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

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

 SciTePress

2 PRELIMINARIES

A T-S fuzzy model of a plant with r rules can be

represented as

Plant Rule i:

IF x

(t)is M

AND ... AND x

(t)is M

THEN

() () (), 1,...,

tAxtButi r

=+ =

where x(t)∈



is the state vector, u(t)∈

\ is the

control input, the matrices

and

B are of

appropriate dimensions,

(j=1,2,…, n) is the j

fuzzy set of the i

rule, and, x

(t) (k=1,2,…,g) are

the premise variables. It should be noted that

() ()

txt



for the continuous-time T-S fuzzy

model and

() ( 1)xt xt

=+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:

x(t) ( ( )){ ( ) ( )}

iii

hxt Axt But

∑

(1)

where

(())

( ( )) ( ( )), ( ( )) ,

(())

() (), ..., () .

iiji

wxt

wxt M xt hxt

wxt

xt x t x t

⎡⎤

⎣⎦

∏

∑

(())

ij j

xt is the grade of membership of

()

. It is assumed that (()) 0

wxt ≥ ,

i=1,…, r,

(()) 0

wxt

∑

for all t. Therefore,

(()) 0

hxt ≥ ,

(()) 1

hxt

∑

for all t. Each linear

consequent equation given by

() ()

xt But+ is

called a subsystem. The free system of Eq.(1) is

defined as

(()) ()

()

(())

wxt Axt

wxt

∑

(2)

In this paper, it is also assumed that A

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

∀

)

AP PA

(3)

(

APA P

−

< ) (4)

Proof: See (Tanaka, 1992).

Note that Eq.(3) and Eq.(4) depends only on

In other words, it does not depend on (())

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) ()(), ()

tAtxtxtx

== (5)

is called uniformly stable if there exists a finite

positive constant γ such that for any t

and x

the

corresponding solution satisfies

() ,

txtt

≤

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

and x

the corresponding solution satisfies

() ,

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

APA-P<0

and

BPB-P<0

where A, B, and P

are n×n matrices, then

APB+BPA-2P<0

(Jamshidi, 1997) .

3 MAIN RESULTS

Theorem: The equilibrium point of the fuzzy system

x(t+1) ( ( )) ( ) ( )

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

()

Pt I

≤≤ (7)

() ( 1) () () , 1,...,

tPt At Pt I i r

+−≤−= (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 and ()

t for any t

and x

, one

obtains the following relations for t≥ t

() () () ()

txtPtxt

−≤−

(9)

( 1)( 1)( 1) ()()() ()

tPtxt xtPtxt xt

+++− ≤−

(10)

Furthermore, by the combination of Eq. (9) and

Eq.(10) one gets

( 1) ( 1) ( 1) (1 ) () () ()

tPtxt xtPtxt

+ + +≤−

(11)

Then, substituting Eq.(6) in Eq.(11) one obtains

() () [ () ( 1) () (1 ) ()]

() ()

ij i j

xxxAtPtAt Ptx

ωω

+−−

≤

∑

(12)

Eq. (12) implies

((1)(1)())

ii i

APt A Pt x

ωω

⎡

⎤

+−−

⎢

⎥

⎣

⎦

∑

[(1) (1)2(1)()]

TT T

ij i j j i

x APt A APt A Pt x

ωω

⎡⎤

++ +−−

⎢⎥

⎣⎦

≤

∑

(13)

Using Eq.(7) and Eq.(8) one obtains the

following inequalities:

[ ()( 1) () (1 )()]() 0

xAtPt At Ptxt

−− ≤ (14)

[ ()( 1) () (1 )()]() 0

xAtPt At Ptxt

−− ≤

(15)

Using Lemma (Jamshidi, 1997), Eq.(14) and Eq.(15)

the following relation

() ( 1) () () ( 1) () 2(1 ) ()

ijji

tPt A t A tPt At Pt

++ +−−

(16)

can be rewritten as

( ( ) ( )) ( 1)( ( ) ( ))

() ( 1) () (1 ) ()

ij ij

tAtPt AtAt

AtPt At Pt

A t Pt A t Pt

−

−+−+

+−− +

+−−

(17)

Since P(t+1) is positive definite, it is obvious that

( ( ) ( )) ( 1)( ( ) ( )) 0

ij ij

At At Pt At At

−

−+−≤

and

() ( 1) () (1 ) () 0, ,

tPt At Pt ij

+−− ≤∀.

It follows from Eq. (17)

() ( 1) () () ( 1) () 2(1 ) () 0

ijji

AtPt At AtPt At Pt

++−−≤

(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

≤

−<

(19)

STABILITY OF TAKAGI-SUGENO FUZZY SYSTEMS

215

Setting

=− in Eq.(11) and iterating it for t≥t

one obtains for t≥t

2( )

() () () ( )

ooo

tPtxt xPt x

−

≤ (20)

Using Eq. (7), the following expression can be

obtained for t≥t

2( )

()

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

()

Pt I

ηρ

≤≤

A (t)P(t+1)A (t) - P(t) 0 , i = 1, ..., r≤

where

and

are finite positive constants.

Theorem: Suppose the fuzzy system given in

Eq.(6) is uniformly exponentially stable. Then there

exists a positive constant

such that if

()

Δ≤for all t and i=1,…, r, then

x(t+1) ( ( ))[ ( ) ( )] ( )

iii

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

≤

( ) ( 1) ( ) ( ) i = 1, ..., r

A t Pt A t Pt I

+−≤−

where

and

are finite positive constants. Suppose

that there exists an integer t

such that P(t

) 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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Tanaka, K., Sugeno, M., 1990. Stability analysis of fuzzy

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