# State Estimation for Standard Neural Network Models with Time-Varying Delays

Authors: Jin Zhu; Tai-Fang Li; Huanqing Wang
Journal: Complexity
 (2022)

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## Abstract

The paper deals with the issue of state estimation for standard neural network models with time-varying delays. A new augmented vector with the derivative of the state is introduced in the Lyapunov–Krasovskii functional. The state estimation criteria are obtained by constructing the suitable Lyapunov–Krasovskii functional; meanwhile, the observer gain and the controller gain are derived in terms of linear matrix inequality. The free matrix-based integral inequality is utilized to handle the integral terms, and the zero equation is added to the derivative of the Lyapunov–Krasovskii functional, which decreases the conservatism. The effectiveness and feasibility of the proposed methods are demonstrated by two numerical examples.


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## Body

## 1. Introduction

In the past decades, recurrent neural networks (RNNs) have been investigated extensively due to their wide applications, such as pattern recognition, signal processing, combinatorial optimization, and so on [1–5]. In addition, time delays are inevitably existing due to the limited switching speed of the amplifiers and the transmission time between neurons, which may cause oscillation and instability of systems [6–8]. The dynamic output feedback control problem has been discussed in [6]. The dynamic compensator and the output feedback controller were designed for a class of multiple time-delay systems. In [7], the issue of the output feedback-based leader-following consensus was investigated. The proposed consensus algorithms can handle the consensus problem of nonidentical nonlinear multiagent systems. The leader-following consensus issue for nonlinear time-delay multiagent systems was studied in [8]. The output feedback distributed consensus control algorithm has been proposed, which can be applied to the consensus problem of the chemical reactor systems. The delay-dependent stability analysis of recurrent neural networks has been researched widely [9–28] and the references therein.On the other hand, the neuron states are unknown or not completely available in the network outputs. Therefore, it is essential to estimate the neuron state through the available output measurements. Recently, the delay-dependent state estimation problem for neural networks has received much significant achievements [29–37]. In [29], delay-dependent state estimation criteria were obtained by employing free matrix-based integral inequality. The criteria were proposed in terms of matrix inequality, not an linear matrix inequality(LMI). The L2-L∞ performance state estimation for neural networks was studied in [30, 31]. The criteria were given to guarantee error system is asymptotically stable; meanwhile, the estimator gain and L2-L∞ performance index have been obtained. The extended dissipative state estimation issue was investigated in [32–34]. The system considered is generalized neural networks in [32, 33]. In [34], the existences of the estimator gain matrices for static neural networks were investigated by constructing a new delay-product-type functional. In [35], the problem of the asynchronous controller design has been presented for discrete-time periodic singularly perturbed Lur’e systems. The hidden Markov model detector is applied to observe the fading channel model. State estimation for switched memristive neural networks with nonhomogeneous sojourn probabilities has been investigated in [36]. The dynamic event-triggered H∞ state estimation was proposed to alleviate the computation burden by using a higher-level deterministic switching signal. In [37], the issue of asynchronous proportional-integral observer for singularly perturbed complex networks has been studied. A nonhomogeneous Markov switching topology was investigated to represent the asynchronous situation. In [38], the authors introduced a novel model named delayed standard neural network models (DSNNMs). The DSNNM represents a neural network model as the interconnection of a linear dynamic system and static delayed nonlinear operators consisting of bounded activation functions. Most recurrent neural networks or nonlinear system modeled by neural networks can be transformed into the DSNNM. To the best of authors’ knowledge, the problem of state estimation has not been touched for delayed standard neural network models.According to the above analysis, the state estimation problem for time-varying delays in standard neural network models is investigated. Firstly, the state estimation problem is discussed for constant delay standard neural network models without input. Secondly, the state-feedback controller is designed such that the close-loop error system is asymptotically stable. The observer gain and the controller gain are derived by solving LMI. Then, the state estimation criteria with less conservatism for standard neural network models with time-varying delay are derived by employing the orthogonal polynomials-based integral inequality. Finally, two numerical examples are provided to illustrate the feasibility and improvement of the proposed method.


## 2. System Description and Preliminaries

The standard neural network models [38] with time-varying delays can be described in the following form:(1)x˙t=A1xt+B1fξt+C1fξt−dt+D1ut,ξt=A2xt+B2fξt+C2fξt−dt+D2ut,yt=A3xt+B3fξt+C3fξt−dt+D3ut,where n is the number of neurons in neural networks, xtϵRn is the neuron state vector, ξtϵRL is the input of nonlinear operator f, utϵRm is the external input, ytϵRr is the measurement output, dt is the transmission delay that satisfies d1≤dt≤d2, d1 and d2 are nonnegative constants, 0<d˙t≤μ, A1ϵRn×n, B1ϵRn×L, C1ϵRn×L, D1ϵRn×m, A2ϵRL×n, B2ϵRL×L, C2ϵRL×L, D2ϵRL×m, A3ϵRr×n, B3ϵRr×L, C3ϵRr×L, and D3ϵRr×m are the corresponding state-space matrices, fϵCRL;RL, satisfies f0=0, LϵR is the number of nonlinear activation functions, (1) with the initial condition function: fξt0+θ=fξt0,∀θϵ−d2,0. Throughout this paper, we always assume that the activation functions satisfy the sector condition:(2)ti−≤fiξitξit≤ti+ti+>ti−≥0.Some useful Lemmas are introduced.Lemma 1 (see [14]).
(Orthogonal polynomials-based integral inequality) Letωs be a differentiable function ω:a,b⟶Rn, for matrices RϵSn+, FiϵRk×ni=1,2,3, and any vector ξϵRk, the following inequality holds:(3)−∫αβω˙TsRω˙sds≤ξT∑i=13SymFiEi+∑i=13β−α2i−1FiR−1FiTξ,where(4)E1ξ=ωβ−ωα,E2ξ=ωβ+ωα−2β−α∫αβωsds,E3ξ=ωβ−ωα+6β−α∫αβωsds−12β−α2∫αβ∫sβωududs,Lemma 2 (see [15]).
LetU, V, W, and M=MT be real matrices with appropriate dimension, then,(5)M+UVW+WTVTUT<0,for all VTV≤I, if and only if there exists a scalar ϵ>0, such that(6)M+ϵ−1UUT+ϵWTW<0.


## 3. Main Results

Firstly, we deal with the problem of state estimation for system (1) with ut=0. In order to estimate the state of neural networks by using the available network output, we construct the state estimator:(7)x^˙t=A1x^t+B1fξ^t+C1fξ^t−dt+Ly˜t,ξ^t=A2x^t+B2fξ^t+C2fξ^t−dt,y^t=A3x^t+B3fξ^t+C3fξ^t−dt,where x^t is the estimation of the state xt, y^t is the estimation of the output yt, LϵRn×r is the observer gain, and KϵRm×n is the controller gain. The system (1) with full-order state estimator (7) can be described:(8)e˙t=A1−LA3et+B1−LB3gξ˜t+C1lgξ˜t−dt,ξ˜t=A2et+B2gξ˜t+C2gξ˜t−dt,(9)et=xt−x^t,y˜t=yt−y^t,ξ˜t=ξt−ξ^t,gξ˜t=fξt−fξ^t,C1l=C1−LC3.Theorem 1.
The error system (8) is asymptotically stable for dt=d, if there exist positive definite matrices P11, P22, Q11, Q22, U11, U22, R, diagonal positive definite matrix Λ, any matrices P12, Q12, U12, X, M1j, M2j, M3j, Nj1, Nj2, Yj1, Yj2, j=1,2,…,7 such that the LMI (10) holds.(10)Ψ=Ω11Ω12Ω13Ω17Ω18Ω19∗Ω14Ω15Ω20Ω21Ω22∗∗Ω16Ω23Ω24Ω25∗∗∗Ω26Ω27Ω27∗∗∗∗Ω28Ω29∗∗∗∗∗Ω30<0,where(11)Ω11=Λ11Λ12Q12+M13T+M23T+M33T+N32T+Y32T+A2TT1+T2∗Λ14−M13T+M23T−M33T−N32T+Y32T∗∗Q22+B2TT1+T2Λ−Λ−ΛT+ΛT1+T2B2,Ω12=M14T+M24T+M34T+N42T+Y42TΛ18−Q12−M14T+M24T−M34T−N42T+Y42TΛ20ΛT1+T2C2Λ22,Ω13=M16T+M26T−12d2M31+M36T+N62T+2dY11+Y62TΛ24−M16T+M26T−12d2M32−M36T−N62T+2dY21+Y62TΛ26−12d2M33+2dY31Λ28,Ω14=−Q22−2dM24+6dM34+N41−Y41−2dY42∗Λ31,Ω15=−12d2M34+2dY41C1−LC3TXTΛ34Λ35,Ω16=−12d2M36T−12d2M36+2dY61+2dY61T−12d2M37T+2dY71T∗dU22+dR−X−XT,Ω17=M11M21M31M12M22M32M13M23M33,Ω18=N11N12N21N22N31N32,Ω19=Y11Y12Y21Y22Y31Y32,Ω20=M14M24M34M15M25M35,Ω21=N41N42N51N52,Ω22=Y41Y42Y51Y52,Ω23=M16M26M36M17M27M37,Ω24=N61N62N71N72,Ω25=Y61Y62Y71Y72,Ω26=−diag1dR,3dR,5dR,Ω27=O3×2,Ω28=−1dU,Ω29=O2×2,Ω30=−3dU,Λ11=P12+P12T+Q11+M11+M11T+M21+M21T+M31+M31T+dU11+N12+N12T+Y12+Y12T,Λ12=−P12−M11+M12T+M21+M22T−M31+M32T−N12+N22T+Y12+Y22T,Λ14=−Q11−M12−M12T+M22+M22T−M32−M32T−N22−N22T+Y22+Y22T,Λ18=P22+M15T−2dM21+M25T+6dM31+M35T+N11+N52T−Y11−2dY12+Y52T,Λ20=−P22T−2dM22−M15T+6dM32+M25T+N21−M35T−Y21−N52T−2dY22+Y52T,Λ22=−2dM23+6dM33+N31−Y31−2dY32,Λ24=P11+M17T+M27T+M37T+dU12+N72T+Y72T+A1−LA3TXT,Λ26=−M17T+M27T−M37T−N72T+Y72T,Λ31=−2dM25−2dM25T+6dM35+6dM35T+N51+N51T−Y51−Y51T−2dY52−2dY52T,Λ34=−2dM26T−12d2M35+6dM36T+N61T+2dY51−Y61T−2dY62T,Λ35=P12T−2dM27T+6dM37T+N71T−Y71T−2dY72T,T−=diagt1−,t2−,…,tL−,T+=diagt1+,t2+,…,tL+.The proofof theorem 1 is detailed in Appendix A.Now, the following observer-based state-feedback controller is proposed to stabilize the system (1):(12)ut=Kx^t.The system (1) with full-order state estimator (7) can be described:(13)e¯˙t=A¯1e¯t+B¯1g¯ξ¯t+C¯1g¯ξ¯t−dt,ξ¯t=A¯2e¯t+B¯2g¯ξ¯t+C¯2g¯ξ¯t−dt,(14)A¯1=A1+D1K−D1K0A1−LA3,B¯1=B100B1−LB3,C¯1=C100C1−LC3,A¯2=A2+D2K−D2K0A2,B¯2=B200B2,C¯2=C200C2,e¯t=xtet,ξ¯t=ξtξ˜t,g¯ξ¯t=fξtgξ˜t.Theorem 2.
If there exist positive definite matricesP^11, P^22, P^33, P^44, Q^11, Q^22, Q^33, Q^44, U^11, U^22, U^33, U^44, S, diagonal positive definite matrix G1, G2, any matrices P^12, P^13, P^14, P^23, P^24, P^34, Q^12, Q^13,Q^14,Q^23, Q^24, Q^34, U^12, U^13,U^14,U^23, U^24, U^34, M^ij, N^jl, Y^jl, i=1,2,…,6,j=1,2,…,7,l=1,2,…,8, such that the inequality (15) hold(15)Ψ=Ω¯11Ω¯12Ω¯13Ω¯17Ω¯18Ω¯19Ω¯41∗Ω¯14Ω¯15Ω¯20Ω¯21Ω¯22Ω¯42∗∗Ω¯16Ω¯23Ω¯24Ω¯25Ω¯43∗∗∗Ω¯26Ω¯27Ω¯27Ω¯51∗∗∗∗Ω¯28Ω¯29Ω¯52∗∗∗∗∗Ω¯30Ω¯52∗∗∗∗∗∗Ψ¯6<0.
Then, the system (13) is asymptotically stable for dt=d, the observer gain matrix is L=1/λL¯, and the controller gain matrix K=K¯S−1.(16)Ω¯11=Λ¯11Λ¯12Q¯12+M¯13T+M¯23T+M¯33T+N¯32T+Y¯32T+Θ¯2∗Λ¯14−M¯13T+M¯23T−M¯33T−N¯32T+Y¯32T∗∗Q¯22+GB¯2TT+TB¯2TG−G−GT,Ω¯12=M¯14T+M¯24T+M¯34T+N¯42T+Y¯42T+Θ¯3Λ¯18−Q¯12−M¯14T+M¯24T−M¯34T−N¯42T+Y¯42TΛ¯20TC¯2GΛ¯22,Ω¯13=M¯16T+M¯26T−12d2M¯31+M¯36T+N¯62T+2dY¯11+Y¯62TΛ¯24−M¯16T+M¯26T−12d2M¯32−M¯36T−N¯62T+2dY¯21+Y¯62TΛ¯26−12d2M¯33+2dY¯31Θ¯5,Ω¯14=−Q¯22−2dM¯24+6dM¯34+N¯41−Y¯41−2dY¯42∗Λ¯31,Ω¯15=−12d2M¯34+2dY¯41Θ¯6−2dM¯26T−12d2M¯35+6dM¯36T+N¯61T+2dY¯51−Y¯61T−2dY¯62TΛ¯35,Ω¯16=−12d2M¯36T−12d2M¯36+2dY¯61+2dY¯61T−12d2M¯37T+2dY¯71T∗dU¯22+dR¯−X¯−X¯T,Ω¯17=M¯11M¯21M¯31M¯12M¯22M¯32M¯13M¯23M¯33,Ω¯18=N¯11N¯12N¯21N¯22N¯31N¯32,Ω¯19=Y¯11Y¯12Y¯21Y¯22Y¯31Y¯32,Ω¯20=M¯14M¯24M¯34M¯15M¯25M¯35,Ω¯21=N¯41N¯42N¯51N¯52,Ω¯22=Y¯41Y¯42Y¯51Y¯52,Ω¯23=M¯16M¯26M¯36M¯17M¯27M¯37,Ω¯24=N¯61N¯62N¯71N¯72,Ω¯25=Y¯61Y¯62Y¯71Y¯72,Ω¯26=−diag1dR¯,3dR¯,5dR¯,Ω¯27=O3×2,Ω¯28=−1dU¯,Ω¯29=O2×2,Ω¯30=−3dU¯,Ω¯51=O6×6,Ω¯52=O4×6,Ω¯41=000000S000000000000000000000000G2T0000,Ω¯42=000000S0G2T000000000000000,Ω¯43=000000S0G2T000000000000λA1−LA3λB1−LB3λC1−LC3,Λ¯11=P¯12+P¯12T+Q¯11+M¯11+M¯11T+M¯21+M¯21T+M¯31+M¯31T+dU¯11+N¯12+N¯12T+Y¯12+Y¯12T−Θ¯1,Λ¯12=−P¯12−M¯11+M¯12T+M¯21+M¯22T−M¯31+M¯32T−N¯12+N¯22T+Y¯12+Y¯22T,Λ¯14=−Q¯11−M¯12−M¯12T+M¯22+M¯22T−M¯32−M¯32T−N¯22−N¯22T+Y¯22+Y¯22T,Λ¯18=P¯22+M¯15T−2dM¯21+M¯25T+6dM¯31+M¯35T+N¯11+N¯52T−Y¯11−2dY¯12+Y¯52T,Λ¯20=−P¯22T−2dM¯22−M¯15T+6dM¯32+M¯25T+N¯21−M¯35T−Y¯21−N¯52T−2dY¯22+Y¯52T,Λ¯22=−2dM¯23+6dM¯33+N¯31−Y¯31−2dY¯32,Λ¯24=P¯12+M¯17T+M¯27T+M¯37T+dU¯12+N¯72T+Y¯72T+Θ¯4,Λ¯26=−M¯17T+M¯27T−M¯37T−N¯72T+Y¯72T,Λ¯31=−2dM¯25−2dM¯25T+6dM¯35+6dM¯35T+N¯51+N¯51T−Y¯51−Y¯51T−2dY¯52−2dY¯52T,Λ¯35=P¯12T−2dM¯27T+6dM¯37T+N¯71T−Y¯71T−2dY¯72T,P¯11=P^11P^12P^12TP^22,P¯12=P^13P^14P^23P^24,P¯22=P^33P^34P^34TP^44,Q¯11=Q^11Q^12Q^12TQ^22,Q¯12=Q^13Q^14Q^23Q^24,Q¯22=Q^33Q^34Q^34TQ^44,U¯11=U^11U^12U^12TU^22,U¯12=U^13U^14U^23U^24,U¯22=U^33U^34U^34TU^44,Θ¯1=A¯1k+A¯1kTA¯1kT−D1K¯∗−D1K¯−K¯TD1T,Θ¯3=C1G10C1G10,Θ¯2=A¯2kTT¯+B1G10−K¯TD2TT¯+B1G1STA2TT¯,Θ¯5=G1TB1T000,Θ¯4=A¯1kT−S0−D1K¯T−S0,Θ¯6=G1TC1T000,X¯=S00S,R¯=R^100R^2,G¯=G100G2,N¯j1=N^j1N^j2N^j3N^j4,N¯j2=N^j5N^j6N^j7N^j8,Y¯j1=Y^j1Y^j2Y^j3Y^j4,Y¯j2=Y^j5Y^j6Y^j7Y^j8,M¯ij=M^2i−12j−1M^2i2j−1M^2i−12jM^2i2j,i=1,2,3,j=1,2,…,7.
The detailed proofof Theorem 2 can be seen in Appendix B.Theorem 3.
The error system (9) is asymptotically stable for given scalars d2≥d1≥0, μ, and d¯=d2−d1, if there exist positive definite matrices P11, P22, Q11, Q22, U11, U22, R, diagonal positive definite matrix Λ, any matrices P12, Q12, U12, X, Eij, E¯ij, Flj, F¯lj, Xlj, X¯lj, i=1,2,3, j=1,2,…,10, l=1,2, such that LMIs (17) and (18) hold(17)Υ=Ξ11Ξ¯12Ξ14Ξ15∗Ξ¯13Ξ16Ξ17∗∗Ξ18Ξ19<0,(18)Υ¯=Ξ11Ξ¯12Ξ14Ξ15∗Ξ¯13Ξ16Ξ17∗∗Ξ18Ξ19<0,(19)Ξ11=Π1Π2Π3Π4Π52X26+X¯26T−2A2TT1ΛT2C2∗Π7Π8Π9Π10Π11∗∗Π12Π13Π14−E16T+E26T+E36T+X26T−F26T∗∗∗Π16Π17E¯16T+E¯26T+E¯36T−X¯26T+F¯26T∗∗∗∗Π19ΛT1+T2C2−2B2TT1ΛT2C2∗∗∗∗∗−1−μQ22−2C2TT1ΛT2C2,Ξ12=Π22Π23Π24Π25P11+2X211T+2X¯211T+d¯U12+A1lTXTΠ27Π28Π29Π30Π31Π32Π33Π34Π35−E111T+E211T+E311T+X211T−F211TΠ37Π38Π39Π40E¯111T+E¯211T+E¯311T−X¯211T+F¯211TΠ42Π43Π44Π45B1lTXTΠ47Π48Π49Π50C1lTXT,Ξ13=Π52Π53Π54Π55Π56∗Π57Π58Π59−2E¯211T+6E¯311T−2X¯211T∗∗Π61Π62−12E311T+2d¯X111T∗∗∗Π64−12E¯311T∗∗∗∗d¯R+d¯U22−X−XT,Ξ14=d¯E11E21E31E12E22E32E13E23E33E14E24E34E15E25E35E16E26E36,Ξ15=d¯F11F21X11X21F12F22X12X22F13F23X13X23F14F24X14X24F15F25X15X25F16F26X16X26,Ξ16=d¯E17E27E37E18E28E38E19E29E39E110E210E310E111E211E311,Ξ17=d¯F17F27X17X27F18F28X18X28F19F29X19X29F110F210X110X210F111F211X111X211,Ξ18=d¯−R000−3R000−5R000000000000,Ξ19=−d¯000000000000U11U1200U12TU2200003U113U12003U12T3U22,Ξ¯12=Π¯22Π23−d¯X¯11+d¯F¯11Π24−2d¯X11Π25+2d¯X¯11Π26Π¯27Π¯28Π29−2d¯X12Π30+2d¯X¯12Π31Π¯32Π33−d¯F13+d¯F¯13Π34−2d¯X13Π35+2d¯X¯13Π36Π¯37Π¯38Π39−2d¯X14Π40+2d¯X¯14Π41Π¯42Π43−d¯X¯15+d¯F¯15−12E35−12E¯35+2d¯X¯15Π46Π¯47Π48−d¯X¯16+d¯F¯16−12E36Π50+2d¯X¯16Π51,Ξ¯13=Π¯52Π¯53Π¯54Π55+d¯X110T−d¯F110T+2d¯X¯17TΠ¯56∗Π¯57Π¯58Π59−d¯X¯110T+d¯F¯110T+2d¯X¯18TΠ¯60∗∗Π¯61−12E¯39−12E310T+6E37+2d¯X¯19Π¯63∗∗∗−12E¯310−12E¯310T+2d¯X¯110+2d¯X¯110TΠ65∗∗∗∗Π66,Π1=Q11+2X21+2X21T+2X¯21+2X¯21T−2A2TT1ΛT2A2+d¯U11,Π2=E11+E21+E31−E¯11+E¯21+E¯31−X21+2X22T+X¯21+X¯22T+F21−F¯21,Π3=−E11+E21+E31+X21+2X23T+2X¯23T−F21−P12,Π4=E¯11+E¯21+E¯31−X¯21+2X24T+2X¯24T+F¯21+P12,Π5=Q12+2X25+X¯25T+Y32T+A2TT1+T2Λ−2A2TT1ΛT2B2,Π7=−1−μQ11+E12+E12T+E22+E22T+E32+E32T−E¯12−E¯12T+E¯22+E¯22T+E¯32+E¯32T−X22−X22T+X¯22+X¯22T+F22+F22T−F¯22−F¯22T,Π8=−E12+E13T+E22+E23T+E32+E33T−E¯13T+E¯23T+E¯33T+X22−X23T+X¯23T−F22+F23T−F¯23T,Π9=E14T+E24T+E34T+E¯12−E¯14T+E¯22+E¯24T+E¯32+E¯34T−X24T−X¯22−F¯24T+X¯24T+F24T+F¯22,Π10=E15T+E25T+E35T−E¯15T+E¯25T+E¯35T−X25T+X¯25T+F25T−F¯25T,Π11=−1−μQ11+E16T+E26T+E36T−E¯16T+E¯26T+E¯36T−X26T+X¯26T+F26T−F¯26T,Π12=−E13−E13T+E23+E23T+E33+E33T+X23+X23T−F23−F23T,Π13=−E14T+E24T+E34T+E¯13−E¯23T+E¯33+X24T−X¯23−F24T+F¯23,Π14=−E15T+E25T+E35T+X25T−F25T,Π16=E¯14+E¯14T+E¯24+E¯24T+E¯34+E¯34T−X¯24−X¯24T+F¯24+F¯24T,Π17=E¯15T+E¯25T+E¯35T−X¯25T+F¯25T,Π19=Q22+2ΛT1+T2B2−2Λ−2B2TT1ΛT2B2,Π22=−2E21+6E31−2X21+2X27T+2X¯27T−d¯X11+d¯F11,Π23=−2E¯21+6E¯31+2X28T−2X¯21+2X¯28T,Π24=−12E31+2X29T+2X¯29T+2d¯X11,Π25=−12E¯31+2X210T+2X¯210T,Π27=E17T−2E22+E27T+6E32+E37T−E¯17T+E¯27T+E¯37T−2X22−X27T+X¯27T−d¯X12+F27T−d¯F12−F¯27T,Π28=E18T+E28T+E38T−E¯18T+E¯28T+E¯38T−2E¯22−X28T+6E¯32−2X¯22+X¯28T+F28T−F¯28T,Π29=E19T+E29T+E39T−12E32−E¯19T+E¯29T+E¯39T−X29T+2d¯X12−2X¯22+X¯29T−F¯29T,Π30=E110T+E210T+E310T−E¯110T+E¯210T+E¯310T−12E¯32−X210T+X¯210T+F210T−F¯210T,Π31=E111T+E211T+E311T−E¯111T+E¯211T+E¯311T−X211T+X¯211T+F211T−F¯211T,Π32=−E17T+E27T+E37T−2E23+6E33−2X23+X¯27T−d¯X13−F27T+d¯F13−d¯P22T,Π33=−E18T+E28T+E38T−2E¯23+6E¯33−2X¯23+X28T−F28T+d¯F13−d¯P22T,Π34=−E19T+E29T+E39T−12E33+X29T+2d¯X13−F29T,Π35=−E110T+E210T+E310T−12E¯33+X210T−F210T,Π37=−2E24+6E34+E¯17T+E¯27T+E¯37T−2X24−X¯27T−d¯X14+d¯F14+d¯P22T+F¯27T,Π38=E¯18T+E¯28T+E¯38T−2E¯24+6E¯34−X¯28T−2X¯24+F¯28T,Π39=−12E34+E¯19T+E¯29T+E¯39T−X¯29T+2d¯X14+F¯29T,Π40=−12E¯34+E¯110T+E¯210T+E¯310T−X¯210T+F¯210T,Π42=−2E25+6E35−d¯X15+d¯F15−2X25,Π43=−2E¯25+6E¯35−2X¯25,Π44=−12E35+2d¯X15,Π45=−12E¯35,Π46=B1lTXT,Π47=−2E26+6E36−d¯X16+d¯F16−2X26,Π48=−2E¯26+6E¯36−2X¯26,Π49=−12E36+2d¯X16,Π52=−2E27−2E27T+6E37+6E37T−2X27−2X27T−d¯X17−d¯X17T+d¯F17+d¯F17T,Π53=−2E28T+6E38T−2E¯27+6E¯37−2X28T−2X¯27T−d¯X18T+d¯F18T,Π54=−2E29T−12E37+6E39T−2X29T+2d¯X17−d¯X19T+d¯F19T,Π55=−2E210T+6E310T−12E¯37−2X210T−d¯X110T+d¯F110T,Π56=−2E211T+6E311T−2X211T−d¯X111T+d¯F111T+d¯P12T,Π57=−2E¯28−2E¯28T+6E¯38+6E¯38T−2X¯28−2X¯28T,Π58=−12E38−2E¯29T+6E¯39T−2X¯29T+2d¯X18T,Π59=−12E¯38−2E¯210T+6E¯310T−2X¯210T,Π64=−12E¯310−12E¯310T,Π¯22=Π22+d¯X11−d¯F11,Π¯27=Π27+d¯X12−F27T+d¯F12,Π¯28=Π28−d¯X¯12+F27T−d¯F¯12,Π¯32=Π32+d¯X13+F27T−d¯F13+d¯P22T,Π¯37=Π32+d¯X14−d¯F14−d¯P22T,Π¯38=Π38−d¯X¯14+d¯F¯14+d¯P22T,Π¯42=Π42+d¯X15−d¯F15,Π¯47=Π47+d¯X16−d¯F16,Π¯52=Π52+d¯X17+d¯X17T−d¯F17−d¯F17T,Π¯53=Π53+d¯X18T−d¯F18T−d¯X17T+d¯F¯17T,Π¯54=Π54−2d¯X17+d¯X19T−d¯F19T,Π¯56=−2E211T+6E311T−2X211T,Π¯57=Π57−d¯X¯18T+d¯F¯18T−d¯X¯18−d¯F¯18,Π¯58=Π58−2d¯X18−d¯X¯19+d¯F¯19,Π¯60=−2E¯211T+6E¯311T−2X¯211T−d¯X¯111T+d¯F¯111T+d¯P12T,Π¯61=6E37−12E39−12E39T,Π¯64=−12E¯310−12E¯310T+2d¯X¯110+2d¯X¯110T.The proof is similar to that of Theorem1, and hence, it is omitted.Remark 1.
In most existing results, the observer gainL is usually designed first and then the controller gain can be solved based on the value of the observer gain. However, in this paper, if we take X=S1 in non-LMI (B2) in Theorem 2 and pre- and postmultiplying the left-hand side matrix of (B2) by Σ−1, the equivalent LMI (15) can be obtained, in which the observer gain L and the controller gain K can be solved simultaneously. This not only simplifies the design process but also decreases the computation complexity in a great extent.Remark 2.
According to Theorem3, the asymptotically stable criterion for the time-varying delay standard neural network models can be obtained if the state estimator gain L=0. The stability criterion was with less conservatism by introducing the derivative of the state in the augmented vector and using the free matrix-based integral inequality.


## 4. Numerical Examples

Example 1.
Consider the nonlinear system.(20)x˙1t=−0.7x1t+0.3x2t,x˙2t=0.1x1t−0.8x2t+F1xt+f2ut+2ut,yt=0.5x1t−0.2x2t,where(21)F1xt=f1xt+f1xt−2,f1xt=exp−x1t−x2tcosx1t+x2t−1,f2ut=1.5cos2utsin2ut.
As referred to in [38], the system (20) approximated by two multilayer perceptrons can be converted into the form:(22)x˙1t=−0.7x1t+0.3x2t,x˙2t=0.1x1t−0.8x2t+g¯+2ut,yt=0.5x1t−0.2x2t,where(23)g¯=tanhW2tanhW1xt−2+tanhW2tanhW1xt+tanhV2tanhV1ut.
The system (22) can be transformed into the standard form of the system (1):(24)A1=−0.70.30.1−0.8,B1=0001×301×21101×301×2,C1=0001×301×21001×301×2,D1=02,A2=00W102×2,A3=0.5−0.2,T−=07×7,T+=I7×7,B2=00W201×20001×3V20003×303×20002×302×2,D2=000003×103×102×1V1,C2=07×7,D3=0,B3=C3=01×7,W1=−0.59−0.14×10−10.4×10−20.1×10−30.18×10−10.4×10−3,V1=−0.27×10−12−0.31×10−2,V2=0.27×10−120.31×10−2,W2=−0.3×10−20.12×10−50.1×10−3.
By applying Theorem2 to the above system, we can obtain the feasible solutions of the state observer gain matrix and the controller gain matrix:(25)L=−1.35130.8611,K=−0.11490.2361.
State trajectories ofxt is depicted in Figure 1 with the initial condition xt=0.7−0.6T. Trajectories of estimation error et are plotted in Figure 2 under the initial condition et=−0.40.6T. It can be seen from Figures 1 and 2 that the designed state estimator and the state-feedback controller are feasible for the system (22).Figure 1
The trajectories ofx(t).Figure 2
The trajectories ofe(t).Example 2.
Consider the local field neural networks with the parameters as referred to in [32]:(26)A1=1.276900000.623100000.92300000.448,A2=A3=I,D1=D2=0,B1=−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,B2=B3=0,C1=0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775,C2=C3=D3=0,T2=0.113700000.127900000.799400000.2368,T1=diag−0.4,0.1,0,−0.3.
By applying Theorem3, the maximum upper bounds d2 for d1=1 and various μ are listed in Table 1. It can be seen from Table 1 that the criteria in this paper show less conservatism.Table 1
The maximum upper bounds ofd2 for d1=1 and various μ in Example 2.
Methodμ=0.1μ=0.5μ=0.9Theorem 4.1 [25]2.78842.43252.1347Theorem34.92874.16522.554


## 5. Conclusions

In this paper, we have investigated the problem of state estimation for standard neural network models with time-varying delays. By introducing the new state in the augmented vector and constructing the appropriate Lyapunov–Krasovskii functional, sufficient conditions have been obtained which can guarantee error system is asymptotically stable. The criteria proposed have less conservatism. Numerical examples illustrate the validity and benefit of our results. In the future, the issue of dissipative analysis will be investigated for the DSNNM and we will try to improve the integral inequality and make a good balance between conservatism and complexity.



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