ThB18.3
2005 American Control Conference June 8-10, 2005. Portland, OR, USA
Stability of linear neutral systems with linear fractional norm-bounded uncertainty Qing-Long Han, Member, IEEE
Abstract — This paper is concerned with the stability problem of uncertain linear neutral systems using a discretized Lyapunov functional approach. The uncertainty under consideration is linear fractional norm-bounded uncertainty which includes the routine norm-bounded uncertainty as a special case. A delay-dependent stability criterion is derived and is formulated in the form of linear matrix inequalities (LMIs). The criterion can be used to check the stability of linear neutral systems with both small and non-small delays. For nominal systems, the analytical results can be approached with fine discretization. For uncertainty systems with small delay, numerical examples show significant improvement over approaches in the literature. For uncertainty systems with non-small delay, the effect of the uncertainty on the maximum time-delay interval for asymptotic stability is also studied.
I. INTRODUCTION the past few years, delay-dependent stability of linear neutral systems has attracted considerable attention, see for example, [1, 4-7, 9-10] and references therein. The goal is to obtain the maximum allowed upper bound on the delay that guarantees the stability of a linear neutral system. Therefore, the admissible allowed upper bound on the delay is the main “performance index” for measuring the conservatism of the conditions obtained. There are two kinds of linear neutral systems. One kind of system is that the discrete delay lies in a given interval [0, rmax ] , where the delay is called a small delay. The other kind of system is one which is stable with some nonzero discrete delay, but is unstable without the delay, see Example 2 in this paper; in this case the discrete delay lies in an interval [rmin , rmax ] , where rmin ! 0 , and it is called a non-small delay. In the time-domain, the direct Lyapunov method is a powerful tool for studying the stability of linear neutral URING
D
This work was supported in part by the Central Queensland University for the 2004 Research Advancement Awards Scheme Project “Analysis and Synthesis of Networked Control Systems” and the strategic research project “Delay Effects: Analysis, Synthesis and Applications” (20032006). Qing-Long Han is with the Faculty of Informatics and Communication, Central Queensland University, Rockhampton, Qld 4702, Australia (phone: +61749309729; fax: +61749309729; e-mail:
[email protected]).
0-7803-9098-9/05/$25.00 ©2005 AACC
systems. In the recent years, Lyapunov-Krasovskii approach is widely employed to consider the delaydependent stability problem for the systems. In the following some more recent results in the time-domain using Lyapunov-Krasovskii approach are mentioned. For linear neutral systems with small discrete delay, there exist many stability results in the literature. For example, based on a neutral model transformation, some sufficient conditions were obtained in [4, 9]. In [5], using the neutral model transformation and the decomposition technique for the discrete delay term, a delay-dependent stability criterion was proposed. Employing a descriptor model transformation and the bounding technique for cross product terms, some sufficient criteria were given in [1]. Recently, He et al. [7] proposed a new method that employed some free weighting matrices (relaxation matrices) to express the relationship between the terms in the Leibniz-Newton formula which was used to derive a delay-dependent stability criterion. It was shown through some examples that the new method improved the results in [1, 5]. However, the results mentioned in this paragraph can only handle the small discrete delay case, they fail to make any conclusion for the non-small discrete delay case. Moreover, the results are more conservative, which can be seen by applying these results to a constant time-delay system without uncertainty and compare with analytical results. For linear neutral systems with non-small discrete delay, simple Lyapunov-Krasovskii functionals are no longer to be valid to judge the stability problem since they require that the corresponding system without delay needs to be stable. Therefore, one has to find some more general Lyapunov-Krasovskii functionals to handle the problem. For a linear system of retarded type with a constant delay, it has been proven that the existence of a more general quadratic form Lyapunov-Krasovskii functional is necessary and sufficient for the stability of an uncertaintyfree time-delay system [8]. A discretized Lyapunov functional approach has been proposed to enable one to write the stability criterion in an LMI form [2]. The criteria have shown significant improvements over the existing results even under very coarse discretization [2, 6]. In this paper, a more general quadratic form LyapunovKrasovskii functional will be introduced and the discretized Lyapunov functional approach will be employed to derive a
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delay-dependent stability criterion for linear neutral systems. The criterion is effective for linear neutral systems with both small and non-small delays. Two numerical examples will be given to illustrate the effectiveness of the criterion. Notation. For a symmetric matrix W, “ W ! 0 ” denotes that W is a positive definite matrix. Similarly, “ t ”, “ ” and “ d ” denote positive semi-definiteness, negative definiteness and negative semi-definiteness. Use W to denote the derivative of W with respect to time t while W (D ) denotes the derivative of W with respect to the argument and evaluated at D . Oi (C ) is the ith eigenvalue of matrix C. Use I to stand for the identity matrix of appropriate dimensions.
unit circle, i.e. Oi (C ) 1 ( i 1, 2," , n ). Choose a Lyapunov-Krasovskii functional V (t , I ) of a quadratic form V (t , I ) : \ u 6 \ V (t , I )
0 1 ( I )T P( I ) ( I )T ³ Q([ )I ([ )d [ r 2 1 0 0 ³ ª ³ I T ([ ) R ([ ,K )I (K )dK º d [ ¼» 2 r ¬« r
1 0 T I ([ ) S ([ )I ([ )d [ 2 ³ r
where P \ nu n , P
(6)
PT ; Q : [r , 0] o \ nun
S : [r , 0] o \ nun , S T ([ )
S ([ )
nu n
II. PROBLEM STATEMENT Consider an uncertain linear neutral system x (t ) Cx (t r ) [ A 'A(t )]x(t ) [ B 'B(t )]x(t r ) (1) with initial condition x(t ) M (t ) , x (t ) M (t ) , t [r , 0] (2) where x(t ) \ n is the state, r ! 0 is a constant time delay,
M (t ) is the initial condition, C \ nun is a constant matrix, A \ nu n and B \ nu n are constant matrices, 'A(t ) and 'B (t ) are uncertain matrices representing the system’s time-varying parameter uncertainties and satisfy 'A(t ) 'B(t ) L( I F (t ) D)1 F (t ) Ea Eb (3)
where L, D, Ea , and Eb are known constant matrices, F (t ) is an unknown matrix satisfying F T (t ) F (t ) d I (4) In order to guarantee the uncertainty in (3) to be well defined, we assume that I DT D ! 0 (5) Remark 1: It should be noted that when D { 0 , the linear fractional norm-bounded uncertainty (3) reduces to the routine norm-bounded uncertainty. Therefore, the linear fractional norm-bounded uncertainty is more general than the routine norm-bounded uncertainty. Define as the set of continuous \ n valued function on
the interval [r , 0] , and let xt be a segment of system trajectory defined as xt (T ) x(t T ) , r d T d 0 . (6) In this paper, we will attempt to formulate a practically computable criterion for robust stability of uncertain system described by (1) to (5). Define the operator I I (0) CI (r ) . (7) Throughout this paper, we assume that A1: All the eigenvalues of matrix C are inside the open
R : [ r , 0] u [r , 0] o \ , R (K , [ ) RT ([ ,K ) and Q, R and S are Lipschitz matrix functions with piecewise continuous derivatives. By Theorem 8.1 in Chapter 9 in [3], for asymptotic stability of system (1)-(5), we have the following result. Theroem 1: Under A1, the system (1)-(5) is asymptotically stable if there exist H1 ! 0 , H 2 ! 0 and a quadratic Lyapunov-Krasovskii functional V of the form (6) which staifies H1 ( I )T ( I ) d V (t , I ) (9) and its derivative along the solution of (1) satisfies V (I ) d H 2I T (0)I (0) (10) d V (t , xt ) xt I . dt Choose Q, R and S to be continuous piecewise linear [2], i.e.,
for any I , where V (t , I )
'
Qi (D ) Q(G i 1 D h) '
S i (D ) S (G i 1 D h) R (G i 1 D h, G j 1 K h)
'
(1 D )Qi 1 D Qi (1 D ) Si 1 D Si
(11)
ij
R (D ,K )
°(1 D ) Ri 1, j 1 K Rij (D K ) Ri , j 1 , D t K ® °¯(1 K ) Ri 1, j 1 D Rij (K D ) Ri 1, j , D K for 0 d D d 1 , 0 d K d 1 , where G i rm ih , i 0, 1, 2, " , N , h r / N . i.e., N is the number of divisions of the interval [r , 0] , and h is the length of each division. It is convenient to write I i (D ) I (G i 1 D h) .
III. MAIN RESULTS With the choice of piecewise linear functions, the Lyapunov-Krasovskii functional condition (9) can be written in the form of a linear matrix inequality. Using similar argument to the proof of Proposition 3 in [2] yields
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the following result. Proposition 1: For piecewise linear Q, S and R as described by (11), there exists an H1 ! 0 such that the Lyapunov-Krasovskii functional satisfies (9) if S ! 0 (12) and ªP Q º »!0 « (13) «Q T 1 Sˆ R » h ¬« ¼» where S diag ( S0 S1 " S N 1 S N ) R
Q
ª R00 «R « 10 « # « «¬ RN 0
R01 " R0 N º R11 " R1N »» # % # » » RN 1 " RNN »¼ Q0 , Q1 , " , QN .
System described by (1), (3)-(5) can be rewritten as x (t ) Cx (t r ) Ax(t ) Bx(t r ) Lu (t ) y (t ) Ea x(t ) Eb x(t r ) Du (t ) subject to uncertain feedback u (t ) F (t ) y (t ) or equivalently, in view of (4) and (15), 1 T 1 T u (t )u (t ) d Ea x(t ) Eb x(t r ) Du (t ) 2 2 u Ea x(t ) Eb x(t r ) Du (t )
G01
G02
H 0s
G11
G12
H1s
T G12
G22
H 2s
H1sT
H 2sT
1 Sd Rd h
H1aT
H 2aT
0
G02
LT PC DT Eb
G11
PA AT P S N QN QNT EaT Ea
G12
PB AT PC QNT C Q0 EaT Eb
G22
C T PB BT PC C T Q0 Q0T C S0 EbT Eb
1 ( Si Si 1 ) h
Rd
ª Rd 11 «R « d 21 « # « ¬« RdN 1
(16)
Rd 22 #
RdN 2
" Rd 1N º " Rd 2 N »» % # » » " RdNN ¼»
1 ( Rij Ri 1, j 1 ) ; i, j 1, 2, " , N h
H sj
[ H sj1 H sj 2 " H sjN ] , j
H aj H 0ai H1ai
(17)
Rd 12
Rdij
H 2si
(14) (15)
H 2ai
0,1, 2;
1 LT (Qi Qi 1 ) 2 1 T 1 A (Qi Qi 1 ) (Qi Qi 1 ) 2 h 1 ( RiT, N RiT1, N ) , i 1, 2, " , N 2 1 1 BT (Qi Qi 1 ) C T (Qi Qi 1 ) 2 h 1 ( RiT,0 RiT1,0 ) , i 1, 2, " , N 2 [ H aj1 H aj 2 " H ajN ] , j 0,1, 2; 1 T L (Qi Qi 1 ) 2 1 T 1 A (Qi Qi 1 ) ( RiT, N RiT1, N ) , i 1, 2, " , N 2 2 1 T 1 B (Qi Qi 1 ) ( RiT,0 RiT1,0 ) , i 1, 2, " , N . 2 2
Proof. Using (1), one obtains V (t , I ) ( I )T P[ AI (0) BI ( r ) Lu (t )] 0
[ AI (0) BI (r ) Lu (t )]T ³ Q([ )I ([ )d [ r
0
( I )T ³ Q([ )I([ )d [ r
0
0
r
r
³ d [ ³ I T ([ ) R ([ ,K )I(K )dK 0
³ I T ([ ) S ([ )I([ )d [ r
(19)
Through integration by parts, we have V (t , I ) uT (t ) LT PI (0) uT (t ) LT PCI ( r )
where G00 I DT D LT P DT Ea
Sdi
H1si
H 0a · ¸ H1a ¸ ¸ H 2a ¸ ¸ ! 0 (18) 0 ¸ ¸ 3 ¸ Sd ¸ h ¹
G01
diag ( Sd 1 Sd 2 " SdN )
H 0si
In the following we employ (14) and (17) to formulate the Lyapunov-Krasovskii derivative condition in the form of a linear matrix inequality. We have the following proposition. Proposition 2: For piecewise linear Q, S and R as described by (11), equation (10) is satisfied for some H 2 ! 0 , and arbitrary I if § G00 ¨ T ¨ G01 ¨ T ¨ G02 ¨ sT ¨ H0 ¨ ¨ aT ¨ H0 ©
Sd
1 I T (0)[ PA AT P Q(0) QT (0) S (0)]I (0) 2 I T (0)[ PB AT PC QT (0)C Q( r )]I ( r ) 1 I T (r )[C T PB BT PC S ( r ) 2 C T Q( r ) QT (r )C ]I (r ) 0
uT (t ) ³ LT Q([ )I ([ )d [ r
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0
I T (0) ³ [ AT Q([ ) Q ([ ) RT ([ , 0)]I ([ )d [ r
0 I T (r ) ³ [ BT Q([ ) C T Q ([ ) RT ([ , r )]I ([ )d [ r
0 § wR ([ ,K ) wR ([ ,K ) · 1 0 d [ I T ([ ) ¨ ¸ I (K )dK 2 ³ r ³ r wK ¹ © w[
1 0 T I ([ ) S ([ )I ([ )d [ . (20) 2 ³ r With the piecewise linear Q, S, and R chosen, (20) can be written as V (t , I ) uT (t ) LT PI (0) uT (t ) LT PCI ( r )
1 I T (0)G110 I (0) I T (0)G120 I (r ) 2 1 T 0 I (r )G22 I ( r ) 2 h N 1 ¦ ³ I iT (D )SdiI i (D )dD 2i1 0
h2 2
N
N
T
¦¦ ³0 I i (D )dD 1
Rdij
i 1 j 1 N
j
0
1
huT (t )¦ ³ [ H 0si (1 2D ) H 0ai ]I i (D )dD N
T
1
[ H1si 0
hI (0)¦ ³ i 1
N
(1 2D
) H1ai ] i (
I D )dD
1
hI T ( r )¦ ³ [ H 2si (1 2D ) H 2ai ]I i (D )dD . i 1
0
(21)
where 0 G110 G11 EaT Ea , G120 G12 EaT Eb , G22 G22 EbT Eb Rewrite (21) as 1 1 V (t , I ) ³ uT (t ) I T (0) I T ( r ) hIT (D ) 2 0 § 0 LT P LT PC H 0s (1 2D ) H 0a · ¨ ¸ § u (t ) · 0 G120 H1s (1 2D ) H1a ¸ ¨ ¨ * G11 I (0) ¸¸ s a ¸¨ 0 u¨ * G22 H 2 (1 2D ) H 2 ¸ ¨ I ( r ) ¸ dD * ¨ ¸¸ ¨ ¸ ¨¨ 1 Sd * * ¨* ¸ © hI (D ) ¹ h © ¹ 1 1 1 h 2 ³ IT (D )dD Rd ³ I (D )dD . (22) 0 0 2 where * stands for symmetric part of the matrix and I (D ) [I1T (D ) I 2T (D ) " I NT (D )]T . Noting that (17), we have 1 1 V (t , I ) d ³ uT (t ) I T (0) I T ( r ) hIT (D ) 2 0
§ H 0s · ¨ s¸ a ¨ H1 ¸ , H ¨¨ s ¸¸ © H2 ¹
§ H 0a · ¨ a¸ ¨ H1 ¸ . ¨¨ a ¸¸ © H2 ¹
(23) can be written as 1 1 V (t , I ) d IuT0 r GI0 r hIuT0 r ³ [ H s (1 2D ) H a ]I (D )dD 0 2 1 1 1 h 2 ³ IT (D ) Sd I (D )dD 0 2 h 1 1 1 h 2 ³ IT (D )dD Rd ³ I (D )dD . (24) 0 0 2 Rewrite (24) as 1 1 V (t , I ) d ³ IuT0 r [ H s (1 2D ) H a ] hIT (D ) 2 0 I · §W s a T ¨ ¸ § [ H (1 2D ) H ] Iu 0 r · dD u ¨ ¸ 1 ¨¨ I Sd ¸¸ © hI (D ) ¹ h © ¹ 1 1 IuT0 r [ H sWH sT H aWH aT ]Iu 0 r 2 3 1 IuT0 r GIu 0 r 2 1 1 1 h 2 ³ IT (D )dD Rd ³ I (D )dD (25) 0 0 2 for arbitrary W. If W is chosen to satisfy I º ªW « »!0 (26) « I 1 Sd » «¬ » h ¼ then, use Lemma 1 in [2] in the first term on the right hand in (25) to obtain 1 1 1 V (t , I ) d ³ IuT0 r [ H s (1 2D ) H a ]dD ³ hIT (D )dD 0 0 2 I · § 1[ H s (1 2D ) H a ]T I dD · §W u 0r ¸ ¸ ¨ ³0 u¨ ¸ 1 ¨¨ I 1 Sd ¸¸ ¨ ¸ h ¹ ¨© ³0 hI(D )dD © ¹ 1 T 1 1 Iu 0 r [ H sWH sT H aWH aT ]Iu 0 r IuT0 r GIu 0 r 2 3 2
0
i 1
Hs
³ I (D )dD 1
§ G00 G01 G02 H 0s (1 2D ) H 0a · ¨ ¸ § u (t ) · G11 G12 H1s (1 2D ) H1a ¸ ¨ ¨ * I (0) ¸¸ s a ¸¨ u¨ * G22 H 2 (1 2D ) H 2 ¸ ¨ I (r ) ¸ dD * ¨ ¸¸ ¨ ¸ ¨¨ 1 Sd * * ¨ * ¸ © hI (D ) ¹ h © ¹ 1 1 1 h 2 ³ IT (D )dD Rd ³ I (D )dD (23) 0 0 2 Introducing the following notation § G00 G01 G02 · § u (t ) · ¨ T ¸ G ¨ G01 G11 G12 ¸ , Iu 0 r ¨¨ I (0) ¸¸ , ¨ I (r ) ¸ ¨ GT GT G ¸ © ¹ 12 22 ¹ © 02
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1 h2 2
³ I (D )dD R ³ I(D )dD § H WH H ·§ I · (D )dD ¨ h I ¨¨ H 1 S ¸¸¸ ¨¨© ³ hI(D )dD ¹¸¸ ³ 1 T
1
d
0
s
1 IuT0r 2
1
L
0
sT
s
u 0r
T
sT
0
d
Eb
§1 0· ¨ ¸, ©0 1¹
§d 0· ¨ ¸, 0 d d 1 ©0 d¹ For D 0 , the maximum time-delay for stability as judged by the criteria in [1, 5, 10], and the discretized Lyapunov functional approach, are estimated in Table I, analytical along with the analytical limit rmax . It is to show that the stability limit obtained by the discretized Lyapunov functional approach is much less conservative than the results in [1, 5, 10] and it converges to analytical solution as N increases. D
1
h ¹ 0 © 1 · § 1 G H sWH sT H aWH aT 0 ¸ 1 T T ¨ Iu 0r ³ hI (D )dD 3 0 ¸ ¨¨ 2 0 ¸¹ 0 © § Iu 0r · ¸ u¨ 1 ¨ ³ hI(D )dD ¸ © 0 ¹ · 1 § 0 0 · § Iu 0r 1 ¨ 1 ¸ IuT0r ³ hIT (D )dD ¨ ¸ 0 2 © 0 Rd ¹ ¨© ³0 hI(D )dD ¸¹ 1 § · G H aWH aT Hs ¨ ¸ 1 1 T 3 T ¸ Iu 0 r h ³ I (D )dD ¨ 0 1 2 ¨ sT H Sd Rd ¸¸ ¨ h ¹ © Iu 0 r § · ¸. (27) u¨ 1 ¨ h ³ I (D )dD ¸ 0 © ¹ Therefore, to satisfy (10), it is sufficient for (26) and 1 a § · aT Hs ¨ G 3 H WH ¸ ¨ ¸!0 (28) 1 ¨ ¸ sT H S R ¨ d d ¸ h © ¹ to be satisfied. Use Proposition 2 in [2] to eliminate matrix W in (26) and (28) to obtain (18). Q.E.D. From the above discussion, we now state and establish the following stability criterion. Theorem 2: Under A1, the system (1)-(5) is asymptotically stable if there exist real matrices P PT , Qi , Si , ( i 0, 1, 2, " , N ), and Rij ( i, j 0, 1, 2, " , N
§D 0 · ¨ ¸ , D t 0 , Ea ©0 D¹
TABLE I. BOUND analytical rmax 6.0372 analytical rmax 6.0372
) such that S0 ! 0 , (13), and (18) hold. Proof: It is easy to see that (12) is implied by S0 ! 0 and (18). Q.E.D. Remark 2: When C { 0 and D { 0 , sytem (1) reduces to the following system x (t ) [ A LF (t ) Ea ]x(t ) [ B LF (t ) Eb ] x(t r ) It is easy to see that the main result in [6] is recovered from Theorem 2, which means that the result in [6] is extended to a more general class of systems. IV. NUMERICAL EXAMPLES To illustrate the effectiveness of the approach, two numerical examples are presented. Example 1: Consider system (1)-(5) with 0 · § 2 § 1 0 · § 0.1 0 · A ¨ ¸, B ¨ ¸, C ¨ ¸, © 0 0.9 ¹ © 1 1¹ © 0 0.1¹
rmax CALCULATED USING THE
METHODS IN [1, 5, 10] AND THIS PAPER FOR c Wu rmax 4.35 N 1 rmax 5.9337
Han rmax 4.33 N 2 rmax 6.0308
0.1
Fridman rma x 3.49 N 3 rmax 6.0359
For D ! 0 , d 0 , applying the criteria in [5, 10], and this paper for N 3 . Table II illustrates the numerical results for different D . We can see that rmax decreases as D increases and the stability limit obtained by the discretized Lyapunov functional approach is much less conservative than the results in [5, 10]. TABLE II. BOUND rmax CALCULATED USING THE METHODS IN [5, 10] AND THIS PAPER FOR DIFFERENT D AND d { 0 D 0.00 0.05 0.10 [5] 4.33 3.61 2.90 [10] 4.35 3.64 3.06 This paper 6.03 4.93 4.05 D 0.15 0.20 0.25 [5] 2.19 1.48 0.77 [10] 2.60 2.24 1.94 This paper 3.36 2.83 2.40
For D 0.2 , d t 0 , the effect of linear fractional factor d on the maximum time-delay for stability rmax is studied. The numerical results for N 3 and different d are estimated in Table III. It is easy to see that as d o 0 , the stability limit approaches the routine norm-bounded uncertainty case. As d increases, rmax decreases. As mentioned in the introduction, the most interesting and significant contribution of this paper is that the method in this paper can be used to detect the robust stability of an uncertain linear neutral system with non-small delay. The existing methods on the small time-delay fail to check the stability of such kinds of systems. The following example illustrates the fact.
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TABLE III. BOUND rmax CALCULATED FOR D DIFFERENT d 0.0 0.1
d
0.2 AND 0.2
N 3 max
r
2.83
2.81
2.74
d N 3 rmax
0.3
0.4
0.5
2.64
2.47
2.35
Example 2: Consider system (1)-(5) with 1 · § 0 §0 0· §0 0 · A ¨ ¸, B ¨ ¸, C ¨ ¸, © 2 0.3 ¹ ©1 0¹ © 0 0.2 ¹ §d 0· §1 0· §D 0 · L ¨ ¸, ¸, D ¨ ¸ , D t 0 , Ea Eb ¨ ©0 d¹ ©0 1¹ ©0 D¹ 0 d d 1. For D 0 , the nominal system is asymptotically stable for r (0.4302, 1.4163) . The maximum time-delay interval (rmin , rmax ) for asymptotic stability is listed in Table IV for different N. It is clear that as N increases, the results converge to the analytical solutions. For D ! 0 , d 0 , now we are in a position to study the effect of the uncertainty bound D on the maximum timedelay interval (rmin , rmax ) for asymptotic stability. The numerical results for N 6 and different D are estimated in Table V. It is easy to see that as D o 0 , the stability interval for delay approaches the uncertain-free case. As D increases, rmin increases, while rmax decreases. TABLE IV. BOUNDS rmin AND rmax CALCULATED FOR VARIOUS N N 1 2 3 0.4738 0.440 0.435 rmin
rmax
0.7140
1.111
1.302
N
rmin
4 0.433
5 0.432
6 0.432
rmax
1.389
1.411
1.415
rmin
TABLE V. BOUNDS rmin AND rmax CALCULATED FOR VARIOUS D 0.00 0.01 0.02 0.03 0.432 0.499 0.577 0.675
0.04 0.851
rmax
1.415
1.010
D
1.358
1.287
1.190
For D 0.01 , d t 0 , the effect of linear fractional factor d on the maximum time-delay interval (rmin , rmax ) for asymptotic stability is also studied. Table VI lists the numerical results for N 6 and different d . It is easy to see that as d o 0 , the stability limit approaches the routine norm-bounded uncertainty case. As d increases, rmin increases, while rmax decreases. TABLE VI. BOUNDS rmin AND rmax CALCULATED FOR D 0.01 AND DIFFERENT d
d rmin
0.0 0.499
0.1 0.507
0.2 0.516
rmax
1.358
1.355
1.351
d rmin
0.3 0.529
0.4 0.547
0.5 0.572
rmax
1.345
1.336
1.321
V. CONCLUSION The stability problem of linear neutral systems has been investigated using the discretized Lyapunov functional approach. A delay-dependent stability criterion has been derived and has been applicable to linear neutral systems with both small and non-small delays. For systems with small delay, a numerical example has shown that the results derived by the new criterion significantly improve the estimate of stability limit over the existing results in the literature. For systems with non-small delay, there has been an example to show the effectiveness of the new criterion while the existing methods on the small timedelay has failed to make any conclusion on the stability of such kinds of systems. REFERENCES [1] E. Fridman and U. Shaked, “Delay-dependent stability and H f control: constant and time-varying delays,” Int. J. Control, vol. 76, no. 1, pp. 48-60, 2003. [2] K. Gu, “A further refinement of discretized Lyapunov functional method for the stability of time-delay systems,” Int. J. Contr.,vol. 74, no. 10, pp. 967-976, 2001. [3] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, New York: SpringVerlag, 1993. [4] Q.-L. Han, “On delay-dependent stability for neutral delaydifferential systems,” Int. J. Appl. Math. and Comp. Sci., vol. 11, no. 4, pp. 965-976, 2001. [5] Q.-L. Han, “Robust stability of uncertain delay-differential systems of neutral type,” Automatica, vol. 38, no. 4, pp. 719723, 2002. [6] Q.-L. Han, K. Gu and X. Yu, “An improved estimate of the robust stability bound of time-delay systems with normbounded uncertainty,” IEEE Trans. Automat. Contr., vol. 48, no. 9, pp. 1629-1634, 2003. [7] Y. He, M. Wu, J-H. She and G.-P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Systems & Control Letters, vol. 51, no. 1, pp. 57-65, 2004. [8] W. Huang, “Generalization of Liapunov’s theorem in a linear delay system,” J. Math. Anal. Appl., vol. 142, pp. 83-94, 1989. [9] S.-I. Niculescu, “On delay-dependent stability under model transformations of some neutral linear systems,” Int. J. Control, vol. 74, no. 6, pp. 609-617, 2001. [10] M. Wu, Y. He and J.-H. She, “New delay-dependent stability criteria and stabilizing method for neutral systems,” Accepted for publication in IEEE Trans. Automat. Contr., 2004.
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