- Email: [email protected]
Novel robust stability criteria of uncertain neutral systems with discrete and distributed delays
Chaos, Solitons and Fractals 40 (2009) 771–777 www.elsevier.com/locate/chaos
Novel robust stability criteria of uncertain neutral systems with discrete and distributed delays Lianglin Xiong a
a,*
, Shouming Zhong a, Junkang Tian
b
School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China b College of Science, Southwest Petroleum University, Nanchong, Sichuan 63700, PR China Accepted 13 August 2007
Abstract The robust stability of uncertain linear neutral systems with discrete and distributed delays is investigated. The uncertainties under consideration are norm bounded, and possibly time-varying. A new method based on the equivalent equation of zero and the Leibniz–Newton formula in the derivative of the Lyapunov–Krasovskii function is put forward, the proposed stability criteria are formulated in the form of a linear matrix inequality and it is easy to check the robust stability of the considered systems. Numerical examples illustrate that the proposed criteria are effective and achieve significant improvement over the results proposed in some previous paper. 2007 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, many books and papers have dealt with the delay differential systems of retarded type or neutral type and have obtained a lot of results with respect to stability and stabilization of the systems, for example [1–14], etc. Current efforts on the problem of stability of time-delay systems of neutral type can be divided into two categories, namely delay-independent stability criteria and delay-dependent stability criteria. For linear time-delay systems of neutral type, some delay-independent stability conditions were obtained. They were formulated in terms of matrix measure and matrix norm [6–8] or the existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation [9]. Although these conditions are easy to check, they required matrix measure to be negative or the parameters to be tuned. A model transformation technique is often used to transform the neutral system with discrete delay to a neutral system with a distributed delay, and delay-dependent stability criteria are obtained by employing Lyapunov–Krasovskii functionals [11–14]. Generally speaking, these results are usually less conservative than the delay-independent stability ones. Some of these results are still conservative since some model transformations will introduce additional dynamics as discussed in [15]. Recently, a new descriptor model transformation and a corresponding Lyapunov–Krasovskii functional have been introduced for stability analysis of systems with delays as outlined in [16]. The advantage of this transformation is to *
Corresponding author. E-mail address: [email protected] (L. Xiong).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.023
772
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
transform the original system to an equivalent descriptor form representation and will not introduce additional dynamics in the sense defined in [15]. The results in [16] are less conservative than some existing ones, and they can be improved by employing the decomposition technique to get a larger bound for discrete delays. Without introducing additional dynamics in the sense defined in [15], this paper presents a novel method of dealing with the problem of delay-dependent stability of neutral systems with distributed delays, which is similar to [17]. This method R t employs free weighting matrices to express the influence of, the relationship between the term x(t–r) and xðtÞ tr x_ ðsÞds, and the transformation of the systems to the one side. The new criterion is based on LMIs, which makes the free weighting matrices easy to select. Since this criterion is both discrete delay-dependent and distributed delay-dependent, it is less conservative than some previous methods for the systems. The criterion obtained is then extended to neutral systems with time-varying uncertainties. Finally, numerical examples are presented to illustrate the improvement over some existing methods.
2. Problem statement Consider the following linear neutral system with discrete and distributed delays: ( Rt x_ ðtÞ C x_ ðt hÞ ¼ AðtÞxðtÞ þ BðtÞxðt rÞ þ DðtÞ ts xðsÞds xðt0 þ hÞ ¼ uðhÞ; 8h 2 ½ maxfh; r; sg; 0
ð1Þ
where x(t) 2 Rn is the state, h > 0 is a constant neutral delay, r > 0 is a constant discrete delay and s > 0 is a constant distributed delay, u(Æ) is a continuous vector valued initial function, C 2 Rn·n is a known constant matrix, and A(t) 2 Rn·n, B(t) 2 Rn·n, D(t) 2 Rn·n are uncertain matrices, not known completely, except that they are within a compact set X which we will refer to as the uncertainty set ðAðtÞ; BðtÞ; DðtÞÞ 2 X Rn3n
for all t 2 ½0; 1Þ:
3. Main result Define the operator Dðxt Þ ¼ xðtÞ Cxðt hÞ Throughout this paper, we assume that A1. All the eigenvalues of matrix C are inside the unit circle. It implies that the operator D(xt) is stable. For the stability of systems (1) we have the following result. Theorem 1. Under A1, the systems described by (1) is asymptotically stable if there exist symmetric positive define matrices P, R, Q1, Q2, X66, Y66 and any appropriate dimensional matrices Xij, Yij(i, j = 1, . . ., 6) such that the following LMIs hold: 1 0 /11 /12 /13 /14 /15 C B T B /12 /22 /23 /24 /25 C C B C B T T C B / / / / / ð2Þ / ¼ B 13 33 34 35 C < 0 23 C B T T T C B/ @ 14 /24 /34 /44 /45 A 0
/T15
/T25
/T35
/T45
X 11
X 12
X 13
X 14
X 15
X 22
X 23
X 24
X 25
X T23
X 33
X 34
X 35
X T24
X T34
X 44
X 45
X T25
X T35
X T45
X 55
X T26
X T36
X T46
X T56
B T B X 12 B B T B X 13 B w¼B BXT B 14 B T BX @ 15 X T16
/55 X 16
1
C X 26 C C C X 36 C C CP0 X 46 C C C X 56 C A X 66
ð3Þ
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
0
Y 11 BYT B 12 B T B Y 13 N¼B BYT B 14 B T @ Y 15 Y T
Y 12 Y 22 Y T23 Y T24
Y 13 Y 23 Y 33 Y T34
Y 14 Y 24 Y 34 Y 44
Y 15 Y 25 Y 35 Y 45
Y T25 D ðtÞY T26
Y T35 Y T
Y T45 D ðtÞY T46
Y 55 DT ðtÞY T56
T
16
36
T
773
1 Y 16 Y 26 DðtÞ C C C C Y 36 CP0 Y 46 DðtÞ C C C Y 56 DðtÞ A Y 66
ð4Þ
where /11 ¼ AT ðtÞP þ PAðtÞ þ X 16 þ X T16 þ Y 16 AðtÞ þ AT ðtÞY T16 þ Q1 þ Q2 þ rX 11 þ sY 11 þ sY 66 /12 ¼ PBðtÞ X 16 þ X T26 þ AT ðtÞY T26 þ Y 16 BðtÞ þ rX 12 þ sY 12 /13 ¼ AT ðtÞPC þ X T36 þ AT ðtÞY T36 þ rX 13 þ sY 13 /14 ¼ X T46 Y 16 þ AT ðtÞY T46 þ rX 14 þ sY 14 /15 /22 /23 /24
¼ X T56 þ AT ðtÞY T56 þ Y 16 C þ rX 15 þ sY 15 ¼ Q1 X 26 X T26 þ Y 26 BðtÞ þ BT ðtÞY T26 þ rX 22 þ sY 22 ¼ BT ðtÞPC X T36 þ BT ðtÞY T36 þ rX 23 þ sY 23 ¼ X T46 Y 26 þ BT ðtÞY T46 þ rX 24 þ sY 24
/25 /33 /34 /35
¼ X T56 þ BT ðtÞY T56 þ Y 26 C þ rX 25 þ sY 25 ¼ Q þ rX 33 þ sY 33 ¼ Y 36 þ rX 34 þ sY 34 ¼ Y 36 C þ rX 35 þ sY 35
/44 ¼ R Y 46 Y T46 þ rX 66 þ rX 44 þ sY 44 /45 ¼ Y T56 þ Y 46 C þ rX 45 þ sY 45 /55 ¼ R þ Y 56 C þ C T Y T56 þ rX 55 þ sY 55 Y 16 ¼ PDðtÞ Y 16 DðtÞ Y 36 ¼ C T PDðtÞ Y 36 DðtÞ Proof. Choose a Lyapunov–Krasovskii functional candidate for systems (1) as Z t Z t Z t Z x_ T ðsÞR_xðsÞ ds þ V ðxt Þ ¼ DT ðxt ÞPDðxt Þ þ xT ðsÞQ1 xðsÞ ds þ xT ðsÞQ2 xðsÞ ds þ þ
Z
0
s
Z
th
tr
th
0 r
Z
t
x_ T ðsÞX 66 x_ ðsÞ ds
tþh
t
xT ðsÞY 66 xðsÞ ds
tþh
where P, R, Q1, Q2, X66, Y66 are symmetric positive definite matrices, and all of these matrices need to be determined. Calculating the derivative of V(xt) along the solutions of (1) yields _V ðxt Þ ¼ 2 xT ðtÞAT ðtÞPxðtÞ þ xT ðt rÞBT ðtÞPxðtÞ xT ðtÞAT ðtÞPCxðt hÞ xT ðt rÞBT ðtÞPCxðt hÞ Z t Z t T T T T x ðsÞD ðtÞPCxðt hÞ ds þ x ðsÞD ðtÞPxðtÞ ds þ x_ T ðtÞR_xðtÞ x_ T ðt hÞR_xðt hÞ þ xT ðtÞQ1 xðtÞ ts
ts
xT ðt rÞQ1 xðt rÞ þ xT ðtÞQ2 xðtÞ xT ðt hÞQ2 xðt hÞ þ rx_ T ðtÞX 66 x_ ðtÞ Z t Z t x_ T ðsÞX 66 x_ ðsÞ ds þ sxT ðtÞY 66 xðtÞ xT ðsÞY 66 xðsÞ ds tr
ts
Using the Leibniz–Newton formula, we can write Z t x_ ðsÞ ds ¼ 0 xðtÞ xðt rÞ
ð5Þ
tr
According to (5), for any appropriate dimensional matrices Xi6 (i = 1, . . ., 5) the following equation hold Z t x_ ðsÞ ds ¼ 0 2fxT ðtÞX 16 þ xT ðt rÞX 26 þ xT ðt hÞX 36 þ x_ T ðtÞX 46 þ x_ T ðt hÞX 56 g xðtÞ xðt rÞ tr
ð6Þ
774
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
In addition, according to (1), for any appropriate dimensional matrices Yi6 (i = 1, . . ., 5) one has 2fxT ðtÞY 16 þ xT ðt rÞY 26 þ xT ðt hÞY 36 þ x_ T ðtÞY 46 þ x_ T ðt hÞY 56 g AðtÞxðtÞ þ BðtÞxðt rÞ þ DðtÞ Z t xðsÞ ds x_ ðtÞ þ C x_ ðt hÞ ¼ 0
ð7Þ
ts
any appropriately dimensional matrices Xij, Yij (i, j = 1, . . ., 6), the following equation also
On the other hand, for holds: 1T 0 0 K11 xðtÞ B xðt rÞ C B KT C B 12 B C B B B xðt hÞ C B KT13 C B B @ x_ ðtÞ A @ KT14 KT15 x_ ðt hÞ
K12 K22 KT23 KT24 KT25
K13 K23 K33 KT34 KT35
K14 K24 K34 K44 KT45
1 10 K15 xðtÞ C B K25 C CB xðt rÞ C C CB K35 CB xðt hÞ C ¼ 0 C CB K45 A@ x_ ðtÞ A
ð8Þ
x_ ðt hÞ
K55
where Kij ¼ rðX ij X ij Þ þ sðY ij Y ij Þ;
i ¼ 1; . . . ; 5;
i 6 j 6 5:
Then, we add the terms on the left sides of Eqs. (6)–(8) to V_ ðxt Þ; and consider the fact that, for any m P 0 and any function f(t), Z t f ðtÞ ds ¼ mf ðtÞ tm
V_ ðxt Þ can be expressed as follows: Z t Z nT2 wn2 ds V_ ðxt Þ ¼ nT1 /n1 tr
t
nT3 Nn3 ds
ts
where n1 ¼ ½ xT ðtÞ n2 ¼ ½ xT ðtÞ n3 ¼ ½ xT ðtÞ
xT ðt rÞ xT ðt rÞ xT ðt rÞ
xT ðt hÞ xT ðt hÞ xT ðt hÞ
x_ T ðtÞ x_ T ðtÞ x_ T ðtÞ
T
x_ T ðt hÞ T x_ T ðt hÞ x_ T ðsÞ T T T x_ ðt hÞ x ðsÞ
and /, w, N are defined in (2)–(4). If / < 0; w P 0; N P 0; V_ ðtÞ < 0 for n1 5 0. This implies that the system (1) with stable operator D(xt) is asymptotically stable by theorem 9.8.1 in [1]. This completes the proof. Now we consider the norm bounded uncertainty described by AðtÞ ¼ A þ DAðtÞ;
BðtÞ ¼ B þ DBðtÞ;
DðtÞ ¼ D þ DDðtÞ
ð9Þ
where, ½ DAðtÞ where F(t) 2 R
DBðtÞ p·q
DDðtÞ ¼ LF ðtÞ½ Ea
Eb
Ed
ð10Þ
is an unknown real and possibly time-varying matrix with Lebesgue measurable elements satisfying
rmax ðF ðtÞÞ 6 1
ð11Þ
and L, Ea, Eb, and Ed are known real constant matrices which characterize how the uncertainty enters the nominal matrices A, B, D. h Now we state the following result. Theorem 2. Under A1, the systems described by (1), with uncertainty described by (9)–(11) is asymptotically stable if there ~ 2 ; X~ 66 ; Y~ 66 and any appropriate dimensional matrices ~ 1; Q ~ Q exist symmetric positive define matrices P~ ; R; ~ ~ X ij ; Y ij ði; j ¼ 1; . . . ; 6Þ such that the following LMIs hold: 1 0~ X 11 X~ 12 X~ 13 X~ 14 X~ 15 X~ 16 B X~ T X~ 22 X~ 23 X~ 24 X~ 25 X~ 26 C C B 12 B X~ T X~ T X~ X~ 34 X~ 35 X~ 36 C C 33 13 23 ~¼B ð12Þ w CP0 B ~T ~T ~T B X 14 X 24 X 34 X~ 44 X~ 45 X~ 46 C C B @ X~ T X~ T X~ T X~ T X~ 55 X~ 56 A 15 25 35 45 X~ T X~ T X~ T X~ T X~ T X~ 66 16
26
36
46
56
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
0
~ 11 / B ~T B /12 B B/ B ~ T13 B ~ ~T /¼B B /14 B ~T B /15 B B ~T @/ 0
~ 12 / ~ 22 / ~T / 23
~ 13 / ~ 23 / ~ 33 /
~ 14 / ~ 24 / ~ 34 /
~ 15 / ~ 25 / ~ 35 /
~ 16 / ~ Y 26 L ~ 36 /
~T / 24 ~T /
~T / 34 ~T /
~T / 44 ~T /
~ 45 / ~ 55 /
Y~ 46 L Y~ 56 L
LT Y~ T26
~T / 36
LT Y~ T46
LT Y~ T56
I
ETb
0
0
0
25
16 ETa
Y~ 11 B ~T B Y 12 B B Y~ T B 13 ~¼B N B Y~ T14 B B Y~ T B 15 B ~T @ N16 ~T N
35
Y~ 12 Y~ 22 Y~ T23 Y~ T24 Y~ T 25
DT Y T26 LT Y~ T26
17
45
Y~ 13 Y~ 23 Y~ 33 Y~ T34 Y~ T35 ~T N 36 ~T N 37
Y~ 14 Y~ 24 Y~ 34 Y~ 44 Y~ T
0 Y~ 15 Y~ 25 Y~ 35 Y~ 45 Y~ 55
DT Y T46 LT Y~ T46
DT Y T56 LT Y~ T56
45
1 Ea C Eb C C 0 C C C 0 C C60 C 0 C C C 0 A I ~ 16 N Y~ 26 D ~ 36 N Y~ 46 D Y~ 56 D ~ 66 N 0
775
ð13Þ
1 ~ 17 N C Y~ 26 L C C ~ 37 C N C C Y~ 46 L C P 0 C Y~ 56 L C C C 0 A
ð14Þ
I
where, ~ 11 ¼ AT P~ þ P~ A þ X~ 16 þ X~ T þ Y~ 16 A þ AT Y~ T þ Q ~1 þ Q ~ 2 þ rX~ 11 þ sY~ 11 þ sY~ 66 / 16 16 ~ 12 ¼ P~ B X~ 16 þ X~ T þ AT Y~ T þ Y~ 16 B þ rX~ 12 þ sY~ 12 / 26 26 ~ 13 ¼ AT P~ C þ X~ T þ AT Y~ T þ rX~ 13 þ sY~ 13 / 36
36
~ 14 ¼ X~ T þ AT Y~ T Y~ 16 þ rX~ 14 þ sY~ 14 / 46 46 ~ 15 ¼ X~ T þ AT Y~ T þ Y~ 16 C þ rX~ 15 þ sY~ 15 / 56 56 ~ 16 ¼ Y~ 16 L þ P~ L / ~ 22 ¼ Q ~ 1 X~ 26 X~ T þ Y~ 26 B þ BT Y~ T þ rX~ 22 þ sY~ 22 / 26 26 ~ 23 ¼ BT P~ C X~ T þ BT Y~ T þ rX~ 23 þ sY~ 23 / 36 36 ~ 24 ¼ X~ T Y~ 26 þ BT Y~ T þ rX~ 24 þ sY~ 24 / 46
46
~ 25 ¼ X~ T þ BT Y~ T þ Y~ 26 C þ rX~ 25 þ sY~ 25 / 56 56 ~ 33 ¼ Q ~ þ rX~ 33 þ sY~ 33 / ~ 34 ¼ Y~ 36 þ rX~ 34 þ sY~ 34 / ~ 35 ¼ Y~ 36 C þ rX~ 35 þ sY~ 35 / ~ 36 ¼ Y~ 36 L C T P~ L / ~ 44 ¼ R ~ Y~ 46 Y~ T þ rX~ 66 þ rX~ 44 þ sY~ 44 / 46
~ 45 ¼ Y~ T þ Y~ 46 C þ rX~ 45 þ sY~ 45 / 56 ~ 55 ¼ R ~ þ Y~ 56 C þ C T Y~ T þ rX~ 55 þ sY~ 55 / 56 ~ 66 ¼ Y~ 66 ET Ed N ~ 16 ¼ P~ D Y~ 16 D N d ~ 36 ¼ C T P~ D Y~ 36 D ~ 17 ¼ P~ L Y~ 16 L N N T ~ 37 ¼ C P~ L Y~ 36 L N Remark. The Proof of Theorem 2 is similar to the approach in [16] with lemma 2.4 in [18] and the Schur’s complement in [19]. It’s omitted. 4. Numerical examples Example 1. Consider the following uncertain neutral systems with discrete and distributed delays system: Z t xðsÞ ds x_ ðtÞ C x_ ðt hÞ ¼ AðtÞxðtÞ þ BðtÞxðt rÞ þ DðtÞ ts
ð15Þ
776
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
with 0:9 0 1 0:12 AðtÞ ¼ BðtÞ ¼ 0 0:9 0:12 1 0 0 0:12 0:12 C¼ DðtÞ ¼ 0 0 0:12 0:12 Using the discrete delay-dependent result given in Theorem 2 of [20], it was found that the upper bound of r must be less than 0.9086, while in [21], it is found that the admissible bound of the discrete delay r for the stability of system (15) are 1.8302 when s = 1 is chosen. Applying Theorem 1 of this paper, the upper bound of the delay r has been obtained as 2.4128 in the case of s = 1. Example 2. Consider the following uncertain neutral systems: x_ ðtÞ C x_ ðt rÞ ¼ ðA þ AðtÞÞxðtÞ þ ðB þ BðtÞÞxðt rÞ 0:9 0:2 1:1 0:2 0:2 0 A¼ B¼ C¼ 0:1 0:9 0:1 1:1 0:2 0:1
ð16Þ
For A(t) = 0 and B(t) = 0 (i.e. nominal system), system (16) reduces to the system discussed in [14]. Using the criterion in this paper, the maximum value of rmax for the nominal system to be asymptotically stable is rmax = 1.6524. By the criteria in [12,14,16], the nominal system is asymptotically stable for any r satisfying r 6 0.71, r 6 0.3, r 6 0.74, respectively. This example shows again that the stability criterion in this paper gives a much less conservative result than these in [12,14,16]. 5. Conclusion The robust stability problem for uncertain neutral systems with discrete and distributed delays has been investigated. Stability criteria have been obtained. Numerical examples have shown significant improvements over some existing results. References [1] Hale J, Verduyn Lunel SM. Introduction to functional differential equations. New York: Springer; 1993. [2] Yang Xiao-Song. Liapunov asymptotic stability and Zhukovskij asymptotic stability. Chaos, Solitons & Fractals 2000;13(11):1995–9. [3] Tu Fenghua, Liao Xiaofeng, Zhang Wei. Delay-dependent asymptotic stability of a two-neuron system with different time delays. Chaos, Solitons & Fractals 2006;2(28):437–47. [4] Park JH, Kwon O. LMI optimization approach to stabilization of time-delay chaotic systems. Chaos, Solitons & Fractals 2005;2(23):445–50. [5] Liu Xingwen, Zhang Hongbin. New stability criterion of uncertain systems with time-varying delay. Chaos, Solitons & Fractals 2005;5(26):1343–8. [6] Xiong Wenjun, Liang Jinling. Novel stability criteria for neutral systems with multiple time delays. Chaos, Solitions & Fractals 2007;5(32):1735–41. [7] Li Hong, Li Hou-biao, Zhong Shou-ming. Stability of neutral type descriptor system with mixed delays. Chaos, Solitions & Fractals 2007;5(33):1796–800. [8] Hu G Di, Hu G Da. Some simple stability criteria of neutral delay-differential systems. Appl Math Comp 1996;80:257–71. [9] Slemrod M, Infante EF. Asymptotic stability criteria for linear systems of differential equations of neutral type and their discrete analogues. J Math Anal Appl 1972;38:399–415. [10] Huaicheng Yan, Xinhan Huang, Min Wang, Hao Zhang. New delay-dependent stability criteria of uncertain linear systems with multiple time-varying state delays. Chaos, Solitons & Fractals, in press [available online 27.10.2006]. [11] Kolmanovskii VB, Richard J-P. Stability of some linear systems with delays. IEEE Trans Automat Contr 1999;44:984–9. [12] Niculescu, S.-I. Further remarks on delay-dependent stability of linear neutral systems. Proceedings of MTNS 2000, Perpigan, France. [13] Han Q-L. Robust stability of uncertain delay-differential systems of neutral type. Automatica 2002;38:719–23. [14] Lien C-H, Yu KW, Hsieh JG. Stability conditions for a class of neutral systems with multiple time delays. J Math Anal Appl 2000;245:20–7. [15] Gu K, Niculescu S-I. Further remarks on additional dynamics in various model transformations of linear delay systems. IEEE Trans Automat Contr 2001;46:497–500.
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[16] Qing-lon Han. A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 2004;40:1791–6. [17] He Y, Wu M, She JH, Liu GP. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst Contr Lett 2004;51:57–65. [18] Xie L. Output feedback H control of systems with parameter uncertainty. Int J Contr 1996;63:741–50. [19] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in systems and control theory. Philadelphia: SIAM; 1994. [20] Chen JD, Lien CH, Fan KK, Chou JH. Criteria for asymptotic stability of a class of neutral systems via a LMI approach. IEE Proc Contr Theory Appl 2001;148(6):442–7. [21] Yue D, Won S, Kwon O. Delay-dependent stability of neutral systems with time delay: an LMI approach. IEE Proc Contr Theory Appl 2003;150:23–7.
Novel robust stability criteria of uncertain neutral systems with discrete and distributed delays Lianglin Xiong a
a,*
, Shouming Zhong a, Junkang Tian
b
School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China b College of Science, Southwest Petroleum University, Nanchong, Sichuan 63700, PR China Accepted 13 August 2007
Abstract The robust stability of uncertain linear neutral systems with discrete and distributed delays is investigated. The uncertainties under consideration are norm bounded, and possibly time-varying. A new method based on the equivalent equation of zero and the Leibniz–Newton formula in the derivative of the Lyapunov–Krasovskii function is put forward, the proposed stability criteria are formulated in the form of a linear matrix inequality and it is easy to check the robust stability of the considered systems. Numerical examples illustrate that the proposed criteria are effective and achieve significant improvement over the results proposed in some previous paper. 2007 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, many books and papers have dealt with the delay differential systems of retarded type or neutral type and have obtained a lot of results with respect to stability and stabilization of the systems, for example [1–14], etc. Current efforts on the problem of stability of time-delay systems of neutral type can be divided into two categories, namely delay-independent stability criteria and delay-dependent stability criteria. For linear time-delay systems of neutral type, some delay-independent stability conditions were obtained. They were formulated in terms of matrix measure and matrix norm [6–8] or the existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation [9]. Although these conditions are easy to check, they required matrix measure to be negative or the parameters to be tuned. A model transformation technique is often used to transform the neutral system with discrete delay to a neutral system with a distributed delay, and delay-dependent stability criteria are obtained by employing Lyapunov–Krasovskii functionals [11–14]. Generally speaking, these results are usually less conservative than the delay-independent stability ones. Some of these results are still conservative since some model transformations will introduce additional dynamics as discussed in [15]. Recently, a new descriptor model transformation and a corresponding Lyapunov–Krasovskii functional have been introduced for stability analysis of systems with delays as outlined in [16]. The advantage of this transformation is to *
Corresponding author. E-mail address: [email protected] (L. Xiong).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.023
772
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
transform the original system to an equivalent descriptor form representation and will not introduce additional dynamics in the sense defined in [15]. The results in [16] are less conservative than some existing ones, and they can be improved by employing the decomposition technique to get a larger bound for discrete delays. Without introducing additional dynamics in the sense defined in [15], this paper presents a novel method of dealing with the problem of delay-dependent stability of neutral systems with distributed delays, which is similar to [17]. This method R t employs free weighting matrices to express the influence of, the relationship between the term x(t–r) and xðtÞ tr x_ ðsÞds, and the transformation of the systems to the one side. The new criterion is based on LMIs, which makes the free weighting matrices easy to select. Since this criterion is both discrete delay-dependent and distributed delay-dependent, it is less conservative than some previous methods for the systems. The criterion obtained is then extended to neutral systems with time-varying uncertainties. Finally, numerical examples are presented to illustrate the improvement over some existing methods.
2. Problem statement Consider the following linear neutral system with discrete and distributed delays: ( Rt x_ ðtÞ C x_ ðt hÞ ¼ AðtÞxðtÞ þ BðtÞxðt rÞ þ DðtÞ ts xðsÞds xðt0 þ hÞ ¼ uðhÞ; 8h 2 ½ maxfh; r; sg; 0
ð1Þ
where x(t) 2 Rn is the state, h > 0 is a constant neutral delay, r > 0 is a constant discrete delay and s > 0 is a constant distributed delay, u(Æ) is a continuous vector valued initial function, C 2 Rn·n is a known constant matrix, and A(t) 2 Rn·n, B(t) 2 Rn·n, D(t) 2 Rn·n are uncertain matrices, not known completely, except that they are within a compact set X which we will refer to as the uncertainty set ðAðtÞ; BðtÞ; DðtÞÞ 2 X Rn3n
for all t 2 ½0; 1Þ:
3. Main result Define the operator Dðxt Þ ¼ xðtÞ Cxðt hÞ Throughout this paper, we assume that A1. All the eigenvalues of matrix C are inside the unit circle. It implies that the operator D(xt) is stable. For the stability of systems (1) we have the following result. Theorem 1. Under A1, the systems described by (1) is asymptotically stable if there exist symmetric positive define matrices P, R, Q1, Q2, X66, Y66 and any appropriate dimensional matrices Xij, Yij(i, j = 1, . . ., 6) such that the following LMIs hold: 1 0 /11 /12 /13 /14 /15 C B T B /12 /22 /23 /24 /25 C C B C B T T C B / / / / / ð2Þ / ¼ B 13 33 34 35 C < 0 23 C B T T T C B/ @ 14 /24 /34 /44 /45 A 0
/T15
/T25
/T35
/T45
X 11
X 12
X 13
X 14
X 15
X 22
X 23
X 24
X 25
X T23
X 33
X 34
X 35
X T24
X T34
X 44
X 45
X T25
X T35
X T45
X 55
X T26
X T36
X T46
X T56
B T B X 12 B B T B X 13 B w¼B BXT B 14 B T BX @ 15 X T16
/55 X 16
1
C X 26 C C C X 36 C C CP0 X 46 C C C X 56 C A X 66
ð3Þ
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
0
Y 11 BYT B 12 B T B Y 13 N¼B BYT B 14 B T @ Y 15 Y T
Y 12 Y 22 Y T23 Y T24
Y 13 Y 23 Y 33 Y T34
Y 14 Y 24 Y 34 Y 44
Y 15 Y 25 Y 35 Y 45
Y T25 D ðtÞY T26
Y T35 Y T
Y T45 D ðtÞY T46
Y 55 DT ðtÞY T56
T
16
36
T
773
1 Y 16 Y 26 DðtÞ C C C C Y 36 CP0 Y 46 DðtÞ C C C Y 56 DðtÞ A Y 66
ð4Þ
where /11 ¼ AT ðtÞP þ PAðtÞ þ X 16 þ X T16 þ Y 16 AðtÞ þ AT ðtÞY T16 þ Q1 þ Q2 þ rX 11 þ sY 11 þ sY 66 /12 ¼ PBðtÞ X 16 þ X T26 þ AT ðtÞY T26 þ Y 16 BðtÞ þ rX 12 þ sY 12 /13 ¼ AT ðtÞPC þ X T36 þ AT ðtÞY T36 þ rX 13 þ sY 13 /14 ¼ X T46 Y 16 þ AT ðtÞY T46 þ rX 14 þ sY 14 /15 /22 /23 /24
¼ X T56 þ AT ðtÞY T56 þ Y 16 C þ rX 15 þ sY 15 ¼ Q1 X 26 X T26 þ Y 26 BðtÞ þ BT ðtÞY T26 þ rX 22 þ sY 22 ¼ BT ðtÞPC X T36 þ BT ðtÞY T36 þ rX 23 þ sY 23 ¼ X T46 Y 26 þ BT ðtÞY T46 þ rX 24 þ sY 24
/25 /33 /34 /35
¼ X T56 þ BT ðtÞY T56 þ Y 26 C þ rX 25 þ sY 25 ¼ Q þ rX 33 þ sY 33 ¼ Y 36 þ rX 34 þ sY 34 ¼ Y 36 C þ rX 35 þ sY 35
/44 ¼ R Y 46 Y T46 þ rX 66 þ rX 44 þ sY 44 /45 ¼ Y T56 þ Y 46 C þ rX 45 þ sY 45 /55 ¼ R þ Y 56 C þ C T Y T56 þ rX 55 þ sY 55 Y 16 ¼ PDðtÞ Y 16 DðtÞ Y 36 ¼ C T PDðtÞ Y 36 DðtÞ Proof. Choose a Lyapunov–Krasovskii functional candidate for systems (1) as Z t Z t Z t Z x_ T ðsÞR_xðsÞ ds þ V ðxt Þ ¼ DT ðxt ÞPDðxt Þ þ xT ðsÞQ1 xðsÞ ds þ xT ðsÞQ2 xðsÞ ds þ þ
Z
0
s
Z
th
tr
th
0 r
Z
t
x_ T ðsÞX 66 x_ ðsÞ ds
tþh
t
xT ðsÞY 66 xðsÞ ds
tþh
where P, R, Q1, Q2, X66, Y66 are symmetric positive definite matrices, and all of these matrices need to be determined. Calculating the derivative of V(xt) along the solutions of (1) yields _V ðxt Þ ¼ 2 xT ðtÞAT ðtÞPxðtÞ þ xT ðt rÞBT ðtÞPxðtÞ xT ðtÞAT ðtÞPCxðt hÞ xT ðt rÞBT ðtÞPCxðt hÞ Z t Z t T T T T x ðsÞD ðtÞPCxðt hÞ ds þ x ðsÞD ðtÞPxðtÞ ds þ x_ T ðtÞR_xðtÞ x_ T ðt hÞR_xðt hÞ þ xT ðtÞQ1 xðtÞ ts
ts
xT ðt rÞQ1 xðt rÞ þ xT ðtÞQ2 xðtÞ xT ðt hÞQ2 xðt hÞ þ rx_ T ðtÞX 66 x_ ðtÞ Z t Z t x_ T ðsÞX 66 x_ ðsÞ ds þ sxT ðtÞY 66 xðtÞ xT ðsÞY 66 xðsÞ ds tr
ts
Using the Leibniz–Newton formula, we can write Z t x_ ðsÞ ds ¼ 0 xðtÞ xðt rÞ
ð5Þ
tr
According to (5), for any appropriate dimensional matrices Xi6 (i = 1, . . ., 5) the following equation hold Z t x_ ðsÞ ds ¼ 0 2fxT ðtÞX 16 þ xT ðt rÞX 26 þ xT ðt hÞX 36 þ x_ T ðtÞX 46 þ x_ T ðt hÞX 56 g xðtÞ xðt rÞ tr
ð6Þ
774
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
In addition, according to (1), for any appropriate dimensional matrices Yi6 (i = 1, . . ., 5) one has 2fxT ðtÞY 16 þ xT ðt rÞY 26 þ xT ðt hÞY 36 þ x_ T ðtÞY 46 þ x_ T ðt hÞY 56 g AðtÞxðtÞ þ BðtÞxðt rÞ þ DðtÞ Z t xðsÞ ds x_ ðtÞ þ C x_ ðt hÞ ¼ 0
ð7Þ
ts
any appropriately dimensional matrices Xij, Yij (i, j = 1, . . ., 6), the following equation also
On the other hand, for holds: 1T 0 0 K11 xðtÞ B xðt rÞ C B KT C B 12 B C B B B xðt hÞ C B KT13 C B B @ x_ ðtÞ A @ KT14 KT15 x_ ðt hÞ
K12 K22 KT23 KT24 KT25
K13 K23 K33 KT34 KT35
K14 K24 K34 K44 KT45
1 10 K15 xðtÞ C B K25 C CB xðt rÞ C C CB K35 CB xðt hÞ C ¼ 0 C CB K45 A@ x_ ðtÞ A
ð8Þ
x_ ðt hÞ
K55
where Kij ¼ rðX ij X ij Þ þ sðY ij Y ij Þ;
i ¼ 1; . . . ; 5;
i 6 j 6 5:
Then, we add the terms on the left sides of Eqs. (6)–(8) to V_ ðxt Þ; and consider the fact that, for any m P 0 and any function f(t), Z t f ðtÞ ds ¼ mf ðtÞ tm
V_ ðxt Þ can be expressed as follows: Z t Z nT2 wn2 ds V_ ðxt Þ ¼ nT1 /n1 tr
t
nT3 Nn3 ds
ts
where n1 ¼ ½ xT ðtÞ n2 ¼ ½ xT ðtÞ n3 ¼ ½ xT ðtÞ
xT ðt rÞ xT ðt rÞ xT ðt rÞ
xT ðt hÞ xT ðt hÞ xT ðt hÞ
x_ T ðtÞ x_ T ðtÞ x_ T ðtÞ
T
x_ T ðt hÞ T x_ T ðt hÞ x_ T ðsÞ T T T x_ ðt hÞ x ðsÞ
and /, w, N are defined in (2)–(4). If / < 0; w P 0; N P 0; V_ ðtÞ < 0 for n1 5 0. This implies that the system (1) with stable operator D(xt) is asymptotically stable by theorem 9.8.1 in [1]. This completes the proof. Now we consider the norm bounded uncertainty described by AðtÞ ¼ A þ DAðtÞ;
BðtÞ ¼ B þ DBðtÞ;
DðtÞ ¼ D þ DDðtÞ
ð9Þ
where, ½ DAðtÞ where F(t) 2 R
DBðtÞ p·q
DDðtÞ ¼ LF ðtÞ½ Ea
Eb
Ed
ð10Þ
is an unknown real and possibly time-varying matrix with Lebesgue measurable elements satisfying
rmax ðF ðtÞÞ 6 1
ð11Þ
and L, Ea, Eb, and Ed are known real constant matrices which characterize how the uncertainty enters the nominal matrices A, B, D. h Now we state the following result. Theorem 2. Under A1, the systems described by (1), with uncertainty described by (9)–(11) is asymptotically stable if there ~ 2 ; X~ 66 ; Y~ 66 and any appropriate dimensional matrices ~ 1; Q ~ Q exist symmetric positive define matrices P~ ; R; ~ ~ X ij ; Y ij ði; j ¼ 1; . . . ; 6Þ such that the following LMIs hold: 1 0~ X 11 X~ 12 X~ 13 X~ 14 X~ 15 X~ 16 B X~ T X~ 22 X~ 23 X~ 24 X~ 25 X~ 26 C C B 12 B X~ T X~ T X~ X~ 34 X~ 35 X~ 36 C C 33 13 23 ~¼B ð12Þ w CP0 B ~T ~T ~T B X 14 X 24 X 34 X~ 44 X~ 45 X~ 46 C C B @ X~ T X~ T X~ T X~ T X~ 55 X~ 56 A 15 25 35 45 X~ T X~ T X~ T X~ T X~ T X~ 66 16
26
36
46
56
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
0
~ 11 / B ~T B /12 B B/ B ~ T13 B ~ ~T /¼B B /14 B ~T B /15 B B ~T @/ 0
~ 12 / ~ 22 / ~T / 23
~ 13 / ~ 23 / ~ 33 /
~ 14 / ~ 24 / ~ 34 /
~ 15 / ~ 25 / ~ 35 /
~ 16 / ~ Y 26 L ~ 36 /
~T / 24 ~T /
~T / 34 ~T /
~T / 44 ~T /
~ 45 / ~ 55 /
Y~ 46 L Y~ 56 L
LT Y~ T26
~T / 36
LT Y~ T46
LT Y~ T56
I
ETb
0
0
0
25
16 ETa
Y~ 11 B ~T B Y 12 B B Y~ T B 13 ~¼B N B Y~ T14 B B Y~ T B 15 B ~T @ N16 ~T N
35
Y~ 12 Y~ 22 Y~ T23 Y~ T24 Y~ T 25
DT Y T26 LT Y~ T26
17
45
Y~ 13 Y~ 23 Y~ 33 Y~ T34 Y~ T35 ~T N 36 ~T N 37
Y~ 14 Y~ 24 Y~ 34 Y~ 44 Y~ T
0 Y~ 15 Y~ 25 Y~ 35 Y~ 45 Y~ 55
DT Y T46 LT Y~ T46
DT Y T56 LT Y~ T56
45
1 Ea C Eb C C 0 C C C 0 C C60 C 0 C C C 0 A I ~ 16 N Y~ 26 D ~ 36 N Y~ 46 D Y~ 56 D ~ 66 N 0
775
ð13Þ
1 ~ 17 N C Y~ 26 L C C ~ 37 C N C C Y~ 46 L C P 0 C Y~ 56 L C C C 0 A
ð14Þ
I
where, ~ 11 ¼ AT P~ þ P~ A þ X~ 16 þ X~ T þ Y~ 16 A þ AT Y~ T þ Q ~1 þ Q ~ 2 þ rX~ 11 þ sY~ 11 þ sY~ 66 / 16 16 ~ 12 ¼ P~ B X~ 16 þ X~ T þ AT Y~ T þ Y~ 16 B þ rX~ 12 þ sY~ 12 / 26 26 ~ 13 ¼ AT P~ C þ X~ T þ AT Y~ T þ rX~ 13 þ sY~ 13 / 36
36
~ 14 ¼ X~ T þ AT Y~ T Y~ 16 þ rX~ 14 þ sY~ 14 / 46 46 ~ 15 ¼ X~ T þ AT Y~ T þ Y~ 16 C þ rX~ 15 þ sY~ 15 / 56 56 ~ 16 ¼ Y~ 16 L þ P~ L / ~ 22 ¼ Q ~ 1 X~ 26 X~ T þ Y~ 26 B þ BT Y~ T þ rX~ 22 þ sY~ 22 / 26 26 ~ 23 ¼ BT P~ C X~ T þ BT Y~ T þ rX~ 23 þ sY~ 23 / 36 36 ~ 24 ¼ X~ T Y~ 26 þ BT Y~ T þ rX~ 24 þ sY~ 24 / 46
46
~ 25 ¼ X~ T þ BT Y~ T þ Y~ 26 C þ rX~ 25 þ sY~ 25 / 56 56 ~ 33 ¼ Q ~ þ rX~ 33 þ sY~ 33 / ~ 34 ¼ Y~ 36 þ rX~ 34 þ sY~ 34 / ~ 35 ¼ Y~ 36 C þ rX~ 35 þ sY~ 35 / ~ 36 ¼ Y~ 36 L C T P~ L / ~ 44 ¼ R ~ Y~ 46 Y~ T þ rX~ 66 þ rX~ 44 þ sY~ 44 / 46
~ 45 ¼ Y~ T þ Y~ 46 C þ rX~ 45 þ sY~ 45 / 56 ~ 55 ¼ R ~ þ Y~ 56 C þ C T Y~ T þ rX~ 55 þ sY~ 55 / 56 ~ 66 ¼ Y~ 66 ET Ed N ~ 16 ¼ P~ D Y~ 16 D N d ~ 36 ¼ C T P~ D Y~ 36 D ~ 17 ¼ P~ L Y~ 16 L N N T ~ 37 ¼ C P~ L Y~ 36 L N Remark. The Proof of Theorem 2 is similar to the approach in [16] with lemma 2.4 in [18] and the Schur’s complement in [19]. It’s omitted. 4. Numerical examples Example 1. Consider the following uncertain neutral systems with discrete and distributed delays system: Z t xðsÞ ds x_ ðtÞ C x_ ðt hÞ ¼ AðtÞxðtÞ þ BðtÞxðt rÞ þ DðtÞ ts
ð15Þ
776
L. Xiong et al. / Chaos, Solitons and Fractals 40 (2009) 771–777
with 0:9 0 1 0:12 AðtÞ ¼ BðtÞ ¼ 0 0:9 0:12 1 0 0 0:12 0:12 C¼ DðtÞ ¼ 0 0 0:12 0:12 Using the discrete delay-dependent result given in Theorem 2 of [20], it was found that the upper bound of r must be less than 0.9086, while in [21], it is found that the admissible bound of the discrete delay r for the stability of system (15) are 1.8302 when s = 1 is chosen. Applying Theorem 1 of this paper, the upper bound of the delay r has been obtained as 2.4128 in the case of s = 1. Example 2. Consider the following uncertain neutral systems: x_ ðtÞ C x_ ðt rÞ ¼ ðA þ AðtÞÞxðtÞ þ ðB þ BðtÞÞxðt rÞ 0:9 0:2 1:1 0:2 0:2 0 A¼ B¼ C¼ 0:1 0:9 0:1 1:1 0:2 0:1
ð16Þ
For A(t) = 0 and B(t) = 0 (i.e. nominal system), system (16) reduces to the system discussed in [14]. Using the criterion in this paper, the maximum value of rmax for the nominal system to be asymptotically stable is rmax = 1.6524. By the criteria in [12,14,16], the nominal system is asymptotically stable for any r satisfying r 6 0.71, r 6 0.3, r 6 0.74, respectively. This example shows again that the stability criterion in this paper gives a much less conservative result than these in [12,14,16]. 5. Conclusion The robust stability problem for uncertain neutral systems with discrete and distributed delays has been investigated. Stability criteria have been obtained. Numerical examples have shown significant improvements over some existing results. References [1] Hale J, Verduyn Lunel SM. Introduction to functional differential equations. New York: Springer; 1993. [2] Yang Xiao-Song. Liapunov asymptotic stability and Zhukovskij asymptotic stability. Chaos, Solitons & Fractals 2000;13(11):1995–9. [3] Tu Fenghua, Liao Xiaofeng, Zhang Wei. Delay-dependent asymptotic stability of a two-neuron system with different time delays. Chaos, Solitons & Fractals 2006;2(28):437–47. [4] Park JH, Kwon O. LMI optimization approach to stabilization of time-delay chaotic systems. Chaos, Solitons & Fractals 2005;2(23):445–50. [5] Liu Xingwen, Zhang Hongbin. New stability criterion of uncertain systems with time-varying delay. Chaos, Solitons & Fractals 2005;5(26):1343–8. [6] Xiong Wenjun, Liang Jinling. Novel stability criteria for neutral systems with multiple time delays. Chaos, Solitions & Fractals 2007;5(32):1735–41. [7] Li Hong, Li Hou-biao, Zhong Shou-ming. Stability of neutral type descriptor system with mixed delays. Chaos, Solitions & Fractals 2007;5(33):1796–800. [8] Hu G Di, Hu G Da. Some simple stability criteria of neutral delay-differential systems. Appl Math Comp 1996;80:257–71. [9] Slemrod M, Infante EF. Asymptotic stability criteria for linear systems of differential equations of neutral type and their discrete analogues. J Math Anal Appl 1972;38:399–415. [10] Huaicheng Yan, Xinhan Huang, Min Wang, Hao Zhang. New delay-dependent stability criteria of uncertain linear systems with multiple time-varying state delays. Chaos, Solitons & Fractals, in press [available online 27.10.2006]. [11] Kolmanovskii VB, Richard J-P. Stability of some linear systems with delays. IEEE Trans Automat Contr 1999;44:984–9. [12] Niculescu, S.-I. Further remarks on delay-dependent stability of linear neutral systems. Proceedings of MTNS 2000, Perpigan, France. [13] Han Q-L. Robust stability of uncertain delay-differential systems of neutral type. Automatica 2002;38:719–23. [14] Lien C-H, Yu KW, Hsieh JG. Stability conditions for a class of neutral systems with multiple time delays. J Math Anal Appl 2000;245:20–7. [15] Gu K, Niculescu S-I. Further remarks on additional dynamics in various model transformations of linear delay systems. IEEE Trans Automat Contr 2001;46:497–500.
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