# 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.

## 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 fields of industry and various sciences [1, 2, 3]. Meanwhile, many kinds of synchronization have been proposed [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and various control methods have been reported to achieve the different kinds of synchronization for complex networks [14, 15, 16, 17, 18, 19, 20].

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 [21] Guo studied lag synchronization of complex networks with non-delay coupling by proposing pinning control. On the basis of adaptive control, Ji *et al.* [22] proposed a method with lag synchronization between uncertain complex dynamical networks CDNs with constant delay coupling. Wang *et al.* [23] 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 [24] investigated both projective and lag synchronization between 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 different nodes [25]. In [26] 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 different 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 different 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 effectiveness of the theoretical results.

The rest of this paper is organized as follows: the DRDNs model with delay coupling is introduced in Section 2. 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 3. Section 4 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 5. Finally, the conclusions are presented in Section 6.

## 2 Model description

*N*linearly and diffusively different nodes with both uncertain parameters and disturbance, described as follows:

*i*th node, \(g_{i}:\mathbf {R}^{n}\longrightarrow\mathbf{R}^{n}\) and \(G_{i}:\mathbf {R}^{n}\longrightarrow\mathbf{R}^{n\times m_{i}}\) are the known continuous nonlinear function matrices determining the dynamic behavior of the node, \({\theta_{i}}\) is the unknown constant parameter vector, \(u_{i}\in\mathbf{R}^{n}\) is the control input,

*c*is the coupling strength, and \(d_{i}\geq0\) is an unknown coupling delay. Here \(\Gamma= \operatorname{diag}(\gamma_{1}, \gamma_{2}, \ldots, \gamma_{n})\) is the inner coupling matrix with \(\gamma_{i}=1 \) for the

*i*th state variable,

*i.e.*matrix Γ determines the variables with which the nodes in system are coupled. \(A=(a_{ij})_{N\times N}\in\mathbf{R}^{N\times N}\) is the coupling configuration matrix representing the topological structure of the networks, where \(a_{ij}\) is defined as follows: if there exists a connection between node

*i*and node

*j*(\(j\neq i\)), then \(a_{ij} > 0\), otherwise \(a_{ij}=0\), and the diagonal elements of matrix

*A*are defined by

*d*stand for the drive system; \(x^{d}=(x^{d}_{1},x^{d}_{2},\ldots,x^{d}_{n})^{T}\in\mathbf{R}^{n}\) denotes the state vector of the drive system, \(f:\mathbf {R}^{n}\longrightarrow\mathbf{R}^{n}\) and \(F_{i}:\mathbf {R}^{n}\longrightarrow\mathbf{R}^{n\times m_{i}}\) are the known continuous nonlinear function matrices determining the dynamic behavior of the node; Φ is the unknown constant parameter vector, and \(\Delta_{d}\) contains the mismatched terms.

*α*is the nonzero a scaling factor, \(\tau>0\) is a constant representing time delay or lag. Then the objective of this paper is to design a controller \(u_{i}(t)\) such that the reference nodes (1) and dynamical networks (3) are asymptotically synchronized such that

### Assumption 2.1

[22]

For any positive constant \(\varepsilon_{i}\) the time-varying disturbance \(\Delta_{i}(t)\) is bounded *i.e.* \(\|\Delta _{i}(t)\|\leq\varepsilon_{i}\).

## 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 3.1

*The projective lag synchronization error*(6)

*is asymptotically stable with a given time delay*

*τ*

*and scaling factor*

*α*,

*by using the following control input and adaptive laws*:

*where*\(k_{1}\), \(k_{2}\), \(k_{3}\),

*and*\(k_{4}\)

*are positive constants and*\(\hat{\Phi}(t)\)

*and*\(\hat{\theta}_{i}(t)\)

*are the estimated parameters for the reference node*(1)

*and network*(3),

*respectively*.

### Proof

*V̇*is inferred as follows:

*V*is positive definite and

*V̇*is negative definite, the error \(e_{i}(t)\) is asymptotically stable in the sense of Lyapunov stability theory and the networks (3) projective lag synchronizes the drive system (1) asymptotically by the control (7) and the update laws (8)-(11). 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 4.1

*We assume a given synchronization scaling factor*

*α*

*and propagation delay*\(\tau(t)\).

*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*

*where*\(k_{1}\), \(k_{2}\), \(k_{3}\),

*and*\(k_{4}\)

*are positive constants and*\(\hat{\Phi}(t)\)

*and*\(\hat{\theta}_{i}(t)\)

*are the estimated parameters for the reference node*(1)

*and network*(3),

*respectively*.

### Proof

## 5 Illustrative example

### 5.1 Synchronization with constant delay

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

#### 5.1.1 Synchronization with identical nodes

*i*th networks nodes with unknown parameters and disturbance to realize PLS in DRDNs and verify the effectiveness of the proposed scheme which can be described as follows:

In these numerical simulations, we assume that \(c=0.2\), \(\alpha=2\), \(d_{i}=0.2\), and \(\tau=1\). The gains of the adaptive laws (8)-(11) are \(k_{1}= 9\), \(k_{2}= 8\), \(k_{3}=1\), \(k_{4}=0.8 \). We take the initial states as \(x^{d}(0)= [ 1\ {-}1\ {-}1]^{T} \), \(x^{r}_{i}(0)\) are chosen in \([-5, 5 ]\) randomly, and \(\hat{\Phi}_{0} = \hat{\theta}_{i0}= q_{i0} =\beta_{i0}=0\).

#### 5.1.2 Synchronization with different nodes

In these numerical simulations, we assume that \(c=0.2\), \(\alpha=2\), \(d_{i}=0.2\), and \(\tau=1\). The gains of the adaptive laws (8)-(11) are \(k_{1}= 9\), \(k_{2}= 9\), \(k_{3}=1\), \(k_{4}=2 \), and \(q_{i}=0 \). We take the initial states as \(x^{d}(0)= [1\ 2\ 3]^{T} \), \(x^{r}_{i}(0)\) are chosen in \([-5 , 5 ]\) randomly, and \(\hat {\Phi}_{0} = \hat{\theta}_{i0}= q_{i0} =\beta_{i0}=0\).

### 5.2 Synchronization with varying coupling delay coupling

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

#### 5.2.1 Synchronization with identical nodes

#### 5.2.2 Synchronization with different nodes

In these numerical simulations, we assume the time delay \(d_{i}(t)=1+0.2\sin(t)\) and \(\tau=1\). The gain of the adaptive laws (17)-(20) are \(k_{1}= 7\), \(k_{2}= 9\), \(k_{3}=1\), \(k_{4}=2.5 \). We take the initial states as \(x^{d}(0)= [3\ 2\ 1]^{T} \), \(x^{r}_{i}(0)\) are chosen in \([-5 , 5 ]\) randomly and \(\hat{\Phi}_{0} = \hat{\theta}_{i0}= q_{i0} =\beta_{i0}=0\).

## 6 Conclusion

An adaptive projective lag synchronization (PLS) scheme was proposed in drive-response dynamical networks with delayed coupling consisting of identical and different 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 effectiveness of the proposed approach.

## Notes

### 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 financial support.

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