# New Results on Stability Analysis of Uncertain Neutral-Type Lur’e Systems Derived from a Modified Lyapunov-Krasovskii Functional

Authors: Wenyong Duan; Yan Li; Jian Chen; Lin Jiang
Journal: Complexity
 (2019)

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## Abstract

This paper is concerned with the problem of the absolute and robustly absolute stability for the uncertain neutral-type Lur’e system with time-varying delays. By introducing a modified Lyapunov-Krasovskii functional (LKF) related to a delay-product-type function and two delay-dependent matrices, some new delay-dependent robustly absolute stability criteria are proposed, which can be expressed as convex linear matrix inequality (LMI) framework. The criteria proposed in this paper are less conservative than some recent previous ones. Finally, some numerical examples are presented to show the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In many real systems, time delay is often considered as the main cause of poor performance and even instability. The stability of time-delay systems is always a hot topic for researchers. As a result, to obtain stability criteria of time-delayed systems by using the Lyapunov theorem, the main efforts are concentrated on the following several directions; one is finding an appropriate positive definite functional with a negative definite time derivative along the trajectory of system, for example, LKF with delay partitioning approach [1, 2], LKF with augmented terms [3], LKF with triple-integral and quadruple-integral terms [4, 5], and so on. The other is reducing the upper bounds of the time derivative of LKF as much as possible by developing various inequality techniques, such as Jensen inequality [6], Wirtinger-based inequality [7], auxiliary function based inequality [8], and Bessel-Legendre inequality [9]. Besides, further to increase the freedom of solving LMIs, there are some other methods, for instance, the generalized zero equality [10, 11], the one- or second-order reciprocally convex combinations [12–15], the free-weighting-matrix approach [16], and so on.In practical engineering applications, most systems are nonlinear. As is known to all, Lur’e system, which is composed of the feedback connection of the linear dynamical system and the nonlinearity satisfying the sector-bounded condition, can represent many deterministic nonlinear systems, for example, Chua’s Circuit and the Lorenz system [17]. Therefore, the study on the stability of Lur ’e systems becomes more and more popular [18–21]. Moreover, the paper [22] pointed out that many practical systems can be modeled as neutral time-delayed systems, in which not only the system states or outputs contain time delays, but also the derivative of the system states. Due to the theoretical and practical significance, the analysis of the robust stability of the time-delayed neutral-type Lur’e systems has attached great importance by many scholars [23–29], where many important robust stability criteria were given. However, the main improvement of stability criteria depends on the development of LKF and the update of inequality techniques based on linear systems. For example, recently, [29] improved the stability results of some previous ones by combining the extended double integral with Wirtinger-based inequalities technique; however, the range of delay with nonzero lower bound and the lower bound of the delay derivative are not involved; in [30], some less conservative stability criteria than some recent previous ones were derived for time-delayed Lur’e system via the second-order Bessel-Legendre inequality approach, a novel inequality technique; in [21], some improved stability criteria for time-delayed neutral-type Lur’e system were given by constructing a novel LKF consisting of a quadratic term and integral terms for the time-varying delays and the nonlinearities, and so on. Recently, C. Zhang [31] considered the effect of the LKFs while discussing the relationship between the tightness of inequalities and the conservatism of criteria for linear systems. The results illustrate the integral inequality that makes the upper bound closer to the true value does not always deduce a less conservative stability condition if the LKF is not properly constructed. Particularly, another novel LKF was proposed by C. Zhang et al. [31, 32] with delay-product-type terms h(t)P1 and (h-h(t))P2. Compared with the general LKF, P1 and P2 were just symmetrical, not always positive definite, which can lead to a less conservative stability condition by extending the freedom for checking the feasibility of stable conditions based on LMI. Recently, to fully utilize the information of delay derivative, a new LKF was constructed by W. Kwon et al. [33] with delay-dependent Lyapunov matrices Q1(t) and Q2(t). W. Kwon et al. point that the stability conditions based on an LKF with delay-dependent matrices are less conservative than those based on the LKF without delay-dependent matrices. As mentioned above, the two types of LKFs only improve one class of Lyapunov matrices, respectively, that is, only for the Lyapunov matrix P or the Lyapunov matrix Q. It is natural to wonder about whether can both classes of Lyapunov matrices be improved, simultaneously.Inspired by the above analysis, the following ideas of reducing the conservation of the previous proposed stability criteria should be addressed:(i)
A modified LKF with the above both classes of Lyapunov matrices, that is delay-product-type and delay-dependent matrices, is constructed. Compared with the general LKFs in some previous published papers, such as [21, 28, 30], the Lyapunov matrices of the nonintegral item are just symmetrical, not always positive definite, which can extend the freedom for checking the feasibility of stable conditions based on LMI. And the delay-dependent matrices of the single-integral items are utilized, which can also further improve the utilization of time delay and its derivative information. In addition, the results proposed by [31–33] can be improved via the LKF modified in this paper due to the combination of the two types of LKFs.(ii)
The double integral items of the modified LKF in this paper are decomposed into two subintervals, that is[0,h(t)] and [h(t),h], instead of being considered directly in [33], which further make full use of the information of time-varying delays h(t), h-h(t) and their derivative h˙(t). And the quadratic generalized free-weighting matrix inequality (QGFMI) technique can be used fully in each subinterval, which can further reduce the conservatism of the stability conditions.(iii)
To deal with the delay-derivative-dependent single-integral items feasibly, another double integral items ofV4(t) are also added to the LKF under the above two subintervals, instead of introducing a positive integral item, which is actually difficult to estimate, to the derivative of the LKF like [33].(iv)
Indeed, the main result of [33] was not LMI due to the terms with h2(t) even h3(t). The matrix inequalities of the stability criteria proposed in this paper are converted to LMIs via the properties of quadratic functions application, which can be solved easily by Matlab LMI-toolbox. In conclusion, it is interesting and still challenging problem to address the above issues, which offers motivation to derive less conservative stability criteria for the time-delayed neutral-type Lur’e systems.This paper mainly analyzes and studies the stability of uncertain neutral-type Lur’e systems with mixed time-varying delays. Some less conservative delay-dependent absolute stability criteria and robust absolute stability criteria than some previous ones are derived via a modified LKF application. In the end, four popular numerical examples are given to illustrate that this method improves some existing methods and achieves good results in stability. The structure of this paper is as follows:Section1 describes the research background and research topic status and defines the scope of the study of this article;Section2 describes the main research questions, including some necessary definitions, assumptions, and lemmas;Section3 presents the main results, including theorems and corollaries; inSection4 the discussions and simulations based on numerical examples are given;Section5 summarizes the whole thesis.Notation. P>0 (<0) represents a positive (negative) definite matrix. I and 0 represent an identity matrix and a zero matrix with the corresponding dimensions, respectively. ∗ denotes the symmetric terms in a block matrix and diag⁡⋯ denotes a block-diagonal matrix. ei(i=1,…,m) are block entry matrices with e2T=0I0⋯0︸m-2, where m is the dimension of the vector ξ. F[α(t),β(t)] denotes F is the function of α(t) and β(t). sym⁡{B}=B+BT.


## 2. Problem Formulation

Consider the following neutral-type Lur’e system with mixed time-varying delays:(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⁡h2,τ,0,ht∈C.1s∈-max⁡h,τ,0,ht∈C.2,where x(t)∈Rn and σ(t)∈Rm are the state and output vectors of the system, respectively. A, A1, B, C, and H are real constant matrices with appropriate dimensions; φ(s) is an Rn-valued continuous initial functional specified on [-max⁡(h,τ),0] or [-max⁡(h2,τ),0] with known positive scalars h, h2, and τ. f(σ(t))∈Rm is the nonlinear functional in the feedback path. The time-varying delays h(t) and τ(t) are continuous-time functional and satisfy the following two types of conditions:C. 1.(2)0≤τt≤τ,τ˙t≤τd<1,h1≤ht≤h2,μ1≤h˙t≤μ2∀t≥0,C. 2.(3)0≤τt≤τ,τ˙t≤τd<1,0≤ht≤h,μ1≤h˙t≤μ2,∀t≥0,whereτ≥0, τd<1, h1≥0, h2≥0, h≥0, μ1 and μ2<1 are constants.The nonlinear functionalf(σ(t)) in the feedback path is given by(4)fσt=f1σ1tf2σ2t⋯fmσmtTsatisfying the finite sector condition:(5)fiσit∈K0,ki=fiσit∣fi0=0,0<σitfiσit≤kiσit2,σit≠0or the infinite sector condition:(6)fiσit∈K0,∞=fiσit∣fi0=0,σitfiσit>0,σit≠0,where K=diag⁡{k1,k2,…,km}.Δ A ( t ), ΔA1(t), and ΔB(t) denote real-valued matrix functions representing parameter uncertainties, which are assumed to satisfy(7)ΔAtΔBtΔA1t=DFtEaEbEa1,where D, Ea, Eb, and Ea1 are known constant matrices with appropriate dimensions, and F(t) is an unknown matrix with Lebesgue-measurable elements and satisfies(8)FTtFt≤I,∀t≥0.This paper mainly analyzes and studies the stability of uncertain neutral-type Lur’e system (1) under conditions (2), (3), (5), (6), (7), and (8) based on Lyapunov stability theory. For neutral-type systems, the assumption that ρ(C)<1 [41] is required, where ρ(·) denotes the spectral radius of C. To obtain the main results of this paper, the following definition and lemmas are important.Definition 1 (robustly absolute stability).
The uncertain neutral-type Lur’e system described by (1) is said to be robustly absolutely stable in the sector [0,K] (or [0,∞)), if the system is asymptotically stable for any nonlinear function f(σ(t)) satisfying (5) (or (6)) and all admissible uncertainties.Lemma 2 (see [15]).
For given vectorsα1, α2 and positive real scalars λ satisfying 0<λ<1, symmetric positive definite matrix R1,R2∈Rn×n, and any matrix U01,U02∈Rn×n, the following inequality holds(9)α1TR1α1λ+α2TR2α21-λ≥α1α2TR1+1-λT11-λU01+λU02∗R2+λT2α1α2,where T1=R1-U02R2-1U02T, T2=R2-U01TR1-1U01.Lemma 3 (QGFMI [33]).
For any given matricesX, Y, a positive definite matrix R and a continuous differentiable function {ω(s)∣s∈[a,b]}, the following inequality holds(10)-∫abωTsRωsds≤η0η1Tb-aXR-1XTXI0∗b-a3YR-1YT+sym⁡Y-I2Iη0η1,where η0 is an any vector, and η1T=∫abωT(s)ds1/b-a∫ab∫θbωT(s)dsdθ.Lemma 4 (see [42]).
For a given quadratic functionl(s)=a2s2+a1s+a0, where ai∈R (i=0,1,2), h12=h2-h1, if the following inequalities hold(11)ilh1<0;iilh2<0;iii-h122a2+lh1<0,one has l(s)<0, for all s∈[h1,h2].Proof.
The proof is similar to lemma 2 of [42]. First, in the case of a2≥0, l is a convex function. So, (i) and (ii) guarantee l(s)<0, ∀s∈[h1,h2]. Next, for a2<0, l is a concave function. So, l(s)≤l˙(h2)(s-h2)+l(h2)=(2a2h2+a1)s-a2h22+a0≔g(s). Then g(h1)=-a2h122+a2h12+a1h1+a0=-a2h122+l(h1)<0 from (iii) and g(h2)=l(h2)<0 from (ii) guarantee that l(s)<0, for all l∈[h1,h2]. This completes the proof.Remark 5.
It is interesting to note that, in Lemma4, when h1=0, inequalities (11) can be rewritten in those of lemma 2 in [42]. Hence the established Lemma 4 covers the lemma in [42].Lemma 6 (see [43]).
Given matricesΓ, Ξ, and Ω=ΩT, the following inequality(12)Ω+ΓFσΞ+ΞTFTσΓT<0holds for any F(σ) satisfying FT(σ)F(σ)≤I, if and only if there exists a scalar ε>0 such that(13)Ω+ε-1ΓΓT+εΞTΞ<0.Remark 7.
Recently, [29] improved the stability results of the uncertain neutral-type Lur’e system (1) by combining the extended double integral with Wirtinger-based inequalities technique. In practice, it is known that the range of delay with nonzero lower bound is often encountered, and such systems are referred to as interval time-delay systems. So, both the range of delay with zero lower bound and that with nonzero lower bound are considered in this paper. In addition, the lower bound of the delay derivative is also involved in this paper, which is not mentioned in [29].


## 3. Main Results

### 3.1. Absolute Stability Criteria for Nominal Form

In this section, we will investigate the robustly absolute stability problem of the system (1). First, we give an absolute stability criterion for nominal form of system (1) without uncertainties described as(14)x˙t-Cx˙t-τt=Axt+A1xt-ht+Bfσt,σt=HTxt,∀t≥0,xs=φs,x˙s=φ˙s,s∈-max⁡h2,τ,0,ht∈C.1s∈-max⁡h,τ,0,ht∈C.2.For the sake of simplicity on matrix representation, the notations of several symbols and matrices are defined as Box1 of Appendix A. The following theorem will give an absolute stability criterion for Lur’e system (14) satisfying the conditionsC. 1 and (5).Box 1:Notations of several symbols and matrices in Theorems 8 and 14.
h 1 t = h ( t ) - h 1 ,   h 2 t = h 2 - h ( t ) ,   h 12 = h 2 - h 1, h d = 1 - h ˙ ( t ) ,   μ - 1 = μ 1 + h ˙ ( t ) ,   γ T ( s ) = x T ( s )   x ˙ T ( s ), v 1 ( t ) =  ∫ t - h 1 t x T ( s ) h 1 d s ,   v 2 ( t ) =  ∫ t - h ( t ) t - h 1 x T ( s ) h 1 t d s ,   v 3 ( t ) =  ∫ t - h 2 t - h ( t ) x T ( s ) h 2 t d s, ω 1 ( t ) = h 1 v 1 ( t ) ,   ω 2 ( t ) = h 1 t v 2 ( t ) ,   ω 3 ( t ) = h 2 t v 3 ( t ), ζ T t = x T ( t ) x T ( t - h 1 ) x T t - h t x T ( t - h 2 ) ω 1 ( t ) ω 2 ( t ) ω 3 ( t ), ζ 1 T t = x T ( t ) x T ( t - h 1 ) x T t - h t x T ( t - h 2 ) v 2 ( t ), ζ 2 T t = x T ( t ) x T ( t - h 1 ) x T t - h t x T ( t - h 2 ) v 3 ( t ), Δ T ( t ) = ω 2 T ( t )   x T ( t - h 1 ) - x T ( t - h ( t ) )   ω 3 T ( t )   x T ( t - h ( t ) ) - x T ( t - h 2 ), ξ T t = [ x T t   x T t - h 1   x T t - h t   x T t - h 2   x ˙ T t   x ˙ T t - h 1   x ˙ T t - h t   x ˙ T t - h 2   v 1 t   v 2 t   v 3 t ∫ t - h 1 t ∫ u t x T s / h 1 d u  d s   ∫ t - h t t - h 1 ∫ u t - h 1 x T s / h 1 t d u  d s   ∫ t - h 2 t - h t ∫ u t - h t x T s / h 2 t d u  d s   x ˙ T t - τ t   f T σ t ], η 1 t = ω 2 ( t ) x ( t - h 1 ) - x ( t - h ( t ) ) ,   η 2 t = ω 3 ( t ) x ( t - h ( t ) ) - x ( t - h 2 ) .Theorem 8.
The system (14) satisfying the conditions (2) and (5) is absolutely stable for given values of h2≥h1≥0, μ1, μ2<1, τd<1 and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R7n×7n, (Pa,Pb∈R5n×5n), (Qa,Qb,R0a,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (Q1,R0,R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm} and any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), X0∈R3n×2n, Xi∈R5n×2n, Yλ∈R4n×2n(i=1,…,4;λ=0,…,4) such that the following LMIs hold for [h(t),h˙(t)]∈{[h1,h2]×[μ1,μ2]}:(15)Q¯a>0,Q¯b>0,R¯0a>0,R¯a>0,R¯b>0,Ω1ht>0,Ω2ht>0,(16)Ω4h1+1h12JTΩ3h1JE1U02∗h12Ω2h1>0,Ω4h2+1h12JTΩ3h2JE2U01T∗h12Ω1h2>0,(17)lh1,h˙t,α=Πh1,h˙tΩ01Ω02Ωa1,4μ1¯Ωa1,2hdΩb2,1,3h1Ωb2,1,1h1∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-h12R¯b000∗∗∗∗-μ1¯h12Q¯b00∗∗∗∗∗-3hdh12R¯a0∗∗∗∗∗∗-3h12Q¯a<0,(18)lh1,h˙t,1-α=Πh1,h˙tΩ01Ω02Ωa1,4μ1¯Ωa1,2Ωb3,2,4h1μ1¯Ωb3,2,2h1∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-h12R¯b000∗∗∗∗-μ1¯h12Q¯b00∗∗∗∗∗-3h12R¯b0∗∗∗∗∗∗-3μ1¯h12Q¯b<0,(19)lh2,h˙t,α=Πh2,h˙tΩ01Ω02hdΩa1,3Ωa1,1hdΩb2,1,3h2Ωb2,1,1h2∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-hdh12R¯a000∗∗∗∗-h12Q¯a00∗∗∗∗∗-3hdh12R¯a0∗∗∗∗∗∗-3h12Q¯a<0,(20)lh2,h˙t,1-α=Πh2,h˙tΩ01Ω02hdΩa1,3Ωa1,1Ωb3,2,4h2μ1¯Ωb3,2,2h2∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-hdh12R¯a000∗∗∗∗-h12Q¯a00∗∗∗∗∗-3h12R¯b0∗∗∗∗∗∗-3μ1¯h12Q¯b<0,(21)-a2h122+lh1,h˙t,α<0,-a2h122+lh1,h˙t,1-α<0,where the related notations are defined in Box 3 of Appendix B.Proof.
Construct anLKF candidate as(22)Vt=∑i=15Vitwith(23)V1t=ζTtPζt+h1tζ1TtPaζ1t+h2tζ2TtPbζ2t,V2t=∫t-htt-h1γTsQ1tγsds+∫t-h2t-htγTsQ2tγsds,V3t=∫t-h1tγTsQ1γsds+∫t-τttx˙TsQ2x˙sds+2∑i=1mλi∫0σifiσidσi,V4t=∫t-htt-h1∫θt-h1γTsQ¯aγsdsdθ+μ1∫t-h2t-ht∫θt-htγTsQ¯bγsdsdθV5t=∫t-h1t∫θtγTsR0γsdsdθ+∫t-htt-h1∫θt-h1γTsR1γsdsdθ+∫t-h2t-ht∫θt-htγTsR2γsdsdθ,where notations of several symbols and matrices can be found in Boxes 1 and 3 of Appendixes A and B.
First step, because the positive definiteness of the Lyapunov matricesP, Pa, and Pb is not required, the positive definiteness of the LKF (22) should be proved. The Pa- and Pb- dependent terms can be rewritten as(24)h1tζ1TtPaζ1t+h2tζ2TtPbζ2t=xtxt-h1xt-htxt-h20Th1tPa+h2tPbxtxt-h1xt-htxt-h20+2xtxt-h1xt-htxt-h20TPa0000ω2t+Pb0000ω3t+ω2TtEPaETω2th1t+ω3TtEPbETω3th2t,where E=0000I.
Based onQ1(t)>0, Q2(t)>0 and Jensen’s inequality, the V2(t) term can be estimated as(25)V2t≥η1TtQ1tη1th1t+η2TtQ2tη2th2t.According to Ωi[h(t)]>0(i=1,2) and Lemma 2, we can obtain the following inequality from (24) and (25)(26)V2t+ω2TtEPaETω2th1t+ω3TtEPbETω3th2t≥η1TtEPaET000+Q1tη1th1t+η2TtEPbET000+Q2tη2th2t≥ΔTtΩ3hth12Δt-1-αh12η1TtU02Ω2ht-1U02Tη1t-αh12η2TtU01TΩ1ht-1U01η2t.It follows from (15)-(16), (22), (24), (25), and (26) and Q1>0, Q2>0, Ri>0(i=0,1,2) that(27)Vt>0.Thus, the LKF (22) is positive definite.
Second step, the time derivative ofV(t) with respect to time along the trajectory of the system (14) is as follows:(28)V˙1t=h˙tζ1TtPaζ1t-h˙tζ2TtPbζ2t+2ζTtPζ˙t+2h1tζ1TtPaζ˙1t+2h2tζ2TtPbζ˙2t=h˙tζ1TtPaζ1t-h˙tζ2TtPbζ2t+2ζTtPx˙tx˙t-h1hdx˙t-htx˙t-h2xt-xt-h1xt-h1-hdxt-hthdxt-ht-xt-h2+2h1tζ1TtPax˙tx˙t-h1hdx˙t-htx˙t-h2xt-h1-hdxt-ht-h˙tv2th1t+2h2tζ2TtPbx˙tx˙t-h1hdx˙t-htx˙t-h2hdxt-ht-xt-h2+h˙tv3th2t,(29)V˙2t=γTt-h1Q1tγt-h1+hdγTt-htQ2t-Q1tγt-ht-γTt-h2Q2tγt-h2-h˙t∫t-htt-h1γTsQ11γsds-h˙t∫t-h2t-htγTsQ21γsds,(30)V˙3t≤γTtQ1γt-γTt-h1Q1γt-h1+x˙TtQ2x˙t-1-τdx˙Tt-τtQ2x˙t-τt+2fTσtΛHTx˙t,(31)V˙4t=h1tγTt-h1Q¯aγt-h1+μ1hdh2tγTt-htQ¯bγt-ht-hd∫t-htt-h1γTsQ¯aγsds-μ1∫t-h2t-htγTsQ¯bγsds,(32)V˙5t=h1γTtR0γt+h1tγTt-h1R1γt-h1+hdh2tγTt-htR2γt-ht-∫t-h1tγTsR0γsds-hd∫t-htt-h1γTsR1γsds-∫t-h2t-htγTsR2γsds.For additional symmetric matrices Qa, Qb, R0a, Ra, and Rb the following zero equations are satisfied(33)0=h˙txTt-h1Qaxt-h1-xTt-htQaxt-ht-2∫t-htt-h1xTsQax˙sds+xTt-htQbxt-ht-xTt-h2Qbxt-h2-2∫t-h2t-htxTsQbx˙sds,(34)0=xTtR0axt-xTt-h1R0axt-h1-2∫t-h1txTsR0ax˙sds,(35)0=hdxTt-h1Raxt-h1-xTt-htRaxt-ht-2∫t-htt-h1xTsRax˙sds+xTt-htRbxt-ht-xTt-h2Rbxt-h2-2∫t-h2t-htxTsRbx˙sds.Taking the zero inequalities in V˙2 and V˙4, we have the following integral terms.(36)φ=-∫t-htt-h1γTsQ¯aγsds-μ1+h˙t∫t-h2t-htγTsQ¯bγsds-∫t-h1tγTsR¯0aγsds-hd∫t-htt-h1γTsR¯aγsds-∫t-h2t-htγTsR¯bγsds.It follows from Lemma 3 with an augmented vector γ(s) that (37)-∫t-h1tγTsR¯0aγsds≤W01TW02TTh1X0R¯0a-1X0TX0H01∗h13G0Y0R¯0a-1Y0TG0+sym⁡G0Y0H02W01TW02T,(38)-∫t-htt-h1γTsQ¯aγsds≤W1TW2TTh1tX1Q¯a-1X1TX1H1∗h1t3G1Y1Q¯a-1Y1TG1+sym⁡G1Y1H3W1TW2T,(39)-μ¯1∫t-h2t-htγTsQ¯bγsds≤μ¯1W1TW3TTh2tX2Q¯b-1X2TX2H2∗h2t3G2Y2Q¯b-1Y2TG2+sym⁡G2Y2H4W1TW3T,(40)-hd∫t-htt-h1γTsR¯aγsds≤hdW1TW2TTh1tX3R¯a-1X3TX3H1∗h1t3G1Y3R¯a-1Y3TG1+sym⁡G1Y3H3W1TW2T,(41)-∫t-h2t-htγTsR¯bγsdsW1TW3TTh2tX4R¯b-1X4TX4H2∗h2t3G2Y4R¯b-1Y4TG2+sym⁡G2Y4H4W1TW3T.For any appropriately dimensioned matrices U¯=[U1TU2TU3TU4T]T, it is true that(42)0=2xTtx˙Ttx˙Tt-τtfTσtU¯Axt+A1xt-ht+Bfσt+Cx˙t-τt-x˙t.Letting S=diag⁡{s1,s2,…,sm}>0, it follows from (5) that(43)2fTσtSKHTxt-fTσt≥0.Finally, from the above derivation, we have(44)V˙t≤ξTtΠht,h˙t+h1W01X0R¯0a-1X0TW01T+13W02G0Y0R¯0a-1Y0TG0W02T+h1thdW1X3R¯a-1X3TW1T+W1X1Q¯a-1X1TW1T+h2tW1X4R¯b-1X4TW1T+μ1¯W1X2Q¯b-1X2TW1T+h1thd3W2G1Y3R¯a-1Y3TG1W2T+13W2G1Y1Q¯a-1Y1TG1W2T+h2t13W3G2Y4R¯b-1Y4TG2W3T+μ1¯3W3G2Y2Q¯b-1Y2TG2W3Tξt=ξTtΠht,h˙t+h1W01X0R¯0a-1X0TW01T+13W02G0Y0R¯0a-1Y0TG0W02T+h1thdW1X3R¯a-1X3TW1T+W1X1Q¯a-1X1TW1T+h2tW1X4R¯b-1X4TW1T+μ1¯W1X2Q¯b-1X2TW1T+αh12hd3W2G1Y3R¯a-1Y3TG1W2T+13W2G1Y1Q¯a-1Y1TG1W2T+1-αh1213W3G2Y4R¯b-1Y4TG2W3T+μ1¯3W3G2Y2Q¯b-1Y2TG2W3Tξtwith α=h(t)-h1/h12≥0, 1-α=h2-h(t)/h12≥0.
Therefore, LMIs (17)–(21) hold, which together withSchur complement equivalence, Lemma 4 and the convex function theory imply that V˙(t)<0. Hence, it follows from the Lyapunov stability theory that the nominal system (14) is absolutely stable for any nonlinear function f(σ(t)) satisfying (5). From Definition 1, this completes the proof.The following theorem will give an absolute stability criterion for the Lur’e system (14) satisfying the conditionsC. 2 and (5).Corollary 9.
The system (14) satisfying the conditions (3) and (5) is absolutely stable for given values of h≥0, μ1,μ2<1, τd<1, and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R5n×5n, (Pa,Pb∈R4n×4n), (Qa,Qb,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm} and any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), Xi∈R5n×2n, Yi∈R4n×2n(i=1,…,4) such that LMIs (15) and (16) and the following LMIs hold for [h(t),h˙(t)]∈{[0,h]×[μ1,μ2]}:(45)l0,h˙t,α=Π0,h˙tΩa1,4μ1¯Ωa1,2hdΩb2,1,30Ωb2,1,10∗-hR¯b000∗∗-μ1¯hQ¯b00∗∗∗-3hdhR¯a0∗∗∗∗-3hQ¯a<0,(46)l0,h˙t,1-α=Π0,h˙tΩa1,4μ1¯Ωa1,2Ωb3,2,40μ1¯Ωb3,2,20∗-hR¯b000∗∗-μ1¯hQ¯b00∗∗∗-3hR¯b0∗∗∗∗-3μ1¯hQ¯b<0,(47)lh,h˙t,α=Πh,h˙thdΩa1,3Ωa1,1hdΩb2,1,3hΩb2,1,1h∗-hdhR¯a000∗∗-hQ¯a00∗∗∗-3hdhR¯a0∗∗∗∗-3hQ¯a<0,(48)lh,h˙t,1-α=Πh,h˙thdΩa1,3Ωa1,1Ωb3,2,4hμ1¯Ωb3,2,2h∗-hdhR¯a000∗∗-hQ¯a00∗∗∗-3hR¯b0∗∗∗∗-3μ1¯hQ¯b<0,(49)-a2h2+l0,h˙t,α<0,-a2h2+l0,h˙t,1-α<0,where the related notations are defined in Box 4 of Appendix C.Proof.
TheLKF (22) can be reduced to the following one by taking h1=0, h2=h, Q1=0, and R0=0:(50)V~t=∑i=14V~itwith(51)V~1t=ζTtPζt+htζ1TtPaζ1t+h-htζ2TtPbζ2t,V~2t=∫t-httγTsQ1tγsds+∫t-ht-htγTsQ2tγsds,V~3t=∫t-τttx˙TsQ2x˙sds+2∑i=1mλi∫0σifiσidσi,V~4t=∫t-htt∫θtγTsQ¯aγsdsdθ+μ1∫t-ht-ht∫θt-htγTsQ¯bγsdsdθV~5t=∫t-htt∫θtγTsR1γsdsdθ+∫t-ht-ht∫θt-htγTsR2γsdsdθ,where notations of several symbols and matrices can be found in Boxes 2 and 4 of Appendixes A and C. The proof of Corollary 9 is omitted because of the similarity to Theorem 8.Box 2:Notations of several symbols and matrices in Corollaries 9 and 15.
h d = 1 - h ˙ t ,   γ T s = x T s   x ˙ T s ,   μ - 1 = μ 1 + h ˙ ( t ), v 1 ( t ) =  ∫ t - h ( t ) t x T ( s ) h ( t ) d s ,   v 2 ( t ) =  ∫ t - h t - h ( t ) x T ( s ) h - h ( t ) d s, ω 1 t = h t v 1 t ,   ω 2 t = h - h t v 2 t ,   ζ T t = x T t   x T t - h t   x T t - h   ω 1 t   ω 2 t, ζ 1 T t = x T t   x T t - h t   x T t - h   v 1 t ,   ζ 2 T t = x T t   x T t - h t   x T t - h   v 2 t, Δ T ( t ) = ω 1 T ( t )   x T ( t ) - x T ( t - h ( t ) )   ω 2 T ( t )   x T ( t - h ( t ) ) - x T ( t - h ), ξ T t = [ x T t   x T t - h t   x T t - h   x ˙ T t   x ˙ T t - h t   x ˙ T t - h   v 1 t   v 2 t, ∫ t - h t t ∫ u t x T s / h t d u  d s   ∫ t - h t - h t ∫ u t - h t x T s / h - h t d u  d s   x ˙ T t - τ t   f T σ t ], η 1 t = ω 1 ( t ) x ( t ) - x ( t - h ( t ) ) ,   η 2 t = ω 2 ( t ) x ( t - h ( t ) ) - x ( t - h ) .Remark 10.
Theorem8 and Corollary 9 can reduce the conservatism of stability conditions based LMI via the LKFs (22) and (50) application. For nonintegral item V1(t), the matrices P, Pa, and Pb are just symmetrical, not always positive definite, and Q1(t) and Q2(t) of the single-integral item V2(t) are delay-dependent matrices which can also further improve the utilization of time delay and its derivative information. When proving V(t)>0, we calculate V1(t) and V2(t) as a whole applying Lemma 2, which may expand the feasible regions of LMIs (15) and (16). When bounded the derivative of the LKFs, three additional zero equations (33)–(35) and Lemma 3 have been used to narrow the gap between the upper bound and the true value, which may be another contribution of reducing the conservatism of stability conditions.Remark 11.
It is worth noting that the delay-product-type item was introduced firstly by C. Zhang et al. [31, 44]; another novel LKF with delay-dependent matrix was constructed by W. Kwon et al. [33], where some improved stability conditions for linear systems with time-varying delay were given. C. Zhang and W. Kwon et al. pointed that the LKFs with delay-product-type item or delay-dependent matrix may reduce the conservatism of stability conditions based on the same inequality scaling technique. The LKFs (22) and (50) constructed in this paper, which combine the advantage of the delay-product-type item and delay-dependent matrix, are more general than those given in [31, 33]. In fact, letting Q11=Q21=0, Q20=diag⁡{Q2,0}, Q¯a=Q¯b=0, and R1=R2, the LKF (50) can be reduced to the LKF (18) of [31]. And taking Pa=Pb=0, Q¯a=Q¯b=0, and R1=R2 the LKF (50) can be reduced to the LKF (16) of [33]. However, the derivation method of [31, 33] cannot be applied directly. The positive definiteness of the LKF and negative definiteness of the LKF’s derivative are proved in Theorem 8.Remark 12.
It is worth noting that in [33], to bound the integral item for -μ≤h˙(t)≤μ,(μ>0)(52)-h˙t∫t-httγTQ¯aγds-h˙t∫t-ht-htγTQ¯bγdsvia the QGFMI technique, the following addition zero equation was introduced(53)0=μ∫t-httγTsQ¯aγsds-μ∫t-httγTsQ¯aγsds+μ∫t-ht-htγTsQ¯bγsds-μ∫t-ht-htγTsQ¯bγsds.
Then, the above integral item can be rewritten as the following form:(54)φ~=-μ+h˙t∫t-httγTQ¯aγds-μ+h˙t∫t-ht-htγTQ¯bγds+μ∫t-httγTQ¯aγds+μ∫t-ht-htγTQ¯bγds.
The first two items on the right can be bounded via the QGFMI technique like (39) of this paper; however, there are fewer proper techniques for obtaining a tight upper bound of the integral terms μ∫t-h(t)tγT(s)Q¯aγ(s)ds and μ∫t-ht-h(t)γT(s)Q¯bγ(s)ds due to their positive definiteness. Thus, to avoid introducing the two positive define integral terms, we give the modified LKF (22) with a double integral item V4(t).Remark 13.
In addition, the main result of [33] was not LMI due to the terms h2(t) even h3(t). In this paper, all matrices inequations of Theorem 8 and Corollary 9 are LMI via Lemma 4 application, which can be solved easily by using Matlab LMI-toolbox. Moreover, the double integral items V4(t) and V5(t) of the LKF (22) divide the time-delay interval [h1,h2] into two subintervals, that is, [h1,h(t)] and [h(t),h2], instead of using the item ∫t-htγT(s)Rγ(s)ds directly, which further make full use of the information of time-varying delays h(t)-h1, h2-h(t) and their derivative h˙(t). Thus, the QGFMI technique can be used fully in each subinterval, which can further reduce the conservatism of the stability conditions.


### 3.2. Robustly Absolute Stability Criteria for Uncertain Form

Next, we extend the obtained absolute stability conditions to robustly absolute stability problem for the uncertain neutral-type Lur’e system (1) with time-varying parameter uncertainties satisfying (7) and (8).Theorem 14.
The system (1) satisfying the conditions (2), (5), (7), and (8) is robustly absolutely stable for given values of h2≥h1≥0, μ1,μ2<1, τd<1, and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R7n×7n, (Pa,Pb∈R5n×5n), (Qa,Qb,R0a,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (Q1,R0,R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm}, any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), X0∈R3n×2n, Xi∈R5n×2n, Yλ∈R4n×2n(i=1,…,4;λ=0,…,4) and ε>0 such that LMIs (15)-(16) and the following LMIs hold for [h(t),h˙(t)]∈{[h1,h2]×[μ1,μ2]}:(55)J1lh1,h˙t,αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(56)J1lh1,h˙t,1-αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(57)J1lh2,h˙t,αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(58)J1lh2,h˙t,1-αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(59)-a2h122+J1lh1,h˙t,αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(60)-a2h122+J1lh1,h˙t,1-αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,where(61)Φ1=e1EaT+e2Ea1T+e16EbTT,Φ2=e1U1D+e5U2D+e15U3D+e16U4DT,J1T=IIIIIII0,J2T=I0000000,J3T=0000000I.Proof.
If we replaceA, A1, and B in LMIs (17)–(21) with A+DF(t)Ea, A1+DF(t)Ea1, B+DF(t)Eb, respectively, Theorem 14 can be proved based on Lemma 6 easily.The following corollary will give the robustly absolute stability criterion for the Lur’e system (1) satisfying the conditionC. 2.Corollary 15.
System (1) satisfying the conditions (3) and (5), (7), and (8) is robustly absolutely stable for given values of h≥0, μ1,μ2<1, τd<1, and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R5n×5n, (Pa,Pb∈R4n×4n), (Qa,Qb,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm}, any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), Xi∈R5n×2n, Yi∈R4n×2n(i=1,…,4) and ε>0 such that LMIs (15)-(16) and the following LMIs hold for [h(t),h˙(t)]∈{[h1,h2]×[μ1,μ2]}:(62)J~1l0,h˙t,αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(63)J~1l0,h˙t,1-αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(64)J~1lh,h˙t,αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(65)J~1lh,h˙t,1-αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(66)-a2h2+J~1l0,h˙t,αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(67)-a2h2+J~1l0,h˙t,1-αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,where (68)Φ~1=e1EaT+e2Ea1T+e12EbTT,Φ~2=e1U1D+e4U2D+e11U3D+e12U4DT,J~1T=IIIII0,J~2T=I00000,J~3T=00000I.Remark 16.
If the nonlinear functionf(σ) in the feedback path satisfies the infinite sector conditions (6), for any si≥0,i=1,2,…,m, it follows from (6) that(69)sifiσihiTxt≥0,which is equivalent to(70)2xTtHSfσ≥0,where S=diag⁡s1,s2,…,sm.
Therefore, the corresponding absolute and robustly absolute stability criteria can be obtained by replacing the matrixΘ1 of Theorems 8 and 14 and Corollaries 9 and 15 with Θ2=e1HSe16T and Θ~2=e1HSe12T, respectively.Remark 17.
For one special case, in the absence of the nonlinear function, that is, in case ofB=0, the system (1) is simply written as the following linear neutral system(71)x˙t-Cx˙t-τt=A+ΔAxt+A1+ΔA1xt-ht,xs=φs,x˙s=φ˙s,s∈-max⁡h2,τ,0,ht∈C.1(72)s∈-max⁡h,τ,0,ht∈C.2,and for another special case, in the absence of the nonlinear function and neutral-type item, that is, in case of B=0 and C=0, the system (1) is simply written as the following linear time-delayed system:(73)x˙t=A+ΔAxt+A1+ΔA1xt-ht,xs=φs,s∈-max⁡h2,τ,0,ht∈C.1s∈-max⁡h,τ,0,ht∈C.2.Obviously, take Theorem 8 as an example. Only letting Θ1=0, Π4=e1U1+e5U2+e15U3 and Π5=Ae1T+A1e2T+Ce15T-e5T, the stability criteria proposed in this paper are also applied to this linear neutral system (71) with time-varying delays; taking Θ1=0, Q2=0, Π4=e1U1+e5U2 and Π5=Ae1T+A1e2T, the stability criteria proposed in this paper are also applied to this linear system (73) with time-varying delays. We will not elaborate here due to the limited space available. However, some results of detailed comparisons will be given in the next section directly.Remark 18.
It is worth pointing out that in this paper, the upper and lower bounds constraints of the time-delay derivative areμ1≤h˙(t)≤μ2, where τd<1,μ2<1 due to the diagonal entry -(1-τ˙(t))Q2, -(1-h˙(t))x˙T(t-h(t))(Q1(t)-Q2(t))x˙(t-h(t)) in Π[h(t),h˙(t)], and -(1-h˙(t))∫t-h(t)tγT(s)R1γ(s)ds in V˙5(t). Similar to Remark 5 of [45], one can establish a stability criterion for system (1) in the case τd≥1 and μ2≥1 if ξ(t),ζ(t),ζ1(t),ζ2(t),γ(s) and V5(t) are replaced with(74)ξ¯Tt=xTtxTt-h1xTt-htxTt-h2x˙Ttx˙Tt-h1x˙Tt-h2v1tv2tv3t∫t-h1t∫utxTsh1duds∫t-htt-h1∫ut-h1xTsh1tduds∫t-h2t-ht∫ut-htxTsh2tdudsfTσt,ζ¯Tt=xTtxTt-h1xTt-h2ω1tω2tω3t,ζ¯1Tt=xTtxTt-h1xTt-h2v2t,ζ¯2Tt=xTtxTt-h1xTt-h2v3t,γ¯Ts=xTs,Q2=0,V¯5t=∫t-h1t∫θt-h1xTsR0xsdsdθ+∫t-h2t-h1∫θt-h1xTsR1xsdsdθ.Due to page limitation, this result is omitted.


## 3.1. Absolute Stability Criteria for Nominal Form

In this section, we will investigate the robustly absolute stability problem of the system (1). First, we give an absolute stability criterion for nominal form of system (1) without uncertainties described as(14)x˙t-Cx˙t-τt=Axt+A1xt-ht+Bfσt,σt=HTxt,∀t≥0,xs=φs,x˙s=φ˙s,s∈-max⁡h2,τ,0,ht∈C.1s∈-max⁡h,τ,0,ht∈C.2.For the sake of simplicity on matrix representation, the notations of several symbols and matrices are defined as Box1 of Appendix A. The following theorem will give an absolute stability criterion for Lur’e system (14) satisfying the conditionsC. 1 and (5).Box 1:Notations of several symbols and matrices in Theorems 8 and 14.
h 1 t = h ( t ) - h 1 ,   h 2 t = h 2 - h ( t ) ,   h 12 = h 2 - h 1, h d = 1 - h ˙ ( t ) ,   μ - 1 = μ 1 + h ˙ ( t ) ,   γ T ( s ) = x T ( s )   x ˙ T ( s ), v 1 ( t ) =  ∫ t - h 1 t x T ( s ) h 1 d s ,   v 2 ( t ) =  ∫ t - h ( t ) t - h 1 x T ( s ) h 1 t d s ,   v 3 ( t ) =  ∫ t - h 2 t - h ( t ) x T ( s ) h 2 t d s, ω 1 ( t ) = h 1 v 1 ( t ) ,   ω 2 ( t ) = h 1 t v 2 ( t ) ,   ω 3 ( t ) = h 2 t v 3 ( t ), ζ T t = x T ( t ) x T ( t - h 1 ) x T t - h t x T ( t - h 2 ) ω 1 ( t ) ω 2 ( t ) ω 3 ( t ), ζ 1 T t = x T ( t ) x T ( t - h 1 ) x T t - h t x T ( t - h 2 ) v 2 ( t ), ζ 2 T t = x T ( t ) x T ( t - h 1 ) x T t - h t x T ( t - h 2 ) v 3 ( t ), Δ T ( t ) = ω 2 T ( t )   x T ( t - h 1 ) - x T ( t - h ( t ) )   ω 3 T ( t )   x T ( t - h ( t ) ) - x T ( t - h 2 ), ξ T t = [ x T t   x T t - h 1   x T t - h t   x T t - h 2   x ˙ T t   x ˙ T t - h 1   x ˙ T t - h t   x ˙ T t - h 2   v 1 t   v 2 t   v 3 t ∫ t - h 1 t ∫ u t x T s / h 1 d u  d s   ∫ t - h t t - h 1 ∫ u t - h 1 x T s / h 1 t d u  d s   ∫ t - h 2 t - h t ∫ u t - h t x T s / h 2 t d u  d s   x ˙ T t - τ t   f T σ t ], η 1 t = ω 2 ( t ) x ( t - h 1 ) - x ( t - h ( t ) ) ,   η 2 t = ω 3 ( t ) x ( t - h ( t ) ) - x ( t - h 2 ) .Theorem 8.
The system (14) satisfying the conditions (2) and (5) is absolutely stable for given values of h2≥h1≥0, μ1, μ2<1, τd<1 and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R7n×7n, (Pa,Pb∈R5n×5n), (Qa,Qb,R0a,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (Q1,R0,R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm} and any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), X0∈R3n×2n, Xi∈R5n×2n, Yλ∈R4n×2n(i=1,…,4;λ=0,…,4) such that the following LMIs hold for [h(t),h˙(t)]∈{[h1,h2]×[μ1,μ2]}:(15)Q¯a>0,Q¯b>0,R¯0a>0,R¯a>0,R¯b>0,Ω1ht>0,Ω2ht>0,(16)Ω4h1+1h12JTΩ3h1JE1U02∗h12Ω2h1>0,Ω4h2+1h12JTΩ3h2JE2U01T∗h12Ω1h2>0,(17)lh1,h˙t,α=Πh1,h˙tΩ01Ω02Ωa1,4μ1¯Ωa1,2hdΩb2,1,3h1Ωb2,1,1h1∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-h12R¯b000∗∗∗∗-μ1¯h12Q¯b00∗∗∗∗∗-3hdh12R¯a0∗∗∗∗∗∗-3h12Q¯a<0,(18)lh1,h˙t,1-α=Πh1,h˙tΩ01Ω02Ωa1,4μ1¯Ωa1,2Ωb3,2,4h1μ1¯Ωb3,2,2h1∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-h12R¯b000∗∗∗∗-μ1¯h12Q¯b00∗∗∗∗∗-3h12R¯b0∗∗∗∗∗∗-3μ1¯h12Q¯b<0,(19)lh2,h˙t,α=Πh2,h˙tΩ01Ω02hdΩa1,3Ωa1,1hdΩb2,1,3h2Ωb2,1,1h2∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-hdh12R¯a000∗∗∗∗-h12Q¯a00∗∗∗∗∗-3hdh12R¯a0∗∗∗∗∗∗-3h12Q¯a<0,(20)lh2,h˙t,1-α=Πh2,h˙tΩ01Ω02hdΩa1,3Ωa1,1Ωb3,2,4h2μ1¯Ωb3,2,2h2∗-h1R¯0a00000∗∗-3h1R¯0a0000∗∗∗-hdh12R¯a000∗∗∗∗-h12Q¯a00∗∗∗∗∗-3h12R¯b0∗∗∗∗∗∗-3μ1¯h12Q¯b<0,(21)-a2h122+lh1,h˙t,α<0,-a2h122+lh1,h˙t,1-α<0,where the related notations are defined in Box 3 of Appendix B.Proof.
Construct anLKF candidate as(22)Vt=∑i=15Vitwith(23)V1t=ζTtPζt+h1tζ1TtPaζ1t+h2tζ2TtPbζ2t,V2t=∫t-htt-h1γTsQ1tγsds+∫t-h2t-htγTsQ2tγsds,V3t=∫t-h1tγTsQ1γsds+∫t-τttx˙TsQ2x˙sds+2∑i=1mλi∫0σifiσidσi,V4t=∫t-htt-h1∫θt-h1γTsQ¯aγsdsdθ+μ1∫t-h2t-ht∫θt-htγTsQ¯bγsdsdθV5t=∫t-h1t∫θtγTsR0γsdsdθ+∫t-htt-h1∫θt-h1γTsR1γsdsdθ+∫t-h2t-ht∫θt-htγTsR2γsdsdθ,where notations of several symbols and matrices can be found in Boxes 1 and 3 of Appendixes A and B.
First step, because the positive definiteness of the Lyapunov matricesP, Pa, and Pb is not required, the positive definiteness of the LKF (22) should be proved. The Pa- and Pb- dependent terms can be rewritten as(24)h1tζ1TtPaζ1t+h2tζ2TtPbζ2t=xtxt-h1xt-htxt-h20Th1tPa+h2tPbxtxt-h1xt-htxt-h20+2xtxt-h1xt-htxt-h20TPa0000ω2t+Pb0000ω3t+ω2TtEPaETω2th1t+ω3TtEPbETω3th2t,where E=0000I.
Based onQ1(t)>0, Q2(t)>0 and Jensen’s inequality, the V2(t) term can be estimated as(25)V2t≥η1TtQ1tη1th1t+η2TtQ2tη2th2t.According to Ωi[h(t)]>0(i=1,2) and Lemma 2, we can obtain the following inequality from (24) and (25)(26)V2t+ω2TtEPaETω2th1t+ω3TtEPbETω3th2t≥η1TtEPaET000+Q1tη1th1t+η2TtEPbET000+Q2tη2th2t≥ΔTtΩ3hth12Δt-1-αh12η1TtU02Ω2ht-1U02Tη1t-αh12η2TtU01TΩ1ht-1U01η2t.It follows from (15)-(16), (22), (24), (25), and (26) and Q1>0, Q2>0, Ri>0(i=0,1,2) that(27)Vt>0.Thus, the LKF (22) is positive definite.
Second step, the time derivative ofV(t) with respect to time along the trajectory of the system (14) is as follows:(28)V˙1t=h˙tζ1TtPaζ1t-h˙tζ2TtPbζ2t+2ζTtPζ˙t+2h1tζ1TtPaζ˙1t+2h2tζ2TtPbζ˙2t=h˙tζ1TtPaζ1t-h˙tζ2TtPbζ2t+2ζTtPx˙tx˙t-h1hdx˙t-htx˙t-h2xt-xt-h1xt-h1-hdxt-hthdxt-ht-xt-h2+2h1tζ1TtPax˙tx˙t-h1hdx˙t-htx˙t-h2xt-h1-hdxt-ht-h˙tv2th1t+2h2tζ2TtPbx˙tx˙t-h1hdx˙t-htx˙t-h2hdxt-ht-xt-h2+h˙tv3th2t,(29)V˙2t=γTt-h1Q1tγt-h1+hdγTt-htQ2t-Q1tγt-ht-γTt-h2Q2tγt-h2-h˙t∫t-htt-h1γTsQ11γsds-h˙t∫t-h2t-htγTsQ21γsds,(30)V˙3t≤γTtQ1γt-γTt-h1Q1γt-h1+x˙TtQ2x˙t-1-τdx˙Tt-τtQ2x˙t-τt+2fTσtΛHTx˙t,(31)V˙4t=h1tγTt-h1Q¯aγt-h1+μ1hdh2tγTt-htQ¯bγt-ht-hd∫t-htt-h1γTsQ¯aγsds-μ1∫t-h2t-htγTsQ¯bγsds,(32)V˙5t=h1γTtR0γt+h1tγTt-h1R1γt-h1+hdh2tγTt-htR2γt-ht-∫t-h1tγTsR0γsds-hd∫t-htt-h1γTsR1γsds-∫t-h2t-htγTsR2γsds.For additional symmetric matrices Qa, Qb, R0a, Ra, and Rb the following zero equations are satisfied(33)0=h˙txTt-h1Qaxt-h1-xTt-htQaxt-ht-2∫t-htt-h1xTsQax˙sds+xTt-htQbxt-ht-xTt-h2Qbxt-h2-2∫t-h2t-htxTsQbx˙sds,(34)0=xTtR0axt-xTt-h1R0axt-h1-2∫t-h1txTsR0ax˙sds,(35)0=hdxTt-h1Raxt-h1-xTt-htRaxt-ht-2∫t-htt-h1xTsRax˙sds+xTt-htRbxt-ht-xTt-h2Rbxt-h2-2∫t-h2t-htxTsRbx˙sds.Taking the zero inequalities in V˙2 and V˙4, we have the following integral terms.(36)φ=-∫t-htt-h1γTsQ¯aγsds-μ1+h˙t∫t-h2t-htγTsQ¯bγsds-∫t-h1tγTsR¯0aγsds-hd∫t-htt-h1γTsR¯aγsds-∫t-h2t-htγTsR¯bγsds.It follows from Lemma 3 with an augmented vector γ(s) that (37)-∫t-h1tγTsR¯0aγsds≤W01TW02TTh1X0R¯0a-1X0TX0H01∗h13G0Y0R¯0a-1Y0TG0+sym⁡G0Y0H02W01TW02T,(38)-∫t-htt-h1γTsQ¯aγsds≤W1TW2TTh1tX1Q¯a-1X1TX1H1∗h1t3G1Y1Q¯a-1Y1TG1+sym⁡G1Y1H3W1TW2T,(39)-μ¯1∫t-h2t-htγTsQ¯bγsds≤μ¯1W1TW3TTh2tX2Q¯b-1X2TX2H2∗h2t3G2Y2Q¯b-1Y2TG2+sym⁡G2Y2H4W1TW3T,(40)-hd∫t-htt-h1γTsR¯aγsds≤hdW1TW2TTh1tX3R¯a-1X3TX3H1∗h1t3G1Y3R¯a-1Y3TG1+sym⁡G1Y3H3W1TW2T,(41)-∫t-h2t-htγTsR¯bγsdsW1TW3TTh2tX4R¯b-1X4TX4H2∗h2t3G2Y4R¯b-1Y4TG2+sym⁡G2Y4H4W1TW3T.For any appropriately dimensioned matrices U¯=[U1TU2TU3TU4T]T, it is true that(42)0=2xTtx˙Ttx˙Tt-τtfTσtU¯Axt+A1xt-ht+Bfσt+Cx˙t-τt-x˙t.Letting S=diag⁡{s1,s2,…,sm}>0, it follows from (5) that(43)2fTσtSKHTxt-fTσt≥0.Finally, from the above derivation, we have(44)V˙t≤ξTtΠht,h˙t+h1W01X0R¯0a-1X0TW01T+13W02G0Y0R¯0a-1Y0TG0W02T+h1thdW1X3R¯a-1X3TW1T+W1X1Q¯a-1X1TW1T+h2tW1X4R¯b-1X4TW1T+μ1¯W1X2Q¯b-1X2TW1T+h1thd3W2G1Y3R¯a-1Y3TG1W2T+13W2G1Y1Q¯a-1Y1TG1W2T+h2t13W3G2Y4R¯b-1Y4TG2W3T+μ1¯3W3G2Y2Q¯b-1Y2TG2W3Tξt=ξTtΠht,h˙t+h1W01X0R¯0a-1X0TW01T+13W02G0Y0R¯0a-1Y0TG0W02T+h1thdW1X3R¯a-1X3TW1T+W1X1Q¯a-1X1TW1T+h2tW1X4R¯b-1X4TW1T+μ1¯W1X2Q¯b-1X2TW1T+αh12hd3W2G1Y3R¯a-1Y3TG1W2T+13W2G1Y1Q¯a-1Y1TG1W2T+1-αh1213W3G2Y4R¯b-1Y4TG2W3T+μ1¯3W3G2Y2Q¯b-1Y2TG2W3Tξtwith α=h(t)-h1/h12≥0, 1-α=h2-h(t)/h12≥0.
Therefore, LMIs (17)–(21) hold, which together withSchur complement equivalence, Lemma 4 and the convex function theory imply that V˙(t)<0. Hence, it follows from the Lyapunov stability theory that the nominal system (14) is absolutely stable for any nonlinear function f(σ(t)) satisfying (5). From Definition 1, this completes the proof.The following theorem will give an absolute stability criterion for the Lur’e system (14) satisfying the conditionsC. 2 and (5).Corollary 9.
The system (14) satisfying the conditions (3) and (5) is absolutely stable for given values of h≥0, μ1,μ2<1, τd<1, and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R5n×5n, (Pa,Pb∈R4n×4n), (Qa,Qb,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm} and any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), Xi∈R5n×2n, Yi∈R4n×2n(i=1,…,4) such that LMIs (15) and (16) and the following LMIs hold for [h(t),h˙(t)]∈{[0,h]×[μ1,μ2]}:(45)l0,h˙t,α=Π0,h˙tΩa1,4μ1¯Ωa1,2hdΩb2,1,30Ωb2,1,10∗-hR¯b000∗∗-μ1¯hQ¯b00∗∗∗-3hdhR¯a0∗∗∗∗-3hQ¯a<0,(46)l0,h˙t,1-α=Π0,h˙tΩa1,4μ1¯Ωa1,2Ωb3,2,40μ1¯Ωb3,2,20∗-hR¯b000∗∗-μ1¯hQ¯b00∗∗∗-3hR¯b0∗∗∗∗-3μ1¯hQ¯b<0,(47)lh,h˙t,α=Πh,h˙thdΩa1,3Ωa1,1hdΩb2,1,3hΩb2,1,1h∗-hdhR¯a000∗∗-hQ¯a00∗∗∗-3hdhR¯a0∗∗∗∗-3hQ¯a<0,(48)lh,h˙t,1-α=Πh,h˙thdΩa1,3Ωa1,1Ωb3,2,4hμ1¯Ωb3,2,2h∗-hdhR¯a000∗∗-hQ¯a00∗∗∗-3hR¯b0∗∗∗∗-3μ1¯hQ¯b<0,(49)-a2h2+l0,h˙t,α<0,-a2h2+l0,h˙t,1-α<0,where the related notations are defined in Box 4 of Appendix C.Proof.
TheLKF (22) can be reduced to the following one by taking h1=0, h2=h, Q1=0, and R0=0:(50)V~t=∑i=14V~itwith(51)V~1t=ζTtPζt+htζ1TtPaζ1t+h-htζ2TtPbζ2t,V~2t=∫t-httγTsQ1tγsds+∫t-ht-htγTsQ2tγsds,V~3t=∫t-τttx˙TsQ2x˙sds+2∑i=1mλi∫0σifiσidσi,V~4t=∫t-htt∫θtγTsQ¯aγsdsdθ+μ1∫t-ht-ht∫θt-htγTsQ¯bγsdsdθV~5t=∫t-htt∫θtγTsR1γsdsdθ+∫t-ht-ht∫θt-htγTsR2γsdsdθ,where notations of several symbols and matrices can be found in Boxes 2 and 4 of Appendixes A and C. The proof of Corollary 9 is omitted because of the similarity to Theorem 8.Box 2:Notations of several symbols and matrices in Corollaries 9 and 15.
h d = 1 - h ˙ t ,   γ T s = x T s   x ˙ T s ,   μ - 1 = μ 1 + h ˙ ( t ), v 1 ( t ) =  ∫ t - h ( t ) t x T ( s ) h ( t ) d s ,   v 2 ( t ) =  ∫ t - h t - h ( t ) x T ( s ) h - h ( t ) d s, ω 1 t = h t v 1 t ,   ω 2 t = h - h t v 2 t ,   ζ T t = x T t   x T t - h t   x T t - h   ω 1 t   ω 2 t, ζ 1 T t = x T t   x T t - h t   x T t - h   v 1 t ,   ζ 2 T t = x T t   x T t - h t   x T t - h   v 2 t, Δ T ( t ) = ω 1 T ( t )   x T ( t ) - x T ( t - h ( t ) )   ω 2 T ( t )   x T ( t - h ( t ) ) - x T ( t - h ), ξ T t = [ x T t   x T t - h t   x T t - h   x ˙ T t   x ˙ T t - h t   x ˙ T t - h   v 1 t   v 2 t, ∫ t - h t t ∫ u t x T s / h t d u  d s   ∫ t - h t - h t ∫ u t - h t x T s / h - h t d u  d s   x ˙ T t - τ t   f T σ t ], η 1 t = ω 1 ( t ) x ( t ) - x ( t - h ( t ) ) ,   η 2 t = ω 2 ( t ) x ( t - h ( t ) ) - x ( t - h ) .Remark 10.
Theorem8 and Corollary 9 can reduce the conservatism of stability conditions based LMI via the LKFs (22) and (50) application. For nonintegral item V1(t), the matrices P, Pa, and Pb are just symmetrical, not always positive definite, and Q1(t) and Q2(t) of the single-integral item V2(t) are delay-dependent matrices which can also further improve the utilization of time delay and its derivative information. When proving V(t)>0, we calculate V1(t) and V2(t) as a whole applying Lemma 2, which may expand the feasible regions of LMIs (15) and (16). When bounded the derivative of the LKFs, three additional zero equations (33)–(35) and Lemma 3 have been used to narrow the gap between the upper bound and the true value, which may be another contribution of reducing the conservatism of stability conditions.Remark 11.
It is worth noting that the delay-product-type item was introduced firstly by C. Zhang et al. [31, 44]; another novel LKF with delay-dependent matrix was constructed by W. Kwon et al. [33], where some improved stability conditions for linear systems with time-varying delay were given. C. Zhang and W. Kwon et al. pointed that the LKFs with delay-product-type item or delay-dependent matrix may reduce the conservatism of stability conditions based on the same inequality scaling technique. The LKFs (22) and (50) constructed in this paper, which combine the advantage of the delay-product-type item and delay-dependent matrix, are more general than those given in [31, 33]. In fact, letting Q11=Q21=0, Q20=diag⁡{Q2,0}, Q¯a=Q¯b=0, and R1=R2, the LKF (50) can be reduced to the LKF (18) of [31]. And taking Pa=Pb=0, Q¯a=Q¯b=0, and R1=R2 the LKF (50) can be reduced to the LKF (16) of [33]. However, the derivation method of [31, 33] cannot be applied directly. The positive definiteness of the LKF and negative definiteness of the LKF’s derivative are proved in Theorem 8.Remark 12.
It is worth noting that in [33], to bound the integral item for -μ≤h˙(t)≤μ,(μ>0)(52)-h˙t∫t-httγTQ¯aγds-h˙t∫t-ht-htγTQ¯bγdsvia the QGFMI technique, the following addition zero equation was introduced(53)0=μ∫t-httγTsQ¯aγsds-μ∫t-httγTsQ¯aγsds+μ∫t-ht-htγTsQ¯bγsds-μ∫t-ht-htγTsQ¯bγsds.
Then, the above integral item can be rewritten as the following form:(54)φ~=-μ+h˙t∫t-httγTQ¯aγds-μ+h˙t∫t-ht-htγTQ¯bγds+μ∫t-httγTQ¯aγds+μ∫t-ht-htγTQ¯bγds.
The first two items on the right can be bounded via the QGFMI technique like (39) of this paper; however, there are fewer proper techniques for obtaining a tight upper bound of the integral terms μ∫t-h(t)tγT(s)Q¯aγ(s)ds and μ∫t-ht-h(t)γT(s)Q¯bγ(s)ds due to their positive definiteness. Thus, to avoid introducing the two positive define integral terms, we give the modified LKF (22) with a double integral item V4(t).Remark 13.
In addition, the main result of [33] was not LMI due to the terms h2(t) even h3(t). In this paper, all matrices inequations of Theorem 8 and Corollary 9 are LMI via Lemma 4 application, which can be solved easily by using Matlab LMI-toolbox. Moreover, the double integral items V4(t) and V5(t) of the LKF (22) divide the time-delay interval [h1,h2] into two subintervals, that is, [h1,h(t)] and [h(t),h2], instead of using the item ∫t-htγT(s)Rγ(s)ds directly, which further make full use of the information of time-varying delays h(t)-h1, h2-h(t) and their derivative h˙(t). Thus, the QGFMI technique can be used fully in each subinterval, which can further reduce the conservatism of the stability conditions.


## 3.2. Robustly Absolute Stability Criteria for Uncertain Form

Next, we extend the obtained absolute stability conditions to robustly absolute stability problem for the uncertain neutral-type Lur’e system (1) with time-varying parameter uncertainties satisfying (7) and (8).Theorem 14.
The system (1) satisfying the conditions (2), (5), (7), and (8) is robustly absolutely stable for given values of h2≥h1≥0, μ1,μ2<1, τd<1, and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R7n×7n, (Pa,Pb∈R5n×5n), (Qa,Qb,R0a,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (Q1,R0,R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm}, any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), X0∈R3n×2n, Xi∈R5n×2n, Yλ∈R4n×2n(i=1,…,4;λ=0,…,4) and ε>0 such that LMIs (15)-(16) and the following LMIs hold for [h(t),h˙(t)]∈{[h1,h2]×[μ1,μ2]}:(55)J1lh1,h˙t,αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(56)J1lh1,h˙t,1-αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(57)J1lh2,h˙t,αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(58)J1lh2,h˙t,1-αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(59)-a2h122+J1lh1,h˙t,αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,(60)-a2h122+J1lh1,h˙t,1-αJ1T+εJ2Φ1TΦ1J2T+sym⁡J2Φ2TJ3T-εJ3J3T<0,where(61)Φ1=e1EaT+e2Ea1T+e16EbTT,Φ2=e1U1D+e5U2D+e15U3D+e16U4DT,J1T=IIIIIII0,J2T=I0000000,J3T=0000000I.Proof.
If we replaceA, A1, and B in LMIs (17)–(21) with A+DF(t)Ea, A1+DF(t)Ea1, B+DF(t)Eb, respectively, Theorem 14 can be proved based on Lemma 6 easily.The following corollary will give the robustly absolute stability criterion for the Lur’e system (1) satisfying the conditionC. 2.Corollary 15.
System (1) satisfying the conditions (3) and (5), (7), and (8) is robustly absolutely stable for given values of h≥0, μ1,μ2<1, τd<1, and kj>0(j=1,2,…,m), if there exist symmetric matrices P∈R5n×5n, (Pa,Pb∈R4n×4n), (Qa,Qb,Ra,Rb∈Rn×n), positive definite matrices (Q2∈Rn×n), (R1,R2,Q1(t),Q2(t)∈R2n×2n), S=diag⁡{s1,s2,…,sm}, Λ=diag⁡{λ1,λ2,…,λm}, any matrices (U01,U02∈R2n×2n,U¯∈R(3n+m)×n), Xi∈R5n×2n, Yi∈R4n×2n(i=1,…,4) and ε>0 such that LMIs (15)-(16) and the following LMIs hold for [h(t),h˙(t)]∈{[h1,h2]×[μ1,μ2]}:(62)J~1l0,h˙t,αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(63)J~1l0,h˙t,1-αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(64)J~1lh,h˙t,αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(65)J~1lh,h˙t,1-αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(66)-a2h2+J~1l0,h˙t,αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,(67)-a2h2+J~1l0,h˙t,1-αJ~1T+εJ~2Φ~1TΦ~1J~2T+sym⁡J~2Φ~2TJ~3T-εJ~3J~3T<0,where (68)Φ~1=e1EaT+e2Ea1T+e12EbTT,Φ~2=e1U1D+e4U2D+e11U3D+e12U4DT,J~1T=IIIII0,J~2T=I00000,J~3T=00000I.Remark 16.
If the nonlinear functionf(σ) in the feedback path satisfies the infinite sector conditions (6), for any si≥0,i=1,2,…,m, it follows from (6) that(69)sifiσihiTxt≥0,which is equivalent to(70)2xTtHSfσ≥0,where S=diag⁡s1,s2,…,sm.
Therefore, the corresponding absolute and robustly absolute stability criteria can be obtained by replacing the matrixΘ1 of Theorems 8 and 14 and Corollaries 9 and 15 with Θ2=e1HSe16T and Θ~2=e1HSe12T, respectively.Remark 17.
For one special case, in the absence of the nonlinear function, that is, in case ofB=0, the system (1) is simply written as the following linear neutral system(71)x˙t-Cx˙t-τt=A+ΔAxt+A1+ΔA1xt-ht,xs=φs,x˙s=φ˙s,s∈-max⁡h2,τ,0,ht∈C.1(72)s∈-max⁡h,τ,0,ht∈C.2,and for another special case, in the absence of the nonlinear function and neutral-type item, that is, in case of B=0 and C=0, the system (1) is simply written as the following linear time-delayed system:(73)x˙t=A+ΔAxt+A1+ΔA1xt-ht,xs=φs,s∈-max⁡h2,τ,0,ht∈C.1s∈-max⁡h,τ,0,ht∈C.2.Obviously, take Theorem 8 as an example. Only letting Θ1=0, Π4=e1U1+e5U2+e15U3 and Π5=Ae1T+A1e2T+Ce15T-e5T, the stability criteria proposed in this paper are also applied to this linear neutral system (71) with time-varying delays; taking Θ1=0, Q2=0, Π4=e1U1+e5U2 and Π5=Ae1T+A1e2T, the stability criteria proposed in this paper are also applied to this linear system (73) with time-varying delays. We will not elaborate here due to the limited space available. However, some results of detailed comparisons will be given in the next section directly.Remark 18.
It is worth pointing out that in this paper, the upper and lower bounds constraints of the time-delay derivative areμ1≤h˙(t)≤μ2, where τd<1,μ2<1 due to the diagonal entry -(1-τ˙(t))Q2, -(1-h˙(t))x˙T(t-h(t))(Q1(t)-Q2(t))x˙(t-h(t)) in Π[h(t),h˙(t)], and -(1-h˙(t))∫t-h(t)tγT(s)R1γ(s)ds in V˙5(t). Similar to Remark 5 of [45], one can establish a stability criterion for system (1) in the case τd≥1 and μ2≥1 if ξ(t),ζ(t),ζ1(t),ζ2(t),γ(s) and V5(t) are replaced with(74)ξ¯Tt=xTtxTt-h1xTt-htxTt-h2x˙Ttx˙Tt-h1x˙Tt-h2v1tv2tv3t∫t-h1t∫utxTsh1duds∫t-htt-h1∫ut-h1xTsh1tduds∫t-h2t-ht∫ut-htxTsh2tdudsfTσt,ζ¯Tt=xTtxTt-h1xTt-h2ω1tω2tω3t,ζ¯1Tt=xTtxTt-h1xTt-h2v2t,ζ¯2Tt=xTtxTt-h1xTt-h2v3t,γ¯Ts=xTs,Q2=0,V¯5t=∫t-h1t∫θt-h1xTsR0xsdsdθ+∫t-h2t-h1∫θt-h1xTsR1xsdsdθ.Due to page limitation, this result is omitted.


## 4. Numerical Examples

In this section, we give three types of examples, including the Lur’e system, the linear neutral system, and two linear systems, to show the effectiveness of the criteria proposed in this paper. Moreover, the conservatism of the criteria is checked based on the calculated maximal admissible delay upper bounds (MADUBs). And the index of the number of decision variables (NoVs) is applied to show the complexity of criteria. The stability criteria proposed in this paper are just dependent onτd, h, μ1, and μ2, however, independent on τ and τ(t). So, only τd of the information of the neutral-type time delay τ(t) is given for Examples 1 and 3. For the sake of simplicity, let -μ1=μ2=μ.Example 1.
Consider the nominal neutral-type Lur’e system (14) with the time-varying delays satisfying C.2 and the nonlinearity satisfying (6), and the system parameters are described as (75)A=-20.50-1,A1=10.40.4-1,B=-0.5-0.75,C=0.20.10.10.2,H=0.20.6.Let the nonlinearityf(σ(t))=σ3(t), where σ(t)=HTx(t)=0.2x1(t)+0.6x2(t). Then, it follows from the infinite sector condition (6) and fi(σi(t))/σi(t)=f(σi(t))/σ(t)=σ2(t)>0, that is fi(σi(t))∈K[0,∞). Under the conditionC. 2, in Table 1, the MADUBs h of the Lur’e system (14) for τd=0.1 and different μ by using Remark 16 and recent methods in [26, 27, 29, 34] are compared. From the table, notwithstanding the NoVs of our criteria are bigger than those of the criteria [26, 27, 29, 34], only [29] is less conservative than Remark 16 under μ=0.2. However our results become better and better with the increasing of μ. Figure 1 displays the responses of system states x(t) for h(t)=2.8490/2+2.8490/2sin⁡0.8t/2.8490, τ(t)=0.5+sin0.1t and initial condition x0=0.2-0.2T. It is possible to see that, for this realization, the trajectory converges to the origin, as expected.Table 1
MADUBsh for different μ (Example 1).
τd Methods∖μ 0.2 0.4 0.6 0.8 NoVs 0.1 [26] 2.9962 2.1316 1.7138 1.3204 75 [27] 3.1743 2.1789 1.7467 1.7153 96 [34] 3.4880 2.3787 1.8062 1.4625 93 [29] 3.8557 2.8490 2.4014 2.2130 117 Remark16 3.6076 2.8490 2.5501 2.4526 705Figure 1
The state responses for Example1.Remark 2.
The MADUBsh of the Lur’e system (14) for τd=0.1 and different μ by using our results are less than those of [21] because the slope restrictions for nonlinearity and the lower bound of the derivative of the neutral-type delay τ˙(t) were considered in [21], where these restrictions are more strict than those of this paper. Thus, the related results of [21] were not compared with the ones of this paper.
In addition, the construction of the LKF with delay-dependent matrices is the main reason to reduce the conservatism of the stability criterion in [29], and another one is that the lower bound of the delay derivative is also involved in this paper, which is not mentioned in [29].Example 2.
Consider the following Chua’s circuit(76)x˙=αy-gxy˙=x-y+zz˙=-βy,where m0=-1/7, m1=2/7, α=9, β=14.28, and c=1, and the nonlinear function g(x) is given by g(x)=m1θ1+0.5(m0-m1)(|x+c|-|x-c|). This Chua’s circuit can be expressed as a Lur’e-type system.
In [19], a master-slave synchronization scheme through a time-delayed state error feedback control is devised for the Chua’s circuit (76), which is given as(77)Master:x˙t=Axt+BfCxtpt=Wxt,Slave:y˙t=Ayt+BfCyt+utqt=Wyt,Controller:ut=-Kaxt-yt+Kbpt-ht-qt-ht,where f(θ)=0.5(θ+1-θ-1)∈[0,1], C=W=[100] and(78)A=-αm1α01-110-β0,B=-αm0-m100.Letting e=x-y, then the resultant error system is given by(79)e˙t=A+Kaet-KbWet-ht+BφCet.Suppose the synchronization controller gains are designed by [18](80)Ka=-1000-1000-1,Kb=6.00291.3367-2.1264.In Table2, we calculate the MADUBs h of the error system (79) for different μ and the conditionC. 2 by using our results and methods in [18, 19, 21, 23, 27, 30] are compared. From the table, it is found that notwithstanding the NoVs of Corollary 9 are bigger than those of the criteria [19, 23, 24, 27], the MADUBs computed by Corollary 9 are larger. Compared with the criteria in [21, 30], Corollary 9 obtains less conservative MADUBs but requires less computation complexity. Moreover, for h(t)=0.2707 and initial condition e(0)=[-0.10.4-0.3]T, Figure 2 depicts the error state responses for the error system (79) under control of the synchronization controller. Thus, one can see that the error system is asymptotically stable for the time-varying delay h(t) less than 0.2707.Table 2
MADUBsh for different μ (Example 2).
Methods∖μ 0 0.3 0.6 0.9 NoVs [19] 0.1622 0.1591 0.1566 0.1527 10 [23] 0.1745 0.1698 0.1698 0.1698 160 [24] 0.1747 0.1710 0.1703 0.1703 140 [27] 0.1771 0.1721 0.1715 0.1715 194 [21] 0.1894 0.1894 0.1894 0.1893 627 [30] 0.2638 0.2578 0.2540 0.2510 809 Corollary9 0.2707 0.2700 0.2545 0.2544 1548Figure 2
The error state responses for Example2.Remark 3.
In Example2, the purpose of this paper is to enlarge the MADUBs under the same controller to [19]. At this point, the error system (79) can be seen as a new Lure system within the given gain matrices A+Ka, -KbW and B proposed by [19]. The MADUBs of are calculated by solving the LMIs in Theorem 8. For an example, an MADUB h=0.2700 is obtained for μ=0.3 by Theorem 8, then the stability of the error system (79) must be guaranteed by Theorem 8 under the same controller to [19] for 0≤h(t)≤h and |μ|≤0.3. In other words, this paper does not design any controllers, but analyzes the stability. The same controller gains of [19] are seen as the known system matrices of the error system (79). Therefore, the controller gains obtained in [19] will also stabilize the system considered in Example 2 under a bounds of h(t) obtained by Theorem 8.Example 3.
Consider the linear neutral-type system (71) with the time-varying delays satisfying C.2, and the system parameters are described as(81)A=-110-1,A1=0.500.5-0.5,C=0.4000.4,D=diag⁡1,1,Ea=diag⁡0.05,0.05,Ea1=diag⁡0.1,0.1.Under the conditionC. 2, in Table 3, we calculate the MADUBs h of the linear neutral-type system (71) for τd=0.6 and different μ1, μ2 by using Remark 16 and methods in [25, 27, 28, 35] are compared. From the table, it is found that notwithstanding the NoVs of our criteria are bigger than those of the criteria [25, 27, 28, 35], the MADUBs computed by our criteria are larger. Figure 3 displays the responses of system states x(t) for h(t)=2.0555/2+2.0555/2sin⁡t/2.0555, τ(t)=1.2sin⁡0.5t and initial condition x(0)=[-0.20.2]T. It is possible to see that, for this realization, the trajectory converges to the origin, as expected.Table 3
MADUBsh for τd=0.6 and different μ1 and μ2 (Example 3).
μ1 Methods∖μ2 0.5 0.9 NoVs Ignoreμ1 [25] 1.5572 1.5572 10 [27] 1.6635 1.5742 160 [35] 1.5812 1.5745 140  -0.5 [28] 1.8763 1.7352 120 Remark17 2.0555 1.9357 518  -0.2 [28] 1.8780 1.7453 120 Remark17 2.0671 1.9558 518Figure 3
The state responses for Example3.Example 4.
Consider the linear systems (73) with time-varying delays, where the system parameters are described as(82)A=01-1-2,A1=00-11.As a special case pointed out by Remark16, Corollary 9 removed some redundant terms which can be used to check the stability of the linear systems with time-varying delays. In order to make a comparison with some existing stability criteria, the MADUBs under the conditionC. 2 are listed in Table 4, which shows that notwithstanding the NoVs of Corollary 9 are bigger than those of the criteria [3, 33, 36, 38, 40], the MADUBs computed by Corollary 9 are larger. Compared with the criteria in [30, 37, 39], Corollary 9 obtains less conservative MADUBs but requires less computation complexity.Table 4
MADUBsh for different μ (Example 4).
Methods∖μ 0.1 0.2 0.5 0.8 NoVs [36] 6.6103 4.0034 1.6875 1.0287 23n2+4n [37] 7.1672 4.5179 2.4158 1.8384 142n2+18n [38] 7.1765 4.5438 2.4963 1.9225 114n2+18n [39] 7.2030 4.5126 2.3860 1.8476 203n2+9n [40] 7.1905 4.5275 2.4473 1.8562 70n2+12n [30] 7.2734 4.6213 2.6505 2.0612 185.5n2+21.5n [3] 7.4001 4.7954 2.7175 2.0894 108n2+12n [33] 8.6565 5.8907 3.1754 2.3953 91.5n2+4.5n Remark17 8.9647 7.1866 4.9390 3.8477 165.5n2+19.5n


## 5. Conclusion

In this paper, some improved absolute and robustly absolute stability criteria are proposed for the uncertain neutral-type Lur’e systems with mixed time-varying delays and sector-bounded nonlinearities via a novel LKF combining the delay-product-type function and the delay-dependent matrix. An effective technique, QGFMI, is applied to further reduce the conservatism of the proposed criteria from some existing results. Finally, some numerical examples are used to illustrate the effectiveness of the proposed approaches.



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