T-S模糊系統估測器輸出回授H∞控制之研究(I)

行政院國家科學委員會專題研究計畫 成果報告

T-S 模糊系統估測器輸出回授 H∞ 控制之研究(I)

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 96-2221-E-151-056-

執 行 期 間 ： 96 年 08 月 01 日至 97 年 10 月 31 日

執 行 單 位 ： 國立高雄應用科技大學電機工程系

計 畫 主 持 人 ： 方俊雄

計畫參與人員： 碩士班研究生-兼任助理人員：邱德議

碩士班研究生-兼任助理人員：爐啟銓

碩士班研究生-兼任助理人員：黃信源

碩士班研究生-兼任助理人員：許聖雨

博士班研究生-兼任助理人員：曾冠瑄

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中 華 民 國 97 年 11 月 02 日

行政院國家科學委員會專題研究計畫成果報告

T-S模糊系統估測器輸出回授H

∞

控制之研究

Observer-based output feedback H

∞

control for T-S Control

計畫編號：

NSC-96-2221-E-151-056

執行期限：96年8月1日至97年7月31日

主持人：方俊雄教授 國立高雄應用科技大學 電機系

參與計畫人員：曾冠瑄、邱德議、爐啟銓、黃信源、許聖雨

一、 中文摘要 本計畫針對T-S模糊系統，探討以觀測器為基礎 的H∞控制問題，根據Lyapunov function的精神，配合 LMI觀念，結合本研究室所研發的三指標，改善文獻 中關於觀測器輸出回授H∞控制器設計條件之寬鬆 度，當考慮觀測器含有雜訊項時，本論文的結果不但 比文獻中的條件更寬鬆，也證明現有文獻的結果是本 論文的特例情形。當考慮觀測器不含系統雜訊項時， 條件的型式更加複雜，必須透過兩階段才能使用LMI 求解，如果要簡化成一階段求解，本文引入等式限 制，成功避開兩階段求解的麻煩。 關鍵字：T-S模糊系統，線性矩陣不等式，H∞控制。 Abstract

The observer-based H∞ control problem for T-S fuzzy

systems is solved in the project. By Lyapunov stability and the three index combination technique, a new nonlinear matrix inequality condition for the existence of an

observer-based H∞ controller is derived. To solve

controllers with LMI algorithms, a set of LMI conditions which are necessary and sufficient to the derived nonlinear condition are developed as well. It can be shown that the present result is more relaxed than the existing ones and includes them as special cases.

Keywords—T-S fuzzy systems, linear matrix inequality

(LMI), H∞ control.

二、緣由與目的

Fuzzy sets and systems have gone through substantial development since the introduction of fuzzy set theory by Zadeh in 1965 [18]. There have been a great variety of successful applications in the area of image processing, industrial applications, medicine, finance, control engineering and so on in the literature [1,2,6,10,17,19]. In particular, the T-S fuzzy model, also called the Type-III fuzzy model by Sugeno [11,12], is recognized as a powerful tool. For example, it offers an alternative approach to describing nonlinear systems [20]. By using the fuzzy model, lots of nonlinear control problems can be solved easily [13,15,16]. Therefore, considerable attention has been paid to the analysis and synthesis of T-S fuzzy systems, and various techniques have been developed during the decade [4,14].

The H∞ control problem of T-S fuzzy systems has been

investigated by many authors recently. For example, the

papers [3,7,8,9] and the references therein solved the H∞

control problem via fuzzy observer-based feedback control. Among which, the result of [8] is quite interesting. It introduced a new type of observer and developed a nonlinear condition for the existence of controllers. For obtaining controllers, a sufficient LMI condition was proposed and solved by a two-step procedure. Later on, the reference [7] proposed an improved result, in which an LMI condition equivalent to the nonlinear condition was developed. By the LMI conditions, the controller and observer can be obtained simultaneously. Thus remove the drawback of two-step procedure in [8].In our paper, a new nonlinear condition which is more relaxed than that of [8] is proposed. An LMI condition which is equivalent to our nonlinear condition is also given. The present LMI condition is more relaxed than those of [7] and [8]. Such improvement on relaxization makes possible for finding

a controller achieving a better H∞ performance and

allowing a larger size of uncertainties for robust control. 三、結果與討論

1. Preliminaries

Consider a nonlinear system that is represented by the following T-S fuzzy model：

Plant Rule i : If θ1(t) is Mi1 and … and θs(t) is Mis

Then

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

i wi ui zi ui yi yi

x t

A x t

B w t

B u t

z t

C x t

D u t

y t

C x t

D w t

⎧

=

+

+

⎪

₌

₊

⎨

⎪

=

+

⎩

(1)

In (1), Mij (i = 1,2,…,r, j = 1,2,…,s) is the fuzzy set and

r is the number of If-Then rules. θi(t), i=1,2,…,s are the

premise variables which are measurable. is

the state vector, the exogenous disturbance,

n

x

∈

w m

w

∈

p

z

∈

the controlled output, the output

vector, and the control input vector. Assume

q

y

∈

u m

u

∈

,

n n i

A

∈

× and

,

u n m ui

B

∈

× n mw

,

wi

B

∈

×

,

p n zi

C

_∈

× p mu

_,

ui

D

_∈

× q n

_,

yi

C

_∈

×

w q m yi

D

_∈

× _{. Given a pair of (x(t), u(t)), the final}

output of the fuzzy system is inferred as follows:

(

)(

1

( )

r _i

( )

_i

( )

_wi

( )

_ui

( )

i

)

x t

=

∑

h

θ

t

A x t

+

B w t

+

B u t

(

)(

)

( )

r _i

( )

_zi

( )

_ui

( )

z t

=

∑

h

θ

t

C x t

+

D u t

(

)

(

)

1

( )

( )

( )

r i

( )

yi yi i

C x t

D w t

y t

h

θ

t

=

+

=

∑

or simplicity is replaced by in the

equel.

efinitions: If w(t) = 0, the system (2) is termed to be

e”. A distu s

e quadratically stable if there exists a

m (2) is = 1 i= (2) where F ,

h

i

(

θ

( )

t

)

h

i s D

“disturbance-fre rbance-free fuzzy ystem is

said to b

P

>

0

such that

V



(

x t <

( )

)

0

, where

(

( )

)

T

( )

( )

V

x t =

x t Px t

. If u(t) = 0, the syste

termed to be “unforc scribed scalar

0

ed”. Given a pre

γ

>

, if for any

w t

(

)

square integrable signals), the response z(t) of the unforced fuzzy system

2

( )

∈

L

0, ,

∞

mw _{(the set of}

i satisfie

then the unforced fuzzy system (2) is said to with

(2), under zero init al condition, s 2 0

( ) ( )

0

( ) ( )

T T

z t z t dt

γ

w t w t dt

∞ ∞

<

∫

∫

(3) be stable

γ

-disturbance attenuation.

Consider the observer model same as in [7,8]

(4)

i

u t

=

∑

h K x t

(5)

where and are the controller gains a

observer gains to be determined. Den

errors as and construct an

ted

is

i) The unforced fuzzy system (6) is stable with

Assume the fuzzy controller is

1 i i=

ˆ

( )

r

( )

i

K

L

_i nd

ote the estimation

ˆ

( )

( )

( )

e t

≡

x t

−

x t

augmen system (6)

γ

-disturbance attenuation. The f

for the ease o son with our result

from a theo

L real number

ollowing is the main result of [8]. It is stated here f relaxization compari

retic viewpoint in next section.

emma 1 [8] : For a given

γ

>

0

, the

control objectives can be achieved if there exist matrices

0

X

>

,

G

>

0

,

K

_i ,

L

_i,

Z

_ii,

i

=

1,2,...,

r

,

Z

_ij, T ji ij

Z

=

Z

,

i

=

1, 2,...,

r

−

1

,

j i

= +

1,...,

ch that the

r

su T T ii ii wi wi ui i T T T i ui ii ii V X XV X B X XB K

following matrix inequalities hold

, 1, 2,..., , ii B 2 Z K B X G G − ⎡ + + ⎤ ⎢ _{− Γ + Γ} ⎥ ⎣ ⎦ i r γ − < = (7) ( )

(

)

(

( )

)

( )

(

)

(

( )

)

1

(

( )

)

1 , s i i r i j i i t h t t M t t β θ θ β θ β θ = = =

∏

∑

(

)

(

)

(

)

2 , , T T T ij ij wi wj wj wi ui j uj i _T ij ij T T T T T i uj j ui ij ij V X XV X B B B B X X B K B K

The objectives of the observer-based

H

∞ control are

(i) The disturbance-free fuzzy system (6) quadratically stable. Z Z K B K B X G G j i γ− ⎡ + + + − + ⎤ ⎢ ⎥ ≤ + ⎢ − + Γ + Γ ⎥ ⎣ ⎦ > (8) (9) WhereV_ii =A_i+B K V_{ui i}, _ij =A_i+A_j+B K_{ui j}+B K_uj i, L C , A A L C L C U, ii Ai i yi ij i j i yj j yi ik Γ = + Γ = + + + =

[

C

zi

+

D K

ui k

−

D K

ui k

]

.

Note that the inequalities (7)-(9) are actually nonlinear

with respect to and . In order to solve and

with LMI algorithms, following the idea of [3,9], it

o-s ce la

by

sufficient to our developed

i

K

L

_i

K

_i

i

L

the paper [8] presented another sufficient condition and then proposed a two-step procedure to solve . The

drawback of tw tep pro dure was overcome tterly

[7], in which a set of LMI conditions were developed and was proved to be equivalent to the nonlinear matrix conditions of [7]. In next section, a new nonlinear condition that is more relaxed than Lemma 1 is proposed. A new LMI condition necessary and

nonlinear condition is also derived for obtaining the gains in one step. Actually, we will show the present result is more relaxed than those of [8] and [7] and includes them as special cases.

2. Main results

Lemma 2 : For a given positive number

γ

>

0

, the

control objectives can be achieved if there e

ij θj 1 i= xist matrices

X

>

0

,

G

>

0

,

K

_i ,

L

_i ,

Y

_iii ,

1, 2, ,

i

=

…

r

;

Y

_jii

=

Y

_iijT ,

Y

_iji ,

i

=

1

, 2,

,

r

,

( )

(

)

1 ˆ ( ) ˆ( ) ( ) ( ) ( ) ( ) ˆ r _{i i wi ui i} i t h A tx B w t B u t L y t y t x = = + + −

(

)

ˆ( ) r _{i yi}ˆ( ) _yi ( .) y t =

∑

h C x t +D w t − 1 1 0 ( ) . ( ) i ui j ui j j i i yj zi ui j ui j i j i j B A L C x t K D K e t = = ⎜

∑



…

j i

≠

,

j

=

1, 2, ,

…

r

,

Y

_ji

=

Y

_ijT ,

Y

_{j i}

=

Y

_{i j}T , T

₂

₂

ij ji

Y

=

Y

,

i

=

1, , ,

r

−

1 1 ( ) ( ) ( ) ( ) ( ) 0 ( ) r r wi i i j r r A B K K x t x t B h h w t e t e t C D z t h h = = ⎛⎡ + − ⎤ ⎞ ⎡ ⎤₌ ⎡ ⎤ ⎡₊ ⎤ ⎟ ⎢ ⎜ + ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎟ ⎣ ⎦ ⎝⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎠ ⎡ ⎤

…

,

j i

= +

1, ,

…

r

−

1

,

1, ,

j

r

=

+ − _{⎢ ⎥} ⎣ ⎦ ⎣ ⎦ 11 1 1 1 0, 1, 2, , , T r ik T T r rr rk T k rk Z Z U k r Z Z U U U I ⎡ ⎤ ⎢ ⎥ ⎢ _{⎥ < =} ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ … = ⎡ ⎤

∑∑

∑∑

+ …

such that the llowing matri

inequa old

fo x

,

1, 2, ,

ii

Y i

iii

r

Λ <

=

…

) ii ij

Λ + Λ + Λ

(10

,

ji

≤

Y

iij

+

Y

iji

+

Y

iij

i

=

…

r

,

1, 2, ,

T

,

1, 2, ,

j i j

≠

=

…

r

…

…

,

r

(11)

,

1, 2, ,

2,

1, 2, . ,

ij i ji j i j ij i j ji T T T ij i j ji

Y

Y

Y

Y

Y

Y

i

r

j i

Λ + Λ + Λ + Λ + Λ + Λ ≤

+

+

+

+

+

=

−

= +

1,

1, 2, ,

r

−

= +

j

…

(12) 1 1 1 1 1 1

0

1, 2, ,

T j jr j T rj rjr rj j rj

Y

Y

U

j

r

Y

Y

U

U

U

I

⎡

⎤

⎢

⎥

⎥

⎢

_<

₌

⎢

⎥

⎢

⎥

−

⎢

⎥

⎣

⎦

…

(13) where T T T i i j ui T T yj i i yj A X XA K B X XB K XB K XB B X A G GA K B X C L G GL C

γ

− ⎡ + + ₋ ⎤ ⎢_{+ +} ⎥ ⎢ ⎥ Λ = + + − + ⎢ ⎥ ⎣ ⎦ 2 T T T i i j ui ui j T ui j wi wj ij _{⎢ ⎥} ⎢ ⎥

[

]

ik zi ui k ui k U = C +D K −D K .

In what follows, we are going to show that Lemma 2 is more relaxed than Lemma 1 and includes it as a special case.

ossible for finding a controller achieving a b

Theorem 1: The set of solutions to (7)-(9) in Lem

ma 1 is a subset of solutions to (10)-(13) in Lemm a 2.

Remark 1: Since Lemma 2 is more relaxed than L

emma 1, the controller set obtained by Lemma 2 is lager than that of Lemma 1. This improvement m akes p

etter H∞ control performance and allowing a bigger size of uncertainties for robust control. This will b e demonstrated in the numerical example section.

Theorem 2 : For a given positive number

γ

>

0

,

there exist matrices

X

>

0

,

G

>

0

,

K

_i,

L

_i,

iii

Y

,

i

=

1, 2, ,

…

r

,

Y

_jii

=

Y

_iijT,

Y

_iji,

i

=

1, 2

, , r

…

,

j i

≠

,

j

=

1, 2, ,

…

r

,

Y

_ij ,

Y

_ji

=

Y

_ijT ,

Y

_ji

=

Y

_{i j}T , , T ij ji

Y

=

Y

i

=

1, 2, ,

…

r

−

2

,

j i

= +

1, ,

…

r

−

1

,

1, ,

j

r

= + …

, such that the matrix inequalities (10)

-(13) hold if and only if there exist matrices

X

>

0

,

0

Y

>

,

M

_i , , , , ,

i iii

Q

iii ,

T

jii

=

Q

iij ,

P

iji ,

Q

iji ,

i

=

1, 2, ,

…

r

,

J

P

i

=

1, 2, ,

…

r

T jii iij

P

=

P

Q

j i

≠

,

j

=

1, 2, ,

…

r

,

=

, T T _, T j i i j , T ji ij

P

P

ji ij

Q

=

Q

, , , ji ij

P

=

P

Q

=

Q

T ij ji

P

=

P

T ij ji

Q

=

Q

i

=

1, 2, ,

…

r

−

2

,

j i

= +

1, ,

…

r

−

1

,

1, ,

j

r

= + …

such that the following matri inequalities hold x 2 , 1,2 , T T T T i i i ui ui i wi wi iii XA A X M B B B B P i r γ− + + + + < = … (14) M , T i iJ +J Ci yi<Qiii ₍₁₅₎ 1, 2 , T T i i y A Y YA C i r + + = …

(

)

(

)

(

)

(

)

(

)

2 2 2 1, 2 , , 1, 2 T T T T T i j i j i ui uj T T j ui ui uj i ui j T T T wi wi wi wj wj wi iij X A A A A X M B B M B B B M B M , T iji iij B B B B B B P i r i j j r

γ

− + + + + + + + + + + + + ≤ = … ≠ = … (16) P P + +

(

)

(

)

(

)

(

)

2 2 , 1, 2 , , 1, 2 T T T T T i j i j yi i j T T yj i i j yi i yj T

iij iji iij

A A Y Y A A C J J C J J J C J C Q Q Q i r i j j + + + + + + + + + ≤ + + = … ≠ = …r (17)

(

)

(

)

(

)

2 2 2 1, 2 T T T T T i j i j j ui T T T T T T T T T T uj i uj uj i u j u ui j ui uj i uj u i u j T T T T wi wj wi w wj wi wj w T T w wi w wj T T T ij i j ji ij i j ji X A A A A A A X M B M B M B M B M B M B B M B M B M B M B M B M B B B B B B B B B B B B P P P P P P i r γ− + + + + + + + + + + + + + + + + + + + + + + + ≤ + + + + + = … −2, j i= +1…r−1, = +j 1…r (18)

(

)

(

)

2 2 1, 2 2, 1 1, 1 T T T T T i j i j yj i T T T T T T T T T T y i yi j y j yi yj i i y j yi yi yj T T T ij i j ji ij i j ji yj A A A Y Y A A A C J C J C J C J C J C J J C J C J C J C J C Q Q Q Q Q Q i r j i r j + + + + + + + + + + + + + + + + ≤ + + + + + = … − = + … − = + …r (19) 1 1 1 1 1 1 1 1 0, 1, 2, T T T k kr z k T T T rk rkr zr k ur z u k zr ur k P P XC M P P XC M C X D M C X D M I k r ⎡ + ⎤ ⎢ ⎥ ⎢ _{⎥ <} ⎢ ₊ ⎥ ⎢ ⎥ + + − ⎢ ⎥ ⎣ ⎦ = … u D D (21)

In this case, the controller gains and the observer gains can be chosen as (20) 1 1 1 1 0, 1,2, k kr rk rkr Q Q k r Q Q ⎡ ⎤ ⎢ _{⎥ < =} ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ … . 1

_and

1

_,

_{1, 2, ,}

i i i i

K

₌

M X

−

L

₌

Y J i

−

₌

_…

r

₍₂₂₎

Consider a nonlinear system [13]

(

2

)

1 1 3 2 3

( )

( ) sin ( ) 0.1 ( )

( ) 1 ( )

( )

( ) 2 ( ) 3 ( )

( )

3. A numerical example

x

_{2 3 1} 2 1 2

( ) sin ( )

( )

( )

t

x t

x t

x t

x t

u t

w t

x t

x t

x t

w t

=

+

−

+

+

+

=

−

+

x t

=

x t

−

x t

+

w t

(

1 2

)

1 1 2 2 1 ( ) ( ) 2.3 ( ) ( ) ( ) 1 ( ) 0.5 ( ) ( ) ( ) 0.5 ( ). z t x t u t y t x t x t w t y t x t w t = + = + + = + Assume

x t

₁

( )

∈ −

[

a a

]

,

x t

₂

( )

∈ −

[

b

b

]

,

where a and b are positive numb e two parameters

will be also used later to test the conservativeness of conditions. The premise membership functions and the

nt matrices are ers. Th conseque 2 1 11 2 21

,

x

M

M

a

=

=

M

₃₁

= −

1

M

₁₁

=

M

₄₁

,

2 2

sin

sin

0

(

b

x

x

b

x

x b

M

M

−

⎧

_≠

⎪

=

_{= ⎨}

2 2 12 32 2

sin )

,

0

b

x

−

⎪

₌

⎩

+

⎢

⎥

= ⎢

⎥

⎢

⎥

⎣

⎦

1

0

0

y

a

C

_{= ⎢}

⎡

+

⎤

_⎥

⎣

⎦

b b

A

b b

⎡

⎤

⎢

⎥

= ⎢

⎥

⎢

⎥

⎣

⎦

⎥

⎦

⎥

⎦

1

22

1

12 42

,

M

= −

M

=

M

1

1

1

-0.1

,

2

3

0

0

1

-1

A

⎡

⎤

⎢

⎥

=

_⎢

−

_⎥

⎢

⎥

⎣

⎦

2 1

1

,

0

0

u

a

B

⎡

⎤

2 1

0 1

0

,

-1

2

1 sin( ) /

-0.1

,

2

-3

0

0 sin( ) /

2 2

1

,

0

0

u

a

B

⎡

+

⎤

⎢

⎥

= ⎢

⎥

⎢

⎥

⎣

⎦

2 2

0 1

0

,

1

0

0

y

a

C

_{= ⎢}

⎡

+

⎤

_⎥

⎣

⎦

3

1 1 -0.1

,

2 -3

0

0 1

-1

A

⎡

⎤

⎢

⎥

= ⎢

⎥

⎢

⎥

⎣

⎦

3

1

,

0

0

u

B

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

3

0 1 0

,

1 0 0

y

C

_{= ⎢}

⎡

⎤

⎣

4

1 sin( ) /

-0.1

,

2

-3

0

0 sin( ) /

-1

b b

A

b b

⎡

⎤

⎢

⎥

= ⎢

⎥

⎢

⎥

⎣

⎦

4

1

,

0

0

u

B

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

4

0 1 0

,

1 0 0

y

C

_{= ⎢}

⎡

⎤

⎣

[

]

1 2 3 4

1 1 1

,

T w w w w

B

=

B

=

B

=

B

=

[

]

1 2 3 4

2.5 3 0

,

z z z z

C

=

C

=

C

=

C

=

Du1 = Du2 = u3 = Du4 = 0.001, Dy1 = D = D = D = 0.005. In this

simulation, assume

a

=

1.4 and

b

=

0.7

. By

Theorem 2, choosing D y2 y3 y4

0.2

γ

=

[

]

[

]

[

]

[

3 4 7732 -0.3923 , , -10.8100 -1.1630 0 1041 -10.1789 -1.077 K = K = -3.8066 -0.60 -3.9595 L

]

1 -6.6894 -0.4333 0.0397 , 2 -6. 0.0390 , . 4 0.0990 K = K = 41 -0.5219 -0.3339 0.6601 -0.2907 0.6559 -1.4381 -16.5338 -1.4346 -16.2066 , -4.4155 -2.6641 -3.9927 -2.5747 -0.8670 0.6159 L L L 1 2 3 4 -1.5494 -16.7745 -1.5705 -16.6232 , , ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ =_{⎢ ⎥} =_{⎢ ⎥} ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ =_{⎢ ⎥} = ⎢ ⎥ ⎣ ⎦ . -0.7884 0.6053 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

Finally, we compare the conservativeness of Theorem with [7] and [8]. For the above numerical example, the maximal interval of the positive parameter a such that the conditions are feasible is given Table 1. Our result allows the largest interval for three different cases.

2

Table 1 the maximal allowable interval of a by using

different conditions cases Theorem 2 [7] [8]

0 a

, we obtain

0.2

γ

=

b

=

0.01

7132042

≤

0 a

< ≤ 0 a

< ≤

1931000

1421000

<

0 a

0.2

γ

=

b

=

π

/ 2

1527433

≤

0 a

< ≤

1126768

0

935466

a

< ≤

<

0.1

γ

=

b

=

2

0

2200000

a

< ≤

0

430414

a

< ≤

0

226590

a

< ≤

Since our condition is the mo relaxed, e designed controller may achieve a better control H∞ performance which is demonstrated as follows.

ble 2 the comparison of H∞ control performance by

st th

Ta

using different design approaches

Theorem 2 [7] [8] a=4358774, b=0.01

γ

=0.198

γ

=0.4

γ

=1.5 a=1527433,

/ 2

b

=

π

γ

=0.2

γ

=1.87

γ

=3.2 a=610558, b=2

γ

=0.06

γ

=0.2

γ

=1.2 Fig. 1 show t system when

he state response of the closed-loop fuzzy the initial condition i_{s [0.7 0.5 0.1]}T an

− d

bance gi _{w t} tsin

w(t) is a distur ven by ( ) 0.5e−0.5 (5 t_π )_.

olid line denotes th ariab e dotted line

enotes the observer state.

= le, th

The s e state v

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 Time (Sec) -0.1 0.1 0.4 0.3 0.2 0.5 0.6 0.7 (a)

ˆx

₁and

x

₁ 0 0.5 1 1.5 2 2.5 3 0 Time (Sec) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.5 (b)

ˆx

₂and

x

₂ 0 1 2 3 4 5 6 7 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time (Sec) 0.25 (c)

ˆx

₃and

x

₃

Fig. 1 Responses of the state and its estimation 四、計畫成果自評

In this project, a new nonlinear matrix inequality

condition for the existence observer-base control

of T-S fuzzy systems is developed. The condition is shown to be more relaxed than [

H

∞

8] and include it as a

special case. For solving the observer and the controller via LMI algorithms by one step, a set of LMI conditions necessary and sufficient to the new developed nonlinear condition is also derived. Our LMI condition is more relaxed than that of [7] since the result of [7] is only equivalent to the nonlinear condition of [8].

計畫的成果已經發表在2007年中華民國第十五屆 模糊理論及應用研討會：H.-W. Chen, S.-W. Kau, C.-C

Lu, T.-Y. Chiu, C.-H. Fang, “Observer-based H∞ fuzzy control design,” 中華民國第十五屆模糊理論及應用 研討會, Dec.14-15, pp. 376-381, 2007. 目前正在整理 投稿的國際期刊上

五、參考文獻

[1] J. C. Bezdek, J. M. Keller, R. Krishnapuram, and N.

R. Pal, Fuzzy Models and Algorithms for Pattern Recognition and Image Processing, Bosten, MA:Kluwer, 1999.

[2] Z. Bingul, G. E. Cook, and A. M. Strauss, “Application of fuzzy logic to spatial thermal control in fusion welding,” IEEE Trans. Ind. Appl., vol. 36, no. 6, pp. 1523-1530, Dec. 2000.

[3] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed H2/H∞ fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach,” IEEE Trans. Fuzzy Systems, vol. 8, no. 3, pp. 249-265, June 2000.

[4] G. Feng “A survey on analysis and design of model

based fuzzy control systems,” IEEE Trans. Fuzzy Systems, vol. 14, no. 5, pp. 676-697, Oct. 2006. [5] S.-W. Kau, Y.-S. Liu, C.-H. Lee, L. Hong, and

C.-H. Fang, “A new LMI condition for robust stability of discrete-time uncertain systems,” Systems and Control Letters, vol. 54, pp. 1195-1203, 2005.

[6] C. C. Lee, “Fuzzy logic in control systems: fuzzy logic controllers---Part I,” IEEE Trans. Systems, Man, Cybern., vol. 20, no. 2, pp. 404-418, Mar./Apr. 1990.

[7] Chong Lin, Qing-Guo Wang and Tong Heng Lee, “Improvement on observer-based H∞ control for T-S fuzzy systems,” Automatica, vol. 41, pp. 1651-1656, July. 2005.

[8] X. Liu and Q. Zhang, “New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,” Automatica, vol. 39, no. 9, pp. 1571-1582, Sep. 2003.

[9] J. C. Lo and M. L. Lin, “Observer-based robust H∞ control for fuzzy systems using two-step procedure,” IEEE Trans. Fuzzy Systems, vol. 12, no. 3, pp. 350-359, Jun. 2004.

[10] H. Seker, M. O. Odetayo, D. Petrovic, and R. N. G. Naguib, “A fuzzy logic based-method for prognostic decision making in breast and prostate cancers,” IEEE Trans. Inform. Technol. Biomed., vol. 7, no. 2, pp. 114-122, Jun. 2003.

[11] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Trans. Systems., Man, Cybern., vol. 15, pp. 116-132, Feb. 1985.

[12] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy sets and Systems, vol. 45, pp. 135-156, 1992.

[13] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Trans.

Fuzzy Systems, vol. 6, no. 2, pp. 250-265, May 1998.

[14] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis. John Wiley & Sons, Inc, New York, 2001.

[15] M. C. M. Teixeira, E. Assuncao, and R.G. Avellar, “On relaxed LMI-based designs for fuzzy regulators and fuzzy observers,” IEEE Trans. Fuzzy Systems, vol. 11, no. 5, pp. 613 -623, Oct. 2003.

[16] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: stability and design issues,” IEEE Trans. Fuzzy Systems, vol. 4, no. 1, pp. 14-23, Feb. 1996.

[17] L. X. Yu and Y. Q. Zhang, “Evolutionary fuzzy neural networks for hybrid financial prediction,” IEEE Trans. Systems, Man, Cybern. C, App. Rev., vol. 35, no. 2, pp. 244-249, May 2005.

[18] L. A. Zadeh, “Fuzzy Sets,” Information Control, vol. 8, pp. 338-353, 1965.

[19] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Systems, Man, Cybern., vol. 3, no. 1, pp. 28-44, Jan. 1973.

[20] K Zeng, N. Y. Zhang, and W. L. Xu, “A comparative study on sufficient conditions for Takagi-Sugeno fuzzy systems as universal approximators,” IEEE Trans. Fuzzy Systems., vol. 8, no. 6, pp. 773-780, Dec. 2000.

出席國際學術會議心得報告

計畫編號 NSC

96-2221-E-151-056

計畫名稱

T-S模糊系統估測器輸出回授H

∞

控制之研究

出國人員姓名

服務機關及職稱

方俊雄 國立高雄應用科技大學 教授

會議時間地點 97.09.21-24 中國、成都

會議名稱

IEEE 智能控制與自動化機電國際研討會

（IEEE International Conf. on Cybernetics & Intelligent Systems and Robotics,

Automation & Mechatronics）

發表論文題目 An improved result for observer-based H

∞

fuzzy control

方俊雄 97.09.26

會議名稱：IEEE 智能控制與自動化機電國際研討會

（IEEE International Conf. on Cybernetics & Intelligent Systems and Robotics, Automation &

Mechatronics）

會議日期：97.09.21-24

會議地點：中國、成都（Chengdu, China）

計畫編號：NSC 96-2221-E-151-056

計畫名稱：T-S模糊系統估測器輸出回授H

∞

控制之研究

一 參加會議經過

97 年 09 月 22 日上午七點搭乘華信航空班機由高雄出發，經香港轉機到成都，抵達成都

住宿飯店時為 9/22 下午 2:30，立即趕往開會的喜來登飯店辦理報到，領取會議資料，參加第

一天下午研討會。

本次會議共有 952 篇論文投稿，614 篇論文被接受，接受率約為 64%，就一般國際研討

會而言，屬於錄取率低且相當嚴格的篩選。大會論文來自歐洲、亞洲、美洲、澳洲等 36 國家，

以中國參加的人數最多，台灣僅有三篇文章出現。本次本次會議主辦由成都電子科技大學主

辦，新加坡 IEEE 分會共同合辦，原來會議時間為六月中旬，因為五月十二日發生四川大地

震，大會考量安全因素，延後到九月份舉行。

本研討會每兩年舉辦一次，這是第三次舉辦，前兩次分別在新加坡和曼谷舉行，後年預

定在馬來西亞舉辦。從參加國家分布和投稿的情形來看，本研討會已經確立了大型國際研討

會的地位，本次選在成都舉行，主要考量成都附近有著名的景點，可以吸引更多國際人士前

來，四川古稱天府之國，風景秀麗，物產豐富。因為會議的重要性，加上地點合適，吸引近

900 位的世界各國學者與會。本次會議日期正式從九月二十二日開始，會議分成 6 場地同時

進行，主題包含網路、資訊、智慧型運輸、人機互動、無線感測、機電系統設計、自動化模

擬、生物機械人、醫療機械人、微奈米機械等，每場次兩個小時，一天安排 8 個場次，分別

從 10:30-18:00，可以說相當密集，另有多個場次的海報互動時間。大會在每天上午 9:00-10:00

各安排一場 Keynote speech：邀請 M. Tomizuka (University of California, Berkeley, CA, USA) 講

Mechatronics considerations for assisting humans 和 M. Y. Wang(The Chinese University of Hong

Kong) 講 Compliant mechanisms for MEMS and flexonics，同時 D. Li (National Natural Science

Foundation of China) 講 Knowledge discovery from networks 這三個主題都是目前熱門或者具

有前瞻性研究的題目，由三位世界著名學者演講，對於與會人士獲益甚多。筆者的論文安排

在第三天下午 16:40-17:00，屬於 fuzzy systems 場次，本次研討會屬於智能控制領域，有關 fuzzy

的主題共有 4 個場次，12 篇論文。

二 與會心得

本次研討會學術的安排相當用心，包括茶水的供應，場地的展示佈置和安排，交通車的

安排，住宿的提供和收費等，大會會場提供三天午餐，是個相當不錯的服務，讓研討會的出

席人員不用擔心民生問題，電子科技大學動員師生近百人舉辦活動，整體而言相當成功，晚

宴地點選在一家具有四川特色的餐館，同時搭配表演具有四川特色的變臉，令人印象深刻。

成都雖然經過地震僅有四個月，但是各項生活和設施已經恢復正常，成都市容相當乾淨

漂亮（可能是大陸屬一屬二乾淨的城市）

，街道的設計區分人、自行車、汽車，不同車道，這

是相當好的規劃，值得一提的是行人可用的走路寬度大約是台灣兩個車道，行人在成都非常

有福氣，只是大陸人遵守交通規則的習慣尚未養成，市政府為了維持交通秩序，每個重要路

口都派了 12 交通維護工人，全天候指揮交通，並呼籲市民遵守交通秩序和規則，用心令人感

動和佩服。成都市有將近一千三百萬居民，車輛很多，交通擁擠，但是政府有很好規劃，包

括地鐵的興建，預定 2012 年完工，為了改善空氣和環保考量，成都市內的計程車和公車都是

使用天然氣，而且天然氣的加氣站上午七點到下午六點僅提供上述兩項交通工具加氣，由於

計程車管制開放，經常一車難叫，叫計程車在成都市要有一些技巧。值得一提的是成都對於

摩托車的規定，強制使用電動摩托車，沒有任何加油的摩托車，每個家中停車處都有充電站，

或是可以將電池拔起到家裡充電，因此成都市內空氣保持很好，噪音很低，市民在這方面的

環保觀念相當落實，這一點很值得台灣政府學習。看完成都對於市容的規劃，我想台灣應該

要加把勁，我希望台灣利用太陽能的資源，積極推展電動摩托車的電池充電，或是發展太陽

能自行車，讓環保觀念落實於台灣人心中，不但省能也環保，對於空氣有很大幫助。台灣太

過自由，很多政策在台灣推行困難，在成都，政府一聲令下，徹底執行，看完台灣和成都的

都市規劃，比對兩者之間政府和民眾的效率與配合度，像極了台灣的公立和私立大學的翻版。

成都市民生活步調緩慢，經常有市民在路旁放一張小桌子，四個人喝起茶，打打牌，這種景

象，在台灣是無法想像，有人說成都人很會過生活。四川的食物以辣出名，這幾天的飲食充

分體會的辣的感覺，真的非常辣！

本次出席國際會議經費由國科會計畫資助，核定經費不足涵蓋全部支出，但仍感謝國科

會協助，有此會議經驗，對爾後學術合作及提升均有幫助，對於城市規劃也可以提供一些建

議給台灣。攜回資料包括論文摘要一本、論文 CD 一片及若干研討會徵稿海報。