# Adaptive Synchronization of Complex Dynamical Networks with State Predictor

Authors: Yuntao Shi; Bo Liu; Xiao Han
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper addresses the adaptive synchronization of complex dynamical networks with nonlinear dynamics. Based on the Lyapunov method, it is shown that the network can synchronize to the synchronous state by introducing local adaptive strategy to the coupling strengths. Moreover, it is also proved that the convergence speed of complex dynamical networks can be increased via designing a state predictor. Finally, some numerical simulations are worked out to illustrate the analytical results.


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## Body

## 1. Introduction

In recent years, the synchronization problem of complex networks is a hot topic in many areas, including mathematics, physics, biology, computer, and artificial intelligence [1–37]. A complex dynamical network is composed of a large number of nodes to be steered to arrive synchronization. However, it is so hard and impractical to control all nodes to synchronize in the large complex network. In order to solve this problem, the pinning control is introduced, which can decrease the number of controllers for synchronization of the complex networks. Pinning control scheme is an effective way to control dynamical networks to a desired state. The idea of the pinning control is to steer a small fraction nodes, and all nodes of the complex network can achieve control target via localized feedback of those nodes.A lot of outstanding works about the synchronization problem of complex networks are presented in the recent references [1–37]. In [1], Wang and Chen used specifically and randomly pinning control strategies for scale-free chaotic dynamical networks. The authors [19] considered the synchronization of complex networks through local adaptive coupling. The authors [20] proposed a decentralized adaptive pinning control scheme for synchronization of undirected networks using a local adaptive strategy to both coupling strengths and feedback gains. In [4], Yu et al. investigated the synchronization via pinning control on general complex networks. In [5, 6], second-order consensus problem for multiagent systems was investigated by using pinning control method.The convergence speed of complex dynamical networks is a significant issue. Through limited communication, each agent can forecast the future states of its neighbors and itself; as well as the new control law can be constructed by the predicted states. With this strategy, the complex dynamical network can evolve more quickly to equilibrium. Motivated by it, this paper investigates the synchronization via designing a state predictor. It is proved that all nodes will asymptotically synchronize to the given homogeneous stationary state using the adaptive strategy to the coupling strengths designed, if the complex dynamical network is connected and at least one node is informed. Introducing the state predictor for formation algorithm in multiagent systems, the simulation results show that using the state predictor in multiagent systems can improve the speed of the system to complete the desired task.The rest of this paper is designed as follows. Section2 gives a model of the complex dynamical network. Some preliminaries are introduced to solve the adaptive synchronization. Section 3 provides the theoretical analysis of adaptive synchronization of the complex dynamical network. Furthermore, some more detailed analyses are presented in this section. Section 4 gives some simulations to illustrate our theoretical results. Conclusion is finally summarized in Section 5.


## 2. Preliminaries and Problem Statement

Consider a complex dynamical network described by(1)x˙i(t)=f(xi(t))-∑j∈𝒩i‍aijcij(t)(xi(t)-xj(t))+γ∑j∈𝒩i‍aijcij(t)(x˙ip(t)-x˙jp(t)),
where xi(t)=(xi1(t), xi2(t),…,xin(t))T∈Rn,(i=1,2,…,N) represents the state vector of the node i at time t; Ni is the neighbor set of node i; f(·)∈Rn is continuously differentiable; aij is the coupling weight between any two agents, where aij≥0 and aii=0; cij(t) denotes the coupling strengths between nodes i and node j; the weighted coupling configuration matrix of the system is defined as
(2)U=[uij]=[u11u12⋯u1Nu21u22⋯u2N⋮⋮⋱⋮uN1uN2⋯uNN]∈RN×N,
with uii=aiicii=-∑j=1,j≠iN‍aijcij and uij=aijcij for i≠j. Design the state predictor for the control law as
(3)X˙p=-LX,
where X˙p=(x˙1p,x˙2p,…,x˙Np) and γ is the impact factor of the state predictor.Under state predictor (3), network (1) can be rewritten as
(4)x˙i(t)=f(xi(t))-∑j∈𝒩i‍aijcij(t)(xi(t)-xj(t))-γ(∑j∈𝒩i‍∑k∈𝒩i‍aijaikcij(t)cik(t)(xi-xk)-∑j∈𝒩i‍∑p∈𝒩j‍aijajpcij(t)cjp(t)(xj-xp)).Definition 1.
Network (4) is said to achieve synchronization if
(5)limt→∞∥xi(t)-x-(t)∥=0,i=1,…,N,
where the homogeneous state satisfies
(6)x-˙(t)=f(x-(t),t)=0.The adaptive control at nodei is designed as
(7)c˙ij(t)=∑p∈𝒩j‍aijajpcjp(t)[xi(t)-x-(t)]T[xi(t)-x-(t)]+∑k∈𝒩i‍aijaikcik(t)[xk(t)-x-(t)]T[xk(t)-x-(t)],
where cij(0)≥0.In the following, some necessary assumptions and lemmas are stated.Assumption 2.
The coupling strengths of the network are bounded:(8)∥cij(t)∥≤cij.Assumption 3 (see [5]).
The vector fieldfi:Rn→Rn(i=1,2,…,N) in network (4) satisfies the Lipschitz condition; there exists a positive constant ρ>0, such that
(9)∥f(x)-f(y)∥≤ρ∥x-y∥.Assumption 4.
The weights satisfy the balance condition:(10)∑j∈Ni‍aijcij(t)=∑j∈Ni‍ajicji(t),∀i.Lemma 5 (see [29]).
For any vectorsx, y∈Rn and positive definite matrix G∈Rn×n, the following matrix inequality holds:
(11)2xTy≤xTGx+yTG-1y.Lemma 6 (see [6]).
Supposing thata and b are vectors, then for any positive-definite matrix E, the following inequality holds:
(12)-2aTb≤infE>0{aTEa+bTE-1b}.Lemma 7 (see [38]).
The matrixA of an undirected graph G is irreducible if and only if the undirected graph is connected.


## 3. Main Results

In this section, we will give detailed analysis of the adaptive synchronization of the network with the state predictor. By using the Lyapunov function approach, adaptive synchronization conditions of such work are obtained.Theorem 8.
Considering network (4) with N nodes steered by adaptive control (7), under Assumptions 2–4, then all nodes will asymptotically synchronize to the given homogeneous stationary state; that is,
(13)limt→∞∥xi(t)-x-(t)∥=0.Proof.
Letx~i(t)≜xi(t)-x-(t). Construct the following Lyapunov function:
(14)V(t)=V1(t)+V2(t),
where
(15)V1(t)=12∑i=1N‍x~iT(t)x~i(t),V2(t)=∑i=1N‍∑j∈𝒩i‍[(1-m)cij(t)]22,
where m>0 is sufficiently large.
By the definition of the matrixU, it is to see that U is symmetric and irreducible. By Lemmas 5–7, differentiating V1(t), we can have
(16)V˙1(t)=∑i=1N‍x~iT(t)[f(xi(t))-f(x-(t))]-∑i=1N‍x~iT(t)∑j∈𝒩i‍aijcij(t)(x~i(t)-x~j(t))-γx~iT(t)∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)(x~i(t)-x~k(t))+γx~iT(t)∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)(x~j(t)-x~p(t))≤ρ∑i=1N‍x~iT(t)x~i(t)-∑i=1N‍∑j∈𝒩i‍aijcij(t)x~iT(t)x~i(t)+∑i=1N‍∑j∈𝒩i‍aijcij(t)x~iT(t)x~j(t)-γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~iT(t)x~i(t)+γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~iT(t)x~k(t)+γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~j(t)-γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~p(t)=∑i=1N‍[ρ-∑j∈𝒩i‍aijcij(t)-γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)]x~iT(t)x~i(t)+∑i=1N‍∑j∈𝒩i‍[∑p∈𝒩j‍aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]x~iT(t)x~j(t)+γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~iT(t)x~k(t)-γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~p(t)≤∑i=1N‍[ρ-∑j∈𝒩i‍aijcij(t)-γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)]×x~iT(t)x~i(t)+12∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~iT(t)x~i(t)+12∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~jT(t)x~j(t)+12γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~iT(t)x~i(t)+12γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~kT(t)x~k(t)+12γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~i(t)+12γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~pT(t)x~p(t)=∑i=1N‍[ρ-∑j∈𝒩i‍aijcij(t)-γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)]×x~iT(t)x~i(t)+∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~iT(t)x~j(t)+γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~iT(t)x~k(t)-γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~p(t)≤∑i=1N‍[ρ-∑j∈𝒩i‍aijcij(t)-γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)]×x~iT(t)x~i(t)+12∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~iT(t)x~i(t)+12∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~jT(t)x~j(t)+12γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~iT(t)x~i(t)+12γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~kT(t)x~k(t)+12γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~i(t)+12γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~pT(t)x~p(t)=∑i=1N‍[ρ-∑j∈𝒩i‍aijcij(t)-γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)+12∑j∈𝒩i‍aijcij(t)+12γ∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)+12γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)+12γ∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~iT(t)x~i(t)+12∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~jT(t)x~j(t)+12γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~kT(t)x~k(t)+12γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~pT(t)x~p(t)=∑i=1N‍[ρ-12∑j∈𝒩i‍aijcij(t)-12γ∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)+γ∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~iT(t)x~i(t)+12∑i=1N‍∑j∈𝒩i‍[aijcij(t)+γ∑p∈𝒩j‍aijcij(t)ajpcjp(t)]×x~jT(t)x~j(t)+12γ∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~kT(t)x~k(t)+12γ∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~pT(t)x~p(t).
Therefore,(17)V˙1(t)≤[x~1T(t)⋯x~NT(t)][ρ-12l11-12γl112+γ∑j=1N‍a1jc1j(t)ljj⋯0⋮⋱⋮0⋯ρ-12lNN-12γlNN2+γ∑j=1N‍aNjcNj(t)ljj][x~1(t)x~2(t)⋮x~N(t)]+[x~1T(t)⋯x~NT(t)][12l11+12γl1120⋯0012l22+12γl222⋯0⋮⋮⋱⋮00⋯12lNN+12γlNN2][x~1(t)x~2(t)⋮x~N(t)]+[x~1T(t)⋯x~NT(t)][12γ∑i=1N‍ai1ci1(t)lii0⋯0012γ∑i=1N‍ai2ci2(t)lii⋯0⋮⋮⋱⋮00⋯12γ∑i=1N‍aiNciN(t)lii][x~1(t)x~2(t)⋮x~N(t)]+[x~1T(t)⋯x~NT(t)][12γ∑i=1N‍ai1ci1(t)lii0⋯0012γ∑i=1N‍ai2ci2(t)lii⋯0⋮⋮⋱⋮00⋯12γ∑i=1N‍aiNciN(t)lii][x~1(t)x~2(t)⋮x~N(t)]=[x~1T(t)⋯x~NT(t)][ρ+γ(W1+V1)⋯0⋮⋱⋮0⋯ρ+γ(WN+VN)][x~1(t)x~2(t)⋮x~N(t)],where lii=-uii=∑j=1,j≠iN‍aijcij(t), Wi=∑j=1N‍aijcij(t)ljj, Vj=∑i=1N‍aijcij(t)lii, i,j=1,2,…,N.
DifferentiatingV2(t), we get
(18)V˙2(t)=(1-m)cij(t)c˙ij(t)=∑i=1N‍∑j∈𝒩i‍(1-m)cij(t)×{∑p∈𝒩j‍aijajpcjp(t)[xi(t)-x-(t)]T×[xi(t)-x-(t)]+∑k∈𝒩i‍aijaikcik(t)[xk(t)-x-(t)]T×[xk(t)-x-(t)]∑p∈𝒩j‍}=(1-m)∑i=1N‍∑j∈𝒩i‍∑p∈𝒩j‍aijcij(t)ajpcjp(t)x~iT(t)x~i(t)+(1-m)∑i=1N‍∑j∈𝒩i‍∑k∈𝒩i‍aijcij(t)aikcik(t)x~kT(t)x~k(t)=[x~1T(t)⋯x~NT(t)]×[(1-m)(W1+V1)⋯0⋮⋱⋮0⋯(1-m)(WN+VN)]×[x~1(t)x~2(t)⋮x~N(t)].
Combining V˙1(t) and V˙2(t), and since m>0 is sufficiently large, then we can have(19)V˙(t)=V˙1(t)+V˙2(t)≤[x~1T(t)⋯x~NT(t)]×[ρ+(γ+1-m)(W1+V1)⋯0⋮⋱⋮0⋯ρ+(γ+1-m)(WN+VN)]×[x~1(t)x~2(t)⋮x~N(t)]<0,where lii, Wi, Vj are defined as before.
So,(20)limt→∞∥xi(t)-x-(t)∥=0.
These complete the proof.Theorem 9.
Network (4) solves an agreement problem faster than the same network without the state predictor.Proof.
The proof is similar to that of Theorem 2 in [33].


## 4. Simulations

In this section, we give the numerical simulations to illustrate the analytical results.Consider a small network with the undirected topology described as the following symmetric matrix:(21)A=[00.09640.07570.05700.096400.11990.13960.07570.119900.05810.05700.13960.05810],
where all nodes of the network and their synchronous goal will obey the same nonlinear dynamics described as the Lorenz system:
(22)f(x(t))=f(x1,x2,x3)={x˙1=10(x2-x1)x˙2=28x1-x1x3-x2x˙3=x1x2-83x3,
as shown in Figure 1.Figure 1
The nonlinear dynamics of the network.Figure2 describes the convergence of the state errors on the x-axis, y-axis, and z-axis, respectively. From this figure, we can see that all nodes of the above network can synchronize to the synchronous state gradually. With the same initial state and the same nonlinear dynamics, influenced by the same adaptive strategy, it can be easy to find as in Figure 3 that the network with a state predictor can also synchronize to the synchronous state and be faster than the network without a state predictor.Figure 2
The trajectories of four agents in the dynamical network without a state predictor.Figure 3
The trajectories of four agents in the dynamical network with a state predictor whenγ=200.


## 5. Conclusion

In this paper, we have investigated the adaptive synchronization of complex dynamical networks with nonlinear dynamics. By introducing local decentralized adaptive strategies to the coupling strengths, we have proved that the network with a state predictor can synchronize to the synchronous state.



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