Adaptive projective lag synchronization of uncertain complex dynamical networks with delay coupling

Abstract

This paper investigates the problem of projective lag synchronization behavior with delayed coupling in drive-response dynamical networks model with identical and non-identical nodes. An adaptive control method is designed to achieve the projective lag synchronization with constant time delay and with time-varying coupling delay. In addition the model harbors fully unknown parameters and disturbances. By using Lyapunov stability theory and adaptive laws, the unknown parameters are estimated. In addition, the unknown bounded mismatch and disturbance terms are also overcome by the proposed control. Finally, the simulation results reveal that the states of the dynamical network with delayed coupling can be asymptotically synchronized onto a desired scaling factor under the designed controller. Additionally, the results prove the validity of the proposed method.

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 DOI 10.1186/s13662-015-0693-2

R E S E A R C H Open Access

http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-015-0693-2&domain=pdf

Web End = Adaptive projective lag synchronization of uncertain complex dynamical networks with delay coupling

Ghada Al-mahbashi1*, Mohd Salmi Md Noorani1, Sakhinah Abu Bakar1,2 and Mohammed Mossa Al-Sawalha2

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM, Bangi, Selangor, MalaysiaFull list of author information is available at the end of the article

Abstract

This paper investigates the problem of projective lag synchronization behavior with

delayed coupling in drive-response dynamical networks model with identical and

non-identical nodes. An adaptive control method is designed to achieve the

projective lag synchronization with constant time delay and with time-varying

coupling delay. In addition the model harbors fully unknown parameters and

disturbances. By using Lyapunov stability theory and adaptive laws, the unknown

parameters are estimated. In addition, the unknown bounded mismatch and

disturbance terms are also overcome by the proposed control. Finally, the simulation

results reveal that the states of the dynamical network with delayed coupling can be

asymptotically synchronized onto a desired scaling factor under the designed

controller. Additionally, the results prove the validity of the proposed method. Keywords: drive-response dynamical networks; projective lag synchronization;

adaptive control; disturbance; coupling delay

1 Introduction

In the past few years, synchronization of dynamical systems has shown interesting behaviors which have received increasing attention in various elds of industry and various sciences []. Meanwhile, many kinds of synchronization have been proposed [] and various control methods have been reported to achieve the dierent kinds of synchronization for complex networks [].

In many practical situations, time delay may cause undesirable dynamic behaviors such as oscillation, instability, and poor performance. Therefore, the development of synchronization of complex dynamical networks with time delays is very important.

In [] Guo studied lag synchronization of complex networks with non-delay coupling by proposing pinning control. On the basis of adaptive control, Ji et al. [] proposed a method with lag synchronization between uncertain complex dynamical networks CDNs with constant delay coupling. Wang et al. [] proposed function projective synchronization (FPS)in CDNs having constant delay coupling and non-identical reference nodes and both network nodes and reference have unknown parameters and bounded external disturbances. Zhang and Zhao [] investigated both projective and lag synchronization be-

2015 Al-mahbashi et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 2 of 13

tween general complex networks via impulsive control. Based on an adaptive feedback controller, projective lag synchronization of the general complex dynamical networks was proposed with non-delay coupling and dierent nodes []. In [] Rui-Jin et al. proposed several nonlinear controllers to realize the problem of projective synchronization with non-delayed and constant delayed coupling in drive-response dynamical networks consisting of identical nodes and dierent nodes.

Motivated by the above discussion, the aim of this paper is to deal with the problem of a projective lag synchronization (PLS) scheme in drive-response dynamical networks (DRDNs) model with coupling delayed consisting of identical and dierent nodes. Both the drive and the network nodes have uncertain parameters and disturbance. Based on Lyapunov stability theory, an adaptive control method is designed to achieve the projective lag synchronization in DRDNs with constant and time-varying coupling delay. Adopting adaptive gains laws, the unknown parameters are estimated. In addition, the controller is designed to overcome the unknown bounded disturbance. In conclusion, the network is asymptotically synchronized with the proposed method. Moreover, numerical simulations are performed to verify the eectiveness of the theoretical results.

The rest of this paper is organized as follows: the DRDNs model with delay coupling is introduced in Section . A general method of PLS in a drive-response dynamical networks (DRDNs) model with constant coupling delayed by an adaptive control method is discussed in Section . Section deals with a further investigation of PLS in a drive-response dynamical networks (DRDNs) model with time-varying coupling delayed by using the proposed method. Examples and their simulations are shown in Section . Finally, the conclusions are presented in Section .

2 Model description

Consider a controlled complex dynamical network with delay coupling consisting of N linearly and diusively dierent nodes with both uncertain parameters and disturbance, described as follows:

xri(t) = gi xri(t) + Gi xri(t) i + c

i are the known continuous nonlinear function matrices determining the dynamic behavior of the node, i is the unknown constant parameter vector, ui Rn

is the control input, c is the coupling strength, and di is an unknown coupling delay.

Here = diag(, , . . . , n) is the inner coupling matrix with i = for the ith state variable, i.e. matrix determines the variables with which the nodes in system are coupled. A = (aij)NN RNN is the coupling conguration matrix representing the topological

structure of the networks, where aij is dened as follows: if there exists a connection be

tween node i and node j (j = i), then aij > , otherwise aij = , and the diagonal elements

of matrix A are dened by

aii =

N

j=,j =i

aij, i = , , . . . , N. ()

N

j=

aij xrj(t di) + i(t) + ui(t), i = , , . . . , N, ()

where xri = (xri, xri, . . . , xrin)T Rn denotes the state vector of the ith node, gi : Rn Rn

and Gi : Rn Rnm

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 3 of 13

The reference node is described as follows:

xd(t) = f xd(t) + F xd(t) + d(t), ()

where the superscripts d stand for the drive system; xd = (xd, xd, . . . , xdn)T Rn denotes

the state vector of the drive system, f : Rn Rn and Fi : Rn Rnm

i are the known continuous nonlinear function matrices determining the dynamic behavior of the node;

is the unknown constant parameter vector, and d contains the mismatched terms.The projective lag synchronization error is dened as

ei(t) = xri(t) xd(t ), i = , . . . , N, ()

where is the nonzero a scaling factor, > is a constant representing time delay or lag. Then the objective of this paper is to design a controller ui(t) such that the reference nodes () and dynamical networks () are asymptotically synchronized such that

lim

t

N

j=

aij ej(t di) + ui(t) + i(t)

f xd(t ) + F xd(t ) (t) + d(t) , i = , . . . , N. ()

Assumption . [] For any positive constant i the time-varying disturbance i(t) is

bounded i.e. i(t) i.

3 PLS in DRDNs with constant delay

In this section, we design an adaptive control method to realize projective lag synchronization for uncertain complex dynamical networks with constant delay coupling.

Theorem . The projective lag synchronization error () is asymptotically stable with a

given time delay and scaling factor , by using the following control input and adaptive laws:

ui(t) = qiei(t) i sgn ei(t) gi xr

i (t) Gi xri(t)

i(t)

(t) , i = , . . . , N, ()

i(t) = kGTi xri(t) ei(t), ()

(t) = kFTi xdi(t ) ei(t), ()

qi(t) = kei(t)Tei(t), () i(t) = kei(t)T sgn ei(t) , ()

where k, k, k, and k are positive constants and

(t) and

i(t) are the estimated parameters for the reference node () and network (), respectively.

xri(t) xd(t ) = , i = , . . . , N, ()

which means that the network () is projective lag synchronized with reference node ().

The error dynamics is obtained:

ei = gi xri(t) + Gi xri(t) i(t) + c

+ f xd(t ) + F xd(t )

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 4 of 13

Proof Construct the Lyapunov function candidate as follows:

V(t) =

N

i=

ei(t)Tei(t) +

k

N

i=

Ti(t)

i(t) +

k

N

i=

Ti(t)

i(t)

+

k

N

i= qi

(t) +

k

N

i=

i(t) +

t tdi

N

i=

ei(s)Tei(s) ds, ()

where

i(t) =

i(t) ,

i(t) =

i(t) i, qi(t) = qi(t) qi,

i(t) = i(t) i, where qi and

i are positive constants.

The time derivative of V(t) along the error dynamics () is

V =

N

i=

eTi(t)ei(t) +

k

Ti(t)

i(t) +

k

Ti(t)

i(t) +

k qi qi(t) +

k

i

i(t)

ei(t di)Tei(t di). ()

By application of the control input () to the error dynamics ei(t) we have

V =

N

i=

+

N

i=

ei(t)Tei(t)

N

i=

eT

i (t) qiei(t) i(t) sgn ei(t) Gi xri(t)

i(t)

eTi(t)

+

N

i=

F xd(t )

(t) + c

N

j=

aij ej(t di) + i(t) d(t)

+

N

i=

k

Ti(t)

i(t) +

k

Ti(t)

i(t) +

k qi qi(t) +

k

i

i(t)

ei(t di)Tei(t di). ()

From the adaptation laws ()-(), V is inferred as follows:

V =

N

i=

+

N

i=

ei(t)Tei(t)

N

i=

qeTi(t)ei(t)

N

i=

eTi(t) sgn ei(t) +

N

i=

eTi(t) i(t) d(t)

+ c

N

i=

eTi(t)

N

j=

aij ej(t di) +

N

i=

ei(t)Tei(t)

N

i=

ei(t di)Tei(t di). ()

Let e(t) = (eT(t), eT(t), . . . , eTN(t))T RnN, P = (A ) where represents the Kronecker

product. Then we have

V = ceT(t)Pe(t d)

N

i=

qeTi(t)ei(t)

N

i=

eTi(t) sgn ei(t)

N

i=

+ eTi(t) i(t) d(t) +

N

i=

ei(t)Tei(t)

N

i=

ei(t di)Tei(t di).

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 5 of 13

We use the fact xTy xTSx + yTSy for any vector x, y Rm, and a positive denite

matrix S Rmm. From the assumption , the following inequality is inferred:

i(t) d(t) i(t) d(t) i(t) + d(t) i,

where i is a positive constant. We have

V

ceT(t)PTPe(t) +

eT(t di)e(t di) qeT(t)e(t)

N

i=

+ i ei(t) +

e(t)Te(t)

e(t di)Te(t di)

eT(t)

cPTP q +

I

e(t) +

N

i=

i ei(t)

i ei(t) ,

where = max(cPTP+I ). Therefore, by taking appropriate q and such that

q < ,i < , i = , , . . . , N,

we obtain

V e(t)Te(t).

Since V is positive denite and V is negative denite, the error ei(t) is asymptotically stable

in the sense of Lyapunov stability theory and the networks () projective lag synchronizes the drive system () asymptotically by the control () and the update laws ()-(). This completes the proof.

4 PLS in DRDNs with time-varying delayed coupling

The adaptive control method is designed to realize projective lag synchronization for uncertain complex dynamical networks with time-varying delay coupling.

Theorem . We assume a given synchronization scaling factor and propagation delay

(t). The projective lag synchronization with time-varying delayed coupling in the drive-response dynamical networks can be realized if the control input and adaptive lows are chosen as

ui(t) = qiei(t) i sgn ei(t) gi xri(t) Gi xri(t)

i(t)

q e(t)Te(t) +

N

i=

+ f xd(t ) + F xd(t )

(t) , i = , . . . , N, ()

i(t) = kGTi xri(t) ei(t), ()

(t) = kFTi xdi(t ) ei(t), ()

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 6 of 13

qi(t) = kei(t)Tei(t), () i(t) = kei(t)T sgn ei(t) , ()

where k, k, k, and k are positive constants and

(t) and

i(t) are the estimated parameters for the reference node() and network (), respectively.

Proof Choose the Lyapunov function candidate as follows:

V(t) =

N

i=

ei(t)Tei(t) +

k

N

i=

Ti(t)

i(t) +

k

N

i=

Ti(t)

i(t)

+

k

N

i= qi

(t) +

k

N

i=

i(t)

+

( )

t t

N

i=

ei(s)Tei(s) ds. ()

The rest of the proof is similar to Theorem . According to the Lyapunov stability theory, the error system is asymptotically stable. This completes the proof.

5 Illustrative example

This section presents the drive-response dynamical networks with three identical, dierent nodes systems, unknown parameters, and disturbance, which are used to show the effectiveness of the proposed schemes obtained in the previous sections. We use the Lorenz system as drive system, which is described as follows:

xd

xd

xd

=

xdxd xd

xdxd

+

xd xd

xd xd

+

d(t), ()

where the unknown parameters vector and mismatch terms are chosen as = [ ] = [ ], d(t) = [ cos(t) sin(t) sin(t)].

The inner coupling matrix = I and the coupling conguration matrix A = (aij) is

chosen to be

A =

.

5.1 Synchronization with constant delay

We discuss the problem of PLS in drive-response dynamical networks with identical and dierent nodes consisting of fully unknown parameters, mismatch terms, and disturbance with constant delay coupling.

.. Synchronization with identical nodesWe take a chaotic Chen system as the ith networks nodes with unknown parameters and disturbance to realize PLS in DRDNs and verify the eectiveness of the proposed scheme

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 7 of 13

Figure 1 The synchronization error ei(t) = xri(t) xd(t ).

which can be described as follows:

xri

xri

xri

=

xrixri

xrixri

+

xri xri xri xri + xri

xri

i i

i

aij xrj(t di) + i(t) + ui. ()

Here the unknown parameters vector is = [ ]T. The disturbance signals are chosen as i = [. cos(t) sin(t) . sin(t) . cos(t)].

In these numerical simulations, we assume that c = ., = , di = ., and = . The gains of the adaptive laws ()-() are k = , k = , k = , k = .. We take the initial states as xd() = [ ]T, xri() are chosen in [, ] randomly, and

=

+ c

i = qi = i = .

The numerical results are presented in Figures and . The time evolution of the synchronization errors is illustrated in Figure , which displays e with t . The

identied parameters of the reference node and network nodes are depicted in Figure (a) and Figure (b), which converge to their real values. These results verify that the proposed control () with adaptive laws ()-() makes the network () projective lag synchronized, even if the network and the reference node () have fully unknown parameters, mismatch terms, and disturbances.

.. Synchronization with dierent nodesThe Chen system, the Lu system, and the Rossler system as the response networks with constant delayed coupling, respectively, are described as follows:

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 8 of 13

Figure 2 The estimated parameters

i (a) and

i (b).

xr

xr

xr

=

xrxr

xrxr

+

xr xr xr xr + xr

xr

+ c

aj xrj(t d) + u(t), ()

xr

xr

xr

=

xrxr

xrxr

+

xr xr

xr xr

+ c

aj xrj(t d) + u(t), ()

xr

xr

xr

=

xr xr xr

xrxr + .

+

xr

xr

aj xrj(t d) + u(t). ()

In these numerical simulations, we assume that c = ., = , di = ., and = . The gains of the adaptive laws ()-() are k = , k = , k = , k = , and qi = . We take the initial states as xd() = [ ]T, xri() are chosen in [, ] randomly, and

=

i = qi =

+ c

i = .

The time evolution of the synchronization errors is illustrated in Figure , which displays e with t . The estimated parameters of the reference node and network

nodes are depicted in Figure (a) and Figure (b), respectively, which converge to their real values. These results prove that the proposed control () with adaptive laws ()-() makes the network () projective lag synchronized if the drive system and the network have unknown parameters.

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 9 of 13

Figure 3 The synchronization error of different nodes.

i (a) and

5.2 Synchronization with varying coupling delay coupling

In this subsection, a drive-response dynamical networks with three identical, dierent node systems, fully unknown parameters, mismatch, and disturbance terms are used to show the eectiveness of the proposed schemes obtained in the previous sections.

.. Synchronization with identical nodesThe chaotic Chen system is chosen as three nodes of complex dynamical networks; the complex dynamical networks with time-varying coupling delay can be described as follows:

Figure 4 The estimated parameters

i (b) with constant coupling delay.

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 10 of 13

Figure 5 The synchronization error of identical nodes with time-varying delay.

xri

xri

xri

=

xrixri

xrixri

+

xri xri xri xri + xri

xri

i i

i

aij xrj t di(t) + i(t) + ui. ()

The propagation delay di(t) = + . sin(t) and = . The gains of the adaptive laws ()-() are k = , k = , k = , k = .. We take the initial states as xd() = [. . .]T, xri() are chosen in [, ] randomly, and

=

+ c

j=

i = qi = i = .

The time evolution of the synchronization errors is depicted in Figure , which displays e with t . The estimated parameters of the reference node and network nodes

are depicted in Figure (a) and Figure (b), which converge to their real values. These results verify that the proposed control () with adaptive laws ()-() makes the network () projective lag synchronized.

.. Synchronization with dierent nodesThe chaotic Chen system, the Lu system, and the Rossler system are chosen as nodes of complex dynamical networks; complex dynamical networks with time-varying coupling delay can be described as follows:

xr

xr

xr

=

xrxr

xrxr

+

xr xr

xr xr + xr

xr

+ c

j=

aj xrj t d(t) + u(t), ()

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 11 of 13

Figure 6 The estimated parameters

i (a) and

i (b) with time-varying coupling delay.

xr

xr

xr

=

xrxr

xrxr

+

xr xr xr xr

+ c

j=

aj xrj t d(t) + u(t), ()

xr

xr

xr

=

xr xr xr

xrxr + .

+

xr

xr

aj xrj t d(t) + u(t). ()

In these numerical simulations, we assume the time delay di(t) = + . sin(t) and = . The gain of the adaptive laws ()-() are k = , k = , k = , k = .. We take the initial states as xd() = [ ]T, xri() are chosen in [, ] randomly and

=

+ c

j=

i = qi = i = .

In Figure shows the time evolution of the synchronization errors. The estimated parameters of the reference node and network nodes are depicted in Figure (a) and Figure (b), respectively, which converge to their real values. These results verify that the proposed control () with adaptive laws ()-() makes the network () projective lag synchronized, even though the drive system and the network have unknown parameters.

6 Conclusion

An adaptive projective lag synchronization (PLS) scheme was proposed in drive-response dynamical networks with delayed coupling consisting of identical and dierent nodes. Both of the reference node and network nodes have fully unknown parameters and disturbances. Adaptive control and update laws were designed to achieve the PLS with constant time delay and with time-varying coupling delay. Based on the Lyapunov stability theory and adaptive laws the unknown parameters were estimated. Furthermore, the unknown bounded disturbances were also overcome by the proposed control. The numerical results showed the eectiveness of the proposed approach.

Al-mahbashi et al. Advances in Dierence Equations (2015) 2015:356 Page 12 of 13

Figure 7 The synchronization error of different nodes with time-varying delay.

i (b) with time-varying coupling delay.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to this work. They all read and approved the nal version of the manuscript.

Author details

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM, Bangi, Selangor, Malaysia. 2Department of Mathematics, Faculty of Science, University of Hail, Hail, Saudi Arabia.

Acknowledgements

The authors would like to acknowledge the grant: UKM Grant DIP-2014-034 and Ministry of Education, Malaysia grant FRGS/1/2014/ST06/UKM/01/1 for nancial support.

Received: 30 June 2015 Accepted: 10 November 2015

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