New Delay-Dependent Stability Criteria for Uncertain Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Abstract

The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method.

Full text

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Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 759248, 22 pages
doi:10.1155/2009/759248
Research Article
New Delay-Dependent Stability Criteria for
Uncertain Neutral Systems with Mixed
Time-Varying Delays and Nonlinear Perturbations
Hamid Reza Karimi, Mauricio Zapateiro, and Ningsu Luo
Institute of Informatics and Applications, University of Girona, Campus de Montilivi, Edifici P4,
17071 Girona, Spain
Correspondence should be addressed to Hamid Reza Karimi, hamidreza.karimi@udg.edu
Received 19 July 2008; Revised 8 November 2008; Accepted 1 January 2009
Recommended by Shijun Liao
The problem of stability analysis for a class of neutral systems with mixed time-varying
neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed.
By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model
transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable
change of variables, new sufficient conditions are established for the stability of the considered
system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-
delay-dependent. The conditions are presented in terms of linear matrix inequalities ?LMIs? and
can be efficiently solved using convex programming techniques. Two numerical examples are
given to illustrate the efficiency of the proposed method.
Copyright q 2009 Hamid Reza Karimi et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Delay ?or memory? systems represent a class of infinite-dimensional systems largely used
to describe propagation and transport phenomena or population dynamics ?1–3?. Delay
differential systems are assuming an increasingly important role in many disciplines like
economic, mathematics, science, and engineering. For instance, in economic systems, delays
appear in a natural way since decisions and effects are separated by some time interval.
The presence of a delay in a system may be the result of some essential simplification of
the corresponding process model. The problem of delay effects on the stability of systems
including delays in the state, and/or input is a problem of recurring interest since the
delay presence may induce complex behaviors ?oscillation, instability, bad performances?
for the schemes ?1, 2?. Some improved methods pertaining to the problems of determining
robust stability criteria and robust control design of uncertain time-delay systems have
been reported; see, for example, ?4, 5? and the references cited therein. When dealing with

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2Mathematical Problems in Engineering
time-varying delays and the reduction of the level of design conservatism, one has to
select appropriate Lyapunov-Krasovskii functional ?LKF? with moderate number of terms
?6?.
Neutral delay systems constitute a more general class than those of the retarded type.
Stability of these systems proves to be a more complex issue because the system involves
the derivative of the delayed state. Especially in the past few decades, increased attention
has been devoted to the problem of robust delay-independent stability or delay-dependent
stability and stabilization via different approaches ?e.g., model transformation techniques
?2, 7–9?, the improved bounding techniques ?10, 11?, and the properly chosen Lyapunov-
Krasovskii functionals ?12, 13?? for a number of different neutral systems with delayed state
and/or input, parameter uncertainties, and nonlinear perturbations ?see, e.g., ?14–25? and
the references therein?.
Among the existing results on neutral delay systems, the linear matrix inequality
?LMI? approach is an efficient method to solve many problems such as stability analysis,
stabilization ?9, 15, 26, 27?, H∞ control problems ?28–30?, filter designs ?31, 32?, and
guaranteed-cost ?observer-based? control ?33–39?. Besides, for neutral systems with mixed
neutral and discrete delays, most of the aforementioned methods can only provide neutral-
delay-independent and discrete-delay-dependent results. Furthermore, the subject of the
robust stability and feedback stabilization of continuous- and discrete-time systems ?within
the framework LMI? under additive perturbations which are nonlinear functions in time and
state of the systems are investigated in ?40, 41?, respectively.
In the recent literature on neutral systems, He et al. in ?42? proposed a new approach to
analyze the stability of the systems with mixed delays by incorporating some free-weighting
matrices, and the less conservative criteria, which were both discrete-delay-dependent and
neutral-delay-dependent, were obtained without considering the model transformations.
However,someofthefreematricesdidnotservetoreducetheconservatismoftheresultsthat
were obtained. Moreover, in ?9, 20?, the authors studied the problem of the robust stability
of neutral systems with nonlinear parameter perturbations and mixed time-varying neutral
and discrete delays and presented neutral-delay-independent stability criteria, that cannot be
directly applied to the systems with different time-varying neutral, discrete, and distributed
delays. Furthermore, from the published results, it appears that general results pertaining
to neutral systems with mixed time-varying neutral, discrete, and distributed delays and
nonlinear parameter perturbations are few and restricted; see ?9, 10, 18, 20, 42? where
most of the efforts were virtually neutral-delay-range-independent or were not centered on
distributed delays.
In this paper, we develop new stability criteria for the stability analysis of the neutral
systems with nonlinear parameter perturbations based on a descriptor model transformation.
The dynamical system under consideration consists of time-varying neutral, discrete, and
distributed delays without any restriction on upper bounds of derivatives of time-varying
delays.Byintroducing anovelLyapunov-Krasovskii functionalandcombiningthedescriptor
model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a
suitable change of variables, new sufficient conditions are established for the stability of
the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent,
and distributed-delay-dependent. The conditions are presented in terms of LMIs and can be
easily solved by existing convex optimization techniques. Two numerical examples are given
to demonstrate the less conservatism of the proposed results over some existence results in
the literature.

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Mathematical Problems in Engineering3
Notations. The superscript
dimensional Euclidean space; Rn×mis the set of all real m by n matrices. ? · ? refers to the
Euclidean vector norm or the induced matrix 2-norm. col{···} and diag{···} represent,
respectively, a column vector and a block diagonal matrix, and the operator sym?A?
represents A ? AT. λmin?A? and λmax?A? denote, respectively, the smallest and largest
eigenvalue of the square matrix A. The notation P > 0 means that P is real symmetric and
positive definite; the symbol ∗ denotes the elements below the main diagonal of a symmetric
block matrix.
?T?stands for matrix transposition; Rndenotes the n-
2. Problem Description
Consider a class of linear neutral systems with different time-varying neutral, discrete, and
distributed delays and nonlinear parameter perturbations represented by
˙ x?t? − C ˙ x?t − τ?t?? ? Ax?t? ? Bx?t − h?t?? ? G1f1?t,x?t?? ? G2f2?t,x?t − h?t???
?t
? G3
t−r?t?
x?t? ? φ?t?,
f3?θ,x?θ??dθ ? G4f4
?t, ˙ x?t − τ?t???,
t ∈ ?−κ,0?,
?2.1?
where κ :? max{h2,τ1,r1}, and x?t? ∈ Rnis the state vector. The time-varying vector valued
initial function φ?t? is a continuously differentiable functional, and the time-varying delays
h?t?, τ?t?, and r?t? are functions satisfying, respectively,
0 < h1≤ h?t? ≤ h2,
0 < τ?t? ≤ τ1,
0 < r?t? ≤ r1,
??˙h?t???≤ h3< ∞,
??˙ r?t???≤ r2< ∞.
?2.2a?
??˙ τ?t???≤ τ2< ∞,
?2.2b?
?2.2c?
The time-varying vector-valued functions fi: R?× Rn→ Rni?i ? 1,...,4? are continuous
and satisfy fi?t,0? ? 0,and the Lipschitz conditions, that is,?fi?t,x0?−fi?t,y0?? ≤ ?Ui?x0−y0??
for all t and for all x0,y0∈ Rnsuch that Uiare some known matrices.
Remark 2.1. In this case, h?t? is called an interval-like or range-like time-varying delay ?14?.
It is also noted that this kind of time-delay describes the real situation in many practical
engineering systems. For example, in the field of networked control systems, the network
transmission induced delays ?either from the sensor to the controller or from the controller to
the plant? can be assumed to satisfy ?2.2a? without loss of generality ?43, 44?.
Throughout the paper, the following assumptions are needed to enable the application
of Lyapunov’s method for the stability of neutral systems ?1?:
?A1? let the difference operator D : C??−κ,0?,Rn? → Rngiven by Dxt? x?t? − Cx?t −
τ?t?? be delay-independently stable with respect to all delays. A sufficient condition
for ?A1? is that
?A2? all the eigenvalues of the matrix C are inside the unit circle.

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4Mathematical Problems in Engineering
Before ending this section, we recall the following lemmas, which will be used in the
proof of our main results.
Lemma 2.2 ?see ?9??. For any arbitrary column vectors a?s?,b?s? ∈ Rp, any matrix W ∈ Rp×p,
and positive-definite matrix H ∈ Rp×p, the following inequality holds:
−2
?t
t−r?t?
b?s?Ta?s?ds ≤
?t
t−r?t?
?a?s?
b?s?
?T?H HW
∗
?HW ? I?TH−1?HW ? I?
??a?s?
b?s?
?
ds.
?2.3?
Lemma 2.3 ?see ?45??. Given matrices Y ? YT,D, E, and F of appropriate dimensions with FTF ≤
I, then the following matrix inequality holds:
Y ? sym?DFE? < 0,
?2.4?
for all F if and only if there exists a scalar ε > 0 such that
Y ? εDDT? ε−1ETE < 0.
?2.5?
3. Main Results
In this section, new delay-range-dependent sufficient conditions for the asymptotic stability
of the neutral system ?2.1? are presented. By utilizing the Leibniz-Newton formula, the
following two zero equations hold:
L1x?t? − L1x?t − h?t?? − L1
?t
?t
t−h?t?
˙ x?s?ds ? 0,
?3.1a?
L2x?t? − L2x?t − τ?t?? − L2
t−τ?t?
˙ x?s?ds ? 0,
?3.1b?
then, we can represent the system ?2.1? as
˙ x?t? − C ˙ x?t − τ?t?? ?? Ax?t? ??Bx?t − h?t?? ? G1f1?t,x?t?? ? G2f2?t,x?t − h?t???
− L2x?t − τ?t?? − L1
?t
?t
t−h?t?
?t
˙ x?s?ds
− L2
t−τ?t?
˙ x?s?ds ? G3
t−r?t?
f3?θ,x?θ??dθ ? G4f4?t, ˙ x?t − τ?t???,
?3.2?
with? A :? A ? L1? L2and?B :? B − L1where the matrices L1,L2∈ Rn×nwill be chosen in the
following theorem.

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Mathematical Problems in Engineering5
Theorem 3.1. Under (A1), for given scalars γ,h1,h2,τ1,r1> 0,h3, τ2,r2, the neutral system ?2.1?
is asymptotically stable, if there exist some scalars δ,α1,α2, matrices P2,{Ni}20
definite matrices P1,{Qi}4
i?1,Y1,Y2, and positive-
i?1,{Ri}4
i?1,?
H1,?
Π16
H2, such that the following LMI is feasible:
Π ?
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
Π11 Π12 Π13 Π14
Π15
Π17
Π18
Π19
Π1,10
Π1,11
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Π22 Π23
0
−BTNT
0
17
0Π27
−BTNT
0
19
Π29
Π2,10
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
Π33
00Π37
Π39
00
∗
∗
∗
∗
∗
∗
∗
∗
Π44
00
−NT
15
0
−NT
−CTNT
Π69
16
Π4,10
0
∗
∗
∗
∗
∗
∗
∗
Π55
Π56 −CTNT
Π66
18−N17G3
Π68
20
00
∗
∗
∗
∗
∗
∗
Π67
00
∗
∗
∗
∗
∗
Π77
−N18G3
Π88
−N18G4
−GT
−sym?N20G4
∗
∗
Π7,10
0
∗
∗
∗
∗
3NT
20
00
∗
∗
∗
?
Π9,10
0
Π10,10
0
∗
Π11,11
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
< 0
?3.3?
with
Π11? sym
⎛
⎝
⎡
⎣
PT
2A ??1 ? α1
δ?PT
2B −?1 ? α1
δPT
?Y1??1 ? α2
?Y1??1 ? α2
2− N1− N5? N9
?Y1
14− N13
?Y2
⎤
?Y2
?Y2
P1− PT
−δPT
2
2A ??1 ? α1
?Y1? NT
?
2
⎤
⎦
⎞
⎠? Ω,
⎡
Π12?
⎡
⎣
⎡
PT
2B − δ?1 ? α1
?Y2? NT
−δ?1 ? α2
2G1
PT
⎤
⎦,
⎡
Π13?
⎣
⎤
N5 N9
00
⎤
⎦,
Π14?
⎣
⎡
−?1 ? α2
⎤
⎦,
Π15?
⎣
PT
2C − ATNT
17
δPT
2C ? NT
17
⎦,
Π16?
⎣
?PT
PT
2G2
δPT
2G1 δPT
2G2
⎦,
?
Π17?
⎡
⎣
NT
3? NT
15− ATNT
18
NT
18
⎤
⎦,
Π18?
2G3− ATNT
δPT
19
2G3? NT
19
,
Π19?
?PT
2G4? NT
4? NT
δPT
16− ATNT
20
2G4? NT
20
?
,
Π1,10?
?h12N1 h12N5 h12N9 τ1N13
0000
?
,

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6Mathematical Problems in Engineering
?T⎤
Π1,11?
⎡
⎣h2
2U2−?1 − h3
Π27? −NT
?α1? 1??
H1 τ1
?α1? 1??
?R3− sym?N2? N6− N10
7? NT
H2 h2
?0
I
??Y1
δY1
?T
τ1
?0
I
??Y2
δY2
⎦,
Π22? UT
?,
4− NT
Π23??N6 N10
8? NT
?,
3− NT
11− BTNT
18,
Π29? −NT
12− BTNT
20,
Π2,10??h12N2 h12N6 h12N10 0?,
?NT
−NT
Π55? −?1 − τ2
Π33? diag?− R1,−R2
?,
Π37?
⎡
⎣
NT
7
−NT
11
⎤
⎦,
Π39?
8
12
?
,
Π44? −?1 − τ2
?Q2− sym?N17C?? UT
?Q1− sym?N14
?,
Π4,10??0 0 0 τ1N14
Π56??−N17G1 −N17G2
Π68??N19G1 N19G2
?,
4U4,
?,
Π66? diag{−I,−I},
Π67?
?−GT
−GT
1NT
18
2NT
18
?
,
?T,
Π69?
?−GT
−GT
1NT
20
2NT
20
?
,
Π77? −I ? r2
1Q4,
Π88? −?1 − r2
?Q4− sym?N19G3
?,
?,
Π4,10??0 0 0 τ1N14
Π9,10??h12N4 h12N8 h12N12 τ1N16
Π10,10? diag?− h12R4,−h12R5,−h12R4,−τ1Q3
Π1,11? diag?− h2?
?,
Π7,10??h12N3 h12N7 h12N11 τ1N15
?,
?,
H2,−h2?
H1,−τ1?
H1,−τ1?
H2
?,
?3.4?
where Ω ? diag{Q1??3
Proof. Firstly, we represent ?3.2? in an equivalent descriptor model form as
i?1Ri? UT
1U1? UT
3U3? sym?N1? N13?, Q2? τ1Q3? h12R4? h2R5}.
˙ x?t? ? η?t?,
0 ? −η?t? ?? Ax?t? ? Cη?t − τ?t?? ??Bx?t − h?t?? − L2x?t − τ?t??
? G1f1?t,x?t?? ? G2f2?t,x?t − h?t??? − L1
?t
t−h?t?
η?s?ds − L2
?t
t−τ?t?
η?s?ds
? G3
?t
t−r?t?
f3?θ,x?θ??dθ ? G4f4?t,η?t − τ?t???.
?3.5?

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Mathematical Problems in Engineering7
Define the Lyapunov-Krasovskii functional
V?t? ?
6 ?
i?1
Vi?t?,
?3.6?
where
V1?t? ? x?t?TP1x?t? :? ξ?t?TTPξ?t?,
V2?t? ?
?t
t−τ?t?
?t
?−h1
−h2
?0
−τ1
?0
−h2
?0
?t
?r1
?x?s?TQ1x?s? ? η?s?TQ2η?s??ds
?
t−h?t?
?t
?t
x?s?TR3x?s?ds ?
2 ?
i?1
?t
?0
t−hi
x?s?TRix?s?ds,
V3?t? ?
t?θ
η?s?TR4η?s?dsdθ ?
−h2
?t
t?θ
η?s?TR5η?s?dsdθ,
V4?t? ?
t?θ
η?s?TQ3η?s?dsdθ,
V5?t? ?
?t
t?θ
?t
??t
?t
t−s
η?s?T
?
0
L1
?
?T
H1
?
0
L1
?
η?s?dsdθ
?
−τ1
t?θ
η?s?T
0
L2
?T
H2
?
0
L2
?
η?s?dsdθ,
V6?t? ?
t−r?t?
s
f3?θ,x?θ??Tdθ
?
Q4
??t
s
f3?θ,x?θ??Tdθ
?
ds
?
0
?θ − t ? s?f3?θ,x?θ??TQ4f3?θ,x?θ??dθds,
?3.7?
with ξ?t? :? col{x?t?,η?t?}, T ? diag{I,0}, and P ??P1 0
the Cauchy-Schwarz inequality and after some manipulations, we obtain
P2 P3
?, where P1? PT
1> 0.
On the other hand, noting that V?φ?t?,t? ≥ λmin?P1??φ?0??2. According to ?34?, using
V?φ?t?,t? ≤ V?φ?0?,0? ≤ ρ
?
?φ?0??2?
?0
−κ
?? ˙φ?θ???2dθ
?
,
?3.8?
where ρ :? max?ρ1,ρ2? with
ρ1:? λmax
?P1
?? 2τ1λmax
?R2
?Q1
?? 2h1λmax
?R3
?R1
1λmax
?
? 2h2λmax
?? 2h2λmax
?? 3r2
?UT
3Q4U3
?,

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8Mathematical Problems in Engineering
ρ2:? 2τ2
1λmax
?Q1
?R5
??0
?? λmax
???h1? h2
?T
?Q2
?? 2h2
?λmax
??
2λmax
?R4
? τ1λmax
?R3
?? 2h2
1λmax
?Q3
?T
?R1
?? 2h2
2λmax
?R2
?
? h2λmax
?? τ1λmax
??0
?
? h2λmax
L1
H1
?0
L1
L2
H2
?0
L2
??
?11
3r3
1λmax
?UT
3Q4U3
?.
?3.9?
Differentiating V1?t? along the system trajectory becomes
˙V1?t? ? 2x?t?TP1˙ξ?t?
? 2ξ?t?TPT
?˙ x?t?
?
0
?
? 2ξ?t?TPT
Aξ?t? ?
?0
C
?
η?t − τ?t?? ?
?
0
?B
f2?t,x?t − h?t???
?
x?t − h?t?? −
?0
L2
?
x?t − τ?t??
?
?0
G1
?t
?
f1?t,x?t?? ?
?0
G2
?
?G3
t−r?t?
f3?θ,x?θ??dθ ? G4f4?t,η?t − τ?t???
?
? β1?t? ? β2?t?,
?3.10?
where
A :?
?
0
I
? A −I
?
,β1?t? ? −2
?t
?t
t−h?t?
ξ?t?TPT
?0
?0
L1
?
η?s?ds,
β2?t? ? −2
t−τ?t?
ξ?t?TPT
L2
?
η?s?ds.
?3.11?
Using Lemma 2.2 for a?s? ? col{0,Li}ξ?s? and b ? P col{ξ?t?,η?t?}, we obtain
β1?t? ≤ h2ξ?t?TPT?WT
? 2ξ?t?TPTWT
1H1
1H1? I?TH−1
?
L1
?
L1
1
?WT
1H1? I?Pξ?t?
0
?
?
?x?t? − x?t − h?t???
?
?
?t
t−h2
η?s?T
0
?T
H1
0
L1
η?s?ds,
β2?t? ≤ τ1ξ?t?TPT?WT
? 2ξ?t?TPTWT
2H2? I?TH−1
2H2
L2
η?s?T?0
2
?WT
2H2? I?Pξ?t?
?0
?
?0
?x?t? − x?t − τ?t???
?
?
?t
t−τ1
L2
?T
H2
L2
η?s?ds.
?3.12?

Page 9

Mathematical Problems in Engineering9
Differentiating the second Lyapunov term in ?3.6? gives
˙V2?t? ? x?t?T
?
Q1?
3 ?
i?1
Ri
?
x?t? ? η?t?TQ2η?t? −?1 −˙h?t??xT?t − h?t??R3x?t − h?t??
−?1 − ˙ τ?t??xT?t − τ?t??Q1x?t − τ?t??
−?1 − ˙ τ?t??ηT?t − τ?t??Q2η?t − τ?t?? −
?
i?1
−?1 − τ2
−?1 − τ2
2 ?
i?1
x?t − hi
?TRix?t − hi
?xT?t − h?t??R3x?t − h?t??
?
≤ x?t?T
Q1?
3 ?
Ri
?
x?t? ? η?t?TQ2η?t? −?1 − h3
?xT?t − τ?t??Q1x?t − τ?t??
?ηT?t − τ?t??Q2η?t − τ?t?? −
2 ?
i?1
x?t − hi
?TRix?t − hi
?,
?3.13?
and the time derivative of the third term of V?t? in ?3.6? is
˙V3?t? ? η?t?T?h12R4? h2R5
?η?t? −
?t−h1
t−h2
η?s?TR4η?s?ds −
?t
t−h2
η?s?TR5η?s?ds
≤ η?t?T?h12R4? h2R5
?η?t? −
?t−h?t?
t−h2
η?s?TR4η?s?ds −
?t−h1
t−h?t?
η?s?T?R4? R5
?η?s?ds,
?3.14?
and, similarly,
˙V4?t? ? τ1η?t?TQ3η?t? −
?t
t−τ1
η?s?TQ3η?s?ds ≤ τ1η?t?TQ3η?t? −
?t
t−τ?t?
η?s?TQ3η?s?ds,
?3.15?
and also the time derivative of the fifth and sixth terms of V?t? in ?3.6? are, respectively,
˙V5?t? ? η?t?T
⎛
⎝h2
η?s?T?0
?0
L1
?T
H1
?0
L1
?
? τ1
?0
L2
?T
H2
?0
L2
?⎞
⎠η?t?
η?s?T?0
−
?t
t−h2
L1
?T
H1
?0
L1
?
η?s?ds −
?t
t−τ1
L2
?T
H2
?0
L2
?
η?s?ds,
?3.16?

Page 10

10Mathematical Problems in Engineering
˙V6?t? ? −?1 − ˙ r?t????t
?t
?r1
?t
?r1
−?1 − r2
?t
t−r?t?
f3?θ,x?θ??Tdθ
?
Q4
??t
t−r?t?
?
?t
t−s
f3?θ,x?θ??dθ
?
? 2
t−r?t?
f3?t,x?t??TQ4
??t
s
f3?θ,x?θ??dθds
?
0
sf3?t,x?t??TQ4f3?t,x?t??ds −
?r1
0
f3?θ,x?θ??TQ4f3?θ,x?θ??dθds
≤
t−r?t?
?θ − t ? r?t???f3?t,x?t??TQ4f3?t,x?t?? ? f3?θ,x?θ??TQ4f3?θ,x?θ???dθ
sf3?t,x?t??TQ4f3?t,x?t??ds
?
0
???t
?θ − t ? r1
t−r?t?
f3?θ,x?θ??Tdθ
?
Q4
??t
t−r?t?
f3?θ,x?θ??dθ
?
−
t−r1
?f3?θ,x?θ??TQ4f3?θ,x?θ??dθ
? r2
1f3?t,x?t??TQ4f3?t,x?t?? −?1 − r2
???t
t−r?t?
f3?θ,x?θ??Tdθ
?
Q4
??t
t−r?t?
f3?θ,x?θ??dθ
?
.
?3.17?
For nonlinear functions fi?·?, we have
0 ≤ −f1?t,x?t??Tf1?t,x?t?? ? x?t?TUT
0 ≤ −f2?t,x?t − h?t???Tf2?t,x?t − h?t??? ? x?t − h?t??TUT
0 ≤ −f3?t,x?t??Tf3?t,x?t?? ? x?t?TUT
0 ? −f4?t,η?t − τ?t???Tf4?t,η?t − τ?t??? ? η?t − τ?t??TUT
1U1x?t?,
2U2x?t − h?t??,
3U3x?t?,
4U4η?t − τ?t??.
?3.18?
Moreover, from the Leibniz-Newton formula and the system ?2.1?, the following equations
hold for any matrices {Ni}10
i?1with appropriate dimensions:
2ϑT?t?χ1
?
?
?
?
x?t? − x?t − h?t?? −
?t
t−h?t?
η?s?ds
?
? 0,
2ϑT?t?χ2
x?t − h1
x?t − h?t?? − x?t − h2
x?t? − x?t − τ?t?? −
?− x?t − h?t?? −
?t−h1
?t−h?t?
t−h2
t−h?t?
η?s?ds
?
?
? 0,
2ϑT?t?χ3
?−
η?s?ds
? 0,
2ϑT?t?χ4
?t
t−τ?t?
η?s?ds
?
? 0,

Page 11

Mathematical Problems in Engineering
?
?t
11
2ϑT?t?χ5
η?t? − Cη?t − τ?t?? − Ax?t? − Bx?t − h?t?? − G1f1?t,x?t?? − G2f2
?t,x?t − h?t???
− G3
t−r?t?
f3?θ,x?θ??dθ − G4f4
?t,η?t − τ?t????
? 0,
?3.19?
where
χ1:?
?
?
?
?
?
NT
1, 0,NT
2, 0,...,0
? ?? ?
6, 0,...,0
? ?? ?
10, 0,...,0
? ?? ?
? ?? ?
0,...,0
? ?? ?
x?t?,η?t?,x?t − h?t??,x?t − h1
η?t − τ?t??,f1?t,x?t??, f2?t,x?t − h?t???,f3?t,x?t??,
?t
6elements
,NT
3, 0,NT
4
?T
?T
,
χ2:?
NT
5, 0,NT
6elements
,NT
7, 0,NT
8
χ3:?
NT
9, 0,NT
6elements
,NT
11, 0,NT
12
?T
,
χ4:?
NT
13, 0,...,0
4elements
,NT
14, 0,...,0
? ?? ?
18,NT
3elements
,NT
15, 0,NT
16
?T
,
χ5:?
6elements
,NT
17,0,0,NT
19,NT
20
?T
,
ϑ?t? :? col
?
?,x?t − h2
?,x?t − τ?t??,
t−r?t?
f3?θ,x?θ??dθ,f4?t,η?t − τ?t???
?
.
?3.20?
Using the obtained derivative terms ?3.10?–?3.17? and adding the right- and the left-hand
sides of ?3.18? and ?3.19? into˙V?t?, the following result is obtained:
˙V?t? ?
6 ?
i?1
˙Vi?t?
≤ ϑ?t?TΣϑ?t? −
?t−h1
?t−h1
?t
?t−h1
t−h?t?
?ϑT?t?χ1? ηT?s?R4
?R−1
?R−1
?Q−1
?R−1
4
?ϑT?t?χ1? ηT?s?R4
?Tds
?Tds
?Tds,
?Tds
−
t−h?t?
?ϑT?t?χ2? ηT?s?R5
?ϑT?t?χ3? ηT?s?R4
?ϑT?t?χ4? ηT?s?Q3
5
?ϑT?t?χ2? ηT?s?R5
?ϑT?t?χ3? ηT?s?R5
?ϑT?t?χ4? ηT?s?Q3
−
t−h?t?
4
−
t−τ?t?
3
?3.21?

Page 12

12 Mathematical Problems in Engineering
where Σ :??Π ?h12χ1R−1
4χT
1?h12χ2R−1
5χT
2?h12χ3R−1
4χT
3?τ1χ4Q−1
3χT
4, and the matrix?Π is given
by
?Π ?
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎢
⎢
⎢
⎢
⎢
?Π11
∗
∗
∗
∗
∗
∗
∗
∗
?Π12 Π13
∗
∗
∗
∗
∗
∗
∗
?Π14
0
?Π15
0
?Π16
0
Π17
?Π18
0
?Π19
Π39
Π22 Π23
0
−BTNT
17
0Π27
−BTNT
19
Π29
Π33
Π37
∗
∗
∗
∗
∗
∗
Π44
00
−NT
15
0
−NT
−CTNT
Π69
16
∗
∗
∗
∗
∗
Π55
Π56 −CTNT
Π66
18−N17G3
Π68
20
∗
∗
∗
∗
Π67
∗
∗
∗
Π77
−N18G3
Π88
−N18G4
−GT
−sym?N20G4
∗
∗
3NT
20
∗
?
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎥
⎥
⎥
⎥
⎥
?3.22?
with
?Π11? sym
?
PT
?
A ?
?
WT
1H1
?0
L1
?
? WT
2H2
?0
L2
??
? I 0 ?
??
? PT?h2
? τ1
?WT
2H2? I?TH−1
?⎛
L1
1H1? I?TH−1
1
?WT
2H2? I??P
?
1H1? I?
?WT
2
?WT
?0
L1
?
?0
I
⎝h2
?0
?B
2H2
?0
?T
H1
? τ1
?0
L2
?T
H2
?0
L2
?⎞
⎠
?0
I
?T
? Ω,
?Π12? PT
?0
L2
?
− PTWT
1H1
?0
L1
?
?
?NT
2− N1− N5? N9
0
?
,
?Π14? −PT
?
− PTWT
?0
L2
?
?
?NT
14− N13
0
?
,
?Π15? PT
?−ATNT
?0
C
?
?
?−ATNT
NT
17
17
?
,
?Π16? PT
?0
G1 G2
0
?
,
?Π18? PT
⎡
?0
G3
?
?
19
NT
19
?
,
?Π19? PT
?0
G4
?
?
⎣NT
4? NT
16− ATNT
NT
20
20
⎤
⎦.
?3.23?

Page 13

Mathematical Problems in Engineering13
Now, if Σ < 0 holds, then˙V?t? < 0 which means that the neutral system ?2.1? is asymptotically
stable. By applying the Schur complement, the matrix inequity Σ < 0 results in
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
?Π11
∗
∗
∗
∗
∗
∗
∗
∗
∗
?Π12 Π13
∗
∗
∗
∗
∗
∗
∗
∗
∗
?Π14
0
Π44
?Π15
0
0
Π55
?Π16
0
0
Π56 −CTNT
Π66
Π17
Π27
Π37
−NT
?Π18
0
0
?Π19
Π39
−NT
−CTNT
?Π69
−GT
−sym?N20G4
∗
∗
Π1,10
Π2,10
0
Π4,10
0
?Π1,11
0
0
0
∗
Π22 Π23
0
−BTNT
17
0
−BTNT
19
Π29
0
Π33
∗
∗
∗
∗
∗
∗
∗
∗
1516
∗
∗
∗
∗
∗
∗
∗
18−N17G3
Π68
20
∗
∗
∗
∗
∗
∗
Π67
Π77
00
0
0
0
0
∗
∗
∗
∗
∗
−N18G3
Π88
−N18G4
3NT
Π7,10
0
Π9,10
Π10,10
∗
∗
∗
∗
20
∗
∗
∗
?
∗
?Π11,11
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
< 0
?3.24?
with
?Π11? sym
?
?
PT
?
A ?
?
WT
1H1
?0
L1
?
? WT
2H2
?0
L2
? ??I 0???
?0
?,
? Ω,
?Π1,11?
h2PT?WT
1H1? I?TH1 τ1PT?WT
?Π11,11? diag?− h2H1,−τ1H2,−h2H1,−τ1H2
?i ? 1, 2?.
Following ?34, 35?, we choose P3? δP2, δ ∈ R, where δ is a tuning scalar parameter
?which may be restrictive?. Let
2H2? I?TH2 h2
I
??0
L1
?T
τ1
?0
I
??0
L2
?T?
,
?3.25?
where Hi? H−1
i
ζ ? diag
⎧
⎪
⎪
⎩
⎨
I,...,I
? ?? ?
16elements
,PT,...,PT
????
4elements
⎫
⎪
⎪
⎭
⎬
.
?3.26?
Premultiplying ζ and postmultiplying ζTto the matrix inequality ?3.24? and considering Yi:?
PT
Hi:? PTHiP, and HiWi? αiI ?i ? 1, 2? to eliminate the nonlinearities in the matrix
inequality, we obtain the LMI ?3.3?. Moreover, from ?2.1? and the fact that x?t? is square
integrable on ?0, ∞?, it follows that Dηt∈ Ln
τ?t?? ∈ Ln
asymptotically stable.
2Li,?
2?0, ∞?. Under ?A1?, the later implies that η?t −
2?0, ∞?. Therefore, by ?1, Theorem 1.6?, we conclude that the neutral system ?2.1? is
Remark 3.2. The results given in Theorem 3.1 are derived for system ?2.1? with time-
varying delays h?t?, τ?t?, and r?t? satisfying ?2.2a?, ?2.2b?, and ?2.2c?, where the derivatives

Page 14

14 Mathematical Problems in Engineering
of the time-varying delays are available. However, in many situations, the information
on the derivative of time-varying delays is unknown a prior. In such circumstances,
the corresponding delay-rate-independent stability analysis results for time-delays only
satisfying
0 < h1≤ h?t? ≤ h2< ∞,
0 < τ?t? ≤ τ1< ∞,
0 < r?t? ≤ r1< ∞,
?3.27?
can be easily obtained by setting Q1? Q2? Q4? R3? 0 in Theorem 3.1.
Remark 3.3. The reduced conservatism of Theorem 3.1 benefits from the construction of
the new Lyapunov-Krasovskii functional in ?3.6?, utilizing Leibniz-Newton formula, using
a free-weighting matrix technique, and no bounding technique is needed to estimate the
inner product of the involved crossing terms ?see, e.g., ?12, 20??. It can be easily seen that
results of this paper are quite different from most existing results in the recent literature
in the following perspectives. ?a? Theoretically stability analysis of neutral systems with
different time-varying neutral, discrete, and distributed delays is much more complicated,
especially, for the case where the delays are time-varying and different. ?b? In this paper,
the derived sufficient conditions are convex, neutral-delay-dependent, discrete-delay-range-
dependent, and distributed-delay-dependent, which make the treatment in the present paper
moregeneral withlessconservative incompare tomostexisting resultsintheliterature which
are independent of the neutral or distributed delays; see for instance ?21, 22, 38?.
4. Uncertainty Characterization
In this section, we will discuss the uncertainty characterization for the linear neutral system
?2.1? with different time-varying neutral, discrete, and distributed delays and nonlinear
parameter perturbations.
4.1. Polytopic Uncertainty
The first class of uncertainty frequently encountered in practice is the polytopic uncertainty
?2?. In this case, the matrices of the system ?2.1? are not exactly known, except that they are
within a compact set Ω denoting
Ω ??C A B G1 G2 G3
?.
?4.1?
We assume that
Ω ?
N
?
j?1
sjΩj
?4.2?

Page 15

Mathematical Problems in Engineering15
for some scalars sjsatisfying
0 ≤ sj≤ 1,
N
?
j?1
sj ? 1,
?4.3?
where the N vertices of the polytope are described by
Ωj??C?j?A?j?B?j?G?j?
1
G?j?
2
G?j?
3
?.
?4.4?
In order to take into account the polytopic uncertainty in the system ?2.1?, we derive
the following result from applying the same transformation that was used in deriving
Theorem 3.1.
Theorem 4.1. Under (A1), for given scalars γ,h1,h2,τ1,r1 > 0,h3, τ2,r2, if the uncertainty set
Ω is polytopic with vertices Ωj, j ? 1, 2,...,N, then the system described by ?2.1?, ?2.2a?,
?2.2b?, ?2.2c?, and ?4.2?–?4.4? is asymptotically stable if there exist some scalars δ,α1,α2, matrices
P2,{Ni}20
satisfied for all
i?1,Y1,Y2, and positive-definite matrices P1,{Qi}4
i?1,{Ri}4
i?1,?
H1,?
H2such that LMI ?3.3? is
?C A B G1 G2 G3
???C?j?A?j?B?j?G?j?
1
G?j?
2
G?j?
3
?,j ? 1, 2,...,N.
?4.5?
Proof. It follows directly from the proof of Theorem 3.1 and using properties of ?4.2?–?4.4?.
4.2. Norm-Bounded Uncertainty
There are also other uncertainties that cannot be reasonably modeled by a polytopic
uncertainty set with a number of vertices. In such a case, it is assumed that the deviation
of the system parameters of an uncertain system from their nominal values is norm bounded
?2?. In our case, consider the time-varying structured uncertain neutral system
˙ x?t? − ?C ? ΔC?t?? ˙ x?t − τ?t?? ? ?A ? ΔA?t??x?t? ? ?B ? ΔB?t??x?t − h?t??
??G1? ΔG1?t??f1?t,x?t??
??G2? ΔG2?t??f2?t,x?t − h?t???
??G3? ΔG3?t???t
??G4? ΔG4?t??f4
x?t? ? φ?t?,
t−r?t?
?t, ˙ x?t − τ?t???,
f3?θ,x?θ??dθ
t ∈ ?−κ,0?,
?4.6?