Robust Stabilization for Uncertain Saturated Time-Delay Systems: A Distributed-Delay-Dependent Polytopic Approach

Abstract

This paper investigates the robust stabilization prob- lem for uncertain linear systems with discrete and distributed delays under saturated state feedback. Different from the existing approaches, a distributed-delay-dependent polytopic approach is proposed in this paper, and the saturation nonlinearity is repre- sented as the convex combination of state feedback and auxiliary distributed-delay feedback. Then, by incorporating an appropriate augmented Lyapunov-Krasovskii (L-K) functional and some integral inequalities, the less conservative stabilization and robust stabilization conditions are proposed in terms of linear matrix inequalities (LMIs). The effectiveness and reduced conservatism of the proposed conditions are illustrated by numerical examples.

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Transactions on Automatic Control
1
Robust stabilization for uncertain saturated
time-delay systems: a distributed-delay-dependent
polytopic approach
Yonggang Chen, Shumin Fei, and Yongmin Li
Abstract—This paper investigates the robust stabilization prob-
lem for uncertain linear systems with discrete and distributed
delays under saturated state feedback. Different from the existing
approaches, a distributed-delay-dependent polytopic approach is
proposed in this paper, and the saturation nonlinearity is repre-
sented as the convex combination of state feedback and auxiliary
distributed-delay feedback. Then, by incorporating an appropriate
augmented Lyapunov-Krasovskii (L-K) functional and some
integral inequalities, the less conservative stabilization and robust
stabilization conditions are proposed in terms of linear matrix
inequalities (LMIs). The effectiveness and reduced conservatism
of the proposed conditions are illustrated by numerical examples.
Index Terms—Robust stabilization, time delays, uncertain sat-
urated systems, distributed-delay-dependent, polytopic approach.
I. INT ROD UC TI ON
Time-delay systems have received much attention during the
past decades due to the extensive existences of time delays in
various practical systems [1-13]. Overall, there are three types
of time delays, i.e., discrete delays [8-10], neutral delays [4,6]
and distributed delays [1,3,5]. For the analysis and synthesis of
time-delay systems, delay-dependent conditions are generally
less conservative than delay-independent ones especially when
the sizes of time delays are small. Compared with some early
results obtained by descriptor system approaches [3,5], free-
weighting matrices technique [4], and Jensen integral inequal-
ities [6], some recent proposed delay-dependent conditions
using delay decomposition approach [7], Wirtinger integral
inequalities [8-11], and input-output approaches [12-13] have
shown to be more effective in reducing the conservatism.
On the other hand, actuator saturation is often inevitable
in practical control applications, and its existence can cause
the instability and performance degradation of overall system.
During the past two decades, much effort has been paid to
linear systems with actuator saturation [14-19]. Under the
This work was supported by the National Natural Science Foundations
of China under Grants 61304061, 61374086, 61273119 and 61174076. This
work was also partly supported by the Open Foundation of Key Laboratory
of Control Engineering of Henan Province under Grant KG 2014-02.
Y. Chen is with the School of Mathematical Sciences, Henan Institute of
Science and Technology, Xinxiang 453003, P.R. China, and also with the Key
Laboratory of Control Engineering of Henan Province, Henan Polytechnic
University, Jiaozuo 454000, P.R. China, E-mail: happycygzmd@tom.com.
S. Fei is with the School of Automation, Southeast University, Key
Laboratory of Measurement and Control of CSE, Ministry of Education,
Nanjing 210096, P.R. China.
Y. Li is with the School of Science, Huzhou Teachers College, Huzhou
313000, P.R. China.
assumption that the open-loop poles are located on the closed
left-half plane, the global/semi-global stabilization problem
was widely investigated in [14,17-18]. For the case that the
open-loop systems are unstable, the local stabilization problem
was well discussed in [15-16,19]. As for the local stabilization,
the polytopic approaches [15-16,19] and generalized sector
condition [16] are the dominant approaches to deal with the
saturation nonlinearity. In particular, it is worth mentioning
that the polytopic approach proposed in [19] is less conserva-
tive than that in [15] for linear systems with multiple inputs.
Using some techniques in [15-16] to deal with saturation
nonlinearity, the problems of stability analysis and local sta-
bilization were also investigated for time-delay systems with
actuator saturation [20-24]. Recently, by introducing auxiliary
time-delay feedback, delay-dependent polytopic approach was
proposed in [25] to reduce the conservatism for neutral time-
delay systems. However, it should be pointed out that the
results in [20-25] are concerned with only discrete and neutral
delays. To the best of our knowledge, the local stabilization
problem for distributed delay systems with actuator saturation
has not been well considered, although such kind of systems
have important practical applications [1,3]. In addition, it
should be pointed out that the proposed L-K functionals in [25]
have to contain some integral terms of state derivative to utilize
the delay-dependent polytopic approach. In fact, such terms
are essential for neutral time-delay systems [4-5], while are
unnecessary for systems with discrete/distributed delay when
performing the stability analysis. Noting that the estimate of
the domain of attraction is associated with the L-K functionals,
therefore, for linear systems with discrete/distributed delay, it
is possible that the additional introduction of such terms will
decrease the effectiveness of the proposed approach in [25].
Motivated by the above discussions, this paper is concerned
with the robust stabilization problem for uncertain linear
systems with discrete and distributed delays under saturated
state feedback. Different from the delay-dependent polytopic
approach in [25], a distributed-delay-dependent polytopic ap-
proach is proposed in this paper, and the saturation nonlinearly
is represented by the convex combination of state feedback and
auxiliary distributed-delay feedback. Then, combining with an
appropriate augmented L-K functional and some inequalities,
the improved stabilization and robust stabilization conditions
are obtained in terms of LMIs, which are well shown by
numerical examples. The main contributions of this paper are
as follows: (1) the distributed-delay-dependent polytopic ap-
proach is proposed for the first time to represent the saturation

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Transactions on Automatic Control
2
nonlinearly, which results in some less conservative stabiliza-
tion conditions for saturated time-delay systems; (2) the local
stabilization problem is well investigated for uncertain linear
distributed delay systems with actuator saturation.
Notation: 𝑃 > 0(≥0) denotes 𝑃being a positive definite
(positive semi-definite) matrix; 𝜆𝑀(𝑃)denotes the maximum
eigenvalue of matrix 𝑃;sym(E) denotes the matrix 𝐸+𝐸𝑇;
𝐼denotes an identity matrix with proper dimension; ∥ ⋅ ∥2and
∥⋅∥∞denote the 2-norm and ∞-norm, respectively; 𝐶1[−ℎ, 0]
denotes the space of the continuously differentiable vector
functions 𝜙over [−ℎ, 0], and 𝑚𝑎𝑥
𝑡∈[−ℎ,0]∥𝜙(𝑡)∥2≜∥𝜙∥𝑐;𝔻𝑚
is the set of 𝑚×𝑚diagonal matrices with diagonal elements
either 1 or 0; 𝑒𝑚,𝑘 ∈ℝ1×𝑚denotes a row vector whose 𝑘-
th element is 1 and the others are zero, and ⊗denotes the
Kronecker product; 𝕀[1, 𝑁 ]denotes the set {1,2,⋅ ⋅ ⋅ , 𝑁 }.
II. PRO BL EM F OR MU LATI ON
Consider the following uncertain saturated linear system
with discrete and distributed delays:
˙𝑥(𝑡) =𝐴0(𝑡)𝑥(𝑡) + 𝐴1(𝑡)𝑥(𝑡−ℎ) + 𝐴2(𝑡)𝑡
𝑡−ℎ
𝑥(𝑠)𝑑𝑠
+𝐵(𝑡)𝑠𝑎𝑡(𝑢(𝑡)), 𝑡 > 0(1)
𝑥(𝑡) =𝜙(𝑡),∀𝑡∈[−ℎ, 0] (2)
where 𝑥(𝑡)∈ℝ𝑛is the system state; 𝑢(𝑡)∈ℝ𝑚is the
control input; the time delay ℎis a known constant scalar;
𝜙(𝑡)∈𝐶1[−ℎ, 0] denotes the initial function; 𝑠𝑎𝑡(𝑢) :
ℝ𝑚→ℝ𝑚is a vector valued standard saturation func-
tion with unity saturation level, which is described by
𝑠𝑎𝑡(𝑢) = [𝑠𝑎𝑡(𝑢1)𝑠𝑎𝑡(𝑢2)⋅ ⋅ ⋅ 𝑠𝑎𝑡(𝑢𝑚)]𝑇,where 𝑠𝑎𝑡(𝑢𝑗) =
𝑠𝑔𝑛(𝑢𝑗)𝑚𝑖𝑛{1,∣𝑢𝑗∣};𝐴0(𝑡), 𝐴1(𝑡), 𝐴2(𝑡)and 𝐵(𝑡)are time-
varying matrices and are assumed to satisfy 𝐴0(𝑡) = 𝐴0+
Δ𝐴0, 𝐴1(𝑡) = 𝐴1+Δ𝐴1, 𝐴2(𝑡) = 𝐴2+Δ𝐴2and 𝐵(𝑡) = 𝐵+
Δ𝐵, where 𝐴0, 𝐴1, 𝐴2, 𝐵 are known real constant matrices
with appropriate dimensions, and Δ𝐴0,Δ𝐴1,Δ𝐴2,Δ𝐵are
unknown parameter uncertainties. Here, we assume that the
uncertainties are norm-bounded and can be described by
[Δ𝐴0Δ𝐴1Δ𝐴2Δ𝐵]≜𝑀 𝐹 (𝑡)[𝐸0𝐸1𝐸2𝐸3](3)
where 𝑀, 𝐸0, 𝐸1, 𝐸2and 𝐸3are known real constant ma-
trices, and 𝐹(𝑡)is unknown time-varying matrix satisfying
𝐹𝑇(𝑡)𝐹(𝑡)≤𝐼. The controller employed in this paper is
presented in the following state feedback form:
𝑢(𝑡) = 𝐾𝑥(𝑡)(4)
where 𝐾∈ℝ𝑚×𝑛is the gain matrix to be designed.
Remark 1: Different from the literature [20-25], the con-
sidered model (1) in this paper contains the distributed delay
term, where the term 𝑡
𝑡−ℎ𝑥(𝑠)𝑑𝑠 can be interpreted as the ac-
cumulation of system state 𝑥(𝑠)over the time period [𝑡−ℎ, 𝑡].
Lemma 1: [19] Let 𝑚≥1be a given integer, and 𝑤∈ℝ
↔
𝑚
be such that ∥𝑤∥∞≤1, where ↔
𝑚=𝑚2𝑚−1. Let the elements
in 𝔻𝑚be labeled as 𝐷𝑖, 𝑖 ∈𝕀[1,2𝑚], and the function 𝑓𝑚be
defined as 𝑓𝑚(0) = 0 and
𝑓𝑚(𝑖) = 𝑓𝑚(𝑖−1) + 1, 𝐷𝑖+𝐷𝑗∕=𝐼𝑚,∀𝑗∈𝕀[1, 𝑖]
𝑓𝑚(𝑗), 𝐷𝑖+𝐷𝑗=𝐼𝑚,∃𝑗∈𝕀[1, 𝑖].
Then for any 𝑢∈ℝ𝑚, there holds:
𝑠𝑎𝑡(𝑢)∈𝑐𝑜𝐷𝑖𝑢+𝒟−
𝑖𝑤:𝑖∈𝕀[1,2𝑚]
where “𝑐𝑜”denotes the convex hull, and 𝒟−
𝑖∈ℝ𝑚×
↔
𝑚is
defined as 𝒟−
𝑖=𝑒2𝑚−1,𝑓𝑚(𝑖)⊗𝐷−
𝑖with 𝐷−
𝑖=𝐼−𝐷𝑖.
In this paper, it is assumed that there exist ↔
𝑚×𝑛matrices
𝑈and 𝑉𝑗, 𝑗 ∈𝕀[1, 𝑁 ], such that the following condition holds:
∥𝑣(𝑡)∥∞=



𝑈𝑥(𝑡) +
𝑁

𝑗=1
𝑉𝑗𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
𝑥(𝑠)𝑑𝑠


∞
≤1(5)
where ˜
ℎ=ℎ/𝑁, and 𝑁is an integer. From (5) and Lemma
1, the saturation nonlinearity 𝑠𝑎𝑡(𝑢(𝑡)) can be written as
𝑠𝑎𝑡(𝑢(𝑡)) =
2𝑚

𝑖=1
𝜆𝑡
𝑖(𝐷𝑖𝐾+𝒟−
𝑖𝑈𝑥(𝑡)
+𝒟−
𝑖
𝑁

𝑗=1
𝑉𝑗𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
𝑥(𝑠)𝑑𝑠(6)
where 𝜆𝑡
1≥0, 𝜆𝑡
2≥0,⋅ ⋅ ⋅ , 𝜆𝑡
2𝑚≥0and 2𝑚
𝑖=1 𝜆𝑡
𝑖= 1. Using
(1) and (6), one can deduce the following closed-loop system:
˙𝑥(𝑡) =
2𝑚

𝑖=1
𝜆𝑡
𝑖[𝐴0(𝑡) + 𝐵(𝑡)(𝐷𝑖𝐾+𝒟−
𝑖𝑈)]𝑥(𝑡)
+𝐴1(𝑡)𝑥(𝑡−ℎ) +
𝑁

𝑗=1
[𝐴2(𝑡) + 𝐵(𝑡)𝒟−
𝑖𝑉𝑗]
×𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
𝑥(𝑠)𝑑𝑠≜𝜂(𝑡).(7)
The main objective of this paper is to design the controller (4)
such that the closed-loop system (7) is robustly asymptotically
stable, and meanwhile obtain a larger estimate of the domain
of attraction of the following form:
𝑋𝜌≜𝜙∈𝐶1[−ℎ, 0] : ∥𝜙∥𝑐≤𝜌1,∥˙
𝜙∥𝑐≤𝜌2(8)
where 𝜌1and 𝜌2are some scalars.
Remark 2: As discussed in [25], the delay-dependent poly-
topic approach utilizes delay information when representing
the saturation nonlinearity, and thus is helpful in reducing the
possible conservatism. However, different from the auxiliary
discrete-delay feedback 𝑣(𝑡) = 𝑈𝑥(𝑡)+𝑉 𝑥(𝑡−ℎ)proposed in
[25] for the case of constant delay, the auxiliary distributed-
delay feedback 𝑣(𝑡) = 𝑈𝑥(𝑡) + 𝑁
𝑗=1 𝑉𝑗𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ𝑥(𝑠)𝑑𝑠
is introduced for the first time in this paper to form the
distributed-delay-dependent polytopic representation (6).
III. MAI N RES ULTS
This section first considers the stabilization problem for the
system (1) without uncertainties, i.e., system matrices in (1)
satisfy: 𝐴0(𝑡) = 𝐴0, 𝐴1(𝑡) = 𝐴1, 𝐴2(𝑡) = 𝐴2and 𝐵(𝑡) = 𝐵.
Here, we choose the following augmented L-K functional [9]:
𝑉(𝑡) =𝜉𝑇(𝑡)𝑃 𝜉(𝑡) +
𝑁

𝑗=1 𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
𝑥𝑇(𝑠)𝑄𝑗𝑥(𝑠)d𝑠
+˜
ℎ−(𝑗−1)˜
ℎ
−𝑗˜
ℎ𝑡
𝑡+𝜃
˙𝑥𝑇(𝑠)𝑍𝑗˙𝑥(𝑠)d𝑠d𝜃(9)

0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Transactions on Automatic Control
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where 𝜉(𝑡) = 𝑥𝑇(𝑡)𝑡
𝑡−˜
ℎ𝑥𝑇(𝑠)d𝑠𝑡−˜
ℎ
𝑡−2˜
ℎ𝑥𝑇(𝑠)d𝑠⋅ ⋅ ⋅
𝑡−(𝑁−1)˜
ℎ
𝑡−𝑁˜
ℎ𝑥𝑇(𝑠)d𝑠𝑇, and 𝑃 > 0, 𝑄𝑗>0, 𝑍𝑗>0.
Theorem 1: For given scalars ℎand 𝛿, if there exist (𝑁+1)𝑛
×(𝑁+1)𝑛matrix ¯
𝑃 > 0,𝑛×𝑛matrices ¯
𝑄𝑗>0,¯
𝑍𝑗>0, 𝑋,
𝑚×𝑛matrix 𝑌, and ↔
𝑚×𝑛matrices 𝐺, 𝐻𝑗, 𝑗 ∈𝕀[1, 𝑁 ],such
that for ∀𝑖∈𝕀[1,2𝑚],∀𝑙∈𝕀[1,↔
𝑚], the following LMIs hold:
(¯
Ω𝑖
𝑟𝑠)2×2+sym(Φ𝑇
1¯
𝑃Φ2)<0(10)
1¯
𝐹𝑙
¯
𝐹𝑇
𝑙¯
Σ + ¯
𝑃≥0(11)
where ¯
𝐹𝑙is the 𝑙-th row of ¯
𝐹= [𝐺 𝐻1𝐻2⋅ ⋅ ⋅ 𝐻𝑁], and
¯
Ω𝑖
11 =









¯𝛼𝑖
1−2¯
𝑍10⋅ ⋅ ⋅ 0𝐴1𝑋𝑇
∗¯𝛼2−2¯
𝑍2⋅ ⋅ ⋅ 0 0
∗ ∗ ¯𝛼3⋅ ⋅ ⋅ 0 0
.
.
..
.
..
.
.....
.
..
.
.
∗ ∗ ∗ ∗ ¯𝛼𝑁−2¯
𝑍𝑁
∗ ∗ ∗ ∗ ∗ ¯𝛼𝑁+1









¯
Ω𝑖
12 =









¯
𝛽𝑖
1¯
𝛽𝑖
2⋅ ⋅ ⋅ ¯
𝛽𝑖
𝑁−1¯
𝛽𝑖
𝑁¯
𝛽𝑖
𝑁+1
ˇ
ℎ¯
𝑍1ˇ
ℎ¯
𝑍2⋅ ⋅ ⋅ 0 0 0
0ˇ
ℎ¯
𝑍2⋅ ⋅ ⋅ 0 0 0
.
.
..
.
.....
.
..
.
..
.
.
0 0 0 ˇ
ℎ¯
𝑍𝑁−1ˇ
ℎ¯
𝑍𝑁0
0 0 0 0 ˇ
ℎ¯
𝑍𝑁𝛿𝑋𝐴𝑇
1









¯
Ω𝑖
22 =






−ˆ
ℎ¯
𝑍10⋅ ⋅ ⋅ 0 ¯𝛾𝑖
1
∗ −ˆ
ℎ¯
𝑍2⋅ ⋅ ⋅ 0 ¯𝛾𝑖
2
.
.
..
.
.....
.
..
.
.
∗ ∗ ∗ −ˆ
ℎ¯
𝑍𝑁¯𝛾𝑖
𝑁
∗ ∗ ∗ ∗ ¯𝛾𝑁+1







Φ1=






𝐼0𝑛×𝑁𝑛 0 0 ⋅ ⋅ ⋅ 0 0𝑛×𝑛
0 0𝑛×𝑁𝑛 𝐼0⋅ ⋅ ⋅ 0 0𝑛×𝑛
0 0𝑛×𝑁𝑛 0𝐼⋅ ⋅ ⋅ 0 0𝑛×𝑛
.
.
..
.
..
.
..
.
.....
.
..
.
.
0 0𝑛×𝑁𝑛 0 0 ⋅ ⋅ ⋅ 𝐼0𝑛×𝑛







Φ2=






0 0 0 ⋅ ⋅ ⋅ 0 0 0𝑛×𝑁𝑛 𝐼
𝐼−𝐼0⋅ ⋅ ⋅ 0 0 0𝑛×𝑁 𝑛 0
0𝐼−𝐼⋅ ⋅ ⋅ 0 0 0𝑛×𝑁 𝑛 0
.
.
..
.
..
.
.....
.
..
.
..
.
..
.
.
0 0 0 ⋅ ⋅ ⋅ 𝐼−𝐼0𝑛×𝑁𝑛 0







¯
Σ =









¯
Σ11 ¯
Σ12 ¯
Σ13 ⋅ ⋅ ⋅ ¯
Σ1𝑁¯
Σ1,𝑁+1
∗¯
Σ22 0⋅ ⋅ ⋅ 0 0
∗ ∗ ¯
Σ33 ⋅ ⋅ ⋅ 0 0
.
.
..
.
..
.
.....
.
..
.
.
∗ ∗ ∗ ∗ ¯
Σ𝑁𝑁 0
∗ ∗ ∗ ∗ ∗ ¯
Σ𝑁+1,𝑁+1









with ¯𝛼𝑖
1=𝐴0𝑋𝑇+𝑋𝐴𝑇
0+𝐵(𝐷𝑖𝑌+𝒟−
𝑖𝐺) + (𝐷𝑖𝑌+
𝒟−
𝑖𝐺)𝑇𝐵𝑇+¯
𝑄1−4¯
𝑍1,¯𝛼𝑗=−¯
𝑄𝑗−1+¯
𝑄𝑗−4( ¯
𝑍𝑗−1+
¯
𝑍𝑗),¯𝛼𝑁+1 =−¯
𝑄𝑁−4¯
𝑍𝑁,¯
𝛽𝑖
1=𝐴2𝑋𝑇+𝐵𝒟−
𝑖𝐻1+ˇ
ℎ¯
𝑍1,
¯
𝛽𝑖
𝑗=𝐴2𝑋𝑇+𝐵𝒟−
𝑖𝐻𝑗, 𝑗 ∈𝕀[2, 𝑁 ],¯
𝛽𝑖
𝑁+1 =−𝑋𝑇+𝛿𝑋𝐴𝑇
0+
𝛿(𝐷𝑖𝑌+𝒟−
𝑖𝐺)𝑇𝐵𝑇,¯𝛾𝑖
𝑘=𝛿𝑋𝐴𝑇
2+𝛿(𝒟−
𝑖𝐻𝑘)𝑇𝐵𝑇,¯𝛾𝑁+1 =
−𝛿(𝑋+𝑋𝑇) + ˜
ℎ2𝑁
𝑗=1 ¯
𝑍𝑗,¯
Σ11 = 2˜
ℎ𝑁
𝑗=1 ¯
𝑍𝑗/(2𝑗−1),
¯
Σ1,𝑘+1 =−2¯
𝑍𝑘/(2𝑘−1),¯
Σ𝑘+1,𝑘+1 = (1/˜
ℎ)[2 ¯
𝑍𝑘/(2𝑘−
1) + ¯
𝑄𝑘], 𝑘 ∈𝕀[1, 𝑁 ],ˇ
ℎ= 6/˜
ℎ, ˆ
ℎ= 12/˜
ℎ2,then for any
initial function 𝜙(𝑡)satisfying 𝑉(0) ≤1, the system (1)
without uncertainties can be asymptotically stabilized by the
state feedback controller (4) with 𝐾=𝑌 𝑋−𝑇.
Proof. Differentiating 𝑉(𝑡)in (9) along the system (7) yields
˙
𝑉(𝑡) = 2𝜉𝑇(𝑡)𝑃˙
𝜉(𝑡) +
𝑁

𝑗=1 𝑥𝑇(𝑡−(𝑗−1)˜
ℎ)𝑄𝑗×
𝑥(𝑡−(𝑗−1)˜
ℎ)−𝑥𝑇(𝑡−𝑗˜
ℎ)𝑄𝑗𝑥(𝑡−𝑗˜
ℎ)+
˜
ℎ2˙𝑥𝑇(𝑡)𝑍𝑗˙𝑥(𝑡)−˜
ℎ𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
˙𝑥𝑇(𝑠)𝑍𝑗˙𝑥(𝑠)d𝑠.(12)
Using Wirtinger integral inequality (see [8]), it follows that
−˜
ℎ𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
˙𝑥𝑇(𝑠)𝑍𝑗˙𝑥(𝑠)d𝑠≤
−[𝑥(𝑡−(𝑗−1)˜
ℎ)−𝑥(𝑡−𝑗˜
ℎ)]𝑇𝑍𝑗
×[𝑥(𝑡−(𝑗−1)˜
ℎ)−𝑥(𝑡−𝑗˜
ℎ)] −3Ψ𝑇
𝑗𝑍𝑗Ψ𝑗(13)
where Ψ𝑗=𝑥(𝑡−(𝑗−1)˜
ℎ)+𝑥(𝑡−𝑗˜
ℎ)−2
˜
ℎ𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ𝑥(𝑠) d𝑠.
For any matrices 𝑇1and 𝑇2, it is clear from system (7) that
2[𝑥𝑇(𝑡)𝑇1+ ˙𝑥𝑇(𝑡)𝑇2][𝜂(𝑡)−˙𝑥(𝑡)] = 0.(14)
Adding the left side of (14) to ˙
𝑉(𝑡), and substituting (13) into
(12), then one can obtain the following inequality:
˙
𝑉(𝑡)≤
2𝑚

𝑖=1
𝜆𝑡
𝑖𝜁𝑇(𝑡)(Ω𝑖
𝑟𝑠)2×2+sym(Φ𝑇
1𝑃Φ2)𝜁(𝑡)(15)
where 𝜁(𝑡) = 𝑥𝑇(𝑡)𝑥𝑇(𝑡−˜
ℎ)𝑥𝑇(𝑡−2˜
ℎ)⋅ ⋅ ⋅ 𝑥𝑇(𝑡−𝑁˜
ℎ)
𝑡
𝑡−˜
ℎ𝑥𝑇(𝑠)d𝑠𝑡−˜
ℎ
𝑡−2˜
ℎ𝑥𝑇(𝑠)d𝑠⋅ ⋅ ⋅ 𝑡−(𝑁−1)˜
ℎ
𝑡−𝑁˜
ℎ𝑥𝑇(𝑠)d𝑠˙𝑥𝑇(𝑡)𝑇,
Φ1and Φ2have the same definitions as in (10), and
Ω𝑖
11 =









𝛼𝑖
1−2𝑍10⋅ ⋅ ⋅ 0𝑇1𝐴1
∗𝛼2−2𝑍2⋅ ⋅ ⋅ 0 0
∗ ∗ 𝛼3⋅ ⋅ ⋅ 0 0
.
.
..
.
..
.
.....
.
..
.
.
∗ ∗ ∗ ∗ 𝛼𝑁−2𝑍𝑁
∗ ∗ ∗ ∗ ∗ 𝛼𝑁+1









Ω𝑖
12 =









𝛽𝑖
1𝛽𝑖
2⋅ ⋅ ⋅ 𝛽𝑖
𝑁−1𝛽𝑖
𝑁𝛽𝑖
𝑁+1
ˇ
ℎ𝑍1ˇ
ℎ𝑍2⋅ ⋅ ⋅ 0 0 0
0ˇ
ℎ𝑍2⋅ ⋅ ⋅ 0 0 0
.
.
..
.
.....
.
..
.
..
.
.
0 0 0 ˇ
ℎ𝑍𝑁−1ˇ
ℎ𝑍𝑁0
0 0 0 0 ˇ
ℎ𝑍𝑁𝐴𝑇
1𝑇𝑇
2









Ω𝑖
22 =






−ˆ
ℎ𝑍10⋅ ⋅ ⋅ 0𝛾𝑖
1
∗ −ˆ
ℎ𝑍2⋅ ⋅ ⋅ 0𝛾𝑖
2
.
.
..
.
.....
.
..
.
.
∗ ∗ ∗ −ˆ
ℎ𝑍𝑁𝛾𝑖
𝑁
∗ ∗ ∗ ∗ 𝛾𝑁+1







with 𝛼𝑖
1=𝑇1[𝐴0+𝐵(𝐷𝑖𝐾+𝒟−
𝑖𝑈)] + [𝐴0+𝐵(𝐷𝑖𝐾+
𝒟−
𝑖𝑈)]𝑇𝑇𝑇
1+𝑄1−4𝑍1, 𝛼𝑗=−𝑄𝑗−1+𝑄𝑗−4(𝑍𝑗−1+𝑍𝑗),
𝛼𝑁+1 =−𝑄𝑁−4𝑍𝑁, 𝛽𝑖
1=𝑇1(𝐴2+𝐵𝒟−
𝑖𝑉1) + ˇ
ℎ𝑍1, 𝛽𝑖
𝑗=
𝑇1(𝐴2+𝐵𝒟−
𝑖𝑉𝑗), 𝑗 ∈𝕀[2, 𝑁 ], 𝛽𝑖
𝑁+1 =−𝑇1+[𝐴0+𝐵(𝐷𝑖𝐾+

0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Transactions on Automatic Control
4
𝒟−
𝑖𝑈)]𝑇𝑇𝑇
2, 𝛾𝑖
𝑘= (𝐴2+𝐵𝒟−
𝑖𝑉𝑘)𝑇𝑇𝑇
2, 𝑘 ∈𝕀[1, 𝑁 ], 𝛾𝑁+1 =
−𝑇2−𝑇𝑇
2+˜
ℎ2𝑁
𝑗=1 𝑍𝑗,ˇ
ℎ= 6/˜
ℎ, ˆ
ℎ= 12/˜
ℎ2.
It is clear that if the following matrix inequality holds:
(Ω𝑖
𝑟𝑠)2×2+sym(Φ𝑇
1𝑃Φ2)<0,(16)
then one can obtain from (15) that ˙
𝑉(𝑡)<0,which gives
𝑉(𝑡)< 𝑉 (0), 𝑡 ≥0.(17)
For the functional 𝑉(𝑡)defined in (9), using Jensen integral
inequalities [6], then it is not difficult to obtain that
𝑉(𝑡)≥𝜉𝑇(𝑡)(Σ + 𝑃)𝜉(𝑡)(18)
where
Σ =









Σ11 Σ12 Σ13 ⋅ ⋅ ⋅ Σ1𝑁Σ1,𝑁 +1
∗Σ22 0⋅ ⋅ ⋅ 0 0
∗ ∗ Σ33 ⋅ ⋅ ⋅ 0 0
.
.
..
.
..
.
.....
.
..
.
.
∗ ∗ ∗ ∗ Σ𝑁𝑁 0
∗ ∗ ∗ ∗ ∗ Σ𝑁+1,𝑁+1









and Σ11 = 2˜
ℎ𝑁
𝑗=1 𝑍𝑗/(2𝑗−1),Σ1,𝑘+1 =−2𝑍𝑘/(2𝑘−1),
Σ𝑘+1,𝑘+1 = (1/˜
ℎ)[2𝑍𝑘/(2𝑘−1) + 𝑄𝑘], 𝑘 ∈𝕀[1, 𝑁 ].
Assume that the following matrix inequalities hold:
𝐹𝑇
𝑙𝐹𝑙≤Σ + 𝑃, 𝑙 ∈𝕀[1,↔
𝑚](19)
where 𝐹= [𝑈 𝑉1𝑉2⋅ ⋅ ⋅ 𝑉𝑁], and 𝐹𝑙is the 𝑙-th row of
the matrix 𝐹, then it can be seen that

𝑈𝑙𝑥(𝑡) +
𝑁

𝑗=1
𝑉𝑗𝑙 𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ
𝑥(𝑠)𝑑𝑠
2
=𝜉𝑇(𝑡)𝐹𝑇
𝑙𝐹𝑙𝜉(𝑡)≤𝜉𝑇(𝑡)(Σ + 𝑃)𝜉(𝑡), 𝑙 ∈𝕀[1,↔
𝑚].(20)
For any 𝜙(𝑡)satisfying 𝑉(0) ≤1, it can be seen from (17)-
(18) and (20) that 𝑈𝑙𝑥(𝑡) + 𝑁
𝑗=1 𝑉𝑗𝑙 𝑡−(𝑗−1)˜
ℎ
𝑡−𝑗˜
ℎ𝑥(𝑠)𝑑𝑠≤1
holds for 𝑙∈𝕀[1,↔
𝑚], which implies that the assumption (5)
can be guaranteed. Then, it can be concluded that the closed-
loop system (7) without uncertainties is locally asymptotically
stable for any initial function 𝜙(𝑡)satisfying 𝑉(0) ≤1.
To obtain LMI-based conditions, we set 𝑇2≜𝛿𝑇1, 𝛿 ∕= 0 in
(16), and introduce the following new matrix variables:
𝑇−1
1≜𝑋, 𝑋𝑄𝑗𝑋𝑇≜¯
𝑄𝑗, 𝑋𝑍𝑗𝑋𝑇≜¯
𝑍𝑗(21a)
𝐾𝑋 𝑇≜𝑌, 𝑈𝑋𝑇≜𝐺, 𝑉𝑗𝑋𝑇≜𝐻𝑗, 𝑗 ∈𝕀[1, 𝑁 ](21b)
˜
𝑋𝑃 ˜
𝑋𝑇≜¯
𝑃 , ˜
𝑋=𝑑𝑖𝑎𝑔{𝑋, 𝑋, ⋅ ⋅ ⋅ , 𝑋 }.(21c)
For (16) and (19), performing some congruence transforma-
tions as in [23], respectively, and noting (21a)-(21c), then one
can obtain LMIs (10) and (11). This completes the proof. □
Remark 3: It is clear that the slack variables 𝐻𝑗, 𝑗 ∈𝕀[1, 𝑁 ]
are additionally introduced in LMIs (10)-(11) due to the use of
the distributed-delay-dependent polytopic representation (6),
which results in a less conservative stabilization condition.
Remark 4: Recently, the delay-dependent polytopic ap-
proach was proposed in [25] by introducing the auxiliary
time-delay feedback. However, it should be pointed out that
the proposed L-K functionals 𝑉(𝑡)in [25] have to contain
some integral terms of state derivative to obtain LMI-based
conditions resulting from ∥𝑣(𝑡)∥∞≤1. It is well-known that
such terms are essential for neutral systems [4-5], while are
unnecessary for systems with discrete/distributed delay. Noting
that the estimate of the domain of attraction is associated
with 𝑉(0) ≤1, it is thus possible that the delay-dependent
polytopic approach proposed in [25] is still conservative for
systems with discrete/distributed delay, due to the additional
introduction of some integral terms of state derivative.
From the proof of Theorem 1, it can be seen that some LMI-
based conditions can be directly deduced from the assumption
∥𝑣(𝑡)∥∞≤1, each term in L-K functional (9) contributes to
the stability analysis, and no redundant terms are intentionally
introduced to match the proposed distributed-delay-dependent
polytopic approach. Therefore, it can be concluded that the
auxiliary distributed-delay feedback proposed in this paper is
more appropriate than the auxiliary discrete-delay feedback in
[25] for linear systems with discrete/distributed delay.
Using the routine method of handing the uncertainties [4],
it is easy to obtain the following robust stabilization condition.
Theorem 2: For given scalars ℎand 𝛿, if there exist (𝑁+
1)𝑛×(𝑁+ 1)𝑛matrix ¯
𝑃 > 0,𝑛×𝑛matrices ¯
𝑄𝑗>0,¯
𝑍𝑗>
0, 𝑋,𝑚×𝑛matrix 𝑌,↔
𝑚×𝑛matrices 𝑌, 𝐺, 𝐻𝑗, 𝑗 ∈𝕀[1, 𝑁],
and a scalar 𝜇 > 0, such that for ∀𝑖∈𝕀[1,2𝑚],∀𝑙∈𝕀[1,↔
𝑚],
the LMI (11) and the following LMI hold:


(¯
Ω𝑖
𝑟𝑠)2×2+sym(Φ𝑇
1¯
𝑃Φ2)𝜇¯
𝑀(¯
𝐸𝑖)𝑇
∗ −𝜇𝐼 0
∗ ∗ −𝜇𝐼 
<0(22)
where (¯
Ω𝑖
𝑟𝑠)2×2,Φ1and Φ2are defined in Theorem 1, and
¯
𝑀=𝑀𝑇0𝑛×2𝑁𝑛 𝛿𝑀𝑇𝑇
¯
𝐸𝑖=𝜈𝑖
00𝑛×(𝑁−1)𝑛𝐸1𝑋𝑇𝜈𝑖
1𝜈𝑖
2⋅ ⋅ ⋅ 𝜈𝑖
𝑁0
with 𝜈𝑖
0=𝐸0𝑋𝑇+𝐸3(𝐷𝑖𝑌+𝒟−
𝑖𝐺), 𝜈𝑖
𝑘=𝐸2𝑋𝑇+
𝐸3𝒟−
𝑖𝐻𝑘, 𝑘 ∈𝕀[1, 𝑁 ], then for any 𝜙(𝑡)satisfying 𝑉(0) ≤1,
the uncertain system (1) can be robustly asymptotically stabi-
lized by the state feedback controller (4) with 𝐾=𝑌 𝑋−𝑇.
Remark 5: For non-unity saturation level with 𝑠𝑎𝑡(𝑢𝑗) =
𝑠𝑔𝑛(𝑢𝑗)𝑚𝑖𝑛{∣𝑢𝑗∣,¯𝑢𝑗}, the matrices 𝐵and 𝑌in Theorems 1-
2 should be substituted by ˜
𝐵= [¯𝑢1𝑏1¯𝑢2𝑏2⋅ ⋅ ⋅ ¯𝑢𝑚𝑏𝑚]and
˜
𝑌= [𝑦𝑇
1/¯𝑢1𝑦𝑇
2/¯𝑢2⋅ ⋅ ⋅ 𝑦𝑇
𝑚/¯𝑢𝑚]𝑇, respectively, where 𝑏𝑗is
the 𝑗-th column of 𝐵and 𝑦𝑗is the 𝑗-th row of 𝑌.
Remark 6: The proposed results in this paper are mainly
concerned with state-delayed systems. In some control ap-
plications, such as digital control and network-based control,
input delay is also frequently encountered [8,26-27]. Using the
distributed-delay-dependent polytopic approach in this paper,
we can also establish some stabilization conditions for systems
with time-varying input delay, which is our further work.
To obtain a larger estimate of the domain of attraction 𝑋𝜌
when designing a controller, we now discuss the estimate and
maximization of the domain of attraction. Noting the L-K
functional (9) and (21a)-(21c), it can be seen that the domain
of attraction 𝑋𝜌can be bounded by the following inequality:
𝑉(0) ≤𝜆𝑀(˜
Λ0)𝜌2
1+
𝑁

𝑗=1 [˜
ℎ2𝜆𝑀(˜
Λ𝑗) + ˜
ℎ𝜆𝑀(𝑋−1¯
𝑄𝑗𝑋−𝑇)]

0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2016.2611559, IEEE
Transactions on Automatic Control
5
×𝜌2
1+ 0.5˜
ℎ3(2𝑗−1)𝜆𝑀(𝑋−1¯
𝑍𝑗𝑋−𝑇)𝜌2
2≤1(23)
where ˜
Λ𝑗=𝑋−1Λ𝑗𝑋−𝑇, 𝑗 ∈𝕀[0, 𝑁 ],with the LMI con-
straint ¯
𝑃≤𝑑𝑖𝑎𝑔{Λ0,Λ1,⋅ ⋅ ⋅ ,Λ𝑁}≜Λ. Here, it is worth
mentioning that the Jensen integral inequality is utilized when
estimating the first term 𝜉𝑇(0)𝑃 𝜉(0) of 𝑉(0) in (23).
As in [25], we introduce the matrix inequality 𝑋−1𝑋−𝑇≤
𝑟𝐼, which can be guaranteed by the following LMI:
𝑟𝐼 𝐼
𝐼 𝑋 +𝑋𝑇−𝐼≥0.(24)
Meanwhile, we define the following LMIs:
¯
𝑃≤Λ,Λ𝑘≤𝑝𝑘𝐼, 𝑘 ∈𝕀[0, 𝑁 ](25)
¯
𝑄𝑗≤𝑞𝑗𝐼, ¯
𝑍𝑗≤𝑧𝑗𝐼, 𝑗 ∈𝕀[1, 𝑁].(26)
Let 𝜌1=𝜌2≜𝜌, then it can be seen from (23) that the
maximization of the estimate of the domain of attraction 𝑋𝜌
in Theorems 1-2 can be formulated as follows:
Pb.1.min
¯
𝑃 , ¯
𝑄𝑗,¯
𝑍𝑗,Λ𝑘,𝑋,𝑌,𝐺,𝐻𝑗,𝑟,𝑝𝑘,𝑞𝑗,𝑧𝑗
𝜆, 𝑠.𝑡., ∀𝑖∈𝕀[1,2𝑚],
∀𝑙∈𝕀[1,↔
𝑚],LMIs (10) −(11) and (24) −(26) hold;
Pb.2.min
¯
𝑃 , ¯
𝑄𝑗,¯
𝑍𝑗,Λ𝑘,𝑋,𝑌,𝐺,𝐻𝑗,𝜇,𝑟,𝑝𝑘,𝑞𝑗,𝑧𝑗
𝜆, 𝑠.𝑡., ∀𝑖∈𝕀[1,2𝑚],
∀𝑙∈𝕀[1,↔
𝑚],LMIs (11),(22) and (24) −(26) hold,
where 𝜆=𝑒∗𝑟+𝑝0+𝑁
𝑗=1[˜
ℎ2𝑝𝑗+˜
ℎ𝑞𝑗+ 0.5˜
ℎ3(2𝑗−1)𝑧𝑗],
and 𝑒is a weighting parameter. Correspondingly, the max-
imum 𝜌can be obtained by 𝜌𝑚𝑎𝑥 =1/Δ𝑚𝑖𝑛, where
Δ𝑚𝑖𝑛 =𝜆𝑀(𝑋−1Λ0𝑋−𝑇) + 𝑁
𝑗=1[˜
ℎ2𝜆𝑀(𝑋−1Λ𝑗𝑋−𝑇) +
˜
ℎ𝜆𝑀(𝑋−1¯
𝑄𝑗𝑋−𝑇) + 0.5˜
ℎ3(2𝑗−1)𝜆𝑀(𝑋−1¯
𝑍𝑗𝑋−𝑇)].
IV. NUM ER IC AL E XA MP LE S
Example 1: Consider the time-delay system (1) without
uncertainties, where 𝐴2= 0,¯𝑢1= 15, ℎ = 1, and
𝐴0=1 1.5
0.3−2, 𝐴1=0−1
0 0 , 𝐵 =10
1.
For this example, by solving Pb.1 in this paper with 𝛿= 1
and 𝑒= 4 ∗109, we can obtain the scalars 𝜌𝑚𝑎𝑥 for 𝑁= 1,2
and 3, which are listed in Table 1. From Table 1, it can be
seen that the proposed conditions (𝐻𝑗∕= 0) in this paper can
provide the larger estimate of the domain of attraction 𝑋𝜌
than that in [20-22,25]. For the case that 𝐻𝑗= 0, 𝑗 ∈𝕀[1, 𝑁 ],
Table 1 shows that the scalars 𝜌𝑚𝑎𝑥 become smaller, which
means that our proposed distributed-delay-dependent polytopic
approach is of vital importance in reducing the conservatism.
If we choose the same parameters as in [25], i.e., 𝛿= 1
and 𝑒= 1 ∗109, then we have 𝜌𝑚𝑎𝑥 = 90.7535 (𝑁=
1),92.1949 (𝑁= 2) and 92.5966 (𝑁= 3). It is clear that
the obtained scalars 𝜌𝑚𝑎𝑥 are still larger that in [20-22,25].
For this example, if we set 𝐴2=0−1
0 0 and 𝐴1= 0,
by solving Pb.1 with 𝑁= 1, 𝛿 = 1 and 𝑒= 4 ∗109, one can
obtain the scalar 𝜌𝑚𝑎𝑥 = 109.8915, which is larger than the
scalar 𝜌𝑚𝑎𝑥 = 101.5648 obtained by solving Pb.1 with 𝐻1=
0. Clearly, our proposed distributed-delay-dependent polytopic
approach is also effective for systems with distributed delay.
Example 2: Consider the time-delay system (1) without
uncertainties, where 𝐴2= 0,¯𝑢1= 5, ℎ = 1.854, and
𝐴0=0.5−1
0.5−0.5, 𝐴1=0.6 0.4
0−0.5, 𝐵 =1
1.
For this example, by solving Pb.1 in this paper (𝛿= 3.6,𝑒=
2∗104), the scalars 𝜌𝑚𝑎𝑥 associated with the domain 𝑋𝜌can
be easily obtained for 𝑁= 1,2and 3, which are listed in Table
2. Table 2 shows that the results in this paper can provide the
larger estimate of the domain 𝑋𝜌than that in [20-23,25]. Also,
Table 2 shows that the smaller scalars 𝜌𝑚𝑎𝑥 are obtained if one
sets 𝐻𝑗= 0, 𝑗 ∈𝕀[1, 𝑁 ], which implies that the slack variables
𝐻𝑗, 𝑗 ∈𝕀[1, 𝑁 ]are important in reducing the conservatism.
Using the obtained design parameters for 𝑁= 3, the state
responses of the closed-loop system and auxiliary time-delay
feedback 𝑣(𝑡)are plotted in Fig.1, where 𝜙(𝑡) = [0.7 0.4]𝑇∈
𝑋𝜌. It can be seen from Fig.1 that the closed-loop system is
stable and the assumption ∥𝑣(𝑡)∥∞≤1can be guaranteed.
If the same parameters as in [25] are chosen when solving
Pb.1 (𝛿= 4.8, 𝑒 = 8000), the obtained scalars 𝜌𝑚𝑎𝑥 are
0.7552 (𝑁= 1),0.7738 (𝑁= 2) and 0.7814 (𝑁= 3),
respectively, which are also larger than that in [20-23,25].
Let 𝐵=1 0.5
1−1, and then solve Pb.1 (𝑁= 3, 𝛿 =
0.7, 𝑒 = 3 ∗105), we have 𝜌𝑚𝑎𝑥 = 5.9676, which is larger
than 𝜌𝑚𝑎𝑥 = 4.6597 obtained by solving Pb.1 with 𝐻𝑗= 0.
TABLE I
TH E SC ALA RS 𝜌𝑚𝑎𝑥 AS SO CIAT ED WI TH T HE DO MA IN 𝑋𝜌
[20] [21] [22] [25] Pb.1 𝑁=1
𝐻𝑗∕=0
58.395 67.0618 79.43 84.6074 93.6526
Pb.1 𝑁=2
𝐻𝑗∕=0 Pb.1 𝑁=3
𝐻𝑗∕=0 Pb.1 𝑁=1
𝐻𝑗=0 Pb.1 𝑁=2
𝐻𝑗=0 Pb.1 𝑁=3
𝐻𝑗=0
95.3053 95.8210 81.4534 83.9677 84.6033
TABLE II
TH E SC ALA RS 𝜌𝑚𝑎𝑥 AS SO CIAT ED WI TH T HE DO MA IN 𝑋𝜌
[20-21] [22] [23] [25] Pb.1 𝑁=1
𝐻𝑗∕=0
infeasible 0.091 0.4521 0.6348 0.7882
Pb.1 𝑁=2
𝐻𝑗∕=0 Pb.1 𝑁=3
𝐻𝑗∕=0 Pb.1 𝑁=1
𝐻𝑗=0 Pb.1 𝑁=2
𝐻𝑗=0 Pb.1 𝑁=3
𝐻𝑗=0
0.8140 0.8265 0.7513 0.7731 0.7830
Example 3: Consider the uncertain system (1) with a
distributed delay, where the values of the parameters 𝐴0, 𝐴2,
𝐵, 𝐸0, 𝐸2are borrowed from Example 5 in [5], and
𝐴1=𝐸1=𝐸3= 0, 𝑀 = [0.4000]𝑇, ℎ = 1.
This system can be seen as a linearized model of the feeding
system and combustion chamber of a liquid monopropellant
rocket motor, please see Refs. [3,5]. Here, it is assumed that
the system is subject to the actuator saturation with ¯𝑢1= 1.
For this example, by solving Pb.2 in this paper with 𝑁=
3, 𝛿 = 0.22 and 𝑒= 3 ∗1017 , it is easy to obtain the controller
gain 𝐾= [0.0025 −0.0121 0.0008 0.0018], and the bound
𝜌𝑚𝑎𝑥 = 9.0220 ∗103of the domain of attraction 𝑋𝜌. Using
the proposed controller, the state responses of the closed-loop
system are plotted in Fig. 2, where 𝜙(𝑡) = [20 −20 −20 20]𝑇.
From Fig.2, it can be seen that the closed-loop system is stable,
which shows the effectiveness of the proposed conditions.

0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2016.2611559, IEEE
Transactions on Automatic Control
6
Remark 7: From Tables 1-2, clearly, the larger estimates of
the domain of attraction 𝑋𝜌can be obtained as 𝑁increases.
However, as 𝑁increases, the more decision variables will be
involved when solving Pb.1. In Examples 1-2, the involved
variables are 37, 62 and 91, respectively, for 𝑁= 1,2and 3.
0 5 10 15 20 25 30
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
t/s
x1, x2 and v(t)
x1
x2
v(t)
K=[−2.7930 −0.2348]
U=[−0.3966 0.0452]
V1=[−0.0397 −0.0499]
V2=[−0.0466 −0.0371]
V3=[−0.1010 −0.0519]
Fig.1. State responses and auxiliary feedback 𝑣(𝑡).
0 50 100 150 200
−40
−30
−20
−10
0
10
20
30
40
t/s
System states
x1
x2
x3
x4
Fig.2. State responses of the closed-loop system.
V. CONCLUSION
In this paper, we have proposed the distributed-delay-
dependent polytopic approach to represent the saturation non-
linearly. Then, combining with an augmented L-K functional
and some integral inequalities, the improved stabilization and
robust stabilization conditions have been obtained for uncer-
tain linear systems with discrete and distributed delays under
saturated state feedback. The proposed approach in this paper
can be extended to discrete time-delay systems, switched time-
delay systems [28-29], and Markov jump systems with time
delays [30-31], which are our further studies. Here, it should
be pointed out that the proposed conditions in this paper are
sufficient and may be still conservative to some extent. The
further directions of reducing the conservatism are to develop
some new delay-dependent polytopic approaches and to utilize
some improved stability analysis approaches [10-13].
REF ER EN CE S
[1] J.K. Hale, S.M. Verduyn Lunel. Introduction to functional differential
equations. New York: Springer, 1993.
[2] K. Gu, V.L. Kharitonov, J. Chen, Stability of time-delay systems, Boston:
Birkh¨𝑎user, 2003.
[3] L. Xie, E. Fridman, and U. Shaked, Robust 𝐻∞control of distributed
delay systems with application to combustion control, IEEE Trans. Autom.
Control, vol. 46, no. 12, pp. 1930-1935, 2001.
[4] Y. He, M. Wu, J. H. She, and G. P. Liu, Delay-dependent robust stability
criteria for uncertain neutral systems with mixed delays, Syst. Control
Lett., vol. 51, no. 1, pp. 57-65, 2004.
[5] W.-H. Chen, W.X. Zheng, Delay-dependent robust stabilization for un-
certain neutral systems with distributed delays, Automatica, vol. 43, no.
1, pp. 95-104, 2007.
[6] J. Sun, G.P. Liu, J. Chen, Delay-dependent stability and stabilization of
neutral time-delay systems, Int. J. Robust Nonlin. Control, vol. 19, no. 1,
pp. 1364-1375, 2009.
[7] Q.-L. Han, A discrete delay decomposition approach to stability of linear
retarded and neutral systems, Automatica, vol. 45, no. 2, pp. 517-524,
2009.
[8] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application
to time-delay systems, Automatica, vol. 49, no. 9, pp. 2860-2866, 2013.
[9] T.H. Lee, J.H. Park, H.Y. Jung, et al. Improved results on stability of time-
delay systems using Wirtinger-based inequality, in: Proc. of the 19th IFAC
World Congress, pp. 6826-6830, 2014.
[10] H.-B. Zeng, Y. He, M. Wu, J. She, Free-matrix-based integral inequality
for stability analysis of systems with time-varying delay, IEEE Trans.
Autom. Control, vol. 60, no. 10, pp. 2768-2772, 2015.
[11] M.J. Park, O.M. Kwon, J.H. Park, et al. Stability of time-delay systems
via Wirtinger-based double integral inequality, Automatica, vol. 55, pp.
204-208, 2015.
[12] L. Zhang, Z. Ning, P. Shi, Input-output approach to control for fuzzy
Markov jump systems with time-varying delays and uncertain packet
dropout rate, IEEE Trans. Cybern., vol.45, no. 11, pp. 2449-2460, 2015.
[13] L. Zhang, Z. Ning, Z. Wang, Distributed filtering for fuzzy time-delay
systems with packet dropouts and redundant channels, IEEE Trans. Syst.,
Man, Cybern.: Syst., vol. 46. no. 4, pp. 559-572, 2016.
[14] Z. Lin, Low gain feedback, London: Springer-Verlag, 1999.
[15] T. Hu and Z. Lin, Control systems with actuator saturation: analysis
and design, BirkhVauser, 2001.
[16] S. Tarbouriech, G. Garcia, J.M. Gomes da Silva Jr, I. Queinnec, Stability
and stabilization of linear systems with saturating actuators, London:
Springer-Verlag, 2011.
[17] B. Zhou, Z. Lin, and G. Duan, A parametric Lyapunov equation
approach to the design of low gain feedback, IEEE Trans. Autom. Control,
vol. 53, no. 6, pp. 1548-1554, 2008.
[18] B. Zhou, Z. Lin, and G. Duan, Robust global stabilization of linear
systems with input saturation via gain scheduling, Int. J. Robust Nonlin.
Control, vol. 20, no. 4, pp. 424-447, 2010.
[19] B. Zhou, Analysis and design of discrete-time linear systems with nested
actuator saturations, Syst. Control Lett., vol. 62, no. 10, pp. 871-879, 2013.
[20] S. Tarbouriech, and J.M. Gomes da Silva Jr, Synthesis of controllers for
continuous-time delay systems with saturating controls via LMIs, IEEE
Trans. Autom. Control, vol. 45, no. 1, pp. 105-111, 2000.
[21] Y.-Y. Cao, Z. Lin, and T. Hu, Stability analysis of linear time-delay
systems subject to input saturation, IEEE Trans. Circuits Syst.-I, vol. 49,
no. 2, pp. 233-240, 2002.
[22] E. Fridman, A. Pila and U. Shaked, Regional stabilization and 𝐻∞
control of time-delay systems with saturating actuators, Int. J. Robust
Nonlin. Control, vol. 13, no. 9, pp. 885-907, 2003.
[23] L. Zhang, E.-K. Boukas, A. Haidar, Delay-range-dependent control
synthesis for time-delay systems with actuator saturation, Automatica,
vol. 44, no. 10, pp. 2691-2695, 2008.
[24] J.M. Gomes da Silva Jr, A. Seuret, E. Fridman and J.P. Richard,
Stabilisation of neutral systems with saturating control inputs, Int. J. Syst.
Sci., vol. 42, no. 7, pp. 1093-1103, 2011.
[25] Y. Chen, S. Fei, Y. Li, Stabilization of neutral time-delay systems with
actuator saturation via auxiliary time-delay feedback, Automatica, vol.
45, pp. 242-247, 2015.
[26] D. Yue, Q.-L. Han, J. Lam, Network-based robust 𝐻∞control of
systems with uncertainty, Automatica, vol. 41, no. 6, pp. 999-1007, 2005.
[27] X. Yin, L. Zhang, Y. Zhu, C. Wang, and Z. Li, Robust control of net-
worked systems with variable communication capabilities and application
to a semi-active suspension system, IEEE/ASME Trans. Mechatronics,
vol. 21, no. 4, pp. 2097-2107, 2016.
[28] Y. Chen, S. Fei, K. Zhang, Stabilisation for switched linear systems with
time-varying delay and input saturation, Int. J. Syst. Sci., vol. 45, no. 3,
pp. 532-546, 2014.
[29] L. Zhang, S. Zhuang, and R.D. Braatz. Switched model predictive
control of switched linear systems: feasibility, stability and robustness,
Automatica, vol. 67, pp. 8-21, 2016.
[30] L. Zhang, Y. Leng, and P. Colaneri. Stability and stabilization of
discrete-time Semi-Markov jump linear systems via Semi-Markov Kernel
approach, IEEE Trans. Autom. Control, vol. 61, no. 2, pp. 503-508, 2016.
[31] L. Zhang, Y. Zhu, P. Shi and Y. Zhao. Resilient asynchronous H-infinity
filtering for Markov jump neural networks with unideal measurementsand
multiplicative noises, IEEE Trans. Cybern., vol. 45, no. 12, pp. 2840-
2852, 2015.