delay-dependent stability for neural networks with time-varying delay
TRANSCRIPT
Chaos, Solitons and Fractals 33 (2007) 171–177
www.elsevier.com/locate/chaos
Delay-dependent stability for neural networkswith time-varying delay q
Hailin Liu *, Guohua Chen
Department of Computer, Guangdong Polytechnic Normal University, Guangzhou 510633, China
Accepted 14 December 2005
Abstract
In this paper, delay-dependent stability problem for neural networks with a time-varying delay is studied. The free-weighting matrix method is employed to derive an LMI-based criterion, in which the restriction of the derivative of atime-varying delay is removed. A delay-dependent and rate-independent stability criterion is derived as an easy corol-lary. Finally, the effectiveness of the presented stability criterion and its improvement over the existing results are dem-onstrated in numerical examples.� 2006 Elsevier Ltd. All rights reserved.
1. Introduction
Neural networks have been extensively studied over the past few decades and have found many applications in avariety of areas, such as associative memory, pattern recognition, and combinatorial optimization. In recent years, con-siderable effort has been devoted to analyzing the stability of neural networks without a time delay. However, time-delay is frequently encountered in neural networks, and it is often a source of instability and oscillations in a system.As a result, the stability of delayed neural networks has received considerable attention (see e.g., [1–21]). The criteriaderived in these papers are based on various types of stability, such as asymptotic stability, complete stability, absolutestability, and exponential stability. Among them, delay-dependent asymptotic stability criteria have attracted muchattention recently [5,13,16,20,21] because delay-dependent criteria make use of information on the length of delays,and are less conservative than delay-independent ones. However, only constant time-delay is taken into account in[5,16,20,21]. In addition, some negative terms in the derivative of the Lyapunov functional tend to be ignored whendelay-dependent stability criteria are derived [5,13]. On the other hand, the restriction that the derivative of a time-vary-ing delay be less than 1 is imposed on stability criteria for neural networks with a time-varying delay [10,13,17–19].Recently, a free-weighting matrix approach was proposed in [22–25], in which free-weighting matrices are employedto express the relationship between the terms in the Leibniz–Newton formula, and all the negative terms in the deriv-ative of the Lyapunov functional are retained. This approach avoids the restriction on the derivative of a time-varyingdelay.
0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.01.025
q This project is supported by Natural Science Foundation of Guangdong’s Universities (Z03060).* Corresponding author.
E-mail address: [email protected] (H. Liu).
172 H. Liu, G. Chen / Chaos, Solitons and Fractals 33 (2007) 171–177
In this paper, the method proposed in [22–25] is employed to derive an LMI-based delay-dependent stability crite-rion for neural networks with a time-varying delay. Unlike existing ones, this criterion allows the derivative of a time-varying delay to take any value. Numerical examples are given to demonstrate the effectiveness and the improvement ofthis method over the existing ones.
2. Problem formulation
Consider the following delayed neural network:
_xðtÞ ¼ �CxðtÞ þ AgðxðtÞÞ þ Bgðxðt � dðtÞÞÞ þ u; ð1Þ
where x(Æ) = [x1(Æ),x2(Æ), . . . ,xn(Æ)]T 2 Rn is the neuron state vector, g(x(Æ)) = [g1(x1(Æ)),g2(x2(Æ)), . . . ,gn(xn(Æ))]T 2 Rn de-notes the neuron activation function, u = [u1,u2, . . . ,un]T 2 Rn is a constant input vector. C = diag{c1,c2, . . . ,cn} is adiagonal matrix with ci > 0, and A and B are the connection weight matrix and the delayed connection weight matrix,respectively. The time delay, d(t), is a time-varying differentiable function that satisfies
0 6 dðtÞ 6 h; ð2Þ_dðtÞ 6 l; ð3Þ
where l is a constant. In addition, it is assumed that each neuron activation function in system (1), gk(Æ), k = 1, 2, . . . ,n,satisfies the following condition:
0 6gkðxÞ � gkðyÞ
x� y6 Lk ; 8x; y 2 R; x 6¼ y; k ¼ 1; 2; . . . ; n; ð4Þ
where Lk, k = 1,2, . . . ,n are positive constants.In the following, the equilibrium point x� ¼ ½x�1; x�2; . . . ; x�n�
T of system (1) is shifted to the origin by the transforma-tion z(Æ) = x(Æ) � x*, which converts the system to the following form:
_zðtÞ ¼ �CzðtÞ þ Af ðzðtÞÞ þ Bf ðzðt � dðtÞÞÞ; ð5Þ
where z(Æ) = [z1(Æ),z2(Æ), . . . ,zn(Æ)]T is the state vector of the transformed system; f(z(Æ)) = [f1(z1(Æ)), f2(z2(Æ)), . . . , fn(zn(Æ))]T,and fkðzkð�ÞÞ ¼ gkðzkð�Þ þ z�kÞ � gjðz�kÞ, k = 1,2, . . . ,n. Note that the functions fk(Æ) (k = 1,2, . . . ,n) satisfy the followingcondition:
0 6fjðzkÞ
zk6 Lk ; f kð0Þ ¼ 0; 8zk 6¼ 0; k ¼ 1; 2; . . . ; n; ð6Þ
which is equivalent to
fkðzkÞ½fkðzkÞ � Lkzk � 6 0; f kð0Þ ¼ 0; k ¼ 1; 2; . . . ; n. ð7Þ
3. Delay-dependent stability criteria
In this section, a delay-dependent stability criterion is derived as follows by using the free-weighting matrix method.
Theorem 1. For given scalars h P 0 and d, the origin of system (5) with (6) and a time-varying delay satisfying
conditions (2) and (3) is globally asymptotically stable if there exist P = PT > 0, Q = QT P 0, W = WT P 0,
Z = ZT > 0, D = diag(d1,d2, . . . , dn) P 0, T = diag(t1, t2, . . . , tn) P 0, S = diag(s1, s2, . . . , sn) P 0, X ¼ X T ¼X 11 X 12 X 13 X 14 X 15
X T12 X 22 X 23 X 24 X 25
X T13 X T
23 X 33 X 34 X 35
X T14 X T
24 X T34 X 44 X 45
X T15 X T
25 X T35 X T
45 X 55
266664
377775 P 0, and any appropriately dimensional matrices H ¼ ½HT
1 HT2 �
Tand
N ¼ ½NT1 NT
2 NT3 NT
4 NT5 �
T such that the following LMIs (8) and (9) are feasible:
H. Liu, G. Chen / Chaos, Solitons and Fractals 33 (2007) 171–177 173
U ¼
U11 U12 U13 U14 U15
UT12 U22 U23 U24 U25
UT13 UT
23 U33 U34 U35
UT14 UT
24 UT34 U44 U45
UT15 UT
25 UT35 UT
45 U55
26666664
37777775< 0 ð8Þ
and
W ¼X N
NT Z
� �P 0; ð9Þ
where
U11 ¼ H 1C þ CTH T1 þ N 1 þ N T
1 þ Qþ hX 11;
U12 ¼ P þ H 1 þ CTH T2 þ N T
2 þ hX 12;
U13 ¼ N T3 � N 1 þ hX 13;
U14 ¼ �H 1Aþ LT þ NT4 þ hX 14;
U15 ¼ �H 1Bþ N T5 þ hX 15;
U22 ¼ H 2 þ H T2 þ hZ þ hX 22;
U23 ¼ �N 2 þ hX 23;
U24 ¼ D� H 2Aþ hX 24;
U25 ¼ �H 2Bþ hX 25;
U33 ¼ �ð1� lÞQ� N 3 � N T3 þ hX 33;
U34 ¼ �N T4 þ hX 34;
U35 ¼ LS � NT5 þ hX 35;
U44 ¼ W � 2T þ hX 44;
U45 ¼ hX 45;
U55 ¼ �ð1� lÞW � 2S þ hX 55;
L ¼ diagðL1; L2; . . . ; LnÞ.
Proof. Construct the following Lyapunov–Krasovskii functional:
V ðzðtÞÞ ¼ V 1ðzðtÞÞ þ V 2ðzðtÞÞ þ V 3ðzðtÞÞ;
V 1ðzðtÞÞ ¼ zTðtÞPzðtÞ þ 2Xn
j¼1
dj
Z zj
0
fjðsÞds;
V 2ðzðtÞÞ ¼Z t
t�dðtÞ½zTðsÞQzðsÞ þ f TðzðsÞÞWf ðzðsÞÞ�ds;
V 3ðzðtÞÞ ¼Z 0
�h
Z t
tþh_zTðsÞZ _zðsÞdsdh;
ð10Þ
where P = PT > 0, Q = QT P 0, W = WT P 0, Z = ZT > 0, and D = diag(d1,d2, . . . ,dn) P 0 are to be determined.For any appropriately dimensional matrices H ¼ ½HT
1 HT2 �
T, the following holds:
0 ¼ 2½zTðtÞH 1 þ _zTðtÞH 2�½_zðtÞ þ CzðtÞ � Af ðzðtÞÞ � Bf ðzðt � dðtÞÞÞ�. ð11Þ
Using the Leibniz–Newton formula, for any appropriately dimensional matrix N, the following is also true:
0 ¼ 2fTðtÞN zðtÞ � zðt � dðtÞÞ �Z t
t�dðtÞ_zðsÞds
" #; ð12Þ
where fðtÞ ¼ ½ zTðtÞ _zTðtÞ zTðt � dðtÞÞ f TðzðtÞÞ f Tðzðt � dðtÞÞÞ �T.
174 H. Liu, G. Chen / Chaos, Solitons and Fractals 33 (2007) 171–177
In addition, for any semi-positive definite matrix
X ¼ X T ¼
X 11 X 12 X 13 X 14 X 15
X T12 X 22 X 23 X 24 X 25
X T13 X T
23 X 33 X 34 X 35
X T14 X T
24 X T34 X 44 X 45
X T15 X T
25 X T35 X T
45 X 55
26666664
37777775
P 0;
the following holds:
0 6 hfTðtÞX fðtÞ �Z t
t�dðtÞfTðtÞX fðtÞds
" #. ð13Þ
It is clear from (7) that
fkðzkðtÞÞ½fkðzkðtÞÞ � LkzkðtÞ� 6 0; k ¼ 1; 2; . . . ; n ð14Þ
and
fkðzkðt � dðtÞÞÞ � ½fkðzkðt � dðtÞÞÞ � Lkzkðt � dðtÞÞ� 6 0; k ¼ 1; 2; . . . ; n. ð15Þ
So, for any T = diag(t1, t2, . . . , tn) P 0, and S = diag(s1, s2, . . . , sn) P 0, it follows from (14) and (15) that
0 6 �2Xn
k¼1
tkfkðzkðtÞÞ½fkðzkðtÞÞ � LkzkðtÞ� � 2Xn
k¼1
skfkðzkðt � dðtÞÞÞ½fkðzkðt � dðtÞÞÞ � Lkzkðt � dðtÞÞ�
¼ 2 zTðtÞLTf ðzðtÞÞ � f TðzðtÞÞTf ðzðtÞÞ�
þzTðt � dðtÞÞLSf ðzðt � dðtÞÞÞ � f Tðzðt � dðtÞÞÞSf ðzðt � dðtÞÞÞ�. ð16Þ
Calculating the derivatives of Vi(z(t)) (i = 1,2,3) along the trajectories of the system (5) yields
_V 1ðzðtÞÞ ¼ 2zTðtÞP _zðtÞ þ 2Xn
k¼1
dkfkðzkðtÞÞ_zkðtÞ
¼ 2zTðtÞP _zðtÞ þ 2f TðzðtÞÞD_zðtÞ; ð17Þ_V 2ðzðtÞÞ ¼ ½zTðtÞQzðtÞ þ f TðzðtÞÞWf ðzðtÞÞ�
� ð1� _sðtÞÞ zTðt � dðtÞÞQzðt � dðtÞÞ þ f Tðzðt � dðtÞÞÞWf ðzðt � dðtÞÞÞ� �
6 ½zTðtÞQzðtÞ þ f TðzðtÞÞWf ðzðtÞÞ� � ð1� lÞ zTðt � dðtÞÞQzðt � dðtÞÞ þ f Tðzðt � dðtÞÞÞWf ðzðt � dðtÞÞÞ� �
;
ð18Þ
_V 3ðzðtÞÞ ¼ h_zTðtÞZ _zðtÞ �Z t
t�h_zTðsÞZ _zðsÞds
6 h_zTðtÞZ _zðtÞ �Z t
t�dðtÞ_zTðsÞZ _zðsÞds. ð19Þ
Adding the terms on the right of Eqs. (11)–(13) and (16) to _V ðzðtÞÞ yields:
_V ðzðtÞÞ 6 fTðtÞUfðtÞ �Z t
t�dðtÞgTðt; sÞWgðt; sÞds; ð20Þ
where gðt; sÞ ¼ ½fTðtÞ_zTðsÞ�T and U and W are defined in (8) and (9), respectively. If U < 0 and W P 0,_V ðzðtÞÞ < �ekzðtÞk2 for a sufficiently small e > 0 such that system (5) is asymptotically stable. h
Remark 2. Theorem 1 reduces to Theorem 1 in [19] by setting N = 0, X = 0 and Z = eI where e > 0 is a sufficient smallscalar. So, by choosing suitable N, X, and Z, Theorem 1 could overcome the conservatism of Theorem 1 in [19].
Remark 3. Theorem 1 in [19] is only applicable to systems with l < 1. On the contrary, Theorem 1 is valid to any l. Infact, by setting Q = 0 and W = 0, an easy delay-dependent and rate-independent criterion can be derived as follows.
H. Liu, G. Chen / Chaos, Solitons and Fractals 33 (2007) 171–177 175
Corollary 4. For given scalars h P 0 and d, the origin of system (5) with (6) and a time-varying delay satisfying
condition (2) is globally asymptotically stable if there exist P = PT > 0, Z = ZT > 0, D = diag(d1,d2, . . . , dn) P 0, T =
diag (t1, t2, . . . , tn) P 0, S = diag(s1, s2, . . . , sn) P 0, X ¼ X T ¼
X 11 X 12 X 13 X 14 X 15
X T12 X 22 X 23 X 24 X 25
X T13 X T
23 X 33 X 34 X 35
X T14 X T
24 X T34 X 44 X 45
X T15 X T
25 X T35 X T
45 X 55
266664
377775 P 0, and any appropriately
dimensional matrices H ¼ ½H T1 H T
2 �T
and N ¼ ½N T1 N T
2 N T3 NT
4 NT5 �
Tsuch that the following LMIs (21) and (9) are
feasible:
U ¼
U11 U12 U13 U14 U15
UT12 U22 U23 U24 U25
UT13 UT
23 U33 U34 U35
UT14 UT
24 UT34 U44 U45
UT15 UT
25 UT35 UT
45 U55
26666664
37777775< 0; ð21Þ
where
U33 ¼ �N 3 � N T3 þ hX 33;
U55 ¼ �2S þ hX 55
and Uij, i = 1,2, . . . , 5, i 6 j 6 5 are defined in Theorem 1.
Remark 5. A parameter-dependent Lyapunov functional can be used to studied the robust stability for system (1) withpolytopic-type uncertainties similar to [23]. On the other hand, the criterion is also easy to extend to system (1) withtime-varying structured uncertainties similar to [24].
4. Numerical examples
In this section, two examples are given to demonstrate the benefits of our method.
Example 6. Consider the delayed neural network (1) with
C ¼
1:2769 0 0 0
0 0:6231 0 0
0 0 0:9230 0
0 0 0 0:4480
266664
377775;
A ¼
�0:0373 0:4852 �0:3351 0:2336
�1:6033 0:5988 �0:3224 1:2352
0:3394 �0:0860 �0:3824 �0:5785
�0:1311 0:3253 �0:9534 �0:5015
266664
377775;
B ¼
0:8674 �1:2405 �0:5325 0:0220
0:0474 �0:9164 0:0360 0:9816
1:8495 2:6117 �0:3788 0:8428
�2:0413 0:5179 1:1734 �0:2775
266664
377775;
L1 ¼ 0:1137; L2 ¼ 0:1279; L3 ¼ 0:7994; L4 ¼ 0:2368.
The delay-independent criteria in [9,19] fail to conclude whether this system is asymptotically stable or not. In [20],the obtained upper bound of a constant delay, h, which ensures that the system is asymptotically stable, is 1.4224. Onthe contrary, Theorem 1 yields a larger h = 3.5841. Moreover, Theorem 1 is applicable to l 5 0, even l P 1. The cor-responding values derived by Theorem 1 is listed in Table 1.
Table 2Calculation results for Example 7
l 0 0.1 0.5 0.9 Any l
Theorem 1 2.1423 1.9868 1.4437 0.9895 –Corollary 4 – – – – 0.8969
Table 1Calculation results for Example 6
l 0 0.1 0.5 0.9 Any l
Theorem 1 3.5841 3.2775 2.1502 1.3164 –Corollary 4 – – – – 1.2598
176 H. Liu, G. Chen / Chaos, Solitons and Fractals 33 (2007) 171–177
Example 7. Consider the delayed neural network (1) with
C ¼4:1989 0 0
0 0:7160 0
0 0 1:9985
264
375; A ¼ 0; B ¼
�0:1052 �0:5069 �0:1121
�0:0257 �0:2808 0:0212
0:1205 �0:2153 0:1315
264
375;
L1 ¼ 0:4219; L2 ¼ 3:8993; L3 ¼ 1:0160.
The delay-independent criteria in [9,19] fail to conclude whether this system is asymptotically stable or not. Theobtained upper bounds of a constant delay, h, which ensures that the system is asymptotically stable in [20,21] are1.7484 and 1.7644, respectively. On the contrary, Theorem 1 yields a larger h = 2.1423. Moreover, Theorem 1 is appli-cable to l 5 0, even l P 1. The corresponding values derived by Theorem 1 is listed in Table 2.
5. Conclusion
This paper investigates the delay-dependent stability problem for neural networks with a time-varying delay. Thefree-weighting matrix method proposed in [22–25] is employed to derive an LMI-based criterion which avoids therestriction of the derivative of a time-varying delay. Finally, numerical examples are given to demonstrate the effective-ness of the presented stability criterion and its improvement over the existing results.
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