Available online at www.sciencedirect.com
Journal of the Franklin Institute 351 (2014) 3766–3781
www.elsevier.com/locate/jfranklin
New stability and stabilization criteria for a class
of fuzzy singular systems with time-varying delay
Huijiao Wanga,n,1, Bo Zhoua, Renquan Lub, Anke Xueb
a
Institute of Automation, College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University,
Hangzhou 310018, PR China
b
Institute of Information & Control, Hangzhou Dianzi University, Hangzhou 310018, PR China
Received 22 August 2012; received in revised form 2 January 2013; accepted 16 February 2013
Available online 22 March 2013
Abstract
This paper deals with the stability analysis and fuzzy stabilizing controller design for fuzzy singular
systems with time-varying delay. The time-varying delay is composed of two parts: constant part and timevarying part. Based on the idea of delay partitioning, a new Lyapunov–Krasovskii functional is proposed to
develop the new delay-dependent stability criteria, which ensures the considered system to be regular,
impulse-free and stable. Furthermore, the desired fuzzy controller gains are also presented by solving a set
of strict linear matrix inequalities (LMIs). Some numerical examples are given to show the effectiveness and
less conservativeness of the proposed methods.
& 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In real world, most physical systems and processes are nonlinear. Many researchers have been
seeking the effective approaches to control nonlinear systems. Among these, there are growing
interests in Takagi–Sugeno (T–S) fuzzy-model-based control [1]. It has been proved that
T–S fuzzy models can be used to appropriate a nonlinear system, which are realized by piecewise smoothly connecting a family of local linear models with fuzzy membership functions.
n
Corresponding author.
E-mail addresses: [email protected], [email protected] (H. Wang), [email protected] (B. Zhou),
[email protected] (R. Lu), [email protected] (A. Xue).
1
This work was supported by the National Natural Science Foundation of People's Republic of China under Grant
61104094, the Natural Science Foundation of Zhejiang Province under Grant Y1080690 and Y2110690, and the Science
Technology Department of Zhejiang Province under Grant 2012C21012.
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jfranklin.2013.02.030
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
3767
This “blending” makes the subsystems of T–S fuzzy model similar to linear systems, and the
fruitful results of linear system theories can be directly applied for the stability analysis and
synthesis of nonlinear systems. Moreover, time delays always exist in many dynamical systems
and delays are sources of poor stability and performance of a system [2–5]. So far, lots of results
have been reported for T–S fuzzy systems or T–S fuzzy systems with time-delay, such as,
stability analysis [6–10], H ∞ control [11–13], H ∞ filtering [14–19], fault detection [20], reliable
control [21,22], state estimation [23,24], etc. The maximum allowable delay serves as
“performance index” for measuring the conservatism of the conditions obtained.
On the other hand, singular systems have been extensively studied in the past years due to the
fact that singular systems better describe physical systems than state-space ones. In fact, singular
systems can be found in electrical circuits, economic systems, moving robots and many other
systems. Recently, a wider class of fuzzy systems described by singular form is considered in [25],
where the model is in the extended T–S fuzzy model. Based on parallel distributed compensation,
delay-independent stability and stabilization results for fuzzy descriptor systems were derived
in [26]. The problems of delay-dependent stability and H ∞ control were discussed using model
transformation techniques in [27]. But model transformation may lead to considerable conservativeness. Using free-weight matrix method, [28] discussed the problems of delay-dependent
stability and L2 −L∞ control. As mentioned in [2], some free-weighting matrices (or slack variable)
may be redundant and they will increase the computational burden in case of stability analysis for
deterministic delay systems by constant Lyapunov–Krasovskii functionals. In [29], the problems
of sliding mode control for fuzzy descriptor systems were presented using delay partitioning
approach, but the time-delay is constant, which is ineffective to the time-varying case. To the best
of authors’ knowledge, the problems of delay-dependent stability analysis and stabilizing controller
design for fuzzy singular systems with time-varying delay have not fully been investigated.
Therefore, it is of great significance to investigate the stability and stabilization of fuzzy singular
systems with time-varying delay.
Motivated by the idea of delay partitioning in [10,30], this paper deals with the problems of
delay-dependent stability analysis and stabilization control for fuzzy singular systems with timevarying delay. The purpose is to present some less conservative stability criteria and design the
fuzzy state feedback controller such that the resulted close-loop system is regular, impulse-free and
stable. The time-varying delay τðtÞ considered in this paper is composed of two parts: constant
part τ1 and time-varying part d(t), that is, τðtÞ ¼ τ1 þ dðtÞ,0≤dðtÞ≤τ2 −τ1 , τ2 is the maximum
allowable delay . Based on delay partitioning approach, in which the constant delay τ1 , not τ2 , is
decomposed into N subintervals and each subinterval has different Lyapunov weighted matrices
Qj ðj ¼ 1,2,…,NÞ, a new Lyapunov–Krasovskii functional for fuzzy singular systems is presented
and novel delay-dependent stability conditions are established via LMIs. Based on delay-dependent
stability conditions, a strict LMI-based design method of the desired fuzzy controller is proposed.
The effectiveness of the method is illustrated by some numerical examples.
The remaining portion of the paper is organized as follows. Section 2 formulates the problem
under consideration. Stability analysis and fuzzy controller designs are presented in Section 3.
Illustrative examples are given in Section 4 to demonstrate the effectiveness of the theoretical
results. Finally, this paper is concluded in Section 5.
Notations: Through this paper, the superscripts “T” and “−1” stand for the transpose of a
matrix and the inverse of a matrix; Rn denotes n-dimensional Euclidean space; Rnm is the set of
all real matrices with m rows and n columns; ∥ ∥ stands for the Euclidean norm for a vector.
ρðMÞ denotes the spectral radius of the matrix M. For a symmetric matrix, n denotes the matrix
entries implied by symmetry.
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H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
2. Problem formulation and preliminaries
T–S fuzzy model, which was proposed by Takagi and Sugeno, can be employed to describe
complex nonlinear systems. It is presented by a family of fuzzy IF-THEN rules that represents
the local linear input–output relations of a nonlinear system. In this paper, we consider the
following fuzzy singular system with time-varying delay
Plant form:
Rule i: IF θ1 is H i1 and IF θ2 is H i2 ⋯ IF θg is Hig, THEN
(
E x_ ðtÞ ¼ Ai ðtÞxðtÞ þ Adi ðtÞxðt−τðtÞÞ þ Bi ðtÞuðtÞ, i ¼ 1,2,…,r
ð1Þ
xðtÞ ¼ ϕðtÞ, t∈½−τ2 ,0
where xðtÞ∈Rn is the state vector, uðtÞ∈Rm is the control input. H ij ðj ¼ 1,…,gÞ are fuzzy sets,
θ ¼ ½θ1 ,…,θg is the premise variable vector. In order to avoid a complicated defuzzification
process of fuzzy controllers, we assumed that the premise variables do not depend on the input
variables u(t). The delay τðtÞ is time-varying and satisfies
0oτ1 ≤τðtÞ≤τ2 ,
0o_τ ðtÞ≤d
ð2Þ
The matrix E∈Rnn may be singular and we assume that rankE ¼ n1 ≤n.
Through the use of “fuzzy blending”, the fuzzy singular system (1) can be inferred as follows:
8
r
>
< Ex_ ðtÞ ¼ ∑ μ ðθÞfAi ðtÞxðtÞ þ Adi ðtÞxðt−τðtÞÞ þ Bi ðtÞuðtÞg
i
ð3Þ
i¼1
>
: xðtÞ ¼ ϕðtÞ, t∈½−τ2 ,0
where the fuzzy basis functions are given by
μi ðθÞ ¼
ωi ðθÞ
,
r
∑i ¼ 1 ωi ðθÞ
r
ωi ðθÞ ¼ ∏ H ij ðθj Þ
j¼1
with H ij ðθj Þ represents the grade of membership of θj in Hij. Here ωi ðθÞ has the following basic
property:
ωi ðθÞ≥0,
i ¼ 1,2,…,r
r
∑ ωi ðθÞ40 ∀t
i¼1
therefore
μi ðθÞ≥0,
i ¼ 1,2,…,r
r
∑ μi ðθÞ ¼ 1 ∀t
i¼1
The purpose of this paper is to develop some new stability conditions for the unforced fuzzy
singular system
8
r
>
< Ex_ ðtÞ ¼ ∑ μ ðθÞfAi ðtÞxðtÞ þ Adi ðtÞxðt−τðtÞÞg
i
ð4Þ
i¼1
>
: xðtÞ ¼ ϕðtÞ, t∈½−τ2 ,0
and design fuzzy controllers to stabilize fuzzy singular system (3) with time-varying delay (2).
Particularly, the lower bound of delay is not restricted to 0, which is even more applicable to
networked control systems. Motivated by the approach in [30,10], our approach is to represent
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
3769
the time delay τðtÞ as two parts: constant part τ1 and time-varying part d(t), that is,
τðtÞ ¼ τ1 þ dðtÞ,
0≤dðtÞ≤τ2 −τ1 :
ð5Þ
Then, applying the idea of delay partitioning to the constant part τ1 , a novel Lyapunov–
Krasovskii functional, in which τ1 is decomposed into N subintervals and different energy
functions corresponding to different segments, is introduced. Especially, when τ1 ¼ τ2 , that is,
τðtÞ is a constant delay τ.
For the unforced singular time-delay system described by
(
E x_ ðtÞ ¼ AxðtÞ þ Ad xðt−τðtÞÞ,
ð6Þ
xðtÞ ¼ ϕðtÞ, t∈½−τ2 ,0
we have the following definitions:
Definition 1 ([31,32]).
(1) The pair (E, A) is said to be regular if detðsE−AÞ is not identically zero.
(2) The pair (E, A) is said to be impulse-free if degðdetðsE−AÞÞ ¼ rank E.
Definition 2 ([31]).
(1) The singular system (6) is said to be regular and impulse-free if the pair (E, A) is regular and
impulse free.
(2) The singular system (6) is said to be asymptotically stable for any nonlinear perturbations (6)
if for any ϵ40, there exists a scalar δðϵÞ40 such that for any compatible initial conditions
ϕðtÞ satisfying sup−τðtÞ≤t≤0 ∥ϕðtÞ∥≤δðϵÞ, the solution x(t) of the system (6) satisfies ∥xðtÞ∥≤ϵ
for t≥0. Furthermore, limt-∞ xðtÞ ¼ 0.
Lemma 1 ([33]). If a functional V : C n ½−τ,0-R is continuous and xðt,ϕÞ is a solution to
Eq. (6), we define
1
V_ ðϕÞ ¼ limþ sup ðVðxðt þ h,ϕÞ−VðϕÞÞÞ
h
h-0
Denote the system parameters of Eq. (6) as
#"
"
A11 A12
Ad11
I n1 0
ðE,A,Ad Þ ¼
,
,
Ad21
A21 A22
0 0
Ad12
Ad22
#!
with A22 ,Ad22 ∈Rn2 n2 ,n1 þ n2 ¼ n. Assume that the singular system (6) is regular and impulsefree, A22 is invertible and ρðA−1
22 Ad22 o1Þ. Then the singular system (6) is stable if there exist
positive numbers α,β,υ and a continuous function, V : C n ½−τ,0-R, such that
β∥ϕ1 ð0Þ∥2 ≤VðϕÞ≤υ∥ϕ∥2 ,
V_ ðxt Þ≤−α∥xt ∥2
where xt ¼ xðt þ θÞ with θ∈½−τ2 ,0 and ϕ ¼ ½ϕT1 ,ϕT2 T with ϕ1 ∈Rn1 .
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H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
Lemma 2 ([34]). For any constant matrix X∈Rnn ,X ¼ X T 40, scalar r40, and vector
function x_ : ½−r,0-Rn such that the following integration is well defined, then
#
"
Z 0
xðtÞ
−X
X
−r
:
ð7Þ
x_ T ðt þ sÞX x_ ðt þ sÞ ds≤½xT ðtÞ xT ðt−rÞ
X −X xðt−rÞ
−r
3. Main results
In this section, we perform stability analysis of unforced fuzzy singular system (4) and design
fuzzy stabilization controllers for the fuzzy singular system (3) based on delay partitioning
approach.
3.1. Delay-dependent stability
The following result concerns the stability for the unforced fuzzy singular system (4).
Theorem 1. For a given integer N≥1, the unforced fuzzy singular system (4) is regular,
impulse-free and stable for any time-varying delay τðtÞ satisfying Eqs. (2) and (5), if there exist
symmetric positive-definite matrices Qj, Wj ðj ¼ 1,2,…,NÞ, S1, S2, R and matrix P with
appropriate dimensions such that
E T P ¼ PT E≥0
2
Ψ ð1Þ
6
6 ð2ÞT
Ψ ¼6Ψ
4
Ψ ð3ÞT
where
2
6
6
6
6
6
6
6
ð1Þ
Ψ ¼6
6
6
6
6
6
4
ð8aÞ
Ψ ð2Þ
Ψ ð3Þ
N
− ∑ Wj
j¼1
3
7
0 7
7o0
5
ð8bÞ
0
−R
Ψ ð1Þ
11
ET W 1 E
⋯
0
0
PT Adi
n
Ψ ð1Þ
22
⋯
0
0
0
⋮
⋮
⋱
⋮
⋮
⋮
n
n
⋯
Ψ ð1Þ
NN
E WNE
E RE
T
0
n
n
⋯
n
Ψ ð1Þ
Nþ1 Nþ1
n
n
⋯
n
n
Ψ ð1Þ
Nþ2 Nþ2
n
n
⋯
n
n
n
T
T
T
T
Ψ ð1Þ
11 ¼ Ai P þ P Ai þ Q1 þ S2 −E W 1 E
T
T
Ψ ð1Þ
jj ¼ −Qj−1 −E W j−1 E þ Qj −E W j E,
Ψ ð1Þ
Nþ1 Nþ1
Ψ ð1Þ
Nþ2 Nþ2
j ¼ 2,3,…,N
¼ −QN −E W N E þ S1 −E RE,
T
¼ −ð1−dÞS1 −2E T RE,
T
T
Ψ ð1Þ
Nþ3 Nþ3 ¼ −S2 −E RE
0
3
7
7
7
7
⋮
7
7
7
0
7
7
7
0
7
7
T
E RE 7
5
0
Ψ ð1Þ
Nþ3 Nþ3
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
3
N
T
∑
W
hA
j
7
6 i
j¼1
7
6
7
6
7
6
0
7
6
7
6
0
7
6
7
6
Ψ ð2Þ ¼ 6
7,
⋮
7
6
7
6
0
7
6
7
6
N
7
6 T
6 hAdi ∑ W j 7
5
4
j¼1
2
3
ðτ2 −τ1 ÞATi R
7
6
0
7
6
7
6
7
6
0
7
6
7
6
Ψ ð3Þ ¼ 6
⋮
7,
7
6
7
6
0
7
6
6
T 7
4 ðτ2 −τ1 ÞAdi R 5
0
3771
2
i ¼ 1,2,…,r:
0
Proof. We first show that the unforced fuzzy singular system (4) is regular and impulse−free.
Since rank E ¼ n1 ≤n, there must exist two invertible matrices G and H∈Rnn such that
I n1 0
ð9Þ
E ¼ GEH ¼
0 0
Similar to Eq. (9), we define
"
#
A i,11 A i,12
A i ¼ GAi H ¼
, i ¼ 1,2,…,r
A i,21 A i,22
"
#
"
W 1,11
P 11 P 12
−T
−T
−1
P ¼ G PH ¼
, W 1 ¼ G W 1G ¼
W 1,21
P 21 P 22
W 1,12
#
W 1,22
ð10Þ
T
From Eq. (8a), we have P ¼ ½PP 11 P0 ,P 11 ¼ P 11 40.
21
22
Since Ψ ð1Þ
11 o0 and Q1 40,S2 40, we can formulate the following inequality easily:
Γ ¼ ATi P þ PT Ai −E T W 1 Eo0
Pre- and post-multiplying Γo0 by HT and H, respectively, yields
"
#
Γ 11
Γ 12
T
T
T
T
T
Γ ¼ H ΓH ¼ A i P þ P A i −E W 1 E ¼
o0
T
n
A i,22 P 22 þ P 22 A i,22
ð11Þ
Since Γ 11 and Γ 12 are irrelevant to the results of the following discussion, the real expression of
these two variables are omitted here. From Eq. (11), it is easy to see that
T
T
A i,22 P 22 þ P 22 A i,22 o0
ð12Þ
T
T
Since μi ðθÞ≥0 and ∑r i ¼ 1 μi ðθÞ ¼ 1, we have ∑r i ¼ 1 μi ðθÞðA i,22 P 22 þ P 22 A i,22 Þo0, this implies
that ∑r i ¼ 1 μi ðθÞA i,22 is nonsingular. Therefore, the unforced fuzzy singular system (4) is regular
and impulse-free.
Next, we will show the stability of the system (4). Similar to Eq. (10), we define
"
#
A di,11 A di,12
A di ¼ GAdi H ¼
, i ¼ 1,2,…,r
A di,21 A di,22
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H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
"
Q 1 ¼ H T Q1 H ¼
"
S 2 ¼ H T S2 H ¼
Q 1,11
Q 1,12
Q 1,21
Q 1,22
#
S 2,11
S 2,12
S 2,21
S 2,22
#
"
,S 1 ¼ H T S1 H ¼
,R ¼ G−T RG−1 ¼
From Eq. (8b), it can be seen that
" T
r
Ai P þ PT Ai þ Q1 þ S2 −E T W 1 E
∑ μi ðθÞ
n
i¼1
"
S 1,11
S 1,12
S 1,21
S 1,22
#
R 11
R 12
R 21
R 22
#
PT Adi
−ð1−dÞS1 −2ET RE
,
ð13Þ
#
o0
ð14Þ
Pre-multiplying and post-multiplying Eq. (14) by diagfHT ,Ig and its transpose, respectively,
letting ð1−dÞS1 þ 2E T RE ¼ Z, and then using the expressions in Eqs. (9), (10), (13), we have
3
2
⋆
⋆
⋆
⋆
r
r
7
6
6 ⋆ ∑ μi ðθÞðA Ti,22 P 22 þ P T22 A i,22 Þ þ Q 1,22 þ S 2,22 ⋆ P T22 ∑ μi ðθÞA di,22 7
7
6
i¼1
i¼1
7
6
7o0
6⋆
⋆
⋆
⋆
7
6
7
6
r
T
5
4⋆
⋆
−Z 22
∑ μi ðθÞA di,22 P 22
i¼1
where Z 22 40 is the n2 n2 block in Z, i.e., Z 22 ¼ ½0 I n2 Z½0 I n2 T , and letting Q 1,22 þ
S 2,22 ¼ Z 22 , which implies that
2 r
3
r
T
T
T
∑
μ
ðθÞðA
P
þ
P
A
Þ
þ
Z
P
∑
μ
ðθÞA
22
di,22 7
i
22 i,22
22
i,22 22
6i¼1 i
i¼1
6
7
ð15Þ
6
7o0
r
T
4
5
∑ μi ðθÞA di,22 P 22
−Z 22
i¼1
Then, pre-multiplying and post-multiplying Eq. (15) by
2
3
r
−1 r
!T
4−
∑ μi ðθÞA i,22
∑ μi ðθÞA di,22
I5
i¼1
i¼1
and its transpose, respectively, yields
ΛT Z 22 Λ−Z 22 o0
with Λ ¼ ð∑ri ¼ 1 μi ðθÞA i,22 Þ−1 ð∑ri ¼ 1 μi ðθÞA di,22 Þ, which implies that ρðΛÞo1 holds for all
allowable μi .
Then, we define the following Lyapunov–Krasovskii functional for the unforced fuzzy
singular system (4),
3
Vðxt ,tÞ ¼ ∑ V m ðxt ,tÞ
m¼1
ð16Þ
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
3773
where
V 1 ðxt ,tÞ ¼ xT ðtÞE T PxðtÞ,
Z
N
V 2 ðxt ,tÞ ¼ ∑
j¼1
Z
þ
t−ðj−1Þh
Z
t−τ1
x ðsÞQj xðsÞds þ
T
t−jh
t
xT ðsÞS1 xðsÞ ds
t−τðtÞ
xT ðsÞS2 xðsÞ ds,
t−τ2
Z
N
V 3 ðxt ,tÞ ¼ ∑
j¼1
Z
þ
−ðj−1Þh
−jh
−τ1 Z t
−τ2
Z
t
x_ T ðsÞðhE T W j EÞ_x ðsÞ ds dθ
tþθ
x_ T ðsÞðτ2 −τ1 ÞE T RE x_ ðsÞ ds dθ,
tþθ
where S1 ¼ ST1 40, S2 ¼ ST2 40, R ¼ RT 40, Qj ¼ QTj 40 and W j ¼ W Tj 40 ðj ¼ 1,2,…,NÞ, and h
is the length of each division, h ¼ τ1 =N and N is the number (a positive integer) of divisions of
the interval ½−τ1 ,0.
Taking the derivation of Vðxt ,tÞ with respect to t along the trajectory of Eq. (4) yields
r
T
T
T
_
V 1 ðxt ,tÞ ¼ 2x ðtÞ ∑ μi ðθÞ P Ai xðtÞ þ P Adi xðt−τðtÞ
i¼1
N
N
j¼1
T
j¼1
V_ 2 ðxt ,tÞ≤ ∑ xT ðt−ðj−1ÞhÞQj xðt−ðj−1ÞhÞ− ∑ xT ðt−jhÞQj xðt−jhÞ
þx ðt−τ1 ÞS1 xðt−τ1 Þ−ð1−dÞxT ðt−τðtÞÞS1 xðt−τðtÞÞ
þxT ðtÞS2 xðtÞ−xT ðt−τ2 ÞS2 xðt−τ2 Þ
Z
N
N
j¼1
j¼1
V_ 3 ðxt ,tÞ ¼ ∑ x_ T ðtÞh2 ET W j Ex_ ðtÞ− ∑
Z
þ_x ðtÞðτ2 −τ1 Þ E RE x_ ðtÞ−
2 T
T
t−ðj−1Þh
t−jh
t−τ1
x_ T ðsÞhET W j Ex_ ðsÞ ds
x_ T ðsÞðτ2 −τ1 ÞE T RE x_ ðsÞ ds:
t−τ2
Since τ1 ≤τðtÞ≤τ2 , we have
Z t−τ1
Z t−τðtÞ
T
T
−
x_ ðsÞðτ2 −τ1 ÞE RE x_ ðsÞ ds ¼ −
x_ T ðsÞðτ2 −τ1 ÞET RE x_ ðsÞ ds
t−τZ2 t−τ
t−τ2
Z t−τðtÞ
1
x_ T ðsÞðτ2 −τ1 ÞET RE x_ ðsÞ ds≤−ðτ2 −τðtÞÞ
x_ T ðsÞE T RE x_ ðsÞ ds
−
Z t−τ1
t−τ2
t−τðtÞ
−ðτðtÞ−τ1 Þ
x_ T ðsÞET RE x_ ðsÞ ds
ð17Þ
t−τðtÞ
According to Lemma 2, we have
Z
−
t−τ1
t−τ2
2
−Ω
6
x_ T ðsÞðτ2 −τ1 ÞET RE x_ ðsÞ ds≤ηT ðtÞ4 n
n
Ω
−2Ω
n
where ηT ðtÞ ¼ ½xT ðt−τ1 Þ xT ðt−τðtÞÞ xT ðt−τ2 Þ, Ω ¼ E T RE.
0
3
7
Ω 5ηðtÞ
−Ω
ð18Þ
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H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
According to Eq. (4), the following holds:
N
N
j¼1
j¼1
∑ x_ T ðtÞh2 ET W j Ex_ ðtÞ ¼ ∑ ξT ðtÞLT1 h2 W j L1 ξðtÞ,
x_ T ðtÞðτ2 −τ1 Þ2 E T RE x_ ðtÞ ¼ ξT ðtÞLT1 ðτ2 −τ1 Þ2 RL1 ξðtÞ
with ξT ðtÞ ¼ ½xT ðtÞ xT ðt−hÞ ⋯ xT ðt−NhÞ xT ðt−τðtÞÞ xT ðt−τ2 Þ and L1 ¼ ½Ai 0 ⋯ 0 Adi 0.
Therefore,
!
r
N
2
T
ð1Þ
T
2
T
V_ ðxt ,tÞ≤ ∑ μi ðθÞξ ðtÞ Ψ þ ∑ L1 h W j L1 þ L1 ðτ2 −τ1 Þ RL1 ξðtÞ
i¼1
ð19Þ
j¼1
Then, if Eq. (8b) holds, there exist α40 such that V_ ðxt Þ≤−α∥xt ∥2 . By Lemma 1, we conclude
that the unforced fuzzy singular system (4) is stable. This completes the proof. □
Remark 1. In some existing literature, for example [3], they decomposed ½0,τ2 into N segments,
R t−ðj−1Þh T
x_ ðsÞhET W j E x_ ðsÞ ds in
in order to fully consider the information of τðtÞ, the term ∑N j ¼ 1 t−jh
the time derivative of VðxðtÞ,tÞ was estimated according to which segments τðtÞ is in, and
introducing an additional Ξ k (see Proposition 1 in [3]) to describe τðtÞ in different segments, which
make the analysis complicated. In this paper, the τðtÞ is composed of two parts: constant part τ1
R t−τ
and varying part d(t), which can simplify the estimation of t−τ21 x_ T ðsÞðτ2 −τ1 ÞE T RE x_ ðsÞ ds
(see Eq. (17)).
Remark 2. Motivated by the delay partitioning approach in [10,30], we divide the constant part of
time-varying delay ½0,τ1 into N segments, that is, ½0,ð1=NÞτ1 , ½ð1=NÞτ1 , N2 τ1 ,…,½ðN−1Þ=Nτ1 ,τ1 .
The difference lies in that we define different energy functional Qj , j ¼ 1,2,…,N in each different
segment, while [10,30] employing the same energy functional Q1. So, the results in this paper may
be more universal.
Remark 3. Since piecewise/fuzzy Lyapunov functions are much richer classes of Lyapunov
function candidates than a common Lyapunov function candidate [35], in order to further reduce
the analysis and synthesis conservatism, piecewise/fuzzy Lyapunov functions can be used to deal
with a larger class of fuzzy dynamic systems. This supplies the further research topic for fuzzy
singular systems with time-delay, especially for those fuzzy systems that do not admit a common
Lyapunov function.
3.2. Fuzzy controller design
In this subsection, a state feedback fuzzy controller is designed guaranteeing the regularity,
absence of impulse and stability of the resultant closed-loop fuzzy singular system. Consider the
following fuzzy control law:
Regulator Rule i: IF θ1 is H i1 and IF θ2 is H i2 ⋯ IF θg is Hig, THEN
uðtÞ ¼ −F i xðtÞ,
i ¼ 1,2,…,r
ð20Þ
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
3775
The overall state feedback control law is inferred by
r
uðtÞ ¼ − ∑ μi ðθÞF i xðtÞ
ð21Þ
i¼1
Supposing that Bi ¼ B, i ¼ 1,2,…,r (i.e., the input matrices in all the r fuzzy plant rules are
equal). The aim is to determine the local feedback gain Fi such that the closed-loop system
8
r
>
< E x_ ðtÞ ¼ ∑ μ ðθÞfðAi −BF i ÞðtÞxðtÞ þ Adi ðtÞxðt−τðtÞÞg
i
ð22Þ
i¼1
>
: xðtÞ ¼ ϕðtÞ, t∈½−τ2 ,0
is regular, impulse-free and stable.
According to Theorem 1, we have the following stabilizing controller design result.
Theorem 2. For given an integer N≥1, the closed-loop fuzzy singular system (22) under fuzzy
control (20) is regular, impulse-free and stable for any time-varying delay τðtÞ satisfying Eqs. (2)
~ j,W
~ j , V~ j ðj ¼ 1,2,…,NÞ, S~ 1 , S~ 2 , R,
~ Z~
and (5), if there exist symmetric positive-definite matrices Q
and matrix X with appropriate dimensions such that
X T E T ¼ EX≥0
"
~j
W
n
"
ð23aÞ
#
X T ET
≥0,
V~ j
"
Z~
n
#
X T ET
≥0
R~
ð23bÞ
#
N
ð1Þ ð2Þ ð3Þ ð2ÞT
ð3ÞT
~
~
~
~
~
~
Ψ Ψ Ψ Ψ − ∑ V j 0Ψ 0−R o0
ð23cÞ
j¼1
where
2
6
6
6
6
6
6
6
ð1Þ
Ψ~ ¼ 6
6
6
6
6
6
6
4
ð1Þ
Ψ~ 11
~1
W
⋯
0
0
Adi X
0
⋮
ð1Þ
Ψ~ 22
⋮
⋯
⋱
0
⋮
0
⋮
n
⋯
ð1Þ
Ψ~ NN
0
⋮
n
0
⋮
~
WN
0
0
n
ð1Þ
Ψ~ Nþ1 Nþ1
Z~
0
Z~
ð1Þ
Ψ~ Nþ3 Nþ3
n
n
n
⋯
n
n
⋯
n
n
ð1Þ
Ψ~ Nþ2 Nþ2
n
n
⋯
n
n
n
ð1Þ
~1
Ψ~ 11 ¼ Ai X þ X T ATi −BY i −Y Ti BT þ Q~ 1 þ S~ 2 −W
ð1Þ
~ j −W
~ j−1 þ Q
~ j,
Ψ~ jj ¼ −Q~ j−1 −W
j ¼ 2,3,…,N
ð1Þ
~ N −W
~ N þ S~ 1 −Z~ ,
Ψ~ Nþ1 Nþ1 ¼ −Q
ð1Þ
Ψ~ Nþ2 Nþ2 ¼ −ð1−dÞS~ 1 −2Z~ ,
~ ~
Ψ ð1Þ
Nþ3 Nþ3 ¼ −S 2 −Z
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
3776
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
2
6
6
6
6
6
ð2Þ
6
Ψ~ ¼ 6
6
6
6
6
4
hðX T ATi −Y Ti BT Þ
0
0
⋮
0
hX ATdi
T
3
2
7
7
7
7
7
7
7,
7
7
7
7
5
6
6
6
6
6
ð3Þ
6
Ψ~ ¼ 6
6
6
6
6
4
ðτ2 −τ1 ÞðX T ATi −Y Ti BT Þ
0
0
⋮
0
ðτ2 −τ1 ÞX T ATdi
0
3
7
7
7
7
7
7
7,
7
7
7
7
5
0
In this case, the local feedback gain Fi is given by
F i ¼ Y i X −1 ,
i ¼ 1,2,…,r:
ð24Þ
Proof. According to Theorem 1, if Eq. (8) holds, the closed−loop fuzzy singular system (22) is
regular, impulse−free and stable. Then substitute Ai by A~ i ¼ Ai −BF i in Eq. (8), we have
E T P ¼ PT E≥0
2 ð1Þ
Ψ ð2Þ
Ψ
N
6
6 ð2ÞT − ∑ W j
Ψ ¼6Ψ
4
j¼1
Ψ ð3ÞT
where
2
6
6
6
6
6
6
6
ð1Þ
Ψ ¼6
6
6
6
6
6
4
ð25aÞ
Ψ
3
ð3Þ
7
0 7
7o0
5
ð25bÞ
0
−R
Ψ ð1Þ
11
ET W 1 E
⋯
0
0
PT Adi
n
Ψ ð1Þ
22
⋯
0
0
0
⋮
⋮
n
n
⋱
⋯
⋮
Ψ ð1Þ
NN
⋮
E WNE
⋮
0
n
n
⋯
n
Ψ ð1Þ
Nþ1 Nþ1
ET RE
n
n
⋯
n
n
Ψ ð1Þ
Nþ2 Nþ2
n
n
⋯
n
n
n
T
T~
T
~
Ψ ð1Þ
11 ¼ A i P þ P A i þ Q1 þ S2 −E W 1 E
T
T
T
Ψ ð1Þ
jj ¼ −Qj−1 −E W j−1 E þ Qj −E W j E,
Ψ ð1Þ
Nþ1 Nþ1
Ψ ð1Þ
Nþ2 Nþ2
j ¼ 2,3,…,N
¼ −QN −E W N E þ S1 −E RE,
T
¼ −ð1−dÞS1 −2E T RE,
T
T
Ψ ð1Þ
Nþ3 Nþ3 ¼ −S2 −E RE
0
3
7
7
7
7
⋮
7
7
7
0
7
7
7
0
7
7
ET RE 7
5
0
Ψ ð1Þ
Nþ3 Nþ3
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
3
N
~ T ∑ Wj
h
A
7
6 i
j¼1
7
6
7
6
7
6
0
7
6
7
6
0
7
6
7
6
Ψ ð2Þ ¼ 6
7,
⋮
7
6
7
6
0
7
6
7
6
N
7
6 T
6 hAdi ∑ W j 7
5
4
j¼1
2
T 3
ðτ2 −τ1 ÞA~ i R
7
6
7
6
0
7
6
7
6
0
7
6
7
6
Ψ ð3Þ ¼ 6
7,
⋮
7
6
7
6
0
7
6
7
6
T
4 ðτ2 −τ1 ÞAdi R 5
3777
2
i ¼ 1,2,…,r:
0
0
From the proof of Theorem 1, we know that P is nonsingular. Now, pre−multiplying and post−
multiplying P−T and diagfϒ ,I,Ig and its transpose to both sides of Eqs. (25a) and (25b)
respectively, and define
9
8
Nþ3
>
=
<zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ >
ϒ ¼ diag P−T ,P−T ,…,P−T , X ¼ P−1 ,
>
>
;
:
~ j ¼ P−T Qj P−1 , V~ j ¼ W −1 , ðj ¼ 1,2,…,NÞ,
Q
j
S~ 1 ¼ P−T S1 P−1 , S~ 2 ¼ P−T S2 P−1 , R~ ¼ R−1 ,
and letting X T E T REX≥Z~ and X T E T W j EX≥V~ j , we have Eqs. (23) and (24), which means that the
closed-loop fuzzy singular system (22) is regular, impulse-free and stable under fuzzy control
(20) . This completes the proof. □
Remark 4. As for the general case, that is, the input matrices Bi are different. The resultant
closed-loop fuzzy singular system turns to be
8
r
r
>
< E x_ ðtÞ ¼ ∑ ∑ μi ðθÞμj ðθÞfðAi −Bi F j ÞðtÞxðtÞ þ Adi ðtÞxðt−τðtÞÞg
i ¼ 1j ¼ 1
>
: xðtÞ ¼ ϕðtÞ,
t∈½−τ2 ,0
The explicit expression of Fi can be obtained similar to Theorem 2.
4. Numerical examples
In this section, some numerical examples are presented to show the usefulness and effectiveness
of the results developed in this paper.
Example 1. Consider
parameters [27,28]:
2
1 0 0
60 1 0
6
E¼6
40 0 1
0
0
a fuzzy singular system composed of two rules and the following
3
0
07
7
7,
05
0 0
2
−3
6 0
6
A1 ¼ 6
4 0
0
−4
0
0:1
0
−0:1
0:1
0:1
−0:2
3
0:2
0 7
7
7,
0 5
−0:2
3778
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
2
−2
6 0
6
A2 ¼ 6
4 0
0
0
−0:2
−2:5
−0:2
−0:1
−0:3
0
0
3
7
7
7,
5
2
−0:5
6 0
6
Ad1 ¼ 6
4 0
0:1 0:1 −0:2 −0:2
2
3
−0:5 0
0
0
6 0
−1
0
07
6
7
Ad2 ¼ 6
7
4 0
0:1 −0:5 0 5
0
0
0
0
0
0
−1
0:1
0
−0:2
0
0
0
3
07
7
7,
05
0
0
supposing that the delay τðtÞ satisfies Eqs. (2) and (5) with τ1 ¼ 2. Table 1 presents the comparison
results with various d, which show that the delay-dependent stability condition in Theorem 1 gives
less conservative results than those in [27,28]. Moreover, the lager N, the less conservatism, while
the computational cost increases. This is reasonable since N is related to the decision variables. The
larger N indicates that the solution can be searched in a wider space and a longer maximum
allowable delay τ2 can be obtained.
Example 2. Consider a nonlinear time-delay system borrowed from [26]
3
ð1 þ a cos θðtÞÞ̈θ ðtÞ ¼ −bθ_ ðtÞ þ cθðtÞ þ cτ θðt−τðtÞÞ þ duðtÞ
_ is assumed to satisfy jθðtÞjoϕ,ϕ
_
where the range of θðtÞ
¼ 2,cτ ¼ 0:8,τðtÞ ¼ 1 þ 0:2 sinðtÞ (thus,
τ_ ðtÞ≤0:2). Similar to [26], we introduce new variable xðtÞ ¼ ½x1 ðtÞ,x2 ðtÞ,x3 ðtÞT with x1 ðtÞ ¼ θðtÞ,
_ and x3 ðtÞ ¼ ̈θ ðtÞ. Then, the system is described by
x2 ðtÞ ¼ θðtÞ
2
1 0
60 1
4
0 0
3
2
0
0
7
6
0 5x_ ðtÞ ¼ 4 0
c
0
1
0
−bx22 ðtÞ
3
2
0
0
7
6
1
5xðtÞ þ 4 0
−1−a cos x1 ðtÞ
cτ
0
0
0
2 3
0
7
6
0 5xðt−τðtÞÞ þ 4 0 7
5uðtÞ
0
d
0
3
This can be expressed exactly by the following fuzzy singular system:
8
3
>
< Ex_ ðtÞ ¼ ∑ μ ðθÞA ðtÞxðtÞ þ A ðtÞxðt−τðtÞ þ B uðtÞÞg
i¼1
>
: xðtÞ ¼ ϕðtÞ,
i
i
di
i
ð26Þ
t∈½−τ2 ,0
Table 1
Comparisons of maximum allowed delay τ2 for Example 1.
d
0.1
0.35
0.6
0.85
0.9
0.95
[27]
[28]
Theorem 1 (N¼ 1)
Theorem 1 (N¼ 2)
Theorem 1 (N¼ 3)
3.3623
3.3685
3.5023
3.6761
3.7467
2.981
3.156
3.2915
3.4755
3.6785
2.601
3.151
3.2379
3.3580
3.5567
1.833
3.076
3.1321
3.2425
3.4965
1.038
2.675
2.9775
3.0737
3.2604
–
2.078
2.5863
2.8257
3.0893
H. Wang et al. / Journal of the Franklin Institute 351 (2014) 3766–3781
where
2
3
3779
2
3
3
0
1
0
0 1
0
6
7
6
7
6
0
1 7
1
E ¼ 4 0 1 0 5, A1 ¼ 4 0
5
5, A2 ¼ 4 0 0
c −bðϕ2 þ 2Þ a−1
c 0 −a−1−aϕ2
0 0 0
2
3
2 3
2
3
0 0 0
0 1
0
0
6
7
6 7
6 0 0 07
1 5, Ad1 ¼ Ad2 ¼ Ad3 ¼ 4
A3 ¼ 4 0 0
5, B1 ¼ B2 ¼ B3 ¼ 4 0 5,
cτ 0 0
c 0 a−1
d
μ1 ¼
1
0
x22 ðtÞ
,
ϕ2 þ 2
0
μ2 ¼
2
1 þ cos x1 ðtÞ
,
ϕ2 þ 2
μ3 ¼
ϕ2 −x22 ðtÞ þ 1−cos x1 ðtÞ
ϕ2 þ 2
Let N ¼ 2, solving LMIs in Theorem 2, one set of feasible solutions is computed as
2
3
0:0679 −0:0156
0
6
0 7
X ¼ 4 −0:0156 0:0136
5,
−0:1033 0:0392 0:0424
Y 1 ¼ ½0:0066 0:0072 0:3578,
Y 2 ¼ ½−0:0258 0:0334 0:3463,
Y 3 ¼ ½−0:0062 0:0408 0:5630:
Then from Eq. (24), the local feedback gains Fi are given by
F 1 ¼ ½10:1630 −12:1332 8:4468,
F 2 ¼ ½9:7956 −9:8556 8:1757,
F 3 ¼ ½16:3310 −16:5416 13:2908:
Through this example, we found that our results are effective.
5. Conclusion
The stability analysis and fuzzy stabilize controller design for a class of fuzzy singular systems
with time-varying delay have been discussed in this paper. Based on delay partitioning method,
new stability criteria for unforced fuzzy singular systems have been established. The explicit
expression of the desired fuzzy controller gains are also presented. All the results reported in this
paper are formulated in terms of strict LMIs, which can be readily solved using standard
numerical software. Some numerical examples are provided to show the effectiveness of the
proposed methods. Furthermore, the delay partitioning method used in this paper can also be
extended to deal with the problem of H ∞ control/filtering, L2 −L∞ control/filtering, state
estimation for T–S fuzzy systems, fuzzy singular systems, stochastic neural networks, and so on.
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