c 2005 Institute for Scientific
°
Computing and Information
INTERNATIONAL JOURNAL OF
INFORMATION AND SYSTEMS SCIENCES
Volume 1, Number 3-4, Pages 339–346
STABILITY ANALYSIS FOR DISCRETE T-S FUZZY SYSTEMS
XIAOGUANG YANG1 , XIAODONG LIU2 , QINGLING ZHANG1 , AND PEIYONG LIU1
Abstract. Quadratic stability problems for discrete T-S fuzzy system have
been studied in this paper. New quadratic stability conditions for discrete TS fuzzy system, which are simpler and more relaxed than that in E.Kim and
H.Lee 2000, are proposed. By using LMIs and MATLAB, the controllers can
be directly obtained. The examples show that the design method proposed in
this paper is more relaxed, efficient and optimal than the previous papers such
as E.Kim and H.Lee 2000.
Key Words. discrete T-S fuzzy system, Schur complement, quadratic stability, linear matrix inequality(LMI).
1. Introduction
Since the inception of the fuzzy logic by Zadeh in 1965, the fuzzy logic has found
extensive applications in the areas of industrial systems and consumer products. In
recent years, it becomes quite popular to employ the so-called T-S fuzzy model to
represent or approximate a nonlinear system. It is because the control technique
based on T-S fuzzy model allows the designers to analyze and design nonlinear
systems by taking full advantage of the strength of modern linear control theory,
what’s more, different from the heuristics-based approach to fuzzy control system,
it can provide a systematic design methodology for nonlinear fuzzy control systems.
This permits appliance of conventional linear system in analysis and design of the
fuzzy control systems. Because some researches consider little the interactions
among the fuzzy subsystems, some results such as E.Kim and H.Lee 2000[1], tend
to give a more conservative condition. In this paper, we propose a new quadratic
stability condition which is simpler than that in [1] and prove that if there exist
matrices satisfying the conditions in [1], then there exist matrices satisfying the
conditions in Theorem 1 of this paper and with an example, we illustrated that the
new conditions in Theorem 1 are not equivalent to the conditions in [1]. Finally,
based on the LMIs, the controller designing methods in the example, we show the
effectiveness of the method in this paper.
2. The discrete T-S fuzzy model and quadratic stability
This paper considers the fuzzy dynamic model of T-S described by the following
fuzzy IF-THEN rules:
IF ξ1 is M1i and , and ξp is Mpi , THEN
x(t + 1) = At x(t) + Bt u(t),
z(t) = Ct x(t), i = 1, 2, · · · , r.
Received by the editors June 1, 2004 and, in revised form, January 22, 2005.
2000 Mathematics Subject Classification. 93A30.
This work is supported by National Nature Science Foundation of China under grant No.
60574011 and No. 60174014 and the Science Funds of Dalian Maritime University .
339
340
X. G. YANG, X.D. LIU, Q. L. ZHANG, AND P. Y. LIU
Where x ∈ Rn is the state, z ∈ Rq is the output, u ∈ Rm is the input, At ∈ Rn×n ,
Bt ∈ Rn×m ,ξ1 , · · · , ξp are premise variables. We set ξ = (ξ1 , · · · , ξp )T . It is assumed
that the premise variables do not depend on control variables and the disturbance.
Then the state equation and the output are defined as follows:
(1)
x(t + 1) =
r
X
λt (ξ)(At x(t) + Bt u(t)),
t=1
(2)
z(t) =
r
X
λt (ξ)Ct x(t).
t=1
Here
p
Y
βi (ξ(t))
, βj (ξ(t)) =
λt (ξ(t)) = Pr
Mkj(ξ(t)) .
j=1 βj (ξ(t))
k=1
Mkj (•) is the membership function of fuzzy set Mkj . In the following we always
assume that:
r
X
(3)
λi (ξ(t)) = 1, λk (ξ(t)) ≥ 0, k = 1, · · · , r.
i=1
Definition [2] For discrete T-S fuzzy system (1), when u(t) ≡ 0, if there exists and a positive-definite matrix X such that 4V (x(t)) < 0,where V (x(t)) =
xT (t)Xx(t), then discrete T-S fuzzy system (1) is called quadratically stable.
3. The stability condition
The following lemma is similar to the paper [3].
Lemma 1 If there exist matrices Fi , i = 1, 2, · · · , r and a positive definite
matrix X such that


R11 · · · R1r
 ..
..  < 0,
..
(4)
 .
.
. 
Rr1 · · · Rrr
where
(5)
(6)
Rii = GTii XGii − X, i = 1, 2, · · · , r,
Rij = (
Gij + Gji T
Gij + Gji
) X(
) − X, i < j.
2
2
Gij = Ai + Bi Fj
Then for discrete T-S fuzzy system (1) , the state feedback is
u(t) =
r
X
λi (ξ(t))Fi x(t).
i=1
Stabilizes the closed-loop system
x(t + 1) =
r X
r
X
j=1 i=1
λi (ξ(t))λj (ξ(t))(Ai + Bi Fj )x(t).
STABILITY ANALYSIS FOR DISCRETE T-S FUZZY SYSTEMS
341
Proof :If a Lyapunov function candidate is chosen as V (x(t)) = xT (t)Xx(t),
then
∆V (x(t)) =
V (x(t + 1)) − V (x(t))
= xT (t + 1)Xx(t + 1) − xT (t)Xx(t)

T
r X
r
X
= 
λi (ξ(t))λj (ξ(t))Gij x(t) X
i=1 j=1


r X
r
X

λi (ξ(t))λj (ξ(t))Gij x(t) − xT (t)Xx(t)
i=1 j=1


r X
r
X
= xT (t) 
λi (ξ(t))λj (ξ(t))(GTij XGij − X) x(t)
i=1 j=1
=
r
X
r
X
£
¤
λ2i (ξ(t))xT (t) GTii XGii − X x(t) +
λi (ξ(t))λj (ξ(t))xT (t)
i=1
"µ
Gij + Gji
2

T 
λ1 x
 ..  
≤  .  
λr x
< 0.
¶T
µ
#
¶
R11
..
.
Gij + Gji
−X
2

· · · R1r
λ1 x
..   ..
..
.
.  .
Rr1
···
X
Rrr
i<j
x(t)



λr x
Q.E.D.
In Lemma 1, (5) and (6) are nonlinear matrix inequalities. If for X > 0, define
Fi = Mi X −1 , i = 1, 2, · · · , r, substituting into (5) and (6), they can be converted to
linear matrix inequalities using Schur complement , hence there exists the following
theorem.
Theorem 1 If there exist matrices Mi , Z, Yij , where Z is a positive definite
matrix , Yii are symmetric matrices, Yji = YijT , i 6= j, i, j = 1, 2, · · · , r, satisfy the
following LMIs:
·
¸
Yii − Z
ZATi + MiT BiT
(7)
< 0,
Ai Z + Bi Mi
Z
(8)
·
Yij − Z
1
(A
Z
+
B
M
i
i
j + Aj Z + Bj Mi )
2

(9)
Y11
 ..
 .
Yr1
1
2
¡
···
..
.
···
ZATi + MjT BiT + ZATj + MiT BjT
−Z

Y1r
..  < 0,
. 
Yrr
then for discrete T-S fuzzy system (1) ,when the state feedback
u(t) =
r
X
i=1
λi (ξ(t))Fi x(t),
¢ ¸
≤ 0,
342
X. G. YANG, X.D. LIU, Q. L. ZHANG, AND P. Y. LIU
stabilizes the closed-loop system
x(t + 1) =
r X
r
X
λi (ξ(t))λj (ξ(t))(Ai + Bi Fj )x(t).
i=1 j=1
Here Fi = Mi X −1 , i = 1, 2, · · · , r.
Proof: let X = Z −1 ,Fi = Mi Z −1 .Pre- and postmultiply (7) and (8) with X
and pre- and postmultiply (9) with diag(X, · · · , X), we have
¡ T
¢
(10)
Ai + FiT BiT X (Ai + Fi Bi ) − X < XYii X, i 6= j, i, j = 1, 2, · · · , r,
(11)
¢ 1
1¡ T
Ai + FjT BiT + ATj + FiT BjT X (Ai + Bi Fj + Aj + Bj Fi )−X ≤ XYij X+XYijT X,
2
2
i 6= j


XY11 X · · · XY1r X


..
..
..
(12)

 < 0.
.
.
.
XYr1 X
···
XYrr X
If a Lyapunov function candidate is chosen as V (x(t)) = xT (t)Xx(t), then we
have
∆V (x(t)) =
=
V (x(t + 1)) − V (x(t))
r
X
λ2i (ξ(t))xT (t)[ATi + FiT BiT )X(Ai + Bi Fi ) − X]x(t)
i=1
+2
λi (ξ(t))λj (ξ(t))xT (t)
i=1 j=1
¸
¢ 1
1¡ T
T T
T
T T
A + Fj Bi + Aj + Fi Bj X (Ai + Bi Fj + Aj + Bj Fi ) − X x(t)
2 i
2
r
X
λ2i (ξ(t))xT (t)XYii Xx(t)
·
<
r X
r
X
i=1
+

=
<
r X
r
X
¡
¢
λi (ξ(t))λj (ξ(t))xT (t) XYij X + XYijT X x(t)
i=1 j=1
T 
λ1 x
XY11 X
 ..  
..
 .  
.
λr x
XYr1 X
0.
···
..
.
···


XY1r X
λ1 x
  .. 
..
 . 
.
XYrr X
λr x
Q.E.D.
[1] states the following. The equilibrium of discrete T-S fuzzy system (1)
x(t + 1) =
r
X
hi (ξ) (Ai x(t) + Bi u(t))
i=1
is quadratically stabilizable via the fuzzy control
u(t) =
r
X
i=1
hi (ξ(t))(−Fi x(t))
STABILITY ANALYSIS FOR DISCRETE T-S FUZZY SYSTEMS
343
0
in the large if there exist symmetric matrices Q and Hij sand matrices Nis such
that
Q>0
·
¸
Q − Hii
QATi − NiT BiT
(13)
> 0(i = 1, · · · , r),
Ai Q − Bi Ni
Q
(14)
·
Q − Hij
1
(A
Q
−
B
i
i Nj + Aj Q − Bj Ni )
2
1
2
¡
QATi − NjT BiT + QATj − NiT BjT
Q
¢ ¸
≤ 0,
(i < j ≤ r)





(15)
H11
H12
..
.
H12
H22
..
.
···
···
..
.
H1r
H2r
..
.
H1r
H2r
···
Hrr



 > 0.

Proposition 1 If there exist matrices satisfying the conditions in [1] then there
exist matrices satisfying the conditions in Theorem 1 and they are not equivalent.
0
T
Proof: Since Hij s in [1] are symmetric matrices, hence, 2Hij = Hij + Hij
0
and if matrices Q,Hij s,Ni s satisfy [1], then Z = Q,Yij = −Hij , Mi=−Ni satisfies
Theorem 1 in this paper. In Example 1, the solution matrices of Theorem 1 in
this paper are different from the solution matrices of [1].Therefore, they are not
equivalent. Q.E.D.
Since each Hij , i 6= j in [1] is symmetric matrix and each Yij , i 6= j in Theorem
1 is not necessary symmetric matrix, hence there are more variables in Yij than
Hij .This implies that the conditions in Theorem 1 admit great more freedom (or
dimension) in guaranteeing the stability of the fuzzy control systems than [1].
Lemma 2 Assume that the number of rules that fire for all t is less than or equal
to s where 1 < s ≤ r.The equilibrium of the discrete fuzzy control system described
by (1)is asymptotically stable in the large if there exists a common positive definite
matrix X and a common positive semidefinite matrix Q such that
GTii XGii − X + (s − 1)Q < 0,
(16)
µ
(17)
Gij + Gji
2
¶T
µ
X
Gij + Gji
2
¶
− X − Q ≤ 0, i < j,
where
Gij = Ai + Bi Fj .
Using Schur complements, matrices inequalities in lemma 2 are converted linear
matrices inequalities, finding P > 0,Y > 0 and Mi (i = 1, · · · , r) such that
·
¸
P − (s − 1)Y P ATi + MiT BiT
(18)
> 0,
Ai P + Bi Mi
P
(19)
·
P +Y
1
(A
Z
+
B
M
i
i
j + Aj Z + Bj Mi )
2
1
2
¡
ZATi + MjT BiT + ZATj + MiT BjT
P
i<j
where
P = X −1 , Mi = Ki P, Y = P QP.
¢ ¸
≥ 0,
344
X. G. YANG, X.D. LIU, Q. L. ZHANG, AND P. Y. LIU
Obtained the local feedback gains Ki , a common positive definite matrix P and
a common positive semidefinite matrix Q:Ki = Mi P −1 , P = X −1 , Q = XY X . By
lemma 2, for α < 1, since the condition ∆V (x(t)) ≤ (α2 − 1)V (x(t))is equivalent to
µ
GTii P Gii − α2 P + (s − 1)Q < 0
¶
µ
¶
Gij+Gji T
Gij+Gji
P
− α2 P − Q ≤ 0, i < j
2
2
where
Gij = Ai + Bi Fj
Therefore, attenuation design problem for response speed in discrete fuzzy control system can be indicated:
min β(i = 1, 2, · · · , r)
satisfy
·
(20)
P > 0, Y ≥ 0
P ATi + MiT BiT
P
βP − (s − 1)Y
Ai P + Bi Mi
(21)
·
βP + Y
1
(A
Z
+
B
i
i Mj + Aj Z + Bj Mi )
2
1
2
¡
¸
>0
ZATi + MjT BiT + ZATj + MiT BjT
P
¢ ¸
≥0
i<j
where
β = α2 , 0 < β < 1
Lemma 3[4] If there exists positive definite matrix X and a series of symmetry
matrices Xij such that
GTii XGii − X + Xij < 0
(22)
µ
(23)
(24)
¶T µ
¶
Gij + Gji
Gij + Gji
X
− X + Xij ≤ 0
2
2


X11 X12 · · · X1r
 X12 X22 · · · X2r 


X= .
..
..  > 0
..
 ..
.
.
. 
X1r X2r · · · Xrr
then there is the same conclusion in lemma 2.
Lemma 4 If there exists a positive definite matrix X and a symmetry matrix
Q in lemma 2, then there must be a positive definite matrix P and a symmetry
matrix X − such that (22)—(24), and there is the same conclusion as in lemma 2.
Proof: Assume r = s.
If there exists positive definite matrix X and a symmetry matrix Q satisfying
(16) and (17), then can find small Qε such that
Qε > 0, Qε ≈ 0,
GTii XGii
µ
¶T
Gij + Gji
2
− X + (r − 1)Q + Qε < 0,
µ
¶
Gij + Gji
X
− X − Q ≤ 0, i < j,
2
where
Gij = Ai + Bi Fj .
STABILITY ANALYSIS FOR DISCRETE T-S FUZZY SYSTEMS
345
Since (16) is a inequality, if choice Xij as
Xii = (r − 1)Q + Qε ,
Xij = −Q,
then

X−
=





=




X11
X12
..
.
X12
X22
..
.
···
···
..
.
X1r
X2r
..
.
X1r
X2r
···
Xrr






(r − 1)Q + Q²
−Q
..
.
−Q
(r − 1)Q + Q²
..
.
···
···
..
.
−Q
−Q
..
.
−Q
−Q
···
(r − 1)Q + Q²
Following proof X − > 0. Let z = [z1T , · · · , zrT ]T , since
 T 
z1
(r − 1)Q + Q²
−Q
 z2T  
−Q
(r
−
1)Q
+ Q²


zT X −z =  .  
.
.
..
..
 ..  
T
−Q
−Q
zr
£
¤
z1 z2 · · · zr
r
r
X
X
=
(zi − zj )T Q(zi − zj ) +
ziT Qε zi > 0.
i<j


.

···
···
..
.
−Q
−Q
..
.
···
(r − 1)Q + Q²
i=1
Therefore, lemma 4 holds. Q.E.D.
4. A Simulation Example
Example 1: Consider the following fuzzy system:
IF x1 is M1 , THEN x(t + 1) = A1 x(t) + B1 u(t).
IF x2 is M2 , THEN x(t + 1) = A2 x(t) + B2 u(t).
Where
·
¸
·
¸
· ¸
·
¸
1 −0.5
−1 −0.5
1
−2
A1 =
, A2 =
, B1 =
, B2 =
.
1
0
1
0
1
1
By Theorem 1, we have the following solution matrices:
·
¸
£
¤
29.6816 7.1671
Z=
, M1 = 4.7694 −20.9665 ,
7.1671 46.9120
·
¸
£
¤
−59.6152
0.7011
M2 = −23.0275 −18.4002 , Y11 =
,
0.7011
−37.2818
·
¸
·
¸
−50.5764 −3.2078
−8.1049 −1.4772
Y22 =
, Y21 =
,
−3.2078 −38.5649
−1.4772 −10.5551
£
¤
£
¤
F1 = 0.2789 −0.4895 , F2 = −0.7072 −0.9665 ,
k F1 k= 0.5634, k F2 k= 0.7622.
Best value of t in the LMI program of MATLAB:-1.690431.





346
X. G. YANG, X.D. LIU, Q. L. ZHANG, AND P. Y. LIU
By [1], we have the following solution matrices:
¸
·
£
¤
8.0097 −4.6218
Z=
, M1 = −36.0174 47.1615 ,
−4.6218 21.0470
·
¸
£
¤
33.2869 5.6237
,
M2 = 12.3023 41.0364 , Y11 =
5.6237 57.8572
·
¸
·
¸
39.3299 −8.4022
−7.3565 −4.0307
Y22 =
, Y21 =
,
−8.4022 57.3252
−4.0307 −21.1360
£
¤
£
¤
F1 = −3.6686 1.4352 , F2 = 3.0471 2.6189 ,
k F1 k= 3.9393, k F2 k= 4.0179
Best value of t in the LMI program of MATLAB: -0.775569.
By Proposition and Example 1, we know that the solutions of Theorem 1 in this
paper are more optimal than [1]. The feedback gains F1, F2 obtained by Theorem
1 in Example 1 are much less than those obtained by [1] and the best value t is
also much less that obtain by [1]. In [1], E. Kim and H. Lee proved that its Th.12
is more relaxed than the conditions derived in the previous papers. Therefore by
Proposition 1 and Example 1, we know that the controllers obtained by Theorem
1 are more relaxed, efficient and optimal than the designs derived earlier.
5. Conclusions
In this paper, quadratic stability conditions for discrete T-S fuzzy system have
been studied. First, sufficient conditions for discrete T-S fuzzy system are presented
in terms of a set of nonlinear matrix inequalities, which guarantee the quadratic stability of the closed-loop fuzzy system. And then, the nonlinear matrix inequalities
are converted into LMIs using the Schur complement. Finally, simulation results on
discrete T-S fuzzy system illustrate the effectiveness of the proposed design method.
References
[1] Kim, E. and Lee, H., New Approaches to Relaxed Quadratic Stability Condition of Fuzzy
Control Systems, IEEE Trans. Fuzzy Syst., 8: 523-533, 2000.
[2] Wang, L.X., Fuzzy System and Fuzzy Control, Tsinghua University Publishing Company,
2003.
[3] Liu, X.D. and Zhang, Q.L., Approaches to Quadratic Stability Conditions And H∞ Control
Designs for T-S Fuzzy Systems, IEEE Trans. Fuzzy Syst., 11(6): 830-839, 2003.
[4] Tong, S.C., Wang, T., Wang, Y.P. and Tang, J.T., Design And Stability Analysis of Fuzzy
Control Systems, Since Press, 2004.
1 School
of Science, Northeastern University, Shenyang 110004, China;
2 Department
of Mathematics, Dalian Maritime University, Dalian 116026, China
E-mail: yxg978@sohu.com