# Adaptive Synchronization of Nonlinearly Parameterized Complex Dynamical Networks with Unknown Time-Varying Parameters

Authors: Tengfei Wang; Junmin Li; Shu Tang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

A new adaptive learning control approach is proposed for a class of nonlinearly parameterized complex dynamical networks with unknown time-varying parameters. By using the parameter separation and reparameterization technique, the adaptive learning laws of periodically time-varying and constant parameters and an adaptive control strategy are designed to ensure the asymptotic convergence of the synchronization error in the sense of square error norm. Then, a sufficient condition of the synchronization is given by constructing a composite energy function. Finally, an example of the complex network is used to verify the effectiveness of proposed approach.


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## Body

## 1. Introduction

Since small world network and scale-free network were discovered in the past decade, complex dynamical networks have become a focus issue and have been studied intensively in various disciplines, such as sociology, biology, mathematics, and engineering [1, 2]. As nonlinear dynamics and topological structure of the complex dynamical networks influence its dynamical behaviors, and synchronization is one of the basic forms of cooperative behavior, many of the basic mechanisms have a direct relationship with the synchronization. Research on synchronization of networks has an important significance on understanding of cooperative behavior mechanism and complex phenomena in nature and society. What is more is that it also has a very broad application prospects on nuclear magnetic resonance, wireless sensor network, and multirobot coordination. On the other hand, it can also provide the necessary theoretical basis simultaneously in multiple mobile robot systems, unmanned aircraft systems and distributed sensor array in the field of intelligent autonomous coordination control of complex systems. Up to now, synchronization can be roughly divided into complete synchronization and generalized synchronization. Generalized synchronization includes cluster synchronization, partial synchronization, and functional project synchronization.However, there are many of real-world network systems such as biological networks, mobile communications networks, and social networks, whose structure will change over time, coupling parameter, and other system parameters. For example, the feed-forward loops (FFLs) are typical network motifs in many real world biological networks. The structures, functions, as well as noise characteristics of FFLs are dependent on the network parameters. In [3], a global relative parameter sensitivities (GRPSs) were proposed for FFLs in the genetic networks. The proposed GRPS approach sheds some light on the potential real world applications, such as the synthetic genetic circuits, predicting the effect of interventions in medicine and biotechnology, and so on. The existence of time-varying coupling parameter increases the difficulty of researching the problem of synchronization for complex networks. In the existing results [4–19], the method of synchronization is divided into two categories which are adaptive coupling and adaptive control approach. [20] studied a time-varying complex dynamical network model and its controlled synchronization criteria. It showed that synchronization of such a time-varying dynamical network was completely determined by the inner-coupling matrix and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network. [21] investigated pinning adaptive synchronization of a general complex dynamical network and indeed gave the positive answers to two questions which were how many nodes and how much coupling strength should a network with fixed structure and coupling be chosen. However, in [20, 21], they considered the controlled complex dynamical network with linearly diffusive couplings; [22] researched adaptive synchronization of an uncertain complex dynamical network. Although it considered complex dynamical network with uncertain nonlinear diffusive couplings, the uncertainty of the system with unknown time-varying parameters was not considered. [23] discussed a global payoff-based strategy updating model for studying cooperative behavior of a networked population. It characterized the interactions among individuals by time varying parameter. It is obvious that the coupling of actual complex networks is mostly complex, and the coupling parameters of the network may be not sure or change with some varying factors. Among the above results, none of them is concerning with the adaptive synchronization of complex network systems with unknown time-varying nonlinear coupling.On the other hand, in recent years, adaptive learning control [24], a new adaptive control scheme, has been proposed to deal with the unknown periodic time-varying parameters. It used various methods to estimate the unknown constant or time-varying parameters of the system in order to obtain control input. The convergence of tracking error was proved by using the composite energy function. It is worth mentioning that learning control based on the composite energy function plays an important role in dealing with the estimates of periodically time-varying parameters.Motivated by the above observations, in this paper, the problem of adaptive synchronization is investigated for nonlinearly parameterized complex dynamical networks with unknown time-varying parameters via adaptive learning control method. The main contribution of the paper is that, for the first time, the problem of adaptive synchronization for nonlinearly parameterized complex dynamical networks with unknown time-varying parameters is solved. By combining the parameter separation [25, 26] and reparameterization technique, the adaptive learning laws of periodically time-varying and constant parameters and the proposed adaptive controllers guarantee the asymptotic convergence of the synchronization error in the LT2 norm and that all closed-loop signals are bounded in the LT2 norm. Simulation results are provided to show the effectiveness of the proposed approach.


## 2. Problem Formulation and Preliminaries

In this section, we introduce the network model considered in this paper and give some useful mathematical preliminaries.Consider a dynamical network consisting ofN identical nodes with nonlinear couplings, in which each node is an n-dimensional dynamical system. The state equations of the network are(2.1)ẋi(t)=f(xi(t))+∑j=1NaijΓg(xj(t),φi(t))i=1,2…,N,
where xi(t)=[xi1(t),xi2(t),…,xin(t)]T∈Rn represents the state vector of the ith node. f:Rn×R→Rn is a smooth nonlinear vector-valued function. φi(t) represents the unknown time-varying function. g:Rn×R×R→Rn is an unknown continuous nonlinear vector-valued function. aij=aji is the coupling element. If there exists a connection between node i and node j(i≠j), then aij=aji>0, else aij=aji=0. The inner coupling matrix Γ=diag(γ1,…,γn) is nonnegative definite.Let(2.2)A=(a11a12⋯a1Na21a22⋯a2N⋮⋮⋱⋮aN1aN2⋯aNN),
where AT=A is an unknown irreducible real symmetric matrix. In this model, it is only required that the elements of the coupling matrix A satisfy(2.3)aii=-∑j=1,j≠iNaij<0,i,j=1,2,…,N.
The system (2.1) is said to be asymptotic synchronization in LT2 norm if(2.4)lim⁡t→∞∫t-Tt‖xi(τ)-s(τ)‖2dτ=0,i=1,2,…,N,
where ∥·∥ stands for the Euclidean vector norm. Synchronous evolution s(t) is an arbitrary desired state, which is also an isolated node of the network (2.1) with ṡ(t)=f(s(t)). We assume that for the system ṡ(t)=f(s(t))  there exists stable equilibrium point, stable periodic orbit, or even chaotic attractor.Remark 2.1.
It is well known that multiagent system is a special kind of complex networks. The consensus has similar meaning with synchronization, and consensus is a fundamental natural phenomenon in nature. [27] researched the cluster consensus of discrete-time multiagent systems. Based on Markov chains and nonnegative matrix analysis, two novel cluster consensus criteria were obtained for MAS with fixed and switching topology, respectively. However, in this paper, we consider a new synchronization problem for continuous-time complex dynamic network with nonlinearly time-varying coupling parameter. This problem is quite different from [27], and the challenge problem is due to the time-varying nonlinearly coupling parameter in (2.1).Now, the following assumptions are introduced.Assumption 2.2.
In the network (2.1), suppose that there exists li>0, satisfying
(2.5)[xi(t)-s(t)]T[f(xi(t))-f(s(t))]≤li[xi(t)-s(t)]T[xi(t)-s(t)],
where xi(t) and s(t) are time varying vectors.Lemma 2.3 (separation principle).
For the unknown continuous functiong(·,·), the following inequality holds
(2.6)‖g(xj(t),φi(t))-g(s(t),φi(t))‖2≤‖xj(t)-s(t)‖2h(xj(t),s(t))λ(φi(t)),
where h(·,·) is known nonnegative continuous function and λ(·) is unknown nonnegative continuous function.Assumption 2.4.
In the network (2.1), φi(t) is unknown time-varying function with a known period T, in that way λ(φi(t)) is also a periodic function. Suppose λ(φi(t))=ϕi(t)+θi, where ϕi(t) is unknown continuous function and θi is unknown constant parameter.Assumption 2.5.
In the network (2.1), the inner coupling matrix Γ, coupling matrix A, and h(·,·) satisfy
(2.7)‖Γ‖=γ,h(xj(t),s(t))<H,|aij|≤a,∀i,j=1,2,…,N,
where γ,H,a are positive constants.Lemma 2.6 (Young’s inequality).
For any vectorsx,y∈Rn, and any c>0, the following matrix inequality holds:
(2.8)xTy≤cxTx+14cyTy.


## 3. Adaptive Controller Design

To achieve the control objective (2.4), we need an adaptive control strategy to nodes in the network (2.1). Then the controlled network is given by(3.1)ẋi(t)=f(xi(t))+∑j=1NaijΓg(xj(t),φi(t))+ui(t),i=1,2,…,N,
where ui(t)=[ui1(t),ui2(t),…,uin(t)]T∈Rn, i=1,2,…,N are the adaptive controllers to be designed. Let synchronization error be ei(t)=xi(t)-s(t), i=1,2,…,N and we have the following dynamical error equations:(3.2)ėi(t)=f(xi(t))-f(s(t))+∑j=1NaijΓ(g(xj(t),φi(t))-g(s(t),φi(t)))+ui(t).
Then we design controllers by(3.3)ui(t)=-NaγH(ϕ̂i(t)+θ̂i(t))(xi(t)-s(t)),
where γ,a,H satisfy condition (2.7), N is the node number of the network (2.1), and ϕ̂i(t),θ̂i(t) are estimations to ϕi(t),θi, respectively. For convenience, we denote    (3.4)ϕ̃i(t)=ϕi(t)-ϕ̂i(t),θ̃i(t)=θi-θ̂i(t).The constant parameter update law and the time-varying periodic adaptive learning law are designed as follows:(3.5)θ̂̇i(t)=rieiT(t)ei(t),(3.6)ϕ̂i(t)={ϕ̂i(t-T)+qieiT(t)ei(t),t∈[T,+∞),qi0(t)eiT(t)ei(t),t∈[0,T),0,t∈[-T,0),
where ri,qi>0 are constants and qi0(t) is a continuous and strictly increasing nonnegative function satisfying qi0(0)=0, qi0(T)=qi, which ensure ϕ̂i(t) is continuous in t=iT, i=1,2,…,N.


## 4. Convergence Analysis

In the section, we will give the convergence of the proposed algorithm in the following theorem.Theorem 4.1.
Under Assumptions2.2–2.5, the control law (3.3) with update law (3.5) and the periodic adaptive law (3.6), guarantees the LT2 asymptotic synchronization of the controlled network (3.1), while keeping all closed-loop signals are LT2 bounded.Proof.
To facilitate the convergence analysis, we choose a Lyapunov-Krasovskii function as follows:(4.1)V(t)={12∑i=1NeiTei+NaγH2∑i=1N∫0tqi-1ϕ̃i2(τ)dτ+NaγH2∑i=1Nri-1(θ̃i(t)+Li)2,t∈[0,T),12∑i=1NeiTei+NaγH2∑i=1N∫t-Ttqi-1ϕ̃i2(τ)dτ+NaγH2∑i=1Nri-1(θ̃i(t)+Li)2,t∈[T,+∞),
where Li is sufficiently large positive constant which will be determined.
Firstly, we will prove the finiteness ofV(t) in [0,T). Secondly, we will prove the asymptotical convergence of ei(t). Finally, we will prove all closed-loop signals are LT2 bounded.
First step, let us confirm the finiteness property ofV(t) for the first period [0,T). From the system dynamics (3.1) and the proposed control laws (3.3), (3.5), and (3.6), it can be seen that the right-hand side of (3.1) is continuous with respect to all arguments. In light of the existence theorem of differential equation, (3.2) has unique solution in an interval [0,T1)⊂[0,T), with 0<T1≤T. Therefore, the boundedness of V(t) over [0,T1) can be guaranteed and we need only focus on the interval [T1,T).
For anyt∈[T1,T), the time derivative of V(t) for t∈[T1,T) is given by
(4.2)V̇(t)=∑i=1NeiT(t)ėi(t)+NaγH2∑i=1Nqi-1ϕ̃i2(t)+NaγH∑i=1Nri-1(θ̃i(t)+Li)θ̃̇i(t),t∈[T1,T).
Taking the first term on the right-hand side of (4.2), one obtains
(4.3)∑i=1NeiT(t)ėi(t)=∑i=1NeiT(t)[f(xi(t))-f(s(t))+∑j=1NaijΓ(g(xj(t),φi(t))-g(s(t),φi(t)))+ui(t)].
Let g(xj(t),φi(t))-g(s(t),φi(t))=Λ, and
(4.4)‖Λ‖2≤‖ej(t)‖2h(xj(t),s(t))λ(φi(t)).
Then according to Assumption2.2 and Lemma 2.6, we get
(4.5)∑i=1NeiT(t)ėi(t)≤∑i=1N[lieiTei+∑j=1NaijeiTΓΛ-NaγH(ϕ̂i(t)+θ̂i(t))eiTei]≤∑i=1N[lieiTei+∑j=1N(aij2ceiTei+14cΛTΓTΓΛ)-NaγH(ϕ̂i(t)+θ̂i(t))eiTei]≤∑i=1N[lieiTei+Na2ceiTei+NHγ24ceiTei(ϕi(t)+θi)-NaγH(ϕ̂i(t)+θ̂i(t))eiTei],
where c is a positive constant. If choosing c=γ/4a, we have
(4.6)∑i=1NeiT(t)ėi(t)≤∑i=1N[li+Naγ4+NaγH(ϕ̃i(t)+θ̃i(t))]eiTei.
Taking the third term on the right-hand side of (4.2), we have
(4.7)NaγH∑i=1Nri-1(θ̃i(t)+Li)θ̃̇i(t)=-NaγH∑i=1N(θ̃i(t)+Li)eiTei.
Let us focus on the second term on the right-hand side of (4.2). Since qi0(t) is a continuous function and strictly increasing in [0,T), qi-1≤qi0-1(t)<∞ is ensured in the interval [T1,T), and
(4.8)NaγH2∑i=1Nqi-1ϕ̃i2(t)≤NaγH2∑i=1Nqi0-1ϕ̃i2(t)=NaγH2∑i=1Nqi0-1[ϕi2(t)+2ϕ̂i2(t)-2ϕi(t)ϕ̂i(t)-ϕ̂i2(t)]≤NaγH2∑i=1Nqi0-1[ϕi2(t)+2ϕ̂i2(t)-2ϕi(t)ϕ̂i(t)]=NaγH2∑i=1Nqi0-1ϕi2(t)-NaγH∑i=1Nϕ̃i(t)eiTei.
Substituting (4.6)–(4.8) into (4.2) yields
(4.9)V̇(t)≤∑i=1N(li+Naγ4-NaγHLi)eiTei+NaγH2∑i=1Nqi0-1(t)ϕi2(t).
It is obviously that there exists sufficiently large positive constantsLi such that li+(Naγ/4)-NaγHLi<0. According to (4.9), we have
(4.10)V̇(t)≤NaγH2∑i=1Nqi0-1(t)ϕi2(t).
Sinceϕi(t) is continuous and periodic, its boundedness can be obtained. The boundedness of ϕi(t) leads to the boundedness of V̇(t). For V(T1) is bounded, the finiteness of V(t) is obvious, for all t∈[T1,T).
Next step, we prove the asymptotical convergence ofe(t).
According to (4.1), we can get
(4.11)ΔV(t)=V(t)-V(t-T)=12∑i=1NeiT(t)ei(t)-12∑i=1NeiT(t-T)ei(t-T)+NaγH2∑i=1N∫t-Tt[qi-1ϕ̃i2(τ)-qi-1ϕ̃i2(τ-T)]dτ+NaγH2∑i=1Nri-1(θ̃i(t)+Li)2-NaγH2∑i=1Nri-1(θ̃i(t-T)+Li)2.
Looking into the first two terms on the right-hand side of (4.11), with Newton-Leibniz formula, we obtain
(4.12)12∑i=1NeiT(t)ei(t)-12∑i=1NeiT(t-T)ei(t-T)=∑i=1N∫t-TteiT(τ)ėi(τ)dτ≤∑i=1N∫t-Tt[li+Naγ4+NaγH(ϕ̃i+θ̃i)]eiT(τ)ei(τ)dτ.
Using the algebraic relation(4.13)(a-b)TH(a-b)-(a-c)TH(a-c)=(c-b)TH[2(a-b)+(b-c)],
where a,b,c∈Rp, H∈Rp×p, and taking the third term on the right-hand side of (4.11), one obtains
(4.14)NaγH2∑i=1N∫t-Tt[qi-1ϕ̃i2(τ)-qi-1ϕ̃i2(τ-T)]dτ=-NaγH∑i=1N∫t-Tt[ϕ̃i(τ)+12qieiTei]eiTeidτ.
The last two terms of right-hand side of (4.11) can be simplified as follows:
(4.15)NaγH2∑i=1Nri-1(θ̃i(t)+Li)2-NaγH2∑i=1Nri-1(θ̃i(t-T)+Li)2=NaγH∑i=1N∫t-Ttri-1(θ̃i(τ)+Li)θ̃̇i(τ)dτ=-NaγH∑i=1N∫t-Tt(θ̃i(τ)+Li)eiT(τ)ei(τ)dτ.
Substituting (4.12)–(4.15) into (4.11), we can attain
(4.16)ΔV(t)≤-∑i=1N∫t-Tt(NaγHLi-li-Naγ4)eiT(τ)ei(τ)dτ.
Choosing Li>(li/NaγH)+(1/4H), we can obtain
(4.17)ΔV(t)<0.
Applying (4.16) repeatedly for any t∈[lT,(l+1)T], l=1,2,…, and denoting t0=t-lT, we have
(4.18)V(t)=V(t0)+∑j=0l-1ΔV(t-jT).
Consideringt0∈[0,T) and the positive of V(t), according to (4.18), we obtain
(4.19)lim⁡t→∞V(t)<max⁡t0∈[0,T)V(t0)-lim⁡l→∞∑j=0l-1∑i=1N∫t-(j+1)Tt-jT(NaγHLi-li-Naγ4)eiT(τ)ei(τ)dτ.
SinceV(t0) is bounded in the interval [0,T), according to the convergence theorem of the sum of series and (4.19), the error e(t) converges to zero asymptotically in LT2 norm. That is to say, we have
(4.20)lim⁡t→∞∫t-TteiT(τ)ei(τ)dτ=0.
Finally, we prove that all the closed-loop signals are bounded inLT2 norm sense. For all t∈[T,∞), the derivative of V(t) is
(4.21)V̇(t)=∑i=1NeiT(t)ėi(t)+NaγH2∑i=1Nqi-1ϕ̃i2(t)-NaγH2∑i=1Nqi-1ϕ̃i2(t-T)+NaγH∑i=1Nri-1(θ̃i(t)+Li)θ̃̇i(t)≤-NaγH2∑i=1Nqi(eiTei)2-∑i=1N(NaγHLi-li-Naγ4)eiTei.
By (4.21), one can obtain
(4.22)V(t)≤V(T)-NaγH2∑i=1N∫Ttqi(eiT(τ)ei(τ))2dτ-∑i=1N∫Tt(NaγHLi-li-Naγ4)eiTeidτ.
Choosing Li>(li/NaγH)+(1/4H), we have
(4.23)V(t)<V(T).
From the boundedness ofV(t) and (4.1), we can conclude that ei, ∫t-Ttϕ̃i2(τ)dτ, θ̃i(t) are all bounded. Since ϕi(t) is continuous periodic function, it implies the LT2 boundedness of ϕ̂i(t) and θ̂i(t). According to (3.3), the LT2 boundedness of the control input ui(t) is obtained. Since ei(t) is bounded, the boundedness of xi(t) is received. So the proof is completed.


## 5. Simulation Examples

To demonstrate the theoretical result obtained in Section3, the Chuas chaotic circuit is used as a dynamical node of the network.Consider the Chuas chaos circuit(5.1)ẋ1=p(-x1+x2-g(x1)),ẋ2=x1-x2+x3,ẋ3=-qx2,
where g(x1)=m0x1+(1/2)(m1-m0)(|x1+1|-|x1-1|) with p=10, q=14.7, m0=-0.68 and m1=-1.27.We take the system (5.1) as identical nodes of the network, which is given by (5.2)(ẋi1ẋi2ẋi3)=(p(-xi1+xi2-g(xi1))xi1-xi2+xi3-qxi2)+∑j=1NaijΓexp⁡(-φi(t)(xj12(t)xj22(t)xj32(t)))+ui(t),i=1,2,…,N,
where(5.3)A=(-1100001-2100001-3200002-4110001-5400014-5)Γ=(100010001).
Nonlinearly parameterized function satisfies(5.4)‖exp⁡(-φi(t)(xj12(t)xj22(t)xj32(t)))-exp⁡(-φi(t)(s2(t)s2(t)s2(t)))‖2≤‖ej(t)‖22φi(t)exp⁡(-1).
The parameters are selected as follows:(5.5)N=6,T=6,a=5,γ=1,H=5,ϕ1(t)=0.2sin⁡2πt3,ϕ2(t)=2cos⁡πt,ϕ3(t)=-sin⁡2πt3,ϕ4(t)=cos⁡πt,ϕ5(t)=-2sin⁡2πt3,ϕ6(t)=2sin⁡2πt3,θ=(2,10,8,3,2,5)T.
In the following simulations, we choose(5.6)q1=0.0001,q2=0.0003,q3=0.0002,q4=0.0005,q5=0.0001,q6=0.0002,q10(t)=t6q1,q20(t)=t6q2,q30(t)=t6q3,q40(t)=t6q4,q50(t)=t6q5,q60(t)=t6q6,r1=r2=r3=r4=r5=r6=0.0005. The initially estimated value of the unknown parameter ϕ(t) is ϕ̂(0)=(0,0,0,0,0,0)T, and the initial states are chosen as(5.7)x1(0)=(2,0,-1)T,x2(0)=(3,1,-2)T,x3(0)=(-3,-2,1)T,x4(0)=(-1,0,-2)T,x5(0)=(-1,1,0)T,x6(0)=(3,-2,0.8)T,s(0)=(-0.2,0.2,0.5)T.According to Theorem4.1, the complex dynamical network (5.2) can be synchronized by applying the adaptive controllers (3.3) and the adaptive learning laws of parameters (3.5) and (3.6). Figure 1 shows the error evolutions under the designed controllers. We clearly show that the states of the network (5.2) asymptotically synchronize in LT2 norm with the states of the desired orbit (5.1) showed in Figure 2. Figure 3 depicts the time evolution of the controllers, and Figure 4 shows the evolution of the estimated time-varying parameters. Figure 5 displays the evolution of the estimated constant parameters. Obviously, Figures 3–5 show that all the control signals and estimated parameters are also bounded.Figure 1
The synchronization errors inLT2 norm.Figure 2
The desired orbit.The curve of control inputsui.
(a)(b)(c)(d)(e)(f)Figure 4
The evaluations of parametersϕi.Figure 5
The evaluations of parametersθi.


## 6. Conclusion

In this paper, a new adaptive learning control method applied in nonlinearly parameterized complex dynamical networks with unknown time-varying parameters is presented. By using the parameter separation technique and constructing a composite energy function, the adaptive learning laws of parameters and the adaptive control strategy are designed to ensure the asymptotic convergence of the synchronization error in the sense of square error norm. An example of the complex network is finally used to verify the proposed theoretical result.



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