Finite-time synchronization of drive-response systems via periodically intermittent adaptive control

doi:10.1016/j.jfranklin.2014.01.008

Journal of the Franklin Institute

Volume 351, Issue 5, May 2014, Pages 2691-2710

https://doi.org/10.1016/j.jfranklin.2014.01.008 Get rights and content

Abstract

In this paper, the finite-time synchronization between two complex dynamical networks via the periodically intermittent adaptive control and the periodically intermittent feedback control is studied. The finite-time synchronization criteria are derived based on finite-time stability theory, the differential inequality and the analysis technique. Since the traditional synchronization criteria for some models are improved in the convergence time by using the novel periodically intermittent adaptive control and periodically intermittent feedback control, the results of this paper are important. Numerical examples are finally presented to illustrate the effectiveness and correctness of the theoretical results.

Introduction

For the last decade or so, models of complex networks have been widely used to research real systems in nature, society and engineering. A complex network consists of a large set of interconnected nodes, where a node is a fundamental unit with detailed contents. These nodes may have different meanings in different situations, such as microprocessors, computers, and companies. However, the network structure facilitates and constrains the network dynamical behaviors.

In the past few decades, the control and synchronization of complex dynamical networks has attracted much attention and some relevant theoretical results have been established [1], [2], [3], [4], [5], [6], [7] due to its many potential practical applications. Many effective control methods including adaptive control [8], [9], [10], [11], [12], [13], [14], feedback control [15], [16], [17], [18], observer control [19], [20], pinning control [21], [22], [23], impulsive control [24], [25], [26], [27], [28] and intermittent control [29], [30], [31], [32], [33], [34], [35], [36], [37] have been proposed to drive the network to achieve synchronization. Among these control approaches, the discontinuous control methods which include impulsive control and intermittent control have attracted much interest due to its practical and easy implementation in engineering fields. But, the intermittent control is different from the impulsive control since impulsive control is activated only at some isolated instants, while intermittent control has a nonzero control width, the explanation can be found in the map (see Fig. 1). Obviously, when u=v, the intermittent control becomes the general continuous control, while u=0, the intermittent control becomes the impulsive control. Besides, using intermittent control is more effective and robust [38]. Therefore, in recent years, many synchronization criteria for complex dynamical networks with or without time delays via intermittent control have been presented, see [30], [31], [39], [40].

Recently, most of the current researches were primarily concerned with asymptotical or exponential synchronization of networks via intermittent control [29], [30], [31]. This indicates that the intermittent control can derive the slave system to synchronize the master system after the infinite horizon. However, in reality, the networks might always be expected to achieve synchronization or stability under a finite time, particularly in engineering fields. In order to achieve synchronization in a given time, the useful and efficient intermittent control methods might be finite-time intermittent control. Finite-time synchronization of complex networks via intermittent control means the optimality in convergence time dealing with intermittent control problems. However, to our best knowledge, there is few published paper considering finite-time synchronization of complex dynamical networks via periodically intermittent control. Considering the important role of synchronization of complex networks, it is worth studying the finite-time synchronization of complex dynamical networks with intermittent control.

The main contribution of this paper lies in the following aspects. Firstly, a central lemma is proved by using the analysis method. Additionally, two different intermittent controllers are designed to synchronize the addressed complex networks and some useful finite-time criteria are obtained. Subsequently, some sufficient conditions are also derived in terms of linear matrix inequality, which is very easy to verify. Besides, a very interesting fact is revealed that the more the control gains are, the longer the time to achieve the finite-time synchronization will be.

The rest of this paper is organized as follows. In Section 2, the synchronization problem of a general complex network is formulated, and some useful lemmas and preliminaries are given. Finite-time synchronization of two complex dynamical networks by intermittent adaptive control theory is rigorously derived in Section 3. Section 4 studies the finite-time synchronization of drive-response networks by using the intermittent updated laws of coupling strengths. Some numerical examples are given for supporting the theory results in Section 5. Conclusions are drawn in Section 6.

Section snippets

Preliminaries

In this paper, we firstly introduce a complex network model considered in our work and give some useful mathematical preliminaries.

Consider a complex dynamical network consisting of N nodes, in which each node is an n-dimensional dynamical system. The state equation of the entire network is given as ${\dot{x}}_{i} (t) = f (x_{i} (t)) + c \sum_{j = 1}^{N} a_{ij} Γ x_{j} (t), i = 1, 2, \dots, N,$ where $x_{i} (t) = {(x_{i 1} (t), x_{i 2} (t), \dots, x_{in} (t))}^{T} \in R^{n}$ is the state vector of the ith dynamical node, $f : R^{n} \to R^{n}$ standing for the activity of an individual subsystem is a

Finite-time synchronization of complex networks via intermittent adaptive control

In this section, we consider the finite-time synchronization of general complex dynamical networks via periodically intermittent control.

For simplicity, regarding model (2.1) as the master (or drive) system, and the response (or slave) system is given by ${\dot{y}}_{i} (t) = f (y_{i} (t)) + c \sum_{j = 1}^{N} a_{ij} Γ y_{j} (t) + u_{i} (t), i = 1, 2, \dots, N,$ where $y_{i} (t) = {(y_{i 1} (t), y_{i 2}, \dots, y_{in} (t))}^{T} \in R^{n}$ is the response state vector of the node i. $u (t) = {(u_{1} (t), u_{2} (t), \dots, u_{N} (t))}^{T}$ is an intermittent controller defined by ${\begin{matrix} u_{i} (t) = r_{i} e_{i} - \bar{k} \frac{\sqrt{λ_{\max} (P)}}{λ_{\min} (P)} sign (e_{i}), & lT \leq t < lT + δ, \\ u \end{matrix}$

Finite-time synchronization of complex networks via intermittent feedback control

In this section, we shall focus on analyzing finite-time synchronization between two complex dynamical networks via periodically intermittent feedback control, and found an adaptive law to automatically force the error systems converge to zero within a finite time and remain on it forever.

Now, we consider the following two complex dynamical networks: ${\begin{matrix} {\dot{x}}_{i} = f (x_{i}) + c \sum_{j = 1}^{N} a_{ij} Γ x_{j} - r_{i} e_{i}, & i = 1, 2, \dots, N, \\ {\dot{y}}_{i} = f (y_{i}) + c \sum_{j = 1}^{N} a_{ij} Γ y_{j} + k_{i} e_{i}, & i = 1, 2, \dots, N, \end{matrix}$ where $r_{i}$ and $k_{i}$ are the intermittent feedback gain which can be

Numerical examples

In the previous section, theorems essentially provide the criteria for finite-time synchronization. In this section, we use a representative example to illustrate how these theorems can be applied to achieve finite-time synchronization in a complex network.

We consider two complex networks, in which each subsystem is a Lorenz system. The dynamics of a single Lorenz system is described as $[\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \\ {\dot{x}}_{3} \end{matrix}] = [\begin{matrix} - r_{1} & r_{1} & 0 \\ r_{3} & - 1 & 0 \\ 0 & 0 & - r_{2} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] + [\begin{matrix} 0 \\ - x_{1} x_{3} \\ x_{1} x_{2} \end{matrix}] = f (x),$ where the parameters are selected as $r_{1} = 10$ , $r_{3} = 28$ , $r_{2} =$

Conclusion

In this paper, the finite-time synchronization issue between two complex dynamical networks is discussed. Based on the Lyapunov stability, the periodically intermittent control schemes are proposed to synchronize such a network in finite time by combining the adaptive control method and the feedback control technique. Some sufficient conditions ensuring the global finite-time stability of the synchronization process are derived. Numerical examples are presented to verify the effectiveness of

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their valuable comments and constructive suggestions.

This research was supported by the National Natural Science Foundation of China (Grant Nos. 61174216, 61273183, 6134028 and 61374085) and the Doctoral Scientific Foundation of China Three Gorges University (Grant no. 0620120132).