Dynamic event-triggered approach for cluster synchronization of complex dynamical networks with switching via pinning control

doi:10.1016/j.neucom.2019.02.044

Neurocomputing

Volume 340, 7 May 2019, Pages 32-41

https://doi.org/10.1016/j.neucom.2019.02.044 Get rights and content

Highlights

•
The properties of switched systems are introduced to complex networks, which can model the real world more accurately.
•
Cluster synchronization control problem is addressed for complex networks in which nodes are modeled as switched systems.
•
A set of dynamic event-triggered synchronization controllers combined with pinning strategy is designed for networks.

Abstract

This paper focuses on the cluster synchronization problem for complex dynamical networks with switching signal which is characterized by average dwell-time constraint. To reduce networks burden, a dynamic event-triggered mechanism and pinning control strategy is firstly introduced to the design of cluster synchronization controllers for a class of switched complex networks. By constructing piecewise continuous Lyapunov functions and utilizing stability analysis methods, the cluster synchronization conditions are proposed under which the synchronization error dynamics with the dynamic event-triggered rule is stable. Finally, one numerical example is presented to show the effectiveness of the theoretical results.

Introduction

Complex networks, involved numerous intercommunicating nodes, are all around us in our daily life, such as the Internet, subway systems and electric power grids [1], [2]. As one of the important branches of complex networks, complex dynamical networks in which each node is modeled as a nonlinear dynamic system has caught ever-increasing attention from different disciplines owing to their tremendous applications. In fact, complex dynamical networks play a vital role in describing plenty of physical processes, social and biological systems. Up to now, lots of significant results about the analysis of dynamic networks in the fields of control science and engineering have been reported [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Whereas, parameters of models applied to simulate the actual problems will be varying since the emergence of sudden noise or the aging of components [16], [17], [18], [19], [20], [21], [22], [23], such as traffic management, aerospace vehicle control, biological population. Besides, in large scale networked control systems, network time-delay, packet loss and disorder, caused by the fact that a large number of nodes compete for common communication channels or jamming attacks, will inevitably happen. In virtue of the changing of states, switching in the networks will occur frequently. That is, it seems inappropriate to formulate each node as one fixed model and the overall nodes of complex networks can be demonstrated as switched systems [21], [22], [23]. Therefore, it’s natural and significant to introduce the idea of switching into complex dynamical networks.

On the other hand, synchronization is one of the ubiquitous nonlinear phenomena in nature and also has been widely observed in various areas such as neuroscience, biology and engineering. During the past decades, the massive investigates of complex dynamic networks has produced a multitude of unexpected and interesting fruits, such as global synchronization [4], stochastic synchronization [8], finite-time synchronization [9], output synchronization [11], and so on. It’s worth mentioning that not all synchronization phenomena are beneficial to human and synchronous firing of neural ensemble results in several neurological diseases, such as Parkinsons disease. In practice, cluster discharge is an important firing pattern of neurons. Consequently, complete synchronization isn’t suitable to describe the firing phenomena in brain and even to destroy perfect dynamic behaviors of systems. From the perspective of dynamic systems, such phenomena in brain can be regarded as cluster synchronization of complex dynamical networks in the abstract. That is, nodes synchronize in the same group and don’t synchronize in different groups while partitioning nodes of complex dynamical networks into several groups called clusters. Some efforts have been devoted to the evolution of cluster synchronization of complex dynamical networks [24], [25], [26], [27], [28], [29]. In [24], authors have studied cluster synchronization problems of linearly coupled complex networks via pinning control. [25] and [26] investigated such property of complex dynamical networks by using decentralized adaptive pinning control and intermittent pinning control, respectively. Because of the affection of each node comes from nodes in both the same cluster and different clusters, traditional methods relate to complete synchronization couldn’t be applied to the research of cluster synchronization problems directly. Therefore, finding effective analysis tools to dispose of such problems is extremely important.

In classical approaches, time-triggered mechanism is frequently used to deal with control problems of dynamical systems. Under such mechanism, the controllers will be updated in succession regardless of whether the new control actions required to maintain the perfect performance of systems, which leads to much waste of energy and communication resources. Taking account of a large number of nodes in complex dynamical networks, energy consumption and network burden is especially serious. To lower data transmission and reduce communication load, event-triggered scheme has been proposed [30]. Different from event-triggered scheme, control actions are executed based on the requirement of systems in presence of an event-triggered scheme. Mathematically speaking, for a given triggered condition which can be defined in various forms depending on the controlled system, new control rule will work for a time while the triggering condition is violated. Many pioneering literatures have contributed to event-triggered mechanism [30], [31], [32], [33], [34], [35], [36], [37]. For instance, [31] studied decentralized event-triggered mechanism in consideration of different triggered instants of nodes. [33] designed event-based H_∞ filter for nonlinear systems with fading channels and multiplicative noises. Considering stochastic parameters and incomplete measurements, [35] studied the state estimation problems of discrete-time systems. Furthermore, to decrease the conservatism of event-triggered conditions, [37] proposed a novel integral-based event-triggering control for nonlinear systems. Under integral-based event-triggered mechanism, systems need fewer data communication.

Aimed to further enhance the performance of event-triggered mechanism about reducing networks burden, an original control strategy named as dynamic event-triggered mechanism for the additional internal dynamic variable has been presented in [38]. Such results have been extended to stochastic systems [39], discrete-time complex networks [40] and switched systems [41]. However, from the authors best knowledge, there are few literatures about cluster synchronization problems of complex dynamical networks with switching signal via dynamical event-triggered approach. Due to the presence of numerous coupled nodes and unknown switching signals, it’s difficult to design dynamic event-triggered conditions to command controllers which also should be established carefully for complex networks.

Motivated by the above discussions, this paper investigates cluster synchronization of switched complex networks under dynamic event-triggered strategy and pinning control. Suppose that each node is described as a switched system and split nodes in complex networks into several clusters. For any selected cluster synchronization pattern, controllers are designed for one node in each cluster under dynamic event-triggered mechanism, and then sufficient conditions ensuring that the given cluster synchronization pattern is achieved for any initial values are derived by applying the theory of switched systems and mathematical analysis approaches. A numerical example is finally implemented to show the effectiveness of the theoretical results.

As a whole, the main contributions of this paper are listed as follows: (1) The properties of switched systems are introduced to complex dynamical networks. Compared with complex dynamical networks, switched complex dynamical networks can model the real world more accurately. (2) Cluster synchronization control problem is addressed for complex networks in which nodes are modeled as switched systems. (3) A set of dynamic event-triggered cluster synchronization controllers combined with pinning strategy is designed for switched complex dynamical networks.

The rest of this paper is organized as follows. Section 2 gives some notations, definitions, assumptions and lemmas. In Section 3, main results are stated by applying graph theory and some mathematical analysis techniques. A numerical example is given to show the validity of the results in Section 4. Finally, concluding remarks are made in Section 5.

Section snippets

Notations and graph theory

Denote $N$ and $N^{+}$ as the set of natural numbers and non-negative integers, respectively. ℜⁿ and ℜ^n×m mean the set of n-dimensional real space and n×m dimensional real spaces, respectively. $I_{n}$ represents identity matrix and $I_{n} \in ℜ^{n \times n},$ $n \in N^{+}$ . $λ_{\max} (C)$ ( $λ_{\min} (C)$ ) denotes the maximum (minimum) eigenvalue of matrix $C$ . $C > 0$ ( $C < 0$ ) stands for the matrix $C$ is a positive (negative) definite matrix. The superscript ’T’ indicates matrix transposition and ‘⊗’ refers to the Kronecker product. If there are no special

Main results

The main purpose of this section is to investigate under what type of conditions, especially dynamic event-triggered mechanism and pinning controllers, the nodes in the same cluster synchronize to the desired trajectories.

Illustrative example

In this section, we give a numerical examples and the simulations to show the effectiveness of the developed results.

Example 1

Consider a network of ten nodes together with three clusters $G_{1} = {1, 2},$ $G_{2} = {3, 4, 5},$ $G_{3} = {6, 7, 8, 9, 10},$ and the set of pinned nodes is $K = {2, 5, 10}$ . Moreover, the ith node described as follows: ${\begin{matrix} {\dot{x}}_{i} (t) & = & - C_{i}^{σ (t)} x_{i} (t) + B_{i}^{σ (t)} f (x_{i} (t)) \\ + \sum_{j = 1}^{N} a_{i j} Γ x_{j} (t) + U_{i}^{σ (t)} (t), i \in K, \\ {\dot{x}}_{i} (t) & = & - C_{i}^{σ (t)} x_{i} (t) + B_{i}^{σ (t)} f (x_{i} (t)) \\ + \sum_{j = 1}^{N} a_{i j} Γ x_{j} (t), i \notin K, \end{matrix}$ where $x_{i} (t) = {(x_{i 1} (t), x_{i 2} (t))}^{T},$ $f (x_{i} (t)) = (- 1.5 s i n (x_{i 1} (t)) + 0.5 t a n h (x_{i 2} (t)), 0.5 c$

Concluding remarks

In this paper, we have studied the dynamic event-triggered cluster synchronization control problem for a class of switched complex dynamical networks via pinning control and proposed a dynamic event-triggered mechanism to design controllers for the networks. By constructing appropriate Lyapunov functions and applying some efficient analysis techniques, some sufficient criteria have been developed in form of matrix inequalities to make sure that complex networks is cluster synchronization.

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No. 61573095) and the Natural Science Foundation of Shanghai (under Grants no. 6ZR1446700). Also, we are very grateful to Editors and reviewers for their insightful comments and suggestions.

Ling Liu received the M.S. degree in applied mathematics from Hubei Normal University, China, in 2017. She is currently pursuing the Ph.D. degree in control science and engineering from Donghua University, Shanghai, China. Her current research interests include theory and application of neural networks, switched systems and complex networks.

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Xiaoli Li received the Ph.D. degree in control science and engineering from Shanghai Jiao Tong University, China, in 2010. Now she is an associate professor in Donghua University, Shanghai, China. Her current research interests include distributed cooperative control, stability, and performance optimization for multi-agent systems, wireless sensor networks and complex networks.

Yuqing Sun received the B.S. degrees and she is currently pursuing the Ph.D. degree in control science and engineering from Donghua University, Shanghai, China, in 2014. Her current research interests include synchronization and stability research of stochastic neural network systems and optimal control.

View full text

Dynamic event-triggered approach for cluster synchronization of complex dynamical networks with switching via pinning control

Highlights

Abstract

Introduction

Section snippets

Notations and graph theory

Main results

Illustrative example

Concluding remarks

Acknowledgements

Phys. Rep.

Neurocomputing

Phys. A

ISA Trans.

J. Frankl. Inst.

IEEE Trans. Neural Netw. Learn. Syst.

IEEE Trans. Cybern.

IEEE Trans. Autom. Control

IEEE Trans. Signal Proces.

Synchronization in complex networks of phase oscillators: a survey

Automatica

Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays

IEEE Trans. Neural Netw.

Local consensus of nonlinear multiagent systems with varying delay coupling

IEEE Trans. Syst. Man Cybern. Syst.

Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control

IEEE Trans. Circuits Syst. I Reg. Pap.

Finite-time stabilization and synchronization of complex dynamical networks noindentical nodes of different dimensions

Nolinear Dyn.

Output synchronization of networked passive systems with event-driven communication

IEEE Trans. Autom. Control

Output convergence of fuzzy neurodynamic system with piecewise constant argument of generalized type and time-varying input

IEEE Trans. Syst. Man Cybern. Syst.

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IEEE Trans. Neural Netw. Learn. Syst.

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Neurocomputing

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Neurocomputing