線性不確定離散奇異時延系統之強健性分析

行政院國家科學委員會專題研究計畫 成果報告

線性不確定離散奇異時延系統之強健性分析

計畫類別： 個別型計畫 計畫編號： NSC92-2213-E-151-006- 執行期間： 92 年 08 月 01 日至 93 年 07 月 31 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 報告類型： 精簡報告 處理方式： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢

中 華 民 國 93 年 8 月 26 日

行政院國家科學委員會補助專題研究計畫

v 成 果 報 告

□期中進度報告

線性不確定離散奇異時延系統之強健性分析

計畫類別：

ˇ

個別型計畫 □ 整合型計畫

計畫編號：NSC92

－2213－E－151－006

執行期間：

92 年 8 月 1 日至 93 年 7 月 31 日

計畫主持人：陳信宏

教授

共同主持人：

計畫參與人員：

成果報告類型(依經費核定清單規定繳交)：

ˇ

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

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□國際合作研究計畫國外研究報告書一份

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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

93 年 7 月 31 日

行政院國家科學委員會專題研究計畫成果報告

線性不確定離散奇異時延系統之強健性分析

計畫編號：NSC 92-2213-E-151-006

執行期限：92 年 8 月 1 日至 93 年 7 月 31 日

主持人：陳信宏 教授 國立高雄應用科技大學機械系

一、中英文摘要 本研究成果報告主要是研究具結構型參數不確定量之線性離散廣義時延系統的強健特徵 值叢集問題。提出一些新的充分條件以保證結構型參數不確定量加到標稱離散廣義時延系統 時仍具正則、脈衝免疫及特徵值仍叢集在一特定區域內的特性。對於具結構型參數不確定量 之線性離散廣義時延系統的強健 D 穩定性(特徵值叢集在一特定圓形區域內)而言，將以數學 方法證明本研究成果報告所提的結果比目前文獻上的結果有較低的保守性，並舉一工程例子 來說明所提充分條件的應用性。 關鍵詞：離散廣義時延系統，強健穩定性，強健特徵值叢集，參數不確定量。 Abstract

In this report, the robust regional eigenvalue clustering problem of linear discrete singular time-delay systems with structured parameter uncertainties is investigated. Two new sufficient conditions are proposed to preserve the assumed properties when the structured (elemental) parameter uncertainties are added into the nominal discrete singular time-delay system. For robust D-stability analysis, the presented criterion is mathematically proved to be less conservative than the existing ones reported recently in the literature. An illustrated example is given to demonstrate the applicability of the proposed sufficient conditions.

Keywords: Discrete singular time-delay systems, stability robustness, regional eigenvalue

clustering robustness, structured parameter uncertainties. 二、計畫緣由與目的

To ensure both stability robustness and certain performance robustness, it is important to guarantee that the eigenvalues of a linear time-invariant system under parameter perturbations remain in a specified region. Recently, Fang et al. (1994), Lee and Fang (1994), Fang (1997), Chou and Liao (1998), and Chou et al. (2003) have investigated the robustness problem of eigenvalue-clustering in specified regions for the following singular systems with structured (elemental) parameter perturbations: Ex&(t)=(A+∆A)x(t) and Ex(k+1)=(A+∆A)x(k), where

n n

R

E∈ × , n n

R

A∈ × , x(t)∈Rn, x(k)∈Rn , ∆ denotes the A n×n time-invariant structured parameter uncertain matrix, and ∆A ≤U, in which U is a given nonnegative constant matrix and represents highly structured information, and ∆A denotes the modulus matrix of the matrix ∆ . A

Here the matrix E may be a singular matrix with rank E( )≤ . In many applications, the matrix n E is a structure information matrix rather than a parameter matrix, i.e., the elements of E

contain only structure information regarding the problem considered. Sometimes the singular system is called generalized state-space system, implicit system, semistate system, or descriptor systems (Lewis, 1986). When E= where I denotes the I n×n identity matrix, the singular systems become the well-known standard state-space systems. Here it should be emphasized that the robust regional eigenvalue-clustering analysis of uncertain singular systems should consider not only regional eigenvalue-clustering robustness but also system regularity and impulse-free (or

causal) behaviors. The later two ones need not be considered in the standard state-space systems (Dai, 1989; Weinmann, 1991; Fang, 1997). On the other hand, over the years, the robustness problem of time-delay systems has been explored because time-delay is commonly encountered in various engineering systems (Chu, 1995). The linear discrete singular time-delay systems can be found in many practical applications, such as electrical networks, large-scale systems, constrained robots, economical systems, and so forth (Liu and Xie, 1998). Therefore, very recently, Xu et al. (2002) and Pan and Chen (2002, 2003) have studied the robust D-stability problem (i.e., the robust eigenvalue-clustering in a specified circular region problem) for linear discrete singular time-delay systems with structured (elemental) parameter uncertainties. To the authors’ present knowledge, the research on the regional eigenvalue-clustering robustness analysis for linear discrete singular time-delay systems with parameter uncertainties is considerably rare and almost embryonic.

Under the assumption that the linear discrete nominal singular system with single time delay, ) ( ) ( ) 1 (k A₀x k A x k d

Ex + = + _d − , is regular and causal, and has all its finite eigenvalues lying within a specified circular region, Xu et al. (2002) and Pan and Chen (2002, 2003) presented some sufficient conditions to guarantee the D-stability robustness of the following linear discrete singular system with both single time delay and structured parameter uncertainties:

Ex(k+1)=(A0+∆A0)x(k)+(Ad +∆Ad)x(k−d), Ex( )0 = Ex0, (1) where n n R A0∈ × , n n d R

A ∈ × , x(k)∈Rn , ∆ and A0 ∆Ad denote the n×n time-invariant

structured parameter uncertain matrices, and d >0 is a known positive integer time delay of the system. In their works, Xu et al. (2002) and Pan and Chen (2002, 2003) assumed that the uncertain matrices ∆ and A0 ∆ are bounded by the following inequalities: Ad

∆A₀ ≤U₀ and ∆A_d ≤U_d , (2) in which U and ₀ U are given nonnegative constant matrices and represent highly structured _d

information. For the linear uncertain discrete singular time-delay system (1), if we set

[

T T T T

]

T d k x k x k x k x k x( ) ( ), ( 1), ( 2),..., ( ) ~ _{= − − − , (3)} then the singular time-delay system (1) becomes the following linear uncertain discrete singular system without time delays:

E~~x(k+1)=(A~+∆A~)~x(k), (4) where ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • = I I E E 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ , ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • = 0 0 0 0 0 0 0 0 0 0 0 ~ I I I A A A d o , and ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ∆ • • • ∆ = ∆ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ d o A A A , (5)

in which E~ , A~ and ∆ are of dimension A~ n(d +1)×n(d +1) . Many existing sufficient conditions, which are proposed by Fang et al. (1994), Lee and Fang (1994), Fang (1997), Chou and Liao (1998) and Chou et al. (2003), for analyzing both the robust D-stability and the robust eigenvalue-clustering problems of linear uncertain discrete singular system without time delays can be used to test the regularity, causality and D-stability (or eigenvalue-clustering) of the linear augmented uncertain discrete singular system (4). However, it should be noted that for a d~

increase in the time delay d, the numbers in rows and columns of matrices E~, A~ and A∆ in ~ Eq. (5) will increase by nd~, respectively. Therefore, for large delays, the sizes of matrices E~, A~ and ∆ become large and the existing sufficient conditions for the linear augmented uncertain A~

discrete singular system (4) will become difficult to apply. Thus, in order to get around the mentioned-above difficulty, Xu et al. (2002) and Pan and Chen (2002, 2003) directly adopted the linear uncertain discrete singular time-delay system (1) instead of the linear augmented uncertain discrete singular system (4) to develop some sufficient conditions to analyze the robust D-stability problem of the linear discrete singular time-delay system with structured parameter uncertainties. Xu et al. (2002) proposed a sufficient condition based on the pulse-response sequence matrix (Chou, 1991) of the nominal singular time-delay system to guarantee the D-stability robustness of the linear discrete singular time-delay system (1) with structured parameter uncertainties. Using the maximum modulus principle and the spectral radius of matrices (Fang, 1997), Pan and Chen (2002, 2003) presented a robust D-stability criterion for the linear discrete singular time-delay system (1) with structured parameter uncertainties. Note that the results proposed by Xu et al. (2002) and Pan and Chen (2002, 2003) can only be applied to analyze the robustness problem of eigenvalue-clustering within a specified circular region.

The purpose of this report is to propose two new approaches to study the regional eigenvalue-clustering robustness problem of linear discrete singular time-delay systems with structured parameter uncertainties, where the specified region may be allowed to be any shape. The main results are presented in Section 3. An example is given in Section 4 to illustrate the application of the proposed sufficient conditions. Finally, Section 5 offers some conclusions.

三、研究方法與成果

Consider a linear discrete singular time-delay system with structured parameter uncertainties described by

∑

= − ∆ + = + p i i i A x k i A k Ex 0 ) ( ) ( ) 1 ( , Ex( )0 = Ex₀, (6) where n n R E∈ × , n n i R A ∈ × , n R k

x( )∈ , and ∆Ai ( i=0,1,2,...,p ) denote the n×n

time-invariant structured (elemental) parameter uncertain matrices. Note that the linear uncertain discrete singular time-delay system (6) considered in this paper is more general than the linear uncertain discrete singular time-delay system (1) considered by Xu et al. (2002) and Pan and Chen (2002, 2003). In many interesting problems, we have only a small number of uncertain parameters, but these uncertain parameters may enter into many entries of the system matrix. In particular, the uncertain matrix ∆ (Ai i=0,1,2,...,p) may be of the form

∑

= = ∆ m j ij ij i H A 1 ε , (7) in which H_ij ( i=0,1,2,...,p and j=1,2,...,m ) are given constant matrices, and εij

(i=0,1,2,...,p and j=1 ,2,...,m) are uncertain parameters.

In this report, we assume that the nominal singular time-delay system

∑

= − = + p i ix k i A k Ex 0 ) ( ) 1

( _{denoted by} (E,A) _{is regular and causal. It is also assumed that all the}

finite eigenvalues of the nominal singular time-delay system (E,A) are located within a specified region D which may be allowed to be any shape. Due to inevitable uncertainties, the nominal system (E,A) is perturbed into the uncertain system (E,A+∆A) which denotes

∑

= − ∆ + = + p i i i A x k i A k Ex 0 ) ( ) ( ) 1

( _{. Our problem is to determine the conditions such that, under the} above-mentioned assumptions, the uncertain singular time-delay system (E,A+∆A) is still regular and causal, and has all its finite eigenvalues retained in the same specified region as the nominal singular time-delay system (E,A) does.

( ) ( ) ( ) 0 1 0 G k z J G z A zE _sp k k = + = −

∑

∞ = − − _{, (8)}

where G(k) is the pulse-response sequence matrix of ( 0) 1 − − A zE _, G_sp(z) _{is a strictly proper} matrix part of ( 0) 1 − − A

zE _{, and J =G(0) is a constant matrix part.}

In what follows, we present two new sufficient conditions for ensuring that the uncertain singular time-delay system (E,A+∆A) remains regular and causal, and has all its finite eigenvalues retained inside the same specified region as the nominal singular time-delay system

) ,

(E A _does.

Theorem 1:

Assume that the nominal singular time-delay system (E,A) is regular and causal, and assume that the nominal singular time-delay system (E,A) has all its finite eigenvalues located inside a specified region D . The uncertain singular time-delay system (E,A+∆A) is still regular and causal, and has all its finite eigenvalues retained inside the same specified region as the nominal singular time-delay system (E,A) does, if the following inequalities are satisfied:

) 1 1 0 0 ⎥< ⎦ ⎤ ⎢ ⎣ ⎡

∑

= m j j j JH r ε , (9a) and 1 ) ( 0 1 1 0 < ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −

∑∑

∑

= = − − = − p i m j ij i p i i i ij qE q A q H r ε , (9b) where q∈ and Q denotes the boundary of the specified region D ; Q J is the constant matrix part of 1

0)

(zE− A − and is defined in the following Eq. (10): ( ) ( ) ( ) 0 1 0 G k z J G z A zE sp k k = + = −

∑

∞ = − − _{, (10)}

in which G(k) is the pulse-response sequence matrix of ( 0) 1 − − A zE _, Gsp(z) is a strictly proper matrix part of ( 0) 1 − − A

zE _{, and J =G(0) is a constant matrix part.}

Proof: Here we omit it.

Theorem 2:

Assume that the nominal singular time-delay system (E,A) is regular and causal, and assume that the nominal singular time-delay system (E,A) has all its finite eigenvalues located inside a specified region D . The uncertain singular time-delay system (E,A+∆A) is still regular and causal, and has all its finite eigenvalues retained inside the same specified region as the nominal singular time-delay system (E,A) _{does, if the following inequalities are satisfied:}

1 1 0 0 <

∑

= m j j jφ ε ; (11a) and 1 0 1 <

∑∑

= = p i m j ij ijθ ε , (11b) where 0; < 0; ), ( ), ( 0 0 0 0 0 for for j j j j j JH JH ε ε µ µ φ ≥ ⎩ ⎨ ⎧ − − = ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ − − − ≥ − = − − = − − − = −

∑

∑

0; < ), ) ( ( sup 0; ), ) (( sup

for

for

1 0 1 0 ij ij i p i i i q ij ij i p i i i q ij H q A q qE H q A q qE ε µ ε µ θ

in which J is the constant matrix part of 1 0)

(zE− A − and is defined in Eq. (9); q∈ and Q Q denotes the boundary of the specified region D .

Proof: Here we omit it.

Remark 1: If Q is chosen as the unit circle and D as the inside-unit-circle disc, the

sufficient conditions (9) and (11) would become the robust stability criterion for the linear uncertain discrete singular time-delay systems.

Remark 2: Xu et al. (2002) have shown that the linear uncertain discrete singular time-delay

system (1) is regular and causal, and has all its finite eigenvalues located within the same specified circular region D(e, f) as the nominal system Ex(k+1)= A0x(k)+Adx(k−d) does, if the following inequality holds:

r[f −1T_r(U₀ +_l−dU_d)]<1, (12) where 0 1 0 0 1 0 0 0 H H U A m j j j m j j j ≤ ≤ = ∆

∑

∑

= = ε ε and _d m j dj dj m j dj dj d H H U A = ≤ ≤ ∆

∑

∑

= =1 1 ε ε , (13)

in which U and ₀ U are given nonnegative constant matrices, _d D(e,f) denotes the interior of the disk centered at (e,0) with radius f , _l=min

{

f +e, f −e

}

, and Tr is defined as

∑

∞ = = 0 ) ( k r r G k T , (14) where Gr(k) is the pulse-response sequence matrix (Chou, 1991) of Gpr(v) defined by

∑

∞ = − − − ₌ + − − = 0 1 ) ( ) ) ( 1 ( ) ( k k r d d r pr fv e A G k v f A vE v G (15)

with v=(z−e)/ f and Ar =(A0−eE)/f . For the case of eigenvalue-clustering in a specified circular region D(e,f), and for the linear uncertain discrete singular time-delay system (1), the sufficient condition in Eqs. (9a) and (9b) proposed in Theorem 1 becomes

) 1 1 0 0 ⎥< ⎦ ⎤ ⎢ ⎣ ⎡

∑

= m j j j JH r ε , (16a) and

∑

= − − + − − + m j j d d j e f E A e f A H r 1 0 1 0 0 (( ) ( ) ) [ ε γ γ 1 ] ) ( ) ) ( ) (( 1 1 0 − + ⋅ + < − + +

∑

= − − − m j dj d d d dj e fγ E A e fγ A e fγ H ε , (16b)

where γ =exp(~jθ) , ~j = −1 and θ∈[0 ,2π]. Now, we compare the proposed sufficient condition (16) with the condition (12) given by Xu et al. (2002). From Eqs. (14) and (15), we have

)) ~ (exp( ) ) ( ) ((e+ fγ E−A₀ − e+ fγ −dA_d −1 = f −1G_pr jθ ( ( )exp( ~ )) 0 1

∑

∞ = − _{⋅ −} = k r k jk G f θ ( ( )exp( ~ )) 0 1

∑

∞ = − _{⋅ −} ≤ k r k jk G f θ _r k r k f T G f 1 0 1 ) ) ( ( − ∞ = − _{⋅ =} =

∑

, (17a) and _r k r r f G k f T G f J 1 0 1 1 ) ) ( ( ) 0 ( − ∞ = − − _{≤ =} =

∑

, (17b) for θ∈[0 ,2π]. Since ) ~ exp( θ γ e f j f e+ = + ≥min

{

f +e, f −e

}

, we have d d d e f e f f e+ )− ≤(min{ + , − })− =_l− ( γ . (18) Using Eqs. (13), (17) and (18), we can obtain

∑

= − − + − − + m j j d d j e f E A e f A H r 1 0 1 0 0 (( ) ( ) ) [ ε γ γ (( ) ( ) ) ( ) ] 1 1 0

∑

= − − − _{⋅ +} + − − + + m j dj d d d dj e fγ E A e fγ A e fγ H ε

∑

= − − − _{+ +} + − − + ≤ m j dj dj d j j d d H f e H A f e A E f e r 1 0 0 1 0 ( ) ) ( ( ) )] ) (( [ γ γ ε γ ε

∑

= − − _{⋅ + +} ≤ m j dj dj d j j r H e f H T f r 1 0 0 1 )] ) ( ( [ ε γ ε

∑

= − − _{⋅ +} ≤ m j dj dj d j j r H H T f r 1 0 0 1 )] ( [ ε _l ε ≤r[f−1T_r(U₀ +_l−dU_d)], (19a) and _⎥ ⎦ ⎤ ⎢ ⎣ ⎡

∑

= ) 1 0 0 m j j j JH r ε _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ≤

∑

= ) 1 0 0 m j j j J H r ε [ ( 0 )] 1 d d r U U T f r − + − ≤ _l , (19b)

for γ =exp(~jθ) and θ∈[0 ,2π] . This mathematically proves that, for the case of eigenvalue-clustering in a specified circular region D(e,f), the sufficient condition (16) proposed in this paper is less conservative than the sufficient condition (21) proposed by Xu et al. (2002). On the other hand, using the maximum modulus principle (Churchill and Brown, 1990; Lee and Fang, 1994; Fang, 1997), Pan and Chen (2002, 2003) have also shown that the linear uncertain discrete singular time-delay system (1) remains regular and causal, and all its finite eigenvalues are located within the specified circular region D(e, f), if the following inequality holds:

1 )] (

[H U₀ + −dU_d <

where H =[suph_ik(~jθ)]_n×n and _l= e+ f exp(~jθ) , and hik(~jθ) is the ik-th element of 1 0 ( ) ) ) (( + − − + − d − d A f e A E f

e γ γ , in which γ =exp(~jθ)and θ∈[0 ,2π]. Now, we also compare the proposed sufficient condition (16) with the condition (20) given by Pan and Chen (2002, 2003). Using the definition of H , we can obtain

H j h j h A f e A E f e+ ) − −( + )−d _d)− =[ _ik(~ )]_n×n ≤[sup _ik(~ )]_n×n = (( γ ₀ γ 1 θ θ , (21a)

and we have (Lee and Fang, 1994; Fang, 1997)

J ≤[suph_ik(~jθ)]_n×n =H , (21b) for θ∈[0 ,2π]. Thus, from Eqs. (13) and (21), we can get

∑

= − − + − − + m j j d d j e f E A e f A H r 1 0 1 0 0 (( ) ( ) ) [ ε γ γ (( ) ( ) ) ( ) ] 1 1 0

∑

= − − − _{⋅ +} + − − + + m j dj d d d dj e fγ E A e fγ A e fγ H ε

∑

= − − − _{+ +} + − − + ≤ m j dj dj d j j d d H f e H A f e A E f e r 1 0 0 1 0 ( ) ) ( ( ) )] ) (( [ γ γ ε γ ε ≤r[((e+ fγ)E−A₀ −(e+ fγ)−dA_d)−1(U₀ +(e+ fγ)−dU_d)] ] ) ( ( [H U₀ e f dU_d r + + − ≤ γ )] ( [H U₀ dU_d r + − ≤ _l , (22a) and _⎥ ⎦ ⎤ ⎢ ⎣ ⎡

∑

= ) 1 0 0 m j j j JH r ε _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ≤

∑

= ) 1 0 0 m j j j J H r ε [ ( 0 d)] d U U H r + − ≤ _l , (22b) for γ =exp(~jθ) and θ ∈[0 ,2π] . This also mathematically proves that, for the case of eigenvalue-clustering in a specified circular region D(e, f) , the sufficient condition (16) proposed in this paper is less conservative than the condition (20) proposed by Pan and Chen (2002, 2003).

四、範例說明

Consider a discrete-time closed-loop control model with computational delay for a DC motor in a hydraulic system as following (Shields, 1994; Liu and Xie, 1998):

x1(k+1)=0.4121x1(k)+0.8113x2(k), (23a) 0=−0.345x₁(k)−x₂(k)+0.1x₁(k−1)+0.1x₂(k −1), (23b) where x1(k) is the axis speed and x2(k) is the armature current. In this example, we assume that

the parameter uncertainties of this DC motor arise from the uncertain viscous-friction coefficient

B with 0.1≤ B≤0.4 (N/m/sec), the uncertain torque constant K with t 0.3005≤K_t ≤0.3795

(N-m/Amp), and the uncertain motor back-emf constant K with _b 0.1725≤Kb ≤0.414 (V/rad/sec), due to the fact that many times they are very hard to measure and/or are not always known to a high degree of accuracy (Eletro-Craft Corporation, 1978). Therefore, under this assumption, the model in (23) becomes

) 1 ( ) ( ) ( ) ( ) ( ) 1 (k+ =A₀x k + ₀₁H₀₁x k + ₀₂H₀₂x k + ₀₃H₀₃x k +A₁x k− Ex ε ε ε , (24) with ε₀₁∈

[

−0.17, 0.2894

]

, ε₀₂∈

[

−0.223, 0.3217

]

, ε₀₃∈

[

−0.5, 0.2

]

, and ε₀₁ ,ε₀₂ ,ε₀₃∈R, where ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 0 0 1 E , _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = 1 345 . 0 8113 . 0 4121 . 0 0 A , _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 1 . 0 1 . 0 0 0 1 A ,

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 0 0 1 01 H , _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 0 1 0 02 H , and _⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = 0 345 . 0 0 0 03 H .

All the finite eigenvalues of the nominal singular time-delay system in Eq. (24) are 0, 0.1 and 1322

.

0 _{which are located inside a specified circular region D centered at 0.1}+~_j0 with radius 7

. 0 .

If we apply the sufficient conditions (12) and (20), respectively, proposed by Xu et al. (2002) and Pan and Chen (2002, 2003), we have

1 0394 . 1 ] [f−1TU₀ = </ r r , and r[HU0]=1.0376</1.

Thus, no conclusion can be made. That is, both sufficient conditions of Xu et al. (2002) and Pan and Chen (2002, 2003) can not be applied in this example.

If we use the proposed sufficient condition in Eqs. (9a) and (9b), we can get 1 0 ] [ ₀₁ JH₀₁ + ₀₂ JH₀₂ + ₀₃ JH₀₃ = < r ε ε ε and 1 9316 . 0 ] ) ( ) ( ) ( [ ₀₁ qE−A₀−q−1A₁ −1H₀₁+ ₀₂ qE−A₀−q−1A₁ −1H₀₂ + ₀₃ qE−A₀−q−1A₁ −1H₀₃ ≤ < r ε ε ε .

Next, applying the proposed sufficient condition in Eqs. (11a) and (11b) with the 2-norm-based matrix measure, we have

(i)ε₀₁φ₀₁+ε₀₂φ₀₂+ε₀₃φ₀₃≤0.0345<1,forε₀₁∈

[

−0.17, 0.289

]

, ε₀₂∈

[

−0.223, 0.322

]

and ε₀₃∈

[

0, 0.2

]

; (ii)ε₀₁φ₀₁+ε₀₂φ₀₂+ε₀₃φ₀₃≤0.0863<1, forε₀₁∈

[

−0.17, 0.289

]

, ε~₂∈

[

−0.223, 0.322

]

and ε₀₃∈

[

−0.5, 0

]

; and

(i)ε₀₁θ₀₁+ε₀₂θ₀₂ +ε₀₃θ₀₃ ≤0.9667<1 ,forε01∈

[

0, 0.289

]

, ε02 ∈

[

0, 0.322

]

and ε03∈

[

0, 0.2

]

;

(ii)ε01θ01 +ε02θ02 +ε03θ03 ≤0.7026<1, for ε01∈

[

0, 0.289

]

, ε02∈

[

−0.223, 0

]

and ε03∈

[

0, 0.2

]

;

(iii)ε01θ01+ε02θ02+ε03θ03 ≤0.2105<1, for ε01∈

[

−0.17, 0

]

, ε02∈

[

−0.223, 0

]

and ε03∈

[

0, 0.2

]

;

(iv)ε₀₁θ₀₁ +ε₀₂θ₀₂ +ε₀₃θ₀₃ ≤_0.4746<₁, for ε₀₁∈

[

−_{0.17, 0}

]

, ε₀₂∈

[

0, 0.322

]

and ε03∈

[

0, 0.2

]

; (v)ε₀₁θ₀₁+ε₀₂θ₀₂ +ε₀₃θ₀₃ ≤_0.9264<₁ , for ε₀₁∈

[

_{0, 0.289}

]

, ε₀₂∈

[

0, 0.322

]

and ε03∈

[

−0.5, 0

]

; (vi)ε₀₁θ₀₁+ε₀₂θ₀₂+ε₀₃θ₀₃ ≤0.6623<1, for ε₀₁∈

[

0, 0.289

]

, ε₀₂∈

[

−0.223, 0

]

and ε₀₃∈

[

−0.5, 0

]

; (vii)ε₀₁θ₀₁+ε₀₂θ₀₂+ε₀₃θ₀₃ ≤0.1702<1, for ε₀₁∈

[

−0.17, 0

]

, ε₀₂∈

[

−0.223, 0

]

and ε₀₃∈

[

−0.5, 0

]

; (viii)ε₀₁θ₀₁+ε₀₂θ₀₂+ε₀₃θ₀₃ ≤0.4343<1, for ε₀₁∈

[

−0.17, 0

]

, ε₀₂∈

[

0, 0.322

]

and ε₀₃∈

[

−0.5, 0

]

. Therefore, the proposed sufficient conditions in (8) and (10) are all satisfied. Thus, according to the sufficient conditions (9) and (11), we can conclude that the linear uncertain singular time-delay system (24) is regular and causal, and has all its finite eigenvalues retained inside the same specified circular region D as the nominal singular time-delay system (E,A) does. From this example, we show that our sufficient conditions (9) and (11) are less conservative than those given by Xu et al. (2002) and Pan and Chen (2002, 2003).

五、結論與討論

Under the assumptions that the linear nominal discrete singular time-delay system is regular and causal, and has all its finite eigenvalues lying inside certain specified regions, two new sufficient conditions have been proposed to preserve the assumed properties when the structured parameter uncertainties are added into the linear nominal discrete singular time-delay system. In the presented theorems, all sufficient conditions depend on the boundary of the specified region. This is not restrictive but necessary, because there is no constraint imposed on the shape of the specified

region in our theorems. But, the existing criteria given by Xu et al. (2002) and Pan and Chen (2002, 2003) can only be used to investigate the robustness problem of eigenvalue-clustering within a specified circular region. When all the finite eigenvalues are just required to locate inside the unit circle, the proposed criteria will become the stability robustness criteria. An example has been given to demonstrate the applicability of the proposed sufficient conditions. In Section 3, for the case of eigenvalue-clustering in a specified circular region and only considering structured parameter uncertainties, by mathematical analysis, the proposed sufficient condition in (9) has been proved to be less conservative than those proposed by Xu et al. (2002) and Pan and Chen (2002, 2003). Besides, Example 1 has also shown that the proposed sufficient condition in (11) could get less conservative results than those sufficient conditions given by Xu et al. (2002) and Pan and Chen (2002, 2003).

六、研究成果自評

本成果報告已達成申請計畫書中預期完成的成果目標。本研究計畫案之部份成果的發表 情況如下所列：

1. Chen, S. H., “Robust D-Stability Analysis for Linear Discrete-Time Singular Systems with Structured Parameter Uncertainties and Delayed Perturbations”, Proceedings of the Institution of Mechanical Engineers, Part I, J. of Systems and Control Engineering, Vol.217, No.I1, pp.1-5, 2003.

2. Chen, S. H. and J. H. Chou, “Stability Robustness of Linear Discrete Singular Time-Delay Systems With Structured Parameter Uncertainties”, IEE Proceedings-Control Theory and Applications, Vol.150, No.3, pp.295-302, 2003.

3. Chen, S. H. and J. H. Chou, “Regional Eigenvalue-Clustering Robustness Analysis for Linear Discrete Singular Time-Delay Systems With Structured Parameter Uncertainties”, Proc. of the 2003 Automatic Control Conference, Taiwan, R.O.C., pp.1295-1300, 2003.

4. Chen, S. H. and J. H. Chou, “D-Stability Robustness for Linear Discrete Uncertain Singular Systems with Delayed Perturbations”, International Journal of Control, Vol.77, No.7, pp.685-692, 2004.

5. Chen, S. H. and J. H. Chou, “Robust Eigenvalue-Clustering in a Specified Circular Region for Linear Uncertain Discrete Singular Systems with State Delay”, Proceedings of the Institution of Mechanical Engineers, Part I, Journal of Systems and Control Engineering, 2004/04 (submitted).

6. Chen, S. H. and J. H. Chou, “Robust D-Stability Analysis for Uncertain Discrete Singular Systems with State Delay”, Applied Mathematics Letters, 2004/05 (submitted).

七、參考文獻

1. Chou, J. H., 1991, “Pole-Assignment Robustness in a Specified Disk”, Systems and Control Letters, Vol.16, pp.41-44.

2. Chou, J. H., S. H. Chen and C. H. Hsieh, 2003, “Regional Eigenvalue-Clustering Robustness Analysis for Singular Systems with Both Structured and Unstructured Perturbations”, Int. J. of Control, Vol.76, pp.18-23.

3. Chou, J. H and W. H. Liao, 1998, “Regional Eigenvalue-Clustering Robustness Analysis for Singular Systems with Structured Parameter Perturbations”, Proceedings of the Institution of Mechanical Engineers, Part I, J. of Systems and Control Engineering, Vol.212, pp.467-471. 4. Chu, J., 1995, “A Time-Delay Control Algorithm for Discrete Systems and Its Applications to

5. Churchill, R. V. and J. W. Brown, 1990, Complex Variables and Applications, McGraw-Hill, New York.

6. Dai, L., 1989, Singular Control Systems, Springer-Verlag, Berlin.

7. Desoer, C. A. and M. Vidyasagar, 1975, Feedback Systems: Input-Output Properties, Academic Press, New York.

8. Eletro-Craft Corporation, 1978, DC Motors, Speed Controls, Servo Systems: an Engineering Handbook, Hopkins, Minnesota.

9. Fang, C. H., 1997, “Robust Stability of Generalized State-Space Systems”, Ph.D. Dissertation, Department of Electrical Engineering, National Sun Yat-Sen University, Taiwan.

10. Fang, C. H., L. Lee and F. R. Chang, 1994, “Robust Control Analysis and Design for Discrete-Time Singular Systems”, Automatica, Vol.30, pp.1741-1750.

11. Lee, L. and C. H. Fang, 1994, “Regional Pole-Clustering Robustness for Uncertain Generalized State-Space Systems”, Proc. of the 33rd IEEE Conference on Decision and Control, Florida, pp.587-588.

12. Lewis, F. L., 1986, “A Survey of Linear Singular Systems”, Circuit, Systems and Signal Processing, Vol.5, pp.3-36.

13. Liu, Y. Q. and X. S. Xie, 1998, Stability and Stabilization of Linear Singular Systems with Time Delay, South China University of Technology Press, Guangzhou.

14. Pan, S. T. and C. F. Chen, 2002, “Robust D-Stability for Discrete-delay Singular Systems”, Proc. of the 2002 Conference on Industrial Automatic Control & Power Application, Taiwan, pp.E3:39-43.

15. Pan, S. T. and C. F. Chen, 2003, “Study on the Robust D-Stability Problem of Discrete Singular Time-Delay Systems”, Presented at the Conference on the Project Reports of Control Division, National Science Council, Taiwan, Poster Paper Number: E109.

16. Shields, D. N., 1994, “Observers for Singular Discrete-Time Descriptor Systems”, Control and Computers, Vol.22, pp.58-64.

17. Weinmann, A., 1991, Uncertain Models and Robust Control, Springer-Verlag, Hong Kong. 18. Xu, S., J. Lam and L. Zheng, 2002, “Robust D-Stability Analysis for Uncertain Discrete

Singular Systems with State Delay”, IEEE Trans. on Circuit and Systems I: Fundamental Theory and Applications, Vol.49, pp.551-555.