Adaptive Synchronization of Complex Dynamical Networks with State Predictor

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.

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. Section 2 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 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝛾_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥̇^𝑝_𝑖(𝑡)−𝑥̇^𝑝_𝑗(𝑡), (1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡), 𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛,(𝑖=1,2,…,𝑁) represents the state vector of the node 𝑖 at time 𝑡; 𝑁_𝑖 is the neighbor set of node 𝑖; 𝑓(⋅)∈𝑅^𝑛 is continuously differentiable; 𝑎_𝑖𝑗 is the coupling weight between any two agents, where 𝑎_𝑖𝑗≥0 and 𝑎_𝑖𝑖=0; 𝑐_𝑖𝑗(𝑡) denotes the coupling strengths between nodes 𝑖 and node 𝑗; the weighted coupling configuration matrix of the system is defined as 𝑈=𝑢_𝑖𝑗=𝑢₁₁𝑢₁₂⋯𝑢_1𝑁𝑢₂₁𝑢₂₂⋯𝑢_2𝑁𝑢⋮⋮⋱⋮_𝑁1𝑢_𝑁2⋯𝑢_𝑁𝑁∈𝑅^𝑁×𝑁, (2) with 𝑢_𝑖𝑖=𝑎_𝑖𝑖𝑐_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗 and 𝑢_𝑖𝑗=𝑎_𝑖𝑗𝑐_𝑖𝑗 for 𝑖≠𝑗. Design the state predictor for the control law as 𝑋̇^𝑝=−𝐿𝑋, (3) where 𝑋̇^𝑝=(𝑥̇^𝑝₁,𝑥̇^𝑝₂,…,𝑥̇^𝑝_𝑁) and 𝛾 is the impact factor of the state predictor.

Under state predictor (3), network (1) can be rewritten as 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑎_𝑖𝑘𝑐_𝑖𝑗(𝑡)𝑐_𝑖𝑘(𝑡)𝑥_𝑖−𝑥_𝑘−_{𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑎_𝑗𝑝𝑐_𝑖𝑗(𝑡)𝑐_𝑗𝑝(𝑡)𝑥_𝑗−𝑥_𝑝. (4)

Definition 1. Network (4) is said to achieve synchronization if lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥(𝑡)=0,𝑖=1,…,𝑁, (5) where the homogeneous state satisfies 𝑥̇‌(𝑡)=𝑓(𝑥(𝑡),𝑡)=0. (6)

The adaptive control at node 𝑖 is designed as 𝑐̇_𝑖𝑗(𝑡)=_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥_𝑖(𝑡)−𝑥(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑥(𝑡+_{𝑘∈𝒩𝑖})𝑎_𝑖𝑗𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥_𝑘(𝑡)−𝑥(𝑡)^𝑇𝑥_𝑘(𝑡)−𝑥(𝑡), (7) where 𝑐_𝑖𝑗(0)≥0.

In the following, some necessary assumptions and lemmas are stated.

Assumption 2. The coupling strengths of the network are bounded: 𝑐_𝑖𝑗(𝑡)≤𝑐_𝑖𝑗. (8)

Assumption 3 (see [5]). The vector field 𝑓_𝑖∶𝑅^𝑛→𝑅^𝑛(𝑖=1,2,…,𝑁) in network (4) satisfies the Lipschitz condition; there exists a positive constant 𝜌>0, such that 𝑓(𝑥)−𝑓𝑦≤𝜌𝑥−𝑦. (9)

Assumption 4. The weights satisfy the balance condition: _{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)=_{𝑗∈𝑁𝑖}𝑎_𝑗𝑖𝑐_𝑗𝑖(𝑡),∀𝑖. (10)

Lemma 5 (see [29]). For any vectors 𝑥, 𝑦∈𝑅^𝑛 and positive definite matrix 𝐺∈𝑅^𝑛×𝑛, the following matrix inequality holds: 2𝑥^𝑇𝑦≤𝑥^𝑇𝐺𝑥+𝑦^𝑇𝐺⁻¹𝑦. (11)

Lemma 6 (see [6]). Supposing that 𝑎 and 𝑏 are vectors, then for any positive-definite matrix 𝐸, the following inequality holds: −2𝑎^𝑇𝑏≤inf_𝐸>0𝑎^𝑇𝐸𝑎+𝑏^𝑇𝐸⁻¹𝑏. (12)

Lemma 7 (see [38]). The matrix 𝐴 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, lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥(𝑡)=0. (13)

Proof. Let 𝑥_𝑖(𝑡)≜𝑥_𝑖(𝑡)−𝑥(𝑡). Construct the following Lyapunov function: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (14) where 𝑉₁(𝑡)=12^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡),𝑉₂(𝑡)=^𝑁_{𝑖=1𝑗∈𝒩𝑖}(1−𝑚)𝑐_𝑖𝑗(𝑡)²2, (15) where 𝑚>0 is sufficiently large.
By the definition of the matrix 𝑈, it is to see that 𝑈 is symmetric and irreducible. By Lemmas 5–7, differentiating 𝑉₁(𝑡), we can have 𝑉̇₁(𝑡)=^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡−))_𝑁𝑖=1𝑥^𝑇_𝑖(𝑡)_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝛾𝑥^𝑇_𝑖(𝑡)_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥_𝑖(𝑡)−𝑥_𝑘(𝑡)+𝛾𝑥^𝑇_𝑖(𝑡)_{𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑝(𝑡)≤𝜌^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑗(𝑡)−𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑘(𝑡)+𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑗(𝑡)−𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑝(𝑡)=^𝑁_𝑖=1𝜌−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)−𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑗(𝑡)+𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑘(𝑡)−𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑝(𝑡)≤^𝑁_𝑖=1𝜌−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)−𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑗(𝑡)𝑥_𝑗(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑝(𝑡)𝑥_𝑝(𝑡)=^𝑁_𝑖=1𝜌−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)−𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑗(𝑡)+𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑘(𝑡)−𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑝(𝑡)≤^𝑁_𝑖=1𝜌−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)−𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑗(𝑡)𝑥_𝑗(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑝(𝑡)𝑥_𝑝(𝑡)=^𝑁_𝑖=1𝜌−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)−𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)+12_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+12𝛾_{𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)+12𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)+12𝛾_{𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑗(𝑡)𝑥_𝑗(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑝(𝑡)𝑥_𝑝(𝑡)=^𝑁_𝑖=11𝜌−2_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)−12𝛾_{𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)+𝛾_{𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+12^𝑁_{𝑖=1𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)+𝛾_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)×𝑥^𝑇_𝑗(𝑡)𝑥_𝑗(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡)+12𝛾^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑝(𝑡)𝑥_𝑝(𝑡). (16) Therefore,𝑉̇₁(𝑡)≤𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)1𝜌−2𝑙₁₁−12𝛾𝑙²₁₁+𝛾∑^𝑁_𝑗=1𝑎_1𝑗𝑐_1𝑗(𝑡)𝑙_𝑗𝑗1⋯0⋮⋱⋮0⋯𝜌−2𝑙_𝑁𝑁−12𝛾𝑙²_𝑁𝑁+𝛾∑^𝑁_𝑗=1𝑎_𝑁𝑗𝑐_𝑁𝑗(𝑡)𝑙_𝑗𝑗𝑥₁𝑥₂⋮𝑥_𝑁+𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)12𝑙₁₁+12𝛾𝑙²₁₁010⋯02𝑙₂₂+12𝛾𝑙²₂₂1⋯0⋮⋮⋱⋮00⋯2𝑙_𝑁𝑁+12𝛾𝑙²_𝑁𝑁𝑥₁(𝑡)𝑥₂(𝑡)⋮𝑥_𝑁(𝑡)+𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)12𝛾∑^𝑁_𝑖=1𝑎_𝑖1𝑐_𝑖1(𝑡)𝑙_𝑖𝑖010⋯02𝛾∑^𝑁_𝑖=1𝑎_𝑖2𝑐_𝑖2(𝑡)𝑙_𝑖𝑖1⋯0⋮⋮⋱⋮00⋯2𝛾∑^𝑁_𝑖=1𝑎_𝑖𝑁𝑐_𝑖𝑁(𝑡)𝑙_𝑖𝑖𝑥₁(𝑡)𝑥₂(𝑡)⋮𝑥_𝑁(𝑡)+𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)12𝛾∑^𝑁_𝑖=1𝑎_𝑖1𝑐_𝑖1(𝑡)𝑙_𝑖𝑖010⋯02𝛾∑^𝑁_𝑖=1𝑎_𝑖2𝑐_𝑖2(𝑡)𝑙_𝑖𝑖1⋯0⋮⋮⋱⋮00⋯2𝛾∑^𝑁_𝑖=1𝑎_𝑖𝑁𝑐_𝑖𝑁(𝑡)𝑙_𝑖𝑖𝑥₁(𝑡)𝑥₂(𝑡)⋮𝑥_𝑁(𝑡)=𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)𝜌+𝛾𝑊₁+𝑉₁𝑊⋯0⋮⋱⋮0⋯𝜌+𝛾_𝑁+𝑉_𝑁𝑥₁(𝑡)𝑥₂(𝑡)⋮𝑥_𝑁(𝑡), (𝑡)(𝑡)(𝑡)(17)where 𝑙_𝑖𝑖=−𝑢_𝑖𝑖=∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡), 𝑊_𝑖=∑^𝑁_𝑗=1𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑙_𝑗𝑗, 𝑉_𝑗=∑^𝑁_𝑖=1𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑙_𝑖𝑖, 𝑖,𝑗=1,2,…,𝑁.
Differentiating 𝑉₂(𝑡), we get 𝑉̇₂(𝑡)=(1−𝑚)𝑐_𝑖𝑗(𝑡)𝑐̇_𝑖𝑗(𝑡)=^𝑁_{𝑖=1𝑗∈𝒩𝑖}(1−𝑚)𝑐_𝑖𝑗(𝑡)×_{𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥_𝑖(𝑡)−𝑥(𝑡)^𝑇×𝑥_𝑖(𝑡)−𝑥(𝑡)+_{𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥_𝑘(𝑡)−𝑥(𝑡)^𝑇×𝑥_𝑘(𝑡)−𝑥(𝑡=())1−𝑚^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑝∈𝒩𝑗}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑗𝑝𝑐_𝑗𝑝(𝑡)𝑥^𝑇_𝑖(𝑡)𝑥_𝑖(𝑡)+(1−𝑚)^𝑁_{𝑖=1𝑗∈𝒩𝑖𝑘∈𝒩𝑖}𝑎_𝑖𝑗𝑐_𝑖𝑗(𝑡)𝑎_𝑖𝑘𝑐_𝑖𝑘(𝑡)𝑥^𝑇_𝑘(𝑡)𝑥_𝑘(𝑡)=𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)×(1−𝑚)𝑊₁+𝑉₁()𝑊⋯0⋮⋱⋮0⋯1−𝑚_𝑁+𝑉_𝑁×𝑥₁(𝑡)𝑥₂(𝑡)⋮𝑥_𝑁(𝑡). (18) Combining 𝑉̇₁(𝑡) and 𝑉̇₂(𝑡), and since 𝑚>0 is sufficiently large, then we can have𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)≤𝑥^𝑇₁(𝑡)⋯𝑥^𝑇_𝑁(𝑡)×𝜌+𝛾+1−𝑚𝑊₁+𝑉₁𝑊⋯0⋮⋱⋮0⋯𝜌+𝛾+1−𝑚_𝑁+𝑉_𝑁×𝑥₁(𝑡)𝑥₂(𝑡)⋮𝑥_𝑁(𝑡)<0, (19)where 𝑙_𝑖𝑖, 𝑊_𝑖, 𝑉_𝑗 are defined as before.
So, lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥(𝑡)=0. (20) 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: 𝐴=00.09640.07570.05700.096400.11990.13960.07570.119900.05810.05700.13960.05810, (21) where all nodes of the network and their synchronous goal will obey the same nonlinear dynamics described as the Lorenz system: 𝑓(𝑥(𝑡))=𝑓𝑥¹,𝑥²,𝑥³=𝑥̇¹=10𝑥²−𝑥¹𝑥̇²=28𝑥¹−𝑥¹𝑥³−𝑥²𝑥̇³=𝑥¹𝑥²−83𝑥³, (22) as shown in Figure 1.

Figure 1 

The nonlinear dynamics of the network.

Figure 2 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 when 𝛾=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.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant no. 61304049, no. 61174116, the Beijing Natural Science Foundation Program 4132021, Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011), and Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201108055).