Synchronous Analysis for Fuzzy Coupled Neural Networks with Column Pinning Controllers

+e 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. +e 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. +e 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.


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][2][3][4][5]. e 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]. e 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][11][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][14][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. ere are two categories of synchronization: self-synchronization and forced synchronization. Without any external force, the selfsynchronization can be achieved by the connection of local nodes. However, the networks usually cannot be synchronized by themselves. erefore, 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][21][22][23][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. erefore, 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: N, R n×m , and R n denote the sets of nonnegative integers, n × m real matrices, and n-dimensional Euclidean space, respectively. For real symmetric matrix Υ, Υ > 0(Υ ≥ 0) indicates that Υ is positive definite (respectively, semidefinite). e superscript T stands for the transpose of a matrix. I n denotes the n-dimensional identity matrix. diag(· · ·) represents the block-diagonal matrix.

Problem Formulation
Without the loss of generality, this brief considers the following FCNNs with N identical nodes: ij ) N×N are the outer coupled matrix, and D 1 ∈ R n×n , D 2 ∈ R n×n are the inner coupled matrix.
where μ and τ are known constants. u i (t) are the pinning controllers. e controllers are designed as where k are the control graph weights. Matrices D 3 , D 4 ∈ R n×n represent control gain matrices.
ese 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 G (1) ij and G (2) ij and the communication connection graph topology k (1) ij and k (2) ij are fixed. e design freedom is in the selection of the control gain matrices D 3 and D 4 . System (3) can be rewritten as e initial variables are given as Combining with the sign ⊗ of Kronecker product, system (1) can be rewritten as Complexity From 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][26][27]). e outer-coupling matrix are assumed as Assumption 2 (see [28][29][30]). For j ∈ 1, 2, . . . , N, ∀s 1 , s 2 ∈ R, s 1 ≠ s 2 , the neural activation functions satisfy We define From 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 e controllers of the fuzzy systems are assumed in the form Controller (14) can be rewritten as e sign of ⊗ is used to replace the Kronecker product, and FCNNs system 13 can be expressed as e controllers of a set of fuzzy rules are written as follows.
Rule l: e resulting FCNNs system can be rewritten as en, we will introduce some useful situations, which are very important to prove our main results. (17) is synchronized if the following equation holds: Lemma 2 (Jensen's inequality). For any real matric Θ ∈ R n×n , Θ T � Θ > 0, constant ϱ > 0 and ϖ: [0, ϱ] ⟶ R n , then Lemma 3 (see [32]). For symmetric constant matric

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.

Synchronization for FNNs without Control.
In this section, we first study the synchronization criterions for TFNNS with time-varying delay and hybrid coupling: where Proof. Consider U as Lemma 1; for system (17), we have Deriving time of system (17), , From reference [33] and Assumption 2, for any diagonal matrix J 1 , J 2 , we have According to (29)-(32), we obtain and Θ l ij is the same in eorem 1. From Definition 1, system (25) is synchronized when Θ l ij < 0.

Complexity
Calculating the time derivative of system 29, then F(z(t)), From Lemmas 2 and 3, we can acquire Note that if X is a matrix with zero column sums, then UX � NX; from Lemma 1, we have As the same method, from Lemma 1, we have where Ω l ij is defined as (50). From Definition 1, it implies that system (17) is synchronized.

Synchronization for Fuzzy System with Pinning Control.
is section deals with the pinning synchronization problems for the closed-loop T-S fuzzy neural networks: Theorem 3. For l � 1, 2, . . . , r, dynamical system (46) is synchronized if there is P z > 0(z � 2, 3, 4) and positive diagonal matrix P 1 , J 1 , and J 2 ; then, the following formulas are holding for all 1 ≤ i < j ≤ N: in which Proof. Based on eorem 1, the feedback gains in the fuzzy coupled system are given by D 3k � z 3k X − 1 and D 4k � z 4k X − 1 . Replace NG (1) ij D 3l with NG (1) ij D 1l + NL (1) ij D 3k , NG (2) ij D 2l with NG (2) ij D 4l + NL (2) ij D 4k . Pre-and postmultiply 13 with diag[X; X; X; X; X], where X − 1 � P 1 ; 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 D 3 and D 4 can be computed, which can fix the communication connection graph topology k (1) ij and k (2) ij . Such complex fuzzy pinning controllers are proposed for the first time.

Numerical Examples
is section provides a numerical example to illustrate the effectiveness of the obtained results. Assume the system without control first and then with pinning control.  μ l (θ(t)) − I N ⊗ C l z(t) + I N ⊗ A l F(z(t)) − τ(t))),

Synchronization for Fuzzy Coupled Networks without
where en, 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.
other parameters are the same in system (52). According to eorem 4, system (55) can achieve synchronization by pinning control. Solving the LMIs in eorem 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 e j (t) � (z ij (t) − z 1j (t)), i � 2, 3, 4, 5, 6; j � 1, 2.

Conclusion
is 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. erefore, the traditional controllers have been improved as the eventdependent 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][35][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
e data used to support the findings the study are available from the corresponding author upon request. 12 Complexity