# Novel Delay-Decomposing Approaches to Absolute Stability Criteria for Neutral-Type Lur’e Systems

Authors: Liang-Dong Guo; Sheng-Juan Huang; Li-Bing Wu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The problem of absolute stability analysis for neutral-type Lur’e systems with time-varying delays is investigated. Novel delay-decomposing approaches are proposed to divide the variation interval of the delay into three unequal subintervals. Some new augment Lyapunov–Krasovskii functionals (LKFs) are defined on the obtained subintervals. The integral inequality method and the reciprocally convex technique are utilized to deal with the derivative of the LKFs. Several improved delay-dependent criteria are derived in terms of the linear matrix inequalities (LMIs). Compared with some previous criteria, the proposed ones give the results with less conservatism and lower numerical complexity. Two numerical examples are included to illustrate the effectiveness and the improvement of the proposed method.


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## 1. Introduction

Over the last 30 years, time delay system has been one of the hottest research areas in control engineering for time delay often appears in many control systems either in the state, the control input, or the measurements [1, 2]. Since time delay frequently occurs in practical systems and is often the source of instability, there have been many results for stability of delayed systems [3–9]. Also stabilization [10–12], filtering [13], and adaptive control [14] of time-delay systems have received considerable attention. The neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar [15]. A considerable number of studies related to this topic have been reported (see, for example, [16–21], and the references therein).Lur’e system is originally from a pilot robot [22], which is one of the important classes of nonlinear systems whose nonlinear element satisfies certain sector constraints. Many systems such as Chua’s circuits, Goodwin models, Swarm models, n-scroll attractors, and hyperchaotic attractors can be represented as Lur’e-type systems [23, 24]. In recent years, stability analysis of the neutral-type Lur’e system with time delays has attracted the attention of many researchers [25–35]. The main purpose of stability analysis is to calculate the maximum allowable delay bounds (MADBs), which is a key index for judging the conservatism of stability criteria, such that the Lur’e system maintains absolute stability for any time delay less than the MADBs by using the LKF method. In [25], absolute stability of Lur’e systems with sector-bounded nonlinearities and constant delay was discussed by using a Lur’e-Postnikov function. To avoid involving a considerable number of free-weighting matrices and leading to a computationally expensive stability criterion in [26], the general free weighting matrix method was proposed and some improved delay-range-dependent stability criteria were obtained [27, 28]. By using integral inequality and the Wirtinger integral inequality method instead of free weighting matrix one, some improved delay-dependent robust stability criteria were derived in [29, 30], respectively. Also, by eliminating nonlinearity and reducing the number of free-weighting matrices, Lur’e systems with interval time-varying delays were discussed in [31]. By discretizing the delay interval into two segmentations with an unequal width, some delay-dependent sufficient conditions were given for the robust stability of neutral-type Lur’e systems in [32, 33]. By employing an augmented LKF method, reciprocally convex combination approach, and convex combination technique, several robust absolute stability criteria for uncertain neutral-type Lur’e systems with time-varying delays were presented in [34]. Very recently, by constructing a LKF including both double-integral terms and triple-integral terms, using the piecewise analysis method, Wirtinger-based integral inequality, and the reciprocally convex combination technique, some new stability criteria were obtained in [35]. In fact, one of the term ∫t−τM/2tϱ1TtQϱ1tds in LKFs in [35] is the special case of the delay-partitioning approach, where ϱ1tT=xTt,xTt−τM/2T. The delay-partitioning method is an effective one to reduce a criterion’s conservativeness, which is widely used in the stability analysis for various systems (see details in [36–39]). However, when utilizing the delay-partitioning approach, the term ∫t−τitϱ2tTQϱ2tds is inevitably involved in LKFs, where ϱ2tT=xTt,xTt−τ1,…,xTt−τN−1T,τi=d/N and N is the delay-partitioning number. It is clear that the derived conditions become more complicated and the computational burden grows bigger when N increases. In addition, to relate to the Wirtinger-based integral inequality and deal with the derivative of triple-integral terms introduced in LKFs, it is ineluctable that the extra terms ∫t−τM/2txTsQds, ∫t−τMt−τM/2xTsQds, ∫−τM/20∫t+θtxTsQdsdθ, and ∫−τM−τM/2∫t+θtxTsQdsdθ had to be introduced into the derivation process in [35], which leads to a sharp increase in the dimensions of the LMIs involved.Motivated by this mentioned above, the aim of this work is to revisit the stability analysis for the neutral-type Lur’e system. In this study, novel delay-decomposing approaches are proposed firstly. Different from the delay-partitioning approaches or the delay-decomposing ones in [32, 33, 36–40], the interval of the state time delay 0,τM is divided into three unequal subintervals 0,τα, τα,τβ, and τβ,τM. In particular, to establish the relationship of the vectors such as xt,xt−τα,xt−τβ, and xt−τM, the novel terms ∫t−ταtϱ3TtQϱ3tds and ∫t−τβtϱ4TtQϱ4tds are introduced in LKFs in each of subintervals, which are more general than the ones in [28, 35], where ϱ3Tt=xTt,xTt−τβ and ϱ4Tt=xTt,xTt−τα. It is worth mentioning that the merit of the proposed delay-decomposing method lies in that the dimensions of the LMIs involved are independent of the number of subinterval. In addition, to avoid introducing the extra vectors by Wirtinger-based integral inequality, the reciprocally convex combination method [41] and the integral inequality [42] are utilized to deal with the bounds of integral terms. Some novel LKFs related to the above inequalities are constructed on the obtained three subintervals. The presented stability criteria are given in terms of LMIs. Compared with the related literature, the conclusions of this paper have the advantages of less conservatism conservatism and the dimensions of the LMIs. Finally, two well-known numerical examples are given to demonstrate the effectiveness and less conservatism over the existing results.Notation: in this paper,ℝn denotes n-dimensional Euclidean space and ℝn×m is the set of all n×m real matrices. For symmetric matrices X and Y, the notation X>Y (respectively, X≥Y) means that the matrix X−Y is positive definite (respectively, nonnegative). diag⋯ denotes the block diagonal matrix. The subscript “T” denotes the transpose of the matrix. In denotes the identity matrix.


## 2. Problem Statements and Preliminaries

Consider a class of Lur’e systems of neutral type with time-varying delays and sector-bound nonlinearities described as follows:(1)x˙t−Cx˙t−dt=Axt+Bxt−τt+Dfσt,σt=HTxt,t≥0,xs=ϕs,x˙s=ϕ˙s,s∈−maxτM,dM,0,where A,B,C,andD are constant matrices with appropriate dimensions. xt=x1t,x2t,…,xntT∈ℝn is the state vector, and σt=σ1t,σ2t,…,σmtT∈ℝm is the output vector. The time delay τt is a time-varying continuous function satisfying(2)0≤τt≤τM,τ˙t≤μτ,0≤dt≤dM,0<d˙t≤μd,where dM,τM,μτ,andμd are the known constant scalars.f
⋅
=
f
1
⋅
,
f
2
⋅
,
…
,
f
m
⋅
T
∈
ℝ
m is the nonlinear function and fiσit⋅i=1,2,…,m is assumed to satisfy the finite sector restriction:(3)fiσit∈K0,ki=fiσit∣fi0=0,0<σtfiσit≤kiσ2t,σit≠0,with known positive scalar ki or the infinite sector restriction:(4)fiσit∈K0,∞=fiσit∣fi0=0,σtfiσit>0,σit≠0.The objective of this paper is to formulate the delay-dependent stability conditions of system (1). The following lemmas will play important roles in deriving the criteria.Lemma 1 (see [41]).
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(5)minαiαi>0,∑iαi=1∑i1αifit=∑ifit+maxgi,jt∑i≠jgi,jt,subject to(6)gi,j:Rm⟶R,gj,it=gi,jt,fitgi,jtgj,itfjt≥0.Lemma 2 (see [42]).
For any matricesQ>0,M,N,X with compatible dimensions, any continuous time vector function xtandηt with compatible dimensions, and any scalar τM satisfying 0≤τt≤τM, the following integral inequality holds:(7)−∫t−τMtxTsQxsds≤τMηTtXηt+2ηTtM∫t−τMt−τtxsds+N∫t−τttxsds,if QMTMX≥0 and QNTNX≥0.Remark 1.
LetM=N and ηTt=xTt,xTt−τt,xTt−τM, the integral inequality reduces the one in [43].


## 3. Main Result

In this section, some new delay-dependent stability criteria are proposed for system (1).Noteτα=ατM and τβ=βτM (0≤α≤0.5,β+α=1). It is easy to see that 0≤τα≤τβ≤τM holds. Then, we divide the time interval 0,τM into three subintervals Δ1≕0,τα,Δ2≕τα,τβ, and Δ3≕τβ,τM. In the following, we will propose some criteria for the three subintervals.Now, we give the stability criteria for system (1) with conditions (2) and (4) when τt∈Δ1 as follows.Theorem 1.
For given scalars0≤α≤0.5, τM≥0, 0≤μd<1, μτ≥0, and ki>0, i=1,2,…,m, system (1) with conditions (2) and (4) is absolutely stable if there exist n×n symmetric positive definite matrices P,S1,S2,R1,R2,R3, 2n×2n symmetric positive definite matrices Q1,Q2, 2n×2n symmetric semi-positive definite matrices Y11,Z11, positive diagonal matrices U=diagu1,u2,…,um, L=diagl1,l2,…,lm, any n×n matrix X12, and 2n×n matrices Y12,Z12, such that LMIs (5) and (6) hold when τt∈Δ1:(8)Ξ1+Ξ1T+Ξ2+Ξ3+Ξ4+Ξ4T<0,(9)Φi≥0,i=1,2,3,where(10)Φ1=R1X12X12TR1,Φ2=Y11Y12Y12TR2,Φ3=Z11Z12Z12TR3,Ξ1=Ξ10+e3,e4Y12e3T−e4T+e4,e5Z12e4T−e5TT,Ξ10=e1Pe8T+e7LHTe8T,Ξ20=e1S1e1T−1−μτe2S1e2T−1−μde6S2e6T+e1,e4Q1e1,e4T−e3,e5Q1e3,e5T+e1,e3Q2e1,e3T−e4,e5Q2e4,e5T,Ξ2=Ξ20−2e7Ue7T,Ξ3=e8S2+τα2R1+τβ−ταR2+τM−τβR3e8T−e2−e1,e3−e2Φ1e2−e1,e3−e2T+τβ−ταe3,e4Y11e3,e4T+τM−τβe4,e5Z11e4,e5T,Ξ4=e1HKUe7T,e8=e1AT+e5BT+e6CT+e7DT,K=diagk1,k2,…,km,e1=In,0n×n,0n×n,0n×n,0n×n,0n×mT,e2=0n×n,In,0n×n,0n×n,0n×n,0n×mT,e3=0n×n,0n×n,In,0n×n,0n×n,0n×mT,e4=0n×n,0n×n,0n×n,0n×n,In,0n×mT,e5=0n×n,0n×n,0n×n,0n×n,In,0n×mT,e6=0n×n,0n×n,0n×n,0n×n,0n×m,InT,e7=0m×n,0m×n,0m×n,0m×n,0m×n,ImT.Proof.
For positive diagonal matricesL=diagl1,l2,…,lm and positive definite matrices P,Rj,Si,andQi⋅i=1,2,j=1,2,3, let us consider the following LKF candidates for the case 0≤τt≤τα:(11)Vxt=V1xt+V2xt+V3xt,where(12)V1xt=xTtPxt+∫t−τttxTsS1xsds+∫t−dttx˙TsS2x˙sds+2∑i=1mli∫0σifiσdσ,V2xt=∫t−ταtxsxs−τβTQ1xsxs−τβds+∫t−τβtxsxs−ταTQ2xsxs−ταds,V3xt=ταV31xt+V32xt+V33xt,V31xt=∫−τa0∫t+θtx˙TsR1x˙sdsdθ,V32xt=∫−τβ−τα∫t+θtx˙TsR2x˙sdsdθ,V33xt=∫−τM−τα∫t+θtx˙TsR3x˙sdsdθ.
The time derivative ofVxt along the trajectory of system (4) is given by(13)V.1xt≤2xTtPx˙t+2fTxtLHTx˙t+xTtS1xt+x˙TtS2x˙t−1−μτxt−τtTtS1xt−τt−1−μdx˙t−dtTtS2x˙t−dt=ξTte1Pe8T+e8Pe1T+e1S1e1T−1−μτe2S1e2T−1−μde6S2e6T+e7LHTe8T+e8HLTe7T+e8S2e8Tξt,where(14)ξTt=xTt,xTt−τt,xTt−τα,xTt−τβ,xTt−τM,x˙Tt−dt,fTσt,(15)V.2xt=xtxt−τβTQ1xtxt−τβ−xt−ταxt−τMTQ1xt−ταxs−τM+xtxt−ταTQ2xtxt−τα−xt−τβxt−τMTQ2xt−τβxs−τM=ξTte1,e4Q1e1,e4T−e3,e5Q1e3,e5T+e1,e3Q2e1,e3T−e4,e5Q2e4,e5Tξt,(16)V.3xt=x˙Ttτα2R1+τβ−ταR2+τM−τβR3x˙t−τα∫t−ταtx˙TsR1x˙sds−∫t−τβt−ταx˙TsR2x˙sds−∫t−τMt−τβx˙TsR3x˙sds.
If0≤τt≤τα holds, one can compute out the following according to Lemma 1:(17)−τα∫t−ταtx˙TsR1x˙sds=−τα∫t−τttx˙TsR1x˙sds−τα∫t−ταt−τtx˙TsR1x˙sds≤−xt−τt−xtxt−τα−xt−τtTΦ1xt−τt−xtxt−τα−xt−τt=−ξTte2−e1,e3−e2Φ1e2−e1,e3−e2Tξt,where Φ1 is given in (6).
Based on Lemma2 (in which M=N=Y12 and ηTt=xTt−τα,xTt−τβ), when Φ2=Y11Y12Y12tR2≥0, the following inequality holds:(18)−∫t−τβt−ταx˙TsR2x˙sds≤τβ−ταxt−ταxt−τβTY11xt−ταxt−τβ+2xt−ταxt−τβTY12xt−τα−xt−τβ=ξTtτβ−ταe3,e4Y11e3,e4T+e3,e4Y12e3T−e4T+e3,e4Y12e3T−e4TTξt,and using Lemma 2 again (in which M=N=Z12 and ηTt=xTt−τβ,xTt−τM), when Φ3=Z11Z12Z12tR3≥0, one can conclude(19)−∫t−τMt−τβx˙TsR3x˙sds≤τM−τβxt−τβxt−τMTZ11xt−τβxt−τM+2xt−τβxt−τMTZ12xt−τβ−xt−τM=ξTtτM−τβe4,e5Z11e4,e5T+e4,e5Z12e4T−e5T+e4,e5Z12e4T−e5TTξt.
Under the assumption on nonlinear function (4), the following inequality holds(20)uifiσtkihiTxt−fiσit≥0,i=1,2,…,m.
It is equivalent to(21)2xTtHKUfσt−fTσtUfσt≥0.
Rewrite the above as follows:(22)2e1HKUe7T−e7Ue7T≥0,where(23)U=diagu1,u2,…,um,K=diagk1,k2,…,km.
Then, combining (3)–(15) leads to(24)V˙xt≤ξTtΞ1+Ξ1T+Ξ2+Ξ3+Ξ4+Ξ4Tξt.
Therefore, if LMIs (5) and (6) hold, one has V˙xt<0, which shows the absolute stability of system (1) subject to (2) and (4), when τt satisfies C1. This completes our proof.
For the caseτt∈Δ2, we construct the LKFs as follows:(25)V¯xt=V1xt+V2xt+V¯3xt,where(26)V¯3xt=V31xt+τβ−ταV32xt+V33xt,and V1xt,V2xt,V31xt,andV33xt are defined in (7).
Then, the derivative ofV¯3xt can be obtained as(27)V3¯˙xt=x˙TtταR1+τβ−τα2R2+τM−τβR3x˙t−∫t−ταtx˙TsR1x˙sds−τβ−τα∫t−τβt−ταx˙TsR2x˙sds−∫t−τMt−τβx˙TsR3x˙sds.
According to Lemma1, if τα≤τt≤τβ, the following inequality holds:(28)−τβ−τα∫t−τβt−ταx˙TsR2x˙sds=−τβ−τα∫t−τtt−ταx˙TsR2x˙sds,−τβ−τα∫t−τβt−τtx˙TsR2x˙sds≤−xt−τt−xt−ταxt−τβ−xt−τtTΦ¯2xt−τt−xt−ταxt−τβ−xt−τt=−ξTte2−e3,e4−e2Φ¯2e2−e3,e4−e2Tξt,where Φ¯2 is given in (20).
In addition, by using Lemma2 (in which M=N=X12 and ηTt=xTt,xTt−τα), if Φ¯1=X11X12tX12R1≥0, one can obtain(29)−∫t−ταtx˙TsR1x˙sds≤ταxtxt−ταTX11xtxt−τα+2xtxt−ταTX12xt−xt−τα=ξTtταe1,e3X11e1,e3T+e1,e3X12e1T−e3T+e1,e3X12e1T−e3TTξt.
The other procedure is straightforward from the proof of Theorem 1; we can cope with the situation ofτt∈Δ2 and have the following.Theorem 2.
For given scalars0≤α≤0.5,τM≥0,0≤μd<1,μτ≥0,andki>0i=1,2,…,m, system (1) with conditions (2) and (4) is absolutely stable if there exist n×n symmetric positive definite matrices P,S1,S2,R1,R2,R3, 2n×2n symmetric positive definite matrices Q1,Q2, 2n×2n symmetric semi-positive definite matrices X11,Z11, positive diagonal matrices U=diagu1,u2,…,um, L=diagl1,l2,…,lm, any n×n matrix Y12, and 2n×n matrices X12,Z12, such that LMIs (19) and (20) hold for the case τt∈Δ2:(30)Ξ¯1+Ξ¯1T+Ξ2+Ξ¯3+Ξ4+Ξ4T<0,(31)Φ¯1≥0,Φ¯2≥0,Φ3≥0,where(32)Φ¯1=X11X12X12tR1,Φ¯2=R2Y12Y12tR2,Ξ¯1=Ξ10+e1,e3X12e1T−e3T+e4,e5Z12e4T−e5T,Ξ¯3=e8S2+ταR1+τβ−τα2R2+τM−τβR3e8T−e2−e3,e4−e2Φ¯2e2−e3,e4−e2T+ταe1,e3X11e1,e3T+τM−τβe4,e5Z11e4,e5Tand Φ3,Ξ10,Ξ2,andΞ4ei,i=1,2,…,8 and K are defined in Theorem 1.
For the caseτt∈Δ3, we construct the following LKF:(33)V^xt=V1xt+V2xt+V^3xt,where(34)V^3xt=V31xt+V32xt+τM−τβV33xtand V1xt,V2t,V31t,andV32t are defined in (7).
Taking the derivative ofV^3xt yields(35)V^˙3xt=x˙TtταR1+τβ−ταR2+τM−τβ2R3x˙t−∫t−ταtx˙TsR1x˙sds−∫t−τβt−ταx˙TsR2x˙sds−τM−τβ∫t−τMt−τβx˙TsR3x˙sds.
By using Lemma1, if τβ≤τt≤τM, one can obtain(36)−τM−τβ∫t−τMt−τβx˙TsR3x˙sds=−τM−τβ∫t−τtt−τβx˙TsR3x˙sds−τβ−τα∫t−τMt−τtx˙TsR3x˙sds≤−xt−τt−xt−τβxt−τM−xt−τtTΦ^3xt−τt−xt−τβxt−τM−xt−τt=−ξTte2−e4,e5−e2Φ^3e2−e4,e5−e2Tξt,where Φ^3=R3Z12tZ12R3.
Handle the integral terms−∫t−ταtx˙TsR1x˙sds and −∫t−τβt−ταx˙TsR2x˙sds in the same way as in (13) and (17), respectively. Following the same procedure, we can cope with the situation of τt∈Δ3 and have the following.Theorem 3.
For given scalars0≤α≤0.5,τM≥0,0≤μd<1,μτ≥0,andki>0i=1,2,…,m, system (1) with conditions (2) and (4) is absolutely stable if there exist n×n symmetric positive definite matrices P,S1,S2,R1,R2,R3, 2n×2n symmetric positive definite matrices Q1,Q2, 2n×2n symmetric semi-positive definite matrices X11,Y11, positive diagonal matrices U=diagu1,u2,…,um, L=diagl1,l2,…,lm, any n×n matrices Z12, and 2n×n matrices X12,Y12, such that LMIs (23) and (24) hold for the case τt∈Δ3:(37)Ξ^1+Ξ^1T+Ξ2+Ξ^3+Ξ4+Ξ4T<0,(38)Φ¯1≥0,Φ2≥0,Φ^3≥0,where(39)Φ^3=R3Z12Z12tR3,Ξ^3=e8S2+ταR1+τβ−ταR2+τM−τβ2R3e8T−e2−e4,e5−e2Φ^3e2−e4,e5−e2T+ταe1,e3X11e1,e3T+τβ−ταe3,e4Y11e3,e4T,Ξ^1=Ξ10+e1,e3X12e1T−e3T+e3,e4Y12e3T−e4T,and Φ2,Ξ10,Ξ2,Ξ4,eii=1,2,…,8, K, and Φ¯1 are defined in Theorems 1 and 2, respectively.Remark 2.
Theorems1–3 present the absolute stability criteria for the Lur’e system. Unlike the delay-partitioning approach or delay-decomposing one used in [28, 32, 33, 35, 40], the interval of the state time delay 0,τM has been divided into three subintervals 0,τα, τα,τβ, and τβ,τM. Obviously, the range of the subintervals 0,τα, τα,τβ, and τβ,τM is ατM, 1−2ατM, and ατM. Thus, to know more about the time-varying τt, one can select the value of the parameter α as close as possible to 0 for the case τt∈Δ1 or case τt∈Δ3 and as close as possible to 0.5 for the case τt∈Δ2, respectively. The following examples in the work will illustrate the point.Remark 3.
The terms∫t−ταtϱ3TtQ1ϱ3tds and ∫t−τβtϱ4TtQ2ϱ4tds have been introduced in LKFs in the work instead of the terms ∫t−τM/2tϱ1TtQϱ1tds in [28, 35] and ∫t−τitϱ2tTQϱ2tds in [36–39], where ϱ1tT=xTt,xTt−τM/2Tandϱ2tT=xTt,xTt−τ1,…,xTt−τN−1T. And the relations between xt,xt−τα,xt−τβ, and xt−τM have been effectively expressed by the derivative of V2xt. One can easily see that ∫t−ταtϱ3TtQ1ϱ3tds or ∫t−τβtϱ4TtQ2ϱ4tds deduces into ∫t−τM/2tϱ1TtQϱ1tds when α=β=0.5. This is to say the established LKFs are more general than ones in [28, 35]. The idea is expected to reduce the conservatism of obtained criteria. On the other hand, it can effectively overcome the weakness of the computational burden growing bigger when delay-partitioning number increases in [36–39].Remark 4.
In [35], the Wirtinger-based integral inequality and double-integral inequality were used to deal with bounds of the derivative of double-integral and triple-integral terms in LKFs. Thus, the extra terms ∫t−τM/2txTsQds, ∫t−τMt−τM/2xTsQds, ∫−τM/20∫t+θtxTsQdsdθ, and ∫−τM−τM/2∫t+θtxTsQdsdθ were unavoidably introduced in the derivation process, which leads the dimensions of the LMIs involved to increase sharply. A detailed comparison is given in Example 1.Remark 5.
If functionfσt of system (1) satisfies sector condition (3), for any ui≥0,i=1,2,…,m, one has the following:(40)uifiσithiTxt≥0,i=1,2,…,m,which is equivalent to(41)2xTtHUfσt≥0or2ξTte1HUe7Tξt≥0.
Thus, if functionfσt of system (1) satisfies sector condition (3), we have the following corollary.Corollary 1.
For given scalars0≤α≤0.5,τM≥0,0≤μd<1,μτ≥0,ki>0,i=1,2,…,m, the system (1) with conditions (2) and (3) is absolutely stable if there exist symmetric positive definite matrices P,S1, S2,R1,R2,R3, Q1,Q2, X11,Y11 with appropriate dimensions, positive diagonal matrices U=diagu1,u2,…,um, L=diagl1,l2,…,lm, and matrices X11,Y11,Z11,X12,Y12,Z12 with appropriate dimensions and properties, such that LMIs (6) and (26) hold for the case τt∈Δ1, (3) and (28) hold for the case τt∈Δ2, and (20) and (28) hold for the case τt∈Δ3, respectively:(42)Ξ1+Ξ1T+Ξ20+Ξ3+Ξ40+Ξ40t<0,(43)Ξ¯1+Ξ¯1T+Ξ20+Ξ¯3+Ξ40+Ξ40t<0,(44)Ξ^1+Ξ^1T+Ξ20+Ξ^3+Ξ40+Ξ40t<0,where Ξ40=e1HUe7T, Ξ1,Ξ¯1,Ξ^1, Ξ20,Ξ3,Ξ¯3,Ξ^3 are defined in Theorems 1–3.


## 4. Illustrative Example

In this section, we will use two well-known numerical examples to show the effectiveness and benefits of our results.Example 1.
Consider the following nominal neutral-type Lur’e system (1) subject to (2) and (4) with the parameters:(45)A=−20.50−1,B=10.40.4−1,C=0.20.10.10.2,D=−0.5−0.75,H=0.20.6.
The purpose of this example is to compute MADBs ofτM such that the neutral-type Lur’e system (1) remains stable for different μτ and μd. For given μd=0.9,α=0.45, μd=0.5,α=0.45, and μd=0.1,α=0.4, the acceptable upper bounds of τM is 0.271, 2.277, and 2.652 when μτ≥1 by using Corollary 1 (τt∈Δ1) in our work, respectively. In order to make a comparison with some existing stability criteria, we calculate the MADBs and list them in Tables 1–3. Together with all derived MADBs listed in Tables 1–3, one can check that Corollary 1 can be superior over some present ones. The number of decision variables, maximal order of LMIs between our work, and the criteria in [29, 30, 33, 35] are listed in Table 4. It shows that our proposed method involves smaller decision variables or lower maximal order of LMIs than the relative ones. To confirm the obtained result, a simulation result is shown in Figure 1 when µ is unknown, x0=0,0.4T,τM=5.886,andht=0.5+0.1sint.Table 1
MADBs ofτM for different μτ, when μd=0.1 for Example 1.
Method
μ
τ
=
0.2
μ
τ
=
0.4
μ
τ
=
0.6
μ
τ
=
0.8
[44]
2.6052
1.9538
1.6232
1.1659
[31]
2.6062
1.9730
1.6408
1.1854
[27]
2.6488
2.0056
1.6679
1.2049
[30]
2.8890
2.0678
1.7048
1.6059
[32]
2.9962
2.1316
1.7138
1.3204
[28]
2.9997
2.1012
1.7307
1.3359
[29]
3.0717
2.1171
1.7310
1.3367
[33]
3.1034
2.1579
1.7436
1.7088
[34]
3.1743
2.1789
1.7467
1.7153
[35]
3.4880
2.3787
1.8062
1.4625
Cor 1 (τt∈Δ1,α=0.45)
5.886
4.456
3.705
2.676
Cor 1 (τt∈Δ2,α=0.3)
3.784
2.865
2.385
2.176
Cor 1 (τt∈Δ3,α=0.4)
3.060
2.157
2.000
1.999
Cor 1 (τt∈Δ3,α=0.04)
3.729
3.729
3.729
3.729Table 2
MADBs ofτM for different μτ, when μd=0.5 for Example 1.
Method
μ
τ
=
0.2
μ
τ
=
0.4
μ
τ
=
0.6
μ
τ
=
0.8
[29]
2.5131
1.8146
1.4951
1.1168
[44]
2.1532
1.6986
1.3687
1.0066
[31]
2.1537
1.7041
1.3828
1.0095
[27]
2.1883
1.7322
1.4056
1.0262
[30]
2.3628
1.7752
1.5027
1.4364
[32]
2.4472
1.8289
1.4551
1.1058
[28]
2.4563
1.8009
1.4823
1.1157
[33]
2.5302
1.8461
1.5371
1.5208
[35]
2.8538
2.0236
1.5679
1.2425
Cor 1 (τt∈Δ1,α=0.45)
4.862
3.848
3.119
2.280
Cor 1 (τt∈Δ2,α=0.3)
3.126
2.474
2.101
1.922
Cor 1 (τt∈Δ3,α=0.04)
2.877
2.877
2.877
2.877
Cor 1 (τt∈Δ3,α=0.4)
2.504
1.847
1.738
1.737Table 3
MADBs ofτM for different μτ, when μd=0.9 for Example 1.
Method
μ
τ
=
0.2
μ
τ
=
0.4
μ
τ
=
0.6
μ
τ
=
0.8
[31]
0.1122
0.1088
0.1086
0.1086
[27]
0.1122
0.1089
0.1086
0.1086
[32]
0.1126
0.1102
0.1102
0.1102
[28]
0.1130
0.1105
0.1105
0.1105
[44]
0.1221
0.1087
0.1083
0.1083
[29]
0.1227
0.1197
0.1197
0.1197
[35]
0.1799
0.1697
0.1660
0.1658
[30]
0.2495
0.2366
0.2300
0.2285
[33]
0.2906
0.2745
0.2653
0.2621
Cor 1 (τt∈Δ1,α=0.35)
0.321
0.311
0.310
0.310
Cor 1 (τt∈Δ1,α=0.4)
0.281
0.272
0.271
0.271
Cor 1 (τt∈Δ2,α=0.4)
0.189
0.189
0.189
0.189
Cor 1 (τt∈Δ3,α=0.4)
0.113
0.113
0.113
0.113
Cor 1 (τt∈Δ3,α=0.04)
0.119
0.119
0.119
0.119Table 4
Number of decision variables (NDV) and maximal order of LMIs (MOL) of different methods.
Method
NDV
MOL
[29] Theorem 1
12
n
2
+
7
n
+
2
m
7
n
+
m
[35] Theorem 10
19.5
n
2
+
6.5
n
+
2
m
10
n
+
m
[30] Theorem 1
11.5
n
2
+
7
n
+
2
m
10
n
+
m
[33] Theorem 1
20
n
2
+
6
n
+
2
m
12
n
+
m
Our Cor 1
16
n
2
+
7
n
+
2
m
6
n
+
mFigure 1
State responses of the system considered in Example 1.Example 2.
Consider Chua’s circuit example discussed in [35]:(46)θ˙1t=aθ2−gθ1,θ˙2t=θ1−θ2+θ3,θ˙3t=−bθ2.
The nonlinear functiongθ1=m1θ1+0.5m0−m1θ1+c−θ1−c. Let m0=−1/7,m1=2/7,a=9,b=14.28, and c=1; then, Chua’s circuit can be expressed as a Lur’e-type system with(47)A=−am1−1a01−210−b−1,B=−6.002900−1.3367002.126400,C=000000000,D=−am0−m100,H=100.
The feedback nonlinear function belongs toK0,1.
Now, we calculate MADBs ofτM. For different μτ, the obtained results are given in Table 5. From this table, it is clear to see that Theorems 1 and 2 offer larger MADBs of τM than those methods in existing references.Table 5
MADBsτM for different μτ, when μd=0.1 for Example 2.
Method
μ
τ
=
0
μ
τ
=
0.3
μ
τ
=
0.6
μ
τ
=
0.9
>
1
[24]
0.1622
0.1591
0.1566
0.1541
0.1527
[27]
0.1745
0.1698
0.1698
0.1698
0.1698
[29]
0.1747
0.1710
0.1703
0.1703
0.1703
[32]
0.1771
0.1721
0.1715
0.1715
0.1715
[35]
0.2790
0.2691
0.2553
0.2463
0.2463
Theorem 1 (α = 0.45)
0.361
0.359
0.359
0.359
0.359
Theorem 2 (α = 0.45)
0.295
0.295
0.295
0.295
0.295
Theorem 3 (α = 0.01)
0.163
0.163
0.163
0.163
0.163


## 5. Conclusions

This paper has investigated the absolute stability analysis for neutral-type Lur’e systems with time-varying delays. Based on the new delay-decomposition approaches in combination with the integral inequality and reciprocally convex technique, several improved stability criteria have been derived by constructing some appropriate LKFs on the subintervals. The merit of the obtained stability criteria lies in the significant less conservativeness and lower computational complexity than some existing ones. Finally, two examples have been given to demonstrate the effectiveness and less conservatism of the proposed method. In the future works, we will be dedicated to study the stability analysis for systems with infinite delays and devote to the study of output feedback, tracking control and filtering of the neutral-type Lur’e systems with time-varying delays based on the method proposed in this paper.



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