Fixed-time stochastic outer synchronization in double-layered multi-weighted coupling networks with adaptive chattering-free control

doi:10.1016/j.neucom.2020.02.072

Neurocomputing

Volume 399, 25 July 2020, Pages 8-17

https://doi.org/10.1016/j.neucom.2020.02.072 Get rights and content

Highlights

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Design fixed-time adaptive chattering-free controllers.
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Propose fixed-time synchronization adaptive chattering-free control strategy for different types of networks.
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Construct double-layered multi-weighted coupling networks.
•
Configure complex networks with linear coupling.
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The fixed time is related to the designed controllers, node sizes of networks and dimension of nodes.

Abstract

The problem for fixed-time outer synchronization of double-layered multi-weighted coupled complex networks with stochastic effects is considered in this paper. To suppress chattering in synchronization, an adaptive chattering-free control algorithm is designed. Based upon the Lyapunov stability theory, some sufficient criteria for the adaptive stochastic outer synchronization are proposed. The designed adaptive chattering-free controller and the sufficient conditions can be applicable to not only the fixed-time stochastic synchronization of double-layered multi-weighed undirected networks, but also the fixed-time stochastic synchronization of double-layered multi-weighted directed networks. Our theoretical results indicate that the settling time is related to the size of dynamic networks, the dimension of each node and the designed adaptive controllers in the fixed-time stochastic outer synchronization. The effectiveness of our derived theoretical framework is illustrated via simulation examples.

Introduction

Complex networks have emerged as important research fields, with many theoretical and practical challenges. One of the important behaviors in networks is synchronization, which requires that the states of all nodes have to converge to the same trajectory. The problem of synchronization in complex networks has attracted lots of attentions, such as synchronization of scale-free networks [1], cluster synchronization for the network with external equitable partitions [2], cluster synchronization on sub-networks [3], [4], synchronization of networks with distributed control [5], cluster synchronization and isolated desynchronization of networks [6], synchronization for double-layered and multiple networks [7], synchronization of Markovian jump neural networks [8], synchronization for stochastic neural networks [9]. However, lots of previous works concentrated on synchronization in infinite time. In the applications of synchronization, such as secure communication and encryption, sometimes it is important to synchronize networks in finite time [10].

Finite-time control first appeared in [11]. Soon afterward, synchronization of inertial neural networks was achieved by finite-time control in [12]. Time-controllable combinatorial synchronization for anti-star networks was achieved in [13]. However, the shortcoming of finite-time synchronization is that the settling time is dependent on initial state values of nodes within networks, which are difficult to be measured in the practical networks.

Compared with finite-time control, the advantage of fixed-time control is that the setting time does not depend on the initial values of nodes within networks, which appeared in [14] for the first time. Soon afterward, a few results about synchronization via fixed-time control strategies were reported, such as synchronization of delayed neural networks in fixed time [15], fixed-time synchronization of Cohen-Grossberg neural networks [16], and fixed-time synchronization of complex networks via nonchattering control [17], [18].

The synchronization of double-layered networks are common in real world [10], such as synchronization of double-layered complex networks [7], finite-time outer synchronization of double-layered networks [10], and many networks are coupled via multiple weights [19], [20], [21]. There are a few works about synchronization of multi-weighted networks, such as in [22], [23], [24]. However, there are few literatures on finite-time and fixed-time synchronization of double-layered multi-weighted networks.

Symbolic functions play important roles in the design of controllers for finite-time and fixed-time synchronization [25], [26], [27], [28], [29]. However, chattering phenomenon is introduced by symbolic functions in the states of systems and signals of control [30], which may damage equipment and cause undesirable effects. It is necessary to design finite-time and fixed-time controllers without symbolic functions.

Finite-time and fixed-time controllers usually contain linear feedback, but the strength of the feedback is usually uncertain in practice. Adaptive control technology can be used to solve this problem [31], [32], [33], such as adaptive control for finite-time cluster general projective synchronization of networks [34], adaptive control for overlapping cluster synchronization of networks [35]. However, there are few studies about adaptive fixed-time synchronization.

Stochastic effects are ubiquitous in the synchronization of dynamic networks [36], such as stochastic synchronization of networks in finite time [37], finite-time stochastic synchronization for neural networks [38], fixed-time synchronization for networks with stochastic noise perturbations [39]. However, all of the controllers are not adaptive in [37], [38], [39].

In this paper, the analytical treatment for fixed-time outer synchronization of double-layered multi-weighted networks with stochastic noises is presented, which is achieved via adaptive chattering-free controllers. The main contributions for this paper include:

(1)
Construct double-layered multi-weighted linear coupling networks for fixed-time outer synchronization while considering stochastic noises.
(2)
To determine the feedback strength and reduce chattering, an adaptive chattering-free controller is introduced into the fixed-time outer synchronization of double-layered multi-weighted coupling networks while taking stochastic noises into consideration, and appropriate adaptive update rates are designed. The controller can be used not only for the synchronization of undirected networks, but also for the synchronization of directed networks.
(3)
Based upon Lyapunov stability theory and the adaptive control scheme, the sufficient conditions of fixed-time stochastic outer synchronization for double-layered multi-weighted directed and undirected networks are proposed.
(4)
The obtained fixed settling time is related to the controllers, the number of nodes and the dimension of each node in the fixed-time outer synchronization of networks, which are double-layered multi-weighted directed and undirected networks with and without stochastic noises.

The effectiveness of our derived theoretical framework is illustrated via simulation examples.

Preliminaries and problem formulation are presented in Section 2. The theoretical framework is proposed in Section 3. Simulation examples are derived in Section 4. Conclusions are drawn in Section 5.

Notation. $ℜ^{+}$ represents the positive real number set. Matrix H^T is the transpose matrix of matrix H. $e_{i} (t) = {(e_{i 1} (t), e_{i 2} (t), \dots, e_{i n} (t))}^{T}$ . | · | represents absolute value. ‖ · ‖ is Euclidean norm. $E [\cdot]$ is the expectation.

Section snippets

Preliminaries and problem formulation

An important problem in synchronization of double-layered networks is to design appropriate controllers which drive response networks to synchronize with driving networks. Consider a driving network $\begin{matrix} d x_{i} (t) & = & [- A x_{i} (t) + f_{i} (t, x_{i} (t)) + \sum_{r = 1}^{η} \sum_{j = 1}^{N} σ_{r} d_{i j}^{r} H_{r} x_{j} (t)] d t, \\ i = 1, 2, \dots, N, \end{matrix}$ where $A = d i a g {a_{1}, a_{2}, \dots, a_{n}}$ is a positive diagonal matrix; $x_{i} (t) = {(x_{i 1} (t), x_{i 2} (t), \dots, x_{i n} (t))}^{T} \in ℜ^{n}$ is the state vector for the ith node; $F (t, x_{i} (t)) = - A x_{i} (t) + f_{i} (t, x_{i} (t))$ describes the dynamics for each node, which can be neurons; σ_r represents

Fixed-time synchronization criteria

The sufficient criteria of fixed-time outer synchronization of doubled-layered multi-weighted networks are presented in this section.

Simulation examples

To support our theoretical results, two simulations are given. The double-layered multi-weighted directed networks consist of 20 nodes. Each node is described by the following HR neuron [45]. $\begin{matrix} \underset{{\dot{x}}_{i} (t)}{\underset{︸}{[\begin{matrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{matrix}]}} & = & - \underset{A}{\underset{︸}{[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]}} \underset{{\dot{x}}_{i} (t)}{\underset{︸}{[\begin{matrix} x \\ y \\ z \end{matrix}]}} \\ + \underset{f_{i} (t, x_{i} (t))}{\underset{︸}{[\begin{matrix} y - a_{1} x^{3} + b_{1} x^{2} + x - z + I \\ c_{1} - d_{1} x^{2} \\ r_{1} s_{1} (x + 1.6) + (1 - r_{1}) z \end{matrix}]}} \end{matrix}$ where x, y, z are the membrane potential with initial value $x (0) = 0.1,$ a recovery variable which related with fast current with initial value $y (0) = - 0.1,$ a slowly changing adaptive current with initial value $z (0$

Conclusion

The fixed-time outer synchronization of double-layered multi-weighted networks with or without stochastic effects has been studied. Via applying the Lyapunov functional, some sufficient criteria on fixed-time synchronization have been supposed to guarantee the synchronization. The designed adaptive chattering-free controllers can be used in the synchronization of the double-layered multi-weighted networks with or without stochastic effects, which shows that the designed controller has strong

Author Contributions

In this paper, the analytical treatment for fixed-time outer synchronization of double-layered multiweighted networks with stochastic noises is presented, which is achieved via adaptive chattering-free controllers. The main contributions for this paper include:

(1)
Fei Tan is in charge of the construction of double-layered multi-weighted linear coupling networks; the design of adaptive chattering-free controller and adaptive update rates with the cooperation of Lili Zhou; the derivation of criteria

Declaration of Competing Interest

The authors declare that they do not have any financial or nonfinancial conflict of interests.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 61803322, 11747141 and 61673169, in part by the Natural Science Foundation of Hunan Province under Grant 2018JJ3512, and in part by the Scientific Research Fund of Hunan Provincial Education Department under Grant 17C1534.

Fei Tan received the B.Eng. degree and the M.S. degree in college of mechanical and electrical engineering from the Northeast Forestry University in 2009 and 2014. He is currently working toward the Ph.D. degree in the School of Automation, Nanjing University of Science and Technology. His research interests include complex networks and control of nonlinear systems.

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Lili Zhou received the B.S. degree in information and computing science from the Hunan Institute of Humanities Science and Technology in 2010, the M.S. degree in math from Hunan University in 2012, and the Ph.D. degree in computer science and technology from Hunan University in 2016. She is currently a lecturer at the College of Information Engineering, Xiangtan University, Xiangtan, China. She has published more than 10 SCI journal papers in the fields of complex networks and chaotic systems, and so far, she has presided four national and provincial projects. Her current research interests include the nonlinear dynamical systems, chaos control and synchronization, neural networks, complex systems and complex networks.

Yuming Chu was born in June 3, 1966, Huzhou, Zhejiang, China. He received the B.Sc. degree from the Hangzhou Normal University, Hangzhou, China, in 1988, M.Sc. degree and a Ph.D. degree from the Hunan University, Changsha, China, in 1991 and 1994, respectively. He worked as an Assistant Professor from 1994 to 1996 and as an Associate Professor from 1997 to 2002 at the Department of Mathematics, Hunan Normal University, Changsha, China. Since 2002, he has been a Professor and Dean in the Department of Mathematics at Huzhou Teachers College, Huzhou, China. Dr. Chu’s current research interests include robust filtering and control, special function, quasiconformal mapping and complex dynamic systems

Yongmin Li received his B.S. in Mathematics from Shaanxi Normal University, M.S in Operational Research and Cybernetics from Guizhou University and Ph.D. in Control Theory and Control Engineering from Nanjing University of Science and Technology, in 1992, 2002 and 2008 respectively. He is currently an associate professor of School of Science, Huzhou Teachers College, Huzhou, China. His current research interest includes robust control, anti-windup compensator design and time-delay systems.

View full text

Fixed-time stochastic outer synchronization in double-layered multi-weighted coupling networks with adaptive chattering-free control

Highlights

Abstract

Introduction

Section snippets

Preliminaries and problem formulation

Fixed-time synchronization criteria

Simulation examples

Conclusion

Author Contributions

Declaration of Competing Interest

Acknowledgements

Physica A

Automatica

Neurocomputing

Neurocomputing

Commun. Nonlinear Sci. Numer.Simul.

Neurocomputing

Neurocomputing

IEEE Trans. Cybern.

Neurocomputing

Neurocomputing

Neurocomputing

Fuzzy Sets Syst.

Neurocomputing

J. Frankl. Inst.

Commun. Nonlinear Sci. Numer.Simul.

Neurocomputing

Appl. Math. Model.

Physica A

Phys. Lett. A

Automatica

Cluster synchronization on multiple nonlinearly coupled dynamical subnetworks of complex networks with nonidentical nodes

IEEE Trans. Neural Netw. Learn.Syst.

Cluster synchronization on multiple sub-networks of complex networks with nonidentical nodes via pinning control

Nonlinear Dyn.