不確定TS模糊控制系統之強健控制性及觀測性研究

行政院國家科學委員會專題研究計畫 成果報告

不確定 TS 模糊控制系統之強健控制性及觀測性研究

研究成果報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 98-2221-E-151-048- 執 行 期 間 ： 98 年 08 月 01 日至 99 年 09 月 15 日 執 行 單 位 ： 國立高雄應用科技大學機械工程系 計 畫 主 持 人 ： 陳信宏 共 同 主 持 人 ： 鄭良安 計畫參與人員： 碩士班研究生-兼任助理人員：呂方 碩士班研究生-兼任助理人員：杞文芳 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 99 年 09 月 18 日

1

行政院國家科學委員會補助專題研究計畫

■ 成 果 報 告

□期中進度報告

不確定 TS 模糊控制系統之強健控制性及觀測性研究

計畫類別：▓ 個別型計畫 □ 整合型計畫

計畫編號：

NSC98-2221-E-151-048

執行期間：

98 年 8 月 1 日至 99 年 9 月 15 日

計畫主持人：

陳信宏 教授

共同主持人：鄭良安

教授

計畫參與人員：

成果報告類型(依經費核定清單規定繳交)：▓精簡報告 □完整報告

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

▓出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、列

管計畫及下列情形者外，得立即公開查詢

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執行單位：國立高雄應用科技大學機械系

中 華 民 國 99 年 9 月 15 日

2

行政院國家科學委員會專題研究計畫成果報告

不確定 TS 模糊控制系統之強健控制性及觀測性研究

計畫編號：NSC98-2221-E-151-048

執行期限：98 年 8 月 1 日至 99 年 9 月 15 日

主 持 人：陳信宏 教授 國立高雄應用科技大學機械系

共同主持人：鄭良安

教授 國立高雄應用科技大學機械系

一、中英文摘要 本研究計畫報告是針對 TS 模糊控制系 統之強健控制性及觀測性問題進行研究。在 假設標稱 TS 模糊控制系統是局部性地可控 制與可觀測的條件下(即標稱 TS 模糊控制系 統的每一條模糊規則相對應的控制矩陣及 觀測矩陣的秩數為滿列)，當結構型參數不 確定量加入標稱 TS 模糊控制系統時，推導 出充分條件以保有所假設之控制性與觀測 性。推導出的充分條件在保有假設的性質下 可提供參數不確定量界線的明顯關係。此 外，本研究計畫案也將推導不確定 TS 模糊 控制系統全面性強健控制性及觀測性條件。 關鍵詞：TS 模糊控制系統、強健控制性、 強健觀測性、結構型參數不確定量。 Abstract

The robust controllability and observability problems for the Takagi-Sugeno (TS) fuzzy-model-based control systems are studied. Under the assumption that the nominal TS-fuzzy-model-based control systems are locally controllable and observable (i.e., each fuzzy rule of the nominal TS-fuzzy-model-based control systems has the full row ranks for its controllability and observability matrices), some sufficient conditions are proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS-fuzzy-model-based control systems. The proposed sufficient conditions can provide the explicit relationship of the bounds on parameter uncertainties for preserving the assumed properties. Besides, two robustly global controllability and

observability conditions of the uncertain TS-fuzzy-model-based control system are also presented in this report.

Keywords: Robust controllability, robust

observability, TS-fuzzy-model-based control systems, structured parameter uncertainties. 二、計畫緣由與目的

Recently, it has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno (1985), known as the TS fuzzy model, is a successful approach for dealing with the nonlinear control systems, and there are many successful applications of the TS-fuzzy-model-based approach to the nonlinear control systems (Tanaka and Wang, 2001; Tong et al., 2004; Lee et al., 2005; Ren and Yang, 2005; Lian et al., 2006; Lian and Liou, 2006; Chen et al., 2007; Chen et al., 2008; Lin et al., 2008; Nachidi et al., 2008; Ho et al., 2009; and references therein). All the above-mentioned works regarding successful applications of the TS-fuzzy-model-based approach (Tanaka and Wang, 2001; Tong et al., 2004; Lee et al., 2005; Ren and Yang, 2005; Lian et al., 2006; Lian and Liou, 2006; Chen et al., 2007; Chen et al., 2008; Lin et al., 2008; Nachidi et al., 2008; and references therein) are, under the assumption that the nominal TS-fuzzy-model-based control systems are locally controllable (i.e., each fuzzy rule of the nominal TS-fuzzy-model-based control systems has a full row rank for its controllability matrix), to

design the fuzzy parallel-distributed-compensation (PDC)

controllers.

On the other hand, in fact, in many cases it is very difficult, if not impossible, to obtain

the accurate values of some system parameters. This is due to the inaccurate measurement, unaccessibility to the system parameters, or variation of the parameters. These parametric uncertainties may destroy the controllability property of the TS-fuzzy-model-based control systems. Some researchers have studied the controllability problems of various types of fuzzy systems (Gupta et al., 1986; Farinwata and Vachtsevanos, 1993; Farinwata, 1996; Cai and Tang, 2000; Li and Shi, 2001; Qiu, 2005; Park et al., 2006; Cao et al., 2007; Liu and Li, 2008); but, to the authors’ best knowledge, there are no literatures to study the issue of robust controllability/observability for the uncertain TS-fuzzy-model-based control systems.

2 The purpose of this report is to present some approaches for investigating the robust controllability/observability problems of the TS-fuzzy-model-based control systems with parameter uncertainties. Under the assumption that the nominal TS-fuzzy-model-based control systems are locally controllable or observable, some sufficient conditions are proposed to preserve the assumed properties when the parameter uncertainties are added into the nominal TS-fuzzy-model-based control systems.

三、研究方法與成果

Based on the approach of using the sector nonlinearity in the fuzzy model construction, both the fuzzy set of premise part and the linear uncertain dynamic model of consequent part in the exact TS fuzzy control model with parametric uncertainties can be derived from the given nonlinear uncertain control model (Tanaka and Wang, 2001). The parametric uncertainties can be viewed to take different forms like structured (elemental) and unstructured (norm-bounded). Elemental parametric uncertainties are those for which the elemental information of the uncertain matrix is utilized and bounds on the individual elements of the uncertain matrix are considered, whereas norm-bounded parametric uncertainties are those for which only a norm bound on the uncertain matrix is considered (Chou, 1995; Chen and Ren, 2001; Hsieh and Chou, 2004). However, if the elemental

information of the parametric uncertain matrices is considered, the results will be less conservative than those results that do not utilize the elemental information of the parametric uncertain matrices (Chou, 1995; Chen and Ren, 2001). In this report, we mainly consider the elemental parametric uncertainties. The TS-fuzzy-model-based control system with parametric uncertainties for the nonlinear control system with parametric uncertainties can be obtained as the following form: i R~ : IF z₁ is M_i1 and … and zg is Mig, THEN

( )

t

(

A A

( )

t

) ( )

xt

(

B B

( )

t

) ( )

u t , x& = _i +Δ _i + _i +Δ _i (1a) AND y

( ) (

t = C_i +ΔC_i

) ( )

x t ,

(1b) with the initial state vector x

( )

0 , where R~i

(

i=1,2,_K,N

)

denotes the i-th implication, N

is the number of fuzzy rules,

( )

[

( ) ( )

_{, ,}

( )

]

T t x t K n 2 1 t , x x t x = denotes the

n-dimensional state vector,

( )

[

( ) ( )

_{, ,}

( )

]

T t y t K m 2 1 t , y y t y = denotes the

m-dimensional output vector,

( )

[

( ) ( )

_{t , ,u}

( )

_t

]

T p K 2 1 t , u u t u = denotes the

p-dimensional input vector,

are the premise variables, and i z i , i A

(

,g

)

i C , 2 , 1 _K = i B

(

i=1,2,_K,N

)

are, respectively, the n× n, p

n× and m n%×

,

consequent constant matrices, ΔAi Δ and Bi

are, respectively, the time invariant uncertain matrices existing in the system matrices the input matrices and the output matrices

of the consequent part of the i-th rule due to the inaccurate measurement, unaccessibility to the system parameters, output-sensor position variations, or variation of the parameters, and and

i C i , 2 , 1 = Δ i

(

,N

)

, i A , 2 , 1 _K = N , K i B ij M i C

)

g j 1,2,_K,

(

= are the fuzzy sets.

In this report, we suppose that the time-varying parametric uncertain matrices ΔAi

( )

t ,ΔBi

( )

t and take the forms

( )

t Ci Δ

( )

( )

, (2a) 1

∑

= = Δ m k ik ik i t t A A ε

has rank _n2, _where n n_, R A∈ × B∈Rn×p and denotes the n I n×n identity matrix.

( )

( )

, (2b) 1

∑

= = Δ m k ik ik i t t B B ε and

From Lemma 1, it is known that, for the uncertain TS-fuzzy-model-based control system (1), each fuzzy-rule-uncertain model

( )

( )

3 ik C

(2c) 1 , m i ik k C ε = Δ =

∑

where εik

( )

t are the time-varying elemental

parametric uncertainties, and , and are, respectively, the given ,

ik A n× ik B n ik C n ×p,

and constant matrices which are prescribed a prior to denote the linearly dependent information on the time-varying elemental parametric uncertainties

m n%×

( )

t ik , ε in which i=1,2 K, ,N and k=1, K2, ,m.

( )

{

Ai+ ΔA ti ,Bi+ ΔB ti ,Ci+ ΔC ti

}

) 1 ( 2_{× + −} p n n n in (1) is controllable if and only if the

matrix

For the uncertain TS-fuzzy-model-based control system in Eqs. (1) and (2), each fuzzy-rule-nominal model x&

( )

t =Aix

( )

t +Biu

( )

t

{

and y

( )

t =Cix

( )

t denoted by A B Ci, ,i i

}

{

is assumed to be controllable or observable (i.e., each fuzzy-rule-nominal model A B Ci, ,i i

}

has a full row rank for its controllability or observability matrix). Due to inevitable uncertainties, each fuzzy-rule-nominal model

{

A B Ci, ,i i

}

( )

is perturbed into the

fuzzy-rule-uncertain model

( )

( )

{

A_i+ ΔA t_i ,B_i+ ΔB t_i ,C_i+ ΔC t_i

}

.

∑

= + = m k ik ik i i Q t E Q 1 ) ( ~

_ε

₍₄₎

has full row rank 2_{, where} n ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • − • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • − • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i i n i n i i n i B A I B I A B I Q (5) and

Definition (Tong et al., 2004):

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • − • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • − • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ik ik ik ik ik ik B A B A B E . (6) The TS-fuzzy-model-based control

system in Eq. (1) is locally controllable locally observable, if each fuzzy-rule model

( )

( )

( )

{

Ai+ ΔA ti ,Bi+ ΔB ti ,Ci+ ΔC ti

}

(

i=1,2,_K,N

)

is controllable. Lemma 1 (Rosenbrock, 1970):

The system model is controllable if and only if the

matrix ) ( ) ( ) (t Ax t Bu t x& = + ( 2_{× + −} p n n n 1)

Let the singular value decomposition of be i Q ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • − • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • − • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B A I B I A B I Q n n n (3)

[

0 2 _{( 1)}

]

, H i p n n i i i U S V Q = _{× −} (7) where and

are the unitary

2 2 _n n i R U ∈ × ) 1 − + p ( ) 1 ( + − × ∈ n n p n n i R V

matrices, denotes the complex-conjugate transpose of matrix H i V , [ _i1 , i V ], , diag _in2 i S =

σ

_K

σ

0 2 2 1≥ i ≥ _in > i and ≥

σ

σ

σ

_L . i Q

{

are the singular values of

Theorem 1:

Suppose that each fuzzy-rule-nominal model 4

}

, , _i i i A B C 1

∑

= m k ik

ε

; , is controllable. The uncertain TS-fuzzy-model-based control system in (1) is robustly locally controllable, if the following conditions simultaneously hold

, 1 )

φ

_ik < (t (8) where i =1,2,_K N 0; < 0 ) ( ; ) ( for for ), ] ), ] T ) 1 ( T ) 1 t t ik ik p p

ε

ε

− − 0 , [ ( 0 , [ ( 2 2 2 2 I V E S I V E S n n i ik n n i ik ik

_μ

μ

φ

⎪⎩ ⎪ ⎨ ⎧ − − = , ik E 1 1 U U H i i H i i − − ( n n × × S ≥ i V

the matrices and ( , i Ui N i =1,2, 2 n I ( )) ( ) ( 1

∑

= Δ + =N i i i At x A t x&

(

( ) ( i i y t = z C+ C

[

z1, z2,K , K ( ) ( i z h 1 ( ) N i h =

∑

z=

) are, respectively, defined in (6) and (7), and denotes the

identity matrix. 2 n × 2 n

The resulting TS-fuzzy-model-based control system with parametric uncertainties inferred from (1) is represented as

(

t)+(B_i+ΔB_i(t))u(t)

)

, (t , (9a)

)

]

)) ( ) , i x t Δ (9b) in which denotes the

g-dimensional premise vector,

T g z , ) ( ) ( ) ( 1

∑

= = N i i i i z w z w z h

and are the grades of membership of in the fuzzy sets

and ), ( ) ( 1 ij j g j i z M z w

Π

= = ij M

(

i=1,2,_K,N ) ( _j ij z M j z , 2 , 1

j = K, g

)

. It can be seen that, for

all t, ₀ and _. ) 1 − + N i i Q ) (z ≥ h_i (n+ p n

∑

= N i i z h 1 ) ( 1 ) ( = i z h ik ik i z t E h ( )

ε

( ) 1

∑

= N i

∑∑

= = m k 1 1

From Lemma 1, it is known that the resulting uncertain TS-fuzzy-model-based control system in (9) is robustly globally controllable if and only if the

matrix 2 n × = Q~

∑∑

= = N i m k ik ik i z t E h 1 1 ) ( ) (

ε

∑

= N i i z h 1 )( ( )+ = Q +D_i

∑∑

= = m k ik ik i z t E h 1 1 ) ( ) (

ε

,

∑

= + N i i z h 1 ( + N i 2 n = Q )D_i (10)

has full row rank _{where Q} is any given constant matrix having full row rank, and are given in (5) and (6), and ) 1 − i Q ( 2_{× n} n n+ p ik E . Q Q_i − D_i =

Let the singular value decomposition of

Q be

[

0 2 ₎

]

, H n V S U Q = _{×n( p−1} (11) in which n2 n2 R ∈ U × and ) 1 ( ) 1 ( + − × + − ∈ Rn n p n n p

V are the unitary

matrices, H

V denotes the

], , , [ diag ₁ 2 n S =

σ

K

σ

a nd 0 2 2 1≥

σ

≥ ≥

σ

n >

σ

_L values of 5 are the singular .

Q In what fo

heorem

The resulting uncertain llows, we present a

sufficient criterion for ensuring that the resulting uncertain TS-fuzzy-model-based control system in (17) is robustly globally controllable.

T 2:

TS-fuzzy-model-based control system in (9) is robustly globally controllable, if the following condition holds , 1 ) ( ) (−Λ +

∑

N

μ

1 1 1 <

∑∑

= = = N i m k ik ik i i

ε

t

φ

(12) where [ ,0 ]T; ) 1 ( 1 2 2 _{× −} − = Λ H _{i n n n p} i S U DV I ; ] 0 , [ T ) 1 ( 1 2 2 _{× −} − = Ω_ik S U HE_ikV I_{n n n p}

the matrices _{Eik, Di, S, U} and _V are, (6 (1 d

Lemma 2:

The system model

respectively, defined in ), 0) an (11);

2

n

I denotes the n2×n2 identity matrix.

) ( ) ( ) (t Ax t Bu t x& = + le if and only if and y

( )

t =Cx

( )

t is observab the _n2_×n

(

_n+m~₋1

)

_matrix ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 0 0 0 0 T ⎣ • • • − • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • − • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 0 T T T T C A I C I A C I Q n n n (13) has rank 2, _where

n A∈Rn×n, m~ n,

R

C∈ ×

atrix

and I denotes _n m

om Lemma 2, known that, for th

the

Fr it is e ncertain TS-fuzzy-model-based control yste

n identity n×

u

s m in Eqs. (1) and (2), each

fuzzy-rule-uncertain model

( )

( )

( )

{

A_i+ ΔA t_i ,B_i+ ΔB t_i ,C_i+ ΔC t_i

}

in Eqs.

(1) and (2) is observable if and only if the

(

~ 1

)

2_{× + −} m n n n matrix

∑

+ = m k i i Q E Q = ik ik ~ 1 has full row rank where

ˆ _{ε , (14)} , 2 n ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ . ⎢ ⎢_{− • • •} 0 0 n i n I A ⎣ ⎡ • • • − • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T i i n T i i i C A I C C I Q (15) and T ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ 0 T i C ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • − • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T i ik T i ik ik C A C A E (16) Let the singular value decomposition of

⎢ ⎢− be i Q

[

0 2

]

, H i m n n i i i U S V Q = _{× ( −1)} (17) where and _∈ n(n+m−1)×n(n+m−1) i R V es, 2 2 _n n i R U ∈ ×

are the unitary matric H i

V denotes the

complex-conjugate transpose of matrix V_i, ], , , [ diag _{i1 in}2 i S = σ _Kσ and σi₁≥σi₂≥ 0 2 > ≥σ_in

L are the singular values of Qi. :

Theorem 3

Suppose that each fuzzy-rule-nominal model

{

A_i, ,_{i i}

}

uncer

B C is observable. The

in Eqs. (1) an

tain TS-fuzzy-model-based control system d (2) is robustly locally observable, if the following conditions simultaneously hold , 1 ˆ ) ( 1 <

∑

= m k ik ik tφ ε (18)

6 where i =1,2,_K,N; 0; < ) ( 0; ) ( ), ] 0 , [ ( ), ] 0 , [ ( ˆ 2 2 n n ik φ ⎪⎩ ⎨₋ = for for T ) 1 ~ ( 1 T ) 1 ~ ( 1 2 2 t t I V E U S I V E U S ik ik m n n i ik H i i m n n i ik H i i ε ε μ μ ≥ ⎪⎧ − − × − × − −

the matrices E_ik, S_i, U_i and V_i

(i=1,2,_K,N) are, respectively, defined in Eqs.

and de otes entity matr

resulting uncertain el-based control system in Eq.

(16) and (17), n the 2 n × id ix. Theorem 4: The 2 n I 2 n TS-fuzzy-mod

(9) is robustly globally observable, if the following condition holds

( )

( )

~ 1, 1 1 1 < + Λ −

∑∑

∑

= = = N i m k ik ik N i i ε tφ μ (19) where 2 2 1 T_; Λ H [ , 0 _{( 1)}] i S U DV Ii n n n m − × − = % % % ; T ) 1 ( 1~ ~_{[ ,0 ]} ~ 2 2 _{× −} − = Ω ik _{n n n m} H ik S U E V I 0; < ) ( 0; ) ( ), ( ), ( for for t t ik ik ik ik ik _ε ε μ μ φ ≥ ⎪⎩ ⎪ ⎨ ⎧ Ω − Ω − = . ˆ 0 Q Q Di = i −

the matrices E_ik, S~, and are,

respectively, defined in Eqs. (16) and (17);

U~ V~

0

Q is any give 2 (n p−1) constant

matrix having full row rank,I_n2 denotes the

2 n identity matrix. 究成果自評 n 四、研 本成果報告已達成申請計畫書中預期 研究計畫案之部份成果 的發 n × n + 2 n × 完成的成果目標。本 表情況如下所列：

1. S. H. Chen, W. H. Ho and J. H. Chou,

“Robust Controllability of TS-Fuzzy-Model-Based Control Systems

with Parametric Uncertainties”, IEEE Trans. on Fuzzy Systems, Vol.17, No.6, pp.1324-1335, December 2009.

2. S. H. Chen, W. H. Ho and J. H. Chou, “Robust Observability of Uncertain Fuzzy Model Dynamic Systems”, Proc. of the 16th National Conference on Fuzzy Theory and Its Applications, (Paper Number: P0206), Taiwan, R.O.C., December 2008

.

3. S. H. Chen, W. H. Ho and J. H. Chou, “Robustly Global Observability of Uncertain TS-Fuzzy-Model-Based Control Systems”, Proc. of the CACS IACC 2009, (Paper Number: 104) Taiwan, R.O.C., November 2009.

4. S. H. Chen, W. H. Ho and J. H. Chou, “Observability Robustness of Uncertain

. Cai, Z. and S.

Fuzzy-Model-Based Control Systems”, (submitted to International Journal).

五、參考文獻

1 Tang, 2000, “Controllability ness of T-Fuzzy Control

ed Control of Fuzzy

ed Stabilization of

in Versus Lyapunov Type

, “Automatic Design of

ustness for Linear Discrete

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1

行政院國家科學委員會補助國內專家學者出席國際學術會議報告

計畫編號 NSC98-2221-E151-048

計畫名稱

不確定 TS 模糊控制系統之強健控制性及觀測性研究

報告人姓名

陳信宏

服務機構

及職稱

國立高雄應用科技大學機械工程

系/教授

時間

會議地點

8/29/2010-9/02/2010

Sapporo, Japan (日本札幌)

會議名稱

(中文)2010 年人工智慧、機器人、與自動化國際研討會

( 英 文 )The 2010 International Symposium on Artificial

Intelligence, Robotics and Automation in Space

(i-SAIRAS 2010)

發表論文題目

(中文)應用多目標最佳化基因演算法於求解排班問題

(英文) Multiobjective optimization genetic algorithms for domestic

airline crew pairing problems r

一、參加會議經過：

本屆2010年人工智慧、機器人、與自動化國際研討會(The 2010 International

Symposium on Artificial Intelligence, Robotics and Automation in Space,

i-SAIRAS 2010)是由Japan Aerospace Exploration Agency所主辦的年度盛會，地

點在日本札幌Sapporo Convention Center舉行，自2010年8月29日至9月1日為期

四天。我與研究團隊於2010年8月28日搭乘復興航空公司班機，由高雄小港國

際機場飛抵日本新千歲空港，再搭巴士至札幌Sapporo Convention Center參加

會議。我與研究團隊住宿於札幌後樂園飯店以方便開會。

今年會議接受發表之口頭論文與壁報論文僅有123篇，共有來自美國、加

拿大、德國、西班牙、英國、法國、荷蘭、義大利、瑞士、芬蘭、澳洲、日本

、以及台灣共13個國家的研究學者共同參與。123篇論文中，大都來自歐美國

家的研究學者，台灣研究學者參與發表論文，僅有我與研究團隊所完成的論文

1篇，這是非常難得的機會，能與歐美先進國家的研究學者共同討論研究的成

果。除論文發表外，會議中也有2場邀請演講。會議論文研討課題非常豐富，

包括：排程、機器人、控制系統、決策支援系統、人工智慧、自動控制、人機

介面、與模型建模與模擬……等等，涵蓋了人工智慧與機器學習、決策支援系

統、軟體工程之理論、技術、和應用等各個層面。在會議期間，我們參加的論

文發表場次以人工智慧、排程理論、機器人、以及自動控制等課題為主，經過

參與討論及交換心得，我們深感獲益良多。

2

domestic airline crew pairing problems”被安排在8月31日下午之壁報論文的場次

中發表，在會議中有多位學者提出問題和我們討論，彼此交換心得，使我們獲

益匪淺。

二、與會心得：

各國的學者專家共聚一堂，彼此交換研究心得，發表新的研究成果，參與

此次國際會議以及在日本札幌的所見所聞，讓我們收獲良多，而且也有下列幾

點心得：

(1) 人工智慧對在工程技術科學的應用與影響，將是研究之一大趨勢。台灣在

此方面之研究有很大的發展空間。

(2) 從各國發表之論文，可發現台灣學者在實務研究與產學合作方面均有待加

強。

(3) 歐美國家的研究學者參加國際研討會的人數非常多，台灣學者專家應加強

與歐美國家學術界之研究交流，相互合作以達到真正的國際交流。

三、建議：

建議國科會與教育部提供更多學者出席國際會議之經費與名額(包括提供

經費與名額，給未獲得國科會研究計畫案的學者)，以培養國際觀，綿延台灣

在國際學術界持續的活絡人脈和增進我國的國際地位。

四、攜回資料名稱及內容：

攜回 i-SAIRAS 2010 論文集之隨身碟乙只，內容有此次國際研討會所發表

的全部論文；以及會議議程手冊乙本，內容有全部論文的摘要。

139

Multiobjective Optimization Genetic Algorithms for Domestic Airline Crew

Pairing Problems

Tung-Kuan Liu*, Chiu-Hung Chen*, Ta-Yuan Chou**, Wen-Hsien Ho***, Shinn-Horng Chen**** and Jyh-Horng Chou****

* Institute of Engi. Scie. and Tech., Natl. Kaohsiung First Univ. of Scie. and Tech., Taiwan, R.O.C. e-mail: {tkliu, u9515905}@ccms.nkfust.edu.tw

** Department of Computer Scie. and Engi., Natl. Sun Yat-sen Univ., Kaohsiung, Taiwan, R.O.C. e-mail: [email protected]

*** Department of Medical Info. Management, Kaohsiung Medical University, Taiwan, R.O.C. e-mail : [email protected]

**** Department of Mech. Engi., Natl. Kaohsiung Univ. of Applied Sciences, Taiwan, R.O.C. e-mail : {choujh, shchen}@cc.kuas.edu.tw

Abstract

Airline crew pairing problems involve assigning the required crew members to each flight segment in a given time period, while complying with a variety of work regulations and collective agreements. Traditional researches formulate the pairing problems as integer programming problems, and use deterministic approaches to optimize the solutions. Such these approaches usually suffer from some critical issues such as time-consuming enumeration for possible pairings and difficulty to cover the whole search space. The goal of this paper is to develop a new multiobjective approach to solve the crew pairing problem by formulating it into multiobjective combination optimization equations and employing the inequality-based multiobjective genetic algorithm (MMGA) as the global optimization explorer. Besides, our experimental results for a real-world short-haul domestic airline show that the proposed approach can provide quite good pairing solutions.

Keywords: Multiobjective, Genetic algorithm, Airline

pairing problems

1. Introduction

Airline crew pairing problems mainly assign the required crew members to each flight segment of a given time period. In general, the goal of the airline companies is to maximize the total profit under the lowest cost. One way of decreasing the total cost is to maximize the usage with limited number of aircrafts and crewmembers. Due to the safety and security reasons [1,2], the usage limitation related to personnel and mechanical health should be taken into consideration. Therefore, crew

pairing is indeed a critical problem concerned to large cost in the airline industry. In these decades, many researches ever focused on the crew pairing problems. However, crew pairing is still extremely complicated in both stages of problem modeling and solving.

In the airline crew scheduling, all flights which are assigned to the aircrafts according to the routing schedule require the personnel, such as pilots and crew members. Due to the laws and regulations, the working hours of personnel are limited. Therefore, the flights assigned to one aircraft should be separated to several sets so they can be assigned to several groups of crew members. The crew pairing problem considered in this paper contains the following issues:

1. Minimizing number of groups 2. Minimizing layover number 3. Satisfying the laws and regulations

Most researches used the enumeration way to optimal solutions. The drawbacks of enumeration are the solution space will be limited and time consuming for planners. Hence, we use genetic algorithms (GA) to optimize them. We formulate the crew pairing problems into combination optimization equations and the optimal solutions would be globally searched by using a method of inequality-based multiobjective genetic algorithm (MMGA). A real-world case study would be presented in this paper to show the good pairing results of the proposed approach.

2. Related works

There are many evolutionary researches related to the Airline Crew Paring Problem. Chu et al. [3] applied a graph based branching heuristic to a restricted set

i-SAIRAS 2010

140

partitioning problem representing a collection of best pairings. Desauliniers et al. [4] modeled the aircrew pairing problem as an integer, nonlinear multi-commodity flow network model and used a Dantzig-Wolfe decomposition to solve this model. Pairing generation is performed by the approach of resource constrained shortest path subproblem. Stojkoviü

et al. [5] used the column generation method embedded in a branch and bound search tree to solve the aircrew pairing problem. Barnhart et al. [6] developed a heuristic methodology by using dual solutions determined in solving the linear programming relaxation of the crew pairing problem. Barnhart and Shenoi [7] used the approximation model of the airline crew pairing to be an advanced initial solution for conventional approaches. The method can identify deadheads quickly and improve the solution qualities. Goumopoulos and Housos [9] proposed an efficient trip generation approach with special pruning rules which are defined using a high-level language. The method is applied to a rule-based system in a real European airline company.

3. Mathematical models

In this section, we described the mathematical models and objective functions.

Notations

ߙ: number of group of crew members

Ⱦ: maximal number of daily flights assigned to each group of crewmembers.

ߛ : number of flights

ߤ :number of possible pairings suggested by planners ݂:identifier of the flight.

Also, various associated information of each ݂_ are listed as follows :

݌Ƹ : origin of ݂ ݌ҧ : destination of ݂ ݐƸ : departure time in ݌ҧ ݐҧ : arrival time in ݌ҧ

First, we proposes an improved form of candidate solution to overcome the time-consumming problem is described follows as :

ࡿ ൌ ൛ܵ௜ǡ௝หܵ௜ǡ௝ א ܨ ׫ሼെͳሽൟǤ (1) where S is a two-dimensional matrix of ߙ ൈ ߚ elements, and each ܵ௜ǡ௝ represents a flight which means the ݆௧௛ flight assigned to the ݅௧௛ group of crew member. To keep the number of flights assigned to each group identical, we assign dummy flights with flight identifer -1. The main feature of proposed model is that the number of pairings becomesto a controllable variable instead of unexpected value within the range Ͳ ൑ ߤ ൑  ʹఉെ ͳ. This is useful when performing practical pairing process since the number of pairing iis related to the manpower in the airline company.

3.1 Objectives

The goal of aircrew pairing problem is to minimize the total cost. To minimize the total cost is equivalent to minimizing the following objective functions, such as ground transition time, number of deadheading crew, number of layover, flying time, and flight duty period, are described as follows.

Transition time objective ensures that each aircraft has sufficient ground turn-around time not less than the legal ground turn-around time, denoted as ܶ_௑, to be allowed for the subsequent flight. The objective is defined as : ׎ଵሺܵሻ ൌ σ σ௔௜ୀଵ ఉିଵ௝ୀଵݔ௜ǡ௝ሺଵሻ (2) ™Š‡”‡ݔ_௜ǡ௝ሺଵሻൌ ൜_ܶ Ͳ ௫െ ൫ݐ௜ǡ௝ାଵെݐ௜ǡ௝൯ ݂݅൫ݐ௜ǡ௝ାଵെݐ௜ǡ௝൯ ൒ ܶ௫ ݋ݐ݄݁ݎݓ݅ݏ݁Ǥ Deadheading crew objective ensures that the arrival airport of ܵ௜ǡ௝ is the same with the departure airport of ܵ௜ǡ௝ାଵ for each aircraft in S, for ͳ ൑ ݅ ൑ן, and ͳ ൑ ݆ ൑ ߚ. This object is reduce the extra cost of the nonprofit flight from ܲ_௜ǡ௝ to ܲ_{௜ǡ௝ାଵ}. The objective is defined as :

׎ଶሺܵሻ ൌ σ σ௔ ఉିଵ_௝ୀଵݔ_௜ǡ௝ሺଶሻ

௜ୀଵ (3)

™Š‡”‡ݔ௜ǡ௝ሺଶሻൌ ൜Ͳͳ݂݅ܲ௜ǡ௝ൌ  ݌Ƹ௜ǡ௝ାଵ ݋ݐ݄݁ݎݓ݅ݏ݁Ǥ

Layover objective ensures each group of crewmembers can start from and end to their home bases. However, we consider a different case of multiple home base consideration. In this condition, the start and end station is no need to be identical. Instead, the consideration of generating pairings is to let the start and end stations belong to the set of crew bases. Suppose the first and the last flights of the ݅௧௛ group in S are ܵ_௜ǡଵ and ܵ௜ǡ௟௔௦௧, respectively. Also, let the set of crew bases be ܲ_஻. Then, the evaluation function can be defined as :

׎ଷሺܵሻ ൌ σ ݇௜௔

௜ୀଵ (4) ™Š‡”‡݇௜_{ൌ ൜Ͳͳ}൫݌௜ǡଵא ܲ஻൯ ר ൫ܲ௜ǡ௟௔௦௧א ܲ஻൯

݋ݐ݄݁ݎݓ݅ݏ݁Ǥ

According to the laws and regulations, the flight duty time, which is the total flight time except for the rest time, of each aircrew pair should not be more than a legal time ܶி஽௉. Therefore, the fourth evaluation function ׎ସሺሻ can be defined as follows.

׎ସሺܵሻ ൌ σ ߟ௜ן

௜ୀଵ (5) ™Š‡”‡ߟ௜_{ൌ ൜Ͳͳ}ݐ௜ǡ௟௔௦௧െݐƸ௝ǡଵ൑  ܶி஽௉

݋ݐ݄݁ݎݓ݅ݏ݁Ǥ

In other words, if the total flight duty time of one aircrew pair exceeds the legal time ܶ_ி஽௉, the evaluation function ׎ସሺࡿሻ will be added the exceeding time, or the violation time.

3.2 Definition of Auxiliary Performance Index Vector

In the original formulations of multiobjective optimization, we haven’t consider the set of admissable bounds, and we decide the admissable bounds performance index for multiobjective optimization. The

141

original objective are transformed into the auxiliary performance index vector :

߉ሺࡿǡ ߝሻ ൌ ሾߣଵሺܵǡ ߝଵሻǡ ߣଶሺܵǡ ߝଶሻǡ ߣଷሺܵǡ ߝଷሻǡ ߣସሺܵǡ ߝସሻሿ (6) ™Š‡”‡ߣ௜ሺܵǡ ߝ௜ሻ ൌ ൜_{׎ଵሺܵሻ െ ߝ௜}Ͳ ݂݅׎ଵሺܵሻ ൑ ߝ௜_{݋ݐ݄݁ݎݓ݅ݏ݁Ǥ}

The auxiliary performance index vector related to the inequalities is converted from the MOI problem to a multiobjective optimization problem. The multiobjective formulation using the auxiliary performance index vector is useful for MOI since the admissible bounds can be combined to all objectives. Therefore, each objective can be transformed to the form of inequalities.

3.3 Formulation of Airline Pairing Problems

Instead of combining these objectives into a single scalar, the air crew routing problem with multiple objectives can be formulated as follows.

Minimize ߣ௜ሺܵǡ ߝ௜ሻǡ ͳ ൑ ݅ ൑ Ͷ (7) Subject to ܵ ൌ ൣܵ௜ǡ௝൧_ןൈఉǤ

4. Solution by Using MMGA

We propose the MMGA to solve the airline crew pairing problem. MMGA employs the global search capability of genetic algorithms and the auxiliary vector performance index can always generate tunable parameters belong to a strictly Pareto ranking and optimized the multi-objective problems. A heuristic Pareto algorithm was also provided to lower the Pareto computation costs.

Figure 1. Flow chart diagram

The flow chart of the algorithm can be summarized in Figure 1. Just like the general multi-objective genetic algorithm (MOGA), evolutionary population should be operated by iterations through initialization, fitness computation, multiobjective evaluation, crossover to generate offspring, mutation and selection for elimination.

4.1 Encoding Scheme

The encoding scheme of each individual is a two-dimensional matrix. To make the encoding more

efficiency, we transform the chromosome to a string. To satisfy the objective of working hour, we use a modified approach to reduce the complexity on solving the working hour objective. In each individual, the flights that are earlier than time t are allocated in the left-hand side of the individual. On the other aspect, the flights that are later than time t are put in the right-hand-side of the individual.

4.2 Selection

Better parents are selected for a subsequent crossover operation, and a roulette wheel method, is utilized for the selection.

4.3 Crossover

In the crossover process, we use an order-based crossover. First, a 0-1 random mask string is generated to determine which flights are fixed on original positions, and which flights are selected to be changed. If the ݅୲୦ element of the generated mask is 1, then the ݅୲୦ gene of offspring1 is fixed on original position. Otherwise, it will be replaced. As shown in Fig. 2, the genes to be replaced on each offspring are in the following order:

Children 1: AĺBĺCĺDĺEĺFĺGĺH Children 2: DĺAĺFĺHĺCĺEĺGĺB

After the process of crossover, the orders of the genes are exchanged according to the following order:

Offspring1: DĺAĺFĺHĺCĺEĺGĺB Offspring2: AĺBĺCĺDĺEĺFĺGĺH

Figure 2. Order-based crossover

4.4 Mutation

The mutation operation as the Figure 3. The individual are temporarily transformed to the conceptual model of 2-dimensional matrix. When selecting the genes to be exchanged, only the segments with violations have more chances to be selected. This can prevent extra costs of inefficient search.

142

Figure 4.Gantt chart of MD90 schedule

Figure 5. Crew pairing with 11 groups

Transition time Deadhead Layover Flying time

Flight duty period

Generation(s) 8,000 6,000 4,000 2,000 0 V io la tio n (s ) 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

143 5. Experiment

In this subsection, we apply the MMGA to solving airline crew pairing case. All timetables are real data obtained from a local airline company.

The parameters used in MMGA, such as population size, number of generations, and crossover and mutation probabilities are 100, 10000, 0.95, and 0.03, respectively. Also, all experiments can stop earlier when the algorithm finds out the solutions without violations

For the given number of groups of crewmembers, the proposed algorithm can find out feasible solutions, which can satisfy all objectives with no violations. The pairing results and convergences of test are shown in Figs. 4 to 5.

6. Conclusions

We have demonstrated an approach of using MMGA to solve the aircrew pairing problem both in the formulation and solution stages. In the formulation stage, we propose a novel permutation-based model that can save the overheads in traditional models, such as assigning cost values, and checking the number of coverage. In the solution stage, we apply the MOI-based MGA (MMGA) to solve the problems of aircraft routing and crew pairing.

According to the experimental results, the proposed method can find out the scheduling result of test case. Instead of using the heuristics of the twice number of aircrafts, the proposed method can further find out less group number of crewmembers when the number of flights is small. Hence, the proposed MMGA can not only find out solutions satisfying the given objectives, but also have more chances to find out optimal solutions especially the group number of crewmembers can be decreased so that the cost can be reduced.

7. Acknowledgement

This work was partially supported by the National Science Council, Taiwan, Republic of China, under grant numbers NSC 98-2221-E-327-025, NSC 98-2811-E-327-003, NSC 98-2811-E-327-004 and NSC98-2221-E-151-048.

8. Reference

΀ϭ΁ Chu HD, Gelman E, Johnson EL (1997) Solving large scale crew scheduling problems. European Journal of Operations Research 97:260-268 ΀Ϯ΁ Desaulniers G, Desrosiers J, Dumas Y, Marc S,

Rioux B, Solomon M, Soumis F (1997) Crew pairing at Air France. European Journal of Operational Research 97:245-259

΀ϯ΁ Barnhart C, Hatay L, Johnson EL, (1995) Deadhead selection for long-haul crew pairing problem, Operations Research 43:491-499

΀ϰ΁ Stojkoviü M, Soumis F (1998) The operational airline crew scheduling problem. Transportation Science 32:232-245

΀ϱ΁ Barnhart C, Shenoi RG (1998) An approximate model and solution approach for the long-haul crew pairing problem. Transportation Science 32: 221-231

[6] Leung, Y.W., and Wang, Y., “An Orthogonal Genetic Algorithm with Quantization for Global Numerical Optimization,” IEEE Transaction on Evolutionary Computation, Vol. 5, Issue 1, pp. 41-51, 2001.

[7] Cohn AM, Barnhart C (2003) Improving crew scheduling by incorporating key maintenance routing decisions. Operations Research 51:387-396 [8] Deb, K., Multi-objective Optimization using Evolutionary Algorithms, John Wiley & Sons, 2003.

΀ϵ΁ Guomopoulos C, Housos E (2004) Efficient trip generation with a rule modeling system for crew scheduling problems. The Journal of Systems and Software 69:43-56

[10] Tsai, J. T., T. K. Liu, and J. H., Chou, “Hybrid Taguchi Genetic Algorithm for Global Numerical Optimization”, IEEE Transaction on Evolutionary Computation, Vol. 8, Issue 4, pp. 365 – 377, 2004. [11] Gopalakrishnan B. and E. L. Johnson, “Airline

Crew Scheduling: State-of-the-Art,” Annals of Operations Research, Vol. 140, pp. 305–337, 2005. [12] Lee, L. H., C. U. Lee, and Y. P. Tan, “A multi-objective genetic algorithm for robust flight scheduling using simulation”, European Journal of Operational Research, Volume 177, Issue 3, pp. 1948-1968, 2007

98 年度專題研究計畫研究成果彙整表

計畫主持人：陳信宏 計畫編號： 98-2221-E-151-048-計畫名稱：不確定 TS 模糊控制系統之強健控制性及觀測性研究 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 2 2 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 2 2 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 0 0 100% 研究報告/技術報告 1 2 100% 研討會論文 1 1 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次

其他成果

(

無法以量化表達之成 果如辦理學術活動、獲 得獎項、重要國際合 作、研究成果國際影響 力及其他協助產業技 術發展之具體效益事 項等，請以文字敘述填 列。) 此計畫共發表兩篇期刊論文(一篇刊登一篇投稿中)及四篇研討會論文。 成果項目 量化 名稱或內容性質簡述 測驗工具(含質性與量性) 0 課程/模組 0 電腦及網路系統或工具 0 教材 0 舉辦之活動/競賽 0 研討會/工作坊 0 電子報、網站 0 科 教 處 計 畫 加 填 項 目 計畫成果推廣之參與（閱聽）人數 0

國科會補助專題研究計畫成果報告自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）

、是否適

合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

■達成目標

□未達成目標（請說明，以 100 字為限）

□實驗失敗

□因故實驗中斷

□其他原因

說明：

2. 研究成果在學術期刊發表或申請專利等情形：

論文：■已發表 □未發表之文稿 □撰寫中 □無

專利：□已獲得 □申請中 ■無

技轉：□已技轉 □洽談中 ■無

其他：（以 100 字為限）

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以

500 字為限）

本研究計畫案之部份成果的發表情況如下所列：

1.S. H. Chen, W. H. Ho and J. H. Chou, ＇Robust Controllability of TS-Fuzzy-Model-Based Control Systems with Parametric Uncertainties＇, IEEE Trans. on Fuzzy Systems, Vol.17, No.6, pp.1324-1335, December 2009.

2.S. H. Chen, W. H. Ho and J. H. Chou, ＇Observability Robustness of Uncertain Fuzzy-Model-Based Control Systems＇, (submitted to International Journal).