# Finite-TimeH∞ State Estimation for Markovian Jump Neural Networks with Time-Varying Delays via an Extended Wirtinger’s Integral Inequality

Authors: Saravanan Shanmugam; M. Syed Ali; R. Vadivel; Gyu M. Lee
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study investigates the finite-time boundedness for Markovian jump neural networks (MJNNs) with time-varying delays. An MJNN consists of a limited number of jumping modes wherein it can jump starting with one mode then onto the next by following a Markovian process with known transition probabilities. By constructing new Lyapunov–Krasovskii functional (LKF) candidates, extended Wirtinger’s, and Wirtinger’s double inequality with multiple integral terms and using activation function conditions, several sufficient conditions for Markovian jumping neural networks are derived. Furthermore, delay-dependent adequate conditions on guaranteeing the closed-loop system which are stochastically finite-time bounded (SFTB) with the prescribedH∞ performance level are proposed. Linear matrix inequalities are utilized to obtain analysis results. The purpose is to obtain less conservative conditions on finite-time H∞ performance for Markovian jump neural networks with time-varying delay. Eventually, simulation examples are provided to illustrate the validity of the addressed method.


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## Body

## 1. Introduction

Due to the great significance of neural networks (NNs) for both practical and theoretical purposes, their dynamics have been explored widely in recent years, such as pattern recognition, signal processing, solving optimization problems, static image processing, associative memories, target tracking, and automatic control. Therefore, many research subjects have been studied in a broad spectrum of stability analysis, passivity analysis, control, filtering design, and state estimation and synchronization, concerning to NNs [1–6]. In [4], passive filter design for fractional-order quaternion-valued neural networks with neutral delays and external disturbance has been studied. The authors in [6] investigated stability criteria of quaternion-valued neutral-type delayed neural networks. In many implementations of NNs, time delays are inevitable [7] and can lead NNs to instability and oscillation. Hence, the stability analysis with time delays in the NN models under consideration has attracted considerable attention [8–14].Due to interconnection failures, sudden environment changes, components, and so on, plenty of structural parameters of neural networks may mutate. In general, there are finite modes in the neural network,s switching or jumping from one mode to another mode by a random form. A Markov chain can be used to describe jumping between different modes of neural networks, and the kinds of systems are called Markovian jump neural networks [15–18]. Many practical control systems can be modeled as Markovian jump neural networks, such as air intake systems and economic systems [19]. In an MJNN, hopping among operation modes is specified by a Markov process, so it is important to understand the impacts of its stochastic attributes on the stability analysis of delayed MJNNs. Some previous works [15–18] have discussed certain standard results in relation to MJNN stability analysis. In [20], the authors conducted an asymptotic stability analysis for stochastic and static NNs with time-varying delays that are mode-dependent. The use of linear matrix inequalities (LMIs) has led to important and interesting results concerning various types of NN with MJ parameters [21, 22]. The mode-dependent MJNNs with time-varying delays and incomplete transition rates can be found in [23], wherein some LMI-based conditions are proposed to obtain the required results.In some cases, we are interested in knowing how the modeled system behaves within fixed- and finite-time intervals. In other words, given an initial bounded state, we require the system to remain in a state that is not superior to a particular threshold during a specified time interval. Since this type of stability ensures a faster convergence of the system, it has been widely used in various NNs, such as the MJNNs, and synchronizing neural networks [24]. An important example can be found in controlling the trajectory of a spacecraft between its initial and final locations within a specified time interval. However, because of the lack of other finite-time-bounded operational conditions, it is natural that research interest has shifted to Lyapunov stability in this paper. In addition, based on LMI results, the idea of finite-time boundedness (FTB) has been revisited here. We also studied that finite-time stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in a finite time. Note that the finite-time stability with control frameworks has gained significant attention in recent years [25–28]. In [29], the authors discussed the finite-time L2-gain performance of MJNNs. The design of a finite-time passive controller for uncertain MJ systems is optimized in [28], wherein a robust and fuzzy finite-time passive control is defined along with the finite-time stochastic stability of a nonlinear MJ system. However, finite-time H∞ state estimation of MJ systems has not been studied much for NNs. This is a primary inspiration for this study. The main contributions of this study are listed as follows:(1)
The comprehensive Markovian jump neural networks with state and input constraints are studied.(2)
We have introduced a novel Lyapunov–Krasovskii functional (LKF), including time-varying delays.(3)
Wirtinger’s double integral inequality, introduced by Park et al. [30], and Wirtinger’s integral inequality, extended by Zhang et al. [31], are introduced into the time-derivative of LKF. This time-derivative forms the LMIs which are FTB. These LMIs deliver more effective outcomes in comparison to previous works. The numerical examples are also given.(4)
To show the real-life application, the four-tank water pumping system and network circuit are considered in this paper in terms of the NN model to show feasibility on a benchmark problem.Notations are as follows:ℝn: n-dimensional Euclidean spaceP>0: the matrix P is a symmetric matrixmin (P): minimum eigenvalue of Pmax(P): maximum eigenvalue of PI: identity matrixdiag⋅: diagonal matrix∗: symmetric matrices


## 2. Preliminaries and Problem Formulation

Given a probability spaceΩ,ℱ,P, where Ω,ℱ,andP represent sample space, σ-algebra of events, and probability measure defined on ℱ, respectively. Let parameter rt,t≥0 be a right continuous Markov chain taking values on Ω,ℱ,P a finite set S=1,2,…,N with generator Π=πijN×N given by(1)Prrt+Δt=j|rt=i=πijΔt+oΔt,i≠j,1+πiiΔt+oΔt,i=j,where rt∈S, Δt>0, limΔt⟶0oΔt/Δ=0, and πij denotes the transition probability from modes i to j satisfying πij≥0, for i≠j, with πii=−∑j=1,j≠isπij, i,j∈S.Consider the MJNNs with time-varying delays are as follows:(2)x˙t=−Artxt+Brthxt+Bdrthxt−δt+J+E1rtwt,yt=Crtxt+Drtxt−δt+E2rtwt,zt=Grtxt,xt=ϕt,t∈δ,0,,where x⋅=x1⋅,x2⋅,…,xn⋅T∈ℝn is the state vector, yt∈ℝp is the output measurement, zt∈ℝm denotes the estimated signal, wt∈ℝq represents exogenous disturbance belonging to L20,∞, hxt=h1x1t,h2x2t,…,hnxntT∈ℝn is a neuron activation function, and J∈ℝn denotes an external input constant vector. Art>0 is a diagonal matrix, and Brt,Bdrt,E1rt,E2rt,Crt,Drt, and Grt are connection weight matrices. A time-varying delay is denoted as δt, where 0≤δt≤δ¯ and δ˙t≤μ, such that δ¯ and μ are known constants. For each possible value of rt=i, i∈S, a matrix Art is denoted by Ai and all other matrices with appropriate dimensions are denoted by Bi,Bdi,E1i,E2i,Ci,Di, and Gi.Assumption 1.
Each neuron activation functionhkt (k=1,2,…,n) is continuous and bounded and satisfies the following condition:(3)ϱk−≤hkx1−hkx2x1−x2≤ϱk+,∀x1,x2∈ℝ,x1≠x2,where ϱk− and ϱk+ are constant. Then, we define the followings(4)L1=diagϱ1−,ϱ2−,…,ϱn−,L2=diagϱ1+,ϱ2+,…,ϱn+,Mt=diagϱ1−ϱ1+,ϱ2−ϱ2+,…,ϱn−ϱn+,Mu=diagϱ1−+ϱ1+2,ϱ2−+ϱ2+2,…,ϱn−+ϱn+2.Assumption 2.
The external disturbancewt fluctuates and satisfies the following inequality:(5)∫0TwTtwtdt≤d,d≥0.
For a MJNN defined as (2), a state estimator is constructed as follows:(6)x˜t=−Aix˜t+Bihx˜t+Bdihx˜t−δt+J+Kiyt−y˜t,y˜t=Cix˜t+Dix˜t−δt,z˜t=Gix˜t,xt=0,t∈δ¯,0,where x˜t∈ℝn denotes the estimated state and z˜t∈ℝq is the estimated measurement of zt. Then, estimator gain matrix Ki is to be constructed.
By defining the erroret=xt−x˜t, fet=hxt−hx˜t, and z¯t=zt−z˜t, an error system can be obtained in the following form:(7)e˙t=−Ai+KiCiet−KiDiext−δt+Bifet+Bdifet−δt+E1i−KiE2iwt,z¯t=Giet,where et=e1t,e2t,…,entT∈ℝn denotes the state vector of modeled system and fet=fetf2et,…,fnetT, and fet=gxt−gx˜t is the transformed activation function. From Assumption 1, the neuron activation function satisfies(8)la−≤faρρ≤la+,where ρ∈ℝ and ρ≠0.Definition 1.
(stochastically finite-time stable (SFTS) [27]). Given time constant T>0, an MJNN defined as (7) with wt=0 is SFTS with respect to c1,c2,T,R if there exists a positive matrix R>0 and scalars c1>0 and c2>0, such that the following inequality holds:(9)Ex0TtRx0t<c1⇒ExTtRxt,t∈0,T.Definition 2.
(stochastically finite-time boundedness (SFTB) [27]). Given a time constant T>0, an MJNN defined as (7) is said to be SFTB with respect to c1,c2,T,R,d, where there exist R>0 and scalars c1>0 and c2>0, such that the following inequality holds:(10)Ex0TtRx0t<c1⟹ExTtRxt,t∈0,T.Definition 3.
(see [32, 33]). For T>0, an MJNN defined as (7) is said to be SFTB with respect to c1,c2,T,R,d and with a prescribed level of noise attenuation γ>0 under a zero initial condition if it holds:(11)E∫0TzTszsds≤γ2E∫0TwTtwtdt.Definition 4.
(see [34]). A functional Vxt,rt,t>0=Vxt,r is said to be a stochastic positive functional. Its weak infinitesimal operator can be defined as(12)ℒVxt,rt=limΔt⟶01ΔtEVxt+Δt,rt+Δt,t+Δt|xt,rt=i−Vxt,i,t.Lemma 1 (see [30]).
Let a constant matrixM>0; the following condition can be defined for all differentiable function ϕ in a,b⟶ℝn for scalars a and b with a<b:(13)−b2−a22∫−a−b∫t+θtη˙TsMη˙sdsdθ≤−Ω1TMΩ1−2Ω2TMΩ2,where Ω1=b−aηt−∫t−at−bηsds and Ω2=−b−a/2ηt−∫t−at−bηsds+3/b−a∫−a−b∫t+θtηsdsdθ.Lemma 2 (see [35]).
For any constant matrixM>0, the following inequality holds for all continuously differentiable function φ on a,b⟶ℝn×n:(14)b−a∫abφTsMφsds≥∫abφsdsT×M∫abφsds+3ΩTMΩ,where Ω=∫abφsds−2/b−a∫ab∫asφθdθds.Lemma 3 (see [31]).
For a given symmetric matrixW2=W2T>0, the following inequality holds for all continuously differential function x in α,β⟶ℝn:(15)β−α∫αβx˙TsW2x˙sds≥ξTΞξ,where(16)ξ=xTβxTα2β−α∫αβxTsds12b−a2∫αβ∫αsxTθdsdθT,Ξ=9W2−3W2−36W22W2∗9W215W2−5W2∗∗48W2−15W2∗∗∗5W2.


## 3. Methodologies and Theoretical Results

### 3.1. Finite-Time State Estimation

This section derives the SFTB of the error system in (7).Theorem 1.
For scalarsc1, c2, T, d, δ¯, μ, and α, an MJNN defined as (7) is SFTB in relation to c1,c2,T,R,d if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, N, and X, where Pi, Qs, and Ws are symmetric positive definite (PD), and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(17)Ψ=ψij9×9<0,(18)eαTc1Λ+λ9d<λ1c2,where(19)ψ11=∑j=1NπijPj+Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NAi−AiTNT−LiCi−CiTLiT,ψ12=LiDi,ψ13=3W2,ψ14=2Pi−N−AiTNT−CiTLiT,ψ15=36W2+δ¯W3+δ¯2W3,ψ16=−2W2−32W3,ψ17=MuUt+NBi,ψ18=NBdi,ψ19=NE1i−LiE2i,ψ22=−1−μQ1−MtUu,ψ23=0,ψ24=DiTLiT,ψ25=0,ψ26=0,ψ27=0,ψ28=MuUu,ψ29=0,ψ33=−Q2−9W2,ψ34=0,ψ35=−30W2,ψ36=−5W2,ψ37=0,ψ38=0,ψ39=0,ψ44=δ¯W2+δ¯44W2−N−NT,ψ45=0,ψ46=0,ψ47=NBi,ψ48=NBdi,ψ49=NE1i−LiE2i,ψ55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ56=6δ¯2W1−30W2+3δ¯W3,ψ57=0,ψ58=0,ψ59=0,ψ66=−12δ¯3W1−5W2−9δ¯2W3,ψ67=0,ψ68=0,ψ69=0,ψ77=Q3−Ut,ψ78=0,ψ79=0,ψ88=−Q3−Uu,ψ89=0,ψ99=−X.
In addition, the desired control gain matrices can be calculated byKi=N−1Li.Proof.
Construct LKF for an MJNN defined as (7):(20)Vt=∑i=15Vit,where(21)V1t=eTtPiet,V2t=∫t−δtteTsQ1esds+∫t−δ¯teTsQ2esds,V3t=∫t−δttfTesQ3fesds,V4t=∫−δ¯0∫t+θteTsW1esdsdθ+∫−δ¯0∫t+θte˙TsW2e˙sdsdθ,V5t=δ¯22∫−δ¯0∫β0∫t+θte˙TsW3e˙sdsdθ.
By differentiating the above LKF to obtain its time derivatives along with the trajectory of the MJNN defined as (7), we obtain(22)ℒV1=2eTtPe˙t+eTt∑j=1NπijPjet,(23)ℒV2=eTtQ1+Q2et−1−δDeTt−δt×Q1et−δt−eTt−δ¯Q2et−δ¯,(24)ℒV3=fTetQ3fet−fTet−δtQ3fet−δt,(25)ℒV4=δ¯eTtW1et−∫t−δ¯teTsW1esds+δ¯e˙TtW2e˙t−∫t−δ¯te˙TsW2e˙sds,(26)ℒV5=δ¯222e˙TtW3e˙t−δ¯22∫δ¯0∫t+θte˙TsW3e˙sdsdθ.
Utilizing Lemma2, we obtain(27)−∫t−δ¯teTsW1esds≤−1δ¯∫t−δ¯tesdsT×W1∫t−δ¯tesds−3δ¯Φ1TW1Φ1,where Φ1=∫t−δ¯tesds−2/δ¯∫−δ¯0∫t+θtesdsdθ.
By applying Lemma3, the following inequality can be written as(28)−∫t−δ¯te˙TsW2e˙sds≤etet−δ¯1δ∫t−δ¯tesds12δ¯2∫t−δ¯t∫t−δ¯sesdsdθT−9W23W236W2−2W2∗−9W2−30W25W2∗∗−192W230W2∗∗∗−5W2etet−δ¯1δ¯∫t−δ¯tesds12δ¯2∫t−δ¯t∫t−δ¯sesdsdθ.
By applying Lemma1, we obtain(29)−δ¯22∫δ¯0∫t+θtx˙TsW3x˙sdsdθ≤−Φ2Φ3TW3002W3Φ2Φ3,where Φ2=δ¯et−∫t−δ¯tesds and Φ3=−δ¯/2et−∫t−δ¯tesds+3/δ¯∫−δ¯0∫t+θtesdsdθ.
From Assumption1, we obtain(30)fkekt−Mk−ektfkekt−Mk−ekt≤0,fkekt−δt−Mk−ekt−δt×fkekt−δt−Mk−ekt−δt≤0,where,k=1,2,…,n.
This can be written algebraically as(31)etfetTMt−Mu∗Ietfet≤0,et−δtfet−δtTMt−Mu∗Iet−δtfet−δt≤0.
Then, the following inequality holds for any positive matricesUt=diagu1,u2,…,un and Uu=diagu^1,u^2,…,u^n:(32)etfetTMtUt−MuUt∗Utetfet≤0,(33)et−δtfet−δtTMtUu−MuUu∗Uuet−δtfet−δt≤0.
For convenience, consider a matrixN with appropriate dimension, and the following zero equality holds:(34)0=2eTt+e˙TtN−e˙Tt−Ai+KiCiet+KiDiet−δt+Bifet+Bdifet−δt+E1i−KiE2iwt.
Therefore, from (22)–(34), given that α>0, we obtain(35)ℒVet−αVxt−αwTtXwt≤ϑTtΨϑt<0,where(36)ϑTt=eTteTt−δteTt−δ¯e˙Tt∫t−δ¯tesdsT∫−δ¯0∫t+θtesdsdθTfTetfet−δtwTt.
We can write this as(37)ℒVet≤αVxt+αwTtXwt.
Multiplying both sides of (37) by e−αt and then integrating from 0 to t, where t∈0,T, we obtain(38)e−αtEVet≤EVe0+α∫0teαswTsXwsds.
It can be simplified as(39)EVet<eαtEVe0+α∫0teαswTsXwsds,(40)EVet<eαTEVe0+λ9d.
LetP¯i=R−1/2PiR−1/2, Q¯1=R−1/2Q1R−1/2, Q¯2=R−1/2Q2R−1/2, Q¯3i=R−1/2Q3R−1/2, W¯1=R−1/2W1R−1/2, W¯2=R−1/2W2R−1/2, and W¯3=R−1/2W3R−1/2.
Conversely,(41)EVe0,0=λmaxP¯ieT0Re0+λmaxQ¯1∫−δ00eTsResds+λmaxQ¯2∫−δ¯0eTsResds+λmaxQ¯3maxMt−,Mu+2×∫−δ00eTsResds+λmaxW¯1∫−δ¯0∫θ0eTsResdsdθ+λmaxW¯2∫−δ¯0∫θ0e˙TsRe˙sdsdθ+λmaxW¯3∫−δ¯0∫β0∫θ0e˙TsRe˙sdsdθ≤λmaxP¯+δ¯λmaxQ¯1+δ¯λmaxQ¯2+δ¯maxMt−,Mu+2λmaxQ¯3+δ¯22λmaxW¯1+δ¯22λmaxW¯2+δ¯512λmaxW¯3sup−δ¯≤s≤0eTsRes,e˙TsRe˙s,Vxt≤eαTΛc1+dλ9,where(42)Λ=λ2+δ¯λ3+δ¯λ4+δ¯maxMt−,Mu+2λ5+δ¯22λ6+δ¯22λ7+δ¯512λ8.
Note that(43)EVxt≥λminP¯ExTtRxt=λ1ExTtRxt.
Then, from (18), we obtain(44)ExTtRxt<c2.
Based on Definition2, an MJNN defined as (7) is SFTB.Corollary 1.
Given scalarsc1, c2, T, δ¯, μ, and α, an MJNN defined as (7) with wt=0 is SFTB in relation to c1,c2,T,R if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, and N, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal such that the following inequality holds:(45)Ψ1=ψi,j8×8<0,(46)eαTc1Λ<λ1c2,where ψij is defined in Theorem 1.Proof.
It can be proved in a similar way as Theorem1. The proof is omitted for brevity.


### 3.2. Finite-TimeH∞ State Estimation

Theorem 2.
Given scalarsc1, c2, T, δ¯, μ, d, and α, an MJNN defined as (7) is SFTB in relation to c1,c2,T,R,d with noise attenuation γ>0 if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, N, and X, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(47)Ψ¯=Ψ1Ψ^Gi∗ψ¯990∗∗−I<0,(48)eαTc1Λ+γ2d<λ1c2,where Ψ1=ψi,j8×8, Ψ^=colψi9,i=1,2,...,8,(49)ψ11=∑j=1NπijPj+Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NAi−AiTNT−LiCi−CiTLiT,ψ12=LDi,ψ13=3W2,ψ14=2Pi−N−AiTNT−CiTLiT,ψ15=36W2+δ¯W3+δ¯2W3,ψ16=−2W2−32W3,ψ17=MuUt+NBi,ψ18=NBdi,ψ19=NE1i−LiE2i,ψ22=−1−μQ1−MtUu,ψ23=0,ψ24=DiTLT,ψ25=0,ψ26=0,ψ27=0,ψ28=MuUu,ψ29=0,ψ33=−Q2−9W2,ψ34=0,ψ35=−30W2,ψ36=−5W2,ψ37=0,ψ38=0,ψ39=0,ψ44=δ¯W2+δ¯44W2−N−NT,ψ45=0,ψ46=0,ψ47=NBi,ψ48=NBdi,ψ49=NE1i−LiE2i,ψ55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ56=6δ¯2W1−30W2+3δ¯W3,ψ57=0,ψ58=0,ψ59=0,ψ66=−12δ¯3W1−5W2−9δ¯2W3,ψ67=0,ψ68=0,ψ69=0,ψ77=Q3−Ut,ψ78=0,ψ79=0,ψ88=−Q3−Uu,ψ89=0,ψ¯99=−γ2I.Proof.
In a similar way to the proof in Theorem1, we obtain(50)EℒVet+z¯Ttz¯t−γ2wTtwt<ϑTtΨ¯ϑt.
It can be deduced from (47) and (50) that(51)EℒVet+z¯Ttz¯t−γ2wTtwt<0.
By integrating (51) from 0 to T, we obtain(52)EVxt−Vxo+∫0Tz¯Ttz¯tdt−γ2∫0TwTtwtdt<0.
Subsequently, the following inequality is obtained:(53)E∫0Tz¯Ttz¯tdt≤γ2E∫0TwTtwtdt.
Hence, we conclude that the MJNN defined as (7) is SFTB.Remark 1.
Consider the following error system from the MJNN defined as (7), with wt=0 and without MJ parameters:(54)e˙t=−A+KCet−KDext−δt+Bfet+Bdfet−δt.Corollary 2.
Given scalarsδ¯ and μ, the error system (54) with wt=0 is said to be stable if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, and N, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(55)Ψ¯=ψ¯i,j8×8<0,where(56)ψ¯11=Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NA−ATNT−LC−CTLT,ψ¯12=LD,ψ¯13=3W2,ψ¯14=2P−N−ATNT−CTLT,ψ¯15=36W2+δ¯W3+δ¯2W3,ψ¯16=−2W2−32W3,ψ¯17=MuUt+NB,ψ¯18=NBdi,ψ¯22=−1−μQ1−MtUu,ψ¯23=0,ψ¯24=DTLT,ψ¯25=0,ψ¯26=0,ψ¯27=0,ψ¯28=MuUu,ψ¯33=−Q2−9W2,ψ¯34=0,ψ¯35=−30W2,ψ¯36=−5W2,ψ¯37=0,ψ¯38=0,ψ¯44=δ¯W2+δ¯44W2−N−NT,ψ¯45=0,ψ¯46=0,ψ¯47=NB,ψ¯48=NBdi,ψ¯55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ¯56=6δ¯2W1−30W2+3δ¯W3,ψ¯57=0,ψ¯58=0,ψ¯66=−12δ¯3W1−5W2−9δ¯2W3,ψ¯67=0,ψ¯68=0,ψ¯77=Q3−Ut,ψ¯78=0,ψ¯88=−Q3−Uu.Proof.
Following similar ideas as in the proof of Theorem1. The proof is omitted for brevity.Remark 2.
Consider a NN from the MJNN defined as (7) with C=0, D=0, wt=0, and no MJ parameters:(57)e˙t=−Aet+Bfet+Bdfet−δt.Corollary 3.
Given scalarsδ¯ and μ, the error system (57) with wt=0 is said to be stable if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, and N, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(58)Ψ˜=ψ˜i,j8×8<0,where(59)ψ˜11=Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NA−ATNT,ψ˜12=0,ψ˜13=3W2,ψ˜14=2P−N−ATNT,ψ˜15=36W2+δ¯W3+δ¯2W3,ψ˜16=−2W2−32W3,ψ˜17=MuUt+NB,ψ˜18=NBdi,ψ¯22=−1−μQ1−MtUu,ψ˜23=0,ψ˜24=0,ψ˜25=0,ψ˜26=0,ψ˜27=0,ψ˜28=MuUu,ψ˜33=−Q2−9W2,ψ˜34=0,ψ˜35=−30W2,ψ˜36=−5W2,ψ˜37=0,ψ˜38=0,ψ˜44=δ¯W2+δ¯44W2−N−NT,ψ˜45=0,ψ˜46=0,ψ˜47=NB,ψ˜48=NBdi,ψ˜55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ˜56=6δ¯2W1−30W2+3δ¯W3,ψ˜57=0,ψ˜58=0,ψ˜66=−12δ¯3W1−5W2−9δ¯2W3,ψ˜67=0,ψ˜68=0,ψ˜77=Q3−Ut,ψ˜78=0,ψ˜88=−Q3−Uu.Proof.
It can be proved in a similar way to Theorem1. The proof is omitted for brevity.Remark 3.
The stability analysis of time-delay systems can be classified into two categories, i.e., delay-dependent stability criteria and delay-independent ones. Also, it is well known that delay-dependent stability criteria, which use the information on the size of time delays, are less conservative than delay-independent ones. Thus, more attention has been paid to the derivation of delay-dependent stability criteria for time-delay systems.Remark 4.
It is important to note that some pioneering works have been done on finite-timeH∞ state estimation for Markovian jump neural networks based on interval time-varying delay with simple LKF techniques. In [29], the authors studied finite-time boundedness for Markovian jump neural networks with L2 gain analysis. Authors in [26] formulated finite-time stabilization of uncertain neural networks. Exponential state estimation problem has been designed for Markovian jumping neural networks in [18]. The model consider in this present study is more practical than that proposed by [18, 26, 29], whereas in this paper, we consider finite-time H∞ state estimation problem with the combination of Markovian jump neural networks’ interval time-varying delay model, which is another advantage. However, the authors in [18, 26, 29] used some simple techniques in LKFs to solve the stability problems to those articles. A new LKF with double and triple integral terms and utilizing extended Writnger’s integral inequality (EWII) techniques has been proposed for the stochastically finite-time bounded analysis of Markovian jump system in this paper. Consider that some less conservative results can occur in our method and can be provided in the numerical example section with real-life examples. Hence, the results presented in this paper are essentially new.Remark 5.
Typically, finite time stability withH∞ control, state estimation approach, and interval time-varying delay is not simply applied to Markovian jump neural networks. Some research publications have handled such issues [17, 18, 26, 29]. As it is, the author utilized some elementary LKFs to deal with the stability problems in those articles. Novel LKF with EWII has been proposed; in addition, the developed stochastic stability criteria tested for feasibility of the benchmark problem to explore the real-world application in this paper. However, the desired control was completely studied for the considered neural network model with the real-world application problem (e.g., four-tank pumping system and network circuit), which is the principle commitment and inspiration of our work.


## 3.1. Finite-Time State Estimation

This section derives the SFTB of the error system in (7).Theorem 1.
For scalarsc1, c2, T, d, δ¯, μ, and α, an MJNN defined as (7) is SFTB in relation to c1,c2,T,R,d if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, N, and X, where Pi, Qs, and Ws are symmetric positive definite (PD), and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(17)Ψ=ψij9×9<0,(18)eαTc1Λ+λ9d<λ1c2,where(19)ψ11=∑j=1NπijPj+Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NAi−AiTNT−LiCi−CiTLiT,ψ12=LiDi,ψ13=3W2,ψ14=2Pi−N−AiTNT−CiTLiT,ψ15=36W2+δ¯W3+δ¯2W3,ψ16=−2W2−32W3,ψ17=MuUt+NBi,ψ18=NBdi,ψ19=NE1i−LiE2i,ψ22=−1−μQ1−MtUu,ψ23=0,ψ24=DiTLiT,ψ25=0,ψ26=0,ψ27=0,ψ28=MuUu,ψ29=0,ψ33=−Q2−9W2,ψ34=0,ψ35=−30W2,ψ36=−5W2,ψ37=0,ψ38=0,ψ39=0,ψ44=δ¯W2+δ¯44W2−N−NT,ψ45=0,ψ46=0,ψ47=NBi,ψ48=NBdi,ψ49=NE1i−LiE2i,ψ55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ56=6δ¯2W1−30W2+3δ¯W3,ψ57=0,ψ58=0,ψ59=0,ψ66=−12δ¯3W1−5W2−9δ¯2W3,ψ67=0,ψ68=0,ψ69=0,ψ77=Q3−Ut,ψ78=0,ψ79=0,ψ88=−Q3−Uu,ψ89=0,ψ99=−X.
In addition, the desired control gain matrices can be calculated byKi=N−1Li.Proof.
Construct LKF for an MJNN defined as (7):(20)Vt=∑i=15Vit,where(21)V1t=eTtPiet,V2t=∫t−δtteTsQ1esds+∫t−δ¯teTsQ2esds,V3t=∫t−δttfTesQ3fesds,V4t=∫−δ¯0∫t+θteTsW1esdsdθ+∫−δ¯0∫t+θte˙TsW2e˙sdsdθ,V5t=δ¯22∫−δ¯0∫β0∫t+θte˙TsW3e˙sdsdθ.
By differentiating the above LKF to obtain its time derivatives along with the trajectory of the MJNN defined as (7), we obtain(22)ℒV1=2eTtPe˙t+eTt∑j=1NπijPjet,(23)ℒV2=eTtQ1+Q2et−1−δDeTt−δt×Q1et−δt−eTt−δ¯Q2et−δ¯,(24)ℒV3=fTetQ3fet−fTet−δtQ3fet−δt,(25)ℒV4=δ¯eTtW1et−∫t−δ¯teTsW1esds+δ¯e˙TtW2e˙t−∫t−δ¯te˙TsW2e˙sds,(26)ℒV5=δ¯222e˙TtW3e˙t−δ¯22∫δ¯0∫t+θte˙TsW3e˙sdsdθ.
Utilizing Lemma2, we obtain(27)−∫t−δ¯teTsW1esds≤−1δ¯∫t−δ¯tesdsT×W1∫t−δ¯tesds−3δ¯Φ1TW1Φ1,where Φ1=∫t−δ¯tesds−2/δ¯∫−δ¯0∫t+θtesdsdθ.
By applying Lemma3, the following inequality can be written as(28)−∫t−δ¯te˙TsW2e˙sds≤etet−δ¯1δ∫t−δ¯tesds12δ¯2∫t−δ¯t∫t−δ¯sesdsdθT−9W23W236W2−2W2∗−9W2−30W25W2∗∗−192W230W2∗∗∗−5W2etet−δ¯1δ¯∫t−δ¯tesds12δ¯2∫t−δ¯t∫t−δ¯sesdsdθ.
By applying Lemma1, we obtain(29)−δ¯22∫δ¯0∫t+θtx˙TsW3x˙sdsdθ≤−Φ2Φ3TW3002W3Φ2Φ3,where Φ2=δ¯et−∫t−δ¯tesds and Φ3=−δ¯/2et−∫t−δ¯tesds+3/δ¯∫−δ¯0∫t+θtesdsdθ.
From Assumption1, we obtain(30)fkekt−Mk−ektfkekt−Mk−ekt≤0,fkekt−δt−Mk−ekt−δt×fkekt−δt−Mk−ekt−δt≤0,where,k=1,2,…,n.
This can be written algebraically as(31)etfetTMt−Mu∗Ietfet≤0,et−δtfet−δtTMt−Mu∗Iet−δtfet−δt≤0.
Then, the following inequality holds for any positive matricesUt=diagu1,u2,…,un and Uu=diagu^1,u^2,…,u^n:(32)etfetTMtUt−MuUt∗Utetfet≤0,(33)et−δtfet−δtTMtUu−MuUu∗Uuet−δtfet−δt≤0.
For convenience, consider a matrixN with appropriate dimension, and the following zero equality holds:(34)0=2eTt+e˙TtN−e˙Tt−Ai+KiCiet+KiDiet−δt+Bifet+Bdifet−δt+E1i−KiE2iwt.
Therefore, from (22)–(34), given that α>0, we obtain(35)ℒVet−αVxt−αwTtXwt≤ϑTtΨϑt<0,where(36)ϑTt=eTteTt−δteTt−δ¯e˙Tt∫t−δ¯tesdsT∫−δ¯0∫t+θtesdsdθTfTetfet−δtwTt.
We can write this as(37)ℒVet≤αVxt+αwTtXwt.
Multiplying both sides of (37) by e−αt and then integrating from 0 to t, where t∈0,T, we obtain(38)e−αtEVet≤EVe0+α∫0teαswTsXwsds.
It can be simplified as(39)EVet<eαtEVe0+α∫0teαswTsXwsds,(40)EVet<eαTEVe0+λ9d.
LetP¯i=R−1/2PiR−1/2, Q¯1=R−1/2Q1R−1/2, Q¯2=R−1/2Q2R−1/2, Q¯3i=R−1/2Q3R−1/2, W¯1=R−1/2W1R−1/2, W¯2=R−1/2W2R−1/2, and W¯3=R−1/2W3R−1/2.
Conversely,(41)EVe0,0=λmaxP¯ieT0Re0+λmaxQ¯1∫−δ00eTsResds+λmaxQ¯2∫−δ¯0eTsResds+λmaxQ¯3maxMt−,Mu+2×∫−δ00eTsResds+λmaxW¯1∫−δ¯0∫θ0eTsResdsdθ+λmaxW¯2∫−δ¯0∫θ0e˙TsRe˙sdsdθ+λmaxW¯3∫−δ¯0∫β0∫θ0e˙TsRe˙sdsdθ≤λmaxP¯+δ¯λmaxQ¯1+δ¯λmaxQ¯2+δ¯maxMt−,Mu+2λmaxQ¯3+δ¯22λmaxW¯1+δ¯22λmaxW¯2+δ¯512λmaxW¯3sup−δ¯≤s≤0eTsRes,e˙TsRe˙s,Vxt≤eαTΛc1+dλ9,where(42)Λ=λ2+δ¯λ3+δ¯λ4+δ¯maxMt−,Mu+2λ5+δ¯22λ6+δ¯22λ7+δ¯512λ8.
Note that(43)EVxt≥λminP¯ExTtRxt=λ1ExTtRxt.
Then, from (18), we obtain(44)ExTtRxt<c2.
Based on Definition2, an MJNN defined as (7) is SFTB.Corollary 1.
Given scalarsc1, c2, T, δ¯, μ, and α, an MJNN defined as (7) with wt=0 is SFTB in relation to c1,c2,T,R if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, and N, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal such that the following inequality holds:(45)Ψ1=ψi,j8×8<0,(46)eαTc1Λ<λ1c2,where ψij is defined in Theorem 1.Proof.
It can be proved in a similar way as Theorem1. The proof is omitted for brevity.


## 3.2. Finite-TimeH∞ State Estimation

Theorem 2.
Given scalarsc1, c2, T, δ¯, μ, d, and α, an MJNN defined as (7) is SFTB in relation to c1,c2,T,R,d with noise attenuation γ>0 if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, N, and X, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(47)Ψ¯=Ψ1Ψ^Gi∗ψ¯990∗∗−I<0,(48)eαTc1Λ+γ2d<λ1c2,where Ψ1=ψi,j8×8, Ψ^=colψi9,i=1,2,...,8,(49)ψ11=∑j=1NπijPj+Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NAi−AiTNT−LiCi−CiTLiT,ψ12=LDi,ψ13=3W2,ψ14=2Pi−N−AiTNT−CiTLiT,ψ15=36W2+δ¯W3+δ¯2W3,ψ16=−2W2−32W3,ψ17=MuUt+NBi,ψ18=NBdi,ψ19=NE1i−LiE2i,ψ22=−1−μQ1−MtUu,ψ23=0,ψ24=DiTLT,ψ25=0,ψ26=0,ψ27=0,ψ28=MuUu,ψ29=0,ψ33=−Q2−9W2,ψ34=0,ψ35=−30W2,ψ36=−5W2,ψ37=0,ψ38=0,ψ39=0,ψ44=δ¯W2+δ¯44W2−N−NT,ψ45=0,ψ46=0,ψ47=NBi,ψ48=NBdi,ψ49=NE1i−LiE2i,ψ55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ56=6δ¯2W1−30W2+3δ¯W3,ψ57=0,ψ58=0,ψ59=0,ψ66=−12δ¯3W1−5W2−9δ¯2W3,ψ67=0,ψ68=0,ψ69=0,ψ77=Q3−Ut,ψ78=0,ψ79=0,ψ88=−Q3−Uu,ψ89=0,ψ¯99=−γ2I.Proof.
In a similar way to the proof in Theorem1, we obtain(50)EℒVet+z¯Ttz¯t−γ2wTtwt<ϑTtΨ¯ϑt.
It can be deduced from (47) and (50) that(51)EℒVet+z¯Ttz¯t−γ2wTtwt<0.
By integrating (51) from 0 to T, we obtain(52)EVxt−Vxo+∫0Tz¯Ttz¯tdt−γ2∫0TwTtwtdt<0.
Subsequently, the following inequality is obtained:(53)E∫0Tz¯Ttz¯tdt≤γ2E∫0TwTtwtdt.
Hence, we conclude that the MJNN defined as (7) is SFTB.Remark 1.
Consider the following error system from the MJNN defined as (7), with wt=0 and without MJ parameters:(54)e˙t=−A+KCet−KDext−δt+Bfet+Bdfet−δt.Corollary 2.
Given scalarsδ¯ and μ, the error system (54) with wt=0 is said to be stable if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, and N, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(55)Ψ¯=ψ¯i,j8×8<0,where(56)ψ¯11=Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NA−ATNT−LC−CTLT,ψ¯12=LD,ψ¯13=3W2,ψ¯14=2P−N−ATNT−CTLT,ψ¯15=36W2+δ¯W3+δ¯2W3,ψ¯16=−2W2−32W3,ψ¯17=MuUt+NB,ψ¯18=NBdi,ψ¯22=−1−μQ1−MtUu,ψ¯23=0,ψ¯24=DTLT,ψ¯25=0,ψ¯26=0,ψ¯27=0,ψ¯28=MuUu,ψ¯33=−Q2−9W2,ψ¯34=0,ψ¯35=−30W2,ψ¯36=−5W2,ψ¯37=0,ψ¯38=0,ψ¯44=δ¯W2+δ¯44W2−N−NT,ψ¯45=0,ψ¯46=0,ψ¯47=NB,ψ¯48=NBdi,ψ¯55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ¯56=6δ¯2W1−30W2+3δ¯W3,ψ¯57=0,ψ¯58=0,ψ¯66=−12δ¯3W1−5W2−9δ¯2W3,ψ¯67=0,ψ¯68=0,ψ¯77=Q3−Ut,ψ¯78=0,ψ¯88=−Q3−Uu.Proof.
Following similar ideas as in the proof of Theorem1. The proof is omitted for brevity.Remark 2.
Consider a NN from the MJNN defined as (7) with C=0, D=0, wt=0, and no MJ parameters:(57)e˙t=−Aet+Bfet+Bdfet−δt.Corollary 3.
Given scalarsδ¯ and μ, the error system (57) with wt=0 is said to be stable if there exist feasible matrices Pi>0, Qs>0, Ws>0 (s=1,2,3), Ut>0, Uu>0, and N, where Pi, Qs, and Ws are symmetric PD, and Ut>0 and Uu>0 are diagonal, such that the following inequality holds:(58)Ψ˜=ψ˜i,j8×8<0,where(59)ψ˜11=Q1+Q2+δ¯W1−9W2−δ¯W3−δ¯24w3−MtUt−NA−ATNT,ψ˜12=0,ψ˜13=3W2,ψ˜14=2P−N−ATNT,ψ˜15=36W2+δ¯W3+δ¯2W3,ψ˜16=−2W2−32W3,ψ˜17=MuUt+NB,ψ˜18=NBdi,ψ¯22=−1−μQ1−MtUu,ψ˜23=0,ψ˜24=0,ψ˜25=0,ψ˜26=0,ψ˜27=0,ψ˜28=MuUu,ψ˜33=−Q2−9W2,ψ˜34=0,ψ˜35=−30W2,ψ˜36=−5W2,ψ˜37=0,ψ˜38=0,ψ˜44=δ¯W2+δ¯44W2−N−NT,ψ˜45=0,ψ˜46=0,ψ˜47=NB,ψ˜48=NBdi,ψ˜55=−1δ¯W1−3δ¯W1−192W2−2W3,ψ˜56=6δ¯2W1−30W2+3δ¯W3,ψ˜57=0,ψ˜58=0,ψ˜66=−12δ¯3W1−5W2−9δ¯2W3,ψ˜67=0,ψ˜68=0,ψ˜77=Q3−Ut,ψ˜78=0,ψ˜88=−Q3−Uu.Proof.
It can be proved in a similar way to Theorem1. The proof is omitted for brevity.Remark 3.
The stability analysis of time-delay systems can be classified into two categories, i.e., delay-dependent stability criteria and delay-independent ones. Also, it is well known that delay-dependent stability criteria, which use the information on the size of time delays, are less conservative than delay-independent ones. Thus, more attention has been paid to the derivation of delay-dependent stability criteria for time-delay systems.Remark 4.
It is important to note that some pioneering works have been done on finite-timeH∞ state estimation for Markovian jump neural networks based on interval time-varying delay with simple LKF techniques. In [29], the authors studied finite-time boundedness for Markovian jump neural networks with L2 gain analysis. Authors in [26] formulated finite-time stabilization of uncertain neural networks. Exponential state estimation problem has been designed for Markovian jumping neural networks in [18]. The model consider in this present study is more practical than that proposed by [18, 26, 29], whereas in this paper, we consider finite-time H∞ state estimation problem with the combination of Markovian jump neural networks’ interval time-varying delay model, which is another advantage. However, the authors in [18, 26, 29] used some simple techniques in LKFs to solve the stability problems to those articles. A new LKF with double and triple integral terms and utilizing extended Writnger’s integral inequality (EWII) techniques has been proposed for the stochastically finite-time bounded analysis of Markovian jump system in this paper. Consider that some less conservative results can occur in our method and can be provided in the numerical example section with real-life examples. Hence, the results presented in this paper are essentially new.Remark 5.
Typically, finite time stability withH∞ control, state estimation approach, and interval time-varying delay is not simply applied to Markovian jump neural networks. Some research publications have handled such issues [17, 18, 26, 29]. As it is, the author utilized some elementary LKFs to deal with the stability problems in those articles. Novel LKF with EWII has been proposed; in addition, the developed stochastic stability criteria tested for feasibility of the benchmark problem to explore the real-world application in this paper. However, the desired control was completely studied for the considered neural network model with the real-world application problem (e.g., four-tank pumping system and network circuit), which is the principle commitment and inspiration of our work.


## 4. Numerical Examples

This section shows our results through some numerical examples on MJNNs with 2 operation modes to demonstrate the effectiveness of the proposed approach.Example 1.
Consider an MJNNs with MJ parameters (i=2):(60)A1=3003,B1=−0.70.30.6−0.8,Bd1=0.800.20.4,E11=0.10.9−0.60.2,C1=0.40.8−0.70.2,D1=−0.400−0.2,E21=0.40.600.8,G1=0.300.20.4,A2=2.5002.5,B2=−0.50.20.4−0.5,Bd2=0.800.20.4,E12=0.070.8−0.50.1,C2=0.50.7−0.80.1,D2=−0.600−0.24,E22=0.40.600.8,G2=0.200.10.2,π=4−4−33,δ¯=2.5,μ=0.3,d=0.02,T=5,c1=1,c2=4,α=0.0002.
The activation functions are given asMt=diag0,0andMu=diag1,1. By solving the LMIs in Theorem 2, we can obtain a feasible solution:(61)P1=180.9128184.5018184.5018198.2708,P2=191.9509182.5133182.5133179.4404,Q1=70.839021.999421.999415.5734,Q2=2.48632.42052.42053.2684,Q3=27.314019.008219.008214.7271,W1=1.74071.53471.53471.9210,W2=0.19520.15750.15750.1754,W3=0.04960.03500.03500.0324.
Then, we obtain state estimator gain matrices as(62)K1=2.84491.1379−1.6262−0.6468,K2=2.26270.9719−1.1607−0.4569.
Thus, the system is SFTB with the external disturbanceγ=0.90.
To demonstrate the capability of the proposed approach, we show the effectiveness of the theoretical results, as shown in Figures1–4 . Figure 1 demonstrates the MJ mode rt. Figures 2 and 3 show the behaviors of the error system and state estimation of the error system, respectively. Figure 4 illustrates that the state xt of the system converges to zero. Furthermore, the superiority of our theoretical results is demonstrated through the simulation result of xTtRxt in Figures 5 and 6. Therefore, the proposed MJNN (7) is STFB.Figure 1
Markovian jumping mode.Figure 2
Estimation errorse1t and e2t.
(a)(b)Figure 3
e(t) and its estimation.
(a)(b)Figure 4
State trajectories of the system.Figure 5
State trajectories ofxTTRxt.Figure 6
Evolution ofxTTRxt.Example 2.
Consider the NN (57) with the following parameters:(63)A=1.276900000.623100000.923000000.4480,B=−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015,Bd=0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775,Mt=diag0,0,0,0,Mu=diag0.1137,0.1279,0.7994,0.2368.
This example shows a comparison of the conservativeness in the stability condition concerning the results in [36–38]. The maximum allowable delay bound (MADB) of δ¯ for various μ can be calculated using the MATLAB LMI toolbox. The MADBs of δ¯ for some values of μ in Example 2 are summarized in Table 1. We found that the outcomes of our proposed method produced better results than the previous research [36–38].Table 1
The maximum delay upper boundsδ¯ for given μ in Example 2.
Method0.50.80.9[36]3.69542.77112.5795[37]3.87093.34423.1291[38]4.27493.19932.9504Corollary34.50083.86213.5402Example 3.
Consider the NN (57) with the following parameters:(64)A=2003.5,B=−10.50.5−1,Bd=−0.50.50.50.5,Mt=0000,Mu=1001.
For some values ofμ, the MADBs of δ¯ are obtained and summarized in Table 2. We compare these results with those of previous studies [36, 39, 40]. As shown in Table 2, the MADB are larger than those obtained from [36, 39, 40]. It shows the superiority that the proposed stability criterion is less conservative than the previous works.Table 2
The maximum allowable delay upper boundsδ¯ for given μ in Example 3.
Method0.80.9[36]0.87840.8484[39]0.88410.8570[40]0.96310.9324Corollary31.36801.1035Remark 6.
We calculated upper bounds with different delta, and they are listed in Tables1 and 2. We provide comparisons with the results obtained in previous studies to show the improvements obtained by our proposed method.Example 4.
The NNs have similar characteristics to the neurons in a biological organism, leading to the nervous system. The NNs can represent not only the nervous systems with neurons but also the engineering systems such as the four-tank water pumping system, as shown in Figure7. The four-tank water pumping system is equipped with 2 water pumps and 4 interconnected tanks with two valves. Voltage ν1 and ν2 are two input processes of two supplying pumps. The four-tank water pumping system can be modeled as a neural network model. Previous studies in [41–43] suggested the state-space equations of this four-tank system which is an application of the neural networks. State feedback controller modeled as follows:(65)x^˙t=A^0x^t+A^1x^t−δ1+B^0u^t−δ2+B^1u^t−δ3,where(66)A^0=−0.00210000−0.00210000−0.04240000−0.0424,A^1=000.042400000.042400000000,B^0=0.1113γ100000.10421−γ200,B^1=0000.11131−γ1000.10421−γ20,γ1=0.333,γ2=0.307,u^=K^x^t,K^=−0.1609−0.1765−0.0795−0.2073−0.1977−0.1579−0.2288−0.0772.
Another control problem of our interests is obtained by adding transport delaysδt through delaying the inlet of incoming water into the tanks. Hence, the proposed approach has been used to study this problem here. Time-varying transport delays between valves and tanks have also been considered in the previous works, but they have not been considered the following aspects. For simplicity, it was assumed that δ1=0, δ2=0, and δ3=δt (since δt≤δ). In this example, the control input u^t indicates the amount of water pumped. Therefore, it is naturally a nonlinear function and can be written as follows:(67)u^t=K^f^x^t,f⌢x^t=f^1x^1t,….,f^4x^4tT,f⌢ix^it=0.1x^it+1−x^it−1,i=1,2,…,4.
The four-tank system (65) can be rewritten to the form of system (57) with K^=1 as follows:(68)e˙t=−Aet+Bfet+Bdfet−δt,where A=A^0−A^1, B=B^0K^, Bd=B^1K^, and f⋅=f^⋅. In addition, Mt=diag0,0,0,0 and Mu=diag0.1,0.1,0.1,0.1 with δ=6.5 and μ=0.5. Using MATLAB LMI toolbox and solving the inequalities in Corollary 2, we are able to obtain feasible solution, which lead to a conclusion that FTPS (68) is stable.Figure 7
Schematic representation of the four-tank water pumping system.Example 5.
A continuous-time artificial NN containingn units can be described as the following well-known differential equations in [44]:(69)deitdt=−eitRiCi+∑j=1nWijyjt+uit,yit=fieit.
This nonlinear system is implemented by a simple resistance-capacitance (RC) network circuit. It is shown in Figure8, where ui=ei and Vi=fieit are input and output voltage of the ith amplifier, where Vi and −Vi are two output terminals of the ith amplifier, and the value Ri is defined as(70)1Ri=1σi+∑j=1n1Rij,Wij=+1Rij,Rijis connected toVj,−1Rij,Rijis connected to−Vj.
Thus, nonlinear system (69) can be rewritten in the following form:(71)e˙t=−Aet+Bfet+Bdu,with(72)A=1R1C10⋯001R2C2⋯0⋮⋮⋱⋮00⋯1RnCn,B=W11C1W12C1⋯W1nC1W21C21W21/C2⋯W2nC2⋮⋮⋱⋮Wn1Cn0⋯WnnCn,Bd=1C10⋯001C2⋯0⋮⋮⋱⋮00⋯1Cn.
The productRiCi  =  δi,i=1,2,…,n, is called as the time constant of the ith neuron. An identical time constant for each neuron would require, that is, Ci=C and Ri=R, for all i. In this case, every individual value for δi would have to be chosen in a way that compensates for Ci and Ri. It is important to note that the time constant δi describes the convergence of the neural state ei of the ith neuron. Because of the high-level gain of the transfer function, the output Vi might be saturated very fast. Thus, even if the state ei is still far from reaching its equilibrium point, the output Vi might already be saturated, and by observing only Vi, it might appear as if the circuit had converged in merely a fraction of the time constant δi.
Consider the delayed neural networks (71) with the following parameters: n=2,C1=C2=R1=R2=w11=1,w22=−1,w12=1.5,w21=−1.5, and s1=s2=0. The neural network equations are, therefore, described as(73)de1tdt=−e1t+fe1t+1.5fe2t,de2tdt=−e2t−1.5fe2t−fe2t,and fet=f1e1t…fnentT∈ℝn, which satisfy(74)la−≤fax1−fax2x1−x2≤la+,∀x1,x2∈ℝ,x1≠x2,
We can choose the valuefekt=tanhet, which implies l1=0 and l2=0.5I, and we now apply Corollary 3 to system (71) by choosing μ=0.5 and δ=0 and Bd=0. Then, we can get (58) is feasible. Figure 9 shows the state responses of the system with the interval 1,−0.5T. Thus, the neural network (71) is asymptotically stable.Figure 8
Circuit diagram for delayed NN.Figure 9
Circuit diagram for delayed NN.


## 5. Conclusion

Herein, we studied the SFTB of MJNNs with time-varying delays. Using an LKF with Wirtinger’s integral inequalities, a sufficient condition was derived such that the MJNNs were SFTB and satisfied a prescribed level ofH∞ disturbance attenuation in a finite-time interval. We illustrated the effectiveness of our main results with five numerical examples. We also compared to show that our results are less conservative than some existing ones. Future works focus on the discrete versions of these inequalities and their applications. [45–47].



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