∈R denotes the control input.Define the output tracking errore1(t)=x1(t)-x1c(t) where x1c(t)∈R is a known smooth enough referenced signal. The paper aims to design a proper controller u(t) such that the tracking error e1(t) tends to an arbitrary small region.


### 3.2. Controller Design

To meet the control objective, a robust command filtered backstepping controller that contains three steps will be constructed.Step 1.
Consider the first dynamical equation in system (4): (6)x˙1t=αx2t+Δg1x1t,where Δg1(x1(t))=-αx1(t)-f(x1(t))+g1(x1(t)) is an unknown nonlinear function. Thus, Δg1(x1(t)) can be approximated through FLS (2) as (7)Δg1x1t=θ1Ttφ1x1t=θ1∗Tφ1x1t+ε1twhere θ1∗T represents the optimal fuzzy parameter and ε1(t) is the optimal approximation error. Then, the virtual input can be designed as (8)z2t=-1αk1e1t+θ1Ttφ1x1t+ε^1tarctane~1tλ1-x˙1ctwhere ε^1(t) is the estimation of the upper bound of the fuzzy approximation error ε1(t), k1 and λ1 are two positive design parameters, and e~1(t) is the compensated tracking error that will be defined later. Then, it follows from (6), (7), and (8) that (9)e˙1t=αx2t+Δg1x1t-x˙1ct=αx2t-z2t+Δg1x1t-x˙1ct+αz2t=-k1e1t+αx2t-z2t+θ1∗Tφ1x1t+ε1t-θ1Ttφ1x1t-ε^1tarctane~1tλ1=-k1e1t+αx2ct-z2t-θ~1Ttφ1x1t+ε1t-ε^1tarctane~1tλ1+αe2twhere θ~1T(t)=θ1T(t)-θ1∗T is the fuzzy parameter estimation error, e2(t)=x2(t)-x2c(t) is the command filtered tracking error, and x2c will be given later. The compensated tracking error signal can be defined as (10)e~1t=e1t-ζ1twhere ζ1(t) is an added term which is the solution of the following filter: (11)ζ˙1t=-k1ζ1t+αx2ct-z2t+αζ2twhere ζ2(t) will be given in Step 2 and x2c(t) is obtained by the filter (12)x˙2ct=-ϖ2x2ct-z2twith ϖ2>0 being a design parameter. The initial condition for x2c(t) is x2c(0)=0. Thus, adaptation laws for θ1(t) and ε^1(t) are designed as (13)θ˙1t=c11e~1tφ1x1t-c11c12θ1tand (14)ε^˙1t=c41e~1ttanhe~1tλ1-c41c42ε^1t,respectively, where c11,c12,c41, and c42 are all positive design parameters.Step 2.
It follows from (4) that (15)x˙2t=x3t+Δg2x-2twhere Δg2(x-2(t))=x1(t)-x2(t)+g2(x-2(t)) is an unknown nonlinear function, which can be approximated through FLS (2) by (16)Δg2x-2t=θ2Ttφ2x-2t=θ2∗Tφ2x-2t+ε2t.
The virtual tracking errors, similar to that in Step1, are defined as (17)e2t=x2t-x2ct,e~2t=e2t-ζ2t,with (18)ζ˙2t=-k2ζ2t+x3ct-z3t+ζ3twhere ζ2(0)=0, ζ3(t) will be given in Step 3, (19)x˙3ct=-ϖ3x3ct-z3t,z3(t) is the virtual control input designed as (20)z3t=-k2e2t-θ2Ttφ2x-2t-ε^2tarctane~2tλ2+x˙2ct-αe~1twith k2 and λ2 being positive design parameters. Adaptation laws for θ2(t) and ε^2(t) can be given as (21)θ˙2t=c21e~2tφ2x-2t-c21c22θ2tand (22)ε^˙2t=c51e~2ttanhe~2tλ1-c51c52ε^2twhere c21,c22,c51, and c52 are all positive design parameters. Based on above discussion, we have (23)e˙2t=x3t+Δg2x-2t-x˙2ct=x3t-z3t+Δg3x-2t-x˙2ct+z3t=-k2e2t+x3t-z3t+θ2∗Tφ2x-2t+ε2t-θ2Ttφ2x-2t-ε^2tarctane~2tλ2-αe~1t=-k2e2t+x3ct-z3t-θ~2Ttφ2x-2t+ε2t-ε^2tarctane~2tλ2-αe~1t+e3t.Step 3.
According to (4), we have (24)x˙3t=Δg3xt+dt+utwith Δg3(x(t))=βx2(t)-γx3(t)+g3(x(t)) being an unknown nonlinear function that can be approximated by (25)Δg3x-t=θ3Ttφ3x-t=θ3∗Tφ3x-t+ε3t.
The control input is designed as(26)ut=-k3e3t-θ3Ttφ3x-t-ε^2t+d^tarctane~3tλ3+x˙3ct-e2twhere k3,λ3 are two positive design parameters and d^(t) is the estimation of external disturbance d(t). The compensated tracking errors are defined by (27)e3t=x3t-x3ct,e~3t=e3t-ζ3t,where (28)ζ˙3t=-k3ζ3t-ζ2t.
We design the following adaptation laws:(29)θ˙3t=c31e~3tφ2x-t-c31c32θ3t,(30)ε^˙3t=c61e~3ttanhe~3tλ3-c61c62ε^3tand (31)d^˙t=c71e~3ttanhe~3tλ3-c71c72d^twhere c31,c32,c61, c62, c71, and c72 are all positive design parameters. Thus, (32)e˙3t=Δg3x-t-x˙3ct+dt+ut=-k3e3t+θ3∗Tφ3x-t+ε3t+dt-d^tarctane~3tλ3-θ3Ttφ3x-t-ε^3tarctane~3tλ3-e2t=-k3e3t-θ~3Ttφ3x-t+ε3t+dt-d^tarctane~3tλ3-ε^3tarctane~3tλ3-e2t.Here d~(t)=d^(t)-d(t) is the estimation error of d(t).To proceed, we need the following assumption and lemma.Assumption 2.
The external disturbance and the fuzzy approximate errors are bounded; i.e., there exist some positive constantsd∗ and εi∗ such that |d(t)|≤d∗ and εi(t)≤εi∗.Lemma 3 (see [49]).
Suppose thatλ>0; then it holds that (33)y-ytanhyλ≤κλwhere κ=0.2785 is a constant.


### 3.3. Stability Analysis

According to the above analysis, the dynamical equations for the tracking errors are obtained as(34)e~˙1t=e˙1t-ζ˙1t=-k1e1t+αx2ct-z2t-θ~1Ttφ1x1t+ε1t-ε^1tarctane~1tλ1+αe2t-ζ˙1t=-k1e1t-θ~1Ttφ1x1t+ε1t-k1ζ1t-ε^1tarctane~1tλ1+αe2t-αζ2t=-k1e~1t+αe~2t-θ~1Ttφ1x1t+ε1t-ε^1tarctane~1tλ1,(35)e~˙2t=e˙2t-ζ˙2t=-k2e2t+x3ct-z3t-θ~2Ttφ2x-2t+ε2t-ε^2tarctane~2tλ2-αe~1t+e3t-ζ˙2t=-k2e2t-θ~2Ttφ2x-2t+ε2t-ζ3t-ε^2tarctane~2tλ2-αe~1t+e3t-k2ζ2t=-k2e~2t-θ~2Ttφ2x-2t+ε2t+e~3t-ε^2tarctane~2tλ2-αe~1t,(36)e~˙3t=e˙3t-ζ˙3t=-k3e3t-θ~3Ttφ3x-t+ε3t-d^tarctane~3tλ3-ζ˙3t-e2t-ε^3tarctane~3tλ3+dt=-k3e~3t-θ~3Ttφ3x-t+ε3t+dt-d^tarctane~3tλ3-e~2t-ε^3tarctane~3tλ3.Remark 4.
In this paper, to cancel the effect of the signalsxic(t)-zi(t), i=1,2,3, three filters, i.e., (11), (18), and (28) have been introduced. It should be pointed out that the proposed filters can guarantee the boundedness of the added signals ζ1(t),ζ2(t),ζ3(t). The stability analysis for these signals is presented in Theorem 5.Theorem 5.
If the input signalsxic(t)-zi(t), i=1,2,3, satisfy |xic(t)-zi(t)|≤B where B is a positive constant, then the filters defined as (11), (18), and (28) have state variables bounded by (37)ζt≤A2k_1-e-2k_twhere ζ(t)=[ζ1(t),ζ2(t),ζ3(t)]T∈R3, k_=1/2min{k1,k2,k3}, and A=B+α.Proof.
The Lyapunov function candidate is chosen asV1(t)=1/2ζ(t)2. According to (11), (18), and (28), the derivative of V1(t) with respect to time can be given as (38)V˙1t=∑i=13ζitζ˙it=-∑i=13kiζi2t+αζ1tx2ct-z2t+ζ2tx3ct-z3t≤-2k_ζt2+Aζt≤-4k_V1t+2AV1t.Thus, (38) implies that (37) holds.The main results of this section are included in the following theorem.Theorem 6.
Consider system (4) under Assumption 2. The virtual control inputs are chosen as (8) and (20). The filters are given as (11), (12), (18), (19), and (28). The fuzzy parameters are updated by (13), (21), and (29). The estimation of the fuzzy approximation errors is updated by (14), (22), and (30). The estimation of d(t) is given as (31). Then, the control input (26) guarantees that the tracking errors e~1(t), e~2(t), and e~3(t) converge to a small region of zero if proper design parameters are chosen.Proof.
Let the Lyapunov function candidate be(39)Vt=12∑i=13e~i2t+∑i=1312ci1θ~iTtθ~it+∑i=1312ci+3,1ε~i2t+12c71d~2twhere ε~i(t)=ε^i(t)-εi∗, d~(t)=d^(t)-d∗ are estimation errors. It follows from (34), (35), (36), Assumption 2, and Lemma 3 that (40)
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According to (40), (41), (42), and (43), the derivative of Lyapunov function (39) can be obtained as (44)
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5where C1=2min{k1,k2,k3}, C2=min{c11c12,c21c22,c31c32}, C3=min{c41c42,c51c52,c61c62}, C4=c71c72, and C5=κλ3d∗+κ∑i=13‍λiεi∗+∑i=13‍ci+3,2/2εi∗2+∑i=13‍ci2/2θi∗Tθi∗+c72/2 are positive constants.
According to (44) one knows that when t→∞, 1/2∑i=13‍e~i2(t)≤C5/C1, ∑i=13‍1/2ci1θ~iT(t)θ~i(t)≤C5/C2, ∑i=13‍1/2ci+3,1ε~i2(t)≤C5/C3, and 1/2c71d~2(t)≤C5/C4. That is to say, all signals in the closed-loop system will keep bounded. The tracking errors e~1(t), e~2(t), and e~3(t) will eventually converge to a small region of zero if proper design parameters are chosen (small C5 and large C1).


## 3.1. Problem Description

The controlled Chua’s system is described as(4)x˙1t=αx2t-αx1t-fx1t+g1x1t,x˙2t=x3t+x1t-x2t+g2x-2t,x˙3t=βx2t-γx3t+g3xt+dt+ut,yt=x1t,with (5)fx1t=ax1t+bx1t+1-x1t-1where α,β,γ,a, and b are system parameters, y∈R represents the system output, g1(x1(t)), g2(x-2(t)), and g(x(t)) are the system uncertainties with x-2(t)=[x1(t),x2(t)]T∈R2, x(t)=[x1(t),x2(t),x3(t)]T∈R3, d(t)∈R is an unknown external disturbance, and u(t)∈R denotes the control input.Define the output tracking errore1(t)=x1(t)-x1c(t) where x1c(t)∈R is a known smooth enough referenced signal. The paper aims to design a proper controller u(t) such that the tracking error e1(t) tends to an arbitrary small region.


## 3.2. Controller Design

To meet the control objective, a robust command filtered backstepping controller that contains three steps will be constructed.Step 1.
Consider the first dynamical equation in system (4): (6)x˙1t=αx2t+Δg1x1t,where Δg1(x1(t))=-αx1(t)-f(x1(t))+g1(x1(t)) is an unknown nonlinear function. Thus, Δg1(x1(t)) can be approximated through FLS (2) as (7)Δg1x1t=θ1Ttφ1x1t=θ1∗Tφ1x1t+ε1twhere θ1∗T represents the optimal fuzzy parameter and ε1(t) is the optimal approximation error. Then, the virtual input can be designed as (8)z2t=-1αk1e1t+θ1Ttφ1x1t+ε^1tarctane~1tλ1-x˙1ctwhere ε^1(t) is the estimation of the upper bound of the fuzzy approximation error ε1(t), k1 and λ1 are two positive design parameters, and e~1(t) is the compensated tracking error that will be defined later. Then, it follows from (6), (7), and (8) that (9)e˙1t=αx2t+Δg1x1t-x˙1ct=αx2t-z2t+Δg1x1t-x˙1ct+αz2t=-k1e1t+αx2t-z2t+θ1∗Tφ1x1t+ε1t-θ1Ttφ1x1t-ε^1tarctane~1tλ1=-k1e1t+αx2ct-z2t-θ~1Ttφ1x1t+ε1t-ε^1tarctane~1tλ1+αe2twhere θ~1T(t)=θ1T(t)-θ1∗T is the fuzzy parameter estimation error, e2(t)=x2(t)-x2c(t) is the command filtered tracking error, and x2c will be given later. The compensated tracking error signal can be defined as (10)e~1t=e1t-ζ1twhere ζ1(t) is an added term which is the solution of the following filter: (11)ζ˙1t=-k1ζ1t+αx2ct-z2t+αζ2twhere ζ2(t) will be given in Step 2 and x2c(t) is obtained by the filter (12)x˙2ct=-ϖ2x2ct-z2twith ϖ2>0 being a design parameter. The initial condition for x2c(t) is x2c(0)=0. Thus, adaptation laws for θ1(t) and ε^1(t) are designed as (13)θ˙1t=c11e~1tφ1x1t-c11c12θ1tand (14)ε^˙1t=c41e~1ttanhe~1tλ1-c41c42ε^1t,respectively, where c11,c12,c41, and c42 are all positive design parameters.Step 2.
It follows from (4) that (15)x˙2t=x3t+Δg2x-2twhere Δg2(x-2(t))=x1(t)-x2(t)+g2(x-2(t)) is an unknown nonlinear function, which can be approximated through FLS (2) by (16)Δg2x-2t=θ2Ttφ2x-2t=θ2∗Tφ2x-2t+ε2t.
The virtual tracking errors, similar to that in Step1, are defined as (17)e2t=x2t-x2ct,e~2t=e2t-ζ2t,with (18)ζ˙2t=-k2ζ2t+x3ct-z3t+ζ3twhere ζ2(0)=0, ζ3(t) will be given in Step 3, (19)x˙3ct=-ϖ3x3ct-z3t,z3(t) is the virtual control input designed as (20)z3t=-k2e2t-θ2Ttφ2x-2t-ε^2tarctane~2tλ2+x˙2ct-αe~1twith k2 and λ2 being positive design parameters. Adaptation laws for θ2(t) and ε^2(t) can be given as (21)θ˙2t=c21e~2tφ2x-2t-c21c22θ2tand (22)ε^˙2t=c51e~2ttanhe~2tλ1-c51c52ε^2twhere c21,c22,c51, and c52 are all positive design parameters. Based on above discussion, we have (23)e˙2t=x3t+Δg2x-2t-x˙2ct=x3t-z3t+Δg3x-2t-x˙2ct+z3t=-k2e2t+x3t-z3t+θ2∗Tφ2x-2t+ε2t-θ2Ttφ2x-2t-ε^2tarctane~2tλ2-αe~1t=-k2e2t+x3ct-z3t-θ~2Ttφ2x-2t+ε2t-ε^2tarctane~2tλ2-αe~1t+e3t.Step 3.
According to (4), we have (24)x˙3t=Δg3xt+dt+utwith Δg3(x(t))=βx2(t)-γx3(t)+g3(x(t)) being an unknown nonlinear function that can be approximated by (25)Δg3x-t=θ3Ttφ3x-t=θ3∗Tφ3x-t+ε3t.
The control input is designed as(26)ut=-k3e3t-θ3Ttφ3x-t-ε^2t+d^tarctane~3tλ3+x˙3ct-e2twhere k3,λ3 are two positive design parameters and d^(t) is the estimation of external disturbance d(t). The compensated tracking errors are defined by (27)e3t=x3t-x3ct,e~3t=e3t-ζ3t,where (28)ζ˙3t=-k3ζ3t-ζ2t.
We design the following adaptation laws:(29)θ˙3t=c31e~3tφ2x-t-c31c32θ3t,(30)ε^˙3t=c61e~3ttanhe~3tλ3-c61c62ε^3tand (31)d^˙t=c71e~3ttanhe~3tλ3-c71c72d^twhere c31,c32,c61, c62, c71, and c72 are all positive design parameters. Thus, (32)e˙3t=Δg3x-t-x˙3ct+dt+ut=-k3e3t+θ3∗Tφ3x-t+ε3t+dt-d^tarctane~3tλ3-θ3Ttφ3x-t-ε^3tarctane~3tλ3-e2t=-k3e3t-θ~3Ttφ3x-t+ε3t+dt-d^tarctane~3tλ3-ε^3tarctane~3tλ3-e2t.Here d~(t)=d^(t)-d(t) is the estimation error of d(t).To proceed, we need the following assumption and lemma.Assumption 2.
The external disturbance and the fuzzy approximate errors are bounded; i.e., there exist some positive constantsd∗ and εi∗ such that |d(t)|≤d∗ and εi(t)≤εi∗.Lemma 3 (see [49]).
Suppose thatλ>0; then it holds that (33)y-ytanhyλ≤κλwhere κ=0.2785 is a constant.


## 3.3. Stability Analysis

According to the above analysis, the dynamical equations for the tracking errors are obtained as(34)e~˙1t=e˙1t-ζ˙1t=-k1e1t+αx2ct-z2t-θ~1Ttφ1x1t+ε1t-ε^1tarctane~1tλ1+αe2t-ζ˙1t=-k1e1t-θ~1Ttφ1x1t+ε1t-k1ζ1t-ε^1tarctane~1tλ1+αe2t-αζ2t=-k1e~1t+αe~2t-θ~1Ttφ1x1t+ε1t-ε^1tarctane~1tλ1,(35)e~˙2t=e˙2t-ζ˙2t=-k2e2t+x3ct-z3t-θ~2Ttφ2x-2t+ε2t-ε^2tarctane~2tλ2-αe~1t+e3t-ζ˙2t=-k2e2t-θ~2Ttφ2x-2t+ε2t-ζ3t-ε^2tarctane~2tλ2-αe~1t+e3t-k2ζ2t=-k2e~2t-θ~2Ttφ2x-2t+ε2t+e~3t-ε^2tarctane~2tλ2-αe~1t,(36)e~˙3t=e˙3t-ζ˙3t=-k3e3t-θ~3Ttφ3x-t+ε3t-d^tarctane~3tλ3-ζ˙3t-e2t-ε^3tarctane~3tλ3+dt=-k3e~3t-θ~3Ttφ3x-t+ε3t+dt-d^tarctane~3tλ3-e~2t-ε^3tarctane~3tλ3.Remark 4.
In this paper, to cancel the effect of the signalsxic(t)-zi(t), i=1,2,3, three filters, i.e., (11), (18), and (28) have been introduced. It should be pointed out that the proposed filters can guarantee the boundedness of the added signals ζ1(t),ζ2(t),ζ3(t). The stability analysis for these signals is presented in Theorem 5.Theorem 5.
If the input signalsxic(t)-zi(t), i=1,2,3, satisfy |xic(t)-zi(t)|≤B where B is a positive constant, then the filters defined as (11), (18), and (28) have state variables bounded by (37)ζt≤A2k_1-e-2k_twhere ζ(t)=[ζ1(t),ζ2(t),ζ3(t)]T∈R3, k_=1/2min{k1,k2,k3}, and A=B+α.Proof.
The Lyapunov function candidate is chosen asV1(t)=1/2ζ(t)2. According to (11), (18), and (28), the derivative of V1(t) with respect to time can be given as (38)V˙1t=∑i=13ζitζ˙it=-∑i=13kiζi2t+αζ1tx2ct-z2t+ζ2tx3ct-z3t≤-2k_ζt2+Aζt≤-4k_V1t+2AV1t.Thus, (38) implies that (37) holds.The main results of this section are included in the following theorem.Theorem 6.
Consider system (4) under Assumption 2. The virtual control inputs are chosen as (8) and (20). The filters are given as (11), (12), (18), (19), and (28). The fuzzy parameters are updated by (13), (21), and (29). The estimation of the fuzzy approximation errors is updated by (14), (22), and (30). The estimation of d(t) is given as (31). Then, the control input (26) guarantees that the tracking errors e~1(t), e~2(t), and e~3(t) converge to a small region of zero if proper design parameters are chosen.Proof.
Let the Lyapunov function candidate be(39)Vt=12∑i=13e~i2t+∑i=1312ci1θ~iTtθ~it+∑i=1312ci+3,1ε~i2t+12c71d~2twhere ε~i(t)=ε^i(t)-εi∗, d~(t)=d^(t)-d∗ are estimation errors. It follows from (34), (35), (36), Assumption 2, and Lemma 3 that (40)
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By using the adaptations laws (13), (14), (21), (22), (29), (30), and (31), we have(41)∑i=131ci1θ~iTtθ~˙it=∑i=131ci1θ~iTtθ˙it=e~2tθ~2Ttφ2x-2t+e~3tθ~3Ttφ3x-t+e~1tθ~1Ttφ1x1t-∑i=13ci2θ~iTtθit=e~2tθ~2Ttφ2x-2t+e~3tθ~3Ttφ3x-t+e~1tθ~1Ttφ1x1t-∑i=13ci2θ~iTtθ~it+θi∗≤e~2tθ~2Ttφ2x-2t+e~3tθ~3Ttφ3x-t+e~1tθ~1Ttφ1x1t-∑i=13ci22θ~iTtθ~it+∑i=13ci22θi∗Tθi∗,(42)∑i=1312ci+3,1ε~itε~˙it=∑i=1312ci+3,1ε~itε^˙it=∑i=13ε~ite~ittanhe~itλi-ci+3,2ε^it=∑i=13ε~ite~ittanhe~itλi-∑i=13ci+3,2ε~itε~it+εi∗≤∑i=13ε~ite~ittanhe~itλi-∑i=13ci+3,22ε~i2t+∑i=13ci+3,22εi∗2,(43)1c71d~td^˙t=d~te~3ttanhe~3tλ3-c72d~td~t+d∗≤d~te~3ttanhe~3tλ3-c722d~2t+c722d∗2.
According to (40), (41), (42), and (43), the derivative of Lyapunov function (39) can be obtained as (44)
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C
5where C1=2min{k1,k2,k3}, C2=min{c11c12,c21c22,c31c32}, C3=min{c41c42,c51c52,c61c62}, C4=c71c72, and C5=κλ3d∗+κ∑i=13‍λiεi∗+∑i=13‍ci+3,2/2εi∗2+∑i=13‍ci2/2θi∗Tθi∗+c72/2 are positive constants.
According to (44) one knows that when t→∞, 1/2∑i=13‍e~i2(t)≤C5/C1, ∑i=13‍1/2ci1θ~iT(t)θ~i(t)≤C5/C2, ∑i=13‍1/2ci+3,1ε~i2(t)≤C5/C3, and 1/2c71d~2(t)≤C5/C4. That is to say, all signals in the closed-loop system will keep bounded. The tracking errors e~1(t), e~2(t), and e~3(t) will eventually converge to a small region of zero if proper design parameters are chosen (small C5 and large C1).


## 4. Simulation Example

Let the system parameters beα=11.25,β=18.6,γ=0,a=-0.68,b=-0.545, and the initial condition be x(0)=[1.5,-1,0.5]T. Then, the uncontrolled system (4) (i.e., gi(·)≡d(t)≡u(t)≡0,i=1,2,3) shows complicated behavior, which is depicted in Figure 1.Figure 1
Dynamical behavior of system (4) in (a) x1-x2-x3, (b) x1-x2 plane, (c) x1-x3 plane, and (d) x2-x3 plane.
(a) (b) (c) (d)In simulation, the system uncertainties are chosen asg1(x1(t))=sin(x1(t)), g2(x-2(t))=sinx12(t)+x22(t), and g3(x(t))=cosx12(t)+x22(t)+x32(t). The disturbance is selected as d(t)=0.2sint+0.1cost. The referenced signal x1c(t) is defined by (45)x1ct=0t∈0,6,1.5t>6.There are three FLSs used. For the first FLS, the input variable isx1(t), and we define 5 Gaussian membership function distributed on interval [-55]. For the second one, the input variables are x1(t) and x2(t). For each input, we define 5 Gaussian membership functions distributed on interval [-55]. For the last one, the input variables are x1(t), x2(t), and x3(t). For inputs x1(t) and x2(t), we define 5 Gaussian membership functions distributed on interval [-55], and for input x3(t), we define 5 Gaussian membership functions distributed on interval [-1010]. The initial conditions for FLSs are θ1(0)=05×1, θ2(0)=025×1, and θ3(0)=0125×1.The simulation results are presented in Figures2–5. It has been shown that in Figure 2, the state x1(t) tracks the referenced signal x1c(t)=0 for t≤5 in about 2.5 seconds and tends to x1c(t)=1.5 for t>5 in a short time. The tracking error tends to zero rapidly. Figure 3 shows the time response of the control input u(t). It should be pointed out that the proposed controller, the commonly used term sign(·), is not used in this paper. That is to say, our controller is smooth and bounded, just as indicated in Figure 3. Tracking errors e2(t) and e3(t) are presented in Figure 4. The fuzzy systems parameters are given in Figure 5. It is to know that these simulation results are matched with Theorem 6, and the proposed method has good robustness.Figure 2
x 1 ( t ) and e1(t).Figure 3
Control inputu(t).Figure 4
e 2 ( t ) and e3(t).Figure 5
θ 1 ( t ), θ2(t), and θ3(t).


## 5. Conclusions

In this paper, a command filtered AFBC method has been proposed for Chua’s chaotic system with system uncertainties and external disturbances. It has been shown that the proposed method works well without the knowledge of any explicit uncertainty detection. One of the distinctive features of the proposed control approach consists in the fact that the problem of “explosion of complexity” in traditional backstepping design procedure is solved by the proposed first-order filter. In the stability analysis, Lyapunov stability criteria are used. The proposed command filtered AFBC can guarantee the convergence of tracking errors. Simulation results have verified our methods.



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*Source: 9635358-2018-09-20.xml*