stability of n-dimensional linear systems with

Jrl Syst Sci & Complexity (2006) 19: 149–156

STABILITY OF N -DIMENSIONAL LINEAR SYSTEMS WITH MULTIPLE DELAYS AND APPLICATION TO SYNCHRONIZATION∗ ¨ · Changpin LI Weihua DENG · Jinhu LU

Received: 13 April 2006 Abstract This paper further investigates the stability of the n-dimensional linear systems with multiple delays. Using Laplace transform, we introduce a definition of characteristic equation for the n-dimensional linear systems with multiple delays. Moreover, one sufficient condition is attained for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. In particular, our result shows that some uncommensurate linear delays systems have the similar stability criterion as that of the commensurate linear delays systems. This result also generalizes that of Chen and Moore (2002). Finally, this theorem is applied to chaos synchronization of the multi-delay coupled Chua’s systems. Key words Chaos synchronization, multi-delay linear systems, stability.

1 Introduction It is well known that time delay lies in everywhere. In our real-world, time delay is inevitable for most practical systems due to their physical essence and the finite spread speed of their information exchange systems[1−9] . The research of delay differential equations can trace back to the eighteenth century. Also, there is a long history for investigating the stability of delay differential systems in applied mathematics and control engineering fields. Over the last decades, there are many results reported in the literature on the stability of delay differential equations (see [1–4, 6, 9–17]). In 2002, Chen and Moore got an analytical stability bound of a special delay second-order system with repeating poles by using Lambert function[1] . In this article, by using Laplace transform, we introduce a definition of characteristic equation for the n-dimensional linear systems with multiple delays and attain a sufficient condition for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. Furthermore, our result shows that some uncommensurate linear delays systems have the Weihua DENG School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China; Department of Mathematics, Shanghai University, Shanghai 200444, China. ¨ Jinhu LU Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China; Department of Ecology and Evolutionary Biology, Princeton University, NJ 08544, USA. Email: [email protected]. Changpin LI Department of Mathematics, Shanghai University, Shanghai 200444, China. ∗ This work was supported by the National Natural Science Foundation of China under Grants 60304017, 20336040, and 60221301, the Scientific Research Startup Special Foundation on Excellent PhD Thesis and Presidential Award of Chinese Academy of Sciences, and the Tianyuan Foundation under Grant A0324651.

150

¨ · CHANGPIN LI WEIHUA DENG · JINHU LU

similar stability criterion as that of the commensurate linear delays systems. Our result also generalizes the result of Chen and Moore[1]. Especially, for second-order systems, we present a very simple stability criterion. Finally, this criterion is applied to chaos synchronization of the multi-delay coupled Chua’s systems[10−14] . This paper is organized as follows. Section 2 introduces some preliminary knowledge. The definition of characteristic equation for the n-dimensional linear systems with multi-delay is introduced and a sufficient condition is obtained for the Lyapunov globally asymptotical stability of the general multi-delay linear systems in Section 3. Section 4 applies the stability criterion to chaos synchronization of the multi-delay coupled Chua’s systems. Some conclusions and comments are finally given in Section 5.

2 Preliminaries This section proposes the formulation of the problem and introduces some background knowledge. Consider the following n-dimensional linear systems with multi-delay:  d x1 (t)   = a11 x1 (t − τ11 ) + a12 x2 (t − τ12 ) + · · · + a1n xn (t − τ1n ),   dt    d x2 (t)    = a21 x1 (t − τ21 ) + a22 x2 (t − τ22 ) + · · · + a2n xn (t − τ2n ), dt (1) ..    .        d xn (t) = an1 x1 (t − τn1 ) + an2 x2 (t − τn2 ) + · · · + ann xn (t − τnn ) , dt

where the initial values xi (t) = φi (t) for −τmax ≤ t ≤ 0 and i = 1, 2, · · · , n, aij are constant numbers for i, j = 1, 2, · · · , n, time delays 0 ≤ τij ≤ τmax are constant numbers for i, j = 1, 2, · · · , n, state variables xi (t), xi (t − τij ) ∈ R and φi (t) ∈ C[−τmax , 0] for i, j = 1, 2, · · · , n. Here, system (1) can be commensurate or uncommensurate delays systems[15−17] . Taking Laplace transform on both sides of (1) gives  Z 0  −st −sτ12 −sτ11   sX (s) − x (0) = a e X (s) + e x (t)dt + a e X2 (s) 12 1 1 11 1 1    −τ11  Z 0 Z 0     −st −sτ1n −st  + e x (t)dt + · · · + a e X (s) + e x (t)dt , 2 1n n n    −τ12 −τ1n   Z 0    −sτ21  sX (s) − x (0) = a e X (s) + e−st x1 (t)dt + a22 e−sτ22 X2 (s)  2 2 21 1   −τ21   Z 0 Z 0 −st −sτ2n −st + e x2 (t)dt + · · · + a2n e Xn (s) + e xn (t)dt , (2)    −τ −τ 22 2n    ..    .   Z 0     −st −sτn2 −sτn1  sX (s) − x (0)= a e X (s) + e x (t)dt + a e X2 (s) n n n1 1 1 n2    −τn1   Z 0 Z 0     + e−st x2 (t)dt + · · · + ann e−sτnn Xn (s) + e−st xn (t)dt ,  −τn2

−τnn

where Xi (s) (1 ≤ i ≤ n) are the Laplace transforms of xi (t) (1 ≤ i ≤ n) with Xi (s) = L(xi (t)).

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STABILITY OF N -DIMENSIONAL LINEAR MULTIPLE DELAYS SYSTEMS

Now we can rewrite (2) as follows:  s − a11 e−sτ11 −a12 e−sτ12 ···  −a e−sτ21 −sτ22 s − a22 e ··· 21   .. ..  .. .  . . −sτn1 −sτn2 −an1 e −an2 e ···

where

−a1n e−sτ1n −a2n e−sτ2n .. . s − ann e−sτnn

     

X1 (s) X2 (s) .. . Xn (s)





    =    

b1 (s) b2 (s) .. . bn (s)



  ,  

(3)

 Z 0 Z 0  −st −sτ  −sτ 12 11  b1 (s) = a11 e e φ1 (t)dt + a12 e e−st φ2 (t)dt    −τ −τ 11 12  Z 0    −sτ1n  + · · · + a e e−st φn (t)dt + φ1 (0),  1n   −τ  1n Z 0 Z 0     −st −sτ22 −sτ21  b2 (s) = a21 e e φ1 (t)dt + a22 e e−st φ2 (t)dt    −τ21 −τ 22  Z 0

+ · · · + a2n e−sτ2n e−st φn (t)dt + φ2 (0),    −τ 2n   ..    .   Z 0 Z 0    −st −sτn2 −sτn1  b (s) = a e e φ (t)dt + a e e−st φ2 (t)dt  n n1 1 n2   −τ −τ  n1 n2 Z 0     −st −sτnn  e φn (t)dt + φn (0) . + · · · + ann e  −τnn

Therefore, for any 0 ≤ τij ≤ τmax and φi (t) ∈ C[−τmax , 0] with i, j = 1, 2, · · · , n, when ℜ(s) ≥ 0, there exists a positive constant M such that k bi (s) k < M , where ℜ(s) is the real part of s. Moreover, bi (s) (i = 1, 2, · · · , n) are continuous functions of s for ℜ(s) ≥ 0.

3 Main Theorem This section introduces the definition of characteristic equation for the n-dimensional linear systems with multi-delay and a sufficient condition for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. Denote ∆(s) be the matrix   ··· −a1n e−sτ1n −a12 e−sτ12 s − a11 e−sτ11  −a e−sτ21 −a2n e−sτ2n  s − a22 e−sτ22 · · · 21   .  ∆(s) =  .. .. ..  .. .   . . . −sτnn −sτn1 −sτn2 · · · s − ann e −an1 e −an2 e According to the theory of linear system and the continuity of det(∆(s)) and bi (s) for ℜ(s) ≥ 0, if det(∆(s)) 6= 0 for ℜ(s) ≥ 0, then we can analytically solve Xi (s) from (3). Moreover, Xi (s) (i = 1, 2, · · · , n) are the continuous functions of s for ℜ(s) ≥ 0. Assume that det(∆(s)) 6= 0 for ℜ(s) ≥ 0. Multiplying s on both sides of (3), it becomes      s − a11 e−sτ11 −a12 e−sτ12 ··· −a1n e−sτ1n sX1 (s) sb1 (s)  −a e−sτ21     s − a22 e−sτ22 · · · −a2n e−sτ2n  21    sX2 (s)   sb2 (s)    =  , (4) .. .. .. .. ..      .. .      . . . . . −an1 e−sτn1

−an2 e−sτn2

···

s − ann e−sτnn

sXn (s)

sbn (s)


152

Since bi (s) is bounded, sbi (s) → 0 as s → 0+ for i = 1, 2, · · · , n. Therefore, we have lim sXi (s) = 0, i = 1, 2, · · · , n.

s→0+

From the final-value theorem of Laplace transform[6], we get lim xi (t) = lim sXi (s) = 0, i = 1, 2, · · · , n.

t→+∞

s→0+

Thus the Lyapunov stability of the zero solution of system (1) is completely determined by ∆(s). Now we have the following definition. Definition 1 ∆(s) is called the characteristic matrix of system (1) and det(∆(s)) = 0 is called the characteristic equation of system (1). Denote   a11 a12 · · · a1n  a   21 a22 · · · a2n    A =  . .. ..  . .. .  .. . .  an1

an2

···

ann

Remark 1 If τij = τ > 0 for i, j = 1, 2, · · · , n, then the characteristic matrix and equation of (1) are sI − Ae−sτ and det(sI − Ae−sτ ) = 0, respectively. It coincides with the usual definitions of the characteristic matrix and characteristic equation of the delay systems. On the other hand, if τij = 0 for i, j = 1, 2, · · · , n, then the characteristic matrix and characteristic equation of (1) are sI − A and det(sI − A) = 0, respectively. It also agrees with the usual definitions of the characteristic matrix and characteristic equation of the linear systems. According to the characteristic equation of system (1), we have the following sufficient condition for the Lyapunov globally asymptotical stability of the zero solution of system (1). Theorem 1 If all the eigenvalues of A have negative real parts and the characteristic equation det(∆(s)) = 0 has no purely imaginary roots for any τij ≥ 0, i, j = 1, 2, · · · , n, then the zero solution of system (1) is Lyapunov globally asymptotically stable. Proof Assume that l(s) = det(sI −A) and φ(s) = det(∆(s))−det(sI −A), then l(s)+φ(s) = det(∆(s)). Obviously, l(s) = sn + c1 sn−1 + · · · , and φ(s) =

X n i=1

dii e

−sτii

X n n−1 −sτij + d0 s + hij e + h0 sn−2 + · · · , i,j=1

where d0 , h0 , c1 , dii , hij , · · · are constants. Clearly, there exists a constant r > 0, such that kl(s)k > kφ(s)k for ksk > r and ℜ(s) ≥ 0. Also, there is no root for l(s) = 0 in the area of ℜ(s) ≥ 0 and ksk > r. According to Rouch´e Theorem, there is no root for l(s) + φ(s) = 0 in the same area as above, either. Note that all the zero points of l(s) are in the area of ℜ(s) < 0. If there is a zero point for l(s) + φ(s) = 0 in the area of ℜ(s) > 0, then there exists a set of parameters τij (i, j = 1, · · · , n) such that s passes through the imaginary axis between −r and r. Obviously, it contradicts to the assumption of this theorem. Thus, det(∆(s)) 6= 0 for ℜ(s) ≥ 0. The proof is thus completed. Consider the general second-order delay differential equations d2 y(t) dy(t) + am + bm y(t) − Kp y(t − τ ) = 0 , 2 dt dt

(5)


153

where am , bm , Kp , τ are real numbers with τ ≥ 0. When am = 2α, bm = α2 , system (5) becomes system (1) of Chen and Moore in [1]. Therefore, system (1) of Chen and Moore in [1] is a special case of system (5). Rewrite (5) as the following equivalent form:  dy(t)   = x(t), dt (6)   dx(t) = −am x(t) − bm y(t) + Kp y(t − τ ) . dt Thus its corresponding coefficient matrix A is given by A=

0

1

K p − bm

−am

!

.

Obviously, when am > 0

and bm − Kp > 0 ,

A has two negative eigenvalues. And its characteristic equation is given by ! s −1 = s2 + am s + bm − Kp e−sτ = 0 . det(∆(s)) = −Kp e−sτ + bm s + am

(7)

(8)

Assume that s = ω i is a root of (8), where ω is a real number. Then one gets −ω 2 + am ωi + bm − Kp e−τ ωi = 0 . That is, −ω 2 + bm − Kp cos(τ ω) = 0

and am ω + Kp sin(τ ω) = 0 .

From the above two equations, one has ω 4 + (a2m − 2bm )ω 2 + (b2m − Kp2 ) = 0 . Thus, when b2m − Kp2 > 0 and a2m − 2bm > 0 ,

(9)

equation (8) has not purely imaginary roots. According to Theorem 1, one has the following corollary. Corollary 1 If am > 0, bm − Kp > 0, b2m − Kp2 > 0, and a2m − 2bm > 0, then the zero solution of system (5) is Lyapunov globally asymptotically stable. Remark 2 Theoretical analysis shows that our result is concordant to that of Chen and Moore[1] . However, our conditions of Corollary 1 are more easier to be verified than that of Chen and Moore[1]. In particular, system (1) of Chen and Moore in [1] is a special case of system (5) with am = 2α, bm = α2 . Therefore, our result is more general than that of Chen and Moore[1] . Furthermore, Corollary 1 is a sufficient condition.

4 Application to Chaos Synchronization This section uses the chaos synchronization of the multi-delay coupled Chua’s circuit as an example to show the effectiveness of the above Theorem 1[14] .

154


It is well known that Chua’s circuit is a piecewise-linear chaotic system which has rich dynamical behaviors. The multi-delay Chua’s circuit is described by  dx1   = α(y1 (t − τ12 ) − x1 (t − τ11 ) − f (x1 (t − τ11 ))),   dt    dy1 (10) = x1 (t − τ21 ) − y1 (t − τ22 ) + z1 (t − τ23 ),  dt       dz1 = −βy1 (t − τ32 ), dt

where α > 0, β > 0, a < b < 0, τij ≥ 0, and f (·) is given by

f (x1 (t − τ11 )) = bx1 (t − τ11 ) + 0.5(a − b)[| x1 (t − τ11 ) + 1 | − | x1 (t − τ11 ) − 1 |]. In the following, one chooses system (10) as the drive system. And the corresponding response system is described by  dx2   = α(y2 (t − τ12 ) − x2 (t − τ11 ) − f (x1 (t − τ11 ))) − k1 (x2 (t − τ11 ) − x1 (t − τ11 )),   dt    dy2 (11) = x2 (t − τ21 ) − y2 (t − τ22 ) + z2 (t − τ23 ) − k2 (y2 (t − τ22 ) − y1 (t − τ22 )),  dt    dz    2 = −βy2 (t − τ32 ) − k3 (z2 (t − τ33 ) − z1 (t − τ33 )). dt Subtracting (10) from (11), then one gets the error system as follows  dex   = α(ey (t − τ12 ) − ex (t − τ11 )) − k1 ex (t − τ11 ),   dt    dey = ex (t − τ21 ) − ey (t − τ22 ) + ez (t − τ23 ) − k2 ey (t − τ22 ),  dt       dez = −βey (t − τ32 ) − k3 ez (t − τ33 ) , dt

(12)

where

ex (t − τ11 ) = x2 (t − τ11 ) − x1 (t − τ11 ),

ex (t − τ21 ) = x2 (t − τ21 ) − x1 (t − τ21 ),

ey (t − τ12 ) = y2 (t − τ12 ) − y1 (t − τ12 ),

ey (t − τ22 ) = y2 (t − τ22 ) − y1 (t − τ22 ),

ey (t − τ32 ) = y2 (t − τ32 ) − y1 (t − τ32 ), ez (t − τ23 ) = z2 (t − τ23 ) − z1 (t − τ23 ),

ez (t − τ33 ) = z2 (t − τ33 ) − z1 (t − τ33 )

are the error variables, k1 , k2 , k3 are the gain constants. Obviously, the synchronization between (10) and (11) is equivalent to the globally asymptotical stability of the zero solution of error system (12)[7−8] . From error system (12), the corresponding coefficient matrix A is described by   −k1 − α α 0   A= 1 −k2 − 1 1    0 −β −k3

155


and its characteristic equation det(∆(s)) is given by s + (k1 + α)e−sτ11 −αe−sτ12 −sτ21 −e s + (k2 + 1)e−sτ22 det(∆(s)) = 0 βe−sτ32

0 −e−sτ23 s + k3 e−sτ33

= 0.

When k1 = k2 = k3 = 5, it is very easy to verify that all eigenvalues of A have negative real parts and det(∆(s)) = 0 has no purely imaginary roots. That is, all conditions of Theorem 1 are satisfied. Fig. 1 shows the multi-delay Chua’s attractors and the corresponding error evolutions of the multi-delay coupled Chua’s systems, where α = 10, β = 14.87, a = −1.27, b = −0.68, and k1 = k2 = k3 = 5. 0.6

5 e1 e2 e3

4

0.4

3 0.2 2

1

1

2

y

3

e (e , e )

0

−0.2

0 −0.4 −1

−0.6

−2

−0.8 −3

(a)

−2

−1

0 x

1

2

−3 0

3

(τ11 , τ12 , τ21 , τ22 , τ23 , τ32 , τ33 ) = (0.03, 0.09, 0.06, 0.06, 0.00, 0.03, 0.00).

(b)

3

6

9 t

12

15

18

(τ11 , τ12 , τ21 , τ22 , τ23 , τ32 , τ33 ) = (0.03, 0.09, 0.06, 0.06, 0.00, 0.03, 0.00). 5

0.8

4

0.4

3

0.2

2

3

2

0

1

1

y

3

e (e , e )

0.6

e 1 e 2 e

−0.2

0

−0.4

−1

−0.6

−2

−0.8 −4

(c)

−3

−2

−1

0 x

1

2

3

(τ11 , τ12 , τ21 , τ22 , τ23 , τ32 , τ33 ) = (0.03, 0.03, 0.06, 0.09, 0.00, 0.03, 0.00). Figure 1

−3 0

4

3

6

9

12

15

18

t

(d)

(τ11 , τ12 , τ21 , τ22 , τ23 , τ32 , τ33 ) = (0.03, 0.03, 0.06, 0.09, 0.00, 0.03, 0.00).

Multi-delay Chua’s attractors and the corresponding error evolutions of the multi-delay coupled Chua’s systems, where α = 10, β = 14.87, a = −1.27, b = −0.68, k1 = k2 = k3 = 5.

156


5 Conclusions This paper has further studied the stability of the general multi-delay n-dimensional linear systems. By using Laplace transform, we propose the definition of characteristic equation for the multi-delay n-dimensional linear systems, which is a natural generalization of the characteristic equation of linear systems. Moreover, our result shows that some uncommensurate linear delays systems have the similar stability criterion as that of the commensurate linear delays systems. Especially, a sufficient condition is given for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. This sufficient condition extends that of Chen and Moore[1] . Finally, this result is applied to the chaos synchronization of the multi-delay coupled Chua’s systems. References [1] Y. Q. Chen and K. L. Moore, Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W , Automatica, 2002, 38(5): 891–895. [2] Y. Q. Chen and K. L. Moore, Analytical stability bound for a class of delayed fractional-order dynamics systems, Nonlinear Dynamics, 2002, 29(1–4): 191–200. [3] M. De Sousa Vieira and A. J. Lichtenberg, Controlling chaos using nonlinear feedback with delay, Phys. Rev. E, 1996, 54(2): 1200–1207. [4] R. He and P. G. Vaidya, Time delayed chaotic systems and their synchronization, Phys. Rev. E, 1999, 59(4): 4048–4051. [5] J. W. Ryu, W. H. Kye, S. Y. Lee, M. W. Kim, M. H. Choi, S. Rim, Y. J. Park, and C. M. Kim, Effects of time-delayed feedback on chaotic oscillators, Phys. Rev. E, 2004, 70(3): 036220. [6] W. H. Deng, Y. J. Wu, and C. P. Li, Stability analysis of differential equations with time-dependent delay, Int. J. Bifurcation and Chaos, 2006, 16(2): 465–472. [7] J. L¨ u and G. Chen, A time-varying complex dynamical network models and its controlled synchronization criteria, IEEE Trans. Auto. Contr., 2005, 50(6): 841–846. [8] J. Zhou, J. A. Lu, and J. L¨ u, Adaptive synchronization of an uncertain complex dynamical network, IEEE Trans. Auto. Contr., 2006, 51(4): 652–656. [9] S. -I. Niculescu and W. Michiels, Stabilizing a chain of integrators using multiple delays, IEEE Trans. Automat. Contr., 2004, 49(5): 802–807. [10] J. L¨ u, F. L. Han, X. H. Yu, and G. Chen, Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method, Automatica, 2004, 40(10): 1677–1687. [11] J. L¨ u and G. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 2002, 12(3): 659–661. [12] W. H. Deng and C. P. Li, Synchronization of chaotic fractional Chen system, Journal of Physical Society of Japan, 2005, 74(6): 1645–1648. [13] W. H. Deng and C. P. Li, Chaos synchronization of the fractional L¨ u system, Physica A, 2005, 353: 61–72. [14] C. P. Li, W. H. Deng, and D. Xu, Chaos synchronization of the Chua system with a fractional order, Physica A, 2006, 360(2): 171–185. [15] S. I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer-Verlag, Heidelberg, Germany, 2001. [16] K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003. [17] K. Gu and S. I. Niculescu, Survey on recent results in the stability and control of time-delay systems, J. Dynam. Syst. Measur. Contr., 2003, 125: 158–165.