Fuzzy stability criterion of a class of nonlinear systems

INFORMATION

SCIENCES

71,3-26

(1993)

3

Fuzzy Stability Criterion of a Class of Nonlinear Systems KAZUO

TANAKA

and MANABU

SAN0

Department of Mechanical Systems Engineering, 2-40-20, Kodatsuno, Kanazawa, 920 Japan

Kanazawa Unir!ersity,

ABSTRACT A fuzzy stability criterion of a class of nonlinear systems is discussed in accordance with the definition of stability in the sense of Lyapunov. First, a sufficient condition that guarantees stability of a fuzzy system is given in terms of Lyapunov’s direct method. Two concepts for stability of fuzzy systems are defined: locally stable and globally table. Second, a construction procedure for Lyapunov functions is presented. Finally, the construction procedure is applied to the fuzzy stability criterion of a class of nonlinear systems that can be approximated by fuzzy systems.

1.

INTRODUCTION

One of the most important concepts concerning the properties of control systems is stability. We already have some studies on the stability of fuzzy control systems [l, 21. It is said that stability analysis of a fuzzy system is difficult because it is essentially a nonlinear system. It is, however, important to analyze stability of fuzzy control systems in the design of fuzzy controllers. One of the authors has discussed stability analysis of fuzzy control systems using Lyapunov’s direct method in previous papers [5, 61. These papers give some theorems for stability of fuzzy systems in accordance with the definition of stability in the sense of Lyapunov. The main disadvantage of Lyapunov’s stability criterion is that it gives only the sufficient conditions for stability, not the necessary conditions. Furthermore, there are no unique methods of determining the Lyapunov function for a wide class of systems. We must find a Lyapunov function in order

to check

stability

of a fuzzy

system.

0 Elsevier Science Publishing Co., Inc. 1993 655 Avenue of the Americas, New York, NY 10010

0020-0255/93/$6.00

4

K. TANAKA AND M. SAN0

In this paper, we present a construction procedure to find Lyapunov functions as effectively as possible. Moreover, we show that the construction procedure can be widely applied not only to a fuzzy system but also to a class of nonlinear systems that can be approximated by fuzzy systems. This paper is organized as follows. Section 2 shows a type of fuzzy system used in this paper. Section 3 gives stability analysis of fuzzy systems. Section 4 discusses a construction procedure for Lyapunov functions. Section 5 shows the fuzzy stability criterion or a class of nonlinear systems that can be approximated by fuzzy systems. 2.

FUZZY

SYSTEM

The fuzzy model (fuzzy free system) proposed by Takagi and Sugeno 141 is of the form L’:IFx(k)

is A; and*++and x(k-n+l)

x’(k+l)=a;x(k)+

*~~+a;X(k--n+l),

is A’,,THEN (I)

where L’ (i= 1, 2 >..*, I) denotes the ith implication, 1 is the number of fuzzy implications, x’(k + 1) is the output from the ith implication, abs (p=O,l,..., n) are consequent parameters, x(k) -x(/~-n + 1) are state variables, and Abs are fuzzy sets. Each linear consequent equation is called a “subsystem.” Given an input (x(k), x(k - l), . . . , x(k -n + l)), the final output of the fuzzy model is inferred by taking the weighted average of the x”(k + 1)s:

where Cf= ,w”(k) > 0, w’(k) 2 0, x’(k + 1) is calculated for the input by the consequent equation of the ith implication, and the weight W’ implies the overall truth value of the premise of the ith implication for the input calculated as

(3)

FUZZY

STABILITY

CRITERION

5

The linear subsystem in the consequent be written in the matrix form

part of the ith implication can

A;x(k), where x(k)=[x(k),x(k-l),...,x(k-n+l)]*

and

Ai=

4

1

a; 0

“* ...

afmz ...

0

..

..

..

0

...

..:

0’

at_, . ..

ai 0

1’

The output of the fuzzy system is inferred as x(k+

1) = f: w’(k)Aix(k) i=

I

i i

w’(k).

(4)

i=l

Equation (4) is equivalent to Eq. (2). 3.

STABILITY

ANALYSIS

We have derived some results on stability of a fuzzy system in the previous papers [5, 61. Theorem 3.1 have been derived as a main stability theorem for fuzzy systems. The proof of Theorem 3.1 is given in the literature [5, 61. THEOREM 3.1. The equilibtium of a fuzzy system described by Eq. (4) is asymptotically stable in the large if there exists a common positive definite matrix P such that A;PA,-P
foriE{1,2,...,

(5)

l}, that is, for all the subsystems.

This theorem is reduced to the Lyapunov stability theorem for linear discrete systems when I = 1. Theorem 3.1 gives, of course, a sufficient

6

K. TANAKA AND M. SAN0

condition for ensuring stability of Eq. (4). Next, we define two concepts for stability of fuzzy systems. DEFINITION 3.1. A fuzzy system such that all the Ajs are stable matrices is said to be locally stable. DEFINITION 3.2. A stable fuzzy system is said to be globally stable.

We may intuitively guess that a fuzzy system is globally stable if it is locally stable. However, this is not the case in general: we shall discuss it in Example 3.1. A fuzzy system is locally stable if there exists a common positive definite matrix P. Conversely, there does not always exist a common positive definite matrix P even if a fuzzy system is locally stable. Of course, a fuzzy system may be asymptotically stable in the large even if there does not exist a common positive definite matrix P. However, we must notice that a fuzzy system is not always asymptotically stable in the large even if it is locally stable as shown in Example 3.1. EXAMPLE

L’:IFx(k-1)

3.1 [5]. Let us consider the following fuzzy free system:

,THENx’(k+l)

is

=x(k)

-0.5x(/k-l).

1

L*:IFx(k-1)

is 2 -1

,THENx*(k+l)=

-.+)-0.5.+-l).

1

From the linear subsystems, we obtain

Al=[; -;‘5],

A*=[-;

-;‘5].

Figure la and lb illustrates the behavior of the following linear systems for the initial condition x(O) = - 0.70 and x(1) = 0.90, respectively: x(k+

1) =A,x(k),

x(k+

1) =A,x(k).

These linear systems are stable because A, and A, are stable matrices. However, the fuzzy system that consists of these linear systems is unstable as shown in Figure lc. Obviously, in this example, there does not exist a common P because the fuzzy system is unstable.

FUZZY

STABILITY

7

CRITERION

(a)

k

Cc)

-5 Fig. 1 system.

(a) Behavior

of A,x(k).

(b) Behavior

4. CONSTRUCTION PROCEDURE LYAPUNOV FUNCTIONS

of A,x(k).

(c) Behavior

of the fuzzy

FOR

We have given a theorem for stability of fuzzy systems in the previous section. In order to guarantee stability of a fuzzy system, we must find a positive definite matrix P such that A;PA,-P
K. TANAKA

8

AND M. SAN0

THEOREM4.1. Assume that Ai is a stable matrix for i = 1,2,. . . ,l. A,A, is a stable matrix for i, j = 1,2,. . . , 1 if there exists a common positive definite matrix P such that A;PA,-P
(6)

for i = 1,2,. . . , 1. Proof. From Eq. (6), we obtain

ArArPA-A. - ArPA < 0’ I I I’ 11 From the preceding inequality and Eq. (6), we obtain A;A;PA,A;-P
AiAj must be a stable matrix for i, j = 1,2,. . . ,I.

Theorem 4.1 there does not common P only Now, assume

n

shows that if one of the AiAjs is not a stable matrix, then exist a common P. It is difficult to find effectively a using Theorem 4.1. that

Moreover, assume that P and Q are positive definite matrices such that ArPA-P=

-Q.

Then, the equation Ar PA - P = - Q is equivalent to

I

a11 alla12 2 2 -1 al2

a1la22

2a, ++a421 ,a21 2a12a22

-

1

a:2-

]I1 II Pll P12

a21a22 41 1

P22

=

-

411 q12 q22

.

(7)

More generally, Eq. (7) can be rewritten by using two mappings 77and 0 as q(A) e(P) = - e(Q),

(8)

FUZZY

STABILITY

CRITERION

9

where A, P and QER”~“, BERGS”, 1)/2. Obviously, Eq. (7) is equivalent 2 all

v(A) =

-1

%,a21

411Q12

I

~(P),~(Q)ER”‘, to Eq. (8) when

a,,a22

4,

+~,,a,,

-

*

2a12a2,

42

~(r)=[$]

and

a21422 a:2-

1

Bo_[~~~].

We notice that 0 is bijective. A common by solving Eq. (5). It is, however, possible matrix P such that P=K’(

and m=n(n+

P cannot be directly obtained to obtain a positive definite

-(n(A))-%(Q))

(9)

from Eq. (8) if n(A) is a nonsingular matrix. Next, we give a necessary and sufficient condition common P using the mappings n and 0.

for the existence

of a

THEOREM 4.2.Assume that A, and A, are stable matrices and $A,) and n(A,) are nonsingular matrices. Then, there exits a positive definite matrix Q such that

@W(Q)) where G = T(A,X~(A,))-’ definite matrix P such that

Proof. ( 3 ) Because matrix P such that

> 0,

( 10)

if and only if there exists a common positive

A;PA,

-P
A;PA2

- P < 0.

A, is a stable matrix,

A?;PA, -P=

-Q

there exists a positive

definite

K. TANAKA AND M. SAN0

10

for a positive definite matrix Q. From Eq. (91,

V’) = -(TM)-‘e(Q). If we let Q’ = ~T’(GB(Q)),

(11)

then

e(Q’> = Ge(Q) =~(A~(vG%))-~~(QL

(12)

where notice from Eq. (10) that Q’ > 0. From Eq. (11) and Eq. (121, we obtain

e(Q’) = - q(Az) V’).

(13)

ATPA,-P=

(14)

Then,

Therefore,

-Q’
we have AT;PA, - P < 0, ATPA

-P < 0.

( = 1 Assume that ATPA, -P= A;PA,-P=

-Q, -Q’,

where Q,Q’ > 0. By using 77and 8, we obtain 77(4)W)

= -e(Q)*

(15)

rl(A,)o(P)

= - e(Q’),

(16)

From Eq. (151,

W’) = +(A,))-'e(Q)).

(17)

FUZZY

STABILITY

11

CRITERION

By substituting Eq. (17) into Eq. (16), we obtain

eta’)

=~tA,)(77(A,))-'etQ)).

Therefore,

= O-‘(GO(Q))

>O.

n

COROLLARY4.1. Assume that Ai is a stable matrix for i = 1,2,. . . ,I and n(Ai) is a nonsingular matrix for i = 1,2,. . . ,l. Then there exists a positive definite matrix Q such that

‘-‘(Gjfl(Q))>O,

(18)

for j=2,3,..., 1, where Gj= ~(AjXn(A,))-’ common positiue definite matrix P such that

if and only if there exists a

A;PA,-P
,..., 1.

Proof. Follows directly from Theorem

4.2.

w

Corollary 4.1 shows that there exists a common positive definite matrix P such that A;PA,-P
0-l (Gje(Q))

>O,

for j=2,3,..., 1. Conversely, there does not exit a common positive definite matrix P if there exits an integer j such that

0-l (Gje(Q)) for any positive definite matrix Q.

G+0,

K. TANAKA

12

AND

M. SAN0

The procedure to find a common positive definite matrix P is given in the following text. This construction procedure consists of five steps. Here, it is assumed that Ais are stable matrices. Step 1. First, we calculate AiAjs for i, j = 1,2,. . .,I and check the existence of a common P by Theorem 4.1. If there does not exit a common P, stop. Step 2. Second,

Pjs such that

we calculate

for i=l , 2 ,.. .,I, where tion (19a) is equivalent

Qis are arbitrary to A;PiAi-Pi=

Step 3. Third,

if there

positive

definite

matrices.

-Qi.

Equa-

( 19b)

exits q in (Pi I i = 1,2,. . . , I} such that

for i= 1, 2 ,... ,I, then we select succeeded, perform Step 4).

q

as a common

P. If Step

3 has not

Step 4. Next, we calculate

(f-‘(Gjo(Q)), for arbitrary

positive

definite

matrix

j=2,3

,***, 1.

(20)

Q, where

Step 5. Finally, we check whether Eq. (20) is a positive definite matrix for j = 2,3,. . . , I or not. There exists a common P if Eq. (20) is a positive

FUZZY definite another

STABILITY

CRITERION

13

matrix for j = 2,3,. . . , 1. Conversely, go back to Step 41 and choose Q if Eq. (20) is not a positive definite matrix for an integer j.

Next, we give three

examples

for the construction

procedure.

EXAMPLE 4.1. Let us consider stability of the fuzzy system Example 3.1. In Example 3.1, A, and A, were given as -tl’],

A_=[

-;

shown

in

-;‘I.

Step 1. We obtain

A,A,=

[

1;‘;

.

-0.5 -0.5

1 ’

It is seen from Theorem 4.1 that there does not exist a common P because the eigenvalues of both A,A, and A,A, are -0.135 and - 1.865. EXAMPLE

4.2. Assume

that A, and A, are given as -,.5S8],

Then we consider stability subsystems A,x(k) and A,x(k).

of the

A,=

fuzzy

[;

system

-,.36,1

that

consists

of the

Step 1. We obtain

The eigenvalues

of both A, A, and A,A,

Step 2. Next, P, and P, are calculated Eq. (19b), respectively, where I denotes

are 0.277 f 0.3681’.

for Q, = Q2 = I from Eq. (19al or the identity matrix:

K. TANAKA AND M. SAN0

14 Step 3.

1 1

- 0.231 0.0787 11.6 -7.28 Therefore,

a07

<‘.

we can select P, as a common P.

EXAMPLE 4.3. Assume that A, and A, are given as

-;‘],

A*=[ -;

-;‘].

Then we consider stability of the fuzzy system, which consists of the subsystems A,x(k) and A,x(k). Step 1. We obtain -0.1 -0.5 I ’

A,A,=

[ . -;.;

(‘2

.I.

The eigenvalues of both A,A, and A,A, are - 0.931 and - 0.268. Step 2. Next, P, and P2 are calculated for Qr = QZ =I from Eq. (19a) or Eq. (19b), respectively, where I denotes the identity matrix: P, =

2.71 -0.181

Step 3.

Step 4. Assume that

FUZZY

STABILITY

1.5

CRITERION

Then,

K’(GB(Q))

=0-l

-os2oq,,

+ l.l9q,,

- 1.52q,,

-0.814q,,

+ 0.783q,,

- 0.814q,,

.

(21)

q22

/

Step 5. It is clear from Theorem 4.2 that the right side of Eq. (21) must be a positive definite matrix in order that there exists a common P. In other words, the following conditions must be satisfied: a>O,

(22)

axe-bxb>O,

(23)

where a = -0.52Oq,,

+ l.l9q,,

- 1.52q2,,

b = -0.814q,,

+ 0.783q,,

- 0.814q,,,

c=q22.

q,l > 0 and qllqz2 >q12q,2 20 matrix. From Eq. (221, we obtain

Here,

a

Xq,,=

because

Q is a positive definite

From qllqz2 >q12q12 20, and q22:

the following inequality holds for any qll, q12,

Eq. (24) < -o.~2oq,,q,,

+ 1.19q,,q12-

1.52q,,q,,.

The right side of the inequality can be represented

[%I

(24)

-0.520qllql,+1.19q,,q,2-1.52q,,q22.

%2]

0.520 0.595

0.595 -1.52

by the quadratic form

I[ 1 911 q12

.

(25)

16

K. TANAKA AND M. SAN0

It is clear that the square matrix in Eq. (25) is a negative definite matrix. This means that Eq. (25) is a negative value for any qll, q,2, and qz2 such that

Therefore, that

Eq. (24) is also a negative value for qny q,,,

q11q22

qw12,

and qz2 such

-q12q12>0*

That is, aXq,,


We obtain a<0

(26)

because q,l > 0. Equation (26) is contradictory there does not exist a common P. 5.

FUZZY

STABILITY

CRITERION

to Eq. (22). Therefore,

OF NONLINEAR

SYSTEMS

The construction procedure for Lyapunov functions have been shown in Section 4. In this section, we show that the construction procedure can be applied not only to a fuzzy system, but also to a class of nonlinear systems that can be approximated by fuzzy systems of Eq. (1). A piecewise linear model can be described as a special case of Eq. (1) if we use crisp sets instead of fuzzy sets in the premise parts of a fuzzy system. Therefore, the construction procedure can be applied to piecewise linear models. The stability criteria of ordinary fuzzy models and polynomial nonlinear models are discussed in the following text.

FUZZY 5.1.

STABILITY

ORDINARY

17

CRITERION

FUZZY

MODELS

We attempt to apply the construction procedure to two ordinary fuzzy models. For simplicity, all fuzzy sets of the consequent parts of the ordinary fuzzy models are simplified by real values. EXAMPLE 5.1. Figure 2 shows the rule table and the membership functions of an ordinary fuzzy model, where PB, PS, P, Z, N, NS, and NB denote “positive big,” “positive small,” “positive,” “zero,” “negative,” “negative small,” and “negative big,” respectively. Let us consider stability of the ordinary fuzzy system. We can read the following IF-THEN rule from the rule table. IF .x(k)

is “negative”

THENx(k+l)

and x( k - 1) is “positive

big,”

= -10.

PB -10 _ -5 ; 0 . .... . . .._......_................ PS -7.5 ;-2.5 ;2.5 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z _5/0;5 .. ... . . . . . . . . . . . . . . . . . . . . . . . . . . .

a.

I.,,

0 .’

: Tn

.

...::.:

.

a,

‘......

N

.:~~:~~_::;

.

.

-

.

:..I lo o

:

.._.’ : ..

7.Q ,:.’ ‘... ~ NS -1;4jg z ‘Y.. .:.. I ...... . _._...._....................

NB

Fig. 2.

3 i 8 i 13 /

M

....

..-I. ....

2;

Ordinary fuzzy model (Example 5.1).

z

‘I

K. TANAKA AND M. SAN0

18

This ordinary fuzzy model must be approximated by the fuzzy model shown in Eq. (1) in order to check stability. The approximation is realized as follows. (1) First, we divide the premise space x(k) X.X(/C-1) of the ordinary fuzzy model into some fuzzy subspaces. After a structure of fuzzy subspaces is assumed, each fuzzy subspace is approximated by a linear model as shown in Figure 3. (2) Second, we check whether a good approximation is obtained. If this is not the case, go back to (1) and select another structure of fuzzy subspaces. A good structure of fuzzy subspaces must be found by the method of trial and error. It is, however, possible to obtain a better approximation if the number of fuzzy subspaces increases. The following fuzzy model is derived by the preceding procedure.

L’:IFx(k-1)

is -5

0’

THENx’(k+l)=OSx(k)-0.8x(&l),

L2: IFx(k-1)

is LZ-5

0’

THENx2(k+1)=0.5x(k)-OSx(k-1). From L, and L,, we obtain A, and A, as

-;-“1,

A*=[;.5 -;‘I.

Step 1. We obtain

A,A, =

-;‘;” [ .

-0.25 -0.5 I ’

-0.4 -0.8 I ’

The eigenvalues of both A, A, and A, A, are 0.525 + 0.353i.

FUZZY

STABILITY

CRITERION

19

xktl)=O, 5x(k)

0

-0.5x(H)

IL’> 0

x (k-U)=O, 5x(k) -0,8x

(k-

Fig. 3.

..

Lo I

1)

A structure

0

of fuzzy subspaces and approximated

linear models.

Step 2. Next, P, and P2 are calculated for Q, = Q2 =Z from Eq. (19a) or Eq. (19b), respectively, where Z denotes the identity matrix:

P, =

6.02 - 1.34

-1.34 4.85

Step 3.

AT,P,A, -P2 =

1

1;‘;; .

1

0V51
we can select P, as a common P.

K. TANAKA AND M. SAN0

20

EXAMPLE 5.2. Let us consider stability of the ordinary fuzzy system shown in Figure 4. The following fuzzy model is obtained in the same manner as Example 5.1.

is 1

L’:IFx(k--1)

-5

0’ THENx’(k+1)=0.5x(k)-0.8x(&l).

is Aandx(k)isl -5 0

L*:IFx(k-1)

-5

THENx*(ki-1)

is A -5

L3: IFx(k-1)

=0.9x(k)

and x(k) 0

0’ -O.Sx(k-1).

is A -5

THENx3(k+1)=0.5x(k)-0.5x(&l).

x(ktl)=O.5x(k) -0.5xW1) x(ktl)=O.9x(k) -0,8x(k-1) x(ktl)=O.5x(k) -0,8x(k-1)

Fig. 4.

. . . . . . . . . . . ..~........~...__.~._.._........

N$

3

i

5.5

; f3

i

10.5; 131

Ordinary fuzzy model (Example 5.2).

0’

FUZZY

STABILITY

CRITERION

From L,, L,, and L,, we obtain A,, A,, and A, as

Step 1. We obtain

1

1

-0.80 0.72 ’

--0.80 0.40 ’

The eigenvalues of both A, A, and A, A, are - 0.575 f 0.556i: -0.25 -0.5

1 ’

The eigenvalues of both A, A, and A, A, are - 0.525 + 0.3531’:

-0.50 -0.45

1 ’

The eigenvalues of both A, A, and A,A,

-0.80 -0.40

1 .

are - 0.425 + 0.4681’.

Step 2. Next, Pi, P2, and P, are calculated for Qi = Q2 = Q3 = I from Eq. (19a) or Eq. (19b), respectively, where I denotes the identity matrix. P, =

P3 =

3.00 -0.50 ]

-0.50 1.75

1.

Step 3. -1.92 -1.00

2.36 -0.96

22

K. TANAKA AND M. SAN0

Step 4, There does not exist a Lyapunov function if there does not exist a common positive definite matrix P such that A’;PA, -P
Assume that

Then,

~-‘(Go)

-O.l51q,, - 1.084&l.l51q,, 0.963~~, + l.S35q,, - 0.963q2,

= B-’

*

1271

q22

i

Step 5. It is clear from Theorem 4.2 that the right side of Eq. (27) must be a positive definite matrix in order that there exists a common positive definite matrix P such that ATPA, -P
In other words, the following conditions must be satisfied: a > 0, axe-bxb>O,

(28) (29)

FUZZY

STABILITY

23

CRITERION

where a=

-0.520q,,+1.19q,,-1.52q,,, + 0.783q,,

b = -0.814q,,

- 0.814q,,

,

c=q**. From Eq. (291, we obtain aXc-bXb=

-0.927qf,

-2.005q,,q22-2.956q,,q,2

- 4.04q,,q,,

- 2.078qf,

- 2.356q&.

> 0 because Q is a positive definite 9llq22>ql2q12 holds: any qll, q12, and q22, the following inequality

matrix.

Here,

Eq. (30) < -0.927&

- 2.956q,,q,,

(30) So, for

- 4.04q,,q2,

- 2.078qi2 - 4.361qf2. The right side of the inequality

can be represented

I[911 I

by the quadratic

-1.478

0

- 4.361 - 2.020

-2.020 - 2.078

q12

form

(31)

q22

It is clear that the square matrix in Eq. (31) is a negative definite matrix. This means that Eq. (31) is a negative value for any q,l, q12, and q22 such that 411'0,

Therefore, that

Eq. (301 is also a negative

value for any ql,, q12, and qz2 such

K. TANAKA

24

AND

M. SAN0

That is, axe-bxb
(32) is contradictory P.

A POLYNOMUL

EXAMPLE

to Eq. (29). Therefore,

NONLINEAR

We shall discuss 5.3.

stability

(32)

MODEL

of a polynomial

5.3. Let us consider

there does not exist a

nonlinear

the following

model

in Example

polynomial:

x(k+I)={-0.384x3(k)-2.39x3(k-l)-0.369x2(k)x(k-I) +2.23x(k)x2(k-1)+9.49x2(k)+35.8x2(k-1)-29.lx(k)x(k-1) -26.9x(k)

+ 380x( k - 1)) x 10p3.

This polynomial can be approxiamted by the following fuzzy model in the same manner as Example 5.1. We must notice that the fuzzy model is equivalent to a piecewise linear model because all the premise parts are represented by crisp sets:

L’:IFx(k-1)

L2:IFx(k-1)

is 1 ,THENx’(k+l)=O.Olx(k)+0.3x(k-1). 2.5 is 1 ,THENx*(k+l)= 2.5

-0.OBx(k)+0.5x(k-1).

Here, we must check whether a certain degree of approximation is obtained or not. In this example, it is possible to analyze stability of the polynomial nonlinear model because a fairly good approximation is obtained. From L, and L,, we obtain A, and A, as

A,=[,*”

;‘“I,

A*=[ -;-OS ;*‘I.

FUZZY

STABILITY

25

CRITERION

Step 1. We obtain 0.299

ALA,=

0.005 0.5

-008 [

’

The eigenvalues

1’

=

;‘;‘:” [ .

-;.y].

of both A, A, and A, A, are 0.3 and 0.5.

Step 2. Next, P, and P, are calculated Eq. (19b), respectively, where I denotes

p = 1

A,A,

2.20 [ 0.0094

0.00941, 1.20

for Q, = QZ = I from Eq. (19al or the identity matrix:

P1=[

-_;:;;

-::;;I.

Step 3. 0.162 -1.44

A;P, A2 -P,

= ]

Therefore, nonlinear 6.

I;‘;;;, .

-0.0927 -0.649

1* 1’ <,I

<0

we can select P, and P, as a common P. That is, this polynomial model is asymptotically stable in the large.

CONCLUSION

We have shown some theorems for stability of fuzzy systems in accordance with the definition of stability in the sense of Lyapunov. Next, two definitions for stability have been given: local stability and global stability. We have shown that a fuzzy system is not always globally stable even if its all subsystems are stable, i.e., locally stable. The construction procedures have been presented. It is possible to find a Lyapunov function by the procedure as effectively as possible. We have demonstrated that the procedure can be applied to other nonlinear systems if the nonlinear systems can be approximated by fuzzy systems of Eq. (1). The stability criteria of two ordinary fuzzy models and a polynomial nonlinear model have been illustrated. The stability criterion proposed in this paper may be applicable to many other nonlinear systems.

26

K. TANAKA

AND M. SAN0

REFERENCES 1. A. A. Kania et al., On stability of formal fuzziness systems, Inform. Sci. 22:51-68 (1980). 2. J. B. Kiszka et al., Energetic stability of fuzzy dynamic systems, IEEE Syst., Man, Cybemet. SMC-15:783-792 (1985). 3. B. C. Kuo, Digital Control Systems, Holt-Saunders, New York, 1980. 4. T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Syst. Man, Cybernet. SMC-15(1):116-132 (1985). 5. K. Tanaka and M. Sugeno, Stability analysis of fuzzy systems using Lyapunov’s direct method, in Proceedings of NAFIPS’90, 1990, pp. 133-136. 6. K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45(2):135-156 (1992). Received 1 November

1991; revised 4 July 1992