Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 836814, 18 pages
doi:10.1155/2012/836814
Review Article
Robust Stabilization for Continuous Takagi-Sugeno
Fuzzy System Based on Observer Design
Yassine Manai and Mohamed Benrejeb
LA.R.A. Automatic, National Engineering School of Tunis, University of Tunis-ElManar, BP 37 Belvedere,
1002 Tunis, Tunisia
Correspondence should be addressed to Yassine Manai, yacine.manai@gmail.com
Received 30 June 2012; Revised 19 September 2012; Accepted 19 September 2012
Academic Editor: Mohammed Chadli
Copyright q 2012 Y. Manai and M. Benrejeb. 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.
This paper investigates the influence of a new parallel distributed controller PDC on the
stabilization region of continuous Takagi-Sugeno T-S fuzzy models. Using a nonquadratic
Lyapunov function, a new sufficient stabilization criterion is established in terms of linear matrix
inequality. The criterion examines the derivative membership function; an approach to determine
state variables is given based on observer design. In addition, a stabilization condition for
uncertain system is given. Finally, numeric simulation is given to validate the developed approach.
1. Introduction
Fuzzy control systems have experienced a big growth of industrial applications in the recent
decades, because of their reliability and effectiveness. Many researches have investigated the
Takagi-Sugeno models 1–3 during the last decades. Two classes of Lyapunov functions are
used to analyze these systems: quadratic Lyapunov functions and nonquadratic Lyapunov
ones which are less conservative than the first class. Many researches have investigated with
nonquadratic Lyapunov functions 4–10.
As the information about the time derivatives of membership function is considered by
the PDC fuzzy controller, it allows the introduction of slack matrices to facilitate the stability
analysis. The relationship between the membership function of the fuzzy model and the fuzzy
controllers is used to introduce some slack matrix variables. The boundary information of
the membership functions is brought to the stability condition and thus offers some relaxed
stability conditions 6. In order to determine the state variables many approaches of observer
design are given 9–11.
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Mathematical Problems in Engineering
In this paper, new stabilization conditions for Takagi-Sugeno uncertain fuzzy models
based on the use of fuzzy Lyapunov function are presented. This criterion is expressed in
terms of linear matrix inequalities LMIs which can be efficiently solved by using various
convex optimization algorithms 12, 13. The presented method is less conservative than
existing results.
The organization of the paper is as follows. In Section 2, we present the system
description and problem formulation and we give some preliminaries which are needed
to derive results. Section 3 will be concerned with stabilization analysis for continuous T-S
fuzzy systems by the use of new PDC controller based on derivative membership functions.
An observer approach design is derived to estimate state variables. In Section 4, a new
stabilization condition for uncertain system is given. Next, a new robust PDC controller
design approach is presented. Illustrative examples are given in Section 5 for a comparison
of previous results to demonstrate the advantage of the proposed method. Finally Section 6
makes the conclusion.
Notation. Throughout this paper, a real symmetric matrix S > 0 denotes S being a positive
definite matrix. The superscript “T” is used for the transpose of a matrix.
2. System Description and Preliminaries
Consider an uncertain T-S fuzzy continuous model for a nonlinear system as follows:
IF z1 t is Mi1 , . . . , zp t is Mip ,
THEN ẋt Ai ΔAi xt Bi ΔBi ut
i 1, . . . , r,
2.1
where Mij i 1, 2, . . . , r, j 1, 2, . . . , p is the fuzzy set and r is the number of model rules,
xt ∈ n is the state vector, ut ∈ m is the input vector, Ai ∈ n×n , Bi ∈ n×m are constant
real matrices, and z1 t, . . . , zp t are known premise variables. ΔAi , and ΔBi are time-varying
matrices representing parametric uncertainties in the plant model. These uncertainties are
admissibly norm-bound and structured.
The final outputs of the fuzzy systems are:
ẋt r
hi zt{Ai ΔAi xt Bi ΔBi ut},
i1
2.2
where
zt z1 tz2 t · · · zp t ,
wi zt
,
hi zt r
i1 wi zt
wi zt p
j1
Mij zj t
2.3
∀t.
Mathematical Problems in Engineering
3
The term Mi1 zj t is the grade of membership of zj t in Mi1
Since
⎧
r
⎪
⎨wi zt 0
i1
⎪
⎩wi zt ≥ 0,
we have
i 1, 2, . . . , r
⎧
r
⎪
⎨hi zt 1,
i1
⎪
⎩hi zt ≥ 0,
i 1, 2, . . . , r
2.4
for all t.
We have the following property:
r
ḣk zt 0.
k1
2.5
This study investigates the PDC controller influence on the closed-loop stability region and
gives robustness analysis of uncertain Takagi-Sugeno fuzzy system. Thus, we consider a PDC
fuzzy controller which examines the derivative membership function and it is given by
ut −
r
r
hi ztFi xt − ḣρ zt Kρ R xt.
i1
ρ1
2.6
The fuzzy controller design consists to determine the local feedback gains Fi , Kρ , and R in
the consequent parts. The state variables are determined by an observer and are detailed in
the next section.
The open-loop system is given by
ẋt r
hi ztAi ΔAi xt.
i1
2.7
By substituting 2.6 into 2.2, the closed-loop fuzzy system can be represented as:
⎫
⎧
r
r r
⎨
⎬
hi zthj zt AΔi − BΔi Fj − ḣρ ztBΔi Kρ R xt,
ẋt ⎭
⎩
i1 j1
ρ1
2.8
where AΔi Ai ΔAi and BΔi Bi ΔBi .
Assumption 2.1. The time derivative of the premises membership function is upper bound
such that |ḣk | ≤ φk , for k 1, . . . , r, where, φk , k 1, . . . , r are given positive constants.
Assumption 2.2. The matrices denote the uncertainties in the system and take the form of
ΔAi Dai Fai tEai ,
ΔBi Dbi Fbi tEbi ,
2.9
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Mathematical Problems in Engineering
where Dai , Dbi , Eai , and Ebi are known constant matrices and Fai t and Fbi t are unknown
matrix functions satisfying:
FaTi tFai t ≤ I,
∀t,
FbTi tFbi t ≤ I,
∀t,
2.10
where I is an appropriately dimensioned identity matrix.
Lemma 2.3 Boyd et al. Schur complement 12. Given constant matrices Ω1 , Ω2 , and Ω3 with
appropriate dimensions, where Ω1 ΩT1 and Ω2 ΩT2 , then
Ω1 ΩT3 Ω−1
2 Ω3 ≺ 0
2.11
if and only if
Ω1 ΩT3
0
−Ω2
or
−Ω2 Ω3
0.
Ω1
2.12
Lemma 2.4 Peterson and Hollot 2. Let Q QT , H, E, and Ft satisfying F T tFt ≤ I are
appropriately dimensional matrices then the following inequality
Q HFtE ET F T tH T ≺ 0
2.13
is true, if and only if the following inequality holds for any λ 0
Q λ−1 HH T λET E ≺ 0.
2.14
The aim of the next section is to find conditions for the stabilization of the closed-loop
T-S fuzzy system by using the Lyapunov theory.
3. Main Results
Consider the closed-loop system without uncertainties
ẋt r
hi zt{Ai xt Bi ut}.
i1
3.1
In order to give stability conditions, the slack matrix variables and the membership function
boundary Mozelli et al. 14 are used. Consider the following null product that will serve
stability analysis purposes:
r
T
T
2 x tM ẋ tμM × ẋt − hi zt{Ai xt Bi ut} 0.
i1
3.2
Mathematical Problems in Engineering
5
3.1. A PDC Controller with Derivative Membership Function
By substituting 2.6 into 3.1, the closed-loop fuzzy system can be represented as:
r
r r
hi zthj zt Ai − Bi Fj −
ẋt ḣm ztBi Km R xt.
i1 j1
m1
3.3
The next Theorem gives sufficient conditions to guarantee stability of system 3.3.
Theorem 3.1. Under Assumptions 2.1 and 2.2, and for μ 0, ε ≥ 0, the Takagi-Sugeno fuzzy system
3.1 is stabilizable with the PDC controller 2.6, with gains given by Fi STi H −T , Kρ VρT H −T ,
and R V T H −T , if there exist positive definite symmetric matrices Tk , k 1, 2, . . . , r, Y , and any
matrices H, Si , Vρ , and V with appropriate dimensions such that the following LMIs hold.
Ti 0,
Ti Y 0
i 1, 2, . . . , r,
Λii ≺ 0,
3.4
Λij ≺ 0,
Λij Λij Λji ,
3.5
where
⎤
⎡
T
Tφ − Ai H T − HATi Bi STj Sj BiT Bi V φ V φ BiT
∗
Λij ⎣
⎦,
T
Ti − μ Ai H T − Bi STj − Bi V φ H
μ HT H
Vφ r
φρ VρT V T ,
3.6
ρ1
Tφ r
φk Tk Y .
k1
Proof of Theorem 3.1. Let’s consider the fuzzy weighting-dependent Lyapunov-Krasovskii
functional as:
V xt r
hk zt · Vk xt,
k1
3.7
with Vk xt xT tPk εXxt, k 1, 2, . . . , r, where Pk PkT , X X T , ε ≥ 0, and
Pk εX ≥ 0.
6
Mathematical Problems in Engineering
This candidate Lyapunov function satisfies
i V xt is C1 ,
ii V 0 0 and V xt ≥ 0 for xt /
0,
iii xt
→ ∞ ⇒ V xt → ∞.
The time derivative of V xt is given by:
V̇ xt r
r
ḣk ztVk xt hk ztV̇k xt.
k1
k1
3.8
Adding the null product, then
V̇ xt r
r
ḣk ztVk xt hk ztV̇k xt 2 xT tM ẋT tμM
k1
k1
⎫
⎧
⎤ 3.9
r
r r
⎨
⎬
hi zthj zt Ai − Bi Fj − ḣρ ztBi Kρ R xt⎦.
× ⎣ẋt −
⎭
⎩
i1 j1
ρ1
⎡
Equation 3.9 can be rewritten as,
V̇ xt Υ1 x, z Υ2 x, z,
3.10
where
r
ḣk zt · Pk εX xt 2 xT tM ẋT tμM
Υ1 x, z x t
T
k1
⎤
⎡
r
r hi ztḣρ ztBi Kρ R xt⎦,
×⎣
i1 ρ1
r
!
Υ2 x, z hi zt 2xT tPi εXẋt 2 xT tM ẋT tμM
3.11
i1
⎤
r
r "
#
hi zthj zt Ai − Bi Fj xt⎦.
× ⎣ẋt −
⎡
i1 j1
Then, based on Assumption 2.1, an upper bound of Υ1 x, z is obtained as:
⎧
r
r
⎨
Υ1 x, z ≤
hi zt
φk · xtT Pk εXxt 2 xT tM ẋT tμM
⎩ k1
i1
⎤⎫
⎡
r
⎬
×⎣ φρ Bi Kρ R xt⎦ .
⎭
ρ1
3.12
Mathematical Problems in Engineering
7
Based on 2.5, it follows that rk1 ḣk zt1 − εX 0 where X is any symmetric
matrix of proper dimension.
Suppose that X rk1 ḣk zt1 − εX and adding X to 3.12, then
⎧
r
r
⎨
Υ1 x, z ≤
hi zt
φk · xtT Pk Xxt 2 xT tM ẋT tμM
⎩ k1
i1
⎤⎫
⎡
r
⎬
×⎣ φρ Bi Kρ R xt⎦
⎭
ρ1
3.13
r
!
hi zt xtT P φ xt 2xT tM · Bi K φ xt 2ẋT tμM · Bi K φ xt ,
i1
where
Pφ r
φk · Pk X,
Kφ r
φρ Kρ R .
3.14
ρ1
k1
Then,
V̇ xt ≤
r
r hi zthj zt
i1 j1
× xtT P φ xt 2xT tM · Bi K φ xt 2ẋT tμM · Bi K φ xt
3.15
2xT tPi εRẋt 2xT tMẋt 2ẋT tμMẋt
!
−2xT tM Ai − Bi Fj xt − 2ẋT tμM Ai − Bi Fj xt .
T
Using vector ηT xT t ẋT t , 3.15 can be rewritten as
V̇ xt ≤
r
r hi zthj ztηT Ξij η
i1 j1
r
i1
h2i ztηT Ξii η
r hi zthj ztη
T
3.16
Ξij Ξji η,
i1 i≺j
where
⎡%
⎤
&
T
T
−
A
P
−
M
A
−
B
F
−
B
K
−
B
F
−
B
K
M
∗
i
i j
i φ
i
i j
i φ
⎢ φ
⎥
Ξij ⎣
⎦.
T
T
μ MM
Pi εX − μM Ai − Bi Fj − Bi K φ M
3.17
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Mathematical Problems in Engineering
If Ξii ≺ 0 and Ξij Ξji ≺ 0, then V̇ xt ≺ 0 and 3.3 is stable. Pre- and postmultiplying Ξii ≺ 0 and Ξij Ξji ≺ 0 by nonsingular matrices diagM−1 , M−1 and
diagM−T , M−T , respectively, and pre- and postmultiplying by M−1 and M−T , respectively,
then we obtain
M−1 Pi εXM−T 0 i 1, 2, . . . , r,
M−1 Pi XM−T 0
Λii ≺ 0
Λij ≺ 0
i 1, 2, . . . , r,
3.18
i 1, 2, . . . , r,
i ≺ 1, 2, . . . , r,
where Λij Λij Λji , and
⎡
⎢
⎢
⎢
Λij ⎢
⎢
⎣
⎤
%
−1
−T
−T
M P φ M − Ai − B i F j − B i K φ M
⎥
⎥
T &
⎥
−1
⎥
−M Ai − Bi Fj − Bi K φ
∗
⎥
! ⎦
M−1 Pi εXM−T − μ Ai − Bi Fj − Bi K φ M−T M−1 μ M−T M−1
3.19
for the following variables definition:
H M−1 ,
Ti HPi εXH T ,
Vj HKρT ,
Tφ HP φ H T ,
V HRT ,
Y HXH T .
Sj HFjT ,
3.20
If LMI in 3.4 holds then the closed-loop continuous fuzzy system 3.3 is asymptotically
stable. The control gains are given by Fi STi H −T , Kρ VρT H −T , and R V T H −T . This
completes the proof.
Remark 3.2. The selection of φk given in Assumption 2.1 is performed by using a simple
procedure given in 15 .
Remark 3.3. The major contribution of the Theorem 3.1 is represented by the proposed PDC
controller given by 2.6. The contribution appears in the gains Kρ R introduced in
the controller term based on derivative membership functions. The stabilization condition
proposed is less conservative than some of those in the literature, as is shown in the example
below.
3.2. Observer Design
In order to determine state variables of the system, this section gives a solution by the means
of fuzzy observer design. The following condition is to be satisfied by the observer:
xt − xt
( −→ 0
as t −→ ∞,
3.21
Mathematical Problems in Engineering
9
where xt
( denotes the state vector estimated by a fuzzy observer. This condition guarantees
that the steady-state error between xt and xt
( converge to 0 and we denote this error by
et xt − xt.
(
A stabilizing observer-based controller can be formulated as follow:
( ẋt
r
#
"
hi zt Ai xt
( Bi ut Lj Ci xt
( − yt ,
j1
yt −
r
3.22
hi ztCi xt,
(
i1
where yt denotes the output vector.
We consider the proposed PDC controller given by 2.6:
ut −
r
r
hi ztFi xt
( − ḣρ zt Kρ R xt.
(
i1
ρ1
3.23
Replacing the fuzzy controller 2.6 in fuzzy observer 3.22 we obtain the closed-loop
fuzzy system as:
ẋt r
r hi zthj zt
i1 j1
⎧
⎨
⎩
Ai − Bi Fj
⎫
⎬
− ḣρ zt Kρ R xt
⎭
ρ1
r
⎫
r
⎬
hi zthj zt Bi Fj ḣρ zt Kρ R et
⎭
⎩
i1 j1
ρ1
⎧
⎨
r
r ėt 3.24
r
r "
#
hi zthj zt Ai − Li Cj et.
i1 j1
The augmented system is represented as follows:
ẋa t r
r ) ij xa t
hi zthj ztG
i1 j1
r
r ) ji
) ij G
G
) ii xa t 2
xa t,
hi zthj ztG
hi zthj zt
2
j1
i1 i≺j
r
3.25
10
Mathematical Problems in Engineering
where
xa t xt
,
et
⎡
r
r
⎤
Ai − Bi Fj − ḣρ Bi Kρ R Bi Fj ḣρ Bi Kρ R
) ij ⎣
⎦.
G
ρ1
ρ1
0
Ai − Li Cj
3.26
By applying Theorem 1 in 16 in the augmented system 3.25 we derive the following
Theorem.
Theorem 3.4. Under Assumptions 2.1 and 2.2, and for 0 ≤ μ ≤ 1, the Takagi-Sugeno fuzzy system
3.1 is stable if there exist positive definite symmetric matrices Pk , k 1, 2, . . . , r, and R, matrices
F1 , . . . , Fr such that the following LMIs holds.
Pk R 0,
k ∈ {1, . . . , r},
Pj μR ≥ 0,
j 1, 2, . . . , r,
!
) ii ≺ 0, i, k ∈ {1, . . . , r},
) T Pk μR Pk μR G
Pφ G
ii
) ij G
) ji
G
2
T
) ij G
) ji
G
≺ 0,
Pk μR Pk μR
2
for i, j, k 1, 2, . . . , r such that i ≺ j,
3.27
where
⎡
r
r
⎤
Ai − Bi Fj − ḣρ Bi Kρ R Bi Fj ḣρ Bi Kρ R
) ij ⎣
⎦,
G
ρ1
ρ1
0
Ai − Li Cj
3.28
r
Pφ φk Pk R.
k1
Proof of Theorem 3.4. The result follows immediately from the proof of Theorem 1 in 16.
4. Robust Stability Condition with PDC Controller
Consider the uncertain closed-loop system 2.8. A sufficient robust stability condition is
given as follows.
Theorem 4.1. Under Assumptions 2.1 and 2.2, and for μ 0, ε ≥ 0, the Takagi-Sugeno fuzzy system
2.2 is stabilizable with the PDC controller 2.6, with gains given by Fi STi H −T , Kρ VρT H −T ,
Mathematical Problems in Engineering
11
and R V T H −T , if there exist positive definite symmetric matrices Tk , k 1, 2, . . . , r, Y , and any
matrices H, Si , Vρ , and V with appropriate dimensions such that the following LMIs hold.
Ti 0,
Ti Y 0
i 1, 2, . . . , r,
≺ 0,
4.1
ii
≺ 0,
ij
ij
Tφ ij
r
,
ji
4.2
φk Tk Y ,
k1
where
⎡
T ⎤
T
Φ
F
Φ
MD
MD
E
−
E
K
11
12
ai
bi
bi
i
φ
a
i
⎢
⎥
⎢
⎥
Φ22 μMDai μMDbi
0
0
⎢
⎥
⎢
⎥
∗
−λI
0
0
0
⎥,
Σij ⎢
⎢
⎥
⎢
⎥
∗
∗
−λI
0
0
⎢
⎥
−1
⎣
⎦
0
∗
∗
∗
−λ I
−1
∗
∗
∗
∗
−λ I
4.3
with
T
Φ11 P φ M · Bi K φ K φ BiT M − MGii Gii T MT ,
T
Φ12 Pi εX − μ Gii − Bi K φ MT M,
Φ22 μ M MT ,
where Gij Ai − Bi Fj , Gii Ai − Bi Fi , and Pφ r
k1
φk Pk R.
4.4
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Mathematical Problems in Engineering
Proof. Proof of Theorem 4.1 The result follows immediately from the proof of Theorem 3.1
by replacing in the matrix inequality Ai with Ai Dai FtEai and Bi with Bi Dbi FtEbi , we
obtain the following inequality:
r
r V̇ xt ≤
hi zthj zt
i1 j1
× xtT P φ xt 2xT tM · Bi K φ xt 2ẋT tμM · Bi K φ xt
2xT tPi εRẋt 2xT tMẋt 2ẋT tμMẋt
− 2xT tM Ai − Bi Fj xt − 2xT tM
Eai
Fai t
0
xt
×
Dai Dbi
−Ebi Fj K φ
0
Fbi t
− 2ẋT tμM Ai − Bi Fj xt − 2ẋT tμM
Eai
Fai t
0
× Dai Dbi
xt .
−Ebi Fj K φ
0
Fbi t
4.5
T
Using vector ηT xT t ẋT t , 4.5 can be rewritten as
V̇ xt ≤
r
r ) ij η
hi zthj ztηT Ξ
i1 j1
r
i1
) ii η
h2i ztηT Ξ
r hi zthj ztη
T
4.6
) ji η,
) ij Ξ
Ξ
i1 i≺j
where
⎛
⎞ ⎫ ⎤
⎡⎧
E
⎪
ai
0
F
t
⎪
⎪
ai
⎠ ⎪
⎪
⎪ M⎝ Dai Dbi
⎪
⎥
⎢⎪
⎪
⎨
⎬ ⎥
−Ebi Fj K φ
0
Fbi t
⎢⎪
⎢
∗⎥
⎥
⎢
T
⎪
⎥
⎢⎪
⎪
⎪
E
ai
⎪
⎪
F
0
t
⎥,
⎢
)
a
i
Ξij Ξij ⎢⎪
⎪
⎪
⎥
Dai Dbi
MT ⎪
⎩
⎭
⎥
⎢
F
−E
K
0
F
t
bi
bi
j
φ
⎥
⎢
⎥
⎢
E
Fai t
ai
⎦
⎣
0
0
μM Dai Dbi
−Ebi Fj K φ
0
Fbi t
) ij Ξij Ξ
0
M Dai Dbi Fai t
Eai
0
μM Dai Dbi
0
Fbi t −Ebi Fj K φ 0
⎡
⎤
⎡
T ⎤
T DaT MT μDaT MT
T
i
i
0
E − Ebi Fj K φ
⎦ Fai t
⎣ T T
⎦.
⎣ ai
Dbi M μDbTi MT
0
Fbi t
0
0
4.7
Mathematical Problems in Engineering
13
) ij obtained as:
Then, based on Lemma 2.4, an upper bound of Ξ
⎡
⎤
DaT μDaT
i
i
D
D
ai
bi ⎣
) ij Ξij λ−1 M
⎦M T
Ξ
μDai μDbi DbTi μDbTi
⎡
T ⎤
T
Eai
0
F
−
E
K
E
bi
j
φ
a
⎦
⎣
i
≺0
λ
−Ebi Fj K φ 0
0
0
4.8
by Schur complement, we obtain,
⎡
T ⎤
T
Φ
F
Φ
MD
MD
E
−
E
K
11
12
ai
bi
bi
j
φ
a
i
⎢
⎥
⎢
⎥
Φ22 μMDai μMDbi
0
0
⎢
⎥
⎢
⎥
∗
−λI
0
0
0
⎢
⎥ ≺ 0,
⎢
⎥
⎢
⎥
∗
∗
−λI
0
0
⎢
⎥
−1
⎣
⎦
0
∗
∗
∗
−λ I
−1
∗
∗
∗
∗
−λ I
4.9
with
T
T
Φ11 P φ M · Bi K φ K φ BiT M − M Ai − Bi Fj Ai − Bi Fj MT ,
T
Φ12 Pi εX − μ Ai − Bi Fj − Bi K φ MT M,
4.10
Φ22 μ M MT .
If the LMI 4.1 holds then the system 2.8 is stable. This completes the proof.
The following theorem gives sufficient conditions for robust PDC controller design.
Theorem 4.2. Under Assumptions 2.1 and 2.2, and for μ 0, ε ≥ 0, the Takagi-Sugeno robust
.−T ,
fuzzy system 2.2 is stabilizable with the PDC controller 2.6, with gains given by Fi S)Ti H
14
Mathematical Problems in Engineering
.−T , and R V) T H
.−T , if there exist positive definite symmetric matrices T)k , k 1, 2, . . . , r,
Kρ V)ρT H
. S)i , V)ρ , and V) with appropriate dimensions such that the following LMIs hold.
Y) , and any matrices H,
T)i 0,
T)i Y) 0
i 1, 2, . . . , r,
4.11
) ii ≺ 0,
Λ
) ij ≺ 0,
Λ
) ij Λ
) ji ,
) ij Λ
Λ
T)φ r
φk T)k Y) ,
4.12
k1
r
)
Vφ φρ V)ρT V) T ,
ρ1
where
⎡
T
) T ) T
)
)
⎢ Φ11 Φ12 Dai Dbi HEai −Sj Ebi − V φ Ebi
⎢
) 22 μDai μDbi
0
0
Φ
⎢
⎢
∗
−λI
0
0
0
) ij ⎢
Λ
⎢
∗
∗
−λI
0
0
⎢
⎢
⎣
0
∗
∗
∗ −λ−1 I
∗
∗
∗
∗
−λ−1 I
⎤
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎦
4.13
with
T
. T Bi S)T S)j BT Bi V) φ V) φ BT ,
.T − HA
) 11 T)φ − Ai H
Φ
i
j
i
i
/
0
T
.
.T − Bi S)T − Bi V) φ H,
) 12 T)i − μ Ai H
Φ
j
4.14
) 22 μ H T H .
Φ
Proof of Theorem 4.2. Consider 4.8 and pre- and postmultiplying by M−1 and M−T ,
respectively, then we obtain
) ij Λij λ−1
Λ
⎡
⎤
DT μDaT
i
Dai Dbi ⎣ ai
⎦
μDai μDbi DbTi μDbTi
⎡
T ⎤
T
Eai
0
−1 ⎣Eai − Ebi Fj K φ
⎦
M−T ≺ 0,
λM
−Ebi Fj K φ 0
0
0
with Λij defined by 3.5.
4.15
Mathematical Problems in Engineering
15
By Schur complement, we obtain,
⎡
T
) 11 Φ
) 12 Dai Dbi M−1 EaT −M−1 Ebi Fj K φ
Φ
i
⎢
⎢
) 22 μDai μDbi
0
0
Φ
⎢
⎢
⎢
∗ −λI
0
0
0
⎢
⎢
∗
∗
−λI
0
0
⎢
⎣
0
∗
∗
∗
−λ−1 I
∗
∗
∗
∗
−λ−1 I
⎤
⎥
⎥
⎥
⎥
⎥ ≺ 0,
⎥
⎥
⎥
⎦
4.16
with
T
) 11 M−1 P φ M−T − Ai − Bi Fj − Bi K φ M−T − M−1 Ai − Bi Fj − Bi K φ ,
Φ
) 12 M−1 Pi εXM−T − μ Ai − Bi Fj − Bi K φ M−T M−1 ,
Φ
4.17
) 22 μ M−T M−1
Φ
for the following variables definition:
. M−1 ,
H
. T,
S)j HF
j
. i εXH
.T ,
T)i HP
. ρT ,
V)j HK
. T,
V) HR
. φH
.T ,
T)φ HP
. H
.T .
Y) HX
4.18
If LMI in 4.11 holds then the closed-loop continuous fuzzy system 2.8 is
asymptotically stable.
.−T , Kρ V)ρT H
.−T , and R V) T H
.−T . This
The control gains are given by Fi S)Ti H
completes the proof.
5. Numerical Examples
In order to show the improvements of the proposed approaches over some existing results, in
this section, we present a numerical example in which we present the feasible area for a T-S
fuzzy system. Indeed, we compare the proposed fuzzy Lyapunov approaches Theorem 3.4
with result provided by 17, and in 14, Theorem 6. A second example is given to improve
the given gains of robust PDC controller.
Example 5.1. Consider the following continuous T-S fuzzy system:
ẋt r
i1
hi ztAi xt,
5.1
16
Mathematical Problems in Engineering
2
1.8
1.6
1.4
b
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
a
Figure 1: Feasible area provided by 17 , 14 and Theorem 3.1 •.
with
r 2;
3.6 −1.6
−a −1.6
,
A2 ,
6.2 −4.3
6.2 −4.3
−0.45
−b
,
B2 ,
B1 −3
−3
A1 5.2
where a ∈ 0, 25, b ∈ 0, 2, considering μ 0.04 and φ1,2 1.
The proposed approach Theorem 3.4 gives less conservative stabilization conditions
Figure 1 than some recent results provided by 14, 17.
Example 5.2. Consider the uncertain continuous T-S fuzzy system given by 2.8 with
3.6 −1.6
−a −1.6
,
A2 ,
6.2 −4.3
6.2 −4.3
−0.45
−b
B1 ,
B2 ,
−3
−3
0.3 0
0.3
Da1 Ea1 ,
Db1 Eb1 ,
0 0
0
−0.2 0
−0.2
,
Db2 Eb2 Da2 Ea2 0 0.3
0.3
r 2;
A1 5.3
Mathematical Problems in Engineering
17
for a 1, b 0.5, considering μ 0.04, λ 0.1 and φ1,2 1, we find the following gains
values:
F1 −5.03401.5230,
K1 −180.233465.7330,
F2 3.7868 − 0.8082,
K2 −180.949465.9073,
R 201.9216 − 71.7345.
5.4
6. Conclusion
This paper provided new conditions for the stabilization with a class of PDC controller
of Takagi-Sugeno fuzzy systems in terms of a combination of the LMI approach and the
use of nonquadratic Lyapunov function as fuzzy Lyapunov function. In addition, the time
derivative of membership function is considered by the PDC fuzzy controller and the slack
matrix variables are introduced in order to facilitate the stability analysis. An approach to
design an observer is derived in order to estimate variable states. In addition, a new condition
of the stabilization of uncertain system is given in this paper.
The stabilization condition proposed in this paper is less conservative than some of
those in the literature, which has been illustrated via examples.
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