Abstract
In this paper, we propose and study a global synchronization of delayed complex networks with hybrid coupling, which is composed of constant coupling, discrete-delay coupling, and unbounded distributed-delay coupling, and its actuator saturation control design is then further investigated. Several effective sufficient conditions of global synchronization are attained based on the Lyapunov function and a linear matrix inequality (LMI), which can be easily computed by the interior-point method. In addition, we established the control design of actuator saturation of the addressed stochastic delayed complex networks. More relaxed conditions by employing the new type of augmented matrices by using multitude Kronecker product terms can be handled, which can be introduced. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed synchronization scheme.
1. Introduction
In the past decades, complex networks have been gaining increasing research attention due to their potential applications to many real-world systems in various fields of science and engineering. The studies of complex networks have been an active field of research in many scientific and technical disciplines [1–3]. One of the most significant and interesting phenomena in complex networks is the synchronization of all dynamical nodes or vertices connected by links or edges. Examples of such complex networks can be found everywhere in our daily life, from physical objects such as the Internet, multiagent systems, social networks the world wide web (WWW), and electricity distribution network, telephone cell graphs, food webs, neural networks, electrical power grids, cellular and metabolic networks, scientific citation web, living organisms, etc. Complex network is a large number of highly interconnected fundamental units and therefore exhibit very complicated dynamics [4–12].
In real-world situation, time delay is ubiquitous in many physical systems due to the finite switching speed of amplifiers, finite signal propagation time in networks, finite reaction times, memory effects, and so on. Furthermore, the time delay may cause undesirable dynamic behaviors such as oscillation, instability, and poor performance. Therefore, the synchronization problem of complex dynamical networks with time delays has become a topic of both theoretical and practical importance. In this regard, some synchronization criteria for the general complex networks with coupling delays which are classified into two types: delay-independent and delay-dependent conditions are derived (see [10, 12–15]).
In real-time systems, the complex networks are often largely affected by the noisy environment. During the communications among the nodes of complex networks, the network-induced delays and external disturbances are simply unavoidable. In particular, the ’random’ phenomena often appear due to connections over communication channels, which includes random communication delay, random measurements, and random packet losses, which have lately attracted a great deal attention in the networked control society. As is widely recognized an extensive elegance of practical systems is influenced by additive nonlinear disturbances which might be caused by environmental circumstances. The randomly occurring nonlinearities, also called stochastic nonlinearities, have recently received some interest in the literature; see [16–18]. For example, in [19], the filtering and control problems for continuous-time systems with stochastic nonlinearities have been investigated. Unfortunately, to the best of the authors’ knowledge, the global synchronization problem for complex networks experiencing randomly occurring nonlinearities with actuator saturation and stochastic disturbances has acquired very little interest for time delays, and the purpose of this paper is therefore to fill in such a gap.
Nowadays, in digitalized world, control systems are often controlled by digital controllers that make a limitation on the power of the actuator in practical systems. Thus, the analysis and synthesis of controllers of dynamic systems subject to actuator saturation have attracted a considerable attention among control community [20–22]. As for the stabilization of stochastic systems with actuator saturation, only a few works have been done on this subject. To the best of the author’s knowledge, however, the problems of analysis and design for stochastic systems with both state delay and actuator saturation remain open.
Motivated by the above observations, in this paper, we investigate the synchronization problem of delayed complex networks (SDCN) with hybrid coupling, actuator saturation and stochastic nonlinearities, multiple stochastic disturbances, time-varying delays, and continuously distributed delays. By employing the properties of Kronecker product [23] and stochastic analysis techniques [24, 25]. We construct a novel Lyapunov functional and introducing several new lemmas; some sufficient conditions are obtained to analyze the synchronization criteria, which are formulated in the form of linear matrix inequalities. A numerical example is presented to illustrate the effectiveness of the proposed control design method.
The novelties of this paper are as follows:
(1) The global synchronization of the complex networks with hybrid coupling, actuator saturation, randomly occurring nonlinearities, and continuously distributed delays is considered.
(2) Novel Lyapunov functional consisting of Kronecker product is constructed to derive the main results.
(3) Various types of inequality lemmas are applied in the derivation process which leads to the feasible criteria.
(4) Simulation results are provided to show the effectiveness of the proposed results.
The rest of this paper is organized as follows. In Section 2, a stochastic complex network model with time delays, stochastic nonlinearities, and multiple stochastic disturbances with hybrid couplings and time delays is proposed, and some preliminaries are briefly outlined. In Section 3, the Lyapunov functional method combined with the LMI technique, so as to ensure that the considered complex networks with multiple stochastic disturbances and stochastic nonlinearities are globally synchronized in the mean square, is proposed.
Section 4 is devoted to deriving the results on controller design of actuator saturation. In Section 5, a numerical example is provided to show the applicability of the obtained results. The conclusions are finally drawn in Section 6.
Notations. Throughout this paper, and denote, respectively, the n-dimensional Euclidean space and the set of all real matrices. means that matrix P is real, symmetric, and positive definite. represents the n-dimensional identity matrix, stands for a block-diagonal matrix, and denotes a matrix column with blocks given by the matrices in. If A is a matrix, the notation means the largest eigenvalue of A. The Kronecker product of matrices and is denoted as ; matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.
2. Problem Formulation and Preliminaries
Consider the following array of delayed complex networks with multiple stochastic disturbances and stochastic nonlinearities consisting of coupled nodes of the formwhere is the state vector of the node at time t; , , , and are the known real constant matrices; represents the inner coupling between the subsystems at time and , respectively, where is time-varying delay and satisfy ; and are the outer-coupling configuration matrices representing the coupling strength and the topological structure of the complex networks; is the vector-valued standard saturation function defined as , where , furthermore, is the noise intensity function vector, and are mutually independent scalar Brownian motions defined on withMoreover, is the distributed time-delay kernel; and the discrete time delay is a time-varying differentiable function satisfyingwhere are constants. Finally , , and are continuous nonlinear functions, and is a stochastic variable.
Define the following random events for system (1):
Letting be a Bernoulli distributed sequence defined byit follows that satisfies , Prob, where the constant reflects the occurrence probability of the event of the nonlinear functions and . It is further assumed that the variables and are mutually independent.
The controller is given as
Assumption 1 (see [26]). The outer-coupling configuration matrices of the complex networks satisfy
Assumption 2 (see [27]). Nonlinear functions and are bounded functions satisfying and for all , and some fixed constants , the active function satisfiesWe denote
Assumption 3 (see [28]). The distributed time-delay Kernel is continuous and integrable and also satisfies
Assumption 4 (see [29]). The noise intensity function vector . There exist constant matrices and of appropriate dimensions such that the inequalityholds for all and .
Combining with the sign of Kronecker product, model (1) can be rewritten aswhere
Next, we give some useful definition and lemmas.
Definition 5. Model (1) is said to be globally synchronized for any initial conditions , if the following holds: in which stands for the Euclidean norm.
Lemma 6 (see [30]). For , if , then can be represented as where and .
Lemma 7 (see [31]). For any constant matrix , , scalar , and vector function , one has
Lemma 8 (see [27]). For any diagonal matrices , , it follows that
It is equivalent to the following equation:
Lemma 9 (see [32]). Let denote the notation of Kronecker product. Then, the following relationships hold:
Lemma 10 (see [33]). Let , , and , , , and , with , then
Lemma 11 (see [34]). Let , be any real matrix, , , , , with , and and are functions and defined in system (2). Then for any vectors and with appropriate dimensions, the following matrix inequality holds:
3. Synchronization Stability Criteria of Complex Networks
In this section, we deal with globally synchronization problem of the complex network (1) or (12) with stochastic nonlinearities.
Rewrite system (12) asDefinewhere , , , , ,
Then network (4) can be expressed aswhere , , .
Theorem 12. For given scalars and , the complex networks (23) are said to be globally synchronized if there exist positive definite matrices , and positive diagonal matrices , such that the following linear matrix inequalities hold for all :where
Proof. Choose the Lyapunov functional candidate:where Let be the weak infinitesimal generator of random process . Then, we can obtain [25]According to Assumption 4 and condition (26), it is clear that From Assumption 2, by the methods in Lemma 8, for any positive diagonal matrices , one hasAccording to (31)–(40) and Lemma 11, we can obtain where From Definition 5, we conclude that system (23) has global synchronization. The proof is completed.
4. Actuator Saturation and Control Design
In this section, by use of pre- and postmultiplying (27), some matrices and nonlinear matrix inequality (27) can be changed into linear matrix inequality and the final optimization problem can be obtained Then, the optimization problem can be given assuch that inequality (27) holds;Let
From the value , we havewhere denotes the qth row of . Pre- and postmultiplying (27) by lead toTherefore, the optimization problem (45) can be rewritten assuch that inequalities (44)-(49) hold, where with then
the controller gain is given as .
5. Numerical Example
In this section, we present a numerical example so as to illustrate the advantage and usefulness of our main results.
Example 1. Consider the following complex networks with time delays.
Choose the coupling matrices , and the linking matrices , as Let the nonlinear function and the noise intensity function vector be given by Then it is easy verify that , , and We solve the LMIs in Theorem 12. we obtain the feasible solutions as follows: Moreover, the corresponding control gain matrix is obtained; .
The initial condition is selected as . In order to obtain the simulated curve of the system . Simulation results are shown in Figures 1–7. Synchronization of the complex networks with attractor is shown in Figures 1 and 2. Then Figures 3 and 4 state that trajectories are complex networks. It is shown that the trajectory reaches practical randomly nonlinearity. After , the trajectory is going to be synchronized, which is the same as the theoretical value presented in the paper. Figure 5 shows the control input with stochastic disturbance. The event time , its getting 1 occurs, and , its getting 2 occurs, which is shown in Figures 6 and 7 and it is clear from the figure that once the trajectories reach practical actuator saturation, the interevent time increases; i.e., the control is being updated after long time gaps. There is a maximum time gap between two consecutive control updates after reaching the optimization obtained as . The average value of the interevent time is a big multiple of sampling interval.
6. Conclusion
This paper studies the global synchronization of complex networks with hybrid coupling and stochastic disturbance. By defining appropriate Lyapunov functional and by employing integral inequalities, new delay-dependent globally synchronization criteria have been derived in terms of LMIs. Based on the synchronization criterion and the control design, a sufficient condition is provided to guarantee the globally synchronization of considered system with actuator saturation of randomly nonlinearity. Then method of control gain has also been given. The theoretical results were validated by numerical example and simulations. It should be pointed out that the proposed method in this paper can also be extended to the Markovian jumping complex dynamical systems with parameter uncertainties, partially unknown transition probabilities and random packet loss. This work will be done in the near future.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Quanxin Zhu’s work was jointly supported by the National Natural Science Foundation of China (61773217 and 61374080), the Natural Science Foundation of Jiangsu Province (BK20161552), and the Construct Program of the Key Discipline in Hunan Province. M. Syed Ali’s work was supported by NBHM Grant 2—48(5)/2016/NBHMR.P/-R-D II/14088.