Adaptive Synchronization via State Predictor on General Complex Dynamic Networks

This paper considers the adaptive synchronization of general complex dynamic networks via state predictor based on the fixed topology for nonlinear dynamical systems. Using Lyapunov stability properties, it is proved that the complex dynamical networks with state predictor are asymptotically stable.Moreover, it is also shown that the rate of convergence of complex dynamical networks with state predictor is faster than the complex dynamical networks without state predictor.


Introduction
In recent years, the synchronization of complex dynamical networks has received more and more attention.The synchronous research can be applied in many fields, such as biology, smart city, computer, and the traffic .
It is well known that the complex network has a lot of nodes; however, in order to save an increasing number of energies, the pinning control is introduced to study the synchronization of complex dynamical networks.So far, the pinning control is a main tool by controlling a small number of nodes to steer the whole network.In [2], the pinning control of a continuous-time complex dynamical network with general coupling topologies was researched.The speed of synchronization is a significant issue, so, in [6], a state predictor was introduced.In [7], the adaptive synchronization of complex dynamical networks with state predictor was studied, therefore, this paper studies the problem using the pinning control.
This paper considers the adaptive synchronization of general complex dynamic networks via state predictor based on the fixed topology for nonlinear dynamical systems.With the limited information, state predictor can predict the future state of the nodes and its neighbors; therefore, general complex dynamic networks via state predictor can be faster to achieve synchronization.This paper is organized as follows.Section 2 gives a model of the complex dynamical network.In addition, some preliminaries are introduced to prove the adaptive synchronization.Section 3 gives the main results and the theoretical analysis.The simulations of the theoretical results are given in Section 4. Finally, the conclusion is drawn in Section 5.
Under the state predictor (3), network (1) can be written as The control input is designed as where ℎ  is a binary number; if the th agent is controlled, ℎ  = 1; otherwise ℎ  = 0.   is the feedback gain of position.
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.
Lemma 3 (see [8]).For any vectors ,  ∈   and positivedefinite matrix  ∈  × , the following matrix inequality holds: Lemma 4 (see [9]).Suppose that  and  are vectors; then for any positive-definite matrix , the following inequality holds: Lemma 5 (see [10]).The following equation holds: Lemma 6 (see [18]).For a connected graph which is undirected, the Laplace matrix is positive semidefinite matrix, and the minimum nonzero eigenvalue is the algebraic connectivity of , as follows: Lemma 7 (see [18]).For a system which is similar to ẋ  =   ( = 1, 2, . . ., ), the evolution rate associated with the minimum nonzero eigenvalue  2 . 2 describes the lower bound of convergence rate.Generally, the bigger the  2 is, the faster the system converges.

Main Results
In the following, we will give the main result.Theorem 8. Consider network (4) with the state predictor (3) and  nodes steered by adaptive control (8), under Assumption 2, and at least one node is selected to be controlled.Then, all nodes asymptotically synchronize with the given homogeneous stationary state: Proof.Let x () ≜   () − ().Construct the following Lyapunov function: where Then where   = ∑  =1, ̸ =     (), ,  = 1, 2, . . ., .Consider the following: Since the positive constant m is sufficiently large, V() < 0.
Theorem 9. Network (4) with the state predictor (3) is faster to achieve synchronization than the network without the state predictor.
Proof.For the system with state predictor, the main difference is whether the system contains  ∑ ∈N      ()( ẋ   () − ẋ   ()).We consider the minimum nonzero eigenvalue of state predictor.The Laplace matrix is positive semidefinite matrix, so there is a nonsingular matrix  that can make the Laplace matrix expressed as Obviously, under the same conditions, a system with state predictor has greater minimum nonzero eigenvalue.According to Lemma 7, network (4) with the state predictor (3) is faster to achieve synchronization than the network without the state predictor.

Simulations
In this section, a numerical simulation is given to illustrate the analytical results.Consider a network with the undirected topology described as follows: where each node is a Lorenz system: Figure 1 describes the error states on the -axis, -axis, and -axis, respectively.From Figure 1, we can see that all nodes can synchronize with the synchronous state by degrees.In particular, under the same conditions, the network with a state predictor can be synchronized faster.It is shown in Figure 2.

Conclusion
In this paper, we have investigated the state predictor problem for synchronization of complex dynamical networks in fixed topology.By introducing local adaptive strategies for the coupling strengths, we have proved that the complex dynamical networks are asymptotically stable.It is obvious that the rate of convergence of the network with a state predictor is faster than the network without a state predictor.

Figure 1 :Figure 2 :
Figure 1: The dynamical network without a state predictor.