# Parameter Identification and Hybrid Synchronization in an Array of Coupled Chaotic Systems with Ring Connection: An Adaptive Integral Sliding Mode Approach

Authors: Nazam Siddique; Fazal ur Rehman
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This article presents an adaptive integral sliding mode control (SMC) design method for parameter identification and hybrid synchronization of chaotic systems connected in ring topology. To employ the adaptive integral sliding mode control, the error system is transformed into a special structure containing nominal part and some unknown terms. The unknown terms are computed adaptively. Then the error system is stabilized using integral sliding mode control. The controller of the error system is created that contains both the nominal control and the compensator control. The adapted laws and compensator controller are derived using Lyapunov stability theory. The effectiveness of the proposed technique is validated through numerical examples.


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## Body

## 1. Introduction

Ever since the classic effort by Pecora and Carroll [1], the synchronization in chaotic systems became an active research area because of its useful applications [2–5]. Different synchronization schemes for chaotic systems were studied and investigated [6–10]. Among these schemes, hybrid synchronization [11–19] is one in which some of the chaotic systems are synchronized whereas others are antisynchronized. Due to its importance, hybrid synchronization has been the subject of many research works. These works include study of synchronization/antisynchronization for permanent magnet synchronous motors connected in ring topology [11], function projective type synchronization for complex dynamical networks [12], investigation of complete synchronization and antiphase synchronization together [13], hybrid synchronization of networks having heterogeneous systems [14], study of synchronization for fractional-order systems [15], investigation of synchronization for systems with hyperchaotic nature [16, 17], and study of synchronization for complex networks [18, 19]. Study of hybrid synchronization involves multiple chaotic systems like complete, adaptive, global, projective, and antisynchronization systems [17, 20–27], and synchronization systems in multiple coupled complex networks [28, 29]. Currently, hybrid synchronization of several connected chaotic systems is a hot topic of research and the work includes investigation of complex network synchronization for perturbations and delays [30] and a study of neural networks for synchronization [31].In this paper, chaotic systems connected in ring topology are considered which can be shown in Figure1. This arrangement was studied in [32] for the known parameter case. We extend this work and consider that all the parameters of all the systems in the ring connected network are unknown. To reach hybrid synchronization in this system we use adaptive integral SMC.Figure 1
Chaotic systems connected in ring topology [32].SMC is a distinct nonlinear control method. The objective of the SMC method is to drive the states of the system to a specific surface, called sliding manifold. When the surface is touched, the dynamic system is required to persist on it afterward. The main disadvantage of SMC is discontinuous control law. In practice, this creeps towards an undesirable occurrence called “chattering.” The closed-loop dynamics of the system in SMC depend only on the design parameters of the switching sliding manifold. Sliding mode control also offers several benefits like simplicity, robustness to external disturbance, and parameter variation and quick response. The integral SMC, a variant of SMC, guarantees the robustness [33] because of eliminating the reaching-phase. Since reaching-phase is eradicated therefore robustness of the dynamic system can be sure all the way through the system response, beginning from initial conditions. The integral sliding mode control combines both the nominal control which steadies the nominal system and the discontinuous control which discards uncertainties.In this research, we use adaptive integral sliding mode control technique to identify the unknown parameters and to achieve the hybrid synchronization of many chaotic systems connected in the ring topology. By using this technique, the error system is transformed into a special structure containing nominal part and some unknown terms. The unknown terms are computed adaptively. Then the error system is synthesized using integral sliding mode control. The controller of the error system is created that contains both the nominal control and the compensator control. The adaptive laws and compensator controller are derived using Lyapunov stability theory. The effectiveness of the proposed technique is validated through numerical examples.The remainder of the paper is arranged as follows; Section2 presents preliminaries and system description. Section 3 presents proposed control strategies for the general case of hybrid synchronization. Section 4 presents application examples. Section 5 presents discussion and simulation results and in the last section, paper is concluded.


## 2. System Description and Preliminaries

The structure of chaotic systems connected in ring topology is described as(1)x˙1=f1x1+F1x1θ1+D1xM-x1x˙2=f2x2+F2x2θ2+D2x1-x2⋮x˙M=fMxM+FMxMθM+DMxM-1-xM,where x1,x2,…,xM are vectors and xo=(xo1,xo2,…,xoM)T,fo:Rm→Rm,o=1,2,…,M are the nonlinear continuous function, and θo∈Rg,o=1,2,3,4,…,M are vectors of unknown parameters. Fo(xo)∈Rm×g,o=1,2,3,4,…,M are matrices, Do=diag{do1,do2,…,doM},o=1,2,3,4,…,M are m dimensional diagonal matrices, and dop≥0 represent diagonal matrix parameters. When fo·≠fp(·),o,p=1,2,3,4,…,M, then the above arrangement has nonidentical systems.Remark 1.
In (1), all the chaotic systems are connected in ring topology [17], because the first system is connected with Mth and second is connected with first; and so the final chaotic system is connected with M-1 system.So the hybrid synchronization for the above system (1) may be expressed as follows: (2)x˙1=f1x1+F1x1θ1+D1xM-x1x˙2=f2x2+F2x2θ2+D2x1-x2+u1⋮x˙M=fMxM+FMxMθM+DMxM-1-xM+uM-1.Definition 2.
For system (2), we define that it exhibits hybrid synchronization if controllers uo,o=1,2,3,4,…,M-1 exist such that all trajectories x1(t),x2(t),x3(t),x4(t),…,xM(t) for any initial condition (x1(0),x2(0),x3(0),x4(0),…,xM(0)) satisfy the following:
(
1
) For the antisynchronization errors ero=(ero1,ero2,…,erom)T,o=1,2,3,4,…,M-1, we have that(3)limt→∞⁡ero=limt→∞⁡xot+xo+1t=0,o=1,2,3,4,…,M-1.
(
2
) The Synchronization errors erp′=(erp1′,erp2′,…,erpm′)T and erq′=(erq1′,erq2′,…,erqm′)T if M(M≥3) is odd; then erp′ and erq′ satisfy(4)limt→∞⁡erp′=limt→∞⁡xp+2t-xpt=0,p=1,3,5,7,9,…,M-2limt→∞⁡erq′=limt→∞⁡xq+2t-xqt=0,q=2,4,6,8,10,…,M-3and if M(M≥4) is even, then erp′ and erq′ satisfy(5)limt→∞⁡erp′=limt→∞⁡xp+2t-xpt=0,p=1,3,5,7,9,…,M-3limt→∞⁡erq′=limt→∞⁡xq+2t-xqt=0,q=2,4,6,8,10,…,M-2.We notice from this definition that the error systems ero, erp′, and erq′ are globally and asymptotically stable. This shows that, to ensure hybrid synchronization, we have to design controllers uo(t) to make ero, erp′, and erq′ converge to zero.


## 3. Parameter Identification and Hybrid Synchronization for the General Case

For antisynchronization, the error vectors are(6)er1=x2+x1,er2=x3+x2,…,eM-1=xM+xM-1.Let θ^o be estimates of θo,o=1,2,3,4,…,M, respectively, and let θ~o=θo-θ^o,o=1,2,3,4,…,M be errors while estimating the parameters θo,o=1,2,…,M, respectively. Derivative of (6) leads to the following:(7)e˙r1=x˙2+x˙1=f2x2+F2x2θ^2+F2x2θ~2+D2x1-x2+u1+f1x1+F1x1θ^1+F1x1θ~1+D1xM-x1e˙r2=x˙3+x˙2=f3x3+F3x3θ^3+F3x3θ~3+D3x2-x3+u2+f2x2+F2x2θ^2+F2x2θ~2+D2x1-x2+u1e˙r3=x˙4+x˙3=f4x4+F4x4θ^4+F4x4θ~4+D4x3-x4+u3+f3x3+F3x3θ^3+F3x3θ~3+D3x2-x3+u2⋮e˙rM-1=x˙M+qx˙M-1=fMxM+FMxMθ^M+FMxMθ~M+DMxM-1-xM+uM-1+fM-1xM-1+FM-1xM-1θ^M-1+FM-1xM-1θ~M-1+DM-1xM-2-xM-1+uM-2which can be written as(8)e˙r1e˙r2⋮e˙rM-1=f2x2+f1x1+F2x2θ^2+F1x1θ^1+D2x1-x2+D1xM-x1f3x3+f2x2+F3x3θ^3+F2x2θ^2+D3x2-x3+D2x1-x2⋮fMxM+fM_1xM_1+FMxMθ^M+FM-1xM-1θ^M-1+DMxM-1-xM+DM-1xM-2-xM-1+F2x2θ~2+F1x1θ~1F3x3θ~+F2x2θ~2⋮FMxMθ~M+FM-1xM-1θ~M-1+100⋯0110⋯0011⋯0⋮⋮⋮⋮⋮00⋯11u1u2⋮uM-1.By choosing(9)u1u2⋮uM-1=100⋯0110⋯0011⋯0⋮⋮⋮⋮⋮00⋯11-1+er2er3⋮erM-1v-A,where v is the new input vector and(10)A=f2x2+f1x1+F2x2θ^2+F1x1θ^1+D2x1-x2+D1xM-x1f3x3+f2x2+F3x3θ^3+F2x2θ^2+D3x2-x3+D2x1-x2⋮fMxM+fM_1xM_1+FMxMθ^M+FM-1xM-1θ^M-1+DMxM-1-xM+DM-1xM-2-xM-1then system (7) becomes(11)e˙r1=er2+F2x2θ~2+F1x1θ~1e˙r2=er3+F3x3θ~3+F2x2θ~2e˙r3=er4+F4x4θ~4+F3x3θ~3⋮e˙rM-2=erM-1+FM-1xM-1θ~M-1+FM-2xM-2θ~M-2e˙rM-1=v+FMxMθ~M+FM-1xM-1θ~M-1.To employ the integral sliding mode control, choose the nominal system for (11) as(12)e˙r1=er2e˙r2=er3e˙r3=er4⋮e˙rM-2=erM-1e˙rM-1=v0.To stabilize the system (11), we choose the Hurwitz sliding surface for system (12) as σ0=(1+d/dt)M-2er1=er1+c1er2+⋯+cM-3erM-2+erM-1, where the coefficients co are chosen in such a way that σ0 becomes Hurwitz polynomial. Then σ˙0=er2+c1er3+c2er4+⋯+cM-3erM-1+v0.By choosingv0=-er2-c1er3-c2er4-⋯-cM-3erM-1-kσ0-ksign(σ0),k>0, we have σ˙0=-ksign(σ0)-kσ0; therefore σ0→0, which gives er1,er2,…,erM-1→0. Therefore, system (12) becomes asymptotically stable.The sliding surface for system (9) is σ=σ0+z. The term z in the sliding surface is an integral term computed later. To avoid the reaching-phase, choose z(0) such that σ(0)=0. By choosing v=v0+vs where v0 is the nominal input vector and vs is compensator terms computed later, then the time derivative of the sliding surface becomes as(13)σ˙=σ˙0+z˙=e˙r1+c1e˙r2+c2e˙r3+c3e˙r4+⋯+cM-3e˙rM-2+e˙rM-1+z˙=er2+c1er3+c2er4+⋯+cM-3erM-1+v0+vs+z˙+F1x1θ~1+1+c1F2x2θ~2+c1+c2F3x3θ~3+c2+c3F4x4θ~4+⋯+cM-4+cM-3FM-2xM-2θ~M-2+cM-3+1FM-1xM-1θ~M-1+FMxMθ~M.By choosing a Lyapunov function: V=(1/2){σTσ+θ~1Tθ~1+θ~2Tθ~2+∑o=1M-4θ~o+2Tθ~o+2+θ~M-1Tθ~M-1+θ~MTθ~M}, designing the adaptive laws for θ~o,θ^o,o=1,2,…,M and computing vs such that V˙<0.Theorem 3.
Consider a Lyapunov functionV=(1/2){σTσ+θ~1Tθ~1+θ~2Tθ~2+∑o=1M-4θ~o+2Tθ~o+2+θ~M-1Tθ~M-1+θ~MTθ~M}. Then V˙<0 if the adaptive laws for θ~o,θ^o,o=1,2,3,4,…,M and the value of vs are chosen as(14)z˙=-er2-∑o=3M-1co-2ero-v0,vs=-kσθ~˙1=-F1Tx1σ-k1θ~1,θ^˙1=-θ~˙1θ~˙2=-1+c1F2Tx2σ-k2θ~2,θ^˙2=-θ~˙2θ~˙o+2=-co+co+1Fo+2Txo+2σ-ko+2θ~o+2,θ^˙o+2=-θ~˙o+2,o=1,…,M-4θ~˙M-1=-cM-3+1FM-1TxM-1σ-kM-1θ~M-1,θ^˙M-1=-θ~˙M-1θ~˙M=-FMTxMσ-kMθ~M,(15)θ^˙M=-θ~˙M.Proof.
Since(16)V˙=σTσ˙+θ~1Tθ~˙1+θ~2Tθ~˙2+∑o=1M-4θ~o+2Tθ~˙o+2+θ~M-1Tθ~˙M-1+θ~MTθ~˙M=σTer2+∑o=3M-1co-2ero+v0+vs+z˙+θ~1Tθ~˙1+F1Tx1σ+θ~2Tθ~˙2+1+c1F2Tx2σ+∑o=1M-4θ~o+2Tθ~˙o+2+co+co+1Fo+2Txo+2σ+θ~M-1Tθ~˙M-1+cM-3+1FM-1TxM-1σ+θ~MTθ~˙M+FMTxMσby using (14) we have(17)V˙=-kσ2-∑o=1Mkoθ~oTθ~o.From this we conclude that σ,θ~o→0. Since σ→0, therefore eo→0,o=1,2,3,…,M-1. Thus the antisynchronization is achieved.
The controllers designed for the antisynchronization are used for the complete synchronization. For this we consider two cases:
Case 1. For odd number of systems and M(M≥3), synchronization error is expressed like(18)erp′t=xp+2t-xpt,p=1,3,5,7,9,…,M-2erq′t=xq+2t-xqt,q=2,4,6,8,10,…,M-3.Then(19)limt→∞⁡erp′=limt→∞⁡xp+2t-xptlimt→∞⁡xp+2t+xp+1t-xp+1t-xpt≤limt→∞⁡xp+2t+xp+1t+limt→∞⁡xp+1t+xpt=0due  to  antisynchronization.
Thus(20)limt→∞⁡erp′=0,p=1,3,5,7,9,…,M-2.Similarly(21)limt→∞⁡erp′=limt→∞⁡xq+2t-xqt=limt→∞⁡xq+2t+xq+1t-xq+1t-xqt≤limt→∞⁡xq+2t+xq+1t+limt→∞⁡xq+1t+xqt=0limt→∞⁡erp′=0,q=2,4,6,8,10,…,M-3.
Case 2. For even number of systems and M(M≥4), synchronization error is expressed like(22)erp′=xp+2t-xpt,erq′=xq+2t-xqtlimt→∞⁡erp′=limt→∞⁡xp+2t-xpt=limt→∞⁡xp+2t+xp+1t-xp+1t-xpt≤limt→∞⁡xp+2t+xp+1t+limt→∞⁡xp+1t+xpt=0limt→∞⁡erp′=0,p=1,3,5,7,9,…,M-3.Similarly(23)limt→∞⁡erq′=limt→∞⁡xq+2t-xqt=0,q=2,4,6,8,10,…,M-2.So complete synchronization is too achieved.


## 4. Application Examples

To illustrate the control design procedure for reaching the hybrid synchronization behavior, we present two examples withM=3 and M=4.Example 1 (M=3).
Chen chaotic system, Lorenz chaotic system, and Lü chaotic system, in that order, are selected for this numerical example and are expressed below:(24)x˙11=-35x11+35x12+d11x31-x11x˙12=-7x11+28x12+x11x13+d12x32-x12x˙13=-3x13+x11x12+d13x33-x13x˙21=-36x21+36x22+d21x11-x21+u11x˙22=20x22-x21x23+d22x12-x22+u12x˙23=-3x23+x21x22+d23x13-x23+u13x˙31=-10x31+10x32+d31x21-x31+u21x˙32=28x31-x32-x31x33+d32x22-x32+u22x˙33=-83x33+x31x32+d33x23-x33+u23.By assuming that the systems parameters are unknown, we write systems (24) as(25)x˙11=a1x11+b1x12+d11x31-x11x˙12=c1x11+d1x12-x11x13+d12x32-x12x˙13=e1x13+x11x12+d13x33-x13x˙21=a2x21+b2x22+d21x11-x21+u11x˙22=c2x22-x21x23+d22x12-x22+u12x˙23=e2x23+x21x22+d23x13-x23+u13x˙31=a3x31+b3x32+d31x21-x31+u21x˙32=c3x31-x32-x31x33+d32x22-x32+u22x˙33=e3x33+x31x32+d33x23-x33+u23.Let a^o,b^o,c^o,e^ro,d^1,o=1,2,3 be estimates of ao,bo,co,ero,d1,o=1,2,3, respectively, and let a~o=ao-a^o,b~o=bo-b^o,c~o=co-c^o,e~ro=ero-e^ro,d~1=d1-d^1,o=1,2,3 be errors in estimation of ao,bo,co,eo,d1,o=1,2,3, respectively.
Defining(26)x1=x11x12x13,x2=x21x22x23,x3=x31x32x33,u1=u11u12u13,u2=u21u22u23,θ^=a^b^c^d^,θ~=a~b~c~d~then the systems (25) in vector form are written as(27)x˙1=f1x1+F1x1θ^1+F1x1θ~1x˙2=f2x2+F2x2θ^2+F2x2θ~2+u1x˙3=f3x3+F3x3θ^3+F3x3θ~3+u2,where(28)f1x1=d11x31-x11x11x13+d12x32-x12x11x12+d13x33-x13,F1x1=x11x1200000c^1d^100000x13,θ^1=a^1b^1c^1d^1e^1,θ~1=a~1b~1c~1d~1e~1,f2x2=d21x11-x21-x21x23+d22x12-x22x21x22+d23x13-x23,F2x2=x21x220000x220000x23,θ^2=a^2b^2c^2e^2,θ~2=a~2b~2c~2e~2,f3x3=d31x21-x31-x32-x31x33+d32x22-x32x31x32+d33x23-x33,F3x3=x31x320000x310000x33,θ^3=a^3b^3c^3e^3,θ~3=a~3b~3c~3e~3.For the antisynchronization, the errors(29)er1=er11er12er13,er2=er21er22er23are(30)er1=x2+x1,er2=x3+x2.Therefore(31)e˙r1e˙r2=f2x2+F2x2θ^2+f1x1+F1x1θ^1F2x2θ^2+f3x3+F3x3θ^3+F1x1θ~1+F2x2θ~2F2x2θ~2+F3x3θ~3+1011u1u2.By choosing(32)u1u2=1011-1-f2x2+F2x2θ^2+f1x1+F1x1θ^1f3x3+F3x3θ^3+F2x2θ^2+er2v,where v=v1v2v3T is the new input vector, then system (31) becomes(33)e˙r1=e2+F2x2θ~2+F1x1θ~1e˙r2=v+F3x3θ~3+F2x2θ~2.The nominal system for (33) is(34)e˙r1=e2e˙r2=v0,where v0=v01v02v03T is the nominal input vector.
The sliding surface for nominal system (34) is σ0=er1+er2; that is,(35)σ01σ02σ03=er11+er21er12+er22er13+er23.The nominal system (34) is asymptotically stable if v0=-er2-kσ0-ksign(σ0),k>0. The sliding surface for system (33) is σ=σ0+z=er1+er2+z, where z=z1z3z3T is some integral term and z(0) is chosen in such a way that σ(0)=0. Defining v=v0+vs where v0=v01v02v03T is the nominal input and vs=vs1vs2vs3T is compensator, then system (33) can be rewritten as(36)e˙r11=er21+a~2x21+b~2x22+a~1x11+b~1x12e˙r12=er22+c~2x22+c~1x11+d~1x12e˙r13=er23+e~r2x23+e~r1x13e˙r21=v1+a~3x31+b~3x32+a~2x21+b~2x22e˙r22=v2+c~3x31+c~2x22e˙r23=v3+e~r3x33+e~r2x23.Then σ˙=e˙r1+e˙r2+z˙ gives(37)σ˙1=er21+v01+vs1+z˙1+a~1x11+b~1x12+2a~2x21+2b~2x22+a~3x31+b~3x32σ˙2=er22+v02+vs2+z˙2+c~1x11+d~1x12+2c~2x22+c~3x31σ˙3=er23+v03+vs3+z˙3+2e~r2x23+e~r1x13+e~r3x33.By choosing a Lyapunov function(38)V=12σ12+σ22+σ32+a~12+a~22+a~32+b~12+b~22+b~32+c~12+c~22+c~32+e~12+e~22+e~32+d~12then V˙<0 if the adaptive laws for a~o,a^o,b~o,b^o,c~o,c^o,e~ro,e^ro,d~1,d^1,o=1,2,3 and the values of vso,o=1,2,3 are chosen as(39)z˙1=-er21-v01,vs1=-k1s1z˙2=-er22-v02,vs2=-k2s2z˙3=-er23-v03,vs3=-k3s3a~˙1=-σ1x11-k4a~1,a^˙1=-a~˙1a~˙2=-2σ1x21-k5a~2,a^˙2=-a~˙2a~˙3=-σ1x31-k6a~3,a^˙3=-a~˙3b~˙1=-σ1x12-k7b~1,b^˙1=-b~˙1b~˙2=-2σ1x22-k8b~2,b^˙2=-b~˙2b~˙3=-σ1x32-k9b~3,b^˙3=-b~˙3c~˙1=-σ2x11-k10c~1,c^˙1=-c~˙1c~˙2=-2σ2x22-k11c~2,c^˙2=-c~˙2c~˙3=-σ2x31-k12c~3,c^˙3=-c~˙3e~˙r1=-σ3x13-k13e~r1,e^˙r1=-e~˙r1e~˙r2=-2σ3x23-k14e~r2,e^˙r2=-e~˙r2e~˙r3=-σ3x33-k15e~r3,e^˙r3=-e~˙r3d~˙1=-σ2x12-k16d~1,d^˙1=-d~˙1,ko>0,o=1,2,…,16.We have(40)V˙=-k2σ12-k2σ22-k3σ32-k4a~12-k5a~22-k6a~32-k7b~12-k8b~22-k9b~32-k10c~12-k11c~22-k12c~32-k13e~r12-k14e~r22-k15e~r32-k16d~12.From this we conclude that σo,e~ro,d~1,a~o,b~o,c~o→0,o=1,2,3. Since σo→0,o=1,2,3, therefore er1,er2→0. Therefore, antisynchronization is realized. For complete synchronization, the error is er1′=x3-x1, which can be written as er1′=x3-x1=x3+x2-x2-x1=er2-er1. As er1,er2→0 therefore er1′→0. Thus complete synchronization is achieved. Results of simulations are depicted in Figures 2–4.Figure 2
(a)
Convergence of error systemer11, er12, er13, er21, er22, er23
(b)
Convergence of error systemer31′, er32′, er33′Figure 3
(a)
Time history of system statesx11, x21, and x31
(b)
Time history of system statesx12, x22, and x32
(c)
Time history of system statesx13, x23, and x33Figure 4
(a)
Parameter estimationa1, b1, c1, d1, and er1
(b)
Parameter estimationa2, b2, c2, and er2
(c)
Parameter estimationa3, b3, c3, and er3Example 2 (M=4).
In this example, only Lorenz systems with unknown parameters are chosen to examine the effectiveness of the proposed method.
These can be expressed as(41)x˙11=-10x11+10x12+d11′x41-x11x˙12=28x11-x12-x11x13+d12′x42-x12x˙13=-83x13+x11x12+d13′x43-x13x˙21=-10x21+10x22+d21′x11-x21+u11′x˙22=28x21-x22-x21x23+d22′x12-x22+u12′x˙23=-83x23+x21x22+d23′x13-x23+u13′x˙31=-10x31+10x32+d31′x21-x31+u21′x˙32=28x31-x32-x31x33+d32′x22-x32+u22′x˙33=-83x33+x31x32+d33′x23-x33+u23′x˙41=-10x41+10x42+d41′x31-x41+u31′x˙42=28x41-x42-x41x43+d42′x32-x42+u32′x˙43=-83x43+x41x42+d43′x33-x43+u33′.With unknown parameters above four systems can be represented as(42)x˙11=ax11+bx12+d11′x41-x11x˙12=cx11-x12-x11x13+d12′x42-x12x˙13=dx13+x11x12+d13′x43-x13(43)x˙21=ax21+bx22+d21′x11-x21+u11′x˙22=cx21-x22-x21x23+d22′x12-x22+u12′x˙23=dx23+x21x22+d23′x13-x23+u13′(44)x˙31=ax31+bx32+d31′x21-x31+u21′x˙32=cx31-x32-x31x33+d32′x22-x32+u22′x˙33=dx33+x31x32+d33′x23-x33+u23′(45)x˙41=ax41+bx42+d41′x31-x41+u31′x˙42=cx41-x42-x41x43+d42′x32-x42+u32′x˙43=dx43+x41x42+d43′x33-x43+u33′.Let a^,b^,c^,d^ be estimates of a,b,c,d, respectively, and let a~=a-a^,b~i=b-b^,c~=c-c^,d~1=d-d^ be the errors in estimation of a,b,c,d, respectively.
Defining(46)x1=x11x12x13,x2=x21x22x23,x3=x31x32x33,x4=x41x42x43,u1′=u11′u12′u13′,u2′=u21′u22′u23′,u3′=u31′u32′u33′,θ^=a^b^c^d^θ~=a~b~c~d~systems (43)–(45) are represented as(47)x˙1=f1x1+F1x1θ^+F1x1θ~x˙2=f2x2+F2x2θ^+F2x2θ~+u1′x˙3=f3x3+F3x3θ^+F3x3θ~+u2′x˙4=f4x4+F4x4θ^+F4x4θ~+u3′,where(48)f1x1=d11′x41-x11-x12-x11x13+d12′x42-x12x11x12+d13′x43-x13,F1x1=x11x120000x110000x13f2x2=d21′x11-x21-x22-x21x23+d22′x12-x22x21x22+d23′x13-x23,F2x2=x21x220000x210000x23f3x3=d31′x21-x31-x32-x31x33+d32′x22-x32x31x32+d33′x23-x33,F3x3=x31x320000x310000x33f4x4=d41′x31-x41-x42-x41x43+d42′x32-x42x41x42+d43′x33-x43,F4x4=x41x420000x410000x43.For the antisynchronization, the error vectors(49)er1=er11er12er13,er2=er21er22er23,er3=er31er32er33are(50)er1=x2+x1,er2=x3+x2,er3=x4+x3.Therefore(51)e˙r1e˙r2e˙r3=F1x1+F2x2θ^+f1x1+f2x2F2x2+F3x3θ^+f2x2+f3x3F3x3+F4x4θ^+f3x3+f4x4+F1x1+F2x2F2x2+F3x3F3x3+F4x4θ~+100110011u1′u2′u3′.Choose(52)u1′u2′u3′=100110011-1-f2x2+f1x1+F2x2+F1x1θ^f3x3+f2x2+F3x3+F2x2θ^f4x4+f3x3+F4x4+F3x3θ^+er2er3v,where v is the new input vector.
We have(53)e˙r1e˙r2e˙r3=er2er3v+F2x2+F1x1F3x3+F2x2F4x4+F3x3θ~or(54)e˙r1=er2+F2x2+F1x1θ~e˙r2=er3+F3x3+F2x2θ~e˙r3=v+F4x4+F3x3θ~.The nominal system for (54) is(55)e˙r1=er2e˙r2=er3e˙r3=v0.The Hurwitz sliding surface vector for nominal system (55) can be defined as(56)σ0=er1+2er2+er3,where(57)σ0=σ01σ02σ03=er11+2er21+er31er12+2er22+er32er13+2er23+er33.Then the derivative of the above system is σ˙0=e˙r1+2e˙r2=er2+2er3+v0. By choosing v0=-er2-2er3-kσ0-ksign(σ0),k>0, we have σ˙0=-ksign(σ0)-kσ0. So the error dynamics (55) are asymptotically stable.
Sliding surface for the system (55) is described as σ=σ0+z=er1+2er2+er3+z, where z=z1z3z3T is some integral term, by choosing z(0) in such a way that σ(0)=0. Select v=v0+vs, where v0=v01v02v03T is the nominal input and vs=vs1vs2vs3T is compensator term computed later.
The system (54) can be rewritten as(58)e˙r11=er21+a~x21+b~x22+a~x11+b~x12e˙r12=er22+c~x21+c~x11e˙r13=er23+d~x23+d~x13e˙r21=er31+a~x31+b~x32+a~x21+b~x22e˙r22=er32+c~x31+c~x21e˙r23=er33+d~x33+d~x23e˙r31=v01+vs1+a~x41+b~x42+a~x31+b~x32e˙r32=v02+vs2+c~x41+c~x31e˙r33=v03+vs3+d~x43+d~x33.Then σ˙=e˙r1+2e˙r2+e˙r3+z˙ gives(59)σ˙1=e˙r11+2e˙r21+e˙r31+z˙1=er21+a~x21+b~x22+a~x11+b~x12+2er31+2a~x31+2b~x32+2a~x21+2b~x22+v01+vs1+a~x41+b~x42+a~x31+b~x32+z˙1σ˙2=e˙r12+2e˙r22+e˙r32+z˙2=er22+c~x21+qc~x11+2er32+2c~x31+2c~x21+v02+vs2+c~x41+c~x31+z˙2σ˙3=e˙r13+2e˙r23+e˙r33+z˙3=er23+d~x23+d~x13+2er33+2d~x33+2d~x23+v03+vs3+d~x43+d~x33+z˙3or(60)σ˙1=er21+2er31+a~x11+3x21+3x31+x41+b~x12+3x22+3x32+x42+v01+vs1+z˙1σ˙2=er22+2er32+c~x11+3x21+3x31+x41+v02+vs2+z˙2σ˙3=er23+2er33+d~x13+3x23+3x33+x43+v03+vs3+z˙3.Choose the Lyapunov function: V=1/2σ12+σ22+σ32+a~2+b~2+c~2+d~2.
ThenV˙<0 if the adaptive laws for a~,b~,c~,d~,a^,b^,c^,d^ and vso,o=1,2,3 are chosen as(61)z˙1=-er21-2er31-v01,vs1=-k1s1z˙2=-er22-2er32-v02,vs2=-k2s2z˙3=-er23-2er33-v03,vs3=-k3s3a~˙=-σ1x11+3x21+3x31+x41-k4a~,a^˙=-a~˙b~˙=-σ1x12+3x22+3x32+x42-k5b~,b^˙=-b~˙c~˙=-σ2x11+3x21+3x31+x41-k6c~,c^˙=-c~˙d~˙=-σ3x13+3x23+3x33+x43-k7d~,d^˙=-d~˙,ko>0,o=1,…,7.From this we conclude that a~,b~,c~,e~r,σ1,σ2,σ3,d~→0. Since σo→0,o=1,2,3, therefore er1,er2,er3→0. Thus antisynchronization is achieved.
For complete synchronization, the errors areer1′=x3-x1,er2′=x4-x2 which can be written as(62)er1′=x3-x1=x3+x2-x2-x1=er2-er1er2′=x4-x2=x4+x3-x3-x2=er3-er2.As er1,er2,er3→0 therefore er1′,er2′→0; thus complete synchronization is achieved. Simulation results are shown in Figures 5–7.Figure 5
(a)
Convergence of error systemer11, er12, er13, er21, er22, er23, er31, er32, and er33
(b)
Convergence of error systemer11′, er12′, and er13′
(c)
Convergence of error systemer21′′, er22′′, and er23′′Figure 6
(a)
Time history of system statesx11, x21, x31, and x41
(b)
Time history of system statesx12, x22, x32, and x42
(c)
Time history of system statesx13, x23, x33, and x43Figure 7
Parameter estimationa, b, c, and d.


## 5. Simulation Results and Discussion

Figures2–4 show simulation results for Example 1 with M=3. The initial (starting point) conditions are chosen as (x11(0)=10, x12(0)=20, x13(0)=30), (x21(0)=-5.8, x22(0)=8, x23(0)=10), and (x31(0)=11, x32(0)=15, x33(0)=26). The coupling parameters are chosen as d11=d21=d13=d23=d31=d33=0,d12=10,d22=11 and d32=1. Figure 2(a) shows that the errors er11, er12, er13, er21, er22, and er23 asymptotically go to zero. Figure 2(b) displays the errors er31′, er32′, and er33′, converging to origin. Figure 3(a) shows system states x11, x21, x31, Figure 3(b) shows system states x12, x22, and x32, and Figure 3(c) shows system states x13, x23, and x33. From these figures we can see that systems x1t and x2t, and systems x2t and x3t achieve the antisynchronization, and systems x1t and x3t attain complete synchronization and therefore milestone is attained, that is, hybrid synchronization. Figure 4 shows the adaptive estimation of parameters a, b, c, d, and er for the three systems. Figure 4(a) shows the estimation of a1, b1, c1, d1, and er1, the parameters of the first system, which converge to their true values of −35, 35, −7, 28, and −3, respectively. Figure 4(b) shows the estimation of a2, b2, c2, and er2, and the parameters of the second system, which converges to their true values of −36, 36, 20, and 3, respectively. Figure 4(c) shows the estimation of a3, b3, c3, and er3, the parameters of the third system, which converges to their true values −10, 10, 28, and −8/3, respectively.Figures5–7 show simulation results for Example 2 with M=4. The initial (starting point) conditions are chosen as (x11(0)=4, x12(0)=5, x13(0)=-3), (x21(0)=5, x22(0)=2, x23(0)=-5), (x31(0)=11, x32(0)=15, x33(0)=10), and (x41(0)=4, x42(0)=5, x43(0)=-3). The coupling parameters are chosen as d12=d22=d32=d13=d23=d33=d42=d43=0,d11=d21=1 and d31=d41=-1. Figure 5(a) shows the errors er11, er12, er13, er21, er22, er23, er31, er32, and er33 asymptotically converge to zero. Figure 5(b) shows the errors er11′, er12′, and er13′ asymptotically converge to zero. Figure 5(c) shows the errors er21′′, er22′′, and er23′′ asymptotically converge to zero. Figure 6(a) shows system states x11, x21, x31, and x41, Figure 6(b) shows system states x12, x22, x32, and x42, and Figure 6(c) shows system states x13, x23, x33, and x43. From these figures we can see that systems x1t and x2t, systems x2t and x3t, and systems x3t and x4(t) achieve the antisynchronization, and systems x1t and x3t and systems x2t and x4(t) attained the complete synchronization and hence hybrid synchronization is attained. Figure 7 shows an adaptive estimation of parameters a, b, c, and d for four systems which converge to their true values of −10, 10, 28, and −8/3 respectively.


## 6. Conclusion

In this article, we presented control design method for parameter identification and synchronization/antisynchronization of many attached chaotic systems connected in the ring topology. The methodology is created using adaptive integral SMC. The error system was transformed into a special structure containing nominal part and some unknown terms. The unknown terms were computed adaptively. Then the error system was stabilized using integral sliding mode control. The stabilizing controller for the error system is created that contains the nominal control and the compensator control. Simulation results show that antisynchronization and complete synchronization were achieved with the proposed control laws and that the uncertain parameters converge to their actual values.



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