模糊控制系統穩定度分析與控制器設計之改良---應用多重李雅普諾夫函數法

行政院國家科學委員會專題研究計畫 成果報告

模糊控制系統穩定度分析與控制器設計之改良--應用多重

李雅普諾夫函數法

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 95-2221-E-151-022-

執 行 期 間 ： 95 年 08 月 01 日至 96 年 07 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：楊閎智、陳宏瑋、謝文建

協同主持人：葛世偉

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 96 年 10 月 07 日

行政院國家科學委員會專題研究計畫成果報告

模糊控制系統穩定度的分析與設計－多重李雅普諾夫函數法

Stabilization of T-S Fuzzy Control Systems

via the Multiple Lyapunov Function Approach

計畫編號：NSC 95-2221-E-151-022

執行期限：95 年 8 月 1 日至 96 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：楊閎智、陳宏瑋、謝文建、葛世偉

一、中文摘要 本計畫針對連續及離散 T-S 模糊系統，利用多重 李雅普諾夫函數法，並引進三指標組合技術設計狀 態回授控制器，建立更寬鬆的二次可穩定化條件， 所有的結果以線性矩陣不等式型式表示。由於連續 與離散系統使用的李雅普諾夫函數型式不同，因此 本計畫使用不同的狀態回授控制器之型式對應推 導。尤其是連續系統的控制器型式，比現有文獻更 簡單且易於實現。最後，精簡報告後面附上一些數 值模擬的例題，驗證本計畫所提出方法，實際應用 的可行性。 關鍵詞：T-S 模糊系統，多重李雅普諾夫函數法， 三指標組合技術，線性矩陣不等式，可穩定化條件 Abstract

This project proposes a more relaxed stabilization condition and then designs a state feedback controller via the multiple Lyapunov function approach for continuous time and discrete time T-S fuzzy systems. All the results are represented in the form of linear matrix inequalities (LMIs). Due to the different types of Lyapunov functions for continuous-time and discrete-time systems, the forms of state feedback controllers for both systems are also different. Particularly, for continuous-time cases, the form of the proposed controller is much simpler and more realizable. At the end this report, some practical examples are given to illustrate the proposed ideas. Keywords ： T-S fuzzy systems, linear matrix inequality (LMI), three-index combination, state feedback control.

二、緣由與目的

Most plants in industry are usually nonlinear. It’s difficult to analyze and design controllers for such systems. In order to solve this kind of control

problems, various methods have been developed. In which, fuzzy control is an interesting and popular approach for this problem. Fuzzy control has attracted much attention because it can deal with complicated nonlinear systems easily [6,7,8,15].

In recent years, the issue of stabilization of fuzzy control systems has been discussed extensively. Most of design approaches are based on the T-S fuzzy model [9,21]. The T-S fuzzy model, which is described by many fuzzy IF-THEN rules, has local linear models in its consequent parts. Lots of nonlinear control problems can be easily solved by using the T-S fuzzy model approach.

Based on the T-S fuzzy model, many papers tried to relax the conservatism of stabilization conditions and then used the conditions to design the fuzzy controllers [10,11]. Recently, using the so-called three-index combination technique, a new LMI-based stabilization conditions were developed [2] and proved to be more relaxed than those in [3] and [25]. Dealing with the stability issue of T-S fuzzy control systems, most approaches were based on a single Lyapunov function. These methods basically converted the stabilization problem to solve a single Lyapunov matrix variable in a set of stabilization conditions. Usually, the conditions are expressed in the form of linear matrix inequalities (LMIs) [19,20]. It is known that the stabilization conditions based on a single Lyapunov function is quite conservative, especially when a large number of subsystems are involved.

Recently, the concept of multiple Lyapunov function is introduced to relax the conservativeness of stabilization conditions. In the existing literature, two kinds of approaches have been proposed： (A)Membership function partition ： Divide the

membership function into several state spaces [4,5,16,18,24,27]. According to the state spaces, different positive definite matrices P are solved i and make sure the system is stable in every state space.

(B)Lyapunov stability：Directly choose a multiple Lyapunov function to investigate the stability of closed-loop T-S fuzzy systems and derive the stabilization conditions in an LMI formulation [1,12,13,14,22,23,26].

The drawback of method (A) is that its state-space is multidimensional and usually an appropriate partition of membership function is difficult when the number of premise variables is more than three. In this thesis, we will use method (B) to develop our stabilization conditions and express them in the LMI formulation.

三、結果與討論

In this section, we derive a more relaxed stabilization condition for continuous-time and discrete-time T-S fuzzy systems via the state feedback controller. The concept of multiple Lyapunov function is introduced to derive the stabilization condition. All the conditions are represented in the form of linear matrix inequalities.

1. Takagi-Sugeno fuzzy systems

In recent years, the T-S fuzzy model is widely used in the fuzzy control. The model was proposed by Takagi and Sugeno [21], which can approximate most of the nonlinear systems easily. In the general, the model is represented as

1 1 1 1

Rule IF ( ) isi: z t Mi( ( )) and andz t " z ts( ) isMis( ( ))z t

Then ( ) ( ) ( ) ( ) ( ) i i i x t A x t B u t y t C x t δ = + ⎧ ⎨ ₌ ⎩

(1)

where δ represented an operator. For continuous-time cases δx t( ) means ( )x t , for

discrete-time cases δx t( ) represents x t( +1). In (1),

( ( )), ij j

M z t i=1, 2, ," r , j=1, 2,",s is fuzzy set and is the number of If-Then rules.

are measurable premise variables. is the state vector,

is the control input, and is the measured output. r ( ), 1, 2, , j z t j= " s ( ) n x t ∈ \ _{u t}( )∈ \mu ( ) q y t ∈ \ , i i,

A B and are real constant matrices

that describe the local system. i

C

The final output of the fuzzy system can be inferred as follows：

(

1 ( ) r i( ( )) i ( ) i ( ) i

)

x t h z t A x t B u t δ = =

∑

+ (2) 1 ( ) r i( ( )) i ( ) i y t h z t C x t = =

∑

(3) where 1 ( ( )) ( ( )) ( ( )) i i _r i i w z t h z t w z t = =

∑

represents the weighting of the rule, in which is

defined as . Note that

and . th i w z ti( ( )) 1 ( ( )) s ( ( )) i _j ij w z t =

∏

₌M z tj ( ( )) 0, 1, 2, , i w z t ≥ i= " r is r 1 ( ( )) 1 r i i h z t = =

∑

Definition 1：The fuzzy system (2) is said to be state-feedback stabilizable if there exists a fuzzy state-feedback controller such that the closed-loop fuzzy system is stable.

2. Stabilization of continuous-time T-S fuzzy systems

Consider the fuzzy state feedback control：

1 1 IF ( ) is and and ( ) is Then ( ) ( ). i s i Rule i z t M z t M u t = −K x t " : (4) where i=1, 2, ," . The final output of the fuzzy controller is 1 ( ) r i i ( ) i u t h K x t = = −

∑

(5)

which induces to the closed-loop system

(

)

1 1 ( ) r r i j i i j ( ) i j x t h h A B K = = =

∑∑

−  x t . (6)

Theorem 1 ： Assume that h z tρ( ( )) ≤φρ , where

,ρ=1, 2, , r" . φρ' s are given scalars. The fuzzy system (2) is state-feedback stabilizable via (5) if there exist Pj >0, j=1, 2, ," r

,

Kj, j=1, 2,"r

,

, iii Y 1, 2, , , T, 1, 2, , , jii iij i r Y Y i r T , ij ji YA=YA = " = = " , T i j j i YA =YA YjiA=YA jTi, i=1, 2, ," r−2, j= + "i 1,

,r−1,A= +j 1, ," r and scalars ε >ijA 0, , ,i j A 1, , r = " such that 1 ( ) * 0 0, * * r T T iii i i i iii i iii iii P Y A B K P i r ρ ρ ρ φ ε ε ε = ⎡ _{− −} ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ − ⎥< = ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

∑

" 1, 2, , * * (7) 1 3 ( ) ( ) ( ) * 0 * * * * * * r T iij i i j _T i i i T j j i T iji iij jii iij P Y A B K A B K A B K Y Y ρ ρ ρ φ ε ε = ⎡⎛ ₋ ⎞ ⎛ − ⎞ ⎢⎜ ⎟ ⎜ ⎟ − ⎢⎜ _{⎟ ⎜ ⎟} + − ⎢⎜ _{− −} ⎟ ⎝ ⎠ ⎝ ⎠ ⎢ ⎢ ₋ ⎢ − ⎢ ⎢ ⎢ ⎢⎣

∑

0 0 0 0 0 0 * T T iij j jii i iij jii P P ε ε ε ε ⎤ ⎥ ⎥ ⎥ ⎥_< ⎥ ⎥ ⎥ − _⎥ ⎥ − _⎥⎦ 1, 2, , ; 1, 2, , ; i= " r j= " r i≠ j ) 0 , (8) ( ) ( ) ( ) ( * 0 * * * * * * * * * * * * * * T T i i j i i T T j j i i ij ij A B K A B K A B K A B K ε ε ⎡ ⎛ − ⎞ ⎛ − ⎞ ⎢Ω ⎜ ⎟ ⎜_⎜ ⎟_⎟ ⎜ ⎟ ⎢ _⎝+ − _{⎠ ⎝}+ − _⎠ ⎢ − ⎢ ⎢ ₋ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ A A A A A ( ) ( ) 0 0 0 0 0 0 0 0 ₀ 0 0 0 * 0 * * 0 * * * T j _{T T T} ij ij j j i i T j j j i ij ij j i A B K P P P A B K ε ε ε ε ε ε ε ⎤ ⎛ − ⎞ ⎥ ⎜ ⎟ ⎜_{+ −} ⎟ _⎥ ⎝ ⎠ _⎥ ⎥ ⎥ < ⎥ ⎥ − ⎥ − _⎥ ⎥ − ⎥ − _⎥⎦ A A A A A A A A A A A 1, 2; 1, , 1; 1, , i= "r− j= +i " r− A= +j " , (9) r 1 1 1 2 1 2 1 2 2 2 1 1 2 0, 1, 2, , i i ir i i ir ri i rir Y Y Y Y Y Y i r Y Y Y ⎡ ⎤ ⎢ ⎥ ⎢ _{⎥ ≤ =} ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ " " " # # % # " . (10) where 1 6 r T T T ij i j ji ij i j ji P Y Y Y Y Y Y ρ ρ ρ φ = Ω =

∑

− A− A − A− A− A − A.

Remark 1：In [12], The controller is not suitable for implementation since it contains a derivative term

. Furthermore, the stabilization conditions are not in LMI of

( ( ))

i h z t

ε and γ . To solve them with the LMI toolbox, it needs to guess ε and γ in advance, which increases the complexity of feasibility.

2. Stabilization of discrete-time T-S fuzzy systems －non PDC control law

Consider the fuzzy controller described by

1 1 1 ( ) r j j r j j ( ) j j u t h F h G x t − = = ⎛ ⎞⎛ ⎞ = −⎜ ⎟⎜ ⎟ ⎝

∑

⎠⎝

∑

⎠ . (11) Then the closed-loop fuzzy system can be expressed as 1 1 1 1 ( 1) r i i i r j j r j j ( ) i j j x t h A B h F h G − = = = ⎛ _{⎛ ⎞⎛ ⎞} ⎞ ⎜ ⎟ + = _{− ⎜ ⎟⎜ ⎟} ⎜ _{⎝ ⎠⎝ ⎠} ⎟ ⎝ ⎠

∑

∑

∑

x t . (12)

Theorem 2：The fuzzy system (2) is state-feedback stabilizable via (11) if there exist Pj >0, j= " , 1, ,r

i

F , Gi, i=1, 2, ," r, Yiii m( ), i=1, 2, , ," r Yjii m( ) = ( ),

T iij m

Y i=1, 2, , ;" r Yijk m( ) =Ykji mT( ), Yjik m( ) =Ykij mT( ),

( ) T( ), ikj m jki m Y =Y i=1, 2, ," r−2, j= +i 1, ," r−1, k = j+ "1, , ,r m=1, 2,", r such that ( ) ( ); 1, 2, , ; 1, 2, , ii m Yiii m i r m r Λ < = " = " r ; (13) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 T

ii m ij m ji m Yiji m Yiij m Yiij m Λ + Λ + Λ < + + 1, 2, ; 1, 2, , ; ; 1, 2, , i= "r j= " r i≠ j m= " ; (14) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ij m 2 ik m 2 ji m 2 jk m 2 ki m 2 kj m T T T

jik m jik m ijk m ijk m ikj m ikj m

Y Y Y Y Y Y Λ + Λ + Λ + Λ + Λ + Λ < + + + + + 1, 2, - 2; 1, , -1; i= "r j= +i " r 1, , ; 1, 2, , k= +j " r m= " r r ; (15) 1 1( ) 1 2( ) 1 ( ) 2 1( ) 2 2( ) 2 ( ) 1( ) 1 2( ) ( ) 2 2 0, 2 i m i m ir m i m i m ir m ri m i m rir m Y Y Y Y Y Y Y Y Y ⎡ ⎤ ⎢ ⎥ ⎢ _{⎥ <} ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ " " # # % # " ; 1, 2, , ; 1, 2, , i= " r m= " ; (16) where ( ) ( ) j ij m T i j i j m m m P A G B F G G P − ∗ ⎡ ⎤ Λ _{= ⎢ ⎥} − − + − ⎣ ⎦.

Remark 2: In [1,23], two indices ij are used. This is because the condition is derived from two-index combination approach. Most literature adopted such idea in the past [1,23]. Although the method is simple, the stabilization conditions are too conservative. To overcome the drawbacks, the “ three-index combination ＂is developed.

Example：Consider the following T-S fuzzy model

1 1 1 1

Plant Rule : If ( ) is ( ( )) then ( 1) i ( ) i ( ). i x t F x t x t+ =A x t +B u t where 1 2 3 2 2 0.9 , , 0.1 0.1 0.1 1.7 a a A A A a a − − − 0.5 ⎡ ⎤ ⎡ ⎤ ⎡ =_{⎢ ⎥} =_{⎢ ⎥} =_⎢ ⎤ − − − − ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦, 1 2 3 3 , , 4.8 4.8 0.1 b b B =⎡_{⎢ ⎥}⎤ B =_⎢⎡ ⎤_⎥ B =⎡_⎢ ⎤_⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦.

The parameters and are adjusted to compare the relaxation of [

a b

1,23] and Theorem 2. Fig.1 shows the parameter regions where the fuzzy state feedback stabilization controller of the above system can be found based on [1,23] and Theorem 2. In Fig.1, the symbol denotes the region space of

and where the LMI conditions in [ ∗

a b 1,23], and

Theorem 2 are feasible.

(a) [1] (b) [23]

(c) Theorem 2

Fig. 1 Stabilization regions based on (a) [1], (b) [23], (c) Theorem 2

Remark 3：In view of Theorem 3, The relation of [23] and Theorem 2 can be depicted in Fig. 2.

Fig. 2 Relation of [23] and Theorem 2

3. Stabilization of discrete-time T-S fuzzy systems －PDC control law

Although the fuzzy controller

1 1 1 ( ) r j j r i i ( ) i i u t h F h G x t − = = ⎛ ⎞⎛ ⎞ = −⎜ ⎟⎜ ⎟ ⎝

∑

⎠⎝

∑

⎠

is not in the PDC control form, it can be reduced to a PDC form by simply setting Gi =I, i.e.

1 ( ) r j j ( ) j u t h F x t = ⎛ ⎞ = −⎜ ⎟ ⎝

∑

⎠ . (17)

In this case, the closed-loop system becomes

1 1 ( 1) r i i i r j j ( ) i j x t h A B h F = = ⎛ ⎛ ⎞⎞ + = ⎜_⎜ − ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

∑

∑

⎟⎟x t . (18)

Theorem 3 : The fuzzy system (2) is state -feedback stabilizable via (17) if there exist symmetric matrices

0, j P > j=1, ", r , G and Mj , Yiii m( ) , i=1, 2, ," r, Yjii m( ) = Yiij mT( ), i=1, 2, ", ;r ( ) T( ), ijk m kji m

Y =Y Yikj m( ) =Yjki mT( ), Yjik m( ) =Ykij mT( ), i=1,

2, ," r−2, j= i+ " ,1, 1, k= +j 1, b b r− ", ,r m= 1, 2," such that , r ( ) ( ); 1, 2, , ; 1, 2, , ii m Yiii m i r m r Λ < = " = " r ; (19) a a ( ) ( ) ( ) ( ) ( ) T( )

ij m ii m ji m Yiji m Yiij m Yiij m

Λ + Λ + Λ < + + 1, 2, ; 1, 2, , ; ; 1, 2, , i= "r j= " r i≠ j m= " T ; (20) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ij m ik m ji m jk m ki m kj m T T

jik m jik m ijk m ijk m ikj m ikj m

Y Y Y Y Y Y Λ + Λ + Λ + Λ + Λ + Λ < + + + + + 1, 2, - 2; 1, , -1; i= "r j= +i " r ; b 1, , ; 1, 2, , k= +j " r m= " r r

;

(21) 1 1( ) 1 2 ( ) 1 ( ) 2 1( ) 2 2 ( ) 2 ( ) 1( ) 2 ( ) ( ) 0, i m i m ir m i m i m ir m ri m ri m rir m Y Y Y Y Y Y Y Y Y ⎡ ⎤ ⎢ ⎥ ⎢ _{⎥ >} ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ " " # # % # " , a 1, 2, , ; 1, 2, , i= " r m= "

;

(22) where ( ) ( T ) j ij m i i j m G G P A G B M P ⎡− + − ∗ ⎤ Λ _{= ⎢ ⎥} − − ⎣ ⎦.

The fuzzy state feedback gains are

1_{, 1, 2,} j j F =M G− j= "r. Theorem2 [23]

四、計畫成果自評 本 計 畫 已 依 原 來 之 計 畫 書 之 進 度 完 成 100%，目前相關成果已整理三篇論文在下列研討 會及期刊：

2006 CACS Automatic Control Conference ：

Stabilization of Discrete-Time Nonlinear Control Systems – Multiple Fuzzy Lyapunov Function Approach, Nov. 10-11, pp.626-630,聖約翰科技大學.

2006 The 14th National Conference on Fuzzy Theory and Its Applications： Stabilization of Continuous

-Time T-S Fuzzy Control Systems via the Multiple Fuzzy Lyapunov Function Approach, Dec.14-15, pp. A1-2-1-A1-2-6, 義守大學.

IEEE Transactions on Systems, Man, and Cybernetics Part B：Stabilization of Discrete-Time

Nonlinear Control Systems ─ Multiple Fuzzy Lyapunov Function Approach.

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出席國際會議心得報告

計畫編號

NSC 95-2221-E-151-022

計畫名稱

模糊控制系統穩定度的分析與設計－多重李雅普諾夫函數法

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 電機系教授

出國時間地點 96.07.03-96.07.05 希臘、科斯島（Kos, Greece）

國外研究機構 歐洲控制研討會（European Control Conference）

工作記要：

96 年 7 月 1 日下午三點搭乘華航班機由高雄出發，經泰國曼谷到雅典，再由雅典轉機往

柯斯島，抵達科斯島飯店時為 7/2 下午 8:30，休息一晚，準備隔天的開會。

本次會議共有 1156 篇論文投稿，834 篇論文被接受，接受率約為 72%，論文來自歐洲、

亞洲、美洲、澳洲等 60 國家，其中法國 120 篇最多，美國 100 篇居次，台灣大約已有 10 篇，

包括台大電機系張帆人教授、台大機械系王富正教授、台北科大車輛系黃哲緒主任、大同大

學電機系游文雄教授和他的博士班學生、成功大學化工系張鈺庭教授、義守大學化工系黃奇

教授、中山大學電機系李立教授的博士班學生以及筆者共有 10 人與會，和大陸參加的人數相

當，比日本少，但是比韓國多。

歐洲控制研討會每兩年舉辦一次，這是第七次舉辦，從參加國家分布和投稿的情形來看，

歐洲控制研討會已經確立了大型國際研討會的地位，本次選在希臘的柯斯島，位於雅典南方

大約 400 公里的渡假小島，雖然地點不起眼，因為會議的重要性，仍然吸引來自近 1500 位的

世界各國學者，可見研討會的重要性。本次會議日期正式從七月三日開始，會議進行分成 15

場次同時進行，會議主題包含智慧型控制、非線性控制、控制應用、交通控制、適應控制、

強韌控制、網路控制等，每場次兩個小時，一天安排 4 個場次，分別從 9:00-11:00，11:30-13:30，

15:30-17:30，18:00-20:00，可以說相當密集，另有兩個場次的海報互動時間。大會在第二、

三天中午 11:00-12:00 各安排一場 Keynote speech：邀請 K. Glove 講 Control challenges in

automotive engine management 和 J. S. Baras 講 Security and trust for wireless automatic networks

位大師級的學者演講，對於與會的學者獲益甚多，為了吸引學者，大會在 keynote speech 後面

並安排兩場小型演講：包括 Tempo 講 Randomized algorithms for systems and control；Schmidt

講 Walking primitive data bases for perception based guidance control of biped robot；Willems 講

Dissipative dynamical systems；Morari 講 Multiparametric linear programming with applications to

control。理論與實務的題目都有，這四位都是國際知名的學者，所演講的題目也都具有很高

的吸引力，大會在這方面的安排相當用心。筆者的論文安排在第一天上午 11:50-12:10，屬於

fuzzy systems 場次，本次研討會屬於控制領域，有關 fuzzy 的主題共有 2 個場次，12 篇論文。

今年論文的分布，並沒有特別突出潮流或主題，倒是研討會中有許多是歐洲的汽車公司研發

人員，發表對於汽車引擎控制、油量控制、車流控制等的相關論文，顯見歐洲汽車工業的技

術確實有獨到之處。此外本次大會還安排了許多控制的應用實例論文，例如燃料電池熱氣控

制、電氣閥控制、機器人控制等，聽了之後對於應用的方法和經驗確實提升不少。

本次研討會學術的安排相當用心，其他相關的行政搭配則令人失望，包括現場網路連線

功能的提供，茶水的供應，場地的展示佈置和安排，餐點的提供，交通車的安排，住宿的配

合，旅遊行程的收費等，比起研討會高額的註冊費（500 歐元）

，實在不成正比，大會會場並

沒有提供充足的茶水服務，休息時間的茶點陽春到不行，機場飯店的交通車的收費讓人有被

敲竹槓的感覺，科斯地點隨然偏僻，各種消費並不便宜，本次會議的機票和註冊費早已經超

過國科會補助的 7 萬元，其他的住宿費用和當地的生活所需就不用談了，前後相加本次參加

會議已經透支 2 萬元以上，出國參加會議花時間，又花自己的錢，實在不划算，不過為了學

術的進步，也只能忍痛吸收。

科斯島最繁榮的是科斯鎮，比台灣的一個小村還小，當地人民生活沒有壓力，靠著觀光

收入維生，因此生活步調很慢，符合度假的環境需求，由於人口少，土地顯得非常夠用，因

此民房最多只有兩層樓高，中午因為酷熱，很少人在外面，中飯時間都在兩點以後，下午九

點太陽才下山，碗飯才開始，因此六點以後各店面就坐滿的當地居民和觀光客，喝咖啡啤酒

聊天，同樣是人，在台灣就辛苦太多了。科斯距離土耳其邊界很近，坐船只要 20 分鐘，兩國

因為觀光經濟的關係，衍生出一種特殊的出入境型態，雖然土耳其不是歐盟的一員，在科斯

坐船到土耳其不需要簽證，方便觀光客一次遊覽兩個國家，生意相當不錯，在這裡明顯感受

經濟比政治重要的多，不知國內搞政治的人看了之後有何感想。到歐洲轉機相當不方便，本

次飛行光是等機的時間就超過 24 小時，參加會議可以說是來去匆匆，或許這是影響亞洲國家

參加歐洲會議的原因，反過來很少歐洲國家來亞洲參加國際會議，也就可以理解的。

本次出席國際會議經費由國科會計畫資助，雖然經費無法涵蓋全部支出，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，攜回資料包括論文摘要一本、CD