Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 714525, 10 pages
http://dx.doi.org/10.1155/2013/714525
Research Article
Passivity and Passification for Delay Fuzzy
System Based on Delay Partitioning Approach
Xiangjie Liu,1 Dan Yue,1 and Xiuming Yao2
1
The State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources,
North China Electric Power University, Beijing 102206, China
2
Department of Automation, North China Electric Power University, Baoding 071003, China
Correspondence should be addressed to Xiuming Yao; xiumingyao@gmail.com
Received 17 February 2013; Accepted 28 March 2013
Academic Editor: Constantinos Siettos
Copyright © 2013 Xiangjie Liu 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.
A delay partitioning approach is introduced to solve problems of passivity and passification for continuous T-S fuzzy system with
time delay. Our aim is to design a state feedback controller such that the resulting closed system is passive. By constructing a
Lyapunov-Krasovskii functional, delay-dependent passivity/passification performance conditions are formulated in terms of linear
matrix inequalities (LMIs). Finally, numerical examples are used to illustrate the effectiveness of the proposed approaches which
can further reduce conservatism and become more obvious with partitioning getting thinner.
1. Introduction
The passivity concept was introduced by Willems [1] and
developed further by Hill and Moylan [2]. Passivity of nonlinear systems has attracted great interest in the control area
mainly because of the link between stability and passivity.
The passivity theory provides a nice tool for analyzing the
stability of systems and has found applications in diverse
areas such as stability, signal processing, chaos control, and
synchronization.
Since most physical systems in real world are nonlinear,
researchers have been devoting their efforts to the control of
nonlinear systems. Among the methods, the fuzzy control has
been proven to be effective in dealing with the analysis of
nonlinear systems. especially the T-S fuzzy control [3, 4]. It is
denoted by a group of IF-THEN rules that the conventional
linear system theory can be applied to the analysis of the
class of nonlinear systems, and numerous nonlinear analysis
problems have been studied based on this T-S fuzzy model,
such as [5–7] reported the problem of stability analysis,
and [8–10] investigated the 𝐻∞ control designs. References
[11–13] mentioned the fault detection of the T-S fuzzy systems.
𝐻∞ model reduction is addressed in [14]. The fuzzy controller
was carried out via Parallel Distributed Compensation (PDC)
technique [15]. Based on the PDC technique, the fuzzy
controller can also be designed to guarantee the passivity of
T-S fuzzy systems.
On the other hand, the time delay exists naturally in
various control systems. Time delays often degrade the
system’s performance and even cause instability. Therefore,
time delays have received great attention in recent years
and many researchers have studied various analytical techniques and developed many synthesis methods for timedelay systems. For instance, model reduction is addressed in
[16] and filtering problems are investigated in [17, 18]. So,
the passivity and passification analysis of nonlinear systems
with time delays is worth to be discussed and researched.
To date, researches have gained many results in passivity
control of T-S fuzzy systems; the passivity of delayed neural
networks is considered in [19], passivity of fuzzy time-delay
systems is investigated in [20] which adopt delay moom’s
inequality, and passive controller design for T-S fuzzy systems
is addressed in [21]. The passivity of uncertain fuzzy systems
is considered in [22]. However, the above methods still have
strong conservation, and it is necessary for us to further study.
In this paper, we adopt a delay partitioning approach to
study the passivity and passification of T-S fuzzy systems with
time delay. Based on this idea, we can further reduce the
2
Journal of Applied Mathematics
conservatism, and it becomes even less conservative when
the partitioning goes finer. The results of this paper are given
in terms of LMIs. The rest of the paper is organized as
follows. In Section 2, the problem to be studied is stated and
some preliminaries are presented. Passivity analysis results
are presented in Section 3. Based on the results obtained in
Section 3, we design the controller in Section 4. In Section 5,
numerical examples are given to demonstrate the effectiveness of the theoretical results. Finally, conclusions are drawn
in Section 6.
Notations. Throughout the paper, 𝐴−1 and 𝐴𝑇 denote the
inverse and transpose of a square matrix 𝐴. 𝑅𝑛 denotes
the 𝑛-dimensional Euclidean apace and ‖ ⋅ ‖ refers to the
Euclidean vector norm. The notation 𝐴 > 0 is used to define a
symmetric positive definite matrix and sym (𝐴) is defined as
𝐴+𝐴𝑇 . Matrices are assumed to have compatible dimensions.
with
𝑟
𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [−ℎ, 0] ,
𝑟
𝑟
(2)
where 𝑀𝑖𝑗 is the fuzzy set; 𝑟 is the number of IF-THEN
rules; 𝑧(𝑡) = [𝑧1 (𝑡), 𝑧2 (𝑡), . . . , 𝑧𝑝 (𝑡)] is the premise variables
vector; 𝐾𝑖 , 𝑖 = 1, . . . , 𝑟 are constant matrices representing
state-feedback control gains. Let 𝜆 𝑖 (𝑡) be the normalized
membership function of the fuzzy set
𝑝
𝜆 𝑖 (𝑡) =
,
𝑟
𝐶 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐶𝑖 ,
(5)
𝐶𝑑 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐶𝑑𝑖 ,
𝑖=1
𝑖=1
𝑟
𝐷 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐷𝑖 .
𝑖=1
By applying (2) into (4), we can get the following closed-loop
system:
𝑟
𝑥̇ (𝑡) = ∑ ∑ 𝜆 𝑖 (𝑧 (𝑡)) 𝜆 𝑗 (𝑧 (𝑡))
𝑖=1 𝑗=1
× [(𝐴 𝑖 + 𝐵𝑖 𝐾𝑗 ) 𝑥 (𝑡)
(6)
𝑟
(1)
where 𝑥(𝑡) ∈ 𝑅 is the state vector; 𝑢(𝑡) is the control input
vector; ℎ is a time delay; 𝜑(𝑡) is the initial condition.
Controller Rule 𝑖:
IF 𝑧1 (𝑡) is 𝑀𝑖1 and. . . and 𝑧𝑝 (𝑡) is 𝑀𝑖𝑝 THEN
∏𝑗=1 𝑀𝑖𝑗 (𝑧𝑗 (𝑡))
𝑖=1
+𝐴 𝑑𝑖 𝑥 (𝑡 − ℎ) + 𝐵1𝑖 𝜔 (𝑡) ] ,
𝑛
𝑖 = 1, . . . , 𝑟,
𝑟
𝐵1 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐵1𝑖 ,
𝑖=1
𝑟
𝑖 = 1, . . . , 𝑟,
𝑢 (𝑡) = 𝐾𝑖 𝑥 (𝑡) ,
𝑖=1
𝐵 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐵𝑖 ,
𝑥̇ (𝑡) = 𝐴 𝑖 𝑥 (𝑡) + 𝐴 𝑑𝑖 𝑥 (𝑡 − ℎ) + 𝐵𝑖 𝑢 (𝑡) + 𝐵1𝑖 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶𝑖 𝑥 (𝑡) + 𝐶𝑑𝑖 𝑥 (𝑡 − ℎ) + 𝐷𝑖 𝜔 (𝑡) ,
𝐴 𝑑 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐴 𝑑𝑖 ,
𝑖=1
2. Problem Statement and Preliminaries
Consider the T-S fuzzy system with time delay has the
following form.
Plant Rule 𝑖:
IF 𝑍1 (𝑡) is 𝑀𝑖1 and . . . and 𝑍𝑝 (𝑡) is 𝑀𝑖𝑝 THEN
𝑟
𝐴 (𝑡) = ∑𝜆 𝑖 (𝑡) 𝐴 𝑖 ,
𝑦 (𝑡) = ∑𝜆 𝑖 (𝑧 (𝑡)) [𝐶𝑖 𝑥 (𝑡) + 𝐶𝑑𝑖 𝑥 (𝑡 − ℎ) + 𝐷𝑖 𝜔 (𝑡)] .
𝑖=1
Before formulating the main problem, we first give the
following definition.
Definition 1 (Li et al. [20]). The fuzzy system (1) is called
passive if there exists a scalar 𝛾 ≥ 0 such that
𝑇
𝑇
0
0
2 ∫ 𝜔(𝑠)𝑇 𝑦 (𝑠) 𝑑𝑠 ≥ −𝛾 ∫ 𝜔𝑇 (𝑠) 𝜔 (𝑠) 𝑑𝑠
(7)
for all 𝑇 ≥ 0.
3. Passivity Analysis
(3)
The problems to be addressed in this paper can be expressed
as follows.
where 𝑀𝑖𝑗 (𝑧𝑗 (𝑡)) is the grade of membership function of 𝑧𝑗 (𝑡)
𝑝
in 𝑀𝑖𝑗 (𝑡). It is assumed that ∏𝑗=1 𝑀𝑖𝑗 (𝑧𝑗 (𝑡)) ≥ 0, 𝑖 = 1, . . . , 𝑟,
𝑝
and ∑𝑟𝑖=1 {∏𝑗=1 𝑀𝑖𝑗 (𝑧𝑗 (𝑡))} > 0 for all 𝑡. Therefore, 𝜆 𝑖 (𝑡) ≥ 0
and ∑𝑟𝑖=1 𝜆 𝑖 (𝑡) = 1 for all 𝑡. By applying (3) into (1), the fuzzy
system can be expressed as
Problem 1 (passivity analysis). Given the feedback controller
gain matrices 𝐾𝑖 , 𝑖 = 1, . . . , 𝑟 in (2), determine under what
conditions the closed-loop system (6) is passive for all 𝑇 ≥ 0
in the sense of Definition 1.
𝑝
∑𝑟𝑖=1 {∏𝑗=1 𝑀𝑖𝑗 (𝑧𝑗 (𝑡))}
𝑥̇ (𝑡) = 𝐴 (𝑡) 𝑥 (𝑡) + 𝐴 𝑑 (𝑡) 𝑥 (𝑡 − ℎ)
+ 𝐵 (𝑡) 𝑢 (𝑡) + 𝐵1 (𝑡) 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶 (𝑡) 𝑥 (𝑡) + 𝐶𝑑 (𝑡) 𝑥 (𝑡 − ℎ) + 𝐷 (𝑡) 𝜔 (𝑡)
(4)
Problem 2 (passification). Determine the feedback controller
gain matrices, 𝐾𝑖 , 𝑖 = 1, . . . , 𝑟 in (2), such that the closed-loop
system (6) is passive for all 𝑇 ≥ 0 in the sense of Definition 1.
In this section, we will present a sufficient condition in
terms of LMIs, under which the closed-loop system (6) is
passive.
Journal of Applied Mathematics
3
Theorem 2. Given matrices 𝐾𝑖 , an integer 𝑚 ≥ 1 and a scalar
ℎ > 0, if there exist symmetric positive definite matrices 𝑃, 𝑄𝑖 ,
𝑍𝑖 , 𝑅𝑖 and matrices 𝑆1𝑖 , 𝑆2𝑖 and scalar 𝛾 > 0, satisfying
[
[
[
Θ𝑖𝑖𝑙𝑘 + 𝜎𝑊𝜎𝑇 𝑊𝜎
∗
[
[
𝑉1 (𝑡) = 𝑥𝑇 (𝑡) 𝑃𝑥 (𝑡) ,
𝑆1𝑖
]
𝑚 ] < 0,
− 𝑍𝑖
ℎ ]
Θ𝑖𝑗𝑙𝑘 + 𝜎𝑊𝜎𝑇 𝑊𝜎
∗
[
[
+[
𝑍𝑖 < 𝑅𝑗 ,
∫
−ℎ/𝑚 𝑡+𝛽
𝑉3 (𝑡) = ∫
]
𝑚 ]
− 𝑍𝑖
ℎ ]
1 ≤ 𝑟 < 𝑗 ≤ 𝑟,
𝑡
𝑉2 (𝑡) = ∫
(8)
𝑆1𝑖
𝑡
𝑡−ℎ/𝑚
𝑥̇𝑇 (𝛼) 𝑅 (𝛼) 𝑥̇ (𝛼) 𝑑𝛼 𝑑𝛽,
(12)
𝛾𝑇 (𝛼) 𝑄 (𝛼) 𝛾 (𝛼) 𝑑𝛼,
where
𝑆1𝑗
(9)
]
𝑚 ]<0
− 𝑍𝑗
ℎ ]
∗
[
0
𝑖, 𝑙, 𝑘 = 1, . . . , 𝑟
𝜎𝑊𝜎𝑇 𝑊𝜎
Θ𝑗𝑖𝑙𝑘 +
Proof. Choose a Lyapunov-Krasovskii functional as 𝑉(𝑡) =
𝑉1 (𝑡) + 𝑉2 (𝑡) + 𝑉3 (𝑡) with
𝛾 (𝑡) = [𝑥𝑇 (𝑡)
𝑥𝑇 (𝑡 −
ℎ
)
𝑚
⋅⋅⋅
𝑥𝑇 (𝑡 −
𝑙, 𝑘 = 1, . . . , 𝑟,
𝑖, 𝑗 = 1, . . . , 𝑟,
(10)
𝑇
𝑚−1
ℎ)] .
𝑚
(13)
The time-derivative of 𝑉(𝑡) along the trajectory of the system
in (6) is given by
where
𝑉1̇ = 𝑥̇𝑇 (𝑡) 𝑃𝑥 (𝑡) + 𝑥𝑇 (𝑡) 𝑃𝑥̇ (𝑡) ,
∧
Ω𝑖𝑗𝑙𝑘 = 𝑊𝑝𝑇 𝑃 𝑊𝑝 +
ℎ
𝑊𝑟𝑇 𝑅𝑖 𝑊𝑟 + 𝑊𝑞1𝑇 𝑄𝑖 𝑊𝑞1
𝑚
𝑡
ℎ
𝑉2̇ = 𝑥̇𝑇 (𝑡) 𝑅 (𝑡) 𝑥̇ (𝑡) − ∫
𝑥̇𝑇 (𝛼) 𝑅 (𝛼) 𝑥̇ (𝛼) 𝑑𝛼
𝑚
𝑡−ℎ/𝑚
− 𝑊𝑞2𝑇 𝑄𝑙 𝑊𝑞2 + sym (𝑆𝑖 𝑊𝑠𝑗𝑘 ) ,
ℎ
ℎ
ℎ
𝑉3̇ = 𝛾𝑇 (𝑡) 𝑄 (𝑡) 𝛾 (𝑡) − 𝛾𝑇 (𝑡 − ) 𝑄 (𝑡 − ) 𝛾 (𝑡 − ) .
𝑚
𝑚
𝑚
(14)
Θ𝑖𝑗𝑙𝑘 = Ω𝑖𝑗𝑙𝑘 − (𝑊1 𝐶𝑗𝑇 𝑊2 + 𝑊2𝑇 𝐶𝑗 𝑊1𝑇
+ 𝑊2𝑇 𝐷𝑗𝑇 𝑊2 + 𝑊2𝑇 𝐷𝑗 𝑊2
+ 𝑊2𝑇 𝐶𝑑𝑗 𝑊3
+
𝑇
𝑊3𝑇 𝐶𝑑𝑗
𝑊2
+
𝛾𝑊2𝑇 𝑊2 ) ,
𝜉 (𝑡) = [𝛾𝑇 (𝑡)
𝐼𝑛 𝑂𝑛,𝑚𝑛 𝑂𝑛,𝑛 𝑂𝑛,𝑛
𝑊𝑝 = (𝑂𝑛,𝑛 𝑂𝑛,𝑚𝑛 𝐼𝑛 𝑂𝑛,𝑛 ) ,
𝑂𝑛,𝑛 𝑂𝑛,𝑚𝑛 𝑂𝑛,𝑛 𝑂𝑛.𝑛
𝑊𝜎 = (𝐼𝑛 , 𝑂𝑛,𝑚𝑛+2𝑛 ) ,
𝑊𝑟 = (𝑂𝑛,𝑚𝑛+𝑛 , 𝐼𝑛 , 𝑂𝑛,𝑛 ) ,
𝑊𝑞1 = (𝐼𝑚𝑛 , 𝑂𝑚𝑛,3𝑛 ) ,
𝑆𝑖 = [𝑆1𝑖
⋅⋅⋅
𝑥̇𝑇 (𝑡)
𝜔𝑛 (𝑡)]
𝑇
𝜔𝑇 (𝑡)] ,
𝑇
(15)
and according to the Newton-Leibniz formula and the system
in (6), we have
Π1 = 2𝜉𝑇 (𝑡) 𝑆1 (𝑡)
𝑆2𝑖 ] ,
× [𝑥 (𝑡) − 𝑥 (𝑡 −
𝐼
−𝐼𝑛
𝑂𝑛,𝑚𝑛+𝑛
𝑊𝑆𝑗𝑘 = [𝐴 + 𝑛𝐵 𝐾 𝑂
𝐵1𝑗 ] ,
𝑗
𝑗 𝑘
𝑛,𝑚𝑛−𝑛 𝐴 𝑑𝑗 −𝐼𝑛
𝐼𝑛
𝑊1 = (
),
𝑂𝑚𝑛+2𝑛,𝑛
𝑥𝑇 (𝑡 − ℎ)
𝜔 (𝑡) = [𝜔1 (𝑡)
𝑂𝑛,𝑛 𝑃 𝑂𝑛,𝑛
∧
𝑃 = ( 𝑃 𝑂𝑛,𝑛 𝑂𝑛,𝑛 ) ,
𝑂𝑛,𝑛 𝑂𝑛,𝑛 𝑂𝑛,𝑛
𝑊𝑞2 = (𝑂𝑚𝑛,𝑛 , 𝐼𝑚𝑛 , 𝑂𝑚𝑛,2𝑛 ) ,
Define
𝑡
ℎ
)−∫
𝑥̇ (𝛼) 𝑑𝛼] = 0,
𝑚
𝑡−ℎ/𝑚
Π2 = 2𝜉𝑇 (𝑡) 𝑆2 (𝑡) [ (𝐴 (𝑡) + 𝐵 (𝑡) 𝐾 (𝑡)) 𝑥 (𝑡)
𝑊2 = (𝑂𝑛,𝑚𝑛+2𝑛 , 𝐼𝑛 )
+ 𝐴 𝑑 (𝑡) 𝑥 (𝑡 − ℎ)
+𝐵1 (𝑡) 𝜔 (𝑡) − 𝑥̇ (𝑡)] = 0,
𝑊3 = (𝑂𝑛,𝑚𝑛 , 𝐼𝑛 , 𝑂𝑛,2𝑛 ) .
(11)
Then the fuzzy system (1) is passive in the sense of Definition 1
for the time delay 0 ≤ 𝜏 ≤ ℎ.
Π3 =
ℎ 𝑇
𝜉 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡)
𝑚
𝑡
−∫
𝑡−ℎ/𝑚
𝜉𝑇 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡) 𝑑𝛼 = 0.
(16)
4
Journal of Applied Mathematics
Therefore
𝑉̇ (𝑡) ≤ 𝑉1̇ (𝑡) + 𝑉2̇ (𝑡) + 𝑉3̇ (𝑡) + Π1 + Π2 + Π3
ℎ 𝑇
𝑥̇ (𝑡) 𝑅 (𝑡) 𝑥̇ (𝑡)
𝑚
Λ (𝑡) = 𝑥̇𝑇 (𝑡) 𝑃𝑥 (𝑡) + 𝑥𝑇 (𝑡) 𝑃𝑥̇ (𝑡) +
𝑥̇𝑇 (𝛼) 𝑅 (𝛼) 𝑥̇ (𝛼) 𝑑𝛼 + 𝛾𝑇 (𝑡) 𝑄 (𝑡) 𝛾 (𝑡)
+ 𝛾𝑇 (𝑡) 𝑄 (𝑡) 𝛾 (𝑡) − 𝛾𝑇 (𝑡 −
= 𝑥̇𝑇 (𝑡) 𝑃𝑥 (𝑡) + 𝑥𝑇 (𝑡) 𝑃𝑥̇ (𝑡) +
−∫
𝑡
𝑡−ℎ/𝑚
where
− 𝛾𝑇 (𝑡 −
ℎ
ℎ
ℎ
) 𝑄 (𝑡 − ) 𝛾 (𝑡 − ) + 2𝜉𝑇 (𝑡) 𝑆1 (𝑡)
𝑚
𝑚
𝑚
× [𝑥 (𝑡) − 𝑥 (𝑡 −
× 𝛾 (𝑡 −
𝑡
ℎ
𝑥̇ (𝛼) 𝑑𝛼]
)−∫
𝑚
𝑡−ℎ/𝑚
ℎ 𝑇
𝑥̇ (𝑡) 𝑅 (𝑡) 𝑥̇ (𝑡)
𝑚
ℎ
ℎ
) 𝑄 (𝑡 − )
𝑚
𝑚
ℎ
ℎ
) + 2𝜉𝑇 (𝑡) 𝑆1 (𝑡) [𝑥 (𝑡) − 𝑥 (𝑡 − )]
𝑚
𝑚
+ 2𝜉𝑇 (𝑡) 𝑆2 (𝑡)
× [ (𝐴 (𝑡) + 𝐵 (𝑡) 𝐾 (𝑡)) 𝑥 (𝑡)
+ 2𝜉𝑇 (𝑡) 𝑆2 (𝑡)
+ 𝐴 𝑑 (𝑡) 𝑥 (𝑡 − ℎ) + 𝐵1 (𝑡) 𝜔 (𝑡) − 𝑥̇ (𝑡) ] .
× [(𝐴 (𝑡) + 𝐵 (𝑡) 𝐾 (𝑡)) 𝑥 (𝑡) + 𝐴 𝑑 (𝑡) 𝑥 (𝑡 − ℎ)
(22)
+ 𝐵1 (𝑡) 𝜔 (𝑡) − 𝑥̇ (𝑡)]
+
ℎ 𝑇
𝜉 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡)
𝑚
−∫
𝑡
𝑡−ℎ/𝑚
= Λ (𝑡) +
−∫
𝜉𝑇 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡) 𝑑𝛼
𝑥̇ (𝑡) = (𝑂𝑛,𝑚𝑛+𝑛 , 𝐼𝑛 , 𝑂𝑛,𝑛 ) 𝜉 (𝑡) ,
𝑥 (𝑡) = (𝐼𝑛 , 𝑂𝑛,𝑚𝑛+2𝑛 ) 𝜉 (𝑡) ,
ℎ 𝑇
𝜉 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡)
𝑚
𝑡
𝑡−ℎ/𝑚
𝑡
𝑡−ℎ/𝑚
𝑡
𝑡−ℎ/𝑚
𝛾 (𝑡 −
Then
(17)
If
𝑍 (𝑡) < 𝑅 (𝛼) .
(18)
Then
−1
𝑍 (𝑡) > 𝑅 (𝛼) ,
𝑡
−∫
𝑡−ℎ/𝑚
(19)
𝜉𝑇 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡) 𝑑𝛼
< −∫
(20)
𝑡
𝑡−ℎ/𝑚
𝜉𝑇 (𝑡) 𝑆1 (𝑡) 𝑅−1 (𝛼) 𝑆1𝑇 (𝑡) 𝜉 (𝑡) 𝑑𝛼.
ℎ
𝑉̇ (𝑡) ≤ Λ (𝑡) + 𝜉𝑇 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡)
𝑚
𝑡
𝑡−ℎ/𝑚
∧
Ω (𝑡) = 𝑊𝑝𝑇 𝑃 𝑊𝑝 +
ℎ 𝑇
𝑇
𝑄 (𝑡) 𝑊𝑞1
𝑊 𝑅 (𝑡) 𝑊𝑟 + 𝑊𝑞1
𝑚 𝑟
𝑇
− 𝑊𝑞2
𝑄 (𝑡 −
ℎ
) 𝑊𝑞2 + sym (𝑆 (𝑡) 𝑊𝑆 (𝑡)) ,
𝑚
𝑂𝑛,𝑛 𝑃 𝑂𝑛,𝑛
𝑃̂ = ( 𝑃 𝑂𝑛,𝑛 𝑂𝑛,𝑛 ) ,
𝑂𝑛,𝑛 𝑂𝑛,𝑛 𝑂𝑛,𝑛
𝑊𝑟 = (𝑂𝑛,𝑚𝑛+𝑛 , 𝐼𝑛 , 𝑂𝑛,𝑛 ) ,
(𝑥̇𝑇 (𝛼) 𝑅 (𝛼) + 𝜉𝑇 (𝑡) 𝑆1 (𝑡)) × 𝑅−1 (𝛼)
× (𝑅 (𝛼) 𝑥̇ (𝛼) +
Λ (𝑡) = 𝜉𝑇 (𝑡) Ω (𝑡) 𝜉 (𝑡) ,
𝑆 (𝑡) = [𝑆1 (𝑡)
𝑆2 (𝑡)]
𝐼𝑛 𝑂𝑛,𝑚𝑛 𝑂𝑛,𝑛 𝑂𝑛,𝑛
𝑊𝑝 = (𝑂𝑛,𝑛 𝑂𝑛,𝑚𝑛 𝐼𝑛 𝑂𝑛,𝑛 ) ,
𝑂𝑛,𝑛 𝑂𝑛,𝑚𝑛 𝑂𝑛,𝑛 𝑂𝑛.𝑛
So, we can obtain
−∫
ℎ
) = (𝑂𝑚𝑛,𝑛 , 𝐼𝑚𝑛 , 𝑂𝑚𝑛,2𝑛 ) 𝜉 (𝑡) .
𝑚
𝑥̇ (𝛼) 𝑑𝛼
𝜉𝑇 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡) 𝑑𝛼.
−1
(23)
𝛾 (𝑡) = (𝐼𝑚𝑛 , 𝑂𝑚𝑛,3𝑛 ) 𝜉 (𝑡) ,
𝑥̇𝑇 (𝛼) 𝑅 (𝛼) 𝑥̇ (𝛼) 𝑑𝛼
− 2𝜉𝑇 (𝑡) 𝑆1 (𝑡) ∫
−∫
Besides
𝑆1𝑇 (𝑡) 𝜉 (𝑡)) 𝑑𝛼,
(21)
𝑊𝑞1 = (𝐼𝑚𝑛 , 𝑂𝑚𝑛,3𝑛 ) ,
𝑊𝑞2 = (𝑂𝑚𝑛,𝑛 , 𝐼𝑚𝑛 , 𝑂𝑚𝑛,2𝑛 ) ,
−𝐼𝑛
𝑂𝑛,𝑚𝑛+𝑛
𝐼𝑛
].
𝑊𝑆 (𝑡) = [
−𝐼
𝐴
𝐵
𝐾
𝑂
𝐴
𝐵
𝑛
1 (𝑡) ]
[ (𝑡)+ (𝑡) (𝑡) 𝑛,𝑚𝑛−𝑛 𝑑 (𝑡)
(24)
Journal of Applied Mathematics
5
Besides
We can obtain
2𝜔𝑇 (𝑡) 𝑦 (𝑡) + 𝛾𝜔𝑇 (𝑡) 𝜔 (𝑡)
𝑉̇ (𝑡) < 2𝑦𝑇 (𝑡) 𝜔 (𝑡) + 𝛾𝜔𝑇 (𝑡) 𝜔 (𝑡) .
= 𝑥𝑇 (𝑡) 𝐶𝑖𝑇 𝜔 (𝑡) + 𝜔𝑇 (𝑡) 𝐶𝑖 𝑥 (𝑡)
(31)
+ 𝜔𝑇 (𝑡) (𝐷𝑖𝑇 + 𝐷𝑖 ) 𝜔 (𝑡) + 𝜔𝑇 (𝑡) 𝐶𝑑𝑖 𝑥 (𝑡 − ℎ)
𝑇
+ 𝑥𝑇 (𝑡 − ℎ) 𝐶𝑑𝑖
𝜔 (𝑡) + 𝛾𝜔𝑇 (𝑡) 𝜔 (𝑡)
= 𝜉𝑇 (𝑡) [𝑊1 𝐶𝑖𝑇 𝑊2 + 𝑊2𝑇 𝐶𝑖 𝑊1𝑇 + 𝑊2𝑇 (𝐷𝑖𝑇 + 𝐷𝑖 ) 𝑊2
𝑇
+ 𝑊2𝑇 𝐶𝑑𝑖 𝑊3 + 𝑊3𝑇 𝐶𝑑𝑖
𝑊2 + 𝛾𝑊2𝑇 𝑊2 ] 𝜉 (𝑡) ,
(25)
where
𝐼𝑛
),
𝑊1 = (
𝑂𝑚𝑛+2𝑛,𝑛
𝑊2 = (𝑂𝑛,𝑚𝑛+2𝑛 , 𝐼𝑛 ) ,
It follows by integrating (31) with respect to 𝑡 over the time
period 0 ∼ 𝑇 that (7) holds, and hence the delayed fuzzy
system (1) is passive in the sense of Definition 1. Our next
objective is to convert the inequalities in (18) and (29) to some
finite LMIs, then (29) can be rewritten as
𝜂𝑖𝑗𝑙𝑘 = 𝜂𝑖𝑖𝑙𝑘 + (𝜂𝑖𝑗𝑙𝑘(𝑖<𝑗) + 𝜂𝑗𝑖𝑙𝑘(𝑖<𝑗) ) < 0,
𝑟
𝑟
𝑖=1
𝑗=1
𝜂𝑖𝑗𝑙𝑘 = ∑𝜆 𝑖 (𝜃 (𝑡)) ∑ 𝜆 𝑗 (𝜃 (𝑡))
(26)
𝑟
𝑊3 = (𝑂𝑛,𝑚𝑛 , 𝐼𝑛 , 𝑂𝑛,2𝑛 ) .
× ∑𝜆 𝑙 (𝜃 (𝑡 −
𝑙=1
So, we have
𝑉̇ (𝑡) − 2𝜔𝑇 (𝑡) 𝑦 (𝑡) − 𝜆𝜔𝑇 (𝑡) 𝜔 (𝑡)
≤ 𝜉𝑇 (𝑡) Θ (𝑡) 𝜉 (𝑡) +
−∫
𝑡
𝑡−ℎ/𝑚
𝑟
ℎ
)) ∑ 𝜆 𝑘 (𝜃 (𝑡))
𝑚
𝑘=1
× [Ω𝑖𝑗𝑙𝑘 − (𝑊1 𝐶𝑗𝑇 𝑊2 + 𝑊2𝑇 𝐶𝑗 𝑊1𝑇
ℎ 𝑇
𝜉 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡)
𝑚
+ 𝑊2𝑇 (𝐷𝑗 + 𝐷𝑗𝑇 ) 𝑊2
(𝑥̇𝑇 (𝛼) 𝑅 (𝛼) + 𝜉𝑇 (𝑡) 𝑆1 (𝑡)) × 𝑅−1 (𝛼)
𝑇
+ 𝑊2𝑇 𝐶𝑑𝑗 𝑊3 + 𝑊3𝑇 𝐶𝑑𝑗
𝑊2 + 𝛾𝑊2𝑇𝑊2 )
+ 𝜎𝑊𝜎𝑇 𝑊𝜎
× (𝑅 (𝛼) 𝑥̇ (𝛼) + 𝑆1𝑇 (𝑡) 𝜉 (𝑡)) 𝑑𝛼
(27)
+
with
ℎ
𝑆 𝑍 −1 𝑆𝑇 ] .
𝑚 1𝑖 𝑖 1𝑖
Θ (𝑡) = Ω (𝑡) − (𝑊1 𝐶𝑖𝑇 𝑊2 + 𝑊2𝑇 𝐶𝑖 𝑊1𝑇
+ 𝑊2𝑇 (𝐷𝑖 + 𝐷𝑖𝑇 ) 𝑊2
+ 𝑊2𝑇 𝐶𝑑𝑖 𝑊3
+
𝑇
𝑊3𝑇 𝐶𝑑𝑖
𝑊2
Equation (18) can be rewritten as
+
𝛾𝑊2𝑇 𝑊2 ) .
(28)
𝑍𝑖 < 𝑅𝑗 .
If
Θ (𝑡) + 𝜎𝑊𝜎𝑇 𝑊𝜎 +
(32)
ℎ
𝑆 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) < 0
𝑚 1
(29)
(33)
From the Schur complement, we can get the inequalities (8),
(9), and (10), and the proof is completed.
then
Θ (𝑡) +
ℎ
𝑆 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) < −𝜎𝑊𝜎𝑇 𝑊𝜎 ,
𝑚 1
4. Controller Design
𝑉̇ (𝑡) − 2𝜔𝑇 (𝑡) 𝑦 (𝑡) − 𝛾𝜔𝑇 (𝑡) 𝜔 (𝑡)
< 𝜉𝑇 (𝑡) Θ (𝑡) 𝜉 (𝑡) +
ℎ 𝑇
𝜉 (𝑡) 𝑆1 (𝑡) 𝑍−1 (𝑡) 𝑆1𝑇 (𝑡) 𝜉 (𝑡)
𝑚
< −𝜉𝑇 (𝑡) 𝜎𝑊𝜎𝑇 𝑊𝜎 𝜉 (𝑡) < −𝜎‖𝑥 (𝑡)‖2 < 0.
(30)
In this section, fuzzy state feedback controllers will be
designed based on the result developed in the previous
section.
Theorem 3. Given an integer 𝑚 > 1 and scalars ℎ > 0, 𝜏1 ,
𝜏2 , . . . , 𝜏(𝑚+3) , there exists a fuzzy state feedback controller such
that the closed-loop system (4) is passive if there exist symmetric
6
Journal of Applied Mathematics
positive definite matrices 𝑃, 𝑄𝑖 , 𝑍𝑖 , 𝑅𝑖 , and matrices 𝑋, 𝑆1𝑖 , 𝑀𝑖
and scalars 𝛾 > 0, 𝜎 > 0, satisfying
𝑆1𝑖
𝜑𝑖𝑖𝑙
𝑊𝜎𝑇 𝑋𝑇
𝑊2𝑇 𝑋𝑇
𝑊2𝑇 𝑋𝑇
[
[
[ ∗ − 𝑚 𝑍𝑖
0
0
0
[
ℎ
[
[
[∗
∗
−𝜎−1 𝐼𝑛
0
0
[
[
[
[∗
∗
∗
𝐷𝑖−𝑇 + 𝐷𝑖−1
0
[
[
[
∗
∗
∗
∗
𝛾−1 𝐼𝑛
[
If the above conditions are feasible, the gains of the controller
are given by
𝐾𝑖 = 𝑀𝑖 𝑋−1 ,
]
]
]
]
]
]
]
]<0
]
]
]
]
]
]
(34)
𝐺1 = diag {𝑆, . . . , 𝑆, 𝐼𝑛 , 𝑆, 𝑆, 𝑆, 𝑆} ∈ 𝑅(𝑚𝑛+7𝑛)×(𝑚𝑛+7𝑛) ,
𝐺2 = diag {𝑆, . . . , 𝑆, 𝐼𝑛 } ∈ 𝑅(𝑚𝑛+3𝑛)×(𝑚𝑛+3𝑛) ,
]
[
[
[∗
[
[
[
[
+[∗
[
[
[∗
[
[
[
∗
[
−
Premultiplying and postmultiplying (34) with 𝐺1 and 𝐺1𝑇 ,
then we obtain
𝑊𝜎𝑇 𝑋𝑇
𝑊2𝑇 𝑋𝑇
𝑊2𝑇 𝑋𝑇
0
0
0
∗
−𝜎−1 𝐼𝑛
0
0
∗
∗
𝐷𝑗−𝑇 + 𝐷𝑗−1
0
∗
∗
∗
𝛾−1 𝐼𝑛
𝑚
𝑍
ℎ 𝑗
1 ≤ 𝑖 < 𝑗 ≤ 𝑟,
]
]
]
]
]
]
]
]<0
]
]
]
]
]
]
Δ 𝐺2 𝑆1𝑖 𝑆𝑇 𝐺2 𝑊𝜎𝑇 𝑋𝑇 𝑆𝑇
𝐺2 𝑊2𝑇 𝑋𝑇 𝑆𝑇
𝐺2 𝑊2𝑇 𝑋𝑇 𝑆𝑇
[
]
[
]
𝑚
[
]
𝑇
[∗ − 𝑆𝑍𝑖 𝑆
]
0
0
0
[
]
ℎ
[
]
[
]
−1 𝑇
[∗
]
∗
−𝜎
𝑆𝑆
0
0
[
] < 0,
[
]
[
]
[
]
−𝑇
−1
𝑇
∗
∗
𝑆 (𝐷𝑖 + 𝐷𝑖 ) 𝑆
0
[∗
]
[
]
[
]
[
]
−1
𝑇
[∗
]
∗
∗
∗
𝛾 𝑆𝑆
[
]
(40)
where
Δ = 𝐺2 𝜙𝑖𝑖𝑙 𝐺2𝑇 ,
]
𝐺2 𝜙𝑖𝑖𝑙 𝐺2𝑇
𝑙 = 1, . . . , 𝑟,
=
𝑂𝑛,𝑛
𝑊𝑝𝑇 [𝑆𝑃𝑆𝑇
[ 𝑂𝑛,𝑛
(35)
𝑍𝑖 < 𝑅𝑗 ,
(39)
𝐺3 = diag {𝑆, . . . , 𝑆} ∈ 𝑅𝑚𝑛×𝑚𝑛 .
𝜙𝑖𝑗𝑙 𝑆1𝑖 𝑊𝜎𝑇 𝑋𝑇 𝑊2𝑇 𝑋𝑇 𝑊2𝑇 𝑋𝑇
]
[
]
[
[ ∗ −𝑚𝑍
0
0
0 ]
]
[
𝑖
]
[
ℎ
]
[
−1
[∗
∗
−𝜎 𝐼𝑛
0
0 ]
]
[
]
[
]
[
−𝑇
−1
]
[∗
∗
∗
𝐷
+
𝐷
0
]
[
𝑖
𝑖
]
[
−1
∗
∗
∗
∗
𝛾
𝐼
𝑛 ]
[
𝑆1𝑗
(38)
Proof. Assume that 𝑋 is invertible, define 𝑆 = 𝑋−𝑇 , and
𝑖, 𝑙 = 1, . . . , 𝑟,
𝜙𝑗𝑖𝑙
𝑖 = 1, . . . , 𝑟.
𝑖, 𝑗 = 1, . . . , 𝑟,
(36)
+
𝑆𝑃𝑆𝑇 𝑂𝑛,𝑛
𝑂𝑛,𝑛 𝑂𝑛,𝑛 ] 𝑊𝑝
𝑂𝑛,𝑛 𝑂𝑛,𝑛 ]
ℎ 𝑇
𝑊 𝑆𝑅 𝑆𝑇 𝑊𝑟
𝑚 𝑟 𝑖
(41)
𝑇
𝑇
+ 𝑊𝑞1
𝐺3 𝑄𝑖 𝐺3𝑇 𝑊𝑞1 − 𝑊𝑞2
𝐺3 𝑄𝑙 𝐺3𝑇 𝑊𝑞2
where
𝜑𝑖𝑗𝑙 =
∧
𝑊𝑝𝑇 𝑃
+ sym (𝑉𝑖 𝑊𝑆𝑖𝑗 −
𝑊1 𝑋𝑇 𝐶𝑖𝑇 𝑊2
𝑂𝑛,𝑛 𝑃 𝑂𝑛,𝑛
𝑃̂ = ( 𝑃 𝑂𝑛,𝑛 𝑂𝑛,𝑛 ) ,
𝑂𝑛,𝑛 𝑂𝑛,𝑛 𝑂𝑛,𝑛
𝑈 = [𝜏1 𝐼𝑛
𝑊𝑆𝑖𝑗 = [
+ sym (𝐺2 𝑉𝑖 𝑊𝑆𝑖𝑖 𝐺2𝑇 ) − 𝑊1 𝐶𝑖𝑇 𝑊2
ℎ
𝑇
𝑇
𝑊𝑝 + 𝑊𝑟𝑇 𝑅𝑖 𝑊𝑟 + 𝑊𝑞1
𝑄𝑖 𝑊𝑞1 − 𝑊𝑞2
𝑄𝑙 𝑊𝑞2
𝑚
𝜏2 𝐼𝑛
𝐼𝑛
𝐴 𝑖 𝑋 + 𝐵𝑖 𝑀𝑗
⋅⋅⋅
− 𝐼𝑛
𝑂𝑛,𝑚𝑛−𝑛
−
𝑊2𝑇 𝐶𝑑𝑖 𝑋𝑊3 ) ,
𝑉𝑖 = [𝑆1𝑖
𝐺2 𝑊𝜎𝑇 𝑋𝑇 𝑆𝑇 = 𝑊𝜎𝑇 𝑆𝑇 ,
𝑈] ,
𝑇
−𝑋
𝑃 = 𝑆𝑃𝑆𝑇 ,
𝐵1𝑖
𝐺2 𝑊2𝑇 𝑋𝑇 𝑆𝑇 = 𝑊2𝑇 𝑆𝑇 .
We defining
𝜏(𝑚+3) 𝐼𝑛 ] ,
𝑂𝑛,𝑚𝑛+𝑛
𝐴 𝑑𝑖 𝑋
𝑇
− 𝑊2𝑇 𝐶𝑖 𝑊1𝑇 − 𝑊3𝑇 𝐶𝑑𝑖
𝑊2 − 𝑊2𝑇 𝐶𝑑𝑖 𝑊3 ,
].
(37)
𝑅𝑖 = 𝑆𝑅𝑖 𝑆𝑇 ,
𝑆1𝑖 = 𝐺2 𝑆1𝑖 𝑆𝑇 ,
𝑆2𝑖 = [𝜏1 𝑆𝑇
𝜏2 𝑆𝑇
⋅⋅⋅
𝑄𝑖 = 𝐺3 𝑄𝑖 𝐺3𝑇 ,
𝑍𝑖 = 𝑆𝑍𝑖 𝑆𝑇 ,
𝜏(𝑚+2) 𝑆𝑇
(42)
𝑇
𝜏(𝑚+3) 𝐼𝑛 ] .
Journal of Applied Mathematics
7
10
Then
9.5
𝐺2 𝑉𝑖 𝑊𝑆𝑖𝑖 𝐺2𝑇
9
− 𝐼𝑛
+ 𝐺2 𝑈 [𝐴 𝑖 𝑋 + 𝐵𝑖 𝑀𝑖
= 𝐺2 𝑆1𝑖 𝑆𝑇 [𝐼𝑛
− 𝐼𝑛
+ 𝑆2𝑖 [𝐴 𝑖 + 𝐵𝑖 𝐾𝑘
= [𝑆1𝑖
𝑆2𝑖 ] [
𝑂𝑛,𝑚𝑛−𝑛
8.5
𝐴 𝑑𝑖 𝑋
𝐵1𝑖 ] 𝐺2𝑇
−𝑋
𝑏
= 𝐺2 𝑆1𝑖 [𝐼𝑛
𝑂𝑛,𝑚𝑛+𝑛 ] 𝐺2𝑇
7.5
𝑂𝑛,𝑚𝑛+𝑛 ]
𝑂𝑛,𝑚𝑛−𝑛
𝐼𝑛
𝐴 𝑖 + 𝐵𝑖 𝐾𝑘
𝐴 𝑑𝑖
− 𝐼𝑛
𝑂𝑛,𝑚𝑛−𝑛
8
− 𝐼𝑛
7
𝐵1𝑖 ]
𝑂𝑛,𝑚𝑛+𝑛
𝐴 𝑑𝑖 − 𝐼𝑛
𝐵1𝑖
6.5
]
6
−2
(43)
−1
0
1
2
3
𝑎
Figure 1: The feasible region based on Theorem 2.
then
10
𝑂𝑛,𝑛 𝑃 𝑂𝑛,𝑛
ℎ
𝐺2 𝜑𝑖𝑖𝑙 𝐺2𝑇 = 𝑊𝑝𝑇 [ 𝑃 𝑂𝑛,𝑛 𝑂𝑛,𝑛 ] 𝑊𝑝 + 𝑊𝑟𝑇 𝑅𝑖 𝑊𝑟
𝑚
[𝑂𝑛,𝑛 𝑂𝑛,𝑛 𝑂𝑛,𝑛 ]
+
𝑇
𝑊𝑞1
𝑄𝑖 𝑊𝑞1
−
𝑇
𝑊𝑞2
𝑄𝑙 𝑊𝑞2
+
9.5
9
sym (𝐺2 𝑉𝑖 𝑊𝑆𝑖𝑖 𝐺2𝑇 )
𝑇
− 𝑊3𝑇 𝐶𝑑𝑖
𝑊2 − 𝑊2𝑇 𝐶𝑑𝑖 𝑊3 .
𝑏
− 𝑊1 𝐶𝑖𝑇 𝑊2 − 𝑊2𝑇 𝐶𝑖 𝑊1𝑇
8.5
8
7.5
(44)
7
6.5
Thus, by the Schur complement, we can obtain (40) is
equivalent to (8). Pre- and postmultiplying (35) with 𝐺1 and
𝐺1𝑇 , we obtain (9), pre- and postmultiplying (36) with 𝑆 and
𝑆𝑇 , we obtain (10). The proof is completed.
6
−2
−1
0
1
2
3
𝑎
Figure 2: The feasible region based on [22].
5. Numerical Example
Without delay and uncertainty, Example 1 designs different
passive controllers by applying the theorem of our paper and
the literature [22], respectively. We can compare the region of
feasible solution.
Example 1. Consider a fuzzy system of the following form.
Plant Rules:
Rule 1: IF 𝑥1 (𝑡) is 𝑀1 , THEN
𝑥̇ (𝑡) = 𝐴 1 𝑥 (𝑡) + 𝐵1 𝑢 (𝑡) + 𝐵11 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶1 𝑥 (𝑡) + 𝐷1 𝜔 (𝑡) .
(45)
𝑦 (𝑡) = 𝐶2 𝑥 (𝑡) + 𝐷2 𝜔 (𝑡)
𝑎 −0.02 1
0 5] ,
𝐴 1 = [1
1
2 3]
[
1
𝐵1 = [0] ,
[𝑏]
(46)
𝐴2 = [
[
1
𝐵2 = [0] ,
[2]
1 0 0
𝐵12 = [1 2 3] ,
[0 1 2]
2 0 0
𝐷1 = 𝐷2 = [0 2 0] ,
[1 1 1 ]
Rule 2: IF 𝑥1 (𝑡) is 𝑀2 , THEN
𝑥̇ (𝑡) = 𝐴 2 𝑥 (𝑡) + 𝐵2 𝑢 (𝑡) + 𝐵12 𝜔 (𝑡) ,
with
−0.225 −0.02 0
1
0 3] ,
1
1 1]
1 0 0
𝐵11 = [1 2 0] ,
[0 1 0]
2.5 0 0
𝐶1 = 𝐶2 = [ 0 2.5 0] ,
[ 1 2 1]
𝑎 ∈ [−2, 3] , 𝑏 ∈ [6, 10] .
(47)
We can obtain Figure 1 by applying our method and obtain
Figure 2 by applying method in [22]. Symbol “∗ ” shows that
8
Journal of Applied Mathematics
feasible solution exists at that point; symbol “o” shows that
feasible solution does not exist at that point.
It is obvious that Theorem 3 can relax the conditions in
[22], and our method futher reduced the conservatism.
Example 2 (Li et al. [20]). Consider a delayed fuzzy system of
the following form.
Plant Rules:
Rule 1: IF 𝑥1 (𝑡) is 𝑀1 , THEN
𝑥̇ (𝑡) = 𝐴 1 𝑥 (𝑡) + 𝐴 𝑑1 𝑥 (𝑡 − ℎ) + 𝐵1 𝑢 (𝑡) + 𝐵11 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶1 𝑥 (𝑡) + 𝐶𝑑1 𝑥 (𝑡 − ℎ) + 𝐷1 𝜔 (𝑡) .
(48)
Table 1: Allowable maximum time delay ℎ.
Rule 2: IF 𝑥2 (𝑡) is 𝑀2 , THEN
𝑥̇ (𝑡) = 𝐴 2 𝑥 (𝑡) + 𝐴 𝑑2 𝑥 (𝑡 − ℎ) + 𝐵2 𝑢 (𝑡) + 𝐵12 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶2 𝑥 (𝑡) + 𝐷2 𝜔 (𝑡)
Rule 2: IF 𝑥1 (𝑡) is 𝑀2 , THEN
𝑥̇ (𝑡) = 𝐴 2 𝑥 (𝑡) + 𝐴 𝑑2 𝑥 (𝑡 − ℎ) + 𝐵2 𝑢 (𝑡) + 𝐵12 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶2 𝑥 (𝑡) + 𝐶𝑑2 𝑥 (𝑡 − ℎ) + 𝐷2 𝜔 (𝑡)
−1 0.2
],
0 −0.1
𝐴2 = [
𝐴 𝑑1 = [
0.1 0.6
],
0.2 −0.1
𝐴 𝑑2 = [
𝐵1 = [
0.2
],
−0.5
𝐵2 = [
𝐵12 = [
0.3
],
0.3
−0.2 0
],
0 0
0.1 1
𝐶2 = [
],
0 0
𝐶𝑑2 = [
0.1 −0.1
],
0
0
𝐷2 = [
−1 0
],
0.1 −0.5
𝐴1 = [
−0.1125 −0.02
],
1
0
𝐴 𝑑1 = [
−0.0125 −0.005
],
0
0
𝐶1 = [
𝐶𝑑1
(50)
29.0804 1.8737
],
1.8737 3.7272
𝑥 (0) = [−1
1.8020 0.0328
],
0.1385 0.4057
(51)
𝛾 = 237.5809.
By using the delay-dependent criterion Theorem 3 of [20], we
can obtain 𝛾 = 604.9523 when 𝜏 = 7.5. And applying our
result, 𝛾 = 237.5809 is obtained. We can see that our results
is not conservative since 𝛾 is much smaller than the result of
[20].
Example 3. Consider a delayed fuzzy system of the following
form.
Plant Rules:
Rule 1: IF 𝑥2 (𝑡) is 𝑀1 , THEN
𝑥̇ (𝑡) = 𝐴 1 𝑥 (𝑡) + 𝐴 𝑑1 𝑥 (𝑡 − ℎ) + 𝐵1 𝑢 (𝑡) + 𝐵11 𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶1 𝑥 (𝑡) + 𝐷1 𝜔 (𝑡) .
𝑥22 (𝑡)
,
2.25
𝑀21 (𝑥2 (𝑡)) =
𝑥22 (𝑡)
. (55)
2.25
The initial state is
0.5 0
],
0 0
0.1 0
]
0 0
𝑋=[
𝐵11 = 𝐵12 = 𝐼2 ,
The membership function is
𝑀11 (𝑥2 (𝑡)) = 1 −
we let ℎ = 7.5, 𝑚 = 2, 𝜏1 = 10, 𝜏2 = 0, 𝜏3 = 5, 𝜏4 = 10, 𝜏5 = 10
and we can obtain
𝑃=[
−0.0125 −0.23
𝐴 𝑑2 = [
],
0
0
(54)
−0.1 0.2
=[
],
0
0
𝐷1 = [
−0.1125 −1.527
],
1
0
𝐶1 = 𝐶2 = 𝐷1 = 𝐷2 = 𝐼2 .
1 0
],
0.2 0
1 0.2
],
0 0
𝐴2 = [
1
𝐵1 = 𝐵2 = [ ] ,
1
−0.2 0.1
],
0.1 −0.2
𝐵11 = [
(53)
with
(49)
with
𝐴1 = [
The upper bound of ℎ
1.5
2
3.16
8.8
Methods
[21]
Theorem 3, 𝑚 = 1
Theorem 3, 𝑚 = 4
Theorem 3, 𝑚 = 5
(52)
− 1.2]𝑇 ,
𝜔 (𝑡) = [
sin (3𝑡)
exp (𝑡)
cos (3𝑡) 𝑇
] .
exp (𝑡)
(56)
First, we should find the allowable maximum time delay
ℎ which let the fuzzy system passive. Table 1 shows the
maximum time delay obtained by [21] and our method.
It clearly shows that the method in our paper can get
larger upper bound than before. It also shows that conservatism is further reduced when 𝑚 increases.
Then we let ℎ = 1.5, 𝑚 = 4, and we can obtain
𝑃=[
9.4980 9.1547
],
9.1547 8.8849
𝑋=[
0.6351 0.4997
] . (57)
0.5807 0.4826
The fuzzy controller gains by our method are given by
𝐾1 = [150.7358
− 218.3595] ,
𝐾2 = [150.9502
− 219.0519] .
(58)
Figure 3 shows the state response 𝑥1 (𝑡) of the closed-loop
system with the controller gains in (58), and Figure 4 shows
the state response 𝑥2 (𝑡) of the closed-loop system.
Journal of Applied Mathematics
9
1.5
the Doctoral Fund of Ministry of Education of China under
Grant 20120036120013.
1
References
𝑥1
0.5
0
−0.5
−1
0
5
10
15
Time (s)
Figure 3: State response ×1 of the system.
1.5
1
𝑥2
0.5
0
−0.5
−1
−1.5
0
5
10
15
Time (s)
Figure 4: State response ×2 of the system.
6. Conclusion
This paper has adopted the delay partitioning approach to
analyse the passivity and passification of delay fuzzy system
based on T-S model. The theorems given in this paper are all
in terms of LMIs. Examples have illustrated the effectiveness
of our results. The method in our paper has further reduced
the conservatism and the effect has been more apparent when
𝑚 increases. In addition, the results obtained in this paper
also can be extended to the fuzzy system with time-varying
delay and uncertainties.
Acknowledgments
This work was supported in part by the National Natural
Science Foundation of China under Grant 61273144 and
61203041, the Natural Science Foundation of Beijing under
Grant 4122071, the Chinese National Post-doctor Science
Foundation under Grants 2011M500217 and 2012T50036, and
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