Abstract
The synchronous research for fuzzy coupled neural networks (FCNNs) is studied by a new strategy of column pinning controllers. In this paper, the Lyapunov Krasovskii functional (LKF) is taken as an important element for the pinning control laws. The networks are interconnected by coupling gains that define a physical interaction graph. Different from the preset technique in traditional intermittent control, a novel additional communication control graphs of pinning control law are introduced, which has not been investigated before. The proposed control laws can achieve the control objectives of being introduced as an array of vector with Kronecker produce operation. Under the proposed framework of intermittent control, numerical simulations via MATLAB are used to confirm the availability of the suggested control laws.
1. Introduction
In recent decades, the investigation on neural networks (NNS) has aroused ever-increasing interest of researchers due to their strong application in various fields [1–5]. The coupled neural networks (CNNs) are seen as a special type of complex networks, which consist of a large set of interconnected single NNs with each individual being called node. Usually, the CNNs exhibit more unpredictable and complicated behaviors than the single NNs. Synchronization of CNNs describes a typical collective behavior and has many applications. For example, the complex oscillatory patterns were stored and retrieved as the synchronization states by presenting an architecture of CNNs in [6, 7]. A secure communication system was introduced by utilizing the coupled cellular NNs in [8]. The research on synchronization of CNNs not only opens up new opportunities in the understanding of brain science but also makes an important step forward to the practical applications [9].
However, the aforementioned results are valid when for the structures and the parameters of coupled neural networks are exactly known. In many practical models of the real world, uncertainty or vagueness is unavoidable. Fuzzy theory [10–12] is considered an efficient tool to solve vagueness problems of the complex systems. Compared with the traditional NNS, the FCNNS have advantages for their capabilities in handling uncertain information and representing nonlinear dynamics [13–15].
In practice, many feasible control schemes can be adopted to study synchronous research of complex networks, such as the sampled-data control [16] and intermittent control [17]. Among them, the sampled-data control and impulsive control are two schemes with low control cost because their controllers are updated only at some discrete times. Besides, intermittent control is also an economic choice. In such scheme, the controller is only imposed on the systems at work time. Hence, the notion of intermittent control came into researchers’ vision and has stimulated many renewed results. In [18], the quasi-synchronization of delayed chaotic systems was investigated by periodically intermittent control. In [19], the synchronization issues of complex networks were visited by designing an intermittent controller equipped with two switched periods. There are two categories of synchronization: self-synchronization and forced synchronization. Without any external force, the self-synchronization can be achieved by the connection of local nodes. However, the networks usually cannot be synchronized by themselves. Therefore, it is more desirable to force the networks to synchronize. Due to the high dimension and complex topology, it will be expensive and literally infeasible to add controllers to all nodes. Hinted by such consideration, the strategy so-called pinning control is proposed which only controls a small location of the nodes, such as [20–24].
As far as we know, there are no pining control results for FCNNs. So, how to solve the pinning synchronization problems for FCNNs is still challenging. Motivated by the foregoing discussion, this brief explores the synchronization of FCNNs by proposing the concept of column pinning control law. In the developed control scheme, the work conditions are decided by the dynamic relationships among the Lyapunov–Krasovskii functional (LKF) and some other column vectors. Namely, the pinning controller is imposed on the systems when the trajectory of LKF goes into the column regions. Our scheme changes the intrinsic characteristic of the existing control methods that the work conditions are predetermined in prior. From the events’ point of view, whether the controller is imposed or not is decided by the dynamic of LKF. Therefore, our scheme can be understood as a class of event-dependent column controllers. Under the framework of the proposed scheme, several simple criterions are developed to study the synchronization for the considered FCNNs.
Notations: , , and denote the sets of nonnegative integers, real matrices, and -dimensional Euclidean space, respectively. For real symmetric matrix , indicates that is positive definite (respectively, semidefinite). The superscript stands for the transpose of a matrix. denotes the -dimensional identity matrix. represents the block-diagonal matrix.
2. Problem Formulation
Without the loss of generality, this brief considers the following FCNNs with N identical nodes:in the formula, , , , , are diagonal positive matrix, are the outer coupled matrix, and are the inner coupled matrix.
satisfiedwhere and are known constants.
are the pinning controllers.
The controllers are designed aswhere , , and , for , and are the control graph weights. Matrices represent control gain matrices. These gain matrices are the control parameters designed to guarantee synchronization of the coupled neural networks.
Remark 1. It is noted that the physical coupling graphs combined with the communication control graphs together form a cyber-physical system, where in the physical connection graph topology and the communication connection graph topology are fixed. The design freedom is in the selection of the control gain matrices .
System (3) can be rewritten aswhere matrix , satisfiedThe initial variables are given asLetCombining with the sign of Kronecker product, system (1) can be rewritten asFrom equation (15), we have
Remark 2. It is the first introduction of the pinning control laws as an array of vector with Kronecker produce operation.
Assumption 1 (see [25–27]). The outer-coupling matrix are assumed as
Assumption 2 (see [28–30]). For , , , the neural activation functions satisfyWe defineFrom T-S fuzzy model concept, for the first time, a class of FNNS with pinning controllers is described here. Model 1 with T-S theory is described.
Rule : if is , is is , thenThe controllers of the fuzzy systems are assumed in the formController (14) can be rewritten aswhere matrix are defined asThe sign of is used to replace the Kronecker product, and FCNNs system 13 can be expressed as, , is the grade of membership of in . satisfiedThe controllers of a set of fuzzy rules are written as follows.
Rule : If is , is is , thenThe resulting FCNNs system can be rewritten asThen, we will introduce some useful situations, which are very important to prove our main results.
Definition 1. System (17) is synchronized if the following equation holds:
Lemma 1 (see [31]). Define , where , and , , , and with , then
Lemma 2 (Jensen’s inequality). For any real matric , constant and , then
Lemma 3 (see [32]). For symmetric constant matric , , , and vector function , then we havewhere .
3. Synchronization Results for Fuzzy System
First, we consider the synchronization results of FCNNs without control. Whereafter, we will establish some concise sufficient conditions which ensure synchronization of FCNNs.
3.1. Synchronization for FNNs without Control
In this section, we first study the synchronization criterions for TFNNS with time-varying delay and hybrid coupling:
Theorem 1. For , system 25 is synchronized if and , then the following formulas are holding for all :where
Proof. Consider as Lemma 1; for system (17), we haveDeriving time of system (17),From reference [33] and Assumption 2, for any diagonal matrix , we haveAccording to (29)–(32), we obtainin which , , and is the same in Theorem 1. From Definition 1, system (25) is synchronized when .
Note that, in Theorem 1, we did not introduce free-weighting matrix. Next, we will choose other Lyanpunov–Krosovskii functional and introduce more free-weighting matrices, which can add more useful conditions.
Theorem 2. For , system 25 is synchronized if there exist , , , , and ; then, the following formulas are holding for all :where
Proof. From Assumptions 1 and 2, consider the following LKF for model (17):whereCalculating the time derivative of system 29, thenFrom Lemmas 2 and 3, we can acquirewhere .
Note that if is a matrix with zero column sums, then ; from Lemma 1, we haveAs the same method, from Lemma 1, we haveFor any matrix , from system (17), we can easily obtainLet , from (32) and (41)–(44), we can obtainwhere is defined as (50). From Definition 1, it implies that system (17) is synchronized.
3.2. Synchronization for Fuzzy System with Pinning Control
This section deals with the pinning synchronization problems for the closed-loop T-S fuzzy neural networks:
Theorem 3. For , dynamical system (46) is synchronized if there is and positive diagonal matrix ; then, the following formulas are holding for all :in which
Proof. Based on Theorem 1, the feedback gains in the fuzzy coupled system are given by and . Replace with , with . Pre- and postmultiply 13 with , where ; then, we can obtain the above criteria.
Theorem 4. For , system (46) is synchronized if there exists , , , , and ; then, the following formulas are holding for all :where
Proof. Based on Theorem 2, let , and the feedback gains in the fuzzy system are given by and . Replace with , with . Pre- and postmultiply (35) with , where ; then, we can obtain the above criteria.
Remark 3. From these two pinning synchronized results, it is noted that the fuzzy pinning control gain matrices can be computed, which can fix the communication connection graph topology . Such complex fuzzy pinning controllers are proposed for the first time.
4. Numerical Examples
This section provides a numerical example to illustrate the effectiveness of the obtained results. Assume the system without control first and then with pinning control.
4.1. Synchronization for Fuzzy Coupled Networks without Control
Consider the following TNNs model, the parameters in which are defined aswhereThe outer-coupling matrix , are defined as
Then, we plot the states of network (52) without control in Figure 1. It is easy to see that the system cannot be synchronized by itself.
4.2. Pinning Synchronization for Fuzzy Neural Networks with Hybrid Coupling
Now, we consider the system with pinning control:
The controllers’ parameters are as follows:other parameters are the same in system (52).
According to Theorem 4, system (55) can achieve synchronization by pinning control. Solving the LMIs in Theorem 4, we can obtain the fuzzy pinning control gain matrices as follows:
From the examples, network (55) without control is shown in Figure 1, and the system with pinning control is shown in Figure 2. It is easy to see that the results are very good by our methods. We also show the synchronization errors in Figure 3, where .
5. Conclusion
This paper has investigated the synchronization of T-S FNNs by proposing a novel pinning control scheme. Instead of the presenting technique in prior, the proposed scheme regulates the column controllers by some events which are yielded by the relationships among the LKF and three nonnegative regions. Therefore, the traditional controllers have been improved as the event-dependent one in this paper. A concise criterion has been presented to guarantee the pinning synchronization of the considered CNNs. Simulations are finally provided to display the feasibility and improvements of the proposed pinning control scheme. Our results can only be studied as theoretical research now. We expect that the innovations of this paper can shed further light on the more problems (such as ([34–36]) under column controllers law. By the similar mechanism, our further directions include (1) design an intermittent output feedback controller and (2) design an intermittent adaptive controller.
Data Availability
The data used to support the findings the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Defense Pre-Research Foundation of China (1126170104A, 1126180204B, and 1126190508A).