# Synchronization of Chaotic Systems with Dead Zones via Fuzzy Adaptive Variable-Structure Control

Authors: Yongbing Huangfu; Kaijuan Xue
Journal: Complexity
 (2021)

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## Abstract

This work is devoted to solving synchronization problem of uncertain chaotic systems with dead zones. Based on the Lyapunov stability theorems, by using fuzzy inference to estimate system uncertainties and by designing effective fuzzy adaptive controllers, the synchronization between two chaotic systems with dead zones is realized and a fuzzy variable-structure control is implemented. The stability is proven strictly, and all the states and signals are bounded in the closed-loop system. A simulation example is presented to test the theoretical results finally.


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## Body

## 1. Introduction

It is widely known that chaos is almost everywhere in the domain of engineering and science. It is a kind of complex dynamic behavior of nonlinear dynamic systems, and chaos has many applications in mechanical, electronics, and biochemistry fields. There are many common chaotic systems, such as Chen system [1, 2], Lorenz system [3], Genesio–Tesi system [4, 5], Rössler system [6], and Lur’e system [7]. Chaotic systems, as everyone knows, have deterministic behavior; for example, they are extremely sensitive when the initial conditions alter to a small extent and difficult to predict, but their trajectories are bounded in the phase space [8, 9].People always think that chaotic systems cannot be synchronized due to the characteristics of chaos. It was realized first on electronic circuit by Pecora et al. [10, 11]. They found that if the chaotic system can be decomposed into two subsystems, and, in the response system, all the conditional Lyapunov exponents are less than zero, there will be chaotic synchronization effect in the drive and response system [1, 12]. Chaos synchronization means that, for two chaotic systems starting from different initial points, their trajectories gradually tend to be consistent with each other over time, and this synchronization is structurally stable [13–15]. After that, synchronization study in chaotic dynamical systems has received much attention. There are some way to achieve the synchronization of some chaotic systems, for instance, time-delay feedback control [7], active control synchronization [16], impulsive synchronization and synchronization [17, 18], sliding mode synchronization [19, 20], and projective synchronization [15].The input nonlinearities, such as dead zones and saturation, may destroy the control performance of the system, and the control problem of uncertain nonlinear systems with nonlinear input is getting more complicated and receiving more considerable attention [21, 22]. Control performance of the system may be degraded or even cause the instability of the control system due to input nonlinearities, and their synchronization problem becomes more challenging [23]. Please refer to [15, 24–30] for some works about synchronization study in chaotic dynamical systems, which are subject to input nonlinearity.Dead zone is that the value of the output variable does not change with the change of the input variable value. The range of the input variable can be understood as dead zone [31]. Previously, many researchers have used various methods to settle the synchronization problem of the nonlinear systems, that is, the control input with dead zones. For instance, using Laplace transform approach to settle synchronization of nonlinear systems with disturbances subjected to dead-zone and saturation characteristics in control input was studied in [32], projective synchronization of Chua’s chaotic systems that control input has dead zones was studied in [33, 34], and using adaptive fuzzy sliding mode control to deal with unknown nonlinear chaotic gyros synchronization with unknown dead-zone input was studied in [35].In this paper, we put forward a fuzzy adaptive variable-structure synchronization scheme to manage the dead-zone nonlinearity and analyze the synchronization properties of chaotic systems. Comparing the related works, for example, [26, 32, 34], the contribution of this works consists in the following: (1) Input nonlinearity is considered in this paper. However, it is not considered in the above literature. (2) Compared with [26, 32], the assumptions of this work are more realistic.This paper is organized as follows. In Section2, the notation, problem statement, and preliminaries are raised, including the description of the uncertain chaotic MIMO systems, fuzzy logic system, and input nonlinearity. In Section 3, we present a fuzzy adaptive controller based on the universal approximator property. In Section 4, the effectiveness of the approach is tested and verified by a simulation. In addition, conclusions are contained in Section 5.


## 2. Preliminaries

Throughout this paper,ℝ represents the real numbers, ℝn represents the real n-vectors, and ℝm×n represents the real m×n matrices. ⋅ represents any suitable vector norm.Two uncertain chaotic MIMO systems are given as follows. The driving system is expressed by(1)x˙1=f1x,⋮x˙n=fnx,and the response system is given as(2)y˙1=f1y+∑j=1ng1jΦjuj,⋮y˙n=fny+∑j=1ngnjΦjuj,where x=x1,…,xnT∈ℝn and y=y1,…,ynT∈ℝn, respectively, are the overall state vector of the driving system and the response system, which are measurable. u=u1,…,unT∈ℝn means the controller, fix, fiy,i=1,…,n are unknown nonlinear functions, gij,i,j=1,…,n is unknown gain matrix, and Φu=Φ1u1,Φ2u2,…,ΦnunT is input nonlinearity.Let us denoteFx=f1x,…,fnxT, Fy=f1y,…,fnyT, and G=g11…g1n⋮⋱⋮gn1…gnn; then, driving system (1) is rewritten as(3)x˙=Fx,and (2) can be written as(4)y˙=Fy+GΦu,where F⋅ and Φ⋅∈ℝnandG⋅∈ℝn×n.Some simple assumptions are set forth.Assumption 1.
G is an unknown positive-definite matrix, and one can find an unknown positive constant σ0 such that G≥σ0In, with In being the identity matrix.Remark 1.
In fact, the above assumption is not restrictive, because many physical systems satisfy it. This assumption that devoted to adaptive control of MIMO systems is ubiquitous in the literature, and without exaggeration, we can say that the controllability of the system is assured by it.The response system is driving by controller; however, it does not affect the behavior of the driving system. The control purpose of this paper is to put forward a control inputu in response system to synchronize the driving systems; that is, all signals of the driving and response systems must be bounded under the constraint.First, we define the synchronization errors between driving and response systems as(5)ei=yi−xi.The filtered synchronization errors is given by(6)S=e1,…,enT.Thus, we have(7)S˙=y˙−x˙.Afterwards, (7) will be used to develop the controller and conduct the stability analysis.


### 2.1. Description of the Fuzzy Logic System

By and large, a fuzzy system contains four aspects, that is, fuzzifier, fuzzy rules, fuzzy inference, and defuzzifier, which is depicted in Figure1. By using proper fuzzy rules, a fuzzy inference drives the input y¯T=y1,y2,…,yn∈ℝn becoming an output signal f^∈ℝ. The i-th fuzzy rule has the following form:Figure 1
Fuzzy system.ℝi: if y1 is A1i and … and yn is Ani then f^ is fi, with A1i,A2i,…, and Ani being fuzzy sets. The output of a fuzzy system is given by(8)f^y¯=∑i=1mfi∏j=1nμAjiyj∑i=1n∏j=1nμAjiyj,=θTψx¯,where μAJIyj means the membership function of yj to Aji; suppose that there are m fuzzy rules involved in the fuzzy system. θT=f1,…,fmT can be adjusted online, and(9)ψiy¯=∏j=1nμAjiyj∑i=1m∏j=1nμAjiyj,that being the fuzzy basis function (FBF) with ∑i=1m∏j=1nμAjiyj>0.It can be noted that the fuzzy system (8) is ubiquitous in control applications. In light of the universal approximation results, fuzzy system (8), on a compact operating space, is capable of approximating any nonlinear smooth function f to an arbitrary degree of accuracy. It is extremely important to assume that the membership function parameters need the designer to be prespecified. That being said, the structure of the fuzzy system needs the designer decision for determination, namely, the pertinent inputs. However, the parameters θ have to be defined by learning algorithms.


### 2.2. Input Nonlinearity

We conduct the input nonlinearityΦiui as(10)Φiui=ϕi+uiui−ui+,ui>ui+,0,−ui−≤ui≤ui+,ϕi−uiui+ui−,ui<−ui−,where ϕi+ui>0 and ϕi−ui>0 are nonlinear functions with respect to ui and ui+,ui− are positive constants.Here,Φiui has some significant properties as follows:(11)ui−ui+Φiui≥mi+∗ui−ui+2,ui>ui+,ui+ui−Φiui≥mi−∗ui+ui−2,ui<−ui−,where mi+∗ and mi−∗ are constants known as “gain reduction tolerances.” We can make ηi=minmi+∗,mi−∗.Then, we give some reasonable assumptions to study the properties of input nonlinearity in control problems.Assumption 2.
(a)
mi+∗,mi−∗, namely, the gain reduction tolerances, are unknown, so ηi is unknown.(b)
The explicit mathematical equation ofΦiui is unknown, but we know the properties (11) and assume that ui+ and ui− are known constants.Remark 2.
(1)
It can be seen from (10) and (11) that the input nonlinearity Φiui can be reduced to the special sector nonlinear function if ui+=ui−=0. Consequently, the MIMO system with the input nonlinearities (10), which we considered, is more universal.(2)
It can be noted that the model (10) has been widely used in the past, but it has some limitations, and we made some improvements. The limitations are as follows:(i)
The chaotic system, which they considered, is a simple SISO system, which is input with sector nonlinearities and/or dead zones(ii)
They assumed thatmi+∗ and mi−∗ or ηi=minmi+∗,mi−∗ are known in their control scheme


## 2.1. Description of the Fuzzy Logic System

By and large, a fuzzy system contains four aspects, that is, fuzzifier, fuzzy rules, fuzzy inference, and defuzzifier, which is depicted in Figure1. By using proper fuzzy rules, a fuzzy inference drives the input y¯T=y1,y2,…,yn∈ℝn becoming an output signal f^∈ℝ. The i-th fuzzy rule has the following form:Figure 1
Fuzzy system.ℝi: if y1 is A1i and … and yn is Ani then f^ is fi, with A1i,A2i,…, and Ani being fuzzy sets. The output of a fuzzy system is given by(8)f^y¯=∑i=1mfi∏j=1nμAjiyj∑i=1n∏j=1nμAjiyj,=θTψx¯,where μAJIyj means the membership function of yj to Aji; suppose that there are m fuzzy rules involved in the fuzzy system. θT=f1,…,fmT can be adjusted online, and(9)ψiy¯=∏j=1nμAjiyj∑i=1m∏j=1nμAjiyj,that being the fuzzy basis function (FBF) with ∑i=1m∏j=1nμAjiyj>0.It can be noted that the fuzzy system (8) is ubiquitous in control applications. In light of the universal approximation results, fuzzy system (8), on a compact operating space, is capable of approximating any nonlinear smooth function f to an arbitrary degree of accuracy. It is extremely important to assume that the membership function parameters need the designer to be prespecified. That being said, the structure of the fuzzy system needs the designer decision for determination, namely, the pertinent inputs. However, the parameters θ have to be defined by learning algorithms.


## 2.2. Input Nonlinearity

We conduct the input nonlinearityΦiui as(10)Φiui=ϕi+uiui−ui+,ui>ui+,0,−ui−≤ui≤ui+,ϕi−uiui+ui−,ui<−ui−,where ϕi+ui>0 and ϕi−ui>0 are nonlinear functions with respect to ui and ui+,ui− are positive constants.Here,Φiui has some significant properties as follows:(11)ui−ui+Φiui≥mi+∗ui−ui+2,ui>ui+,ui+ui−Φiui≥mi−∗ui+ui−2,ui<−ui−,where mi+∗ and mi−∗ are constants known as “gain reduction tolerances.” We can make ηi=minmi+∗,mi−∗.Then, we give some reasonable assumptions to study the properties of input nonlinearity in control problems.Assumption 2.
(a)
mi+∗,mi−∗, namely, the gain reduction tolerances, are unknown, so ηi is unknown.(b)
The explicit mathematical equation ofΦiui is unknown, but we know the properties (11) and assume that ui+ and ui− are known constants.Remark 2.
(1)
It can be seen from (10) and (11) that the input nonlinearity Φiui can be reduced to the special sector nonlinear function if ui+=ui−=0. Consequently, the MIMO system with the input nonlinearities (10), which we considered, is more universal.(2)
It can be noted that the model (10) has been widely used in the past, but it has some limitations, and we made some improvements. The limitations are as follows:(i)
The chaotic system, which they considered, is a simple SISO system, which is input with sector nonlinearities and/or dead zones(ii)
They assumed thatmi+∗ and mi−∗ or ηi=minmi+∗,mi−∗ are known in their control scheme


## 3. Design of the Fuzzy Adaptive Controller

In this part, for the class of unknown chaotic MIMO systems, we will develop a fuzzy adaptive variable-structure control plan (3).Substituting (4) into the (7), we get(12)S˙=Fy+GΦu−x˙.Now posingG1=G−1, we have(13)G1S˙=G1−x˙+Fy+Φu.Further, for the stability analysis and controller design, (13) can be arranged as(14)G1S˙=G1−x˙+Fy+Φu,=αy,ν+Φu,where αy,ν=α1y,ν,α2y,ν,…,αny,ν=G1ν+Fy and ν=−x˙.Assumption 3.
There is an unknown continuous positive functionα¯iy satisfying αiy,ν≤ηα¯iy, where η=miniηi.Remark 3.
The reasons why the above Assumption3 is not restrictive are as follows:(i)
We assume that the upper boundηα¯iy is unknown(ii)
Asν is a function about y,yd,yd∈L∞ and αiy,ν is continuous, α¯iy is always thereThe unknown continuous positive functionα¯iy over compact set Ωy can be approximated by the fuzzy system (8) as follows:(15)α¯^iy,θ=θiTψiy,where ψiy is the FBF vector, which is determined in advance by the designer, and θi is the adjustable parameter vector in the fuzzy system.Let(16)θi∗=argminθisupy∈Ωyα¯iy−α¯^iy,θibe the optimal value of θi.It is worth noting that, for the sake of analysis, we put forward artificial constant quantitiesθi∗, and when implementing the controller, their values are not needed.Fix the parameter estimate error as(17)θ˜i=θi−θi∗,and the fuzzy approximation error as(18)εiy=α¯iy−α¯^iy,θi∗,where α¯^iy,θi∗=θi∗Tψiy.In this work, assume the compact setΩy and the fuzzy systems we used do not infringe the universal approximator property, and Ωy is supposed to be large enough, so that it can contain the input vector of the fuzzy system in a closed-loop control system. So, it is rational to suppose that εiy is bounded for all y∈Ωy, i.e., εiy≤ε¯i, ∀y∈Ωy, where ε¯i is an unknown constant.Then, we have(19)α¯^iy,θi−α¯iy=α¯^iy,θi−α¯^iy,θi∗+α¯^iy,θi∗−α¯iy,=α¯^iy,θi−α¯^iy,θi∗−εiy=θi˜Tψiy−εiy.In order to achieve the control objective, let us propose a suitable fuzzy adaptive variable-structure controller:(20)ui=−ρitsignSi−ui−,Si>0,0,Si=0,−ρitsignSi+ui+,Si<0,with ρit=k0i+k1iSi+θiTψiy,∀i=1,…,n and(21)k˙0i=−γ0iσ0ik0i+γ0iSi,k0i0≥0,(22)θ.i=−γ1iσ1iθi+γ1iSiψiy,θij≥0,where γ0i, γ1i, σ0i, σ1i, k1i>0 are design constants and k0i and θi are the online estimates of the uncertain terms k0i∗=ε¯i and θi∗, respectively.Remark 4.
From adaptive laws (21) and (22), we can get their solutions satisfied k0it≥0 and θit≥0, for t>0 so that k0i0≥0 and θi0≥0.Multiplying (9) by 1/ηST and using Assumption 3, we have(23)1ηSTG1xS˙=1ηSTαy,ν+1ηSTΦu,≤∑i=1nSiα¯iy+1ηSTΦu.From (19) and (23), we get(24)1ηSTG1xS˙≤∑i=1nSiα¯iy+1ηSTΦu,≤−∑i=1nSik˜0i−∑i=1nSiθ˜iTψiy+∑i=1nSik0i+∑i=1nSiθiTψiy+1ηSTΦu,where θ˜i=θi−θi∗ and k˜0i=k0i−k0i∗=k0i−ε¯i.Theorem 1.
For system (3), if Assumptions 1–3 are satisfied, the control law (20)–(22) can guarantee the following properties:(i)
It is no exaggeration to say that, in the closed-loop system, all signals are uniformly ultimately bounded(ii)
The system enclosed is asymptotically stableProof.
of Theorem 1. Let the Lyapunov function be(25)V=12ηSTG1S+12∑i=1n1γ0ik˜0i2+12∑i=1n1γ1iθ˜iTθ˜i.
The time derivative ofV is(26)V˙=1ηSTG1S˙+∑i=1n1γ0ik˜0ik˙0i+∑i=1n1γ1iθ˜iTθ.i,with G˙1=0.
It can be noticed from (24) that ui<−ui− for Si>0 and ui>ui+ for Si<0. Thus, from (20) and (24), we can get that for Si>0,(27)ui+ui−Φiui=−ρitsignSiΦiui≥mi−∗ρi2t≥ηρi2t,and for Si<0,(28)ui−ui+Φiui=−ρitsignSiΦiui≥mi+∗ρi2t≥ηρi2t.
Then, forSi>0 and Si<0, we have(29)−ρitsignSiΦiui≥ηρi2t.
SinceSi2>0 and SisignSi=Si, then from (29), we have(30)−ρitSi2signSiΦiui≥ηρi2tSi2=ηρi2tSi2.
For allSi, we have(31)SiΦiui≥−ηρitSi.
Using (21), (22), (24), and (26), (31) becomes(32)V˙≤∑i=1nSik0i+∑i=1nSiθiTψiy+1η∑i=1nSiΦiui−∑i=1nσ0ik˜0ik0i−∑i=1nσ1iθ˜iTθi≤∑i=1nSik0i+∑i=1nSiθiTψiy+∑i=1n−ρitSi−∑i=1nσ0ik˜0ik0i−∑i=1nσ1iθ˜iTθi=−∑i=1nk1iSi2−∑i=1nσ0ik˜0ik0i−∑i=1nσ1iθ˜iTθi.
Obviously, we have(33)−σ0ik˜0ik0i≤−σ0i2k˜0i2+σ0i2k0i∗2,−σ1iθ˜iTθi≤−σ1i2θ˜i2+σ1i2θi∗2.
Then, (32) becomes(34)V˙≤−∑i=1nk1iSi2−∑i=1nσ0i2k˜0i2−∑i=1n−∑i=1nσ1i2θ˜i2+∑i=1nσ0i2k0i∗2+∑i=1nσ1i2θi∗2.
SinceG≥σg0In, then we have(35)STG−1S=STG1S≤1σg0S2.
From (34) and (35), we have(36)V˙≤−μV+π,where(37)π=∑i=1nσ0i2k0i∗2+∑i=1nσ1i2θi∗2,μ=minmini2ησg0k1i,miniγ0iσ0i,miniγ1iσ1i.
Multiplying (36) by eμt yields(38)ddtVeμt≤πeμt.
Integrating (38) over 0,t, we have(39)0≤Vt≤πμ+V0−πμe−μt.
From the above analysis, we can getk0i, θi, Si, E and y that are uniformly ultimately bounded. Thus, ui is bounded. Then, by using (25), V0 can be written as(40)V0=12ηS0TG1S0+12∑i=1n1γ0ik0i0−k0i∗2+12∑i=1n1γ1iθi0−θi∗Tθi0−θi∗.
AsG1 is symmetric positive-definite (i.e., there is an unknown positive constant σg1, such that G1≥σg1In), from (25) and (39), we have(41)Si≤2ησg1πμ+V0−πμe−μt1/2.
We can get that the solution ofSi exponentially converges to a bounded region ΩSi=Si|Si≤2η/σg1π/μ1/2. This completes the proof of the theorem.;Remark 5.
IfΦiui=ui (or when ui+=ui−=0, and ϕi+ui=ϕi−ui=1), namely, there are neither dead zones, nor sector nonlinearities in the input, we can prove that the controller is still applicable for these MIMO chaotic systems.Remark 6.
(1)
There is a special case thatui+=ui−=ui0, and (20) can be simplified to the following form:(42)ui=−ρit+ui0signSi,whereρit=k0i+k1iSi+θTψiy.(2)
It is worth mentioning that the functionsign⋅ can be replaced by any equivalent smooth function: tanh⋅, arctan⋅, Sat⋅, and so on. The chattering effect caused by the discontinuous control term in (20) and (42) can be removed.


## 4. Simulation Results

In order to demonstrate the effectiveness of the proposed adaptive fuzzy controller for uncertain chaotic MIMO systems, we consider the Lotka–Volterra system: the driving system is given as(43)x˙1=x1−x1x2+2x12−2.7x3x12,x˙2=−x2+x1x2,x˙3=−3x3+2.7x3x12,and the response system is(44)y˙1=y1−y1y2+2y12−2.7y3y12+Φ1u1,y˙2=−y2+y1y2+Φ2u2,y˙3=−3y3+2.7y3y12+Φ3u3,where x1, x2, x3 and y1, y2, y3 are state variables of the driving system and the response system, respectively, and Φiui, i=1, 2, 3, are the inevitable input nonlinear models. Let x=x1,x2,x3T, y=y1,y2,y3T, u=u1,u2,u3T, and Φu=Φ1u1,Φ2u2,Φ3u3T.Then, system (43) can be written as(45)x˙=Fx,and system (44) can be written as(46)y˙=Fy+GΦu,The input nonlinearitiesΦiui, for i=1, 3, are described by(47)Φiui=ui−21−0.3sinui,ui>2,0,−2≤ui≤2,ui+20.8−0.3cosui,ui<−2,and the input nonlinearity Φ2u2 is supposed to be(48)Φ2u2=u2−51−0.3sinu2,u2>5,0,−5≤u2≤5,u2+50.8−0.3cosu2,u2<−5.It is worth noting thatFx, G, Φu are unknown, except for some structural natures, for example:(i)
The symmetry and sign ofG(ii)
The natures (11) of Φiui and an understanding of ui+ and ui−, with i=1, 2, 3The initial conditions of the driving system arex0=x10,x20,x30T=1,1.4,1T, and the initial conditions of the response system are y0=y10,y20,y30T=2,2,1T.The adaptive fuzzy systems,θiTψix, with i=1,2,3, have the vector y=y1,y2,y3T as input, and three triangular membership functions that uniformly distribute on the intervals −2,2 are given for each entry variable of these fuzzy systems. The design parameters are taken as q=−61/44, v=3/4, γ01=γ03=30, γ02=80, γ11=γ12=γ13=4000, σ01=σ02=σ03=0.001, σ11=σ12=σ13=0.0005, λ1=λ2=λ3=2, k11=k12=k13=2. The initial conditions are given as k010=k020=k030=0, θ1j0=θ2j0=θ3j0=0, j=1,…,27.It is worth noting thatsignSi, which is a discontinuous function, has been replaced with tanhksiSi, which is a smooth function, with ksi=20, i=1, 2, 3.Finally, the simulation results are shown in Figures2–4. Figure 2 shows the chaotic phenomenon of the driving system. Figure 3 shows that the driving system and the response system are basically synchronized after 0.2 seconds, and the results indicate the effectiveness of our method. The transient behaviors of controller are presented in Figure 4.Figure 2
Simulation results in (a)x1 and x2; (b) x1 and x3; (c) x2 and x3; and (d) x1, x2, and x3.
(a)(b)(c)(d)Figure 3
Simulation results in (a) synchronizing ofx1 and y1; (b) synchronizing of x2 and y2; (c) synchronizing of x3 and y3; and (d) e1, e2, and e3.
(a)(b)(c)(d)Figure 4
Φ1u1, Φ2u2, and Φ3u3.


## 5. Conclusion

This paper proposes a fuzzy adaptive variable-structure controller for the synchronization of the MIMO unknown chaotic system that has sector nonlinearities and dead zones. Based on the Lyapunov stability theory, the whole system can achieve asymptotic stability; namely, all the closed-loop signals and states are bounded, and the synchronization performance of two systems can be achieved. To be specific, a smooth function can reduce the chattering phenomena in the process of control. The validity of the approach has been tested and verified by means of example and simulation. The approach can be applied to settle the synchronization of a large class of chaotic system with dead zones. How to obtain accurate control performance is one of our future research directions.



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