- Email: [email protected]
Improvement on delay dependent absolute stability of Lurie control systems with multiple time delays
Applied Mathematics and Computation 216 (2010) 1024–1027
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Improvement on delay dependent absolute stability of Lurie control systems with multiple time delays Phan T. Nam a,*, Pubudu N. Pathirana b a b
Department of Mathematics, Quynhon University, Binhdinh, Vietnam School of Engineering, Deakin University, Geelong, Australia
a r t i c l e
i n f o
a b s t r a c t Absolute stability of Lurie control systems with multiple time-delays is studied in this paper. By using extended Lyapunov functionals, we avoid the use of the stability assumption on the main operator and derive improved stability criteria, which are strictly less conservative than the criteria in [2,3]. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Lurie control system Absolute stability Delay-dependent Linear matrix inequality
1. Introduction Consider the following Lurie control system with multiple time-delays:
8 m P > > _ > < xðtÞ ¼ AxðtÞ þ Bi xðt si Þ þ bf ðrðtÞÞ; i¼1
ð1:1Þ
> rðtÞ ¼ cT xðtÞ; > > : xðhÞ ¼ /ðhÞ; h 2 ½ maxfsi g; 0; where xðtÞ 2 Rn is the state, A; Bi ði ¼ 1; 2; . . . ; mÞ 2 Rnn ; b; c 2 Rn ; x0 ðhÞ ¼ /ðhÞ 2 Cð½maxfsi g; 0; Rn Þ. The nonlinearity f() satisfies
si P 0 ði ¼ 1; 2; . . . ; mÞ and initial condition is
f ðÞ 2 K½0; 1 ¼ ff ðÞjf ð0Þ ¼ 0; 0 < rf ðrÞ < 1; r – 0g: In the paper [2], the authors considered absolute stability of Lurie control system with multiple time-delays (1.1). The proposed result is less conservative due to decomposing the matrix Bi ¼ Bi1 þ Bi2 ði ¼ 1; 2; . . . ; mÞ and using the operator Rt P Dðxt Þ ¼ xðtÞ þ m i¼1 Bi1 tsi xðsÞds to represent the system in the form of a descriptor system with discrete and distributed delays. However, their condition is still dependent on the stability of Dðxt Þ. Therefore, the main purpose of this paper is to reduce the stability of Dðxt Þ. Hence, we will get an improvement on the result in [2]. The following lemma is needed for our main result. Lemma 1.1 [1]. Assume that S 2 Rnn is a symmetric positive definite matrix. Then for every Q 2 Rnn ,
2hQy; xi hSy; yi 6 hQS1 Q T x; xi;
8x; y 2 Rn :
If we take S ¼ I then we have j2hQy; xij 6 kyk2 þ kQxk2 . We also need the main theorem in [2] and its proof for our main result.
* Corresponding author. E-mail address: [email protected] (P.T. Nam). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.090
P.T. Nam, P.N. Pathirana / Applied Mathematics and Computation 216 (2010) 1024–1027
1025
Theorem 1.1 [2]. Assuming that Dðxt Þ is stable. The system (1.1) is absolutely stable if there exist P ¼ P T > 0; Q i ¼ Q Ti > 0; Ri ¼ RTi > 0 ði ¼ 1; 2; . . . ; mÞ and a > 0; b > 0 such that the following LMI holds:
0 B B B B B B B B B X¼B B B B B B B B B @
PA0 þ AT0 P
N1
Pb þ bAT C þ ac
N2
Q
R1
R2
Rm
H
Q0
NT3
0
0
0
0
0
H
H
2bcT b
NT4
0
0
0
0
H
H
H
N5
BT Q
BT R1
BT R2
BT Rm
H
H
H
H
Q
0
0
0
H
H
H
H
H
s1 1 R1
0
0
H .. .
H .. .
H .. .
H .. .
H .. .
H .. .
s1 2 R2 .. .
.. .
0 .. .
H
H
H
H
H
H
H
1 C C C C C C C C C C: C C C C C C C C A
ð1:2Þ
s1 m Rm
Proof. Consider the following Lyapunov functional
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ;
ð1:3Þ
where
V 1 ðtÞ ¼ DT ðxt ÞPDðxt Þ; m Z t X xT ðsÞRi xðsÞds; V 2 ðtÞ ¼ i¼1
V 3 ðtÞ ¼
m Z X i¼1
V 4 ðtÞ ¼ 2b
tsi 0
dh si
Z r
Z
t
xT ðsÞRi xðsÞds;
tþh
f ðrÞdr:
0
By proof in [2], if (1.2) holds then we have
V P0 and
V_ 6 yT ðtÞNyðtÞ;
ð1:4Þ
where
yðtÞ ¼ ½DT ðxt Þ; zT1 ðtÞ; f T ðrðtÞÞ; zT2 ðtÞT ; z1 ðtÞ ¼ ½xT ðt s1 Þ; xT ðt s2 Þ; . . . ; xT ðt sm ÞT ; Z t T Z t Z t xT ðsÞds; xT ðsÞds; . . . ; xT ðsÞds z2 ðtÞ ¼ ts1
ts2
tsm
and N < 0. This implies that there exists a positive number k such that
_ VðtÞ < k kDðxt Þk2 þ
m X i¼1
Z m X kxðt si Þk þ 2
t
tsi
i¼1
! 2 2 x ðsÞds þ kf ðrðtÞk < kkDðxt Þk2 : T
Since Dðxt Þ is stable, the system (1.1) is absolutely stable. Remark 1. Since xðtÞ ¼ Dðxt Þ
Pm
i¼1 Bi1
Z m X Bi1 kxðtÞk 6 kDðxt Þk þ i¼1
t
tsi
Rt
tsi
h
xðsÞds, we have
xðsÞds :
Applying the Bunhiakovski’s inequality, we have
kxðtÞk2 6 ðm þ 1Þ kDðxt Þk2 þ
m X i¼1
Z kBi1 k2
t
tsi
2 ! xðsÞds :
ð1:5Þ
1026
P.T. Nam, P.N. Pathirana / Applied Mathematics and Computation 216 (2010) 1024–1027
This implies that
kDðxt Þk2 6 If
Pm
2 i¼1 kBi1 k
2 Z t m X 1 : kBi1 k2 xðsÞds kxðtÞk2 þ mþ1 t s i i¼1
< 1 then we have,
kDðxt Þk2 6
2 Z m X t 1 : xðsÞds kxðtÞk2 þ mþ1 tsi i¼1
Hence,
V_ < If
Pm
2 i¼1 kBi1 k
! m X k kxðt si Þk2 þ kf ðrðtÞk2 : kxðtÞk2 þ mþ1 i¼1
ð1:6Þ
> 1 then we have,
1
2
kDðxt Þk 6 Pm
2
2 i¼1 kBi1 k
kDðxt Þk 6 Pm
2 ! Z t m X 1 2 2 kBi1 k xðsÞds kxðtÞk þ mþ1 tsi i¼1 t 2 xðsÞds :
1
i¼1 kBi1 k
2
Z m X 1 2 6 kxðtÞk þ Pm 2 ðm þ 1Þ i¼1 kBi1 k tsi i¼1
ð1:7Þ
Hence,
V_ <
ðm þ 1Þ
k Pm
i¼1 kBi1 k
kxðtÞk2 þ
2
m X
! kxðt si Þk2 þ kf ðrðtÞk2 :
ð1:8Þ
i¼1
Thus if (1.2) holds then there exists a positive number
k0 ¼
8 k < mþ1 ; : ðmþ1ÞPkm
i¼1
kBi1 k2
;
such that
V_ < k0 kxðtÞk2 þ
m X
! kxðt si Þk2 þ kf ðrðtÞk2 :
ð1:9Þ
i¼1
2. Main result Using Remark 1, we have an improved criterion for absolute stability of system (1.1) as follows: Theorem 2.1. The system (1.1) is absolutely stable if there exist P ¼ P T > 0; Q i ¼ Q Ti > 0; Ri ¼ RTi > 0 ði ¼ 1; 2; . . . ; mÞ and
a > 0; b > 0 such that the LMI (1.2) holds. Proof. Consider the Lyapunov functional
V ðtÞ ¼ VðtÞ þ V 5 ðtÞ;
ð2:1Þ
2
where V 5 ðtÞ ¼ kxðtÞk , is a positive number that will be chosen later. Since VðtÞ P 0; V ðtÞ P kxðtÞk2 . The derivative of V 5 ðtÞ is given by
" #T m X T _V 5 ðtÞ ¼ 2xðtÞ _ xðtÞ ¼ 2 AxðtÞ þ Bi xðt si Þ þ bf ðrðtÞÞ xðtÞ:
ð2:2Þ
i¼1
By Lemma 1.1, we have
V_ 5 ðtÞ 6 2kAk þ
m X
! 2
kBi k þ kbk
2
2
kxðtÞk þ
i¼1
Choosing
( ) k0 1 ¼ min ;1 ; P 2 2 2 2kAk þ m i¼1 kBi k þ kbk
m X i¼1
! 2
kxðt si Þk þ kf ðrðtÞÞk
2
:
P.T. Nam, P.N. Pathirana / Applied Mathematics and Computation 216 (2010) 1024–1027
1027
we have
V_ ðtÞ < 0: This implies that system (1.1) is absolutely stable. The proof is completed. h Remark 2. By the same way, we also reduce the stability of main operator of the criterion in [3]. Hence, we will get an stability criterion, which is less conservative than the criterion in [3].
3. Conclusion Absolute stability of Lurie control systems with multiple time-delays is studied in this paper. By improving the estimation of derivative of the Lyapunov functionals in [2,3], we propose improved Lyapunov functionals and obtain stability criteria, which are strictly less conservative than the criteria in [2,3]. Acknowledgements This work was supported by the National Foundation for Science and Technology Development, Vietnam and by the Australian Research Council under the Discovery Grant DP0667181. References [1] S. Boyd, El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix and Control Theory, SIAM Studies in Appl. Math., vol. 15, SIAM, PA, 1994. [2] Junkang Tian, Shouming Zhong, Lianlin Xiong, Delay-dependent absolute stability of Lurie control systems with multiple time-delays, Applied Mathematics and Computation 188 (2007) 379–384. [3] Jiuwen Cao, Shouming Zhong, New delay-dependent condition for absolute stability of Lurie control systems with multiple time-delays and nonlinearities, Applied Mathematics and Computation 194 (2007) 250–258.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Improvement on delay dependent absolute stability of Lurie control systems with multiple time delays Phan T. Nam a,*, Pubudu N. Pathirana b a b
Department of Mathematics, Quynhon University, Binhdinh, Vietnam School of Engineering, Deakin University, Geelong, Australia
a r t i c l e
i n f o
a b s t r a c t Absolute stability of Lurie control systems with multiple time-delays is studied in this paper. By using extended Lyapunov functionals, we avoid the use of the stability assumption on the main operator and derive improved stability criteria, which are strictly less conservative than the criteria in [2,3]. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Lurie control system Absolute stability Delay-dependent Linear matrix inequality
1. Introduction Consider the following Lurie control system with multiple time-delays:
8 m P > > _ > < xðtÞ ¼ AxðtÞ þ Bi xðt si Þ þ bf ðrðtÞÞ; i¼1
ð1:1Þ
> rðtÞ ¼ cT xðtÞ; > > : xðhÞ ¼ /ðhÞ; h 2 ½ maxfsi g; 0; where xðtÞ 2 Rn is the state, A; Bi ði ¼ 1; 2; . . . ; mÞ 2 Rnn ; b; c 2 Rn ; x0 ðhÞ ¼ /ðhÞ 2 Cð½maxfsi g; 0; Rn Þ. The nonlinearity f() satisfies
si P 0 ði ¼ 1; 2; . . . ; mÞ and initial condition is
f ðÞ 2 K½0; 1 ¼ ff ðÞjf ð0Þ ¼ 0; 0 < rf ðrÞ < 1; r – 0g: In the paper [2], the authors considered absolute stability of Lurie control system with multiple time-delays (1.1). The proposed result is less conservative due to decomposing the matrix Bi ¼ Bi1 þ Bi2 ði ¼ 1; 2; . . . ; mÞ and using the operator Rt P Dðxt Þ ¼ xðtÞ þ m i¼1 Bi1 tsi xðsÞds to represent the system in the form of a descriptor system with discrete and distributed delays. However, their condition is still dependent on the stability of Dðxt Þ. Therefore, the main purpose of this paper is to reduce the stability of Dðxt Þ. Hence, we will get an improvement on the result in [2]. The following lemma is needed for our main result. Lemma 1.1 [1]. Assume that S 2 Rnn is a symmetric positive definite matrix. Then for every Q 2 Rnn ,
2hQy; xi hSy; yi 6 hQS1 Q T x; xi;
8x; y 2 Rn :
If we take S ¼ I then we have j2hQy; xij 6 kyk2 þ kQxk2 . We also need the main theorem in [2] and its proof for our main result.
* Corresponding author. E-mail address: [email protected] (P.T. Nam). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.090
P.T. Nam, P.N. Pathirana / Applied Mathematics and Computation 216 (2010) 1024–1027
1025
Theorem 1.1 [2]. Assuming that Dðxt Þ is stable. The system (1.1) is absolutely stable if there exist P ¼ P T > 0; Q i ¼ Q Ti > 0; Ri ¼ RTi > 0 ði ¼ 1; 2; . . . ; mÞ and a > 0; b > 0 such that the following LMI holds:
0 B B B B B B B B B X¼B B B B B B B B B @
PA0 þ AT0 P
N1
Pb þ bAT C þ ac
N2
Q
R1
R2
Rm
H
Q0
NT3
0
0
0
0
0
H
H
2bcT b
NT4
0
0
0
0
H
H
H
N5
BT Q
BT R1
BT R2
BT Rm
H
H
H
H
Q
0
0
0
H
H
H
H
H
s1 1 R1
0
0
H .. .
H .. .
H .. .
H .. .
H .. .
H .. .
s1 2 R2 .. .
.. .
0 .. .
H
H
H
H
H
H
H
1 C C C C C C C C C C: C C C C C C C C A
ð1:2Þ
s1 m Rm
Proof. Consider the following Lyapunov functional
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ;
ð1:3Þ
where
V 1 ðtÞ ¼ DT ðxt ÞPDðxt Þ; m Z t X xT ðsÞRi xðsÞds; V 2 ðtÞ ¼ i¼1
V 3 ðtÞ ¼
m Z X i¼1
V 4 ðtÞ ¼ 2b
tsi 0
dh si
Z r
Z
t
xT ðsÞRi xðsÞds;
tþh
f ðrÞdr:
0
By proof in [2], if (1.2) holds then we have
V P0 and
V_ 6 yT ðtÞNyðtÞ;
ð1:4Þ
where
yðtÞ ¼ ½DT ðxt Þ; zT1 ðtÞ; f T ðrðtÞÞ; zT2 ðtÞT ; z1 ðtÞ ¼ ½xT ðt s1 Þ; xT ðt s2 Þ; . . . ; xT ðt sm ÞT ; Z t T Z t Z t xT ðsÞds; xT ðsÞds; . . . ; xT ðsÞds z2 ðtÞ ¼ ts1
ts2
tsm
and N < 0. This implies that there exists a positive number k such that
_ VðtÞ < k kDðxt Þk2 þ
m X i¼1
Z m X kxðt si Þk þ 2
t
tsi
i¼1
! 2 2 x ðsÞds þ kf ðrðtÞk < kkDðxt Þk2 : T
Since Dðxt Þ is stable, the system (1.1) is absolutely stable. Remark 1. Since xðtÞ ¼ Dðxt Þ
Pm
i¼1 Bi1
Z m X Bi1 kxðtÞk 6 kDðxt Þk þ i¼1
t
tsi
Rt
tsi
h
xðsÞds, we have
xðsÞds :
Applying the Bunhiakovski’s inequality, we have
kxðtÞk2 6 ðm þ 1Þ kDðxt Þk2 þ
m X i¼1
Z kBi1 k2
t
tsi
2 ! xðsÞds :
ð1:5Þ
1026
P.T. Nam, P.N. Pathirana / Applied Mathematics and Computation 216 (2010) 1024–1027
This implies that
kDðxt Þk2 6 If
Pm
2 i¼1 kBi1 k
2 Z t m X 1 : kBi1 k2 xðsÞds kxðtÞk2 þ mþ1 t s i i¼1
< 1 then we have,
kDðxt Þk2 6
2 Z m X t 1 : xðsÞds kxðtÞk2 þ mþ1 tsi i¼1
Hence,
V_ < If
Pm
2 i¼1 kBi1 k
! m X k kxðt si Þk2 þ kf ðrðtÞk2 : kxðtÞk2 þ mþ1 i¼1
ð1:6Þ
> 1 then we have,
1
2
kDðxt Þk 6 Pm
2
2 i¼1 kBi1 k
kDðxt Þk 6 Pm
2 ! Z t m X 1 2 2 kBi1 k xðsÞds kxðtÞk þ mþ1 tsi i¼1 t 2 xðsÞds :
1
i¼1 kBi1 k
2
Z m X 1 2 6 kxðtÞk þ Pm 2 ðm þ 1Þ i¼1 kBi1 k tsi i¼1
ð1:7Þ
Hence,
V_ <
ðm þ 1Þ
k Pm
i¼1 kBi1 k
kxðtÞk2 þ
2
m X
! kxðt si Þk2 þ kf ðrðtÞk2 :
ð1:8Þ
i¼1
Thus if (1.2) holds then there exists a positive number
k0 ¼
8 k < mþ1 ; : ðmþ1ÞPkm
i¼1
kBi1 k2
;
such that
V_ < k0 kxðtÞk2 þ
m X
! kxðt si Þk2 þ kf ðrðtÞk2 :
ð1:9Þ
i¼1
2. Main result Using Remark 1, we have an improved criterion for absolute stability of system (1.1) as follows: Theorem 2.1. The system (1.1) is absolutely stable if there exist P ¼ P T > 0; Q i ¼ Q Ti > 0; Ri ¼ RTi > 0 ði ¼ 1; 2; . . . ; mÞ and
a > 0; b > 0 such that the LMI (1.2) holds. Proof. Consider the Lyapunov functional
V ðtÞ ¼ VðtÞ þ V 5 ðtÞ;
ð2:1Þ
2
where V 5 ðtÞ ¼ kxðtÞk , is a positive number that will be chosen later. Since VðtÞ P 0; V ðtÞ P kxðtÞk2 . The derivative of V 5 ðtÞ is given by
" #T m X T _V 5 ðtÞ ¼ 2xðtÞ _ xðtÞ ¼ 2 AxðtÞ þ Bi xðt si Þ þ bf ðrðtÞÞ xðtÞ:
ð2:2Þ
i¼1
By Lemma 1.1, we have
V_ 5 ðtÞ 6 2kAk þ
m X
! 2
kBi k þ kbk
2
2
kxðtÞk þ
i¼1
Choosing
( ) k0 1 ¼ min ;1 ; P 2 2 2 2kAk þ m i¼1 kBi k þ kbk
m X i¼1
! 2
kxðt si Þk þ kf ðrðtÞÞk
2
:
P.T. Nam, P.N. Pathirana / Applied Mathematics and Computation 216 (2010) 1024–1027
1027
we have
V_ ðtÞ < 0: This implies that system (1.1) is absolutely stable. The proof is completed. h Remark 2. By the same way, we also reduce the stability of main operator of the criterion in [3]. Hence, we will get an stability criterion, which is less conservative than the criterion in [3].
3. Conclusion Absolute stability of Lurie control systems with multiple time-delays is studied in this paper. By improving the estimation of derivative of the Lyapunov functionals in [2,3], we propose improved Lyapunov functionals and obtain stability criteria, which are strictly less conservative than the criteria in [2,3]. Acknowledgements This work was supported by the National Foundation for Science and Technology Development, Vietnam and by the Australian Research Council under the Discovery Grant DP0667181. References [1] S. Boyd, El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix and Control Theory, SIAM Studies in Appl. Math., vol. 15, SIAM, PA, 1994. [2] Junkang Tian, Shouming Zhong, Lianlin Xiong, Delay-dependent absolute stability of Lurie control systems with multiple time-delays, Applied Mathematics and Computation 188 (2007) 379–384. [3] Jiuwen Cao, Shouming Zhong, New delay-dependent condition for absolute stability of Lurie control systems with multiple time-delays and nonlinearities, Applied Mathematics and Computation 194 (2007) 250–258.