# Improved Robust Stability Criteria of Uncertain Neutral Systems with Mixed Delays

Authors: Zixin Liu; Shu Lü; Shouming Zhong; Mao Ye
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

The problem of robust stability for a class of neutral control systems with mixed delays is investigated. Based on Lyapunov stable theory, by constructing a new Lyapunov-Krasovskii function, some new stable criteria are obtained. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.


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## Body

## 1. Introduction

As one of important dynamical systems, neutral system has been received considerable attention in past years. Large numbers of monographs and papers on the stability of neutral type system with or without time delays have been published. A wide variety of methods disposing the stability problems of neutral system have been proposed [1–6]. It is well known that, because of the finite switching speed, memory effect, and so on, time delays are unavoidable in nature and technology. They can make important effects on the stability of concerned dynamical systems. Thus, the studies on stability of time-delayed neutral system are of great significance. In recent years, all kinds of delays such as time-varying delay [7–10], distributed delay [11–13], and mixed delay [14–16] were considered, and corresponding stable criteria have been derived. In practical, during the design of control system and its hardware implementation, the convergence of a control system may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, the deviation of vital data, and so on. Therefore, the studies on robust convergence of delayed control system have been a hot research topic, and many sufficient conditions have been derived to guarantee the robust asymptotic or exponential stability for different class of delayed systems (see [2, 7, 8, 10, 12, 13, 17–20]). General speaking, these criteria can be divided into two categories [21]: that is, delay-independent criteria and delay-dependent criteria. As pointed out in [12], when the size of time delay is small, delay-dependent criteria may be less conservative than those of delay-independent criteria, and the more free-weighting matrices are introduced in criterion, the less conservative it may be. On the other hand, compared with traditional matrix measure, matrix norm, and Riccati matrix criteria, linear matrix inequality (LMI) technique can be easily checked by LMI toolbox in MATLAB software and can make free weighting matrices easy to select. Thus, it becomes one of the most extensively used techniques in control system. In addition, the admissible allowed upper bound on the delay is usually regarded as the performance index for measuring the conservatism of the conditions obtained.Motivated by the afore-mentioned analysis, in this paper, based on the equivalent equation of the zero which is similar to [2] in the derivative of a Lyapunov-Krasovskii functional, we will focus on deriving some improved robust stable criteria for a class of neutral control systems with mixed delays. By constructing a new Lyapunov function, some new delay-dependent stable criteria are derived via sufficiently employing Newton-Leibniz formula to introduce large numbers of free weighting matrices. These free weighting matrices express the influence of the relationship among terms x(t),ẋ(t),x(t-τ1),x(t-τ2), ẋ(t-τ2), ∫t-τ1tẋ(s)ds, ∫t-τ2tẋ(s)ds. Since these criteria are both discrete delay-dependent and distributed delay-dependent, they are less conservative than some previous methods for the concerned systems. When norm-bounded parameter uncertainties appear in the concerned system, delay-dependent robust asymptotic stability criteria are also presented. All of these criteria are expressed in the forms of linear matrix inequalities (LMIs), which can be easily solved. Finally, numerical examples are given to illustrate the improvement of the main results. Simulations show that our results are valid.


## 2. Preliminaries

Consider uncertain neutral system with mixed delays [4] as follows:(2.1)
Σ
:
{
x
̇
(
t
)
-
C
x
̇
(
t
-
τ
2
)
=
(
A
+
Δ
A
(
t
)
)
x
(
t
)
+
(
B
+
Δ
B
(
t
)
)
x
(
t
-
τ
1
)
+
(
D
+
Δ
D
(
t
)
)
∫
t
-
h
t
x
(
s
)
d
s
,
t
>
0
,
x
(
t
)
=
ϕ
(
t
)
,
t
∈
[
-
τ
,
0
]
,
where x(t)∈ℛn is the state vector; A,B,C∈ℛn×n represent the weighting matrices; τ1,τ2 are discrete delays; h is distributed delay; ϕ(t) is initial condition which is continuous on interval [-τ,0], where  τ=max{τ1,τ2,h}; ΔA(t),ΔB(t),ΔD(t) denote the time-varying structured uncertainties which are of the following form:(2.2)
[
Δ
A
(
t
)
,
Δ
B
(
t
)
,
Δ
D
(
t
)
]
=
K
F
(
t
)
[
E
a
,
E
b
,
E
d
]
,
where K,Ea,Eb,Ed are the known constant matrices with appropriate dimensions; F(t) is unknown continuous time-varying matrix function satisfying FT(t)F(t)≤I,forallt≥0.The nominalΣ0 of Σ can be defined as(2.3)
Σ
0
:
{
x
̇
(
t
)
-
C
x
̇
(
t
-
τ
2
)
=
A
x
(
t
)
+
B
x
(
t
-
τ
1
)
+
D
∫
t
-
h
t
x
(
s
)
d
s
,
t
>
0
,
x
(
t
)
=
ϕ
(
t
)
,
t
∈
[
-
τ
,
0
]
.For further discussion, we first introduce the following lemmas.Lemma 2.1 (see [22]).
Given constant symmetric matricesΣ1,Σ2,Σ3 where Σ1T=Σ1 and 0<Σ2=Σ2T, then Σ1+Σ3TΣ2-1Σ3<0 if and only if
(2.4)(Σ1Σ3TΣ3-Σ2)<0,or(-Σ2Σ3Σ3TΣ1)<0.Lemma 2.2 (see [23]).
For given matricesQ=QT,H,E, and R=RT>0 with appropriate dimensions, then
(2.5)Q+HFE+ETFTHT<0,
for all F satisfying FTF≤R if and only if there exists a positive number ε>0, such that
(2.6)Q+ε-1HHT+εETRE<0.Lemma 2.3.
For any real vectorX,Y and positive definite matrix Σ>0 with appropriate dimensions, it follows that
(2.7)2XTY≤XTΣX+YTΣ-1Y.


## 3. Main Results

In this section, we will analyse the stability problem of uncertain neutral systems with mixed delays described by (2.1). First, we consider the stability problem for the nominal system (2.3) with ΔA(t)=0,ΔB(t)=0,ΔD(t)=0. In order to introduce free-weighting matrix, we can use the following fact:(3.1)
M
[
x
(
t
)
-
x
(
t
-
τ
1
)
-
∫
t
-
τ
1
t
x
̇
(
s
)
d
s
]
=
0
,
where M is an arbitrary matrix with appropriate dimensions. Substituting zero equation (3.1) into system (2.3), the original system can be transformed into the following form:(3.2)
x
̇
(
t
)
-
C
x
̇
(
t
-
τ
2
)
=
(
A
-
M
)
x
(
t
)
+
(
B
+
M
)
x
(
t
-
τ
1
)
+
D
∫
t
-
h
t
x
(
s
)
d
s
+
M
∫
t
-
τ
1
t
x
̇
(
s
)
d
s
,
t
>
0
,
x
(
t
)
=
ϕ
(
t
)
,
t
∈
[
-
τ
,
0
]
.For the asymptotic stability of system (3.2), we can obtain the following results.Theorem 3.1.
For any given matrixM, scalars τ1>0,τ2>0,h>0, the nominal system Σ0 is asymptotically stable if ∥C∥<1, and there exist positive definite matrices P1,Q1,Q2,Q3,Q4,Q5, arbitrary matrices Pi(i=2,3,…,25), Fi(i=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible:
(3.3)Ξ1=(Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ17Ξ18Ξ19*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28Ξ29**Ξ33Ξ34Ξ35Ξ36Ξ37Ξ38Ξ39***Ξ44Ξ45Ξ46Ξ47Ξ48Ξ49****Ξ55Ξ56Ξ57Ξ58Ξ59*****Ξ66Ξ67Ξ68Ξ69******Ξ77Ξ78Ξ79*******Ξ88Ξ89********Ξ99)<0,
where
(3.4)Ξ11=P2T(A-M)+P10T+P18T+(A-M)TP2+P10+P18+hQ3+Q1,Ξ12=P1T-P2T+(A-M)TP3+P11+P19,Ξ13=P2T(B+M)-P18T+(A-M)TP4+P12+P20,Ξ14=-P10T+(A-M)TP5+P13+P21,Ξ15=P2TC+(A-M)TP6+P14+P22,Ξ16=P2TM-P18T+(A-M)TP7+P15+P23-F1T,Ξ17=-P10T+(A-M)TP8+P16+P24-F1T,Ξ18=P2TD+(A-M)TP9+P17+P25-F1T,Ξ19=F1T,Ξ22=-P3T+P3T+τ1Q4+τ2Q5+Q2,Ξ23=P3T(B+M)-P19T-P4,Ξ24=-P11T-P5T,Ξ25=P3TC-P6T,Ξ26=P3TM-P19T-P7-F2T,Ξ27=-P11T-P8-F2T,Ξ28=P3TD-P9-F2T,Ξ29=F2T,Ξ33=P4T(B+M)-P20T+(B+M)TP4-P20-Q1,Ξ34=-P12T+(B+M)TP5-P21,Ξ35=P4TC+(B+M)TP6-P22,Ξ36=P4TM-P20T+(B+M)TP7-P23-F3T,Ξ37=-P12T+(B+M)TP8-P24-F3T,Ξ38=P4TD+(B+M)TP9-P25-F3T,Ξ39=F3T,Ξ44=-P13T-P13,Ξ45=P5TC-P14,Ξ46=P5TM-P21T-P15-F4T,Ξ47=-P13T-P16-F4T,Ξ48=P5TD-P17-F4T,Ξ49=F4T,Ξ55=P6TC+CTP6-Q2,Ξ56=P6TM-P22T+CTP7-F5T,Ξ57=-P14T+CTP8-F5T,Ξ58=P6TD+CTP9-F5T,Ξ59=F5T,Ξ66=P7TM-P23T+MTP7-P23-F6T-F6,Ξ67=-P15T+MTP8-P24-F6T-F7T,Ξ68=P7TD+MTP9-P25-F6T-F8T,Ξ69=F6T,Ξ77=-P16T-P16-F7T-F7,Ξ78=P8TD-P17-F8T-F7T,Ξ79=F7T,Ξ88=P9TD+DTP9-F8T-F8T,Ξ89=F8T,Ξ99=-(1hQ3+1τ1Q4+1τ2Q5).Proof.
Constructing a new Lyapunov functional candidate for system (3.2) as follows:
(3.5)V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t)+V6(t),
where
(3.6)V1(t)=YT(t)PY(t),Y(t)=[xT(t),ẋT(t),xT(t-τ1),xT(t-τ2),ẋT(t-τ2),(∫t-τ1tẋ(s)ds)T,(∫t-τ2tẋ(s)ds)T,(∫t-htx(s)ds)T]T,V2(t)=∫t-τ1txT(s)Q1x(s)ds,V3(t)=∫t-τ2tẋT(s)Q2ẋ(s)ds,V4(t)=∫t-ht∫stxT(ξ)Q3x(ξ)dξds,V5(t)=∫t-τ1t∫stẋT(ξ)Q4ẋ(ξ)dξds,V6(t)=∫t-τ2t∫stẋT(ξ)Q5ẋ(ξ)dξds,(3.7)P=(P1000000000000000000000000000000000000000000000000000000000000000).
Set
(3.8)P̃=(P10000000P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16P17P18P19P20P21P22P23P24P25),
along the trajectories of system (3.2), the derivative of V(t) is given by
(3.9)V̇(t)=V̇1(t)+V̇2(t)+V̇3(t)+V̇4(t)+V̇5(t)+V̇6(t),
where
(3.10)V̇1(t)=2YT(t)PẎ(t)=2YT(t)P̃T×(ẋ(t)-ẋ(t)+Cẋ(t-τ2)+(A-M)x(t)+(B+M)x(t-τ1)+D∫t-htx(s)ds+M∫t-τ1tẋ(s)dsx(t)-x(t-τ2)-∫t-τ2tẋ(s)dsx(t)-x(t-τ1)-∫t-τ1tẋ(s)ds)=2YT(t)P̃T(0I000000A-M-IB+M0CM0DI00-I00-I0I0-I00-I00)Y(t),(3.11)V̇2(t)=xT(t)Q1x(t)-xT(t-τ1)Q1x(t-τ1),V̇3(t)=ẋT(t)Q2ẋ(t)-ẋT(t-τ2)Q2ẋ(t-τ2),V̇2(t)+V̇3(t)=YT(t)(Q10000000*Q2000000**-Q100000***00000****-Q2000*****000******00*******0)Y(t).
By Lemma 2.3, similar to the disposal route in [34], we have
(3.12)V̇4(t)=hxT(t)Q3x(t)-∫t-htxT(s)Q3x(s)ds≤hxT(t)Q3x(t)+∫t-ht[-2xT(s)FY(t)+YT(t)FTQ3-1FY(t)]ds=hxT(t)Q3x(t)-2(∫t-htx(s)ds)TFY(t)+hYT(t)FTQ3-1FY(t),
where F=[F1,F2,F3,F4,F5,F6,F7,F8]. Similarly, we have
(3.13)V̇5(t)=τ1ẋT(t)Q4ẋ(t)-∫t-τ1tẋT(s)Q4ẋ(s)ds≤τ1ẋT(t)Q4ẋ(t)+∫t-τ1t[-2ẋT(s)FY(t)+YT(t)FTQ4-1FY(t)]ds=τ1ẋT(t)Q4ẋ(t)-2(∫t-τ1tẋ(s)ds)TFY(t)+τ1YT(t)FTQ4-1FY(t),(3.14)V̇6(t)=τ2ẋT(t)Q5ẋ(t)-∫t-τ2tẋT(s)Q5ẋ(s)ds≤τ2ẋT(t)Q5ẋ(t)+∫t-τ2t[-2ẋT(s)FY(t)+YT(t)FTQ5-1FY(t)]ds=τ2ẋT(t)Q5ẋ(t)-2(∫t-τ2tẋ(s)ds)TFY(t)+τ2YT(t)FTQ5-1FY(t).
From (3.12)–(3.14), we get
(3.15)V̇4(t)+V̇5(t)+V̇6(t)≤YT(t)(hQ30000-F1T-F1T-F1T*τ1Q4+τ2Q5000-F2T-F2TF2T**000-F3T-F3T-F3T***00-F4T-F4T-F4T****0-F5T-F5T-F5T*****-F6T-F6-F7T-F6T-F8T-F6T******-F7T-F7T-F8T-F7T*******-F8T-F8T)Y(t)+YT(t)FT(hQ3-1+τ1Q4-1+τ2Q5-1)FY(t).
Hence,(3.16)V̇(t)≤YT(t)[Ξ′+FT(hQ3-1+τ1Q4-1+τ2Q5-1)F]Y(t),
where
(3.17)Ξ′=(Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ17Ξ18*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28**Ξ33Ξ34Ξ35Ξ36Ξ37Ξ38***Ξ44Ξ45Ξ46Ξ47Ξ48****Ξ55Ξ56Ξ57Ξ58*****Ξ66Ξ67Ξ68******Ξ77Ξ78*******Ξ88).
In views of Lemma 2.1 and (3.3), we have V̇(t)<0, namely, there exists a positive scalar λ>0 such that V̇(t)≤-λ∥Y(t)∥2. According to [35], system (2.3) is asymptotically stable, this completes the proof.Remark 3.2.
Motivated by the results obtained in [34], free-weighting matrices Fi(i=1,2,…,8) are introduced in Theorem 3.1 so as to reduce the conservatism of the delay-dependent result. Moreover, more free-weighting matrices are introduced by the construction and disposal of V1(t), which may make the conservatism reduce further.Remark 3.3.
The transformation from system (2.3) to (3.2) enables us to utilize the information of the relationship among terms x(t),x(t-τ1),∫t-τ1tẋ(s)ds. Combined with the arbitrariness of matrix M, the conservatism of stability criterion is reduced further.Remark 3.4.
WhenM=I, we can obtain the following simplified corollary.Corollary 3.5.
For given positive scalarsτ1>0,τ2>0,h>0, the nominal system Σ0 is asymptotically stable if ∥C∥<1, and there exist positive definite matrices P1,Q1,Q2,Q3,Q4,Q5 and arbitrary matrices Pi(i=2,3,…,25), Fi(i=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible:
(3.18)Ξ2=(Ξ11′Ξ12′Ξ13′Ξ14′Ξ15′Ξ16′Ξ17′Ξ18′Ξ19*Ξ22Ξ23′Ξ24Ξ25Ξ26′Ξ27Ξ28Ξ29**Ξ33′Ξ34′Ξ35′Ξ36′Ξ37′Ξ38′Ξ39***Ξ44Ξ45Ξ46′Ξ47Ξ48Ξ49****Ξ55Ξ56′Ξ57Ξ58Ξ59*****Ξ66′Ξ67′Ξ68′Ξ69******Ξ77Ξ78Ξ79*******Ξ88Ξ89********Ξ99)<0,
where
(3.19)Ξ11′=P2T(A-I)+P10T+P18T+(A-I)TP2+P10+P18+hQ3+Q1,Ξ12′=P1T-P2T+(A-I)TP3+P11+P19,Ξ13′=P2T(B+I)-P18T+(A-I)TP4+P12+P20,Ξ14′=-P10T+(A-I)TP5+P13+P21,Ξ15′=P2TC+(A-I)TP6+P14+P22,Ξ16′=P2T-P18T+(A-I)TP7+P15+P23-F1T,Ξ18′=P2TD+(A-I)TP9+P17+P25-F1T,Ξ23′=P3T(B+I)-P19T-P4,Ξ26′=P3T-P19T-P7-F2T,Ξ33′=P4T(B+I)-P20T+(B+I)TP4-P20-Q1,Ξ34′=-P12T+(B+I)TP5-P21,Ξ35′=P4TC+(B+I)TP6-P22,Ξ36′=P4T-P20T+(B+I)TP7-P23-F3T,Ξ37′=-P12T+(B+I)TP8-P24-F3T,Ξ38′=P4TD+(B+I)TP9-P25-F3T,Ξ46′=P5T-P21T-P15-F4T,Ξ56′=P6T-P22T+CTP7-F5T,Ξ66′=P7T-P23T+P7-P23-F6T-F6,Ξ67′=-P15T+P8-P24-F6T-F7T,Ξ68′=P7TD+P9-P25-F6T-F8T.Remark 3.6.
Based on Theorem3.1 and Corollary 3.5, by using Lemmas 2.1 and 2.2, we can perform the robust asymptotic stability analysis for system (2.1) with uncertainty ΔA(t),ΔB(t),ΔD(t) as follows.Theorem 3.7.
For any given matrixM, scalars τ1>0,τ2>0,h>0, the original system Σ is robustly and asymptotically stable if ∥C∥<1, and there exist positive definite matrices P1,Q1,Q2,Q3,Q4,Q5, positive scalar δ, and arbitrary matrices Pi(i=2,3,…,25), Fi(i=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible:
(3.20)Ξ3=(Ξ11+δEaTEaΞ12Ξ13Ξ14Ξ15Ξ16Ξ17Ξ18Ξ19P2TK*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28Ξ29P3TK**Ξ33+δEbTEbΞ34Ξ35Ξ36Ξ37Ξ38Ξ39P4TK***Ξ44Ξ45Ξ46Ξ47Ξ48Ξ49P5TK****Ξ55Ξ56Ξ57Ξ58Ξ59P6TK*****Ξ66+δEcTEcΞ67Ξ68Ξ69P7TK******Ξ77Ξ78Ξ79P8TK*******Ξ88Ξ89P9TK********Ξ990*********-δI)<0.Proof.
ReplacingA,B,D in (3.3) with A+KF(t)Ea, B+KF(t)Eb, and D+KF(t)Ed, respectively, (3.3) for system (2.1) is equivalent to the following form:
(3.21)Ξ+Π1TFT(t)Π2+Π2TF(t)Π1<0,
where Π1=[KTP2,KTP3,KTP4,KTP5,KTP6,KTP7,KTP8,KTP9], Π2=[Ea,0,Eb,0,0,Ec,0,0,0]. From Lemma 2.2, a sufficient condition for (3.20) is that there exists a positive scalar δ>0 such that
(3.22)Ξ+δ-1Π1TΠ1+δΠ2TΠ2<0.
In views of Lemma 2.1, we can easily obtain this conclusion, this completes the proof.Corollary 3.8.
For given positive scalarsτ1>0,τ2>0,h>0, the original system Σ is robustly and asymptotically stable if ∥C∥<1, and there exist positive definite matrices P1,Q1,Q2,Q3,Q4,Q5, positive scalar δ, and arbitrary matrices Pi(i=2,3,…,25), Fi(i=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible:
(3.23)Ξ4=(Ξ11′+δEaTEaΞ12′Ξ13′Ξ14′Ξ15′Ξ16′Ξ17′Ξ18′Ξ19P2TK*Ξ22Ξ23′Ξ24Ξ25Ξ26′Ξ27Ξ28Ξ29P3TK**Ξ33′+δEbTEbΞ34′Ξ35′Ξ36′Ξ37′Ξ38′Ξ39P4TK***Ξ44Ξ45Ξ46′Ξ47Ξ48Ξ49P5TK****Ξ55Ξ56′Ξ57Ξ58Ξ59P6TK*****Ξ66′+δEcTEcΞ67′Ξ68′Ξ69P7TK******Ξ77Ξ78Ξ79P8TK*******Ξ88Ξ89P9TK********Ξ990*********-δI)<0.


## 4. Numerical Examples

In this section, some numerical examples will be presented to show the validity and improvement of the main results derived earlier.Example 4.1.
Consider the following neutral  system presented in Park and Kwon [28]:
(4.1)ẋ(t)-Cẋ(t-τ2)=Ax(t)+Bx(t-τ1)+D∫t-htx(s)ds,
with
(4.2)A=(-3-210),B=(-0.50.10.30),C=0,D=0.
Set M=(-100-2). For comparisons, we calculate the allowable upper bound of τ1 for which the asymptotic stability is guaranteed. For this example, Table 1 shows that Theorem 3.1 obtained in this paper is less conservative than the related results obtained in [12, 24–28].Table 1
Stability bounds of time delays (Example4.1).
[24]
[25]
[26]
[27]
[28]
[12]
Ours
τ
1
0
<
τ
1
<
0.4991
0
<
τ
1
<
0.7602
0
<
τ
1
<
0.4991
0
<
τ
1
<
1.6965
τ
1
>
0
τ
1
>
0
τ
1
>
0Example 4.2.
Consider the following neutral system studied in He et al. [32]:
(4.3)ẋ(t)-Cẋ(t-τ2)=Ax(t)+Bx(t-τ1)+D∫t-htx(s)ds,
with
(4.4)A=(-0.90.20.1-0.9),B=(-1.1-0.2-0.1-1.1),C=(-0.200.2-0.1),D=0.
Set M=(-100-2).
For this example, we calculate the allowable upper bound ofτ1 for which the asymptotic stability is guaranteed. Table 2 shows that Theorem 3.1 obtained in this paper is less conservative than the related results obtained in [12, 13, 24, 28–33].Table 2
Some comparison for allowable upper bounds onτ1 (Example 4.2).
[29]
[30]
[31]
[24]
[28]
[32]
[13]
[12]
[33]
Ours
τ
1
,
τ
2
0.3
0.71
0.74
0.8844
1.3718
1.6525
1.6525
1.6525
2.2254
2.2254Example 4.3.
Consider the neutral system studied in Liu et al. [12]:
(4.5)ẋ(t)-Cẋ(t-τ2)=(A+ΔA(t))x(t)+(B+ΔB(t))x(t-τ1)+(D+ΔD(t))∫t-htx(s)ds,
with
(4.6)A=(-200-15),B=(13-31),C=(cc00cc),D=(10.5-0.51),0≤cc<1.
For the convenience of comparison, let ΔA(t)=ΔB(t)=ΔD(t)=0. Set M=(-100-2). The comparative results between Theorem 3.1 with the result obtained in [12] are given in Table 3. When
(4.7)[ΔA(t),ΔB(t),ΔD(t)]=KF(t)[Ea,Eb,Ed],
where K=I,Ea=Eb=Ed=0.2I, Table 4 shows that the robust stability criterion obtained in this paper is also less conservative than the related result obtained in [12]. From the simulation figures (see Figures 1–4), one can see that the results derived in this paper are valid.Table 3
Calculated allowable size of distributed delayh (ΔA(t)=ΔB(t)=ΔD(t)=0) (Example 4.3).
Liu [12]
cc
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
τ
1
=
τ
2
=
h
0.67
0.62
0.56
0.51
0.46
0.41
0.36
0.32
0.28
τ
1
=
τ
2
=
0.3
0.79
0.78
0.73
0.66
0.58
0.50
0.41
0.37
0.21
Ours
τ
1
=
τ
2
=
h
2.15
2.13
2.11
2.09
2.07
2.04
2.02
1.99
1.97
τ
1
=
τ
2
=
0.3
2.61
2.61
2.61
2.61
2.61
2.61
2.61
2.61
2.61Table 4
Calculated allowable size of distributed delayh (K=I,Ea=Eb=Ed=0.2I) (Example 4.3).
Liu [12]
cc
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
τ
1
=
τ
2
=
h
0.67
0.62
0.56
0.51
0.46
0.41
0.36
0.32
0.28
τ
1
=
τ
2
=
0.3
0.79
0.78
0.73
0.66
0.58
0.50
0.41
0.37
0.21
Ours
0.3
τ
1
=
τ
2
=
h
1.82
1.79
1.77
1.75
1.72
1.70
1.67
1.63
1.60
τ
1
=
τ
2
=
0.3
1.94
1.94
1.94
1.94
1.94
1.94
1.94
1.94
1.94Figure 1
State trajectories of system (12) withcc=0.4,h=τ1=τ2=1.60,Ea=Eb=Ed=0.2I,K=I.Figure 2
State trajectories of system (12) withcc=0.4,τ1=τ2=0.3,h=1.94,Ea=Eb=Ed=0.2I,K=I.Figure 3
State trajectories of system (12) withcc=0.4,τ1=τ2=0.3,h=2.61,Ea=Eb=Ed=0.Figure 4
State trajectories of system (12) withcc=0.4,τ1=τ2=h=1.97,Ea=Eb=Ed=0.Example 4.4.
Consider the following uncertain neutral system [32, 33, 36]:
(4.8)ẋ(t)-Cẋ(t-τ)=[A+KF(t)Ea]x(t)+[B+KF(t)Eb]x(t-τ),
where A=(-200-0.9), B=(-10-1-1), C=(c00c),0≤c<1, L=(0.2000.2), Ea=Eb=(1001). Table 5 gives out the related comparative results, from which one can see that Theorem 3.7 obtained in this paper is less conservative than those established in [32, 33, 36].Table 5
Comparison ofτmax using different methods (Example 4.4).
cc
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Han [33]
3.13
2.98
2.83
2.66
2.49
2.31
2.12
1.93
Han [36]
1.77
1.63
1.48
1.33
1.16
0.98
0.79
0.59
He et al. [32]
2.39
2.05
1.75
1.49
1.27
1.08
0.91
0.76
Theorem3.7
3.45
3.21
3.02
2.88
2.62
2.54
2.51
2.23


## 5. Conclusion

By constructing a new Lyapunov function, some new robust stable criteria for a class of neutral control systems with mixed delays are obtained. These criteria are formulated in the forms of linear matrix inequalities. Compared with some previous publications, our results are less conservative. Numerical examples and simulations show that our results are valid.



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*Source: 294845-2009-11-22.xml*