# Robust Adaptive Output Feedback Control Scheme for Chaos Synchronization with Input Nonlinearity

Authors: Xiaomeng Li; Zhanshan Zhao; Jing  Zhang; Meixia Zhu
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

This paper proposes a robust adaptive output feedback control strategy which can automatically regulate control gain for chaos synchronization. Chaotic systems with input nonlinearities, delayed nonlinear coupling, and external disturbance can achieve synchronization by applying this strategy. Utilizing Lyapunov method and LMI technique, the conditions ensuring chaos synchronization are obtained. Finally, simulations are given to show the effectiveness of our control strategy.


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## Body

## 1. Introduction

Chaos synchronization has attracted a lot of interest due to its wide engineering application in various areas like secure communication [1–4], neural networks [5, 6], electronic engineering [7], and so on [8, 9]. Consider the fact that chaotic system is a class of nonlinear dynamical system which sensitively depends on initial conditions. It is necessary to solve the problem of chaos synchronization.In the practical physical systems, physical limitations will lead to state nonlinearity and input nonlinearity, such as sector [10–12], saturation [13–15], and dead zones [16–19]. Considering state nonlinearity, [20] proposed an adaptive control strategy for multirate networked nonlinear systems. In [21], fuzzy control method has been applied to a class of nonlinear system. Filter design and H∞ performance for nonlinear networked systems have been researched in [22]. A novel sliding mode observer approach has been proposed for a class of stochastic systems in [23]. Moreover, it should not be ignored that effect of nonlinear control inputs can cause serious degradation of synchronization performance even nonsynchronous. Thus, nonlinear control inputs should be considered in synchronization controller design of chaotic systems. Unfortunately, in [24, 25], input nonlinearity has not been considered.As a source of nonsynchronism, time-varying delay has to be faced in many engineering synchronization systems, such as chemical processes [26, 27] and pneumatic systems [28]. Therefore, designing a controller for time-varying delay systems is necessary [29].Recently, considering nonlinearly coupled chaotic systems, [30] proposed a state feedback controller to achieve synchronization. However, the input nonlinearity was not considered. To the authors’ knowledge, synchronization for coupled chaotic systems with input nonlinearity and time delays has been rarely mentioned. Furthermore, in real application, only the output state is available. Therefore, it is necessary to design a synchronization controller in output feedback form.Motivated by the previous discussions, we propose an adaptive output feedback controller to make chaotic systems synchronize. Input nonlinearities, nonlinear coupling, and time-varying delay have been taken into account. By utilizing Lyapunov method and LMI technique, the conditions ensuring synchronization are obtained.In the rest of this paper, Section2 provides systems and problem description. Then a robust adaptive output feedback control strategy is proposed for chaos synchronization in Section 3. In Section 4, simulations are given to demonstrate effectiveness of this control strategy. Conclusion are collected in Section 5.


## 2. Problem Formulation

Consider chaotic systems as follows:(1)x˙mt=Axmt+Acxmt-d1t+Bft,xmt+Bcgt,xmt-d1t-Bdkt,xst-d2t,ymt=Cxmt(2)x˙st=Axst+Acxst-d1t+Bft,xst+Bcgt,xst-d1t-Bdkt,xmt-d2t+Dωt+EΛut,yst=Cxst,where xm∈Rn, xs∈Rn, ym∈Rm, and ys∈Rm are the state vector and output vector for the drive and response system, respectively. f(t,x),g(t,x),k(t,x):Rn×Rn→Rm represent nonlinear vectors. u(t)=[u1(t)⋯um(t)]T∈Rm is the control input vector; ω(t) denotes the external disturbance; time-varying delay d1(t) and d2(t) satisfy(3)d11≤d˙1t≤d12,(4)d21≤d˙2t≤d22.Λ(u(t))=[λ1(u1(t))⋯λm(um(t))]T is representing the nonlinear control input vector which satisfies the following inequality:(5)νitλiνit≥χiνit2.χi is an unknown positive constant satisfying χ∗=minχi. Constant matrices A,Ac,B,Bc,Bd,C,D,E have appropriate dimensions.Synchronization error can be defined ase(t)=xs(t)-xm(t). Using (1) and (2), synchronization error can be obtained:(6)e˙t=Aet+Acet-d1t+BΨt+BcΥt,d1+BdΞt,d2+Dωt+EΛut,yet=Cet,where(7)Ψt=ft,xst-ft,xmt,Υt,d1=gt,xst-d1t-gt,xmt-d1t,Ξt,d2=kt,xst-d2t-kt,xmt-d2t.The objective is to make drive and response systems synchronize. Obviously, ife(t)→0, then xs(t)-xm(t)→0 and it means that system (1) and (2) is synchronized.To obtain the synchronization conditions, the following lemma and assumptions will be used during the proof.Lemma 1.
If matrixH=H11H12H21H22, where H11 and H22 are square matrices, then the following inequalities are equivalent:(1)
H<0;(2)
H11<0, H22-H12TH11-1H12<0;(3)
H22<0, H11-H12H22-1H12T<0.Assumption 2.
The nonlinear functionΨ(t), Υ(t,d1), and Ξ(t,d2) satisfy the global Lipschitz condition:(8)Ψt≤L1xst-xmt,Υt,d1≤L2xst-d1t-xmt-d1t,Ξt,d2≤L3xst-d2t-xmt-d2t.Assumption 3.
MatrixP>0 and satisfies the following equation:(9)ETP=C.


## 3. Robust Adaptive Controller Design Based on LMI

In order to make drive and response systems synchronize, the following adaptive controller is considered:(10)ut=-ν2Cet,where ν is an adaptive control gain and is adjusted by the following adaptation law:(11)ν˙=χ∗ρCet2,ν0>0,where ρ is a positive parameter.By applying of the adaptive controller, synchronization errors will converge to zero asymptotically. In Theorems4 and 5 the main results will be presented.Theorem 4.
Consider the drive and response system (1) and (2) under ω(t)=0. By application of the adaptive control law (10) and (11), if existing symmetric and positive definite matrices P, W1, W2, and a scalar α>0, satisfying the following LMI:(12)ΔPAc0PBPBcPBdL1T00∗-1-d12W100000L2T0∗∗-1-d22W200000L3T∗∗∗-I00000∗∗∗∗-I0000∗∗∗∗∗-I000∗∗∗∗∗∗-I00∗∗∗∗∗∗∗-I0∗∗∗∗∗∗∗∗-I<0,where(13)Δ=PA+ATP-αPEETP+W1+W2,the system (1) and (2) is synchronized.Proof.
We consider the following Lyapunov-Krasovskii functional:(14)V=eTtPet+∫t-d1tteTvW1evdv+∫t-d2tteTvW2evdv+12ρ-1ν~2,where ν~=ν∗-ν. ν∗ and ρ are positive constants.
The derivative ofV can be calculated as follows:(15)V˙=2eTtPe˙t+eTtW1et-1-d˙1teTt-d1tW1et-d1t+eTtW2et-1-d˙2teTt-d2tW2et-d2t-ρ-1ν˙ν~=2eTtPAet+Acet-d1t+BΨt+BcΥt,d1+BdΞt,d2+EΛut+eTtW1et-1-d˙1teTt-d1tW1et-d1t+eTtW2et-1-d˙2teTt-d2tW2et-d2t-ρ-1ν˙ν~.Incorporating (3) and (4), we can get(16)V˙≤eTtPA+ATP-αPEETP+W1+W2et+αETPet2+2eTtPAcet-d1t+2eTtPBΨt+2eTtPBcΥt,d1+2eTtPBdΞt,d2+2eTtPEΛut-1-d12eTt-d1tW1et-d1t-1-d22eTt-d2tW2et-d2t-ρ-1ν˙ν~.Using (8) leads to(17)V˙≤eTtPA+ATP-αPEETP+W1+W2et+αETPet2+2eTtPAcet-d1t+2eTtPBΨt+2eTtPBcΥt,d1+2eTtPBdΞt,d2+2eTtPEΛut-1-d12eTt-d1tW1et-d1t-1-d22eTt-d2tW2et-d2t-ρ-1ν˙ν~+eTtL1TL1et+eTt-d1tL2TL2et-d1t+eTt-d2tL3TL3et-d2t-ΨTtΨt-ΥTt,d1Υt,d1-ΞTt,d2Ξt,d2.Assume Ce(t)=Z(t)m×1; Zn(t) is the nth element of Z(t). By using (11), it is easy to prove that ν>0. Based on (5), we need to consider the following two cases:(1)
WhenZn(t)>0, we have un(t)<0. Multiplying Zn(t)>0 and dividing un(t)<0 by both sides of (5), we can obtain that Zn(t)λn(νn(t))≤χnZn(t)νn(t).(2)
WhenZn(t)<0, we have un(t)>0. Multiplying Zn(t)<0 and dividing un(t)>0 by both sides of (5), we can obtain that Zn(t)λn(νn(t))≤χnZn(t)νn(t).
Form the previous discussion, we can prove that the following relation always holds:(18)Zntλnνnt≤χnZntνnt.Using Assumption 3 and χ∗=minχn we have(19)2eTtPEΛνt=2∑n=1mZntλnνnt≤2∑n=1mχnZntνnt≤-χ∗νCet2.Let α=χ∗ν∗; incorporating the previous result (19), we can obtain(20)V˙≤eTtPA+ATP-αPEETP+W1+W2+L1TL1et+χ∗ν∗Cet2+2eTtPAcet-d1t+2eTtPBΨt+2eTtPBcΥt,d1+2eTtPBdΞt,d2-χ∗νCet2-1-d12eTt-d1tW1et-d1t-1-d22eTt-d2tW2et-d2t-χ∗ν∗-νCet2+eTt-d1tL2TL2et-d1t+eTt-d2tL3TL3et-d2t-ΨTtΨt-ΥTt,d1Υt,d1-ΞTt,d2Ξt,d2=eTtPA+ATP-αPEETP+W1+W2+L1TL1et+eTt-d1tL2TL2-1-d12W1et-d1t+eTt-d2tL3TL3-1-d22W2et-d2t+2eTtPAcet-d1t+2eTtPBΨt+2eTtPBcΥt,d1+2eTtPBdΞt,d2-ΨTtΨt-ΥTt,d1Υt,d1-ΞTt,d2Ξt,d2which further can be written as V˙≤ξTΓ1ξ, where(21)ξT=eTteTt-d1teTt-d2tΨTtΥTt,d1ΞTt,d2T,(22)Γ1=Δ1PAc0PBPBcPBd∗Δ20000∗∗Δ3000∗∗∗-I00∗∗∗∗-I0∗∗∗∗∗-I<0,where(23)Δ1=PA+ATP-αPEETP+L1TL1+W1+W2,Δ2=L2TL2-1-d12W1,Δ3=L3TL3-1-d22W2.Using Lemma 1, (22) can be transformed to (12), which completed the proof of Theorem 4.In fact, disturbances always come from surrounding environment and communication channel. So, it is significant to solve the robustness problem in practical application. Theorem5 will tackle this issue.Theorem 5.
Consider the drive and response system (1) and (2) under ω(t)≠0. By application of the adaptive control law (10) and (11), for given γ, if existing symmetric and positive definite matrices P, W1, W2, and a scalar α>0, satisfy the following LMI:(24)ΠPAc0PBPBcPBdPDL1T00∗-1-d12W1000000L2T0∗∗-1-d22W2000000L3T∗∗∗-I000000∗∗∗∗-I00000∗∗∗∗∗-I0000∗∗∗∗∗∗-γ2I000∗∗∗∗∗∗∗-I00∗∗∗∗∗∗∗∗-I0∗∗∗∗∗∗∗∗∗-I<0,where(25)Π=PA+ATP-αPEETP+W1+W2+CTC,the attention rate γ for H∞ synchronization in the disturbance situation can be achieved.Proof.
With zero initial condition, let us introduce(26)J=∫o∞yeTtyet-γ2ωTtωtdt≤0.For ω(t)≠0, the following function can be obtained:(27)J≤∫o∞V˙d+yeTtyet-γ2ωTtωtdt≤0.From the proof of Theorem 4, V˙d can be obtained:(28)V˙d=eTtPA+ATP-αPEETP+L1TL1+W1+W2et+eTt-d1tL2TL2-1-d12W1et-d1t+eTt-d2tL3TL3-1-d22W2et-d2t+2eTtPAcet-d1t+2eTtPBΨt+2eTtPBcΥt,d1+2eTtPBdΞt,d2+2eTtPDωt-ΨTtΨt-ΥTt,d1Υt,d1-ΞTt,d2Ξt,d2.Let(29)ξdT=eTteTt-d1teTt-d2tωTtΨTtΥTt,d1ΞTt,d2T. Equation (28) can be written as(30)J≤ξdTΓ2ξdT,where(31)Γ2=Π1PAc0PBPBcPBdPD∗Π200000∗∗Π30000∗∗∗-I000∗∗∗∗-I00∗∗∗∗∗-I0∗∗∗∗∗0-γ2I,(32)Π1=PA+ATP-αPEETP+L1TL1+W1+W2+CTC,(33)Π2=L2TL2-1-d12W1,Π3=L3TL3-1-d22W2.Using Lemma 1, (31) can be transformed to (24), which completed the proof of Theorem 5.


## 4. Simulation Results

Consider the drive and response system (1) and (2) with parameters:(34)A=-100-1,Ac=0000,B=2-0.1-54.5,Bc=-1.5-0.1-0.2-4,Bd=0.001000.001,C=1.5000.5,E=2002,D=1001.ft,xt=tanh⁡xt,gt,xt-d1t=tanh⁡xt-d1t,ht,xt-d2t=xt-d2tsint,d1t=1+0.3sint,d2t=1-0.02sin10t,λu1tλu2t=1+0.4sin⁡u1tu1t1.2+0.2cos⁡u2tu2t,ωt=0.3sin100t0.5sin110tT.Initial conditions are chosen as(35)xm10,xm20=0.1,-0.3,xs10,xs20=-2,1,χ∗ρ=5,γ=0.6.Whenω(t)=0, using the LMI given in Theorem 4 (12), we can obtain(36)W1=29.11480.29020.29023.7117,W2=27.69710.23990.23992.9992,P=0.7500000.2500,α=46.6242.Whenω(t)≠0, using the LMI given in Theorem 5 (24), we can obtain(37)W1=48.13110.24570.24576.9022,W2=32.26550.22540.22546.8700,P=0.7500000.2500,α=77.6427.The phase trajectory of system (1) and (2) without any control is shown in Figures 1(a) and 1(b). The error between them is shown in Figure 1(c). It is obvious in Figure 1 that the systems are nonsynchronous. After application of the proposed controller, the phase trajectory of system (1) and (2) with nonlinear control inputs is shown in Figures 2(a) and 2(b). The error between them is shown in Figure 2(c). It is obvious in Figure 2 that the systems are synchronous. Figure 3(a) illustrates the phase trajectory of response system with nonlinear control inputs and disturbances after applying the proposed controller. Figure 3(b) illustrates that the error signal tends to zero in a short time, in spite of the disturbances.Figure 1
Behavior of the drive and response systems without any control: (a) phase trajectory of drive system, (b) phase trajectory of response system, and (c) synchronization errors.
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Behavior of the drive and response systems with controller: (a) phase trajectory of drive system, (b) phase trajectory of response system, and (c) synchronization errors.
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Behavior of the drive and response systems with controller and disturbance: (a) phase trajectory of response system and (b) synchronization errors.
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## 5. Conclusion

We investigate the synchronization problem of chaotic systems with nonlinear control inputs. A robust adaptive controller has been established. By applying this controller, control gain can be regulated automatically and the synchronization of chaotic systems can be achieved. Then, considering external disturbances, we propose a newH∞ synchronization method for chaotic systems. From the above simulation results, we can find that the error signal tends to zero in a short time. Therefore our control strategy is effective in synchronizing chaotic systems.



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