STABILITY OF TAKAGI-SUGENO FUZZY SYSTEMS
İlker Üstoğlu
Istanbul Technical University, Faculty of Electrical and Electronic Engineering,
Control Engineering Department, Maslak, TR-34469, Istanbul, Turkey
Keywords: Takagi-Sugeno fuzzy systems, Uniform and Exponential stability, Time varying systems.
Abstract: Takagi-Sugeno (T-S) fuzzy models are usually used to describe nonlinear systems by a set of IF-THEN
rules that gives local linear representations of subsystems. The overall model of the system is then formed
as a fuzzy blending of these subsystems. It is important to study their stability or the synthesis of stabilizing
controllers. The stability of TS models has been derived by means of several methods: Lyapunov approach,
switching systems theory, linear system with modeling uncertainties, etc. In this study, the uniform stability,
and uniform exponential stability of a discrete time T-S model is examined. Moreover, a perturbation result
and an instability condition are given. The subsystems of T-S models that is studied here are time varying
and a new exponential stability theorem is given for these types of TS models by examining the existence of
a common matrix sequence.
1 INTRODUCTION
Fuzzy systems can approximate a wide class of
nonlinear systems as accurately as required with
some number of fuzzy IF-THEN rules. They are
known as universal approximators, and their use
offers many advantages (L.X.Wang, 1996). Stability
is the most important concept for analysis and
design of a control system. Stability analysis of
fuzzy systems has been difficult because fuzzy
systems are essentially nonlinear systems (Tanaka
1996, Calcev, 1998, Kim, 2001). The issue of the
stability of fuzzy control systems has been studied
using nonlinear stability frameworks (Tanaka,
1990).
Takagi-Sugeno (T-S) fuzzy models (Takagi,
1985) are nonlinear systems in nature. In this type of
fuzzy model the consequent part of a fuzzy rule is a
mathematical formula, representing local dynamics
in different state space regions (subsystems) as
linear input-output relations (Tanaka, 1996). Thus,
T-S fuzzy systems are considered as a weighted
average of the values in the consequent parts of the
fuzzy rules. The overall model of the system is
consequently a fuzzy blending of these subsystems.
Recently, fuzzy control and modeling is being
used in many practical industrial applications. One
of the first questions to be answered is the stability
of the fuzzy system. Tanaka and Sugeno (Tanaka,
1992), have provided a sufficient condition for the
asymptotic stability of a fuzzy system in the sense of
Lyapunov through the existence of a common
Lyapunov function for all the subsystems.
A system is said to be stable in the sense of
Lyapunov if its trajectories can be made arbitrarily
close to the origin for any initial starting state. When
a system is stable and initial states that are close to
the region of origin converge to the origin, the
system has asymptotic stability. A stable system in
Lyapunov sense does not guarantee asymptotic
stability because asymptotic stability is stricter than
Lyapunov stability.
Additionally, one needs to know how fast the
system converges to the equilibrium point. On the
other hand, exponential stability is used to estimate
how fast the system trajectory approaches and
converges to the equilibrium point as time goes to
infinity. Since exponential stability is stricter than
asymptotic stability it guarantees both Lyapunov
stability and asymptotic stability but not vice versa.
The preliminaries were presented in Section 2.
Section 3 discusses the main results on the uniform
stability, uniform exponential stability and
instability. Moreover, a perturbation result is
presented. Finally, Section 4 contains some
concluding remarks.
213
Üsto
˘
glu Ä
ˇ
r. (2006).
STABILITY OF TAKAGI-SUGENO FUZZY SYSTEMS.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 213-216
DOI: 10.5220/0001214802130216
Copyright
c
SciTePress
2 PRELIMINARIES
A T-S fuzzy model of a plant with r rules can be
represented as
Plant Rule i:
IF x
1
(t)is M
i1
AND ... AND x
g
(t)is M
ig
THEN
() () (), 1,...,
ii
x
tAxtButi r
δ
=+ =
where x(t)
n
\
is the state vector, u(t)
m
\ is the
control input, the matrices
i
A
and
i
B are of
appropriate dimensions,
ij
M
(j=1,2,…, n) is the j
th
fuzzy set of the i
th
rule, and, x
k
(t) (k=1,2,…,g) are
the premise variables. It should be noted that
() ()
x
txt
δ
=
for the continuous-time T-S fuzzy
model and
() ( 1)xt xt
δ
=+for the discrete time T-S
fuzzy model.
Given a pair of (x(t), u(t)), the resulting fuzzy
system model is inferred as follows:
1
x(t) ( ( )){ ( ) ( )}
r
iii
i
hxt Axt But
δ
=
=+
(1)
where
1
1
1
(())
( ( )) ( ( )), ( ( )) ,
(())
() (), ..., () .
g
i
iiji
r
i
i
i
g
wxt
wxt M xt hxt
wxt
xt x t x t
=
=
==
⎡⎤
=
⎣⎦
(())
ij j
M
xt is the grade of membership of
()
j
x
t
in
ij
M
. It is assumed that (()) 0
i
wxt ,
i=1,…, r,
1
(()) 0
r
i
i
wxt
=
>
for all t. Therefore,
(()) 0
i
hxt ,
1
(()) 1
r
i
i
hxt
=
for all t. Each linear
consequent equation given by
() ()
ii
A
xt But+ is
called a subsystem. The free system of Eq.(1) is
defined as
1
1
(()) ()
()
(())
r
ii
i
r
i
i
wxt Axt
xt
wxt
δ
=
=
=
(2)
In this paper, it is also assumed that A
i
matrices
are time varying, where the coefficients are real
matrix sequences defined for all integer t, from –
to +
. Therefore, the consequent part of each IF-
THEN rule has a linear time varying state equation.
Tanaka and Sugeno (1992) proposed a theorem
on the stability analysis of a T-S fuzzy model, which
was an important breakthrough in the field of fuzzy
control. They proved that finding a common
symmetric positive definite matrix P for all the
subsystems could show the stability of a T-S fuzzy
model. This sufficient condition for ensuring
stability of Eq.(2) is given as follows.
Theorem: The equilibrium of the continuous-
time (discrete-time) T-S fuzzy model (namely, x=0)
is globally asymptotically stable if there exists a
common symmetric positive definite matrix P such
that (
i = 1, ..., r
)
0
T
ii
AP PA
+
<
(3)
(
0
T
ii
APA P
< ) (4)
Proof: See (Tanaka, 1992).
Note that Eq.(3) and Eq.(4) depends only on
i
A
.
In other words, it does not depend on (())
i
wxt . It is
clear that this theorem reduces to Lyapunov stability
theorem for continuous-time (discrete-time) linear
systems when r =1. It should be noted that the total
system might not be stable even if every subsystem
is stable. Eq.(3) and Eq.(4) are sufficient conditions
for stability, but are not necessary conditions. To
satisfy these conditions most of the time a trial and
error type procedure has been used.
In recent years, the stability analysis and control
design problems of fuzzy systems are reduced to
linear matrix inequality (LMI) problems
Numerically, LMI problems can be solved very
efficiently using the interior point algorithms (Boyd,
1994). However, the problem of finding a common
P matrix becomes a very difficult job even by the
LMI method as the number of fuzzy rules increases.
Definition: The discrete-time linear state
equation
(1) ()(), ()
oo
x
tAtxtxtx
+
== (5)
is called uniformly stable if there exists a finite
positive constant γ such that for any t
o
and x
o
the
corresponding solution satisfies
() ,
oo
x
txtt
γ
>
Definition: Eq.(5) is called uniformly
exponentially stable if there exists a finite positive
constant γ and a constant 0≤λ<1 such that for any t
o
and x
o
the corresponding solution satisfies
() ,
o
tt
oo
x
txtt
γλ
>
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214
It is called
uniform when γ does not depend on the
choice of initial time (Rugh, 1996).
Lemma: If P is a positive definite matrix such
that
T
APA-P<0
and
T
BPB-P<0
where A, B, and P
are n×n matrices, then
TT
APB+BPA-2P<0
(Jamshidi, 1997) .
3 MAIN RESULTS
Theorem: The equilibrium point of the fuzzy system
1
x(t+1) ( ( )) ( ) ( )
r
ii
i
hxt Atxt
=
=
(6)
is uniformly exponentially stable if there exists a
common n×n matrix sequence P(t) that for all t is
symmetric and such that
()
I
Pt I
η
ρ
≤≤ (7)
() ( 1) () () , 1,...,
T
ii
A
tPt At Pt I i r
ν
+−= (8)
where η, ρ and ν are positive constants.
Proof: Suppose P(t) satisfies the requirements of
the theorem. Multiplying both sides of Eq. (7) and
Eq. (8) by
()
T
x
t and ()
x
t for any t
o
and x
o
, one
obtains the following relations for t t
0
2
1
() () () ()
T
x
txtPtxt
ρ
−≤
(9)
2
( 1)( 1)( 1) ()()() ()
TT
x
tPtxt xtPtxt xt
ν
+++
(10)
Furthermore, by the combination of Eq. (9) and
Eq.(10) one gets
( 1) ( 1) ( 1) (1 ) () () ()
TT
x
tPtxt xtPtxt
ν
ρ
+ + +≤−
(11)
Then, substituting Eq.(6) in Eq.(11) one obtains
,1
,1
() () [ () ( 1) () (1 ) ()]
0
() ()
r
TT
ij i j
ij
r
ij
ij
xxxAtPtAt Ptx
xx
ν
ωω
ρ
ωω
=
=
+−
(12)
Eq. (12) implies
2
1
,1
((1)(1)())
r
TT
ii i
i
r
ij
ij
x
APt A Pt x
ν
ω
ρ
ωω
=
=
+−
+
,1
[(1) (1)2(1)()]
0
r
TT T
ij i j j i
ij
r
ij
ij
x APt A APt A Pt x
ν
ωω
ρ
ωω
<
=
⎡⎤
++ +
⎢⎥
⎣⎦
+
(13)
Using Eq.(7) and Eq.(8) one obtains the
following inequalities:
[ ()( 1) () (1 )()]() 0
TT
ii
xAtPt At Ptxt
ν
ρ
+
−− (14)
[ ()( 1) () (1 )()]() 0
TT
jj
xAtPt At Ptxt
ν
ρ
+
−−
(15)
Using Lemma (Jamshidi, 1997), Eq.(14) and Eq.(15)
the following relation
() ( 1) () () ( 1) () 2(1 ) ()
TT
ijji
A
tPt A t A tPt At Pt
ν
ρ
++ +
(16)
can be rewritten as
( ( ) ( )) ( 1)( ( ) ( ))
() ( 1) () (1 ) ()
() ( 1) () (1 ) ()
T
ij ij
T
ii
T
jj
A
tAtPt AtAt
AtPt At Pt
A t Pt A t Pt
ν
ρ
ν
ρ
−++
+− +
+−
(17)
Since P(t+1) is positive definite, it is obvious that
( ( ) ( )) ( 1)( ( ) ( )) 0
T
ij ij
At At Pt At At
−+
and
() ( 1) () (1 ) () 0, ,
T
ii
A
tPt At Pt ij
ν
ρ
+− .
It follows from Eq. (17)
() ( 1) () () ( 1) () 2(1 ) () 0
TT
ijji
AtPt At AtPt At Pt
ν
ρ
+
++
(18)
This proves that Eq. (13) is valid. It can be easily
seen from Eq.(7) and Eq.(8) that ρ≥ν, so the
following inequality can be stated
0(1 )1
ν
ρ
−<
(19)
STABILITY OF TAKAGI-SUGENO FUZZY SYSTEMS
215
Setting
2
1
ν
λ
ρ
=− in Eq.(11) and iterating it for tt
o
one obtains for tt
o
,
2( )
() () () ( )
o
tt
TT
ooo
x
tPtxt xPt x
λ
(20)
Using Eq. (7), the following expression can be
obtained for tt
o
,
22
2( )
()
o
tt
o
x
tx
ηρλ
(21)
If one divides both sides of Eq.(21) by
η and takes
the positive square root, the uniform exponential
stability condition is obtained.
Theorem: The fuzzy system given in Eq.(6) is
uniformly stable if there exists a matrix sequence
P(t) that for all t is symmetric and such that
()
I
Pt I
ηρ
≤≤
T
ii
A (t)P(t+1)A (t) - P(t) 0 , i = 1, ..., r
where
η
and
ρ
are finite positive constants.
Theorem: Suppose the fuzzy system given in
Eq.(6) is uniformly exponentially stable. Then there
exists a positive constant
δ
such that if
()
i
At
δ
Δ≤for all t and i=1,…, r, then
1
x(t+1) ( ( ))[ ( ) ( )] ( )
r
iii
i
hxt At At xt
=
=+Δ
is uniformly exponentially stable.
Theorem: Suppose there exists a matrix sequence
P(t) which for all t is symmetric and such that
()Pt
ρ
( ) ( 1) ( ) ( ) i = 1, ..., r
T
ii
A t Pt A t Pt I
η
+−
where
ρ
and
η
are finite positive constants. Suppose
that there exists an integer t
u
such that P(t
u
) is not
positive semidefinite. Then the fuzzy system given
in Eq.(6) is not uniformly stable.
4 CONCLUSIONS
The exponential stability is used to estimate how fast
the system trajectory approaches and converges to
the equilibrium point as time goes to infinity, and it
is stricter than asymptotic stability. Therefore,
exponential stability guarantees both Lyapunov
stability and asymptotic stability but not vice versa.
In this study, some theorems for the stability and
instability of the Takagi-Sugeno fuzzy systems are
introduced. The consequent part of each T-S rule
studied here are time varying. The uniform stability
and uniform exponential stability theorems are given
for these types of T-S models by examining the
existence of a common matrix sequence. Moreover,
a perturbation result is presented.
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