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POWERED BYSClENCEDIRECT.
Applied
ELSEYIER
Mathematics
Letters
17 (2004)
139-144
www.elsevier.com/locate/aml
Sufficient Conditions of Linear Neutral with a Single
for Stability Systems Delay
D. Q.
CA0 Mechanics Chengdu
Department of Applied Southwest Jiaotong University,
and Engineering 610031, P.R.
China
dqcao0263.net
PING
HE*
Department of Applied Mathematics Jiaotong University, Chengdu 610031, heping-ap-swjtuQyahoo.com.cn
Southwest
(Received
May
2002;
accepted
P.R.
China
May 2003)
Abstract-This
paper deals with the asymptotic stability of linear neutral systems with a single delay. Simple delay-independent stability criteria are derived in terms of the measure and norm of the corresponding matrices. The significance of the main criterion is that it takes into consideration the structure information of the system matrices A, B, and C, thus reducing the conservatism found in the existing results. @ 2004 Elsevier Ltd.
Keywords-Neutral
Numerical All rights
examples reserved.
are given
systems,
Algebraic
stability
to demonstrate
the validity
criteria,
independent.
Delay
of our
main
criteria.
1. INTRODUCTION Asymptotic stability of neutral delay-differential systems is playing an increasingly important role in many disciplines such as engineering, science, and mathematics. A number of stability criteria based on the characteristic equation approach, involving the determination of eigenvalues, measures and norms of matrices, or matrix conditions in terms of Hurwitz matrices, have been presented by Stroinski [l], Hale et al. [2], Li [3], Hale and Verduyn Lunel [4], Hu and Hu [5], criteria (delay-independent Bellen et al. [6], Park and Won [7], and Hu et al. [S]. Some stability and/or delay-dependent) are given in terms of the Lyapunov function and matrix inequalities (see, for example, [g-14]). This paper investigates the problem of asymptotic stability of linear neutral systems with a single time delay. Scalar inequalities involving the measures and norms of the corresponding matrices constitute the mathematical foundations of our approach. Based on the characteristic equation of the system, simple delay-independent stability criteria are derived. Involving *Author
to whom
all correspondence
0893-9659/04/$ - see front matter doi: lO.l016/SO893-9659(03)00231-3
should @ 2004
be addressed. Elsevier
Ltd.
All
rights
reserved.
Typeset
by J&@-~
140
D. Q. CAO
P. HE
AND
the structure information of the coefficient matrices A, B, and C, the new criteria can significantly reduce the conservation of the results in the literature. Numerical examples are given to demonstrate the validity of our main criteria and to compare them with the existing ones.
2. NOTATIONS
AND
PRELIMINARIES
Let F(P) denote the n-dimensional real (complex) space and Wxâ (Pxn) denote the set of all real (complex) n by n matrices. I denotes the unit matrix of appropriate order. Xj(A) and p(A) denote the jth eigenvalue and the spectral radius of A, respectively. IAl denotes the modulus matrix of A; A 5 B represents that the elements of A and B satisfy the inequality aij 5 bij for all i and j. llAl/ (:= dw) and p(A) (:= (1/2)X,,,(A + A*)) denote the spectral norm and the matrix measure of A, respectively. Consider the following linear neutral delay-differential system: i(t) where x(t) E Cnxl argument, A, B, and is, all the eigenvalues The following two
LEMMA 2.1. Hurwitz
matrix,
= Ax(t)
+ [Bz(t
- T) + Ck(t
-T)]
,
(1)
is the state vector, the constant parameter 7 2 0 represents the delay C E Fxn. The system matrix A is assumed to be a Hurwitz matrix, that of A have negative real parts. lemmas are cited and will be used in the proof of our main results.
(See Theorem 1 in [5,/.) Th e neutral p(C) < 1 and !RAi [(I - EC)-l(A
+ [B)]
LEMMA 2.2. (See 1151.) Let R E Pxn.
system
(1) is asymptotically
V[ E c such that
< 0,
If IlRll < 1, then
(I - R)-l
stable
if A is a
/[I 5 1. exists,
(2) and (I - R)-l
=
I+R+R2+....
3. MAIN
RESULTS
THEOREM 3.1. Assume that A is a Hurwitz matrix. Then, system if llCl/ < 1 and there exists an invertible matrix T such that EIT 4 p (T-lAT)
+ IIT-~(cA
PROOF. Taking notice of Il&Zl/ 5 I~l/lCIl, have the following inequality: l/(1 - cc,-â11
= III+
+ B)TII
(1) is asymptotically
(IT-l 11 IIC(CA+ B)TII < 0. 1- IICII
+
we have ll[Cll
< 1 for 161 5 1. Using
EC + E2C2 + . . .I/ i 1+
to Lemma
2.1, (1) is asymptotically
!I?& [(I - EC)-l(A By making
stable
+ [B)]
< 0,
(3) 2.2, we
IIECII + llE2C211 + . . .
5 1+ /ICI1 + llC211 +... = (I- llCll)-l, According
Lemma
stable
for IEI51.
(4
if V[ E @ such that
I 1. Thus, for all q 2 1, kg = p(G,) > 1. This shows all the cited criteria are not applicable in this example. EXAMPLE
4.2. Consider system (1) with
Since p(A) = 0.0811 > 0, Corollary 3.1 and Corollary 3.2 are not applicable. Take a similarity transformation by the following similarity matrix: T =
[
â,
1â
1
Then, with simple computation, we have llCll = 0.3618, EAT= 1.1229 > 0, and EZT = -0.0194 < 0 for integer q = 8. Since EAT > 0, the conditions of Theorem 3.1 are not satisfied. However, according to Theorem 3.2, system (1) is asymptotically stable. This shows Theorem 3.2 is shaper than Theorem 3.1. Making the similarity transformation (19), we can calculate kl = 5.4564 > 0, rEg= 6.4858 > 0, k3 = 1.3514 > 0, and k4 = & = 1.0742 > 1. Thus, all the cited criteria are not applicable in this example.
REFERENCES 1. U. Stroinski, Delay-independent stability criteria for neutral differential equations, D@rentiaZ Integral Equations 7, 593-599, (1994). 2. J.K. Hale, E.F. Infante and F.-S.P. Tsen, Stability in linear delay equations, Journal of Mathematical Analysis and Applications 105, 533-555, (1985). 3. L.M. Li, Stability of linear neutral delay-differential systems, Bulletin of the Australian Mathematical Society 38, 339-344, (1988). 4. J. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, (1993). 5. G.-D. Hu and G.-D. Hu, Some simple stability criteria of neutral delay-differential systems, Applied Mathematics and Computation 80, 257-271, (1996). 6. A. Bellen, N. Guglielmi and A.E. Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Transactions on Circuits Systems I CAS-46, 212-216, (1999). 7. J.-H. Park and S. Won, A note on stability of neutral delay-differential systems, Journal of the Fmnklin Institute
8. G.D.
10. 11. 12.
543-548,
(1999).
for stability of linear neutral systems with a single delay, 135, 125-133, (2001). D.Y. Khusainov and E.A. Yunkova, Investigation of the stability of linear-systems of neutral type by the Lyapunov function method, Diffeerential Equations 24, 424-431, (1988). C.-H. Lien, K.-W. Yu and J.-G. Hsieh, Stability conditions for a class of neutral systems with multiple time delays, Journal of Mathematical Analysis and Applications 245, 2G27, (2000). R.P. Agarwal and S.R. Grace, Asymptotic stability of differential systems of neutral type, Appl. Math. Lett. 13 (8), 15-19, (2000). E. Fridman, New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems, Systems and Control Letters 43, 309-319, (2001). Jownal
9.
336,
Hu, G.D.
Hu and B. Cahlon,
of Computational
and
Algebraic
Applied
criteria
Mathematics
-44
D. Q. CAO AND P. HE
13. C.-H.
for a class of uncertain nonlinear neutral time-delay systems, International 215-219, (2001). 14. S.-I. Niculescu, On delay-dependent stability under model transformations of some neutral linear systems, International Journal of Control 74, 609-617, (2001). 15. P. Lancaster apd M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, FL, (1985). 16. E.I. Jury, Inners and Stability of Dynamics Systems, Wiley, New York, (1974). JownaZ
Lien,
New
stability
of Systems
Science
criterion 32,
Applied Mathematics Letters
www.ElsevierMathematics.com
POWERED BYSClENCEDIRECT.
Applied
ELSEYIER
Mathematics
Letters
17 (2004)
139-144
www.elsevier.com/locate/aml
Sufficient Conditions of Linear Neutral with a Single
for Stability Systems Delay
D. Q.
CA0 Mechanics Chengdu
Department of Applied Southwest Jiaotong University,
and Engineering 610031, P.R.
China
dqcao0263.net
PING
HE*
Department of Applied Mathematics Jiaotong University, Chengdu 610031, heping-ap-swjtuQyahoo.com.cn
Southwest
(Received
May
2002;
accepted
P.R.
China
May 2003)
Abstract-This
paper deals with the asymptotic stability of linear neutral systems with a single delay. Simple delay-independent stability criteria are derived in terms of the measure and norm of the corresponding matrices. The significance of the main criterion is that it takes into consideration the structure information of the system matrices A, B, and C, thus reducing the conservatism found in the existing results. @ 2004 Elsevier Ltd.
Keywords-Neutral
Numerical All rights
examples reserved.
are given
systems,
Algebraic
stability
to demonstrate
the validity
criteria,
independent.
Delay
of our
main
criteria.
1. INTRODUCTION Asymptotic stability of neutral delay-differential systems is playing an increasingly important role in many disciplines such as engineering, science, and mathematics. A number of stability criteria based on the characteristic equation approach, involving the determination of eigenvalues, measures and norms of matrices, or matrix conditions in terms of Hurwitz matrices, have been presented by Stroinski [l], Hale et al. [2], Li [3], Hale and Verduyn Lunel [4], Hu and Hu [5], criteria (delay-independent Bellen et al. [6], Park and Won [7], and Hu et al. [S]. Some stability and/or delay-dependent) are given in terms of the Lyapunov function and matrix inequalities (see, for example, [g-14]). This paper investigates the problem of asymptotic stability of linear neutral systems with a single time delay. Scalar inequalities involving the measures and norms of the corresponding matrices constitute the mathematical foundations of our approach. Based on the characteristic equation of the system, simple delay-independent stability criteria are derived. Involving *Author
to whom
all correspondence
0893-9659/04/$ - see front matter doi: lO.l016/SO893-9659(03)00231-3
should @ 2004
be addressed. Elsevier
Ltd.
All
rights
reserved.
Typeset
by J&@-~
140
D. Q. CAO
P. HE
AND
the structure information of the coefficient matrices A, B, and C, the new criteria can significantly reduce the conservation of the results in the literature. Numerical examples are given to demonstrate the validity of our main criteria and to compare them with the existing ones.
2. NOTATIONS
AND
PRELIMINARIES
Let F(P) denote the n-dimensional real (complex) space and Wxâ (Pxn) denote the set of all real (complex) n by n matrices. I denotes the unit matrix of appropriate order. Xj(A) and p(A) denote the jth eigenvalue and the spectral radius of A, respectively. IAl denotes the modulus matrix of A; A 5 B represents that the elements of A and B satisfy the inequality aij 5 bij for all i and j. llAl/ (:= dw) and p(A) (:= (1/2)X,,,(A + A*)) denote the spectral norm and the matrix measure of A, respectively. Consider the following linear neutral delay-differential system: i(t) where x(t) E Cnxl argument, A, B, and is, all the eigenvalues The following two
LEMMA 2.1. Hurwitz
matrix,
= Ax(t)
+ [Bz(t
- T) + Ck(t
-T)]
,
(1)
is the state vector, the constant parameter 7 2 0 represents the delay C E Fxn. The system matrix A is assumed to be a Hurwitz matrix, that of A have negative real parts. lemmas are cited and will be used in the proof of our main results.
(See Theorem 1 in [5,/.) Th e neutral p(C) < 1 and !RAi [(I - EC)-l(A
+ [B)]
LEMMA 2.2. (See 1151.) Let R E Pxn.
system
(1) is asymptotically
V[ E c such that
< 0,
If IlRll < 1, then
(I - R)-l
stable
if A is a
/[I 5 1. exists,
(2) and (I - R)-l
=
I+R+R2+....
3. MAIN
RESULTS
THEOREM 3.1. Assume that A is a Hurwitz matrix. Then, system if llCl/ < 1 and there exists an invertible matrix T such that EIT 4 p (T-lAT)
+ IIT-~(cA
PROOF. Taking notice of Il&Zl/ 5 I~l/lCIl, have the following inequality: l/(1 - cc,-â11
= III+
+ B)TII
(1) is asymptotically
(IT-l 11 IIC(CA+ B)TII < 0. 1- IICII
+
we have ll[Cll
< 1 for 161 5 1. Using
EC + E2C2 + . . .I/ i 1+
to Lemma
2.1, (1) is asymptotically
!I?& [(I - EC)-l(A By making
stable
+ [B)]
< 0,
(3) 2.2, we
IIECII + llE2C211 + . . .
5 1+ /ICI1 + llC211 +... = (I- llCll)-l, According
Lemma
stable
for IEI51.
(4
if V[ E @ such that
I 1. Thus, for all q 2 1, kg = p(G,) > 1. This shows all the cited criteria are not applicable in this example. EXAMPLE
4.2. Consider system (1) with
Since p(A) = 0.0811 > 0, Corollary 3.1 and Corollary 3.2 are not applicable. Take a similarity transformation by the following similarity matrix: T =
[
â,
1â
1
Then, with simple computation, we have llCll = 0.3618, EAT= 1.1229 > 0, and EZT = -0.0194 < 0 for integer q = 8. Since EAT > 0, the conditions of Theorem 3.1 are not satisfied. However, according to Theorem 3.2, system (1) is asymptotically stable. This shows Theorem 3.2 is shaper than Theorem 3.1. Making the similarity transformation (19), we can calculate kl = 5.4564 > 0, rEg= 6.4858 > 0, k3 = 1.3514 > 0, and k4 = & = 1.0742 > 1. Thus, all the cited criteria are not applicable in this example.
REFERENCES 1. U. Stroinski, Delay-independent stability criteria for neutral differential equations, D@rentiaZ Integral Equations 7, 593-599, (1994). 2. J.K. Hale, E.F. Infante and F.-S.P. Tsen, Stability in linear delay equations, Journal of Mathematical Analysis and Applications 105, 533-555, (1985). 3. L.M. Li, Stability of linear neutral delay-differential systems, Bulletin of the Australian Mathematical Society 38, 339-344, (1988). 4. J. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, (1993). 5. G.-D. Hu and G.-D. Hu, Some simple stability criteria of neutral delay-differential systems, Applied Mathematics and Computation 80, 257-271, (1996). 6. A. Bellen, N. Guglielmi and A.E. Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Transactions on Circuits Systems I CAS-46, 212-216, (1999). 7. J.-H. Park and S. Won, A note on stability of neutral delay-differential systems, Journal of the Fmnklin Institute
8. G.D.
10. 11. 12.
543-548,
(1999).
for stability of linear neutral systems with a single delay, 135, 125-133, (2001). D.Y. Khusainov and E.A. Yunkova, Investigation of the stability of linear-systems of neutral type by the Lyapunov function method, Diffeerential Equations 24, 424-431, (1988). C.-H. Lien, K.-W. Yu and J.-G. Hsieh, Stability conditions for a class of neutral systems with multiple time delays, Journal of Mathematical Analysis and Applications 245, 2G27, (2000). R.P. Agarwal and S.R. Grace, Asymptotic stability of differential systems of neutral type, Appl. Math. Lett. 13 (8), 15-19, (2000). E. Fridman, New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems, Systems and Control Letters 43, 309-319, (2001). Jownal
9.
336,
Hu, G.D.
Hu and B. Cahlon,
of Computational
and
Algebraic
Applied
criteria
Mathematics
-44
D. Q. CAO AND P. HE
13. C.-H.
for a class of uncertain nonlinear neutral time-delay systems, International 215-219, (2001). 14. S.-I. Niculescu, On delay-dependent stability under model transformations of some neutral linear systems, International Journal of Control 74, 609-617, (2001). 15. P. Lancaster apd M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, FL, (1985). 16. E.I. Jury, Inners and Stability of Dynamics Systems, Wiley, New York, (1974). JownaZ
Lien,
New
stability
of Systems
Science
criterion 32,
