Synchronization Control of Complex Dynamical Networks Based on Uncertain Coupling

In recent years, the study of complex networks has been a focus subject of technology fields. In this paper, we consider the adaptive control and synchronization of uncertain complex networks. By using the adaptive control techniques with the linear feedback updated law and the well-known LaSalle invariant principle on dynamical system theory, some simple yet generic criteria are derived. Furthermore, the result is applied to typical chaotic cellular neural networks (CNN). Finally, numerical simulations are presented to demonstrate the feasibility and effectiveness of the proposed techniques.

State equation of uncertain complex dynamic network using the adaptive controller is expressed as follows:  (1) and (2) can be divided into the following forms: (1) i g is a linear combination of the nodal state variables, i.e.,  12 ( ), ( ), ( )) ...

Synchronization of Uncertain Complex Dynamic Network
As the synchronization manifold of uncertain complex dynamic network(1 We will analyze complete synchronization problem of uncertain dynamic network. The control objective is to design and implement a suitable controller i u ,so that to make the controlled network (2)  (a) There is a nonnegative constant  making: Theorem1. Suppose that conditions (a) and (b) were established. And there is linear feedback controller.
is the intensity of feedback and has the self-tuning law as follows: Where 0   is a real number which is optional and small enough, and 0 iN  is an arbitrary constant. Synchronization error is expressed as follows: Then the complete synchronization of uncertain dynamic network (2) can be reached and we has:  (15) Where i L is an optional positive constant. Calculating the derivative of () Vt along (16), we have: Where 0  is a constant. That means 0 e  , t , i.e., the complete synchronization of uncertain dynamic network can be reached. By proof, the error satisfies: Corollary 1. If the coupled part of the network is: i.e. the coupling part is linear. Just conditions (a) met, the network can achieve synchronization.
Proof. If the coupling part of the network is linear, we have: i.e., condition (b) is established. So just conditions (A) met, the network can achieve synchronization.
Corollary 2. If the coupled part of the network is: i.e. the coupling part is nonlinear. When i.e., condition (b) is naturally established. So just condition (a) is met, the network can achieve synchronization.

Numerical Simulations
Considered a three-dimensional network model of CNN. The state equation is as follows: Where,  ( ( )) ( ( )), ( ( )), ( ( )) is a piecewise linear function expressed as follows: Obviously, the above network is a typical neural network.
There is a chaotic attractor in the model, as shown in Figure 1, Figure 2, Figure 3. Considered a dynamic system, which is linear dissipative coupled by three cellular neural networks (CNN). The state equation of the whole system is expressed as follows:  Based on the above numerical simulation results, we can see that when the external coupling function of the network is both linear and nonlinear function, the control method applied to the network can make the network achieve synchronization. And the numerical simulation results clearly show that, compared with the method in literature [16], the feedback strength and self tuning law in this paper can make the network achieve complete synchronization in a shorter time. Therefore, the development of the method to further expand the existing literature of ideas and techniques, the results and the theoretical results of this paper are completely consistent.

Conclusion
This paper mainly studies adaptive control and synchronization of uncertain complex networks. The main work is supposed as follows: 1. For the uncertain complex networks with adaptive controller, uncertain complex dynamical network model under the adaptive control. Design the controller to make the network achieve synchronization, and prove its rationality by LaSalle invariance principle.
3. The theoretical results obtained in this paper are verified by MATLAB numerical simulation. The results of this paper have important theoretical significance and reference value in the practical engineering design. 4. In contrast to previous work, the main contribution of this paper is to propose a new self tuning law which can make the complex dynamic networks achieve the synchronization state more quickly.