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Simple criterion for asymptotic stability of interval neutral delay-differential systems
Applied Mathematics F’ERGAMON
Applied Mathematics
Let&s
Letters 16 (2003) 1063-1068
www.elsevier.comjlocate/aml
Simple Criterion of Ititerval Neutral
for Asymptotic Delay-Differential
Stability Systems
J. H. PARK School of Electrical Engineering and Computer Science Yeungnam University, 2141 Dae+Dong Kyongsan 712-749, Korea jessielDyu.ac.kr (Received and accepted October 2002) Communicated
by G. Wake
Abstract-h this paper, the asymptotic stability of interval neutral delay-differential systems is investigated. A delay-independent criterion for the stability of the system is derived in terms of the spectral radius. Numerical computations are performed to illustrate the result. @ 2003 Elsevier Ltd. All rights reserved. Keywords-Interval
neutral system, Asymptotic
stability, Spectral radius.
1. INTRODUCTION Over the decades, the stability analysis of various neutral delay-differential systems has received considerable attention [l-3]. The theory of neutral delay-differential systems is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay-differential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, the neutral systems often appear ‘in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar, etc. In the literature, various analysis techniques utilizing Lyapunov technique, characteristic equation method, or state solution approach have been used to derive stability criteria for asymptotic stability of the systems. The developed stability criteria are often classified into two categories according to their dependence on the size of delays. Several delay-independent sufficient conditions for the asymptotic stability of the systems are presented by some researchers [4-81. Also, a few delay-dependent sufficient conditions have been exploited in [g-11]. In this paper, we are interested in the following linear systems of neutral type described by
2(t) = Arz(t) + B~z(t - h) + cIi(t - h), z(t) = 4(t),
fJE [-hOI,
This work was supported by Korea Research Foundation under Grant KRF-2001~003EO0278. 0893-9659/03/S - see front matter @ 2003 Elsevier Ltd. All rights reserved. doi: 10.1016/S0893-9659(03)00144-7
(1)
J. H. PARK
1064
where x(t) E R” is the state vector, AI, BI, and C, E Rnx” are matrices whose elements vary in prescribed ranges, e.g., AI, BI, and Cl are such that
h is the positive time-delay, and 4(.) is the given continuously differentiable function on [-h, O]. It is well known that interval matrices, which are caused by unavoidable system parametric variations, changes in operating conditions, aging, etc., are real matrices in which all elements are known only to belong to a specified closed interval. In the past, a number of reports have been proposed for the stability analysis of interval systems [12-141. However, up to now, the stability of interval neutral differential systems has not yet received great interest from the researchers. The goal of this paper is to deal with the stability analysis of the system. In this paper, we present a delay-independent criterion for asymptotic stability of the system given in (1). The derived sufficient condition is expressed in terms of the spectral radius of the matrix which is the combination of the modulus matrices.
2. PRELIMINARIES Before we develop our main result, we state some notations and a lemma. Let p[R] denote the largest modulus of the eigenvalues of the matrix R, which is known as the spectral radius of R. ]R] denotes a matrix formed by taking the absolute value of every element of R, and it is called the modulus matrix of R. I denotes the identity matrix of appropriate order. The relation R 5 T represents that all the elements of matrices, R and T, satisfy rij < tij for all i and j. jlRl[ denotes the matrix norm of R. illso, the following lemma is used for the main result.
LEMMA 2.1. Consider any n x n matrices R, T, and V. Part I. (See [Xl.)
If [RI 5 V, then
(4 IRTI I IRIITI 5 VI, (b) IR + TI 5 IRI + ITI L V + ITI, (~1PPI I PPII I P[VI> (4 PWI 5 ~[IRllTll 5 PW% and (e) P[R+ Tl I p[lR + Tll 5 PWI + ITI1I PP’+ PII. Part II. (See 1161.) If p[R] < 1, then det(1 f R) # 0. Part III. (See 1171.) If /[R/l < 1, then (I - R)-l exists, (I - R)-l
= I + R + R2 + ... , and
Il(l- WI1 5 l/O - IIRII)~ 3. MAIN
RESULT
In this section, we derive a sufficient condition for the asymptotic stability of the system given in (1). Denote Al = (&)
,
AZ = (ufj),
III = (bZ> ,
Bz = (b&) ,
Cl = (c2’,),
C, = (c&) ,
(3)
and let A = (uij)
= ; (ufj + ufj) = ;(A1
+ A&
B = (bij) = f (bij + bfj) = ;(B1
+ B2),
c = (Cij) = ; (& + cg = ;(cl
+ Cz),
(4)
Simple Criterion
1065
where A, B, and C are the average matrices between AI and A2, BI and Bz, and Cl and CZ, respectively. In this paper, it is assumed that A is a Hurwitz. Furthermore, we have AA = (a& - aij) = &-A,
AB = (bt - bij) = B1--B,
AC = (cfj - cij) = cr-c,
(5)
where AA, AB, and AC are the bias matrices between AI and A, BI and B, and CI and C, respectively. Also, AM = (c$ - aij)
BM = (bfj - bij) = B2-B,
= A2-A,
CM = (cfj - cij) = C,-C,
(6)
where AM, BM, and CM are the maximal bias matrices between AZ and A, B2 and B, and Cz and C, respectively. Note that
IAAl L
AM,
lABI
IACl
IBM,
< CM.
(7)
Let F(s) = (s1- A)-l, and FM be the matrix formed by taking the maximum magnitude of each element of F(s) for 9s > 0. Then, we have the following theorem. THEOREM 3.1. The interval neutral delay-differential system given in (1) is asymptotically if the following inequalities are satisfied: llcll
+ IicM/l
< 1
stable,
(8)
and P
AM+IBI+BM+
FM. [
1 -
(~Iclll+
(IAl
+
{
+ ICBI +
ICIBM
+ CM
IlcMII)
’ (IcAl
+
(9)
IClAM
< 1.
AM + IBI + BM))
PROOF. The characteristic equation of the system given in (1) is X(s) = det [sl - Al - (BI + Cls) exp(-hs)]
= 0.
(10)
Since det[RT] = det[R] det[T] f or any two n x n matrices R and T, so we have X(s) = det[l - CI exp(-hs)]
det [sl - (I - CI exp(-hs))-’
. (AI + BI exp(-hs)]
.
(11)
In view of IlCll + IICMII < 1, the relation
llC~l/= lIC + AC11I: llcll + IIACIIi llcll + llcMIl < 1 follows. Hence, the matrix (I- CI exp(-hs))-’ llCrexp(-hs)ll I IlC,II < 1 for !Rs 2 0. Therefdre, if we can show that det [s1-
(I - C I exp(-hs))-’
exists and det(l - Cl exp(-hs))
. (AI + Br exp(-hs))]
# 0,
# 0 because
for !J?s> 0,
(12)
then Vs) # 0,
for 9Z.s> 0.
(13)
Here, equations (12) and (13) gu arantee the asymptotic stability of system (1) [5, Theorem l].
J. H.
1066
PARK
For simplicity, let us define [ = exp( -hs) and T = [Cr. Then, using the inequality (I--T)-’ 1+ (I- T)-lT, the left-hand side of (12) becomes det [sl - (I - T)-l(A1+
=
[BI)] = det [sl - (I + (I - T)-*T){A + AA + ((B + AB)}] = det [(sl- A) - AA -{(B-I- AB) - (I - T)-lT(A
+ AA + EB + EAB)]
=det[s1--A]~~~[I-(s~T--AA)-~(AA-+-@?+~AB
(
4
+ (I - T)-lT(A + AA + [B + [AI?)}] = de@1 - A] det [I - F(S) (AA + [B + [AB -I- (I - T)-lT(A
+ AA + EB + [AB)}]
.
Therefore, equation (12) can be rewritten as
A] det [I - F(s) {AA + [B + CAB -t (I - T)-lT(A
det[s1-
+ AA + EB + (AB)}]
# 0. (15)
Since A is a Hurwitz matrix, det[sl - A] # 0 for !Rs 2 0. So, equation (15) is further simplified as det [I - F(S) {AA $ [B + tAB + (I - T)-lT(A f AA + (B + [AB)}] # 0, (16) for Rs >_0. Thus, if we can show that p[F(s){AA -i- [B + [AB -I- (I - T)-lT(A for !Rs 1 0, equation (16) is satisfied by Part II of Lemma 2.1. .Here, note that using Part I of Lemma 2.1, for !Rs 2 0, we have
+ AA + 6B + (AB)}]
IP’II= Wrll I MI 5 IICII+ IICMIL ITAl = [[(C + AC)Al L IcfCAl + [(ACAl
< 1
(17)
5 ICAl + CMIAI.
Similarly,
IETBII FBI + CdBI,
ITA4 i IWM + CMAM,
IETABI
I
ICIBM
-I- CMBM.
(18)
Now, using Parts I and III of Lemma 2.1 and (17),(18), we obtain
p [F(s) {AA + JB + CAB + (I - T)-lT(A
+ AA + EB + tAB)}]
5 p [IF(
(1~~1 + IEB~+ I~ABI + [(I - T)-~(TA
L p [IF(s
{ [AAl + (B( + (AB( + ((I - T)-I( n(TA + TAA + VB
5 p [FM.
5~
[
FM. (CBI
{AM
1
+ IBI
+ BM
AM+IBI+BM+
+ CMIBI
+
(CPM
[ { FM.
+
]](I
-
T)-‘1)
. (ITAl
+
+ tTAB(}]
ITA + ItTBl + ItTABl)}]
1 -.
1 - IITII
+ CMBM)
((CA1
+ CMIAI
+
ICIAM
+ CMAM
(19)
11
1
A,v+IBI-~BM+
. (ICAl 1 -
(Ilc/
+ CMAM + ICBl + CMMJBJ -t- ICJBM Thus, this completes the proof.
+ TAA + ETB + ~TAB)(}]
+
IhI
+ CMBM)
+ CM\A)
+
ICI&
11<.l. I
REMARK 3.1. Since the matrix A is a Hurwitz, the matrix FM always exists and it can obtained for some s on imaginary axis by the maximum modulus principle.
Simple Criterion COROLLARY
1067
3.1. If CI = 0, system (1) becomes a interval retarded delay-differential
system,
i.e., i(t)
= A&)
+ BIz(t - h).
(20)
Then, by Theorem 3.1, sufficient condition for asymptotic stability of system (20) is simplified as
P [F&h
+ lB( + BM)] -=c1.
(21)
3.2. The sufficient conditions for the stability of the interval retarded delay-differential system (20) by Tissir and Hmamed [14] are as follows: REMARK
/-4A+ zB) + II&III + PMII < 0, ~(4 + P(zB) + IIAMII+ IPMII < 0, PW + IPII + IlAvll + IPMII < 0,
for all z,
with Izj = 1,
for all z,
with 1.~1= 1,
(22)
where z = ej”’ with w E [0,2n] and j2 = -1. To apply the above sufficient conditions, it needs the requirement that /.L(A + zB) < 0 or p(A) < 0 while Corollary 3.1 allows a more relaxed requirement that A is a Hurwitz matrix. Furthermore, the stability criterion (21) of Corollary 3.1 is expressed in terms of the spectral radius of the matrices which is the combination of the modulus matrices. Therefore, there is a better possibility that the proposed criterion is less conservative than those in (22), which uses the matrix norms and matrix measure. To demonstrate the applications of the result, we give the following examples. EXAMPLE
3.1. Consider the interval neutral differential system described by (1) where
where u is a parameter scalar for which we shall find the upper bound that guarantees the stability of the systems. J%om (4) and (6), the average matrices are
A= [i6
:],
B= [;;”
o-5(uo+o’2)],
and the matrices AM, BM, and CM are
AM =
[ 1 0’ o’ 15 ’
BM =
1
0.3 0.5V 0.1 ’
0.1 0.1
The rational function matrices F(s) and FM axe computed as
r F(s) =
1
l (s+6)
0
1
(s+4)(s+6)
1
’
FM =
-Is+4
Then, by simple computation of inequality asymptotic stability of the system is
(8) and (9), the bound of v for guaranteeing the
0.2 < u < 1.4.
J. H. PARK
1068
It is interesting to note that when Cr = 0, the bound of u in [14] for the stability is 0.2 < Y < 3.96. However, by applying our criterion (21), the stability bound is 0.2 < v < 60.6, which gives a less conservative result than in [14]. 3.2. Consider the interval retarded delay-differential
EXAMPLE
Al =
[
1”
i3
1 ,
A2 =
[
i2
;’
system described by (20) where
1 ,
From (4) and (6), we obtain A.=
[
y3
i2
1 ,
Since p(A + zB) = p(A) = 0.0811 > 0, the criteria (22) in [14] are not applicable. Hence, one cannot state the stability from the result of Tissir and Hmamed [14]. However, the matrix A is a Hurwitz, and therefore, Corollary 1 can be applied. Then, the rational function matrices F(s) and FM are as follows: 1 l 1 F(s) = s2+3s+2
[
’1 sf3 -2
1 ’
FM=
[ 1 0.5 !i
1.5
Then, by checking criterion (21) of Corollary 1, we obtain p [FM
(AM + jB( + BM)] = 0.9833 < 1.
This gives the asymptotic stability of the system.
REFERENCES 1. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Boston. MA, (1992). 2. J. Hale and SM. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, (1993). 3. V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Di@entiaZ Equations, Kluwer Academic, Dordrecht, (1992). 4. J.K. Hale, E.F. Infante and F.S.P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl. 105, 533-555, (1985). 5. G.-D. Hu and G.-D. Hu, Some simple stability criteria of neutral delay-differential systems, Appl. Math. and Comput. 80, 257-271, (1996). 6. J.X. Kuang, J.X. Xiang and H.J. Tian, The asymptotic stability of oneparameter methods for neutral differential equations, BIT 34, 400-108, (1994). 7. L.M. Li, Stability of linear neutral delay-differential systems, Bull. Austml. Math. Sot. 38, 339-344, (1988). 8. J.H. Park and S. Won, Asymptotic stability of neutral systems with multiple delays, J. Optim. Theory Appl. 103, 187-200, (1999). 9. R.K. Brayton and R.A. Willoughby, On the numerical integration of a symmetric system of difference differential equations of neutral type, J. Math. Anal. Appl. 18, 182-189, (1967). 10. D.Ya. Khusainov and E.V. Yun’kova, Investigation of the stability of linear systems of neutral type by the Lyapunov function method, Difl Chvn. 24, 613-621, (1988). 11. J.H. Park and S. Won, Stability analysis for neutral delay-differential systems, J. Fran/&n Inst. 33’7, l-9, (2000). 12. P.L. Liu and T.J. Su, Robust stability of interval time-delay systems with delay-dependence, Sys. d Lett. 33, 231-239, (1998). 13. M.H. Shih, Y.Y. Lur and C.T. Peng, An inequality for the spectral radius of an interval matrix, Linear Algebra E4 Appl. 274, 27-36, (1998). 14. E. Tissir and A. Hmamed, Stability tests of interval time delay systems, Sys. d Con. Lett. 23, 263-270, (1994). 15. J.M. Ortega, Numerical Analysis, Academic Press, New York, (1972). 16. K.H. Chen, Robust analysis and design of multiloop control systems, Ph.D. Dissertation, National Tsing-Hua University, R.O.C., (1989). 17. P. Lancaster and M. Timenetsky, The Theory of Matrices, Academic Press, Orlando, FL, (1985).
Applied Mathematics
Let&s
Letters 16 (2003) 1063-1068
www.elsevier.comjlocate/aml
Simple Criterion of Ititerval Neutral
for Asymptotic Delay-Differential
Stability Systems
J. H. PARK School of Electrical Engineering and Computer Science Yeungnam University, 2141 Dae+Dong Kyongsan 712-749, Korea jessielDyu.ac.kr (Received and accepted October 2002) Communicated
by G. Wake
Abstract-h this paper, the asymptotic stability of interval neutral delay-differential systems is investigated. A delay-independent criterion for the stability of the system is derived in terms of the spectral radius. Numerical computations are performed to illustrate the result. @ 2003 Elsevier Ltd. All rights reserved. Keywords-Interval
neutral system, Asymptotic
stability, Spectral radius.
1. INTRODUCTION Over the decades, the stability analysis of various neutral delay-differential systems has received considerable attention [l-3]. The theory of neutral delay-differential systems is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay-differential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, the neutral systems often appear ‘in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar, etc. In the literature, various analysis techniques utilizing Lyapunov technique, characteristic equation method, or state solution approach have been used to derive stability criteria for asymptotic stability of the systems. The developed stability criteria are often classified into two categories according to their dependence on the size of delays. Several delay-independent sufficient conditions for the asymptotic stability of the systems are presented by some researchers [4-81. Also, a few delay-dependent sufficient conditions have been exploited in [g-11]. In this paper, we are interested in the following linear systems of neutral type described by
2(t) = Arz(t) + B~z(t - h) + cIi(t - h), z(t) = 4(t),
fJE [-hOI,
This work was supported by Korea Research Foundation under Grant KRF-2001~003EO0278. 0893-9659/03/S - see front matter @ 2003 Elsevier Ltd. All rights reserved. doi: 10.1016/S0893-9659(03)00144-7
(1)
J. H. PARK
1064
where x(t) E R” is the state vector, AI, BI, and C, E Rnx” are matrices whose elements vary in prescribed ranges, e.g., AI, BI, and Cl are such that
h is the positive time-delay, and 4(.) is the given continuously differentiable function on [-h, O]. It is well known that interval matrices, which are caused by unavoidable system parametric variations, changes in operating conditions, aging, etc., are real matrices in which all elements are known only to belong to a specified closed interval. In the past, a number of reports have been proposed for the stability analysis of interval systems [12-141. However, up to now, the stability of interval neutral differential systems has not yet received great interest from the researchers. The goal of this paper is to deal with the stability analysis of the system. In this paper, we present a delay-independent criterion for asymptotic stability of the system given in (1). The derived sufficient condition is expressed in terms of the spectral radius of the matrix which is the combination of the modulus matrices.
2. PRELIMINARIES Before we develop our main result, we state some notations and a lemma. Let p[R] denote the largest modulus of the eigenvalues of the matrix R, which is known as the spectral radius of R. ]R] denotes a matrix formed by taking the absolute value of every element of R, and it is called the modulus matrix of R. I denotes the identity matrix of appropriate order. The relation R 5 T represents that all the elements of matrices, R and T, satisfy rij < tij for all i and j. jlRl[ denotes the matrix norm of R. illso, the following lemma is used for the main result.
LEMMA 2.1. Consider any n x n matrices R, T, and V. Part I. (See [Xl.)
If [RI 5 V, then
(4 IRTI I IRIITI 5 VI, (b) IR + TI 5 IRI + ITI L V + ITI, (~1PPI I PPII I P[VI> (4 PWI 5 ~[IRllTll 5 PW% and (e) P[R+ Tl I p[lR + Tll 5 PWI + ITI1I PP’+ PII. Part II. (See 1161.) If p[R] < 1, then det(1 f R) # 0. Part III. (See 1171.) If /[R/l < 1, then (I - R)-l exists, (I - R)-l
= I + R + R2 + ... , and
Il(l- WI1 5 l/O - IIRII)~ 3. MAIN
RESULT
In this section, we derive a sufficient condition for the asymptotic stability of the system given in (1). Denote Al = (&)
,
AZ = (ufj),
III = (bZ> ,
Bz = (b&) ,
Cl = (c2’,),
C, = (c&) ,
(3)
and let A = (uij)
= ; (ufj + ufj) = ;(A1
+ A&
B = (bij) = f (bij + bfj) = ;(B1
+ B2),
c = (Cij) = ; (& + cg = ;(cl
+ Cz),
(4)
Simple Criterion
1065
where A, B, and C are the average matrices between AI and A2, BI and Bz, and Cl and CZ, respectively. In this paper, it is assumed that A is a Hurwitz. Furthermore, we have AA = (a& - aij) = &-A,
AB = (bt - bij) = B1--B,
AC = (cfj - cij) = cr-c,
(5)
where AA, AB, and AC are the bias matrices between AI and A, BI and B, and CI and C, respectively. Also, AM = (c$ - aij)
BM = (bfj - bij) = B2-B,
= A2-A,
CM = (cfj - cij) = C,-C,
(6)
where AM, BM, and CM are the maximal bias matrices between AZ and A, B2 and B, and Cz and C, respectively. Note that
IAAl L
AM,
lABI
IACl
IBM,
< CM.
(7)
Let F(s) = (s1- A)-l, and FM be the matrix formed by taking the maximum magnitude of each element of F(s) for 9s > 0. Then, we have the following theorem. THEOREM 3.1. The interval neutral delay-differential system given in (1) is asymptotically if the following inequalities are satisfied: llcll
+ IicM/l
< 1
stable,
(8)
and P
AM+IBI+BM+
FM. [
1 -
(~Iclll+
(IAl
+
{
+ ICBI +
ICIBM
+ CM
IlcMII)
’ (IcAl
+
(9)
IClAM
< 1.
AM + IBI + BM))
PROOF. The characteristic equation of the system given in (1) is X(s) = det [sl - Al - (BI + Cls) exp(-hs)]
= 0.
(10)
Since det[RT] = det[R] det[T] f or any two n x n matrices R and T, so we have X(s) = det[l - CI exp(-hs)]
det [sl - (I - CI exp(-hs))-’
. (AI + BI exp(-hs)]
.
(11)
In view of IlCll + IICMII < 1, the relation
llC~l/= lIC + AC11I: llcll + IIACIIi llcll + llcMIl < 1 follows. Hence, the matrix (I- CI exp(-hs))-’ llCrexp(-hs)ll I IlC,II < 1 for !Rs 2 0. Therefdre, if we can show that det [s1-
(I - C I exp(-hs))-’
exists and det(l - Cl exp(-hs))
. (AI + Br exp(-hs))]
# 0,
# 0 because
for !J?s> 0,
(12)
then Vs) # 0,
for 9Z.s> 0.
(13)
Here, equations (12) and (13) gu arantee the asymptotic stability of system (1) [5, Theorem l].
J. H.
1066
PARK
For simplicity, let us define [ = exp( -hs) and T = [Cr. Then, using the inequality (I--T)-’ 1+ (I- T)-lT, the left-hand side of (12) becomes det [sl - (I - T)-l(A1+
=
[BI)] = det [sl - (I + (I - T)-*T){A + AA + ((B + AB)}] = det [(sl- A) - AA -{(B-I- AB) - (I - T)-lT(A
+ AA + EB + EAB)]
=det[s1--A]~~~[I-(s~T--AA)-~(AA-+-@?+~AB
(
4
+ (I - T)-lT(A + AA + [B + [AI?)}] = de@1 - A] det [I - F(S) (AA + [B + [AB -I- (I - T)-lT(A
+ AA + EB + [AB)}]
.
Therefore, equation (12) can be rewritten as
A] det [I - F(s) {AA + [B + CAB -t (I - T)-lT(A
det[s1-
+ AA + EB + (AB)}]
# 0. (15)
Since A is a Hurwitz matrix, det[sl - A] # 0 for !Rs 2 0. So, equation (15) is further simplified as det [I - F(S) {AA $ [B + tAB + (I - T)-lT(A f AA + (B + [AB)}] # 0, (16) for Rs >_0. Thus, if we can show that p[F(s){AA -i- [B + [AB -I- (I - T)-lT(A for !Rs 1 0, equation (16) is satisfied by Part II of Lemma 2.1. .Here, note that using Part I of Lemma 2.1, for !Rs 2 0, we have
+ AA + 6B + (AB)}]
IP’II= Wrll I MI 5 IICII+ IICMIL ITAl = [[(C + AC)Al L IcfCAl + [(ACAl
< 1
(17)
5 ICAl + CMIAI.
Similarly,
IETBII FBI + CdBI,
ITA4 i IWM + CMAM,
IETABI
I
ICIBM
-I- CMBM.
(18)
Now, using Parts I and III of Lemma 2.1 and (17),(18), we obtain
p [F(s) {AA + JB + CAB + (I - T)-lT(A
+ AA + EB + tAB)}]
5 p [IF(
(1~~1 + IEB~+ I~ABI + [(I - T)-~(TA
L p [IF(s
{ [AAl + (B( + (AB( + ((I - T)-I( n(TA + TAA + VB
5 p [FM.
5~
[
FM. (CBI
{AM
1
+ IBI
+ BM
AM+IBI+BM+
+ CMIBI
+
(CPM
[ { FM.
+
]](I
-
T)-‘1)
. (ITAl
+
+ tTAB(}]
ITA + ItTBl + ItTABl)}]
1 -.
1 - IITII
+ CMBM)
((CA1
+ CMIAI
+
ICIAM
+ CMAM
(19)
11
1
A,v+IBI-~BM+
. (ICAl 1 -
(Ilc/
+ CMAM + ICBl + CMMJBJ -t- ICJBM Thus, this completes the proof.
+ TAA + ETB + ~TAB)(}]
+
IhI
+ CMBM)
+ CM\A)
+
ICI&
11<.l. I
REMARK 3.1. Since the matrix A is a Hurwitz, the matrix FM always exists and it can obtained for some s on imaginary axis by the maximum modulus principle.
Simple Criterion COROLLARY
1067
3.1. If CI = 0, system (1) becomes a interval retarded delay-differential
system,
i.e., i(t)
= A&)
+ BIz(t - h).
(20)
Then, by Theorem 3.1, sufficient condition for asymptotic stability of system (20) is simplified as
P [F&h
+ lB( + BM)] -=c1.
(21)
3.2. The sufficient conditions for the stability of the interval retarded delay-differential system (20) by Tissir and Hmamed [14] are as follows: REMARK
/-4A+ zB) + II&III + PMII < 0, ~(4 + P(zB) + IIAMII+ IPMII < 0, PW + IPII + IlAvll + IPMII < 0,
for all z,
with Izj = 1,
for all z,
with 1.~1= 1,
(22)
where z = ej”’ with w E [0,2n] and j2 = -1. To apply the above sufficient conditions, it needs the requirement that /.L(A + zB) < 0 or p(A) < 0 while Corollary 3.1 allows a more relaxed requirement that A is a Hurwitz matrix. Furthermore, the stability criterion (21) of Corollary 3.1 is expressed in terms of the spectral radius of the matrices which is the combination of the modulus matrices. Therefore, there is a better possibility that the proposed criterion is less conservative than those in (22), which uses the matrix norms and matrix measure. To demonstrate the applications of the result, we give the following examples. EXAMPLE
3.1. Consider the interval neutral differential system described by (1) where
where u is a parameter scalar for which we shall find the upper bound that guarantees the stability of the systems. J%om (4) and (6), the average matrices are
A= [i6
:],
B= [;;”
o-5(uo+o’2)],
and the matrices AM, BM, and CM are
AM =
[ 1 0’ o’ 15 ’
BM =
1
0.3 0.5V 0.1 ’
0.1 0.1
The rational function matrices F(s) and FM axe computed as
r F(s) =
1
l (s+6)
0
1
(s+4)(s+6)
1
’
FM =
-Is+4
Then, by simple computation of inequality asymptotic stability of the system is
(8) and (9), the bound of v for guaranteeing the
0.2 < u < 1.4.
J. H. PARK
1068
It is interesting to note that when Cr = 0, the bound of u in [14] for the stability is 0.2 < Y < 3.96. However, by applying our criterion (21), the stability bound is 0.2 < v < 60.6, which gives a less conservative result than in [14]. 3.2. Consider the interval retarded delay-differential
EXAMPLE
Al =
[
1”
i3
1 ,
A2 =
[
i2
;’
system described by (20) where
1 ,
From (4) and (6), we obtain A.=
[
y3
i2
1 ,
Since p(A + zB) = p(A) = 0.0811 > 0, the criteria (22) in [14] are not applicable. Hence, one cannot state the stability from the result of Tissir and Hmamed [14]. However, the matrix A is a Hurwitz, and therefore, Corollary 1 can be applied. Then, the rational function matrices F(s) and FM are as follows: 1 l 1 F(s) = s2+3s+2
[
’1 sf3 -2
1 ’
FM=
[ 1 0.5 !i
1.5
Then, by checking criterion (21) of Corollary 1, we obtain p [FM
(AM + jB( + BM)] = 0.9833 < 1.
This gives the asymptotic stability of the system.
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