Adaptive Synchronization of Complex Dynamical Networks with State Predictor

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.


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 .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 .In [1], Wang and Chen used specifically and randomly pinning control strategies for scale-free chaotic dynamical networks.De Lellis et al. [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
Under state predictor (3), network (1) can be rewritten as where the homogeneous state satisfies The adaptive control at node  is designed as where   (0) ≥ 0.
In the following, some necessary assumptions and lemmas are stated.

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. where where  > 0 is sufficiently large.By the definition of the matrix , it is to see that  is symmetric and irreducible.By Lemmas 5-7, differentiating  1 (), we can have Therefore, ] where   = −  = ∑ Combining V 1 () and V 2 (), and since  > 0 is sufficiently large, then we can have where   ,   ,   are defined as before.
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].

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: where all nodes of the pagebreak network and their synchronous goal will obey the same nonlinear dynamics described as the Lorenz system: as shown in Figure 1. 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.

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.