# Nonfragile Estimator Design for Fractional-Order Neural Networks under Event-Triggered Mechanism

Authors: Xiaoguang Shao; Ming Lyu; Jie Zhang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper is concerned with the nonfragile state estimation for a kind of delayed fractional-order neural network under the event-triggered mechanism (ETM). To reduce the bandwidth occupation of the communication network, the ETM is employed in the sensor-to-estimator channel. Moreover, in order to reflect the reality, the transmission delay is taken into account in the model establishment. Sufficient criteria are supplied to make sure that the augmented system is asymptotically stable by using the fractional-order Lyapunov indirect approach and the linear matrix inequality method. In the end, the theoretical result is shown by means of two numerical examples.


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## Body

## 1. Introduction

The past several decades have witnessed that artificial neural network (ANN) has attracted particular research attention. Because of the outstanding performance, ANN has been extensively applied in image recognition, signal processing [1], fault diagnosis [2], and so on. With the rapid developments of artificial intelligence, the ANN has received considerable attention again by the scholars, which relates to synchronization, dissipativity, attractivity, stability, and state estimation (SE) for various kinds of ANNs [3–7].As we all know, ANNs are composed of plenty of artificial neurons and the SE problem of the neurons plays a vital role in practical applications. As such, quite a lot of results have been reported on the SE issue (see [8–11] and the reference therein). For the practical systems, the parameter uncertainties are often considered. So far, a lot of research studies regarding uncertain systems have been conducted [12–14]. It is worth noting that the existing results assumed that parameter of the estimator is accurate, which, however, is unrealistic. To solve this problem, we aim to design a nonfragile estimator so as to alleviate the effects induced by the uncertainty of the estimator parameter on the system performance. Till now, some initial results have been published on the nonfragile controller design problems [14–17].In the networked systems, the network bandwidth is always limited which therefore may result in network congestion when a large amount of data is transmitted. Up to now, the network-induced phenomena including transmission delay, packet loss, and quantification have been discussed adequately. In recent years, much attention has been focused on the ETM and many communication protocols, which aims to avoid the occurrence of the network-induced phenomena. Based on the ETM, plenty of literature has been available on stability analysis, event-triggered condition design, controller/filter design, and so on [18]. Noting that compared with time-triggered mechanism, the ETM exhibits better performance because the necessary sampling depends on the “event” rather than the “time” [19, 20].In addition, by applying the fractional calculus to the ANNs, the researchers have found that the performance of the fractional-order models is better than integer-order ones, especially in the aspect of memory and hereditary. Till now, some novel fractional-order theories and methods concerning the ANNs have been proposed. For example, a nonfragile nonlinear fractional-order observer is designed in [21] and an adaptive event-triggered scheme has been developed in [22]. But these existing fractional-order systems employed ETM are introduced with single delay or without only. Especially, it is a challenge in a fractional system. However, the problem of multiple time delays in real systems is often encountered. Nevertheless, there are few related studies on the nonfragile SE for fractional-order neural network based on ETM with multiple time delays, which motivates us to shorten this gap.Inspired by the aforementioned lines, a nonfragile state estimator is designed for a class of fractional-order neural networks (FNNs) based on ETM. The advantages in this paper are as follows: (1) compared with the existing estimators, a fractional-order nonfragile estimator is first constructed; (2) to save bandwidth resources, an ETM is applied in the SE problem of the fractional-order neural network; (3) the LMI method and the fractional Lyapunov indirect method are adopted to design the state estimator.The remaining content is outlined as follows. In Section2, some preliminary knowledge is recalled. In Section 3, state estimation criteria are voiced. In Section 4, two numerical examples are given with some simulation figures to support the theorems.Notation. Throughout this paper, ZT and the symbol ∗ in matrix Z represent matrix transposition and the symmetric term, respectively. ℝ is the set of integers, and ℝn denotes the n-dimensional Euclidean space. In means n-dimensional identity matrix. P>0 (P<0) is defined as a positive-definite (negative-definite) matrix. z is the Euclidean norm of a vector z in Rn. λmaxR (λminR) represents the maximum (minimum) eigenvalue of R and symY means Y+YT.


## 2. Preliminaries and Problem Formulation

Some fractional definitions and model descriptions are presented firstly. In addition, some important lemmas that will be used inSection 3 are also presented.Definition 1.
(see [23, 24]). For ht, the fractional integral form is defined as(1)Iαhq=1Γα∫q0qq−θα−1fθdθ,α∈ℝ+,where q≥q0 and Γ⋅ is a gamma function.Definition 2.
(see [23, 24]). Caputo’s derivative of hq is denoted by(2)Dqαq0Chq=1Γz−α∫q0qq−θz−α−1hzθdθ,α∈z−1,z,ht∈ιzt0,∞,ℝ,where q≥q0 and z is a positive integer.
In what follows,Dα stands for Dqαq0C for the convenience of presentation. In this paper, let us consider the following FNN model:(3)Dαυt=−Cυt+Aℏυt+V,yt=Dυt,where α∈0,1 is the predetermined fractional order, C=diagc1,c2,…,cn with ci>0i=1,2,…,n, A=aijn×n is the connection matrix, the vector υt=υ1t,υ2t,…,υntT∈ℝn stands for the neuron state, ℏυt=ℏ1υ1,ℏ2υ2,…,ℏnυnT denotes the activation function of the neurons, yt is the measurement output, V is the system input, and D is a known constant matrix.
In what follows, ETM is introduced in order to reduce the communication burden. The event-triggered condition is predesigned as follows:(4)eyTteyt≤σyTtkh+jhyTtkh+jh,where eyt=ytkh+jh−ytkh, σ is a given constant, jh and tkh are the sampling instant and the release instant, respectively, ytkh+jh stands for the latest sampled signal, and nih=tk+1h−tkh denotes the release period.Remark 1.
The sensor is time-driven at discrete instants, which can avoid the Zeno behavior. Moreover, whenσ=0, ETM becomes a time-triggered one.
In this paper, the transmission delaydk∈0,d¯ between sensor and estimator is considered, where d¯ is a positive scalar. Therefore, tkh+dk is the arrival time of the transmitted data from sensor to estimator.
In view of [25], the holding interval can be rewritten as tkh+dk,tk+1h+dk+1∪j=0j=dMIj, where Ij=tkh+jh+d¯,tkh+jh+h+d¯. For the convenience of analysis, denote dt=t−tkh−jh, and then we have 0≤dt≤h+d¯≜dM. Then, the measurement outputs arrived at the estimator can be rewritten as(5)y¯t=eyt+yt−dt=Dekt+yt−dt,where ekt is the error vector.
Design a nonfragile state estimator for system (3) as follows:(6)Dαυ^t=−Cυ^t+Aℏ˜υ^t+V+K+ΔKy¯t−Dυ^,where υ^t∈ℝn stands for the estimate of υt, K∈ℝn×q is the gain matrix to be determined, and ΔK represents the gain variation that satisfies ΔK=MℱtN, in which M and N are known real matrices and ℱt is an unknown satisfying ℱTtℱt≤I.
Defininget=υt−υ^t, the estimation error dynamics can be obtained from (3) and (5) as follows:(7)Dαet=−C−K+ΔKDet+Aℏet−K+ΔKDekt−K+ΔKDυt−dt+K+ΔKDυt,where ℏet≜ℏ˜υt−ℏ˜υ^t.
For notation simplicity, we defineηt≜υTteTtT. An augmented system model from (3) and (8) is given in the following form:(8)Dαηt=C˜ηt+A¯φηt+E˜ekt+K˜ηt−dt,where(9)C¯=−C0K+ΔKD−C−k+ΔKD,A¯=A00A,K˜=00−K+ΔKD0,E˜=0−K+ΔKD,φηt≜ℏTυtℏ˜TetT.Lemma 1 (see [26]).
Forξ1 and ξ2∈ℝn and any positive scale ϵ>0, one has(10)ψ1Tψ2+ψ2Tψ1≤ϵψ1Tψ1+ϵ−1ψ2Tψ2.Lemma 2 (see [27]).
For∀α∈0,1 and t≥0, if υt∈ℝn is continuous and differential, then(11)DαυTtQυt≤2υTtQDαυt.Lemma 3 (see [28]).
For matricesϖ,E,ℋ, where ϖ is symmetric, the inequality(12)ϖ+ℰℱℋ+ℰℱℋT<0holds if and only if(13)ϖ+ξEET+ξ−1ℋTℋ<0,in which ξ>0 refers to a scalar and ℱTℱ<I.Lemma 4 (see [29]).
Consider a class of fractional-order nonlinear systems:(14)Dαηt=ht,ηt−dt,and the initial condition is ηt0 = ϕ∈Ct0−τ,t0,ℝn. Suppose that ωisi=1,2:ℝ⟶ℝ are positive functions, and ω10=ω20=0,ω2s1<ω2s2∀0<s1<s2. If there exist two constants 0<μ<ϵ and a continuous differential function V:ℝ×ℝn such taht ω1≤Vt,x≤ω2 satisfying(15)DαVt,ηt≤−εVt,ηt+μsup−dM≤−dt≤0Vt+dt,xt+ηt−dt,then the fractional-order system is globally uniformly asymptotically stable.Lemma 5 (see [30]).
The matrix(16)ϕ=ϕ11ϕ12ϕ12Tϕ22<0,holds if and only if aϕ22<0,ϕ11−ϕ12ϕ22−1ϕ12T<0, or bϕ11<0,ϕ22−ϕ12Tϕ11−1ϕ12<0.Lemma 6 (see [31]).
V is continuous on t0,+∞ and bounded on t0−ρ,t0. If there exist p,q such that(17)DtαVt≤−pVt+∑k=1mqhsup−ρh≤ω≤0Vt+ω,t≥t0,where 0<α<1,qk>0,p>∑k=1mqk, and ρ=maxρ1,ρ2,…,ρm, then limt⟶∞Vt=0.


## 3. Main Results

Theorem 1.
For the given positive scalarsε>μ>0, system (8) is globally asymptotically stable if there exist a symmetric matrix P=diagP1,P2>0 and four scalars βii=1,2,3>0,γ>0 satisfying the following LMI:(18)Φ=Φ11Φ12Φ13Φ14∗Φ2200∗∗Φ330∗∗∗Φ44<0,(19)β2σI˜TI˜+β3I¯1TI¯1−μP<0,where(20)Φ11=Λ11DTXT−γDTNTND∗Λ2,Φ12=P1A00P2A,Φ14=000P2M¯,Φ13=−γDTNTND−γDTNTND−XD−γDTNTND−XD−γDTNTND,Φ22=−β1I¯100−β1I¯1,Φ33=−β2I¯1+γDTNTNDγDTNTND∗−β3I¯1+γDTNTND,Φ44=−β3I¯100−γI¯1,Λ11=−P1C−C1TP+β1M11TM11+ϵP1+γDTNTND,I¯1=diagI,I,Λ22=sym−P2C−XD+β1M12TM12+ϵP2+γDTNTND.Furthermore, the nonfragile estimator gainK of (8) is designed as K=P2−1X.Proof.
First, we denote(21)C˜=−C0KD−C−KD,E¯=0−KD,P=P100P2,K¯=00−KD0.
Considering system (8), design the following Lyapunov function:(22)Vt=ηTtPηt.
From Lemma2, one obtains(23)DαVt≤2ηTtPDαηt=2ηTtPC˜ηt+A¯φηt+E˜ekt+K˜ηt−d.
It follows from Lemma3 that(24)2ηTtPA¯φηt≤β1−1ηTtPA¯A¯TPηt+β1ηTtGTGηt,(25)2ηTtPE˜ekt≤β2−1ηTtPK¯E˜TPηt+β2ekTtekt≤β2−1ηTtPE˜E˜TPηt+β2σηTt−dtI˜TI˜ηt−dt,(26)2ηTtPK¯ηt−dt≤β3−1ηTtPK¯K¯TPηt+β3ηTt−dtηt−dt,where I¯=−I,I and I˜=I,0.
Combining (24)–(36), we can get(27)DαVt≤ηTtPC˜+PE˜1+C˜TP+E˜1TP+β1−1PA¯A¯TP+β1GTG+β2−1PE˜E˜TP+β3−1PK˜K˜TPηt+ηTt−dtβ2σI˜TI˜+β3I¯1TI¯1ηt−dt.
By employing Lemma5, Φ<0 is equivalent to(28)Φ¯+γN¯TN¯+γ−1M¯M¯T<0,where(29)Φ¯=PC¯+C¯TP+β1M1TM1+εP1PA¯PE¯PE¯I˜∗−β1I¯100∗∗−β2I¯10∗∗∗−β3I¯1,M¯T=0,MTP2,0n×5nT,N¯=ND,−ND,0,0,−ND,−ND,0.
Defining an equation as follows:(30)ΔΦ=M¯FtN¯+N¯TFTtM¯T,we have(31)ΔΦ≤γ−1M¯M¯T+γN¯TN¯.
Furthermore, from (28) and (31), we arrive at Φ¯+ΔΦ<0.
Based on Lemma3, we can obtain(32)PC˜+PE˜1+C˜TP+E˜1TP+β1−1PA¯A¯TP+β1GTG+β2−1PE˜E˜TP+β3−1PK˜K˜TP<−εP.
Combining (27) and (32), we have(33)DαVt≤−ϵηTtPηt+μsup0≤dt≤dMηTt−dtPηt−dt=−εVt+μsup0≤dt≤dMVt−dt.
It follows from Lemma4 that (4) is an asymptotical estimator and system (3) is globally asymptotically stable. The proof is completed.
It is worth noting that the parameter uncertainties are often unavoidable resulting from the inaccuracy of modeling or the changing environment. In addition, the network output is composed of linear and nonlinear parts. Therefore, the following model of FNN is established:(34)Dαυt=−C+ΔCυt+A+ΔAfυt+B+ΔBfυt−τt+J,yt=Dυt+g˜υt,where τt is a time-varying delay satisfying 0≤τt≤τM. Here, τM is a constant. ΔA,ΔB,ΔC are parameter uncertainties which satisfy the following condition:(35)ΔAΔBΔC=M2¯F1tN1N2N3,where M¯2,N1,N2,N3 are known matrices and F1t is an unknown matrix function which satisfies F1tTF1t≤I.Assumption 1.
g˜:ℝn⟶ℝq stands for the nonlinear disturbance which satisfies the Lipschitz condition:(36)g˜t,a−g˜t,b≤Fa−b,where g˜t,0=0 and F is a known constant matrix.
The estimator and estimation error dynamics are obtained as follows:(37)Dαυ^t=−C+ΔCυ^t+A+ΔAfυ^t+B+ΔBfx^t−τt+J+K+ΔKyt¯−Dυt^,Dαet=−C−K+ΔKDet+−ΔC+ΔKDυt+Af˜et+ΔAfυt+Bf˜et−τt+ΔBfxt−τt−K+ΔKeyt−K+ΔKDxt−dt−K+ΔKgxt−dt.
The augmented system is derived as follows:(38)Dαηt=C˜ηt+A˜φηt+B˜φηt−τt+H˜ηt−dt−G˜gI¯ηt−dt−L˜eyt,where(39)C˜=−C−ΔC0−ΔC+K+ΔKD−C−K+ΔKD,A˜=A+ΔA0ΔAA,B˜=B+ΔB0ΔBB,H˜=00−K+ΔKD0,I¯=00I0,G˜=00K+ΔK0,L˜=0K+ΔK.
The following theorem is given to ensure the above system is asymptotically stable.Theorem 2.
For given positive scalarsϕ,ϑ1,ϑ2 satisfying ϕ>ϑ1+ϑ2, the augmented system (48) is asymptotically stable if there exist a symmetric matrix P=diagP1,P2>0 and seven scalars γii=1,2,3>0, βii=1,2,…,4>0 satisfying the following LMI:(40)Π=Π11Π12Π13Π14Π15Π16∗Π220Π24Π250∗∗Π33000∗∗∗Π44Π450∗∗∗∗Π550∗∗∗∗∗Π66<0,(41)Θ=Λ20∗β1I−ϑ1P2<0,(42)Φ=β2M11TM11−ϑ2P10∗β2M12TM12−ϑ2P2<0,where(43)Π11=Λ1DTXT−γ3DTN4TN4D∗Q¯22+γ3DTN4TN4D,Π12=−γ3DTN4TN4Dγ2M41TN2TN3−XD+γ3DTN4TN4D0,Π13=P1B00P2B,Π14=γ3DTN4TN40X−γ3DTN4TN4D0,Π66=−γ2I0∗−γ3I,Π15=−γ3DTN4TN4−P1M¯2X−γ3DTN4TN4D−P2M¯2,Π22=−β1I+γ3DTN4TN4D0∗−β1I,Π16=P1M¯20P2M¯2P2M¯3,Π24=−γ3DTN4TN4000,Π45=γ3N4TN4000,Π33=−β2I+γ2N3TN30∗−β2I,Π44=−β3I+γ3N4TN40∗−β3I,Π25=−γ3DTN4TN4000,Π55=−β4I+γ3N4TN40∗−γ1I,Λ1=Q¯11+γ2M41TN2TN2M41+γ3DTN4TN4D,Λ2=β1I+β3M22TM22+β4σDTD+β4σsymDTM3+β4σM3TM3−ϑ2P2.
Furthermore, the nonfragile estimator gainK is designed as K=P2−1X.Proof.
Construct the following Lyapunov functional:(44)Vt=ηTtPηt.
From Lemma2, the following inequality is obtained:(45)DαVt≤2ηTtPDαηt=2ηTtPC˜ηt+H˜ηt−dt+A˜φηt+B˜φηt−τt−G˜1gI¯ηt−dt−L˜eyt.
By using Lemma1 and Lipschitz condition, one gets(46)2ηTtPH˜ηt−dt≤β1−1ηTtPH˜H˜TPηt+β1ηt−dtTηt−dt,(47)2ηTtPB˜φηt−τt≤2ηtTPB˜M1ηt−τt≤β2−1ηtPB˜B˜TPη+β2ηTt−τtM1TM1ηt−τt,(48)−2ηTtPG˜gI¯ηt−dt≤β3−1ηTtPG˜G˜TPηt+β3ηTt−dtI¯TM2TM2I¯ηt−dt,(49)−2ηTtPL˜eyt≤β4−1ηTtPL˜L˜TPηt+β4eytTeyt.
From event-triggered condition (5), we can obtain(50)eytTeyt≤σyTtkh+jhytkh+jh≤σyTt−dtyt−dt≤σηTt−dtD¯TD¯ηt−dt+σηTt−dtD¯TM3I˜+σI˜TM3TD¯ηt−dt+σηTt−dtI˜TM3TM3I˜ηt−dt.
Combining (45)–(50), one can get(51)DαVt≤2ηTtPDαηt≤ηTtPC˜ηt+C˜TP+PA˜M4+M4TA˜TP+β1−1PH˜H˜TP+β2−1PB˜B˜TP+β3−1PG˜G˜TP+β4−1PL˜L˜TPηt+ηTt−dtβ1Iˇ+β3I¯TM2TM2I¯+β4σD¯TM3I˜+β4σI˜TM3TD¯+I˜TM3TM3I˜ηt−dt+ηTt−τtβ2M1TM1ηTt−τt,and therefore(52)DαVt+ϕVt−ϑ1sup−τM≤ω≤0Vt+ω,xt+ω−ϑ2sup−dM≤ω≤0Vt+ω,xt+ω≤ηTtPC˜ηt+C˜TP+PA˜M4+M4TA˜TP+β1−1PH˜H˜TP+β2−1PB˜B˜TP+β3−1PG˜G˜TP+β4−1PL˜L˜TPηt+ηTt−dtβ1Iˇ+β3I¯TM2TM2I¯+β4σD¯TM3I˜+β4σI˜TM3TD¯+I˜TM3TM3I˜ηt−dt+ηTt−τtβ2M1TM1ηTt−τt+ϕVt−ϑ1ηTt−τtPηt−τt−ϑ2ηTt−dtPηt−dt.
Based on the above inequations, we have(53)DαVt+ϕVt−ϑ1sup−τM≤ω≤0Vt+ω,xt+ω−ϑ2sup−dM≤ω≤0Vt+ω,xt+ω≤ηTtQ1ηTt+ηTt−dtQ2ηTt−dt+ηTt−τtQ3ηTt−τt,where(54)Q1=symPC˜+PA˜M4+β1PH˜H˜TP+β2PB˜B˜TP+β1PG˜G˜TP+β4PL˜L˜TP+ϕP,Q2=β1Iˇ+β3I¯TM2TM2I¯+β4σD¯TD¯+symβ4σD¯TM3I˜+β4σI˜TM3TM3I˜−ϑ1P,Q3=β2M1TM1−ϑ2P.
By Lemma5, Π<0 is equivalent to(55)Q¯+γ1S1TS1+γ2S2TS2+γ3S3TS3+γ1−1R1R1T+γ2−1R2R2T+γ3−1R3R3T<0,where(56)Q¯=symPC1+PA1M4PH1PB1PG1PL1∗−β1Iˇ000∗∗−β2Iˇ00∗∗∗−β3Iˇ0∗∗∗∗−β4Iˇ,R1=−P1M2¯−P2M2¯07n×n,R2=P1M2¯P2M2¯07n×n,R3=0P2M2¯07n×n,S1=N1,0n×8n,S2=N2M41,0n×3n,N3,0n×4n,S3=N4D,−N4D,−N4D,0n×3n,N4,0,N4.
Let(57)Q¯′=R1F1tS1+R2F1tS2+R3F2tS3+S1TF1TtR1T+S2TF1TtR2T+S3TF2TtR3T≤γ1S1TS1+γ2S2TS2+γ3S3TS3+γ1−1R1R1T+γ2−1R2R2T+γ3−1R3R3T.
Then, one getsQ¯+Q¯′<0. Based on Lemma 5, (55) and (57) imply that Q1≤0. According to (40) and (42), we know that Q2<0, and Q3<0. Therefore, we can obtain(58)DαVt+ϕVt−ϑ1sup−τM≤ω≤0Vt+ω,υt+ω−ϑ2sup−dM≤ω≤0Vt+ω,υt+ω≤0.
Therefore, system (37) is an asymptotical estimator of (34) by using Lemma 6. This completes the proof.


## 4. Numerical Example

To illustrate the theoretical results, two numerical examples are shown in this section.Example 1.
(59)A=−1.2−0.511−0.21.60.20.8−1,C=0.30000.40000.3,D=0.20000.280000.1,where M¯=diag0.4,0.4,0.4, N = diag0.3,−0.3,0.3, and M11=M12= diag0.2,0.2,0.2.
The other parameters are given asα=0.98, τ=1,V=0.5,0.2,−0.8T. The activation function is given as follows:(60)ℏυ1,υ2,υ3=sin0.4πυ1,cos0.2πυ2,sin0.6πυ3.
By using Matlab to solve LMIs (26) and (27), the gain matrix K can be obtained as(61)K=1.56630.1752−1.06550.14751.9113−0.8569−0.7003−0.71381.6980.
The simulation results are shown in Figures1–3, where υ1t,υ2t,υ3t represent the true states and their estimates υ^1t,υ^2t,υ^3t and the initial conditions are (∀t∈−1,0): υt=0.3,0.7,−0.3T,υ^t=−0.5,−0.4,0.2T.
Figure4 shows the estimate error eit⟶0 as t⟶∞. According to the simulation results, we can see the effectiveness of the estimator design method. Figure 5 shows the release instants and intervals with the threshold parameter σ=0.06.Figure 1
Trajectories ofυ1t and υ^1t.Figure 2
Trajectories ofυ2t and υ^2t.Figure 3
Trajectories ofυ3t and υ^3t.Figure 4
The estimation errors.Figure 5
Event-triggered release instants and intervals of Theorem1.Example 2.
To verify that the estimator contains uncertain terms in (34), the following fractional-order model is shown, and the corresponding parameters are as follows:(62)A=−0.6−0.510.1−0.210.2.0.3−1,B=0.2−0.20.10.3−0.20.1−0.4−0.10.3,C=0.20000.20000.3,where M¯2=diag0.4,0.2,0.4, M¯3=diag0.1,0.1,0.1, M11=M12=M41=M42=diag0.2,0.2,0.2, N1=N2=diag0.1,0.1,0.1, and N3=N4=0.2,0.2,0.2. Besides, α=0.92, and the time delays are set as τM=1,dM=0.1,J=0.01,0.02,−0.01T. The measurement output is g˜t,υ=tanh0.4πυ1,tanh0.2πυ2,tanh0.6πυ3. Let the activation function be ℏυ1,υ2,υ3=tanh0.4πυ1,tanh0.3πυ2,tanh0.2πυ3.
By using Matlab to solve the LMI (37), the gain matrix K can be obtained as(63)K=1.41990.2566−1.40640.08081.2616−0.6868−0.3186−0.37232.8109.
The simulation results are shown in Figures6–8, where υ1t,υ2t,υ3t represent the true states and their estimates υ^1t,υ^2t,υ^3t are depicted, respectively, with the initial condition (∀t∈−1,0.): υt=0.7;1.2;−0.6,υ^t=−1;−0.7;0.6.
Figure9 shows the estimate error e1t,e2t,e3t, as t⟶∞. In Figure 8, we can see clearly the error states eit⟶0. Figure 10 shows the release instants and intervals with σ=0.06. According to the figures, we can see that the simulation results voiced the effectiveness of the estimator design.Figure 6
Trajectories ofυ1t and υ^1t.Figure 7
Trajectories ofυ2t and υ^2t.Figure 8
Trajectories ofυ3t and υ^3t.Figure 9
The estimation errors.Figure 10
Event-triggered release instants and intervals of Theorem2.


## 5. Conclusions

This paper has investigated the nonfragile SE issue under the ETM for the FNNs with time delays. Sufficient conditions have been obtained to ensure the asymptotic stability of the considered system by means of the fractional-order Lyapunov functions and the LMI method. The gain matrix of the nonfragile estimator has been characterized by a LMI. At last, two numerical results have confirmed the validity of the designed estimator. In addition, the results could be extended to the SE issue of discrete FNNs with fading measurements and so on.



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