# Synchronous Analysis for Fuzzy Coupled Neural Networks with Column Pinning Controllers

Authors: Dawei Gong; Shijie Song; Michel Lopez; Edgar N. Sanchez
Journal: Complexity
 (2020)

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## 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.


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## Body

## 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:N, Rn×m, and Rn denote the sets of nonnegative integers, n×m real matrices, and n-dimensional Euclidean space, respectively. For real symmetric matrix ϒ, ϒ>0ϒ≥0 indicates that ϒ is positive definite (respectively, semidefinite). The superscript T stands for the transpose of a matrix. In denotes the n-dimensional identity matrix. diag⋯ represents the block-diagonal matrix.


## 2. Problem Formulation

Without the loss of generality, this brief considers the following FCNNs withN identical nodes:(1)z˙it=−Czit+Afzit+Bfzit−τt+∑j=1NGij1D1zjt+∑j=1NGij2D2zjt−τt+uit,i=1,2,…,h,z˙it=−Czit+Afzit+Bfzit−τt+∑j=1NGij1D1zjt+∑j=1NGij2D2zjt−τt,i=h+1,h+2,…,N,in the formula, zit=zi1t,zi2t,…,zintT∈Rn, A∈Rn×n, B∈Rn×n, fzit=f1zi1t,f2zi2t,…,fnzintT, C=diagc1,c2,…,cn are diagonal positive matrix, G1=Gij1N×N,G2=Gij2N×N are the outer coupled matrix, and D1∈Rn×n,D2∈Rn×n are the inner coupled matrix.τt satisfied(2)0≤τ˙t≤μ<1,0≤τt≤τ,where μ and τ are known constants.uit are the pinning controllers.The controllers are designed as(3)uit=∑j=1,j≠iNkij1D3zjt−zit+∑j=1,j≠iNkij2D4zjt−τt−zit−τt,where kijq>0,q=1,2, i=1,2,…,h, and kijq=0, for i=h+1,h+2,…,N, and kijq are the control graph weights. Matrices D3,D4∈Rn×n 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 topologyGij1andGij2 and the communication connection graph topology kij1andkij2 are fixed. The design freedom is in the selection of the control gain matrices D3andD4.
System (3) can be rewritten as(4)uit=∑j=1NLij1D3zjt+∑j=1NLij2D4zjt−τt,where matrix Lq=LijqN×N,q=1,2, satisfied(5)Lijq=kijq,i≠j,Liiq=−∑j=1,j≠iNkijq,i,j=1,2,…,N.
The initial variables are given as(6)zis=Πi0s∈Θ−τ,0,Rn,i=1,2,…,N.
Let(7)zt=z1Tt,z2Tt,…,zNTtT,Fzt=fTz1t,fTz2t,…,fTzNtT,U˜t=u1Tt,u2Tt,…,uNTtT.
Combining with the sign⊗ of Kronecker product, system (1) can be rewritten as(8)z˙t=−IN⊗Czt+IN⊗AFzt+IN⊗BFzt−τt+G1⊗D1zt+G2⊗D2zt−τt+U˜t.
From equation (15), we have(9)U˜t=L1⊗D3zt+L2⊗D4zt−τt.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(10)Gijq=Gjiq≥0,i≠j,q=1,2,Giiq=−∑j=1,j≠iNGijq,i,j=1,2,…,N.Assumption 2 (see [28–30]).
Forj∈1,2,…,N, ∀s1,s2∈R, s1≠s2, the neural activation functions satisfy(11)σr−≤fjs1−fjs2s1−s2≤σr+.
We define(12)Δ1=diagσ1+σ1−,…,σn+σn−,Δ2=diagσ1++σ1−2,…,σn++σn−2.
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.
Rulel: if θ1t is Fl1, θ2t is Fl2,…,θgt is Flg, then(13)z˙it=−Clzit+Alfzit+Blfzit−τt+∑j=1NGij1D1lzjt+∑j=1NGij2D2lzjt−τt+uit,i=1,2,…,h,z˙it=−Clzit+Alfzit+Blfzit−τt+∑j=1NGij1D1lzjt+∑j=1NGij2D2lzjt−τt,i=h+1,h+2,…,N.
The controllers of the fuzzy systems are assumed in the form(14)uit=∑j=1,j≠iNkij1D3lzjt−zit+∑j=1,j≠iNkij2D4lzjt−τt−zit−τt.
Controller (14) can be rewritten as(15)uit=∑j=1NLij1D3lzjt+∑j=1NLij2D4lzjt−τt,where matrix Lq=LijqN×N,q=1,2 are defined as(16)Lijq=kijq,i≠j,Liiq=−∑j=1,j≠iNkijq,i,j=1,2,…,N.
The sign of⊗ is used to replace the Kronecker product, and FCNNs system 13 can be expressed as(17)z˙t=∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt+U˜t,θt=θ1t,θ2t,…,θgt, μlθt=ωlθt/∑l=1rωlθt, Fljθjt is the grade of membership of θjt in Flj. μlθt satisfied(18)μlθt≥0,∑l=1rμlθt=1.
The controllers of a set of fuzzy rules are written as follows.
Rulel: If θ1t is Fl1, θ2t is Fl2,…,θgt is Flg, then(19)U˜t=L1⊗D3lzt+L2⊗D4lzt−τt,l=1,2,…,r.
The resulting FCNNs system can be rewritten as(20)z˙t=∑l=1r∑k=1rμlθtμkθt−IN⊗Clzt+IN⊗AlFzt+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt+L1⊗D3kzt+L2⊗D4kzt−τt.
Then, 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:(21)limt⟶∞zit−zjt=0,i,j=1,2,…,N.Lemma 1 (see [31]).
DefineEN=eeT, where e=1,1,…,1T, and U=NIN−EN, M∈Rn×n, z=z1T,z2T,…,zNTT, and x=x1T,x2T,…,xNTT with zk,xk∈Rn,k=1,2,…,N, then(22)zTU⊗Mx=∑1≤i<j≤NNzi−xjTMzi−xj.Lemma 2 (Jensen’s inequality).
For any real matricΘ∈Rn×n,ΘT=Θ>0, constant ϱ>0 and ϖ:0,ϱ⟶Rn, then(23)ϱ∫0ϱϖTsΘϖsds≥∫0ϱϖsdsTΘ∫0ϱϖsds.Lemma 3 (see [32]).
For symmetric constant matricΘ=Θ11Θ12Θ13∗Θ22Θ23∗∗Θ33>0, Θkq∈Rn×n,1≤q≤3,1≤k≤3, 0<τt≤τ, and vector function z˙t:−τt,0⟶Rn×N, then we have(24)−τt∫t−τttzsFzsz˙sTU⊗Θ11U⊗Θ12U⊗Θ13∗U⊗Θ22U⊗Θ23∗∗U⊗Θ33zsFzsz˙sds,≤−ψTtU⊗Θ11U⊗Θ12U⊗Θ13−U⊗Θ13∗U⊗Θ22U⊗Θ23−U⊗Θ23∗∗U⊗Θ33−U⊗Θ33∗∗∗U⊗Θ33ψt,where ψTt=∫t−τttzsdsT,∫t−τttFzsdsT,zTt,zTt−τt.


## 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:(25)z˙t=∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt.Theorem 1.
Forl=1,2,…,r, system 25 is synchronized if Pz>0z=1,2,3 and J1>0,J2>0, then the following formulas are holding for all 1≤i<j≤N:(26)Θijl=Π11−NGij2P1D2lP1Al+J1Δ2P1Bl0∗−1−μP2−J2Δ10J2Δ20∗∗−J100∗∗∗−J20∗∗∗∗−1−μτP3<0,where(27)Π11=−P1Cl−ClP1−J1Δ1−NGij1P1D1l−NGij1D1lTP1T+P2+τP3.Proof.
ConsiderU as Lemma 1; for system (17), we have(28)V1t=zTtU⊗P1zt,V2t=∫t−τttzTsU⊗P2zsds,V3t=∫t−τtt∫θtzTsU⊗P3zsdsdθ.
Deriving time of system (17),(29)V.1t=2zTtU⊗P1z˙t,=2zTtU⊗P1∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt,(30)V.2t≤zTtU⊗P2zt−1−μzTt−τtU⊗P2zt−τt,(31)V.3t≤τzTtU⊗P3zt−1−μ∫t−τtzTsU⊗P3zsds,≤τzTtU⊗P3zt−1−μτ∫t−τttzsdsTU⊗P3∫t−τttzsds.
From reference [33] and Assumption 2, for any diagonal matrix J1,J2, we have(32)0≤zit−zjtfzit−fzjtT−J1Δ1J1Δ2∗−J1zit−zjtfzit−fzjt,+zit−τt−zjt−τtfzit−τt−fzjt−τtT−J2Δ1J2Δ2∗−J2,×zit−τt−zjt−τtfzit−τt−fzjt−τt.
According to (29)–(32), we obtain(33)V˙t≤∑l=1rμlθt∑i=1N−1∑j=i+1NξijTtΘijlξijt,in which ξijt=zit−zjtT,zit−τt−zjt−τtT,fzit−fzjtT, fzit−τt−fzjt−τtT,∫t−τttzis−zjsdsTT, and Θijl is the same in Theorem 1. From Definition 1, system (25) is synchronized when Θijl<0.
Note that, in Theorem1, 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.
Forl=1,2,…,r, system 25 is synchronized if there exist Pqq>0,Qqq>0,W>0, q=1,2,3, Psq,Qsq,T, 1≤s<q≤3, and J1>0,J2>0; then, the following formulas are holding for all 1≤i<j≤N:(34)P=P11P12P13∗P22P23∗∗P33>0,Q=Q11Q12Q13∗Q22Q23∗∗Q33>0,(35)Ωijl=Ξ11Ξ121−μτQ33Ξ14−1−μτQ13T1−μτQ23TΞ170∗Ξ220001−μτQ23TΞ270∗∗Ξ33Ξ341−μτQ13T0−NGij2D2lTTT−1−μP13∗∗∗Ξ4400BlTT−1−μP23∗∗∗∗−1−μτQ11−1−μτQ1200∗∗∗∗∗−1−μτQ2200∗∗∗∗∗∗Ξ770∗∗∗∗∗∗∗−1−μP33<0,where(36)Ξ11=−WCl−ClTWT+P11+τQ11−1−μτQ33−J1Δ1,Ξ12=P12+τQ12+WAl−NGij1WD1l+J1Δ2,Ξ14=WBl−NGij2WD2l,Ξ17=P13+τQ13−ClTT−NGij1D1lTTT,Ξ22=P22+τQ22−J1,Ξ27=P23+τQ23+AlTTT,Ξ33=−1−μP11−1−μτQ33−J2Δ1,Ξ34=−1−μP12+J2Δ2,Ξ44=−1−μP22−J2,Ξ77=P33+τQ33−T−TT.Proof.
From Assumptions1 and 2, consider the following LKF for model (17):(37)Vt=V1t+V2t+V3t,where(38)V1t=zTtU⊗Wzt,V2t=∫t−τttzsFzsz˙sU⊗P11U⊗P12U⊗P13∗U⊗P22U⊗P23∗∗U⊗P33zsFzsz˙sds,V3t=∫t−τtt∫θtzsFzsz˙sU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33zsFzsz˙sdsdθ.
Calculating the time derivative of system 29, then(39)V˙1t=2zTtU⊗Wz˙t,=2zTtU⊗W∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lFzt,+G2⊗D2lFzt−τt,V˙2t=ztFztz˙tTU⊗P11U⊗P12U⊗P13∗U⊗P22U⊗P23∗∗U⊗P33ztFztz˙t−1−τ˙t,×zt−τtFzt−τtz˙t−τtTU⊗P11U⊗P12U⊗P13∗U⊗P22U⊗P23∗∗U⊗P33zt−τtFzt−τtz˙t−τt.
From Lemmas2 and 3, we can acquire(40)V˙3t=τtztFztz˙tTU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33ztFztz˙t,−1−τ˙t∫t−τttzsFzsz˙sTU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33zsFzsz˙sds,≤τztFztz˙tTU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33ztFztz˙t,−1−μτψTtU⊗Q11U⊗Q12U⊗Q13−U⊗Q13∗U⊗Q22U⊗Q23−U⊗Q23∗∗U⊗Q33−U⊗Q33∗∗∗U⊗Q33ψt,where ψTt=∫t−τttzsdsT,∫t−τttFzsdsT,zTt,zTt−τt.
Note that ifX is a matrix with zero column sums, then UX=NX; from Lemma 1, we have(41)V˙1t=2∑i=1N−1∑j=i+1Nzit−zjtT∑l=1rμlθt−WClzit−zjt,+WAl−NGij1WD1lfzit−fzjt,+WBl−NGij2WD2lfzit−τ1t−fzjt−τ1t,(42)V˙2t≤∑i=1N−1∑j=i+1Nzit−zjtfzit−fzjtz˙it−z˙jtTP11P12P13∗P22P23∗∗P33zit−zjtfzit−fzjtz˙it−z˙jt,−1−μ∑i=1N−1∑j=i+1Nzit−τt−zjt−τtfzit−τt−fzjt−τtz˙it−τt−z˙jt−τtTP11P12P13∗P22P23∗∗P33,×zit−τt−zjt−τtfzit−τt−fzjt−τtz˙it−τt−z˙jt−τt.
As the same method, from Lemma1, we have(43)V˙3t≤τ∑i=1N−1∑j=i+1Nzit−zjtfzit−fzjtz˙it−z˙jtTQ11Q12Q13∗Q22Q23∗∗Q33zit−zjtfzit−fzjtz˙it−z˙jt,−1−μτ∑i=1N−1∑j=i+1N∫t−τttzis−zjsds∫t−τttfzis−fzjsdszit−zjtzit−τt−zjt−τtTQ11Q12Q13−Q13∗Q22Q23−Q23∗∗Q33−Q33∗∗∗Q33,×∫t−τttzis−zjsds∫t−τttfzis−fzjsdszit−zjtzit−τt−zjt−τt.
For any matrixT∈Rn×n, from system (17), we can easily obtain(44)0=2z˙TtU⊗T∑l=1rμlθt−z˙t−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt.
LetζijTt=zit−zjtT,fzt−fzjtT,zit−τt−zjt−τtTfzit−τt−fzit−τtT,∫t−τttzis−zjsdsT∫t−τttfzis−fzjsdsT,z˙it−z˙jtT,z˙it−τt−z˙jt−τtT, from (32) and (41)–(44), we can obtain(45)V˙t≤∑l=1rμlθt∑i=1N−1∑j=i+1NζijTtΩijlζijt,where Ωijl 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:(46)z˙t=∑l=1r∑k=1rμlθtμkθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt,+L1⊗D3kzt+L2⊗D4kzt−τt.Theorem 3.
Forl=1,2,…,r, dynamical system (46) is synchronized if there is Pz>0z=2,3,4 and positive diagonal matrix P1,J1,andJ2; then, the following formulas are holding for all 1≤i<j≤N:(47)Θijlk=Π11Π12AlX+XJ1XΔ2BlX0∗Π220XJ2XΔ20∗∗−XJ1X00∗∗∗−XJ2X0∗∗∗∗−1−μτXP3X<0,in which(48)Π11=−ClX−XCl−XJ1XΔ1−NGij1D1lX−NLij1z3k−NGij1XD1lT−NLij1z3kT+XP2X+τXP3X,Π12=−NGij2D2lX−NLij2z4k,Π22=−1−μXP2X−XJ2XΔ1.Proof.
Based on Theorem1, the feedback gains in the fuzzy coupled system are given by D3k=z3kX−1 and D4k=z4kX−1. Replace NGij1D3l with NGij1D1l+NLij1D3k, NGij2D2l with NGij2D4l+NLij2D4k. Pre- and postmultiply 13 with diagX;X;X;X;X, where X−1=P1; then, we can obtain the above criteria.Theorem 4.
Forl=1,2,…,r, system (46) is synchronized if there exists Pqq>0,Qqq>0, q=1,2,3, Psq,Qsq, 1≤s<q≤3, and J1>0,J2>0,W>0; then, the following formulas are holding for all 1≤i<j≤N:(49)P=P11P12P13∗P22P23∗∗P33>0,Q=Q11Q12Q13∗Q22Q23∗∗Q33>0,(50)Ωijlk=Ξ11Ξ121−μτXQ33XΞ14−1−μτXQ13TX1−μτXQ23TXΞ170∗Ξ220001−μτXQ23TXΞ270∗∗Ξ33Ξ341−μτXQ13TX0Ξ37−1−μXP13X∗∗∗Ξ4400XBl−1−μXP23X∗∗∗∗−1−μτXQ11X−1−μτXQ12X00∗∗∗∗∗−1−μτXQ22X00∗∗∗∗∗∗Ξ770∗∗∗∗∗∗∗−1−μXP33X<0,where(51)Ξ11=−ClX−XClT+XP11X+τXQ11X−1−μτXQ33X−XJ1XΔ1,Ξ12=XP12X+τXQ12X+AlX−NGij1D1lX−NLij1z3k+XJ1XΔ2,Ξ14=BlX−NGij2D2lX−NLij2z4k,Ξ17=XP13X+τXQ13X−XCl−NGij1XD1lT−NLij1z3kT,Ξ22=XP22X+τXQ22X−XJ1X,Ξ27=XP23X+τXQ23X+XAlT,Ξ33=−1−μXP11X−1−μτXQ33X−XJ2XΔ1,Ξ34=−1−μXP12X+XJ2XΔ2,Ξ37=−NGij1XD2lT−NLij2z4kT,Ξ44=−1−μXP22X−XJ2X,Ξ77=XP33X+τXQ33X−X−X.Proof.
Based on Theorem2, let T=W, and the feedback gains in the fuzzy system are given by D3k=z3kX−1 and D4k=z4kX−1. Replace NGij1D1l with NGij1D1l+NLij1D3k, NGij2D2l with NGij2D2l+NLij2D4k. Pre- and postmultiply (35) with diagX;X;X;X;X;X;X;X, where X−1=W; then, we can obtain the above criteria.Remark 3.
From these two pinning synchronized results, it is noted that the fuzzy pinning control gain matricesD3andD4 can be computed, which can fix the communication connection graph topology kij1andkij2. Such complex fuzzy pinning controllers are proposed for the first time.


## 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:(25)z˙t=∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt.Theorem 1.
Forl=1,2,…,r, system 25 is synchronized if Pz>0z=1,2,3 and J1>0,J2>0, then the following formulas are holding for all 1≤i<j≤N:(26)Θijl=Π11−NGij2P1D2lP1Al+J1Δ2P1Bl0∗−1−μP2−J2Δ10J2Δ20∗∗−J100∗∗∗−J20∗∗∗∗−1−μτP3<0,where(27)Π11=−P1Cl−ClP1−J1Δ1−NGij1P1D1l−NGij1D1lTP1T+P2+τP3.Proof.
ConsiderU as Lemma 1; for system (17), we have(28)V1t=zTtU⊗P1zt,V2t=∫t−τttzTsU⊗P2zsds,V3t=∫t−τtt∫θtzTsU⊗P3zsdsdθ.
Deriving time of system (17),(29)V.1t=2zTtU⊗P1z˙t,=2zTtU⊗P1∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt,(30)V.2t≤zTtU⊗P2zt−1−μzTt−τtU⊗P2zt−τt,(31)V.3t≤τzTtU⊗P3zt−1−μ∫t−τtzTsU⊗P3zsds,≤τzTtU⊗P3zt−1−μτ∫t−τttzsdsTU⊗P3∫t−τttzsds.
From reference [33] and Assumption 2, for any diagonal matrix J1,J2, we have(32)0≤zit−zjtfzit−fzjtT−J1Δ1J1Δ2∗−J1zit−zjtfzit−fzjt,+zit−τt−zjt−τtfzit−τt−fzjt−τtT−J2Δ1J2Δ2∗−J2,×zit−τt−zjt−τtfzit−τt−fzjt−τt.
According to (29)–(32), we obtain(33)V˙t≤∑l=1rμlθt∑i=1N−1∑j=i+1NξijTtΘijlξijt,in which ξijt=zit−zjtT,zit−τt−zjt−τtT,fzit−fzjtT, fzit−τt−fzjt−τtT,∫t−τttzis−zjsdsTT, and Θijl is the same in Theorem 1. From Definition 1, system (25) is synchronized when Θijl<0.
Note that, in Theorem1, 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.
Forl=1,2,…,r, system 25 is synchronized if there exist Pqq>0,Qqq>0,W>0, q=1,2,3, Psq,Qsq,T, 1≤s<q≤3, and J1>0,J2>0; then, the following formulas are holding for all 1≤i<j≤N:(34)P=P11P12P13∗P22P23∗∗P33>0,Q=Q11Q12Q13∗Q22Q23∗∗Q33>0,(35)Ωijl=Ξ11Ξ121−μτQ33Ξ14−1−μτQ13T1−μτQ23TΞ170∗Ξ220001−μτQ23TΞ270∗∗Ξ33Ξ341−μτQ13T0−NGij2D2lTTT−1−μP13∗∗∗Ξ4400BlTT−1−μP23∗∗∗∗−1−μτQ11−1−μτQ1200∗∗∗∗∗−1−μτQ2200∗∗∗∗∗∗Ξ770∗∗∗∗∗∗∗−1−μP33<0,where(36)Ξ11=−WCl−ClTWT+P11+τQ11−1−μτQ33−J1Δ1,Ξ12=P12+τQ12+WAl−NGij1WD1l+J1Δ2,Ξ14=WBl−NGij2WD2l,Ξ17=P13+τQ13−ClTT−NGij1D1lTTT,Ξ22=P22+τQ22−J1,Ξ27=P23+τQ23+AlTTT,Ξ33=−1−μP11−1−μτQ33−J2Δ1,Ξ34=−1−μP12+J2Δ2,Ξ44=−1−μP22−J2,Ξ77=P33+τQ33−T−TT.Proof.
From Assumptions1 and 2, consider the following LKF for model (17):(37)Vt=V1t+V2t+V3t,where(38)V1t=zTtU⊗Wzt,V2t=∫t−τttzsFzsz˙sU⊗P11U⊗P12U⊗P13∗U⊗P22U⊗P23∗∗U⊗P33zsFzsz˙sds,V3t=∫t−τtt∫θtzsFzsz˙sU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33zsFzsz˙sdsdθ.
Calculating the time derivative of system 29, then(39)V˙1t=2zTtU⊗Wz˙t,=2zTtU⊗W∑l=1rμlθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lFzt,+G2⊗D2lFzt−τt,V˙2t=ztFztz˙tTU⊗P11U⊗P12U⊗P13∗U⊗P22U⊗P23∗∗U⊗P33ztFztz˙t−1−τ˙t,×zt−τtFzt−τtz˙t−τtTU⊗P11U⊗P12U⊗P13∗U⊗P22U⊗P23∗∗U⊗P33zt−τtFzt−τtz˙t−τt.
From Lemmas2 and 3, we can acquire(40)V˙3t=τtztFztz˙tTU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33ztFztz˙t,−1−τ˙t∫t−τttzsFzsz˙sTU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33zsFzsz˙sds,≤τztFztz˙tTU⊗Q11U⊗Q12U⊗Q13∗U⊗Q22U⊗Q23∗∗U⊗Q33ztFztz˙t,−1−μτψTtU⊗Q11U⊗Q12U⊗Q13−U⊗Q13∗U⊗Q22U⊗Q23−U⊗Q23∗∗U⊗Q33−U⊗Q33∗∗∗U⊗Q33ψt,where ψTt=∫t−τttzsdsT,∫t−τttFzsdsT,zTt,zTt−τt.
Note that ifX is a matrix with zero column sums, then UX=NX; from Lemma 1, we have(41)V˙1t=2∑i=1N−1∑j=i+1Nzit−zjtT∑l=1rμlθt−WClzit−zjt,+WAl−NGij1WD1lfzit−fzjt,+WBl−NGij2WD2lfzit−τ1t−fzjt−τ1t,(42)V˙2t≤∑i=1N−1∑j=i+1Nzit−zjtfzit−fzjtz˙it−z˙jtTP11P12P13∗P22P23∗∗P33zit−zjtfzit−fzjtz˙it−z˙jt,−1−μ∑i=1N−1∑j=i+1Nzit−τt−zjt−τtfzit−τt−fzjt−τtz˙it−τt−z˙jt−τtTP11P12P13∗P22P23∗∗P33,×zit−τt−zjt−τtfzit−τt−fzjt−τtz˙it−τt−z˙jt−τt.
As the same method, from Lemma1, we have(43)V˙3t≤τ∑i=1N−1∑j=i+1Nzit−zjtfzit−fzjtz˙it−z˙jtTQ11Q12Q13∗Q22Q23∗∗Q33zit−zjtfzit−fzjtz˙it−z˙jt,−1−μτ∑i=1N−1∑j=i+1N∫t−τttzis−zjsds∫t−τttfzis−fzjsdszit−zjtzit−τt−zjt−τtTQ11Q12Q13−Q13∗Q22Q23−Q23∗∗Q33−Q33∗∗∗Q33,×∫t−τttzis−zjsds∫t−τttfzis−fzjsdszit−zjtzit−τt−zjt−τt.
For any matrixT∈Rn×n, from system (17), we can easily obtain(44)0=2z˙TtU⊗T∑l=1rμlθt−z˙t−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt.
LetζijTt=zit−zjtT,fzt−fzjtT,zit−τt−zjt−τtTfzit−τt−fzit−τtT,∫t−τttzis−zjsdsT∫t−τttfzis−fzjsdsT,z˙it−z˙jtT,z˙it−τt−z˙jt−τtT, from (32) and (41)–(44), we can obtain(45)V˙t≤∑l=1rμlθt∑i=1N−1∑j=i+1NζijTtΩijlζijt,where Ωijl 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:(46)z˙t=∑l=1r∑k=1rμlθtμkθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt,+L1⊗D3kzt+L2⊗D4kzt−τt.Theorem 3.
Forl=1,2,…,r, dynamical system (46) is synchronized if there is Pz>0z=2,3,4 and positive diagonal matrix P1,J1,andJ2; then, the following formulas are holding for all 1≤i<j≤N:(47)Θijlk=Π11Π12AlX+XJ1XΔ2BlX0∗Π220XJ2XΔ20∗∗−XJ1X00∗∗∗−XJ2X0∗∗∗∗−1−μτXP3X<0,in which(48)Π11=−ClX−XCl−XJ1XΔ1−NGij1D1lX−NLij1z3k−NGij1XD1lT−NLij1z3kT+XP2X+τXP3X,Π12=−NGij2D2lX−NLij2z4k,Π22=−1−μXP2X−XJ2XΔ1.Proof.
Based on Theorem1, the feedback gains in the fuzzy coupled system are given by D3k=z3kX−1 and D4k=z4kX−1. Replace NGij1D3l with NGij1D1l+NLij1D3k, NGij2D2l with NGij2D4l+NLij2D4k. Pre- and postmultiply 13 with diagX;X;X;X;X, where X−1=P1; then, we can obtain the above criteria.Theorem 4.
Forl=1,2,…,r, system (46) is synchronized if there exists Pqq>0,Qqq>0, q=1,2,3, Psq,Qsq, 1≤s<q≤3, and J1>0,J2>0,W>0; then, the following formulas are holding for all 1≤i<j≤N:(49)P=P11P12P13∗P22P23∗∗P33>0,Q=Q11Q12Q13∗Q22Q23∗∗Q33>0,(50)Ωijlk=Ξ11Ξ121−μτXQ33XΞ14−1−μτXQ13TX1−μτXQ23TXΞ170∗Ξ220001−μτXQ23TXΞ270∗∗Ξ33Ξ341−μτXQ13TX0Ξ37−1−μXP13X∗∗∗Ξ4400XBl−1−μXP23X∗∗∗∗−1−μτXQ11X−1−μτXQ12X00∗∗∗∗∗−1−μτXQ22X00∗∗∗∗∗∗Ξ770∗∗∗∗∗∗∗−1−μXP33X<0,where(51)Ξ11=−ClX−XClT+XP11X+τXQ11X−1−μτXQ33X−XJ1XΔ1,Ξ12=XP12X+τXQ12X+AlX−NGij1D1lX−NLij1z3k+XJ1XΔ2,Ξ14=BlX−NGij2D2lX−NLij2z4k,Ξ17=XP13X+τXQ13X−XCl−NGij1XD1lT−NLij1z3kT,Ξ22=XP22X+τXQ22X−XJ1X,Ξ27=XP23X+τXQ23X+XAlT,Ξ33=−1−μXP11X−1−μτXQ33X−XJ2XΔ1,Ξ34=−1−μXP12X+XJ2XΔ2,Ξ37=−NGij1XD2lT−NLij2z4kT,Ξ44=−1−μXP22X−XJ2X,Ξ77=XP33X+τXQ33X−X−X.Proof.
Based on Theorem2, let T=W, and the feedback gains in the fuzzy system are given by D3k=z3kX−1 and D4k=z4kX−1. Replace NGij1D1l with NGij1D1l+NLij1D3k, NGij2D2l with NGij2D2l+NLij2D4k. Pre- and postmultiply (35) with diagX;X;X;X;X;X;X;X, where X−1=W; then, we can obtain the above criteria.Remark 3.
From these two pinning synchronized results, it is noted that the fuzzy pinning control gain matricesD3andD4 can be computed, which can fix the communication connection graph topology kij1andkij2. 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 as(52)z˙t=∑l=12μlθt−IN⊗Clzt+IN⊗AlFzt+IN⊗BlFzt−τt,+G1⊗D1lzt+G2⊗D2lzt−τt,where(53)C1=5002,C2=2001,A1=A2=1.8100.11.8,B1=B2=−1.50.10.1−1.5,D11=2003,D12=1001.5,D21=3121,D22=3002,μ=0,τ=0.97,μ1=11+exp−2z1t,μ2=1−μ1.The outer-coupling matrix GijqN×N,q=1,2, are defined as(54)G1=G2=−5101031−5100301−5013100−5130011−5333333−15.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.Figure 1
States of fuzzy neural networks (8) without pinning control L1=L2=0: zit,i=1,2,3,4,5,6.


### 4.2. Pinning Synchronization for Fuzzy Neural Networks with Hybrid Coupling

Now, we consider the system with pinning control:(55)z˙t=∑l=1r∑k=1rμlθtμkθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt,+L1⊗D3kzt+L2⊗D4kzt−τt.The controllers’ parameters are as follows:(56)L1=L2=−10232302−10233032−10320232−10203232−100000000,other parameters are the same in system (52).According to Theorem4, 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:(57)D31=6.7338009.7243,D32=6.7872009.6380,D41=4.2085005.8983,D42=4.1814005.9448.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 ejt=zijt−z1jt,i=2,3,4,5,6;j=1,2.Figure 2
States of fuzzy neural networks (8) with pinning control: zit,i=1,2,3,4,5,6.Figure 3
Pinning synchronization errors for fuzzy neural networks:ejt,j=1,2.


## 4.1. Synchronization for Fuzzy Coupled Networks without Control

Consider the following TNNs model, the parameters in which are defined as(52)z˙t=∑l=12μlθt−IN⊗Clzt+IN⊗AlFzt+IN⊗BlFzt−τt,+G1⊗D1lzt+G2⊗D2lzt−τt,where(53)C1=5002,C2=2001,A1=A2=1.8100.11.8,B1=B2=−1.50.10.1−1.5,D11=2003,D12=1001.5,D21=3121,D22=3002,μ=0,τ=0.97,μ1=11+exp−2z1t,μ2=1−μ1.The outer-coupling matrix GijqN×N,q=1,2, are defined as(54)G1=G2=−5101031−5100301−5013100−5130011−5333333−15.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.Figure 1
States of fuzzy neural networks (8) without pinning control L1=L2=0: zit,i=1,2,3,4,5,6.


## 4.2. Pinning Synchronization for Fuzzy Neural Networks with Hybrid Coupling

Now, we consider the system with pinning control:(55)z˙t=∑l=1r∑k=1rμlθtμkθt−IN⊗Clzt+IN⊗AlFzt,+IN⊗BlFzt−τt+G1⊗D1lzt+G2⊗D2lzt−τt,+L1⊗D3kzt+L2⊗D4kzt−τt.The controllers’ parameters are as follows:(56)L1=L2=−10232302−10233032−10320232−10203232−100000000,other parameters are the same in system (52).According to Theorem4, 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:(57)D31=6.7338009.7243,D32=6.7872009.6380,D41=4.2085005.8983,D42=4.1814005.9448.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 ejt=zijt−z1jt,i=2,3,4,5,6;j=1,2.Figure 2
States of fuzzy neural networks (8) with pinning control: zit,i=1,2,3,4,5,6.Figure 3
Pinning synchronization errors for fuzzy neural networks:ejt,j=1,2.


## 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.



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*Source: 1397069-2020-10-05.xml*