Robust stability of singularly perturbed discrete-delay ... - IEEE Xplore

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 9, SEPTEMBER 1995

1620

Lemma I 121: For any two matrices X, Y E R n x n

At(-YY)= A,(YX). Lemma 2 [3/: For any two positive definite matrices X, Y E

R" '' Amin(-rY) L

Amin(x)

+

pl(Pate1)

pc(Chen)

0.3820

0.4495

0.4827

0.4841

0.535

Q

Q

eq46)

Q

Q

= €0 = 0.202

F

Amin(lr).

+

and the trivial one is

E

=

60

2 /i' > 0.

= l/&,(Q-'/'P)

(CO

-

eq.(6)

= 0.220

= l/u~ax(Q-l/zP))(Iteration)

F

+

eq.(7)

= 0.075

(Iteration)

then, using = 0.075, we obtained the robust stability bound pk = 0.535 which is improves about 19% the result of Chen and Han. The above results are summarized in Table I.

= X,'!,,K(~FQ - E'PP) =

+-

(5)

Pro08 //A

eq.(6)

F

Proposition 1: Let us consider / i c and pk defined in (2) and (4), respectively. Then one can always find a E > 0 such that ilk

pk(Kim)

~f!,:~ ( ~ l l ~ ( 2 . 1 ~

REFERENCES

-f r ~ - 1 / a ~ ~ ~ - 1 / 2 ) ~ 1 / r )

[I] R. V. Patel and M. Toda, "Quantitative measures of robustness for multivariable systems," in Proc. Joint Automat. Conrr. Cons, San Francisco, CA, 1980, p. TP8-A. [2] T. Kailath,Linear Sysfems. Englewood Cliffs, NJ: Rentice-Hall, 1980, pp. 662. [3] A. R. Amir-Moez, "Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations," Duke Math. J., vol. 23, pp. 463-467, 1956.

= AYl?, (Q(2d,, - F2Q-'/lP"-'"))

2 X~!,~(Q)A,~!~,(2tI,, - C ~ Q - ' / ' P P Q - ' / ~ ) = A:!,t,(Q)(2t - E ~ X , , , , ~ ( Q - ~ ' ~ P ~ Q'/L- ' / ~ ) )

Robust Stability of Singularly Perturbed Discrete-Delay Systems - /lc.

H. Trinh and M. Aldeen

000

2

= CO = € 0 is not the best one for maximizing the pk in (4). The maximized value of pk should be found iteratively with the given &. Remark 2: It also should be remarked that the obtained Q by the result' is not best choice to maximize the /& in (4), and the other matrix Q can maximize it (see the following example). To compare the results, let us consider the following example.' Example:' Consider system (1) with the nominal system Remark 1: In the above, we showed that

pk

1 ~ with ~ .

E

~ / C T ~ ~ , ( Q - ~. 'It' Pshould ) be remarked, however, that this

-3 A = [ 1

-2 01.

Using the matrix (which is obtained iteratively by Chen and Han) 5.2361 2.6180 = [2.6180 2.G180].

(6)

Chen and Han obtained the robust bound p c = 0.4495 by (2) which improved about 17.7% more than the result of Patel and Toda [l]. By adopting r = EO = ~ / c T ~ ~ , ( Q=- 0.202, ~ P ) however, we obtained pk = 0.4827 which shows the improvement about 26.4% more than the result of Patel [l], and 7.4% more than the result of Chen and Han. Now, let us choose E iteratively that maximized the pk, then we obtain E = 0.220 and the robust stability bound /& = 0.4841. To show that the matrix Q in (6) is not best choice that maximize the p k , let us choose (7)

Abstract-This note presents new suflicient conditions for the robust stability of singularly perturbed discrete systems subjected to multiple delayed perturbations. For a given E E [O,EO], where E O < E * , a new criterion for finding the range of allowable bounds on the delayed perturbations is provided. A numerical example is given to illustrate the results.

NOTATION

R" In

Real vector space of dimension n. Identity matrix of order n. det(A) Determinant of the matrix A. Spectral radius of matrix A . p(A) IAl Modulus matrix of A. IlAll Matrix norm of A llAll = g m a x ( ; l ) A [Amax(ATA)]1'2-

I. INTRODUCTION The E-bound stability problem in singularly perturbed systems has attracted the attention of many researchers for the past few years. As a result, explicit upper bounds, E*, for both continuous and discretetime singularly perturbed systems have been derived via various Manuscript received May 31, 1994; revised November 4, 1994 and March 20, 1995. This work was supported by the Australian Research Council Large Grant A49531392. The authors are with the Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, 3052, Australia. IEEE Log Number 9412923.

0018-9286/95$04.00 0 1995 IEEE

1621

TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 9, SEITEMBER 1995

approaches (see, for example, [1]-[6]). As a result, necessary and sufficient conditions for computing E* have been derived [ 11-[3]. More recently, Shao and Sawan [7] studied the robust stability of continuous singularly perturbed systems subjected to parametric perturbations bounded by the H , -norm. Some mechanical and chemical processes such as heat exchangers, flow and pressure control, and econometric systems may be classified as discrete-delay singularly perturbed systems. A literature search indicates that the robust stability problem of such systems has not yet been resolved. In this note, we consider the robust stability of such systems when they are subjected to multiple delayed perturbations. We provide a criterion for finding the range of allowable bounds on the delayed perturbations, for a given E E [0, E ~ ] where , EO < E*. These bounds may then be used in system analysis and controller design.

The approach taken in this note consists of two stages. The first stage involves the derivation of the exact upper bound, E * , of the nominal system. In this regard, the approach reported by Li and Li [2] for discrete singularly perturbed systems will be used. The second stage involves the determination of allowable bounds on the delayed perturbations for EO < E * . In the ensuing analysis, the perturbation matrices A A l g are assumed to be of the following two types: i) Independently structured with JAAfgl 5 U f g , V i = 1, 2 , . . . , m and f.g = 1,2, where U f g are constant matrices whose elements are all positive and ii) Unstructured with llAAfgll 5 Vi = 1.2:..,m and f . g = 1,2, where a{' are positive constant numbers. To proceed with the analysis we first introduce some preliminary results that will be needed in the proofs of the main results of this note.

11. SYSTEMS DESCRIPTION

Let us consider the stability of a perturbed discrete-delay singularly perturbed system described by

A. Preliminary We first introduce Lemma 1, the proof of which can be found in [131. Lemma I : Let A

G ( z )= ( z I n where

o)-'

(34

6 is obtained from 0 in (2) by replacing E

with

EO.

Then

W

IG(Z)I

I G ( ~A ) IH .

I

for

121

2I

(3b)

k=O

(1b) where0 5 8 5 l . x ~ ( k + l )E R n 1 , x 1 ( k )E R " l , s l ( k - i ) E R " ' , z z ( k + l ) E R n 2 , s 2 ( k )E R " 2 , a n d x z ( k - i ) E R"*(nl+nz = n ) are state vectors. Matrices AJg (f,g = 1 , 2 ) are constant with appropriate dimensions, c > 0 is the singular perturbation parameter, and m 2 0 is an integer denoting the number of time delays. Matrices AA:" (f,g = 1,2: / = 1.. . . , In ) are perturbations in the delayed states sl(k - 1 ) and .rL(k - 1 ) . Note that system (1) extends the classes of discrete singularly perturbed systems reported in [2] and [8]-[lo] to include multiple delayed perturbations. Equations (1) can be expressed by the following compact form 7,"

.r(k

+ 1) = Q . r ( k )+

AO,T(I;

- z)

(2)

where G ( k ) is the pulse-response sequence matrix of G ( z )and is obtained from

G ( k )= @-'for k = 1 . 2 , 3 , . '. , 00, where G(0) = 0.

(3c)

The matrix H is obtained from (3b) and (3c) as

Note that matrix 0 is asymptotically stable for E E [ O , E * ) [2]. It follows that for 0 5 50 < E * , matrix 6 is asymptotically stable too. As a result, the convergence of the series is ensured, and H can be directly computed. Lemma 2 /14]: For any (72 x n ) matrix UT,the following statement is true

t=O

{p(W)

where

111. MAIN RESULTS

In this section, we derive new sufficient stability conditions for system (1). Bounds on the perturbations, AAfg (f,g = 1.2;i = 1, . . . ,m),that guarantee the system stability are derived. The derived stability conditions involve computation of matrices of dimension TL only, regardless of the number of the delays in the system. This significantly reduces the amount of computation required as compared to the methods proposed in [ l l ] and [12]. This is because in [ I l l and [12], unlike the approach of this note, the number of eigenvalues is proportional to the number of delays. This number therefore increases as the number of delays increases.

< 1)

implies

{det(In - W)

# O}.

(4)

We now introduce the main results of this note in terms of a theorem. Theorem 1: Let the nominal part of system (2) be asymptotically stable for E E IO,&*) [2], then for a known singular perturbation parameter E O ( O 5 EO < E * ) , system (2) is asymptotically stable for the two types of delayed perturbations, if the following two conditions are satisfied: i) Structured Perturbations r

? ,,

i

where matrix Ut ( i = 0 , 1 , 2 . . . . ,m ) is obtained from matrix JA6,I with its submatrices satisfying the conditions JAAfgl 5 Uf".Vi and f . g = 1.2. i.e.,

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40,NO. 9, SEPTEMBER 1995

1622

n

ii) Unstructured Perturbations

5 Omax ( H )

omax

I)

(

z=O

m

5 gmax(H)

(13)

~z

*=O

where

where 7 %2 O ( i = ( ) , I , . . . ,m ) are as defined in (6). Accordingly, by the above fact, the inequality 7%< implies condition 0 (9). Thus condition ii) of Theorem 1 is proven. Remark I: The results presented in Theorem 1 provide sufficient conditions for computing the bounds on the delayed perturbations when the stability of system (2) is required for a given E = EO. In this regard, the results obtained in this note may be considered as extensions of the results of Chou [ 131 to multiple delayed perturbations case. In addition, this note considers the unstructured delayed perturbations case, whereas in [13] only structured perturbations were considered. Remark 2: Consider the stability of a discrete system with multiple delays

&

and af" ( f , y = 1,2) are positive numbers, defined as

llAAf"1 5 a:". Proof ofTheorem I: The stability of system (2) with E replaced

by

EO

can be expressed as

where matrices 0 and AG, are obtained from matrices 0 and A@, in (2) by replacing c with E O . Using the property of determinant, (7) can be expressed as 7,L

[

tlet I,,

-

(ZIT,

o)-'p

3

,=O

,

z

-1

[

n

( 2 1 , ~-

0)# 0,

det(21,

-

for IzI

2 1.

-1

A G , ~ # 0,

GI-' 1

=O

(8)

for

121

# 0 for

2 1. (9)

Using Lemma 2, condition (9) is satisfied if the following condition is satisfied

Using the properties of the spectral radius, the term on the left-hand side of (10) can be expressed as

7"

+ 1) = C A , z ( k - i) I

As matrix 6 is asymptotically stable, i.e., det(zI,, - (3) I:! 2 1, to prove Theorem 1 it remains to show that det I?&-

rrt

s(k

(14)

=O

w h e r e s ( k + l ) E R " , r ( k - i )ER" arestatevectors,A, ER"'" ( i = 0, 1,2, . . . ,ni ) are constant matrices, and -40 is assumed to be asymptotically stable. It can be seen that system (14) is a special case of system (2) where the delayed perturbations are assumed to be time invariant. Based on the result of Theorem 1, the following result is obvious: The discretedelay system (14) is asymptotically stable if p [ H ~~~1 IA, I] < 1, where H = IAkl. To illustrate the results in this note, let us consider the following numerical example. Example I: Consider a discrete singularly perturbed system reported by Li and Li [2] with added delayed perturbations as described below

ET=,

-c,

(IC

+ 1) = A";cl(k-) + &A'2~2(IC)

+ AA:'-c:1(k - 1)+ &AA:Zsz(k

- 1)

(15a)

m

where B = 0.5. The nominal matrices A f g ( f .g = 1 . 2 ) are given in [2] as (11) Since lA~tllz-'5 l ElfolAo,l for 121 2 1, using the result of Lemma 1 and combining (10) and (11) gives

r

o 0.06 -0.lfi

1 0 -0.47

fi

0 1.2 -O.lJ&

A1 0.2E

By Lemma 2, condition (12) implies condition (9). Thus condition i) For this nominal system, exact upper bound for E , i.e., E*, has been of Theorem 1 is proven immediately. derived in [2] as E* = 0.244. To prove condition ii) of Theorem 1 for the unstructured perIn the following, we compute the bounds of the delayed perturturbations, let us use the following fact: For any ( n x n ) matrix bations, AA{g, when the stability of system (15) is required to be W, if umaX(1i7)< 1, then p ( W * ) < 1 is satisfied automatically. E = E o < E * . Accordingly, the following statement is true: ( ~ ~ ~ ~ ( 1 % '