Exponential stability for cellular neural networks: an LMI approach*

Journal of Systems Engineering and Electronics, Vol. 18, No. 1, 2007, pp.68-71

Exponential stability for cellular neural networks: an LMI approach* Liu Deyou, Zhang Jianhua & Guan Xinping Coll. of Science, Yanshan Univ., Qinhuangdao 066004, P. R. China (Received August 28,2005)

Abstract: A new sufficient conditions for the global exponential stability of the equilibrium point for delayed cellular neural networks (DCNNs) is presented. It is shown that the use of a more general type of Lyapunov-Krasovskii function enables the derivation of new results for an exponential stability of the equilibrium point for DCNNs. The results establish a relation between the delay time and the parameters of the network. The results are also compared with one of the most recent results derived in the literature. Keywords:Delayed cellular neural networks, LMI, Neural networks, Exponential stability.

1. Introduction In recent years. stability of cellular neural networks (CNNs) has been extensively studied and various stability conditions for different types of stability, such as complete stability, asymptotic stability and exponential stability, have been obtained for CNNs with or without delay"-"'. The exponential stability of delayed CNNs (DCNNs). Some criteria on exponential stability are presented by employing a more general type of Layapunov-Krasovskii functional. The delayed neural network model considered here is defined by the following state equations

c ,I

Y[ ( t )= --c, Y , ( t )+

a,, R

J

(v,( t ) )+

J=I

c n

b, g ( Y ( t - h ) )+ S, J

(1)

?=I

or j f t j = -Cy(t)+A g ( y ( t ) )+ E g ( y ( t - h ) )+ S

point of system (1). The equilibrium point of system (1) to the origin. The transformation x(.) = y ( - )- y* puts system (1) into the following form

i ( r ) = -Cx(r) + A f ( x ( t ) )+ E f ( x ( t -%))

(3)

where x(.) = [n,(.j,x~(.);..,x~(.)lT is the state vector of transformed system, and f ( x ( . > ) = [ f i (x<.>>,f*(x(.>),...,fn(x(.))lT

with f,(Xf(.))=gr(Xl(.)+ Y:>-g,(Y;) where f ,(x, (.)) satisfy the condition f,2(x,(9) x,(.>J1(xt(.)) ?f,(O) = o , i =1,2,*..,n

For any initial function pe R([-h,O],R" j there exists the unique solution x(t, p)of Eq.( 1) satisfying the initial condition x(8,p) = p(8),86 [-h,O]. The space of initial functions is provided with the uniform norm Ilpllh= maxB+h,Ol{ ~ ~ p (.@ Throughout ~~l

where Y(.) =[Y,(.),Y~(.),...,~~~(.)~ C = diag(c, ,c2;-., cn),c, > 0

this note, the Euclidean norm for vectors and the induced matrix norm for matrices are used. For t 2 0 denote by x,(p) the segment of trajectory x,(p) =

Let A = {a,,)be the feedback matrix, and B = bl,)be

{ x ( t + @ , p ) l[-h,01}. e~

delayed feedback matrix, S = [s,,s2 ;.-, s,IT be the

Definition 1 System (3) is said to be exponenttially stable if there exist cr > 0 and y 3 1 such that for every solution x ( t , q ) , where p~ C'([-t,O],R"), the following exponential estimate holds

T

constant external input, h be the transmission delay.

go(.)) = k ,(Yl(.j),g2(v~(.)),...,g,(v,, (.)>ITdenotes the cell activations, and the g , (.) are defined by

(2) It is known that the use of bounded functions always guarantees the existence of an equilibrium point for * * system (1 ). Let v* = [ yl .y l , '..,y: IT be an equilibrium g f ( v , ) = 0 . 5 ( l+Il-Iy, ~f -11) i=1,2,...,n

*

(Ix(tjp)II

ye llpllh,t 3 0 ,t 3 0

Definition 2 ,Imax ( M ) denotes the maximum eigenvalue of the matrix M , Amn(M)denotes the minimum eigenvalue of the matrix M , and d,

This project wab wpported in part by the National Natural Science Foundation of China (60404022, 60604004), the Key Scientific Research project of

Education Ministry of China (203014).the National Natural Science Foundation of China for Distinguished Young Scholars (60525303).

Exponential stability for cellular neural networks: an LMI approach denotes the maximum ind,,d,;.-,d,.

D,D,,D,,D,,D, and a positive constant that the inequality E < 0 , then

2. Main stability results In this section, new sufficient conditions are presented for the exponential stability of the equilibrium point for the DCNN defined by system (3). The first result is given in the following. Theorem 1 Let the linear time delay system (3) be given. If there exist real n x n positive definite matrices P , Q , positive definite diagonal matrix

+ Q + 2pP+ 20,

-PC -CP

69

where the positive constant

p

such

a, and a, are defined

as

a; = L A P )

PA- CD + 2pD- Dl + 0, AD + DAT - 2 0 ,

0 0 2 0 , -eZBhQ -D3 + D4 -2 D4

Proof Consider the Lyapunov-Krasovskii fun-

(4)

(5)

so

ctional V(x,) = y +vz+v3 v, = x'(t)Px(t)

is obtained and the conclusion arived at is

V, = f h xT(t + s)eZBhQx(t+ s)ds

i=l

D = diag(d,,d,,...,d,,) Where P and Q are the positive definite matrices, and D is the positive definite diagonal matrices. The following inequalities are obtained for the functional:

Where aland q are given by system (5). Remark 1 Let p = 0 then (4) represents the

1:

the sysem (3). In other words, matrix (6) implies

The time derivative of V(x,) along the trajectories of

global asymptotic stability of system (3) for all h > 0

a,IIX(t)llZ G VO,) G a;

system (3) is V ( t t )= 2xT(t)Pi(t)+ x'(t)Qx(t>-

well-known delay independent stability condition for

Theorem 2

be given. If there exist real n x n positive definite

xT( t - h)ezPhQx(t- h ) -

matrices P ,Q ,a positive definite diagonal matrix D

2pfhxT(+ t s)Qx(t + x)Qx(t +

and a positive constant

such that the inequality

+ L(0,) <0

Clearly, it is can be seen that

Lpl)

V ( x , ) + 2 p V ( x t )d e'(t)Ee(t) e T ( t ) = [ x T ( ft ')( x ( t ) ) x ' ( t - h ) f T ( x ( t - h ) ) l

so V ( x , (PN+ 2 p v ( x ,(PI). (.0

p

L(=,I + .m,(Q< 0

s)ds + 2iT(t)Df( ~ ( t ) )

where

Let the linear time delay system (3)

is obtained. This

inequality leads to the following one

v(xt(q)) G e-'W(q), t 3 o

where

El = -PC - CP + Q + 2 p P + 2P + 2 p D 3, =AD+ DA' -CD+ D

0,=-e2BhQ @,=AD+DA~-CD+D

(6)

70

Liu Deyou, Zhang Jianhua & Guan Xinping

then

Ilx(t,

@-@ Ildlh where

constants a, and a, are defined as a, =

a, = A,- (0+ ~4-( Q )+ 2d,,

f

V(xr)=nT(t)Px(t)+ ~ ' ( t + s ) e ~ ~ ~ Q x ( t + s ) d . s +

D = diag(d,,d,,-.-,d,) > 0 Where P, Q are the positive definite matrices and Dis the positive definite diagonal matrices of. The following inequalities are obtained for the functional

a,llx(t)\r d v(xr ) a;ilxr 1: The time derivative of V ( x , ) along the trajectories of system (3)is V(x,) .Clearly, it is seen that

+ 2/?V(t)d llx(t)11; (Arna (-PC

2PP+2P+2pD)+$,(AD+ ((x(t- h

- CP

&ax

&(-el +

f (A- (-eZBSQ)+ &,=(AD + DA' - CD + D ) )

where A-(-PC-CP+ Q + 2 p P + 2P+ 2/3D)+ ;2,(AD + DA' - CD + D ) )< 0 ; 1 ,( - e Z B+s&,a~ )(AD + D A -~ CD + D ) < o

then (6) looks like

( A D + D A -~ CD + D ) < o

(AD+ D A -~CD + D ) < o

&ax

(8)

This LMI represents the well-known delay independent

stability condition for system (3). In other words, system ( 8 ) implies global asymptotic stability of system ( 3 )for all h > 0

Remark 4 Let P = I,Q = I , D = Z , Eq.(7) looks like

Lax (-2C + 31 + 4 p ) + Am,

( A+ AT - C + I)) < 0 and

-c+z)
(9) These represent exponential stability condition for systems ( 3 ) and ( 9 ) only concludes matrices C,A,B in system ( 3 )without any uncertain matrix. -e2Bh+ & J A + A ~

+Q +

DAT -CD+ D))+

p =0

L a x ( - - P C- CP + Q + 2P) +

(P) (7)

Proof Consider the Lyapunov-Krasovskii functional

V(t)

Remark 3 Let

the positive

3. Numerical examples Ln this section, numerical examples are given to illustrate the differences of the theorems. Example 1 Now, consider system (3)for t > 0 , where t > O

The matrixes are obtained through LMI in Matlab, such as r

1

1 250.2 0 0 879.71

247.807 3 36.024 1 36.024 1 129.234 3

]?a=[

D = [ -48.665 3 0

]

0 ],D1=[l.3659 0 -20.762 3 0 1.495 0

and the conclusion is arrived at that 7

424.313 1 8 1.602 O 9I Where a,and a, are given by Eq47).

( E l )+ &ax ( E 2 )< 0 is easy to Remark 2 Lax realize by LMI, two positive definite diagonal matrices D, ,D4 can be added, so Am ( E l )+ Am ( E ,) < 0 and ~ + D l - D , < 0 , ~ - D l + D 2 < are 0 equivalence relations

211.088 8 135.448 O 3I 387.906 8 285.885 O 5I

,b = 0.49, h d 0.75

Exponential stability for cellular neural networks: an LMI approach

Through theorem 1, the model proves exponential stability. Example 2 Now, consider system (3) for t > 0, where

71

IEEE Int. Con$ Neural Networks & Signal Processing, 2003: 108-111.

[5] Sabri Arik. Global asymptotic stability of a larger class of delayed neural networks: IEEE Circuits and Systems, 2003. ISCAS’O3.Proceedings of the 2003 International Symposium,

The matrixes are obtained through LMI in Matlab, such as

313.935 7 0

D=[

733.148 4 16 587

0 0

D4= [ 7 6 8 . 7 6

]

1374.7

0 772.558 8 ’

1 024.2 0 D5=[ 0 1024.2

1

0 1024.2 024.2 O b = 1.47, satisfies theorem 2, so it is exponentialy stable.

4. Conclusion remarks Exponential estimates for exponentially stable neural networks with time delay have been presented. The results have been obtained using a LyapunovKrasovskii functionnal and the conditions are expressed as feasible LMI conditions. An interesting feature of the approach is that the eigenvalue of the matrix, and it has indicated superiority. The results are illustrated in the example.

Reference Chua L 0, Yang L. Cellular neural networks. IEEE Trans. Circuits Syst, 1988,35: 1257-1272.

Roska T, Boros T, Thiran F? Detecting simple motion using cellular neural networks. Proc. IEEE Int. Workshopon Cellular Neural Networks and TheirApplications, 1999: 127-138. [3] V i a l Singh. A Generalized LMI-Based approach to the global asymptotic stability of cellular neural networks. IEEE Transaction on Neural Networks, 2004, 15:223-225. [4] Zhou Dongming, Zhang Liming, Zhao Dongfeng. Global exponential stability for recurrent neural networks with a general class of activation functions and variable delays.

2003,5: 721-724. [6] Liao Xiaofeng, Yu Juebang, Chen Guanrong. Delay-dependent exponential stability analysis of delayed cellular neural networks. IEEE Communications, Circuits and Systems and West Sin0 Expositions, 2002,2: 1657-1661. [7] Sabri Arik.An analysis of exponential stability of delayed neural networks with time varying delays. Neural Networks, 2003,17: 1027-1031. [8] Zhang Qiang, Wei Xiaopeng, Xu Jin. Global asymptotic stability of Hopfield neural networks with transmission delays. Physics Letter A3 18, 2003:399-405. [9] Cao J, Wang Jun. Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE Transactions, Circuits and Systems I: Fundamental Theory and Applications, 2003,50:34-44. [lo] Zhang Qiang, Ma Runnian, Xu Jin. Global exponential stability for delayed cellular neural networks and estimate of exponential convergence rate. Journal of Systems Engineering and Electronics, 2004,15344-349.

Liu Deyou was born in 1961. He received the B.S. and M.S. degrees from Northeast Heavy Machinery Institute and Harbin institute of technology, 1983 and 1989, respectively. He is currently working towards the Ph.D. degree in control theory and control engineering from Yanshan University. His current research interests include matrix theory and neural networks. E-mail: [email protected]

Zhang Jianhua was born in 1980. He received M.S. degrees from Yanshan University, 2006. He is currently working towards the Ph.D. degree in control theory and control engineering from Yanshan University. His corrent research interests include intelligent control and neural networks.

Gum Xinping was born in 1963. He received the Ph.D. degree from Harbin institute of technology, 1999. His current research interests include nonlinear system and network control.