# Simplified Stability Criteria for Delayed Neutral Systems

Authors: Xinghua Zhang; Xiangyu Gao; Min Su
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

For a class of linear time-invariant neutral systems with neutral and discrete constant delays, several existing asymptotic stability criteria in the form of linear matrix inequalities (LMIs) are simplified by using matrix analysis techniques. Compared with the original stability criteria, the simplified ones include fewer LMI variables, which can obviously reduce computational complexity. Simultaneously, it is theoretically shown that the simplified stability criteria and original ones are equivalent; that is, they have the same conservativeness. Finally, a numerical example is employed to verify the theoretic results investigated in this paper.


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## Body

## 1. Introduction

Some practical systems, such as population ecology, neural networks, heat exchangers, and robots in contract with rigid environments, have been modeled by functional differential equations of neutral type (e.g., [1–6]). Since stability analysis is the primary task of analyzing and synthesizing a system, many researchers have paid more and more attention to establish stability criteria for delayed neutral systems (see [7, 8] and the references therein).For a class of linear time-invariant neutral systems with neutral and discrete constant delays, both neutral- and discrete-delay-dependent stability criteria have been investigated in [9–27]. Many approaches have been provided for obtaining these stability criteria. Here we mention only some critical and representative approaches. Model transformation approach first transforms the neutral system with discrete delay into the one with distributed delays, and then delay-dependent stability criteria can be obtained by constructing a Lyapunov-Krasovskii functional [9]. However, model transformation approach introduces some additional dynamics, which leads still to conservative results [28]. To overcome this shortage, the so-called descriptor model transformation approach is introduced in [15, 24], which transforms the original system to an equivalent descriptor form. The descriptor model transformation approach will not introduce additional dynamics in the sense defined in [28]. Although the results obtained from descriptor model transformation approach may be less conservative than some existing ones, they can be improved by employing some other approaches (e.g., free weighting matrix approach [21, 22] and augmented Lyapunov-Krasovskii functional approach [14, 23, 27]) to get a larger bound for discrete delay.By employing free weighting matrices and the Leibniz-Newton formula, He et al. [21] presented a neutral- and discrete-delay-dependent stability criterion for neutral systems with neutral and discrete constant delays, which reduces the conservativeness of methods involving a fixed model transformation. For the same system models, Qian et al. [27] established neutral- and discrete-delay-dependent stability criteria by constructing a Lyapunov-Krasovskii functional with additional functional parameters and employing free weighting matrices. To the best of our knowledge, the large-scale employment of functional parameters and/or free weighting matrices can cause complex stability criteria, and thereby the computational complexity increases obviously. Therefore, in order to reduce computational complexity, it is necessary to reduce the number of functional parameters and free weighting matrices. This motivates the present study.The aim of this paper is to simplify several stability criteria proposed by He et al. [21] and Qian et al. [27]. The main contributions of the paper are as follows. (i) In the premise of not increasing conservativeness, a pair of stability criteria proposed in [27, Theorem 1 and Corollary 1] are simplified by using matrix analysis techniques; (ii) one of the pair of simplified stability criteria is theoretically presented to be less conservative than [26, Corollary 1], and another is theoretically proven to be equivalent to [21, Theorem 1]; and (iii) by one numerical example, our theoretic results are verified to be more effective than many existing results.


## 2. Problem Statement and Preliminary Results

Consider a class of delayed neutral systems described as(1a)x˙(t)-Cx˙(t-η)=Ax(t)+A1x(t-τ),t≥0,(1b)x(t)=ϕ(t),t∈[-ω,0],wherex(t) is the n-dimensional state vector, A, A1, and C are known real constant matrices with appropriate dimensions, ω=max⁡{η,τ}, ϕ(s) is an ℝn-valued continuous initial function defined on [-ω,0], and positive scalars η and τ represent the neutral and discrete delays, respectively.DefineC([-η,0],ℝn) as the set of continuous ℝn valued function on the interval [-η,0], and let xt∈C([-η,0],ℝn) be a segment of system trajectory defined as
(2)xt(θ)=x(t+θ),θ∈[-η,0].Define an operator𝒟:C([-η,0],ℝn)→ℝn by 𝒟xt=x(t)-Cx(t-η). The definitions on stability of operator 𝒟 and systems (1a) and (1b) can be seen in [29].As mentioned in Introduction, neutral- and discrete-delay-dependent stability criteria for systems (1a) and (1b) have been investigated by He et al. [21] and Qian et al. [27]. In this paper, we will simplify the stability criteria [21, Theorem 1] and [27, Theorem 1 and Corollary 1] in the premise of not increasing conservativeness of stability criteria.In order to present conveniently our main results, the following lemmas are required.Lemma 1 (Schur complement lemma [30]).
Given constant matricesΩ1, Ω2, and Ω3 of appropriate dimensions, where Ω1T=Ω1 and Ω2T=Ω2, then
(3)Ω1+Ω3TΩ2-1Ω3<0
if and only if
(4)[Ω1Ω3TΩ3-Ω2]<0or[-Ω2Ω3Ω3TΩ1]<0.Lemma 2 (see [31]).
Given a real symmetric matrixΨ and a pair of real matrices P and Q, the following LMI problem
(5)Ψ+PTXTQ+QTXP<0
is solvable with respect to decision variable X if and only if
(6)𝒩PTΨ𝒩P<0,𝒩QTΨ𝒩Q<0,
where 𝒩P and 𝒩Q are matrices whose columns form a basis of the right null spaces of P and Q, respectively.


## 3. Several Stability Criteria in the Literature

In this section we introduce several existing stability criteria for systems (1a) and (1b), which will be useful to present conveniently the main results of this paper in the next section.Proposition 3 (see [21, Theorem 1]).
For given scalarsτ>0 and η>0, systems (1a) and (1b) are asymptotically stable, if the operator 𝒟 is stable and there exist real n×n matrices PT=P>0, QiT=Qi>0 (i=1,2), RT=R>0, XˇiiT=Xˇii, Yˇii=Yˇii (i=1,…,5), Xˇij, and Yˇij (1≤i<j≤5) such that the following LMIs are feasible:
(7)[XˇXˇ5TXˇ5Xˇ55]≥0,[YˇYˇ5TYˇ5Yˇ55]≥0,(8)[Φˇ+τXˇ+ηYˇ+ΦˇX+ΦˇYAˇTSSAˇ-S]<0,
where
(9)Xˇ=[Xˇ11Xˇ12Xˇ13Xˇ14Xˇ12TXˇ22Xˇ23Xˇ24Xˇ13TXˇ23TXˇ33Xˇ34Xˇ14TXˇ24TXˇ34TXˇ44],Yˇ=[Yˇ11Yˇ12Yˇ13Yˇ14Yˇ12TYˇ22Yˇ23Yˇ24Yˇ13TYˇ23TYˇ33Yˇ34Yˇ14TYˇ24TYˇ34TYˇ44],Xˇ5=[Xˇ15TXˇ25TXˇ35TXˇ45T],Yˇ5=[Yˇ15TYˇ25TYˇ35TYˇ45T],ΦˇX=Xˇ5TΠ1+Π1TXˇ5,ΦˇY=Yˇ5TΠ2+Π2TYˇ5,Π1=[In-In00],Π2=[In0-In0],S=R+τXˇ55+ηYˇ55,Aˇ=[AA10C],Φˇ=[Φˇ11PA1-ATPC0A1TP-Q1-A1TPC0-CTPA-CTPA1-Q20000-R],Φˇ11=PA+ATP+Q1+Q2.Proposition 4 (see [27, Theorem 1]).
For given scalarsτ>0 and η>0, systems (1a) and (1b) are asymptotically stable, if the operator 𝒟 is stable and there exist real n×n matrices U, PT=P, WT=W, QkT=Qk, NkT=Nk, RkT=Rk, TkT=Tk, Mk, Zk(k=1,2), Ei, Fi, Hi, Yii (i=1,…,5), and Yij (1≤i<j≤5) such that the following LMIs are feasible:
(10)[R2Z2Z2TT2]≥0,(11)Qk≥0,k=1,2,(12)W>0,[R1Z1Z1TT1]>0,(13)[PM1M2M1Tτ-1Q1+N1UTM2TUη-1Q2+N2]>0,(14)Ξ:=[Φ+ηYτG1-τFτG1T-τR1-τZ1-τFT-τZ1T-τT1]<0,(15)Ω:=[Y-G2H-G2TR2Z2HTZ2TT2]≥0,
where
(16)Φ=[Φ11⋯Φ15⋯⋯⋯Φ15T⋯Φ55],Y=[Y11⋯Y15⋯⋯⋯Y15T⋯Y55],F=[F1⋮F5],H=[H1⋮H5],Φ11=PA+ATP+Q1+Q2+τR1+ηR2Φ11=+M1+M1T+M2+M2T+F1+F1TΦ11=+H1+H1T-E1A-ATE1T,Φ12=PA1-M1+H2T-ATE2T+F2T-F1-E1A1,Φ13=-ATPC-M1TC-M2TC-M2Φ13=-H1+H3T+F3T-ATE3T,Φ14=-ATE4T+E1+F4T+H4T+τZ1+ηZ2,Φ15=-ATE5T+F5T+H5T-E1C,Φ22=-Q1-E2A1-A1TE2T-F2-F2T,Φ23=-A1TPC+M1TC-H2-F3T-A1TE3T,Φ24=E2-A1TE4T-F4T,Φ25=-E2C-A1TE5T-F5T,Φ33=-Q2+CTM2+M2TC-H3-H3T,Φ34=E3-H4T,Φ35=-E3C-H5T,Φ44=E4+E4T+W+τT1+ηT2,Φ45=-E4C+E5T,Φ55=-W-E5C-CTE5T,G1=[N1+UT-N1-UTM1T-M1TC]T,G2=[N2+U-U-N2M2T-M2TC]T.Proposition 5 (see [27, Corollary 1]).
Whenη=τ, for a given scalar τ>0, systems (1a) and (1b) are asymptotically stable, if the operator 𝒟 is stable and there exist real n×n matrices WT=W>0, Q1T=Q1>0, PT=P, N1T=N1, R1T=R1, T1T=T1, M1, Z1, Ei, and Fi (i=1,2,4,5) such that the following LMIs are feasible:
(17)[R1Z1Z1TT1]>0,[PM1M1Tτ-1Q1+N1]>0,(18)[Π11Π12Π13Π14τN1-τF1Π12TΠ22Π23Π24-τN1-τF2Π13TΠ23TΠ33Π34τM1-τF4Π14TΠ24TΠ34TΠ44-τCTM1-τF5τN1-τN1τM1T-M1TC-τR1-τZ1-τF1T-τF2T-τF4T-τF5T-τZ1T-τT1]<0,
where(19)Π11=PA+ATP+M1+M1T+Q1+τR1Π11=+F1+F1T-E1A-ATE1T,Π12=PA1-ATPC-M1-M1TCΠ12=-ATE2T-E1A1+F2T-F1,Π13=-ATE4T+E1+F4T+τZ1,Π14=-ATE5T+F5T-E1C,Π22=-Q1-A1TPC-CTPA1+CTM1+M1TCΠ22=-E2A1-A1TE2T-F2-F2T,Π23=E2-A1TE4T-F4T,Π24=-E2C-A1TE5T-F5T,Π33=E4+E4T+τT1+W,Π34=-E4C+E5T,Π44=-E5C-CTE5T-W.Remark 6.
Proposition3 simplifies the notations in [21, Theorem 1].Remark 7.
Propositions4 and 5 correct some slips of the pen in [27, Theorem 1 and Corollary 1]. After [27, Theorem 1] is amended as in Remark 7, [27, Theorem 2] will be correct.


## 4. Simplified Stability Criteria

In this section we will simplify the stability criteria introduced in the previous section. Firstly, the stability criterion presented in Proposition4 can be simplified by the following theorem.Theorem 8.
For given scalarsτ>0 and η>0, systems (1a) and (1b) are asymptotically stable, if the operator 𝒟 is stable and one of the following cases, (i)–(iv), holds. (i)
There exist realn×n matrices U, PT=P, WT=W, QkT=Qk, NkT=Nk, RkT=Rk, TkT=Tk, Mk, Zk(k=1,2), Ei, Fi, Hi, Yii (i=1,…,5), and Yij (1≤i<j≤5) such that the LMIs (10)–(15) hold.(ii)
There exist realn×n matrices U, PT=P, WT=W, QkT=Qk, NkT=Nk, RkT=Rk, TkT=Tk, Mk, Zk(k=1,2), Ei, Fi, Hi, Yii (i=1,…,5), and Yij (1≤i<j≤5) such that the LMIs (11)–(15) and the following LMI (20) hold:
(20)[R2Z2Z2TT2]>0.(iii)
There exist realn×n matrices U, PT=P, WT=W, QkT=Qk, NkT=Nk, RkT=Rk, TkT=Tk, Mk, and Zk (k=1,2) such that the LMIs (11), (12), and (13) and the following LMI (21) hold:
(21)[ΦτG1-τF-ηG2ηHτG1T-τR1-τZ100-τFT-τZ1T-τT100-ηG2T00-ηR2-ηZ2ηHT00-ηZ2T-ηT2]<0.(iv)
There exist realn×n matrices U, PT=P, WT=W, QkT=Qk, NkT=Nk, RkT=Rk, TkT=Tk, Mk, and Zk (k=1,2) such that the LMIs (11), (12), (13), and (20) and the following LMI (22) hold:
(22)Σ~:=[Φ~G~1G~2G~1T-τR10G~2T0-ηR2]<0,where
(23)Φ~=[Φ~11Φ~12Φ~13Φ~14Φ~12TΦ~22Φ~23A1TSCΦ~13TΦ~23TΦ~330Φ~14TCTSA10CTSC-W],Φ~11=PA+ATP+Q1+Q2+τR1+ηR2+M1+M1TΦ~11=+M2+M2T+ATSA+τATZ1T+τZ1AΦ~11=+ηATZ2T+ηZ2A-τ-1T1-η-1T2,Φ~12=PA1-M1+τZ1A1+ηZ2A1+ATSA1+τ-1T1,Φ~13=-ATPC-M1TC-M2TC-M2+η-1T2,Φ~14=τZ1C+ηZ2C+ATSC,Φ~22=-Q1-τ-1T1+A1TSA1,Φ~23=-A1TPC+M1TC,Φ~33=-Q2+CTM2+M2TC-η-1T2,S=W+τT1+ηT2,G~1=[τN1T+τU-Z1T+τATM1-τN1T+Z1T+τA1TM1-τU0],G~2=[-ηN2T-ηUT+Z2T-ηATM2ηUT-ηA1TM2ηN2T-Z2T0].Proof.
Due to Proposition4, it suffices to show that the conditions (i)–(iv) are equivalent.
(i)⇒(ii) Since the eigenvalues of a matrix are continuous functions of its elements, it follows from (i) that there exists a sufficiently small positive scalar ϵ such that
(24)Ξ+diag⁡(ηϵI,0,0,ηϵI,0,0,0)<0,Ω+diag⁡(0,ϵI,ϵI)≥0,[R2+ϵIZ2Z2TT2+ϵI]>0.
Therefore, the LMIs (11)–(15) and (20) are feasible.
(ii)⇒(i) The proof is very easy, and hence it is omitted.
(ii)⇒(iii) By Lemma 1, it follows from (20) and (15) that
(25)Y≥[-G2H][R2Z2Z2TT2]-1[-G2H]T.
This, together with (14) and Lemma 1, implies that (21) holds.
(iii)⇒(ii) It follows from (21) and Lemma 1 that (20) holds and
(26)[Φ+ηYτG1-τFτG1T-τR1-τZ1-τFT-τZ1T-τT1]<0,
where
(27)Y=[-G2H][R2Z2Z2TT2]-1[-G2H]T,
and hence (ii) holds.
(iii)⇔(iv) Let
(28)F~=[FHE],E=[E1TE2TE3TE4TE15]T,K=[e1Te2Te3Te4Te5T]T,L=[(e1-e2-τe7)T(e1-e3+ηe9)TA~T]T,A~=[-A-A10I-C0n×4n],ei=[0n×(i-1)nI0n×(9-i)n],i=1,2,…,9.
Then LMI (21) can be written as
(29)Φ1+KTF~L+LTF~TK<0,
where the matrix Φ1 is obtained from the matrix on the left of (21) by deleting all parts containing one of F, H, and E (e.g., the (5,5)th subblock of Φ1 equals -W, which is obtained from Φ55 by deleting -E5C-CTE5T). Choose
(30)𝒩K=[e6Te7Te8Te9T],𝒩L=[e1+ATe4+τ-1e7-η-1e9e2+A1Te4-τ-1e7e3+η-1e9CTe4+e5e6e8]T.
Noting that𝒩KTΦ1𝒩K<0, one can derive from (12) and Lemma 2 that the LMI (29) (i.e., (21)) is feasible if and only if LMIs (22) and (20) are feasible.
The proof is completed.Clearly, the stability condition (iv) in Theorem8 is more simpler than (i) in Theorem 8 (i.e., Proposition 4). By a process similar to investigating Theorem 8, one can easily obtain the following theorem which simplifies the stability criterion presented in Proposition 5.Theorem 9.
Whenη=τ, for a given scalar τ>0, systems (1a) and (1b) are asymptotically stable, if the operator 𝒟 is stable, and one of the following cases, (i)-(ii), holds.(i)
There exist realn×n matrices WT=W>0, Q1T=Q1>0, PT=P, N1T=N1, R1T=R1, T1T=T1, M1, Z1, Ei, and Fi (i=1,2,4,5) such that the LMIs (17) and (18) hold.(ii)
There exist realn×n matrices WT=W>0, Q1T=Q1>0, PT=P, N1T=N1, R1T=R1, T1T=T1, M1, and Z1 such that (17) and the following LMI (31) hold:
(31)[Π~11Π~12Π~13Π~14Π~12TΠ~22Π~23Π~24Π~13TΠ~23TΠ~330Π~14TΠ~24T0-τR1]<0,where
(32)Π~11=PA+ATP+M1+M1T+Q1+τR1Π~11=+τZ1A+τATZ1T+ATS0A-τ-1T1,Π~12=PA1-ATPC-M1-M1TCΠ~12=+τZ1A1+ATS0A1+τ-1T1,Π~13=τZ1C+ATS0C,Π~14=τN1+τATM1-Z1T,Π~22=-Q1-A1TPC-CTPA1+CTM1Π~22=+M1TC+A1TS0A1-τ-1T1,Π~23=A1TS0C,Π~24=-τN1+τA1TM1+Z1T,Π~33=CTS0C-W,S0=W+τT1.Remark 10.
Theorems8 and 9 indicate that, without increasing conservativeness of stability criteria, the free weighting matrices required in Propositions 4 and 5 (i.e., [27, Theorem 1 and Corollary 1]) can be eliminated. This will obviously reduce the computational complexity, and hence the stability conditions (iv) of Theorem 8 and (ii) of Theorem 9 are more effective.Remark 11.
In the special case thatM1=0, N1=0, Z1=0, and R1=ϵIn (ϵ is a sufficiently small positive number), Theorem 9 reduces to [26, Corollary 1]. So, by choosing suitable M1, N1, Z1, and R1, Theorem 9 could reduce the conservativeness of [26, Corollary 1], by which the improvement over [26, Corollary 1] is theoretically demonstrated.If we chooseU=0, Mi=0, Ni=0, Zi=0, and Ri=ϵIn (ϵ is a sufficiently small positive number) (i=1,2) in Theorem 8, the following corollary can be easily derived.Corollary 12.
For given scalarsτ>0 and η>0, systems (1a) and (1b) are asymptotically stable, if the operator 𝒟 is stable and there exist real n×n matrices PT=P>0, WT=W>0, QkT=Qk>0, and TkT=Tk>0 (k=1,2) such that LMI (33) holds:
(33)Φ^:=[Φ^11Φ^12Φ^13ATSCΦ^12TΦ^22Φ^23A1TSCΦ^13TΦ^23TΦ^330CTSACTSA10CTSC-W]<0,
where
(34)Φ^11=PA+ATP+Q1+Q2+ATSA-τ-1T1-η-1T2,Φ^12=PA1+ATSA1+τ-1T1,Φ^13=-ATPC+η-1T2,Φ^22=-Q1-τ-1T1+A1TSA1,Φ^23=-A1TPC,Φ^33=-Q2-η-1T2,S=W+τT1+ηT2.Next we will show that the pair of stability criteria presented in Proposition3 (i.e., [21, Theorem 1]) and Corollary 12 have the same conservativeness. However, the stability criterion presented in Corollary 12 is more simpler than the one in Proposition 3 (i.e., [21, Theorem 1]), which reduces greatly the computational complexity.Theorem 13.
For given scalarsτ>0 and η>0, LMIs (7) and (8) are feasible if and only if the LMI (33) is feasible.Proof.
The “Only If” Part. It follows from (7) and (8) that
(35)Xˇ≥Xˇ5TXˇ55-1Xˇ5,Yˇ≥Yˇ5TYˇ55-1Yˇ5,Φˇ+τXˇ+ηYˇ+ΦˇX+ΦˇY+AˇTSAˇ<0,
and hence
(36)Φˇ+τXˇ5TXˇ55-1Xˇ5+ηYˇ5TYˇ55-1Yˇ5+ΦˇX+ΦˇY+AˇTSAˇ<0.
Noting that(37)τXˇ5TXˇ55-1Xˇ5+ΦˇX+τ-1Π1TXˇ55Π1≥0,ηYˇ5TYˇ55-1Yˇ5+ΦˇY+η-1Π2TYˇ55Π2≥0,
one can conclude from (36) that
(38)Φˇ-τ-1Π1TXˇ55Π1-η-1Π2TYˇ55Π2+AˇTSAˇ<0,
which becomes LMI (33) by letting W=R, T1=Xˇ55, and T2=Tˇ55. That is, LMI (33) is feasible.
The “If” Part. Since LMI (33) is feasible, there exist real n×n matrices PT=P>0, WT=W>0, QkT=Qk>0, and TkT=Tk>0 (k=1,2) such that LMI (33) is satisfied. Set R=W, Xˇ55=T1, and Yˇ55=T2. Then it follows from (33) and Lemma 1 that
(39)[Φˇ-τ-1Π1TXˇ55Π1-η-1Π2TYˇ55Π2AˇTSSAˇ-S]<0.
Let(40)Xˇ5=-τ-1Xˇ55Π1,Xˇ=Xˇ5TXˇ55-1Xˇ5,Yˇ5=-η-1Yˇ55Π2,Yˇ=Yˇ5TYˇ55-1Yˇ5.
Then LMI (7) holds, and
(41)τXˇ5TXˇ55-1Xˇ5+ΦˇX+τ-1Π1TXˇ55Π1=0,ηYˇ5TYˇ55-1Yˇ5+ΦˇY+η-1Π2TYˇ55Π2=0.
Furthermore, LMI (8) is derived from (39).
The proof is completed.


## 5. Numerical Comparisons of Stability Criteria

In this section we will present one example to demonstrate effectiveness of the theoretic results described above.Example 1 .
Consider the stability of systems (1a) and (1b) with
(42)A=[-0.90.20.1-0.9],A1=[-1.1-0.2-0.1-1.1],C=[-0.200.2-0.1].Case 1 (τ=η).
The upper bounds on the delayτ under which robust stability of this system can be guaranteed using the methods in [9–27] and Theorem 9 of this paper are listed in Table 1. It is clear that our results are significantly better than those in [9–26] because our allowable maximum time delay τ is larger.Table 1
The maximum allowable bounds ofτ=η.
Method
max
⁡
τ
[9]
0.3
[10]
0.5658
[11]
0.5937
[12]
0.6054
[13]
0.6612
[14]
0.7039
[15]
0.74
[16]
0.8844
[17]
1.0402
[18]
1.3718
[19]
1.6014
[20]
1.61
[21]
1.6527
[22]
1.7191
[23]
1.7220
[24]
1.78
[25]
1.7844
[26, Corollary  1]
1.7856
[27] and Theorem 9
1.7890Case 2 (τ≠η).
For different values ofη, Table 2 lists the maximum allowable bounds on τ that guarantee the stability of the system. It can be seen that the maximum allowable bound on τ decreases as η increases when η is small but that τ remains almost unchanged when η increases to certain extent.Table 2
The maximum allowable bounds ofτ for different η.
Method
η
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
10000
100000
[21] and Corollary 12
τ
1.7100
1.6987
1.6883
1.6792
1.6718
1.6664
1.6624
1.6591
1.6563
1.6543
1.6531
1.6527
1.6527
1.6527
1.6527
1.6527
1.6527
[25]
τ
1.7844
1.7757
1.7669
1.7581
1.7495
1.7413
1.7338
1.7273
1.7226
1.7201
1.7193
1.7191
1.7191
1.7191
1.7191
1.7191
1.7191
[27] and Theorem 8
τ
1.8305
1.8174
1.8038
1.7897
1.7755
1.7616
1.7484
1.7366
1.7272
1.7213
1.7202
1.7202
1.7202
1.7202
1.7202
1.7193
1.7193


## 6. Conclusions

By using matrix analysis techniques, several stability criteria (i.e., [21, Theorem 1] and [27, Theorem 1 and Corollary 1]) for delayed neutral systems have been simplified. The numbers of LMI variables in [21, Theorem 1] and [27, Theorem 1 and Corollary 1] are reduced into the simplified ones, which can obviously reduce computational complexity. Furthermore, it is theoretically proven that the simplified stability criteria have the same conservativeness as the original ones. A numerical example is given to illustrate the theoretic results investigated in this paper.Extending the idea of this paper to other system models, including singular delayed systems [32–35], stochastic systems [36], Markovian jump systems [37, 38], and genetic regulatory networks [39–41], is under consideration.



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*Source: 238487-2014-05-13.xml*