# Novel Stability Analysis for Uncertain Neutral-Type Lur’e Systems with Time-Varying Delays Using New Inequality

Authors: Yanmeng Wang; Lianglin Xiong; Yongkun Li; Haiyang Zhang; Chen Peng
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper considers the delay-dependent stability analysis of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities. First of all, using constructed function methods, a new Jensen-like inequality is introduced to obtain less conservative results. Second, a new class of Lyapunov-Krasovskii functional (LKF) is constructed according to the characteristic of the considered systems. Third, combining with the new inequality and reciprocal convex approach and some other inequality techniques, the new less conservative robust stability criteria are shown in the form of linear matrix inequalities (LMIs). Finally, three examples demonstrate the feasibility and the superiority of our methods.


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## Body

## 1. Introduction

Delay phenomenon is often encountered in many practical systems, such as biological systems, chemical systems, electronic systems, and network control systems. However, time-delay is usually the main cause of instability and bad performance. Hence, many authors devote themselves to studying the stability and many effective methods of the time-varying system to gain less conservative delay-dependent stability criteria [1–11], which include linear systems with the delay-fraction theory [2, 3] and nonlinear systems Lur’e systems [6–8]. As is known to all, delay-dependent stability results are less conservative than the delay-independent ones if delay size is very small. Therefore, a lot of articles were published recently which studied the delay-dependent stability for a class of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities, and lots of significant results have been developed [12–30].Delay-dependent stability criteria were presented for nominal and uncertain neutral-type Lur’e systems with constant time delays and sector bounded nonlinearities in [27]. The robust stability problems for neutral-type Lur’e systems with time-varying delays were considered because time delays vary always depending on time in [12–21, 27–30]. The free-weighting matrix method was applied to get less conservative stability criteria and to deal with the robust stability problems for neutral-type Lur’e systems with time-varying delays in [15, 16]. However, the free-weighting matrix method brings more variables which make the computation quite complex. So, it is the right time to improve the disadvantage of the free-weighting matrix method and to get less conservative stability criteria for neutral-type Lur’e systems with time-varying delays. Some new robust stability criteria were proposed without using the general free-weighting matrix method which are less conservative and easier to calculate than some previous ones in [15, 19]. Reference [31] can reduce conservatism by reducing the conservatism of the Jensen-like inequality. Motivated by [31], developing the Jensen-like inequality with double-integral term may reduce the conservativeness. As a result, less conservative criteria may be also got by constructing new integral inequalities used in LKF.This paper studies the stability for a class of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities. To investigate the neutral-type Lur’e system, this paper introduces a new triple-integral inequality used in the following LKFs and gets less conservative criteria. The LKF contains not only double-integral terms but also triple-integral terms. Using some effective techniques, such as a novel integral inequality, a piecewise analysis method, and the reciprocally convex combination inequality, instead of the general free-weighting matrix method, the delay-dependent stability criteria derived in the form of LMIs are less conservative than some existing results in other papers. The effectiveness and the less conservatism of stability criteria proposed in this paper are demonstrated by using numerical examples in Section4.Notation. Rn denotes the n-dimensional Euclidean vector space, and Rn×m denotes the set of all n×m real matrices. For a symmetric matrix P, P>0 (resp., P<0) shows that P is a positive (resp., negative) definite matrix. diag⁡a1,a2,…,an represents a diagonal matrix with diagonal elements a1,a2,…,an. ∗ denotes a symmetric term in a symmetric matrix.


## 2. Problem Statement and Preliminaries

Consider a class of Lur’e systems of uncertain neutral type with time-varying delays and sector-bound nonlinearities described by the following equation:(1)x˙t-Cx˙t-τt=A+ΔAtxt+A1+ΔA1txt-ht+B+ΔBtfσt,σt=HTxt,∀t≥0,xs=ϕs,x˙s=ϕ˙s,s∈-max⁡h,τ,0,where σt∈Rm and xt∈Rn stand for the output and state vectors, respectively. ϕs∈Rn is a real-valued continuous initial function on -max⁡h,τ,0. A, A1, B, C, and H are known real constant matrices with appropriate dimensions. ΔAt, ΔA1t, and ΔBt represent the time-varying uncertainty parameters. fσt∈Rm is the nonlinear function such as(2)fσt=f1σ1tf2σ2t⋯fmσmtT,where σit is the ith component of the output vector σt, and each term fiσiti=1,2,…,m satisfies the finite sector condition shown in Figure 1(a) [32](3)fiσit∈K0,ki=fiσit∣fi0=0,0<σit·fiσit≤kiσi2t,σit≠0with known positive scalars ki or the infinite sector condition shown in Figure 1(b) [32](4)fiσit∈K0,∞=fiσit∣fi0=0,σit·fiσit>0,σit≠0.Figure 1
Two sector bounded nonlinearity.
(a)
Finite sector bounded nonlinearity (b)
Infinite sector bounded nonlinearityΔ A t, ΔA1t, and ΔBt are assumed to satisfy the following condition:(5)ΔAtΔA1tΔBt=DFtEaEa1Eb,where D, Ea, Ea1, and Eb are known constant matrices with appropriate dimensions, and the uncertainty time-varying matrix Ft satisfies(6)FTtFt≤I,∀t≥0.The time-varying delays ht and τt are continuous functions satisfying the following conditions:(7)0≤ht≤h,h˙t≤hd,0≤τt≤τ,τ˙t≤τd<1,∀t≥0.This paper investigates the delay-dependent stability of Lur’e system (1) satisfying conditions (3) (or (4)) and (5)–(7) to gain less conservative robust stability criteria by using a new inequality and constructing a new LKF. Throughout this paper, the results will be acquired by assuming that all the eigenvalues of C are inside the unit circle [33]. The following lemmas are useful in deriving criteria.Lemma 1 (see [34] (reciprocally convex combination)).
Letf1,f2,…,fN:Rm↦R have positive values in an open subset D of Rm. Then, the reciprocally convex combination of fi over D satisfies(8)minαi∣αi>0,∑iαi=1∑i1αifit=∑ifit+maxgi,jt∑i≠jgi,jtsubject  togi,j:Rm⟼R,gj,it≜gi,jt,fitgi,jtgi,jtfjt≥0.Lemma 2 (see [31]).
For a given matrixR>0, any differentiable function ω in a,b→Rn, the following inequality holds: (9)∫abω˙TuRω˙udu≥1b-aωb-ωaTR·ωb-ωa+3b-aΩ1TRΩ1,where(10)Ω1=ωb+ωa-2b-a∫abωudu.Remark 3.
It is clear that this new inequality encompasses the Jensen inequality. It is also worth noting that it plays an important role in getting the derivative of the LKF. Hence, by using optimization theory and improving the Jensen-like inequality, the double-integral inequality would be got in the following lemma.Lemma 4.
For any matrixR>0, scalars h2≥h1≥0, and any continuous function ω in R→Rn the following inequality holds: (11)-∫-h2-h1∫t+θtωTuRωududθ≤-2h22-h12∫-h2-h1∫t+θtωududθTR·∫-h2-h1∫t+θtωududθ-48h24-h14Ω2TRΩ2,where(12)Ω2=h12∫-h2-h1∫t+θtωududθ+12∫-h2-h1∫-h2θ∫t+rtωududrdθ-∫-h2-h1∫t+θt∫t+θuωsdsdudθ.Proof.
Recall that the objectives of the present paper lemma are to acquire new lower bounds of the integral∫-h2-h1∫t+θtωT(u)Rω(u)dudθ and to compare with the Jensen-like inequality. For this reason, the appropriate function z(u) is given as the following form:(13)zu=ωu-2h22-h12∫-h2-h1∫t+θtωududθ-48fu,θh24-h14Ω2,where fu,θ=u-(1/2)θ-t is constructed to satisfy(14)∫-h2-h1∫t+θtfu,θdudθ=0and to get rid of the effect of cross terms. It is easy to obtain(15)∫-h2-h1∫t+θtf2u,θdudθ=h24-h1448,∫-h2-h1∫t+θtωufu,θdudθ=h12∫-h2-h1∫t+θtωududθ-∫-h2-h1∫t+θtωut-ududθ-12∫-h2-h1∫t+θtωuh1+θdudθ=h12∫-h2-h1∫t+θtωududθ+12∫-h2-h1∫-h2θ∫t+rtωududrdθ-∫-h2-h1∫t+θt∫t+θuωsdsdudθ=Ω2.Then, based on the formulas above, the following equality holds:(16)∫-h2-h1∫t+θtzTuRzududθ=∫-h2-h1∫t+θtωTuRωududθ-2h22-h12∫-h2-h1∫t+θtωududθTR∫-h2-h1∫t+θtωududθ+482h24-h142∫-h2-h1∫t+θtf2u,θdudθΩ2TRΩ2-2·48h24-h14∫-h2-h1∫t+θtωufu,θdudθTRΩ2+4·48h24-h14h22-h12∫-h2-h1∫t+θtfu,θdudθ·∫-h2-h1∫t+θtωududθTRΩ2=∫-h2-h1∫t+θtωTuRωududθ-2h22-h12∫-h2-h1∫t+θtωududθTR∫-h2-h1∫t+θtωududθ-48h24-h14Ω2TRΩ2.For a given symmetric positive definite matrix R>0, then, the left-hand side of formula (16) above is positive definite. It is obvious that Lemma 4 is correct.Remark 5.
Compared with Lemma1 in [35], the inequality in Lemma 4 has two advantages. First, since R>0, the second term in the right side of the inequality in Lemma 4 is definite negative. It thus implies that the Jensen-like inequality in [35] has been included in the inequality proposed above. Second, it is also worth noting that this improvement is allowed by using extra signals ∫-h2-h1∫-h2θ∫t+rtωududrdθ,   ∫-h2-h1∫t+θt∫t+θuωsdsdudθ, and ∫-h2-h1∫t+θtωududθ.Remark 6.
It is worth paying attention to the fact that the construction of functionf is the key to Lemma 4. ∫-h0∫t+θtfdudθ=0 must be sure when we construct the function f to get rid of crossing items and to make the second term in the right side of our inequality be definite negative.Remark 7.
In fact, the result of Lemma4 is less conservative than formula (3) in literature [36]. When h1≠0, the result of Lemma 4 is different from inequality (16) in literature [37]. What is more, under the special scaling condition of the left side of the inequality in Lemma 4, the more conservativeness of inequality (16) in literature [37] would be easily concluded. To some extent, the conditions of Lemma 4 are wider than inequality (16) in literature [37] in terms of integral domain. Actually, with the help of delay decomposition technology, ∫-h/20∫t+θtfsdsdθ and ∫-h-h/2∫t+θtfsdsdθ can be scaled properly with our inequality in Lemma 4. In our future research, we will try to use formulas in the literature [37] which may be more flexible than the Lemma 4 to reduce the conservatism of stability criteria in another case.


## 3. Main Results

To accept easily the robust stability problem of system (1), firstly, we investigate the stability criterion of the nominal form:(17)x˙t-Cx˙t-τt=Axt+A1xt-ht+Bfσt,σt=HTxt,∀t≥0,xs=ϕs,x˙s=ϕ˙s,s∈-maxh,τ,0.The stability of system (17) can be analyzed via time-independent functions of the form(18)Vxt=∑i=15Vixt,where(19)V1xt=ξ1TtPξ1t+2∑i=1mλi∫0σitfiσdσV2xt=∫t-h/2txsxs-h2TX11X12∗X22xsxs-h2dsV3xt=∫t-httxTsQ1xsds+∫t-τttx˙TsQ2x˙sdsV4xt=h28∫-h/20∫t+θts-t-θx˙TsQ3x˙sdsdθ+3h28∫-h-h/2∫t+θts-t-θx˙TsQ4x˙sdsdθV5xt=h2∫-h/20∫t+θtx˙TsQ5x˙sdsdθ+h2∫-h-h/2∫t+θtx˙TsQ6x˙sdsdθ,and define(20)ξ1Tt=xTt∫t-h/2txTsds∫t-ht-h/2xTsds∫-h/20∫t+θtxTsdsdθ∫-h-h/2∫t+θtxTsdsdθ,ξTt=xTtxTt-htxTt-h2xTt-hx˙Tt-τtfTσt∫t-h/2txTsds∫t-ht-h/2xTsds∫-h/20∫t+θtxTsdsdθ∫-h-h/2∫t+θtxTsdsdθ.Remark 8.
It is worth noticing that the construction of LKF can be improved by adding(21)h2∫-h/20∫t+θtxTsN1xsdsdθ,h2∫-h-h/2∫t+θtxTsN2xsdsdθto get less conservative stability criteria for any positive definite matrices N1 and N2. However, our methods have reduced the conservatism of the existing results in the instance. Hence, this paper dose not add N1 and N2 to control the number of variables.Remark 9.
A new Lyapunov-Krasovskii functional is constructed in Section3 to obtain a new delay-dependent stability criterion, which includes the relationship among the states, the nonlinear function, and the derivative of the states. In order to fully reduce the conservative of the condition, the constant delay h is decomposed into h/2 and thus (18) is obtained. Particularly, the function V1 is constructed to eliminate the effects among different forms of the states; the function V2 is constructed to eliminate the relationship among the states with different delays; to consider the time-varying delay the function V3 is constructed; and our Lyapunov-Krasovskii functional contains double-integral (function V4) and triple-integral (function V5) terms which yield less conservative delay-dependent stability criteria. Therefore, more information for the state is employed in constructing the more general Lyapunov-Krasovskii functionals, which may lead to reduced conservatism.If nominal system (17) satisfies conditions (3) and (7) the following theorem can be got.Theorem 10.
Nominal system (17) satisfying conditions (3) and (7) is asymptotically stable for given values h≥0, hd, τd<1, and kl>0l=1,2,…,m, if there exist appropriate dimensional matrices X1,X2,X12, positive matrices P=Pij5×5,Qii=1,2,…,6, X11, X22 positive semidefinite diagonal matrices S=diag⁡s1,s2,…,sm, and Λ=diag⁡λ1,λ2,…,λm such that the following LMIs hold:(22)Q5X1∗Q5>0,(23)Q6X2∗Q6>0,(24)X11X12∗X22>0,(25)Ξr=E+Θ1+ΨrηTΩ∗-Ω<0,r=1,2,where(26)E=Eij10×10,Eij=Eji,i,j=1,2,…,10,E11=P11A+P12+12hP14+12hP15+ATP11+P12T+12hP14T+12hP15T+X11+Q1-3740h2Q4-58h2Q3,E12=P11A1,E13=-P12+P13+X12,E14=-P13,E15=P11C,E16=P11B+ATHΛ,E17=-P14+ATP12+P22+h2P24T+h2P25T-9h10Q4-hQ3,E18=-P15+ATP13+P23+12hP34T+12hP35T-1310hQ4,E19=ATP14+P24+12hP44+12hP45T+9Q3,E1,10=ATP15+P25+12hP45+12hP55+275Q4,E22=-1-hdQ1,E26=A1THΛ,E27=A1TP12,E28=A1TP13,E29=A1TP14,E2,10=A1TP15,E33=X22-X11,E34=-X12,E37=-P22+P23T,E38=-P23+P33,E39=-P24+P34,E3,10=-P25+P35,E44=-X22,E47=-P23T,E48=-P33,E49=-P34,E4,10=-P35,E55=-1-τdQ2,E56=CTHΛ,E57=CTP12,E58=CTP13,E59=CTP14,E5,10=CTP15,E66=ΛHTB+BTHΛ,E67=BTP12,E68=BTP13,E69=BTP14,E6,10=BTP15,E77=-P24-P24T-65Q4-7Q3,E78=-P25-P34T-125Q4,E79=-P44+36hQ3,E7,10=-P45+365hQ4,E88=-P35-P35T-295Q4,E89=-P45T,E8,10=-P55+725hQ4,E99=-216h2Q3,E10,10=-2165h2Q4,E23=E24=E25=E35=E36=E45=E46=E9,10=0,Ψ1=-e3-e4TQ6e3-e4-3e3+e4-4he8TQ6·e3+e4-4he8-e2-e3e1-e2TQ5X1∗Q5e2-e3e1-e2,Ψ2=-e1-e3TQ5e1-e3-3e1+e3-4he7TQ5·e1+e3-4he7-e2-e4e3-e2TQ6X2∗Q6e2-e4e3-e2,Θ1=e1THKSe6-e6TSe6+e1THKSe6-e6TSe6T,ei=0⋯01︸i0⋯01×10,η=AA100CB0000,Ω=Q2+h464Q3+9h464Q4+h24Q5+h24Q6.Proof.
The time-derivative of functionsVixt can be calculated, respectively, based on the trajectory of system (17) as follows:(27)V˙1xt=2ξ1TPξ˙1+2fTσtΛHTx˙tV˙2xt=xtxt-h2TX11X12∗X22xtxt-h2-xt-h2xt-hTX11X12∗X22xt-h2xt-hV˙3xt≤xTtQ1xt-1-hdxTt-htQ1·xt-ht+x˙TtQ2x˙t-1-τdx˙Tt-τtQ2x˙t-τtV˙4xt=x˙Tth464Q3+9h464Q4x˙t-h28∫-h/20∫t+θtx˙TsQ3x˙sdsdθ-3h28∫-h-h/2∫t+θtx˙TsQ4x˙sdsdθV˙5xt=h24x˙TtQ5+Q6x˙t-h2∫t-h/2tx˙TsQ5x˙sds-h2∫t-ht-h/2x˙TsQ6x˙sds.Using Lemma 4 and the method of the exchange of integral order, we get(28)-h28∫-h/20∫t+θtx˙TsQ3x˙sdsdθ≤-∫-h/20∫t+θtx˙sdsdθTQ3∫-h/20∫t+θtx˙sdsdθ-96h212∫-h/20∫-h/2θ∫t+rtx˙sdsdrdθ-∫-h/20∫t+θt∫t+θux˙sdsdudθTQ312∫-h/20∫-h/2θ∫t+rtx˙sdsdrdθ-∫-h/20∫t+θt∫t+θux˙sdsdudθ=-h2xt-∫t-h/2txsdsTQ3h2xt-∫t-h/2txsds-96h2h216xt+h4∫t-h/2txsds-32∫-h/20∫t+θtxsdsdθT·Q3h216xt+h4∫t-h/2txsds-32∫-h/20∫t+θtxsdsdθ(29)-3h28∫-h-h/2∫t+θtx˙TsQ4x˙sdsdθ≤-∫-h-h/2∫t+θtx˙sdsdθTQ4∫-h-h/2∫t+θtx˙sdsdθ-965h2h4∫-h-h/2∫t+θtx˙sdsdθ+12∫-h-h/2∫-hθ∫t+rtx˙sdsdrdθ-∫-h-h/2∫t+θt∫t+θux˙sdsdudθTQ4h4∫-h-h/2∫t+θtx˙sdsdθ+12∫-h-h/2∫-hθ∫t+rtx˙sdsdrdθ-∫-h-h/2∫t+θt∫t+θux˙sdsdudθ=-h2xt-∫t-ht-h/2xsdsTQ4h2xt-∫t-ht-h/2xsds-965h23h216xt+h4∫t-h/2txsds+h2∫t-ht-h/2xsds-32∫-h-h/2∫t+θtxsdsdθTQ43h216xt+h4∫t-h/2txsds+h2∫t-ht-h/2xsds-32∫-h-h/2∫t+θtxsdsdθ.For ht∈0,h/2, the following formulas can be got using Lemmas 1 and 2: (30)-h2∫t-h/2tx˙TsQ5x˙sds=-h2∫t-h/2t-htx˙TsQ5x˙sds-h2∫t-httx˙TsQ5x˙sds≤-h/2h/2-htxt-ht-xt-h2TQ5·xt-ht-xt-h2-h/2htxt-xt-htT·Q5xt-xt-ht-3h/2h/2-htxt-ht+xt-h2-hh/2-ht∫t-h/2t-htx˙sdsT·Q5xt-ht+xt-h2-hh/2-ht∫t-h/2t-htx˙sds-3h/2htxt+xt-ht-hht∫t-httx˙sdsT·Q5xt+xt-ht-hht∫t-httx˙sds≤-h/2h/2-htxt-ht-xt-h2TQ5·xt-ht-xt-h2-h/2htxt-xt-htT·Q5xt-xt-ht≤-xt-ht-xt-h2xt-xt-htTQ5X1∗Q5·xt-ht-xt-h2xt-xt-ht-h2∫t-ht-h/2x˙TsQ6x˙sds≤-xt-h2-xt-hTQ6xt-h2-xt-h-3xt-h2+xt-h-4h∫t-ht-h/2xsdsT·Q6xt-h2+xt-h-4h∫t-ht-h/2xsds.At the same time, we can get the following inequation from formula (3):(31)sifiσitkihiTxt-fiσit≥0.It is equivalent to(32)2xTtHKSfσt-fTσtSfσt≥0,where(33)S=diag⁡s1,s2,…,sm,K=diag⁡k1,k2,…,km.According to inequalities (27), (28), (29), (30), and (32), we get(34)V˙xt≤∑i=15V˙ixt+2xTtHKSfσt-fTσtSfσt≤ξTtE+Θ1+Ψ1+ηTΩηξTt.
Obviously, LMI (25) with r=1 suggests that V˙xt<0 using Schur complement.
Forht∈h/2,h, the following formulas can be got by using Lemmas 1 and 2:(35)-h2∫t-h/2tx˙TsQ5x˙sds≤-xt-xt-h2TQ5xt-xt-h2-3xt+xt-h2-4h∫t-h/2txsdsT·Q5xt+xt-h2-4h∫t-h/2txsds-h2∫t-ht-h/2x˙TsQ6x˙sds=-h2∫t-ht-htx˙TsQ6x˙sds-h2∫t-htt-h/2x˙TsQ6x˙sds≤-xt-ht-xt-hxt-h2-xt-htTQ6X2∗Q6·xt-ht-xt-hxt-h2-xt-ht.According to inequalities (27) and (28), (32), and (35), we have(36)V˙xt≤∑i=15V˙ixt+2xTtHKSfσt-fTσtSfσt≤ξTtE+Θ1+Ψ2+ηTΩηξTt.
Obviously, LMI (25) with r=2 suggests that V˙xt<0 using Schur complement. In conclusion, the asymptotic stability of nominal system (17) satisfying (3) has been proven.Remark 11.
If functionfσt of normal system (17) satisfies sector condition (4), for any si≥0,i=1,2,…,m, it follows that(37)sifiσithiTxt≥0,i=1,2,…,m,which is equivalent to(38)2xTtHSfσt≥0.Corollary 12.
Nominal system (17) satisfying conditions (4) and (7) is asymptotically stable for given values h≥0, hd, and τd<1, if there exist appropriate dimensional matrices X1,X2,X12, positive matrices P=Pij5×5,  Qii=1,2,…,6,X11,X22 positive semidefinite diagonal matrices S=diag⁡s1,s2,…,sm, and Λ=diag⁡λ1,λ2,…,λm, such that the following LMIs hold:(39)Ξ~r=E+Θ2+ΨrηTΩ∗-Ω<0,r=1,2as well as (22)–(24), where(40)Θ2=2e1THSe6.Remark 13.
Absolute stability criteria Theorem10 and Corollary 12 proposed in this paper are less conservative than existing results which are shown by the following examples mentioned in the Section 4. The main reasons are the combination of introducing new inequality in Lemma 4 and the method of the exchange of integral order, based on the new constructed Lyapunov functionals.The next theorem and corollary will investigate the robust stability for uncertain neutral-type Lur’e system (1) satisfying conditions (3)–(7) following similar methods of the proof for Theorem 10 and Corollary 12.Theorem 14.
System (1) satisfying conditions (3) and (5)–(7) is robustly absolutely stable for given values h≥0, hd,τd<1, and kl>0l=1,2,…,m, if there exist appropriate dimensional matrices X1,X2,X12, positive matrices P=Pij5×5,Qii=1,2,…,6,X11,X22 positive semidefinite matrices S=diag⁡{s1,s2,…,sm},Λ=diag⁡λ1,λ2,…,λm, and scalars εr>0 such that the following LMIs hold:(41)E+εrΔETΔE+Θ1+ΨrηTΩJ∗-Ω0∗∗DTΩD-εr<0,as well as (22)–(24), where(42)ΔE=EaEa1000Eb0000,J=J1TJ2T00J5TJ6TJ7TJ8TJ9TJ10TT,J1=P11D+ATΩD,J2=A1TΩD,J5=CTΩD,J6=ΛHTD+BTΩD,J7=P12TD,J8=P13TD,J9=P14TD,J10=P15TD,r=1,2.Proof.
If(43)pt=FtEaxt+Ea1xt-ht+Ebfσt,for any ε>0, we have the following inequality:(44)-εpTtpt+εEaxt+Ea1xt-ht+EbfσtT·Eaxt+Ea1xt-ht+Ebfσt≥0.Using A+DFtEa,  A1+DFtEa1,B+DFtEb instead of A,A1,B in formula (18), respectively, LMIs in (25) can be rewritten as (41).Now, if functionfσt of system (1) satisfies sector condition (4), we have Corollary 15.Corollary 15.
System (1) satisfying conditions (4)–(7) is robustly absolutely stable for given values h≥0,hd and τd<1, if there exist appropriate dimensional matrices X1,X2,X12, positive matrices P=Pij5×5,Qii=1,2,…,6,X11,X22 positive semidefinite diagonal matrices(45)S=diag⁡s1,s2,…,sm,Λ=diag⁡λ1,λ2,…,λm,and scalars εr>0 such that the following LMIs hold:(46)E+εrΔETΔE+Θ2+ΨrηTΩJ∗-Ω0∗∗DTΩD-εr<0,as well as (22)–(24), r=1,2.Remark 16.
Clearly, robust stability criteria Theorem14 and Corollary 15 proposed in this paper would be less conservative, because of using the new inequality and some other techniques which are similar to Theorem 10 and Corollary 12.Remark 17.
In order to handle the integral term(47)-∫-h2-h1∫t+θtx˙TsRx˙sdsdθJensen inequality [35] is always presented as(48)-∫-h20∫t+θtx˙TsRx˙sdsdθ≤-1h22∫-h20∫t+θtx˙sdsdθTR∫-h20∫t+θtx˙sdsdθ.Obviously, Lemma 4 not only can obtain a more accurate bound than the Jensen inequality but also can reduce the number of decision variables. What is more, Lemma 4 can dispose the integral terms with h1≠0. Therefore, Lemma 4 is used to derive our criterion, and the effectiveness of our method will be demonstrated by some existing numerical examples.


## 4. Examples

In this section, the effectiveness of our approaches will be shown by the following three numerical examples.Example 1 (see [16, 19–21]).
Consider Chua’s circuit example discussed in [16, 19–21]. Chua’s circuit is a simple nonlinear electronic circuit design. It can show the standard chaos theory. It was published by professor Leo Chua in 1983. The ease of making this circuit makes it a ubiquitous example of chaotic systems in the real world, leading somebody to declare it as a model of a chaotic system. The system equation is given by(49)x˙t=αyt-hxt,y˙t=xt-yt+zt,z˙t=-βyt,with nonlinear characteristic(50)hxt=m1xt+12m0-m1xt+c-xt-c, and parameters m0=-1/7,m1=2/7,   α=9,β=14.28, and c=1. The system can be represented in normal Lur’e system framework (17) with (51)A=-αm1-1α01-210-β-1,A1=-6.002900-1.3367002.126400B=-αm0-m100,H=100,C=0.The feedback nonlinear function belongs to K0,1.
The MADBs ofh of system (49) for different hd have been shown in Table 1 with the existing results in [16, 19–21]. Table 1 shows us that the MADBs obtained by using Theorem 10 in this paper are better than the existing results in [16, 19–21].Table 1
MADBsh for different hd (Example 1).
h d 0 0.3 0.6 0.9 >1.0 [21] 0.1622 0.1591 0.1566 0.1541 0.1527 [16] 0.1745 0.1698 0.1698 0.1698 0.1698 [19] 0.1747 0.1710 0.1703 0.1703 0.1703 [20] 0.1771 0.1721 0.1715 0.1715 0.1715 Theorem10 0.2790 0.2691 0.2553 0.2463 0.2463Example 2 (see [12, 14–20]).
Consider the following nominal neutral-type Lur’e systems:(52)x˙t-Cx˙t-τt=Axt+A1xt-ht+Bfσt,σt=HTxt,∀t≥0,with(53)A=-20.50-1,A1=10.40.4-1,B=-0.5-0.75,C=0.20.10.10.2,H=0.20.6.
The maximum allowable delay bounds (MADBs) ofh of nominal neutral-type Lur’e system (52) for different hd and τd have been shown in Table 2 along with the existing results in [12, 14–20]. Applying Matlab LMI-toolbox, one can get the simulation result in Figure 1. Table 2 shows us that the MADBs obtained by using Corollary 15 in this paper are better than the existing results in [12, 14–20]. What is more, for all the nonlinear function f(σ(t))∈Rm with condition (3) or condition (4), the simulation study is entirely viable, which has been shown in this example (Figure 2).Table 2
MADBsh for different hd and τd (Example 2).
τ d h d 0.2 0.4 0.6 0.8 0.1 [12] 2.2424 1.5434 0.9888 0.7228 [14] 2.6062 1.9730 1.6408 1.1854 [15] 2.6488 2.0055 1.6679 1.2049 [17] 2.9962 2.1316 1.7138 1.3204 [16] 2.9997 2.1012 1.7307 1.3359 [18] 2.6052 1.9538 1.6232 1.1659 [19] 3.0717 2.1171 1.7310 1.3367 [20] 3.1743 2.1789 1.7467 1.7153 Corollary15 3.4880 2.3787 1.8062 1.4625  0.5 [12] 1.8344 1.3029 0.7793 0.6282 [14] 2.1537 1.7041 1.3828 1.0095 [15] 2.1883 1.7322 1.4056 1.0262 [17] 2.4472 1.8289 1.4551 1.1058 [16] 2.4563 1.8009 1.4823 1.1157 [18] 2.1532 1.6986 1.3687 1.0066 [19] 2.5131 1.8146 1.4951 1.1168 Corollary15 2.8538 2.0236 1.5679 1.2425  0.9 [12] 0.1085 0.1019 0.0983 0.0967 [14] 0.1122 0.1088 0.1086 0.1086 [15] 0.1122 0.1089 0.1086 0.1086 [17] 0.1126 0.1102 0.1102 0.1102 [16] 0.1130 0.1105 0.1105 0.1105 [18] 0.1221 0.1087 0.1083 0.1083 [19] 0.1227 0.1197 0.1197 0.1197 Corollary15 0.1799 0.1697 0.1660 0.1658Figure 2
The dynamical behavior for Example2.Example 3 (see [15–17, 19, 20]).
Consider uncertain system (1) with the following parameters:(54)A=-200-0.9,A1=-10-1-1,B=-0.2-0.3,C=0000,H=0.60.8,D=Ea=Ea1=0.1000.1,Eb=0.10,Ec=sin⁡ωt00sin⁡ωt.
The MADBs ofh with hd=0 and τd<1 for different values of K0,ki are listed in Table 3 by using Theorem 14 and Corollary 15 along with the existing results in [15–17, 19, 20]. Table 3 shows us that the MADBs obtained by using Theorem 14 and Corollary 15 in this paper are better than the existing results in [15–17, 19, 20].Table 3
MADBsh for different K (Example 3).
K Methods h f ⋅ ∈ K 0,0.5 [15] 4.3196 [17] 5.1366 [16] 5.1666 [19] 5.3486 [20] 5.4972 Theorem14 5.5513  f ⋅ ∈ K 0,100 [15] 4.1792 [17] 4.9572 [16] 4.9731 [19] 5.1828 Theorem14 5.3557  f ⋅ ∈ K 0 , ∞ [15] 4.1581 [17] 4.9304 [16] 4.9431 [19] 5.1654 [20] 5.2489 Corollary15 5.3186


## 5. Conclusions

In this paper, we have studied the stability for a class of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities by using a new LKF. The LKF contains not only double-integral terms but also triple-integral terms. Using some effective techniques, such as a novel integral inequality, a piecewise analysis method, and the reciprocally convex combination inequality, instead of the general free-weighting matrix method, the delay-dependent stability criteria derived in the form of LMIs are less conservative than some existing results. The effectiveness and the less conservatism of stability criteria proposed in this paper are demonstrated by using standard numerical examples.



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