Elsevier

Physics Letters A

Volume 324, Issues 2–3, 12 April 2004, Pages 166-178
Physics Letters A

Robust adaptive synchronization of uncertain dynamical networks

https://doi.org/10.1016/j.physleta.2004.02.058Get rights and content

Abstract

This Letter studies local and global robust adaptive synchronizations in uncertain dynamical networks. For complex dynamical networks with unknown but bounded nonlinear couplings, with known or unknown bounds, some robust adaptive controllers are designed in this Letter, which can ensure that the state of a dynamical network locally or globally asymptotically synchronize with an arbitrarily assigned state of an isolate node of the network. The Letter also investigates the synchronization problem where the dynamics of the coupled states are different from that of the uncoupled states. The key idea is to suitably combine the Lyapunov stability theory with a good update law for estimating the unknown network coupling parameters in the design of the controllers. Two examples are simulated, using the smooth chaotic Chen system and the piecewise-continuous chaotic Chua circuit, respectively, as the nodes of the dynamical network, which demonstrate the effectiveness of the proposed controllers design methods.

Introduction

Synchronization of complex networks of dynamical systems has received a great deal of attention from the nonlinear dynamics community in the last two decades. In particular, special attention has been focused on the synchronization of chaotic dynamical systems. More recently, many scientists have started to consider synchronization of large-scale networks of coupled chaotic oscillators [2], [3], [4]. Of special interest is the current study of synchronization in different small-word and scale-free dynamical network models [5], [6], [7], [8], [9], [10], which has shed some new light on the synchronization phenomenon in real-world complex networks. In these investigations, an essential requirement is that the structure of the network and the coupling functions are known a priori. Notice also that the final state the network will reach after achieving synchronization is usually not known beforehand and cannot be changed at will. Moreover, one cannot guarantee that all linearly coupled dynamical nodes can synchronize if the network size is sufficient large for some nearest-neighboring coupled networks [6]. Yet, it is very desirable if one can choose what state the network will synchronize to. Likewise, it is preferable if the coupling functions in a dynamical network are not restricted to be linear and completely known, since in applications quite often this requirement is not realistic. Sometimes, even the network structure is only partially known or completely unknown. If there are too many unknown issues, of course, it is very difficult or even impossible to design a controller to achieve the intended network synchronization; however, it would be interesting to see if the task is possible for a general situation where the network coupling is an unknown but bounded nonlinear function, where the bounds may be either known or unknown. By exploring the Lyapunov stability theory and introducing a suitable update law for estimating the unknown network parameters, this Letter shows that some robust adaptive controllers can indeed be designed for the above-described task, which can ensure that the states of the dynamical network locally or globally and asymptotically synchronize with an arbitrarily assigned state of an isolate node of the network. The Letter also investigates the synchronization problem where the dynamics of the coupled states are different from that of the uncoupled states.

The rest of the Letter is organized as follows. The general dynamical model is first described in Section 2. In Section 3, problems of both local and global robust adaptive synchronizations are studied, with two desirable controllers derived. In Section 4, the synchronization problem where the dynamics of the coupled states are different from that of the uncoupled states is investigated. In Section 5, two examples are simulated by using the chaotic Chen system and Chua circuit as the nodes of the dynamical networks, respectively, demonstrating the effectiveness of the proposed controllers design methods. Finally, conclusions are given in Section 6.

Section snippets

Uncertain dynamical network models

Consider a dynamical network consisting of N identical nodes (n-dimensional dynamical systems) with uncertain diffusively nonlinear coupling, in the following form: ẋi=f(xi,t)+gi(x1,x2,…,xN)+vi,t⩾0,i=1,2,…,N, where xi=(xi1,xi2,…,xin)TR, represents the state vector of the ith node, f:Rn×R+→Rn are smooth nonlinear vector functions, gi:Rm→Rn are smooth but unknown nonlinear coupling functions, where m=nN, and viRn are the control inputs.

When the network achieves synchronization, namely, the

Main results on robust adaptive synchronization

Consider two cases where the parameters γij, i,j=1,2,…,N, in Assumption 1 are known and unknown, respectively.

Extension to synchronized coupled state

In the last section, we have studied the synchronized state of an isolated node when the coupling gradually vanishes. In general, however, a synchronized coupled state may have different dynamical behavior from that of the uncoupled (isolated) situation. This issue is addressed in this section.

Consider a synchronized coupled state in a linear connection, y(t)=j=1ldjsi(t), where the individual variables sj(t), j=1,2,…,l, satisfy the dynamics described by (2), and dj≠0, j=1,2,…,l, are unknown

Simulations

The following simulations demonstrate the theoretical results derived in Section 3.

Example 1

The chaotic Chen system [11] is used as nodes of the dynamical network. A single Chen system is described by the following equations: ẋ1=a(x2−x1),ẋ2=(c−a)x1−x1x3+cx2,ẋ3=x1x2−bx3. The system has three equilibrium points if (2ca)b>0: O=(0,0,0),C+=(2c−a)b,(2c−a)b,2c−a,C=(2c−a)b,−(2c−a)b,2c−a. It is known [11] that when a=35,b=3,c=28 system (47) is chaotic. The state equations of the entire network are given by

Conclusions

This Letter has studied both local and global robust adaptive synchronizations of uncertain dynamical networks, with known or unknown bounded nonlinear coupling functions. Some robust adaptive controllers have been designed, analyzed and simulated, which ensure that the state of a dynamical network can locally or globally asymptotically synchronize with an arbitrarily assigned state of an isolate node of the network. The synchronization problem where the dynamics of the coupled states are

Acknowledgements

This research was supported by the Hong Kong Research Grants Council under the CERG Grants CityU 1004/02E and 1115/03E, the China Postdoctoral Science Foundation and the National Natural Science Foundation of China through the grant number 70371066.

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