exponential stability analysis for neural networks ... - Semantic Scholar
Journal of Marine Science and Technology, Vol. 22, No. 2, pp. 146-153 (2014 ) DOI: 10.6119/JMST-013-0207-3
146
EXPONENTIAL STABILITY ANALYSIS FOR NEURAL NETWORKS WITH TIME-VARYING DELAY AND LINEAR FRACTIONAL PERTURBATIONS Chang-Hua Lien and Ker-Wei Yu
Key words: delayed neural network, global exponential stability, delay-dependent criterion, delay-independent criterion, linear fractional perturbation, linear matrix inequality.
ABSTRACT In this paper, the global exponential stability and global asymptotic stability for a class of uncertain delayed neural networks (UDNNs) with time-varying delay and linear fractional perturbations are considered. Delay-dependent and delay-independent criteria are proposed to guarantee the robust stability of UDNNs via linear matrix inequality (LMI) approach. Additional nonnegative inequality approach is used to improve the conservativeness of the stability criteria. Some numerical examples are illustrated to show the effectiveness of our results. From the simulation results, significant improvement over the recent results can be observed.
I. INTRODUCTION The existence of time delays is often a source of oscillation and instability of practical systems. Neural networks has been applied in many mathematical and practical applications, such as approximation, association, diagnosis, decision, generalization, optimization, prediction, and recognition. Many neural networks have been proposed in recent years, such as bidirectional associative memory neural networks [16], cellular neural networks [3], Cohen-Grossberg neural networks [13], and Hopfield neural networks [11]. The delayed neural networks (DNNs) may be used in many areas including the moving images processing and pattern classification. The implementation in hardware for very large scale integration chip, modeling errors, parameters fluctua-
Paper submitted 01/31/12; revised 08/24/12; accepted 02/07/13. Author for correspondence: Chang-Hua Lien (e-mail: [email protected]). Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung, Taiwan, R.O.C.
tion, and external disturbance may destory the stability of DNNs. Hence stability of DNNs is very important and significant in practical applications. In practical analysis for uncertain DNNs, it is reasonable to consider the parameters varying in some prescribed intervals or staisfying some classes of parametric uncertainties. DNNs with interval variations are called the interval delayed neural networks (IDNNs) [2, 5, 8, 9, 11, 12, 15]. In [10] and [18], DNNs with linear fractional parametric perturbations have been investigated. IDNNs and DNNs with general structural perturbation in [4] are speical cases of DNNs with linear fractional parametric perturbations. Hence we will consider the stability analysis of DNNs with linear fractional parametric perturbations in this paper. Depending on whether the stability criterion itself contains the size of delay, criteria for DNN can be classified into two categories, namely delay-independent criteria [2, 5, 9, 12] and delay-dependent criteria [2, 4, 5, 8-10, 15, 18]. Usually the latter is less conservative when the delay is small. In the Lyapunov-based delay-dependent results, the slow-varying constraint τ(t ) < 1 is usually imposed on the time-varying delay [8, 9, 11, 15]. The constraint will be relaxed and delay-dependent result will be proposed in this paper. In [2, 12], algebraic stability, criteria were proposed based on Halanay inequality, Young’s inequality, and Lyapunov functional. It is usually difficult to find feasible solutions for these algebraic criteria. LMI approach is an efficient tool for dealng with these control problems. The LMI problem can be solved quite efficiently by using the toolbox of Matlab [1]. In [4, 5, 7-11, 13-15, 18], stability criteria for DNNs have been proposed via LMI approach. Additional nonnegative inequality approach is used to improve the conservativeness of the obtained results [17]. In this paper, LMI-based delay-dependent and delayindependent criteria are proposed by using new Lyapunov functional. In general, our approach is useful and is easy to generalize to other forms of UDNNs. The notation used throughout this paper is as follows. For a matrix A, we denote the transpose by AT, spectral norm by A , minimal (maximal) eigenvalue by λmin(A) (λmax(A)),
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
symmetric positive (negative) definite by A > 0 (A < 0). A ≤ B means that matrix B – A is symmetric positive semi-definite. For a vector x, we denote the Euclidean norm by x . For
perturbations which satisfy FCT (t ) FC (t ) ≤ I , FAT (t ) FA (t ) ≤ I , FBT (t ) FB (t ) ≤ I .
the state xt of system, we define xt(θ ):= x(t + θ ), ∀θ ∈ [−τM, 0] and denote its norm by xt
s
2
2
x(t + s ) + x (t + s ) .
= sup
−τ M ≤ s ≤0
I denotes the identity matrix. n = {1, 2, ..., n} . diag[ai ] denotes diagonal matrix with the diagonal elements ai. diag[ai ]in=1 denotes block diagonal matrix with diagonal row vector ai.
147
(1h)
The activation functions of DNN (1) given by
f ( x(t )) = [ f1 ( x1 (t ))
f 2 ( x2 (t ))
f n ( xn (t ))]T ,
are bounded and satisfy the following conditions 0≤
V [ A, A]:= { A = (aij ) ∈ℜn×n A ≤ A ≤ A ,i.e., a ij ≤ aij ≤ aij , i, j ∈ n}
fi (ξ1 ) − fi (ξ 2 ) ≤ Li , ξ1 , ξ 2 ∈ ℜ, i ∈ n, ξ1 − ξ 2
(2)
where Li > 0, i ∈ n , are some positive constants. Assume x = [ x1 x2 xn ]T ∈ ℜn is an equilibrium point of system (1), then we can obtain the following system:
with A = ( a ij ) and A = (aij ) .
II. PROBLEM FORMULATION Consider the following uncertain DNN with interval time-varying delay:
z (t ) = −[C + ∆C ]z (t ) + [ A + ∆A] g ( z (t )) + [ B + ∆B ] g ( z (t − τ (t ))),
x (t ) = −[C + ∆C ]x(t ) + [ A + ∆A] y (t ) + [ B + ∆B ] y (t − τ (t )) + J ,
where
t ≥ 0,
(3)
(1a)
y (t ) = f ( x(t )), t ≥ 0 ,
(1b)
x(t ) = φ (t ),
(1c)
t ∈ [−τ M , 0],
z (t ) = [ z1 (t ) z2 (t ) zn (t )]T = x(t ) − x, g ( z (t )) = [ g1 ( z1 (t ))
g 2 ( z2 (t )) g n ( zn (t ))]T ,
gi ( zi (t )) = fi ( xi (t )) − fi ( xi ) = fi ( zi (t ) + xi ) − fi ( xi ),
where x(t ) = [ x1 (t ) x2 (t ) xn (t )] , n ≥ 2 is the number of neurons in the network, 0 ≤ τ (t ) ≤ τ M , τ(t ) ≤ τ D , y(t) is T
the output, J = [ J1 J 2 J n ]T is the external bias vector, C is a positive diagonal matrix, A is the feedback matrix, B is the delay feedback matrix, and φ is the initial continuous function. The linear fractional perturbation matrices ∆C, ∆A, and ∆B are assumed to satisfy the following conditions: ∆C = M C ∆ C (t ) N C , ∆A = M A ∆ A (t ) N A , ∆B = M B ∆ B (t ) N B , (1d)
gi (0) = 0.
(4a)
Let W j = diag[ w ji ] and Y j = diag[ y ji ] , j = 1, 2, be two diagonal matrices with w ji , y ji > 0. From (2) and (4a), we have 0≤
gi ( zi (t )) ≤ Li , 0 ≤ gi ( zi (t )) zi (t ) ≤ Li ⋅ zi2 (t ), zi (t )
0 ≤ g i2 ( zi (t )) ≤ Li ⋅ g i ( zi (t )) zi (t ) ≤ L2i ⋅ zi2 (t ),
(4b) (4c)
where −1
∆ C (t ) = [ I − FC (t )ΘC ] FC (t ), ΘC Θ < I , T C
g T ( z (t ))W1 g ( z (t )) ≤ g T ( z (t ))ΓW1 z (t ), (1e)
g T ( z (t ))ΓW2 z (t ) ≤ z T (t )ΓW2 Γz (t ), −1
∆ A (t ) = [ I − FA (t )Θ A ] FA (t ), Θ AΘ < I , T A
(4d)
(1f) g T ( z (t − τ (t )))Y1 g ( z (t − τ (t ))) ≤ g T ( z (t − τ (t )))ΓY1 z (t − τ (t )),
∆ B (t ) = [ I − FB (t )Θ B ]−1 FB (t ), Θ B ΘTB < I ,
(1g)
where MC, MA, MB, NC, NA, NB, ΘC, ΘA, and ΘB are some given constant matrices with appropriate dimensions. FC(t), FA(t), FB(t) are some unknown matrices representing the parameter
g T ( z (t − τ (t )))ΓY2 z (t − τ (t )) ≤ z T (t − τ (t ))ΓY2 Γz (t − τ (t )),
(4e) where Γ = diag[ Li ] .
Journal of Marine Science and Technology, Vol. 22, No. 2 (2014 )
148
Remark 1. The activation function fi (xi) = 0.5(xi + 1 − xi – 1) is a general form satisfying (2) with Γ = I. Definition 1 [2]. The equilibrium point x of system (1) is said to be the globally exponentially stable (GES) with convergence rate α, if there are two positive constants α and Ψ such that x(t ) − x ≤ Ψ ⋅ e −α t for all t ≥ 0.
Lemma 1 [10, 18]. Suppose ∆ (t ) = [ I − F (t )Θ]−1 F (t ) with unknown matrix F(t) satisfying F T (t ) F (t ) ≤ I , Θ is a given constant matrix and satisfies ΘΘT < I , then for real matrices H, E and X with X = X T, the following statements are equivalent:
(I) The inequality is satisfied X + H ∆ (t ) E + E T ∆T (t ) H T < 0,
(II) There exists a scalar ε > 0, such that X * *
H −ε ⋅ I *
ε ⋅ ΘT < 0.
(5)
−ε ⋅ I
Σ1 + Σ Σ 2 S12 < 0, > 0, R1 > S22 , Σ = * S22 Σ3
0 Σ34
* *
Σ 44 *
Σ16 0 Σ 2 = Σ36 0 0
Σ17 0 Σ37 0 0
Σ18 0 Σ38 0 0
Σ19 0 0 0 0
0 0 0 Σ 410 0
0 0 0 , 0 Σ511
Σ66 * * Σ3 = * * *
0 Σ77 * *
0 0 Σ88 *
Σ69 0 0 Σ99
0 Σ 710 0 0
* *
* *
* *
Σ1010 *
0 0 Σ811 , 0 0 Σ1111
Σ12 = e−2ατ M ⋅ R2 , Σ13 = −C T U T , Σ14 = PA + Γ(W1 − W2 ),
−I
0 0 0]T S12T },
(6)
Σ 25 = Γ(Y1 − Y2 ), Σ33 = −U T − U + τ M ⋅ ( R1 + τ M ⋅ R2 ), Σ34 = UA + V , Σ35 = UB, Σ36 = UM C , Σ37 = UM A , Σ38 = UM B , Σ44 = −2W1 + Q2 , Σ 410 = ε ⋅ N TA , Σ55 = −2Y1 − (1 − τ D ) ⋅ e −2ατ M ⋅ Q2 , Σ511 = ε ⋅ N BT , Σ69 = ε ⋅ ΘTC ,
Σ710 = ε ⋅ ΘTA , Σ811 = ε ⋅ ΘTB , Σ66 = Σ77 = Σ88 = Σ99 = Σ1010 = Σ1111 = −ε ⋅ I . Proof. The Lyapunov functional candidate of the system (1) and (2) is given by V0 ( zt ) = e 2α t ⋅ z T (t ) Pz (t ) + V1 ( zt ) + V2 ( zt ), V1 ( zt ) = ∫
where * is the symmetrical form of matrix,
−I
0 Σ33
Σ15 Σ 25 Σ35 , 0 Σ55
Σ19 = −ε ⋅ NCT , Σ 22 = −e−2ατ M ⋅ [(1 − τ D ) ⋅ Q1 + R2 ] + 2ΓY2Γ,
Theorem 1. The equilibrium point x of system (1) with (2) and τ D ≤ 1 (resp., τ D > 1 or unknown) is unique and globally exponentially stable (GES) with convergence rate α > 0, if there exist some n × n positive definite symmetric matrices P, Q1, Q2 (resp., Q1 = 0, Q2 = 0), R1, R2, S22, a 5n × 5n positive definite symmetric matrix S11, some n × n positive diagonal matrices V, W1, W2, Y1, Y2, some matrices U ∈ ℜn×n, S12 ∈ℜ5n×n, and a positive constant ε, such that the following LMI conditions are satisfied:
+ [I
Σ14
Σ15 = PB, Σ16 = PM C , Σ17 = PM A , Σ18 = PM B ,
In this section, we present a delay-dependent criterion for the global exponential stability of system (1) with (2).
Σ = e−2ατ M ⋅ {τ M ⋅ S11 + S12 ⋅ [ I
Σ13
Σ11 = − PC − C T P + 2α ⋅ P + Q1 + 2α ⋅ΓV + 2ΓW2 Γ − e−2ατ M ⋅ R2 ,
ε ⋅ ET
III. GLOBAL EXPONENTIAL STABILITY ANALYSIS
S11 S= *
Σ11 Σ12 * Σ 22 * Σ1 = * * * * *
0 0 0]
t t −τ ( t )
+∫
(7a)
e 2α s ⋅ [ z T ( s )Q1 z ( s ) + g T ( z ( s ))Q2 g ( z ( s ))]ds
t t −τ M
e 2α s ⋅ ( s − (t − τ M )) ⋅ zT ( s )( R1 + τ M ⋅ R2 ) z ( s )ds,
(7b)
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
n
V2 ( zt ) = 2e 2α t ⋅ ∑ ∫ i =1
zi ( t ) 0
(7c)
vi gi ( s ) ds,
where V = diag[v1 vn ] and the integral term
∫
zi ( t ) 0
From condition in (4c), the time derivative of V2 ( zt ) is bounded by n
vi gi ( s )ds
V2 ( zt ) = 4α ⋅ e2α t ∑ ∫ i =1
is nonnegative in view of (4b). The time derivatives of V0 ( zt ) in (7) along the trajectories of system (3) with (4) satisfy V0 ( zt ) = e
2α t
+e =e
⋅ z (t ) ⋅ 2α Pz (t ) + e T
2α t
2α t
2α t
149
zi ( t ) 0
n
≤ 4α ⋅ e2α t ∑ ∫ i =1
⋅ z (t ) Pz (t )
n
vi gi ( s )ds + 2e 2α t ∑ vi gi ( zi (t )) zi (t ) i =1
zi ( t ) 0
n
vi Li sds + 2e2α t ∑ vi gi ( zi (t )) zi (t ) i =1
T
= e 2α t ⋅ [2α ⋅ z T (t )ΓVz (t ) + 2 g T ( z (t ))Vz (t )].
⋅ z (t ) Pz (t ) + V1 ( zt ) + V2 ( zt )
(8c)
T
Define
⋅ [ z (t )(− PC − C P + 2α ⋅ P ) z (t ) T
T
Z T (t ) = [ z T (t ) z T (t − τ (t )) zT (t ) g T ( z (t )) g T ( z (t − τ (t )))].
+ 2 z T (t ) PAg ( z (t )) + 2 z T (t ) PBg ( z (t − τ (t )))] + V1 ( zt ) + V2 ( zt ),
By Leibniz-Newton formula and LMI (6), the following additional nonnegative inequality can be introduced: (8a) T
where C = C + ∆C , A = A + ∆A, B = B + ∆B. With τ D ≤ 1 , Q1 > 0, Q2 > 0, or τ D > 1, Q1 = 0, Q2 = 0, the time derivative of V1 ( zt ) is bounded by V1 ( zt ) = e2α t [ zT (t )Q1 z (t ) + g T ( z (t ))Q2 g ( z (t ))
0≤e
−2ατ M
S12 Z (t ) ds S22 z ( s )
= e −2ατ M ⋅ {τ (t ) ⋅ Z T (t ) S11Z (t ) + 2Z T (t ) S12 [ z (t ) − z (t − τ (t ))] +∫
− (1 − τ(t )) ⋅ e−2ατ (t ) ⋅ zT (t − τ (t ))Q1 z (t − τ (t ))
Z (t ) S11 ⋅∫ t −τ ( t ) z ( s) * t
t t −τ ( t )
zT ( s ) S22 z ( s )ds}
≤ e −2ατ M ⋅ {τ M ⋅ Z T (t ) S11Z (t ) + 2Z T (t ) S12 [ z (t ) − z (t − τ (t ))]
− (1 − τ(t )) ⋅ e−2ατ (t ) ⋅ g T ( z (t − τ (t )))Q2 g ( z (t − τ (t )))
+∫
+ τ M ⋅ z (t )( R1 + τ M ⋅ R2 ) z (t ) T
t t −τ ( t )
zT ( s ) S 22 z ( s )ds}.
(8d)
From (3), we have −∫
t t −τ M
−τM ⋅ ∫
e
2α ( s −t )
t t −τ M
⋅ z ( s )( R1 + τ M ⋅ R2 ) z ( s )ds T
− zT (t )(U T + U ) z (t )
e 2α ( s −t ) ⋅ z ( s )T R2 z ( s )ds ]
+ zT (t )U [−Cz (t ) + Ag ( z (t )) + Bg ( z (t − τ (t )))] + [Cz (t ) + Ag ( z (t )) + Bg ( z (t − τ (t )))]T U T z (t ) = 0.
≤ e −2α t [ z T (t )Q1 z (t ) + g T ( z (t ))Q2 g ( z (t ))
By the inequality in [4], we have
− (1 − τ D ) ⋅ e −2ατ M ⋅ z T (t − τ (t ))Q1 z (t − τ (t ))
− (1 − τ D ) ⋅ e
−2ατ M
−τ M ⋅ e −2ατ M ⋅ ∫
⋅ g ( z (t − τ (t )))Q2 g ( z (t − τ (t ))) T
t t −τ ( t )
− τ M ⋅ e −2ατ M ⋅ ∫
t
z ( s )T R2 z ( s )ds ].
z ( s )T R2 z ( s )ds
t t −τ ( t )
z ( s )T R2 z ( s )ds T
z ( s )T R1 z ( s )ds
t −τ ( t )
t t −τ ( t )
≤ −τ (t ) ⋅ e −2ατ M ⋅ ∫
+ τ M ⋅ zT (t )( R1 + τ M ⋅ R2 ) z (t )
−e −2ατ M ⋅ ∫
(8e)
t t z ( s )ds R2 ∫ z ( s )ds ≤ −e−2ατ M ⋅ ∫ t −τ (t ) t −τ (t )
(8b)
= −e −2ατ M ⋅ ( z (t ) − z (t − τ (t )))T R2 ( z (t ) − z (t − τ (t ))).
(8f)
Journal of Marine Science and Technology, Vol. 22, No. 2 (2014 )
150
From (4d) and (4e), we have + EC
g T ( z (t ))ΓW1 z (t ) − g T ( z (t ))W1 g ( z (t )) ≥ 0,
z T (t )ΓW2 Γz (t ) − g T ( z (t ))ΓW2 z (t ) ≥ 0,
(9a)
g T ( z (t −τ (t )))ΓY1 z (t −τ (t )) − g T ( z (t −τ (t )))Y1 g ( z (t −τ (t ))) ≥ 0, z T (t − τ (t ))ΓY2 Γz (t − τ (t )) − g T ( z (t − τ (t )))ΓY2 z (t − τ (t )) ≥ 0, (9b)
HC + H A H B
EA
T
0 0 HC ∆ C (t ) EB 0 ∆ A (t ) 0 H A 0 0 ∆ B (t ) H B T
0 0 ∆ C (t ) 0 E ∆ t ( ) 0 A C 0 ∆ B (t ) 0
EA
T
EB , (12)
where
From the inequality R1 > S 22 in (6) and conditions (8)-(9), we have
EC = M CT P 0 M CT U T
0 0 , H C = − N C 0 0 0 0,
V0 ( zt ) + 2e2α t ⋅ [ g T ( z (t ))ΓW1 z (t ) − g T ( z (t ))W1 g ( z (t ))
E A = M AT P 0 M AT U T
0 0 , H A = [ 0 0 0 N A
EB = M BT P 0 M BT U T
0 0 , H B = [ 0 0 0 0 N B ].
+ z T (t )ΓW2 Γz (t ) − g T ( z (t ))ΓW2 z (t )] + 2e 2α t ⋅ [ g T ( z (t − τ (t )))ΓY1 z (t − τ (t ))
− g T ( z (t − τ (t )))ΓY2 z (t − τ (t ))] t −τ ( t )
I − F (t )Θ −1 F (t ) C C C = 0 0
zT ( s )( S 22 − R1 ) z ( s ) ds
≤ e 2α t ⋅ Z T ⋅ Σ1 ⋅ Z ,
(10)
where Σ11 0 * Σ 22 Σ1 = * * * * * *
−C T U
PA + ΓW
0
0
Σ33
UA + V
*
Σ 44
*
*
PB + ΓY 0 UB + Σ , 0 Σ55
(11)
Σ11 = − PC − C T P + 2α ⋅ P + Q1 + 2α ⋅ΓV + 2ΓW2 Γ − e−2ατ M ⋅ R2 , Σ 22 , Σ33 , Σ 44 , Σ55 , and Σ have been defined in (6). From (5), the matrix in (11) can be rearranged as
Σ11 0 * Σ 22 Σ1 = * * * * * *
Σ13
Σ14
0
0
Σ33
Σ34
*
Σ 44
*
*
Σ15 0 Σ35 + Σ 0 Σ55
0] ,
T
0 0 ∆ C (t ) 0 ∆ A (t ) 0 ∆ B (t ) 0 0
+ 2e2α t ⋅ [ z T (t − τ (t ))ΓY2 Γz (t − τ (t ))
t
T
From conditions (1e)-(1g), we have
− g T ( z (t − τ (t )))Y1 g ( z (t − τ (t )))]
≤ e 2α t ⋅ Z T ⋅ Σ1 ⋅ Z + ∫
T
0
[ I − FA (t )Θ A ] 0
FC (t ) 0 0 ΘC FA (t ) 0 0 = I − 0 0 0 FB (t ) 0
FA (t ) 0 −1 [ I − FB (t )Θ B ] FB (t ) 0
−1
0 ΘA 0
0 0 Θ B
−1
0 0 FC (t ) FA (t ) ⋅ 0 0 , 0 0 FB (t ) where 0 0 FC (t ) 0 FA (t ) 0 0 0 FB (t )
T
0 0 FC (t ) 0 FA (t ) 0 ≤ I . 0 0 FB (t )
By using Lemma 1, LMI condition Σ < 0 in (6) will imply Σ1 < 0 in (10). By the S-procedure of [6] with conditions (8)-(10) and Σ1 < 0 , there exists a positive constant ρ > 0 such that 2 V0 ( zt ) ≤ − ρ ⋅ e2α t ⋅ z (t ) .
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
151
From (9) and Σ in (6), we have
From the condition V ( zt ) ≤ 0 , we have
zT [− PC − C T P + 2α ⋅ P + 2α ⋅ ΓV + Q1 − e−2ατ M ⋅ (1 − τ D ) ⋅ Q1
V0 ( zt ) ≤ V0 ( z0 ) , where
+ 2Γ(W2 + Y2 )Γ]z + 2 zT [ PA + PB ]g ( z )
V0 ( z0 ) = z T (0) Pz (0)
+ 2 zT [Γ(W1 − W2 ) + Γ(Y1 − Y2 )]g ( z ) + g T ( z )[−2W1 − 2Y1 ]g ( z )
0
+∫
−τ ( t )
+∫
−τ M
0
e 2α s ⋅ [ z T ( s )Q1 z ( s ) + g T ( z ( s ))Q2 g ( z ( s ))]ds
+ g T ( z )[Q2 − e −2ατ M ⋅ (1 − τ D ) ⋅ Q2 ]g ( z ) ≥ 0,
e 2α s ⋅ ( s + τ M ) ⋅ zT ( s )( R1 + τ M ⋅ R2 ) z ( s )ds
[ zT zT 0 g T ( z ) g T ( z )] Σ [ zT zT 0 g T ( z ) g T ( z )]T ≥ 0.
n
+ 2∑ ∫ i =1
zi (0) 0
This implies
vi gi ( s )ds
[ zT zT 0 g T ( z ) g T ( z )] Σ1 [ zT zT 0 g T ( z ) g T ( z )]T ≥ 0,
≤ [λ max ( P ) + τ M ⋅ λ max (Q1 ) + τ M ⋅ λ max (ΓQ2 Γ)
+ τ M2 ⋅ λ max ( R1 + τ M ⋅ R2 ) + λ max (ΓV ) ⋅ z0
where Σ1 is defined in (11). Note that the condition Σ < 0 in
2 s
(6) is equivalent to Σ1 < 0 in (11), this will imply the result
2
= δ1 ⋅ z 0 s ,
z = g ( z ) = [0 0]T .
Hence the equilibrium point z =
[0 0]T is unique i.e., x is the unique equilibrium point □ of uncertain DNN (1). This completes the proof.
with
δ1 = λ max ( P) + τ M ⋅ λ max (Q1 ) + τ M ⋅ λ max (ΓQ2 Γ) + τ M2 ⋅ λ max ( R1 + τ M ⋅ R2 ) + λ max (ΓV ).
On the other hand, we have
V0 ( zt ) ≥ e2α t ⋅ z T (t ) Pz (t ) ≥ λ min ( P) ⋅ e2α t ⋅ z (t ) . 2
Consequently, we have z (t ) = x(t ) − x ≤
δ1 λ mim ( P)
⋅ z0 s ⋅ e −α t , t ≥ 0.
From Definition 1, this implies that the equilibrium point x of system (1) is globally exponentially stable with convergence rate α. Next we will prove the uniqueness of the equilibrium point x , i.e., the equilibrium point z = [0 … 0]T of (3). Assume z is an equilibrium point of the system (3). Then we have
− Cz + Ag ( z ) + Bg ( z ) = 0. Multiplying both sides of preceding equation by 2 zT P , we have zT (− PC − C T P) z + 2 zT PAg ( z ) + 2 zT PBg ( z ) = 0.
By setting α = 0 and R1 = R2 = U = S = 0 in Theorem 1, we may obtain the following delay-independent asymptotic stability condition (independent of τM) for system (1) with (2). Corollary 1. The equilibrium point x of system (1) with (2) and τD ≤ 1 (resp., τD > 1 or unknown) is unique and globally asymptotically stable (GAS), if there exist some n × n positive definite symmetric matrices P, Q1, Q2 (resp., Q1 = 0, Q2 = 0), some n × n positive diagonal matrices W1, W2, Y1, Y2, and a positive constant ε, such that the following LMI conditions are satisfied: Σˆ 11 0 * Σˆ 22 * * * * * * Σˆ = * * * * * * * * * *
Σˆ 13
Σˆ 14
Σˆ 15
Σˆ 16
Σˆ 17
Σˆ 18
0
0 ˆΣ 33
0
0
0
0
0
0
0
0
0
*
0 ˆΣ 44
0 ˆΣ 39
0
0
*
*
0 ˆΣ 55
0
*
*
*
0 ˆΣ 66
0 ˆΣ 58
*
*
*
*
0 ˆΣ 77
*
*
*
*
*
0 ˆΣ 88
*
*
*
*
*
*
0 ˆΣ 99
*
*
*
*
*
*
*
0
0 0 ˆΣ 69 0
0 0 0 Σˆ 410 0 < 0, 0 Σˆ 710 0 0 Σˆ 1010
(13) where
Journal of Marine Science and Technology, Vol. 22, No. 2 (2014 )
152
Σˆ 11 = − PC − C T P + Q1 + 2ΓW2 Γ, Σˆ 13 = PA + Γ(W1 − W2 ),
Table 1. Some comparisons for system (1) with (2) and (14).
Σˆ 14 = PB, Σˆ 15 = PM C , Σˆ 16 = PM A , Σˆ 17 = PM B , Σˆ 18 = −ε ⋅ N CT , Σˆ 22 = −[(1 − τ D ) ⋅ Q1 ] + 2ΓY2 Γ, Σˆ 24 = Γ(Y1 − Y2 ),
Some upper bounds of the time delay for the stability of system (1) with (2) and (14) Results [10] Our results
τD = 0 (Constant) τD = 0.5
τM < ∞
τM < ∞ (GAS)
τM = 2.9653
τM < ∞ (GAS)
Σˆ 410 = ε ⋅ N BT , Σˆ 58 = ε ⋅ ΘTC , Σˆ 69 = ε ⋅ ΘTA , Σˆ 710 = ε ⋅ ΘTB ,
τD = 0.9
τM = 0.8629
τM < ∞ (GAS)
Σˆ 55 = Σˆ 66 = Σˆ 77 = Σˆ 88 = Σˆ 99 = Σˆ 1010 = −ε ⋅ I .
τD = 1 or unknown
Not provided
Σˆ 33 = −2W1 + Q2 , Σˆ 39 = ε ⋅ N , Σˆ 44 = −2Y2 − (1 − τ D ) ⋅ Q2 , T A
Remark 2. In Corollary 1 with α = 0 and R1 = R2 = U = S = 0, the obtained result is delay-independent of τM. Hence the result “τM < ∞” can be guaranteed when the LMI (13) is feasible.
α = 0, τM = 5000000 (GAS) α = 0.1, τM = 215 (GES)
IV. NUMERICAL EXAMPLES
With α = 0.1, τD = 1, τM = 215, LMI conditions in Theorem 1 still have a feasible solution. The system (1) with (2), (14), τD = 1, and τM = 215, is exponentially stable with convergence rate α = 0.1. In order to show the improvement, we summarize some comparisons in Table 1.
Example 1. Consider the UDNNs in (1) with (2) and the following parameters: (Example 2 of [10])
Example 2. Consider the UDNNs in (1) with (2) and the following parameters: (Example 1 of [14])
1 0 −1 0.5 −2 0.5 C= , A = 0.5 −1.5 , B = 0.5 −2 , 0 1
0 0 0 1.2769 0 0.6231 0 0 C= , 0 0 0.923 0 0 0 0.448 0
0 0 MC = M A = M B = , −0.1 −0.1 1 0 0.5 0 0.1 0.1 NC = , NA = , NB = , 0 0 0.5 0 1 0
0.4 0 Γ= , ΘC = Θ A = Θ B = 0.3. 0 0.8
(14)
With τD = 0.2, LMI conditions in Corollary 1 have a feasible solution: 0.8112 0.1937 0.3581 0.0991 P= , Q1 = , 0.1937 0.9452 0.0991 0.2138 0 2.9551 −0.6585 0.8847 Q2 = , V = , 0.8847 −0.6585 3.1723 0
−0.0373 −1.6033 A= 0.3394 −0.1311
0.4852 0.5988 −0.086 0.3253
−0.3351 0.2336 −0.3224 1.2352 , −0.3824 −0.5785 −0.9534 −0.5015
0.8674 −1.2405 −0.5325 0.022 0.0474 −0.9164 0.036 0.9816 B= , 1.8495 2.6117 −0.3788 0.8428 −2.0413 0.5179 1.1734 −0.2775 0 0 0 L1 0 0 0 0.1137 0 0.1279 0 0 0 L2 0 0 , Γ= = 0 0 0.7994 0 0 0 L3 0 0 0 0.2368 0 0 0 L4 0
0 0 2.3825 0.1643 W1 = , W2 = , 1.9513 0.0368 0 0
M C = M A = M B = N C = N A = N B = 0, ΘC = Θ A = Θ B = 0. (15)
0 0 0.6111 0.3178 Y1 = , Y2 = , ε = 0.2989. 0.165 0.0491 0 0
2.
The system (1) with (2), (14), and τD = 0.2, is asymptotically stable and the equilibrium point x is unique. With τD = 1, LMI conditions in Corollary 1 are not feasible. Hence Theorem 1 should be used to show delay-dependent results.
50, J = 0, x(t ) = [10 −10 5 −5]T , t ∈ [−50, 0] , the system state trajectories are shown in Fig. 1, where 0 is the unique equilibrium point.
Some comparisons of proposed results are shown in Table With fi ( xi ) = Li × 0.5 ( xi + 1 − xi − 1 ) , i = 1, 2, 3, 4, τ (t) =
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
153
Table 2. Some comparisons for system (1) with (2) and (15).
REFERENCES
Some upper bounds of the time delay for the stability of system (1) with (2) and (15) Results [7] [14] Our results τD = 0 1 ≤ τ (t) ≤ 3.5841 1 ≤ τ (t) ≤ 3.8363 τM < ∞ (GAS) τD = 0.5 1 ≤ τ (t) ≤ 2.5802 1 ≤ τ (t) ≤ 2.7299 τM < ∞ (GAS) τD = 0.9 1 ≤ τ (t) ≤ 2.2736 1 ≤ τ (t) ≤ 2.3811 τM < ∞ (GAS) α = 0, τM = 12016155 τD = 1 or 1 ≤ τ (t) ≤ 2.2393 1 ≤ τ (t) ≤ 2.3114 (GAS) unknown α = 0.1, τM = 229 (GAS)
1. Boyd, S. P., Ghaoui, El, Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System And Control Theory, SIAM, Philadelphia, USA (1994). 2. Chen, A., Cao, J., and Huang, L., “Global robust stability of interval cellular neural networks with time-varying delays,” Chaos, Solitons & Fractals, Vol. 23, pp. 787-799 (2005). 3. Chua, L. O. and Roska, T., Cellular Neural Networks and Visual Computting, Cambridge University Press, Cambridge, U.K. (2002). 4. Gau, R. S., Lien, C. H., and Hsieh, J. G., “Global exponential stability for uncertain cellular neural networks with multiple time-varying delays via LMI approach,” Chaos, Solitons & Fractals, Vol. 32, pp. 1258-1267 (2007). 5. Gau, R. S., Lien, C. H., and Hsieh, J. G., “Novel stability conditions for interval delayed neural networks with multiple time-varying delays,” International Journal of Innovative Computing, Information and Control, Vol. 7, pp. 433-444 (2011). 6. Gu, K., Kharitonov, V. L., and Chen, J., Stability of Time-Delay Systems, Birkhauser, Boston, Massachusetts, USA (2003). 7. He, Y., Liu, G. P., and Ress, D., “New delay-dependent stability criteria for neural networks with time-varying delay,” IEEE Transactions on Neural Networks, Vol. 18, pp. 310-314 (2007). 8. Li, C., Liao, X., and Zhang, R., “Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach,” Physics Letters A, Vol. 328, pp. 452-462 (2004). 9. Li, C., Liao, X., Zhang, R., and Prasad, A., “Global robust exponential stability analysis for interval neural networks with time-varying delays,” Chaos, Solitons & Fractals, Vol. 25, pp. 751-757 (2005). 10. Li, T., Guo, L., and Sun, C., “Robust stability for neural networks with time-varying delays and linear fractional uncertainties,” Neurocomputing, Vol. 71, pp. 421-427 (2007). 11. Liao, X. and Wang, J., “Global and robust stability of interval Hopfield neural networks with time-varying delays,” International Journal of Neural Systems, Vol. 13, pp. 177-182 (2003). 12. Liao, X., Wong, K. K., Wu, Z., and Chen, G., “Novel robust stability criteria for interval-delayed Hopfield neural networks,” IEEE Transactions on Circuits and Systems I, Vol. 48, pp. 1355-1359 (2001). 13. Lien, C. H., Yu, K. W., Lin, Y. F., Chang, H. C., and Chung, Y. J., “Stability analysis for Cohen-Grossberg neural networks with time-varying delays via LMI approach,” Expert Systems with Applications, Vol. 38, pp. 6360-6367 (2011). 14. Tian, J. and Zhou, X., “Improved asymptotic stability criteria for neural networks with interval time-varying delay,” Expert Systems with Applications, Vol. 37, pp. 7521-7525 (2010). 15. Xu, S., Lam, J., and Ho, D. W. C., “Novel global robust stability criteria for interval neural networks with multiple time-varying delays,” Physics Letters A, Vol. 342, pp. 322-330 (2005). 16. Yang, X., Li, C., Liao, X., Evans, D. J., and Megson, G. M., “Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays,” Applied Mathematics and Computation, Vol. 171, pp. 108-121 (2005). 17. Yu, K. W., “Global exponential stability criteria for switched systems with interval time-varying delay,” Journal of Marine Science and Technology, Vol. 18, pp. 298-307 (2010). 18. Zheng, C. D., Jing, X. T., Wang, Z. S., and Feng, J., “Further results for robust stability of cellular neural networks with linear fractional uncertainty,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, pp. 3046-3057 (2010).
10
x1 x2 x3 x4
8 6 4 2 0 -2 -4 -6 -8 -10
0
50
100 150 200 250 300 350 400 450 500 t
Fig. 1. The system state trajectories of DNN (1) with (15).
V. CONCLUSIONS In this paper, the global stability and uniqueness of equilibrium point for a class of uncertain delayed neural networks with time-varying delay and linear fractional perturbations has been investigated. Based on the LMI approach and some proposed additional nonnegative inequalities, some delaydependent and delay-independent criteria have been proposed to guarantee the exponential stability and asymptotic stability of UDNN, respectively. Some numerical examples by using the proposed results have shown great improvement over recent published results.
ACKNOWLEDGMENTS The research reported here was supported by the National Science Council of Taiwan, R.O.C. under grant no. NSC 99-2221-E-022-003 and NSC 101-2514-S-022-001. The authors would like to thank the editor and anonymous reviewers for their helpful comments.
146
EXPONENTIAL STABILITY ANALYSIS FOR NEURAL NETWORKS WITH TIME-VARYING DELAY AND LINEAR FRACTIONAL PERTURBATIONS Chang-Hua Lien and Ker-Wei Yu
Key words: delayed neural network, global exponential stability, delay-dependent criterion, delay-independent criterion, linear fractional perturbation, linear matrix inequality.
ABSTRACT In this paper, the global exponential stability and global asymptotic stability for a class of uncertain delayed neural networks (UDNNs) with time-varying delay and linear fractional perturbations are considered. Delay-dependent and delay-independent criteria are proposed to guarantee the robust stability of UDNNs via linear matrix inequality (LMI) approach. Additional nonnegative inequality approach is used to improve the conservativeness of the stability criteria. Some numerical examples are illustrated to show the effectiveness of our results. From the simulation results, significant improvement over the recent results can be observed.
I. INTRODUCTION The existence of time delays is often a source of oscillation and instability of practical systems. Neural networks has been applied in many mathematical and practical applications, such as approximation, association, diagnosis, decision, generalization, optimization, prediction, and recognition. Many neural networks have been proposed in recent years, such as bidirectional associative memory neural networks [16], cellular neural networks [3], Cohen-Grossberg neural networks [13], and Hopfield neural networks [11]. The delayed neural networks (DNNs) may be used in many areas including the moving images processing and pattern classification. The implementation in hardware for very large scale integration chip, modeling errors, parameters fluctua-
Paper submitted 01/31/12; revised 08/24/12; accepted 02/07/13. Author for correspondence: Chang-Hua Lien (e-mail: [email protected]). Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung, Taiwan, R.O.C.
tion, and external disturbance may destory the stability of DNNs. Hence stability of DNNs is very important and significant in practical applications. In practical analysis for uncertain DNNs, it is reasonable to consider the parameters varying in some prescribed intervals or staisfying some classes of parametric uncertainties. DNNs with interval variations are called the interval delayed neural networks (IDNNs) [2, 5, 8, 9, 11, 12, 15]. In [10] and [18], DNNs with linear fractional parametric perturbations have been investigated. IDNNs and DNNs with general structural perturbation in [4] are speical cases of DNNs with linear fractional parametric perturbations. Hence we will consider the stability analysis of DNNs with linear fractional parametric perturbations in this paper. Depending on whether the stability criterion itself contains the size of delay, criteria for DNN can be classified into two categories, namely delay-independent criteria [2, 5, 9, 12] and delay-dependent criteria [2, 4, 5, 8-10, 15, 18]. Usually the latter is less conservative when the delay is small. In the Lyapunov-based delay-dependent results, the slow-varying constraint τ(t ) < 1 is usually imposed on the time-varying delay [8, 9, 11, 15]. The constraint will be relaxed and delay-dependent result will be proposed in this paper. In [2, 12], algebraic stability, criteria were proposed based on Halanay inequality, Young’s inequality, and Lyapunov functional. It is usually difficult to find feasible solutions for these algebraic criteria. LMI approach is an efficient tool for dealng with these control problems. The LMI problem can be solved quite efficiently by using the toolbox of Matlab [1]. In [4, 5, 7-11, 13-15, 18], stability criteria for DNNs have been proposed via LMI approach. Additional nonnegative inequality approach is used to improve the conservativeness of the obtained results [17]. In this paper, LMI-based delay-dependent and delayindependent criteria are proposed by using new Lyapunov functional. In general, our approach is useful and is easy to generalize to other forms of UDNNs. The notation used throughout this paper is as follows. For a matrix A, we denote the transpose by AT, spectral norm by A , minimal (maximal) eigenvalue by λmin(A) (λmax(A)),
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
symmetric positive (negative) definite by A > 0 (A < 0). A ≤ B means that matrix B – A is symmetric positive semi-definite. For a vector x, we denote the Euclidean norm by x . For
perturbations which satisfy FCT (t ) FC (t ) ≤ I , FAT (t ) FA (t ) ≤ I , FBT (t ) FB (t ) ≤ I .
the state xt of system, we define xt(θ ):= x(t + θ ), ∀θ ∈ [−τM, 0] and denote its norm by xt
s
2
2
x(t + s ) + x (t + s ) .
= sup
−τ M ≤ s ≤0
I denotes the identity matrix. n = {1, 2, ..., n} . diag[ai ] denotes diagonal matrix with the diagonal elements ai. diag[ai ]in=1 denotes block diagonal matrix with diagonal row vector ai.
147
(1h)
The activation functions of DNN (1) given by
f ( x(t )) = [ f1 ( x1 (t ))
f 2 ( x2 (t ))
f n ( xn (t ))]T ,
are bounded and satisfy the following conditions 0≤
V [ A, A]:= { A = (aij ) ∈ℜn×n A ≤ A ≤ A ,i.e., a ij ≤ aij ≤ aij , i, j ∈ n}
fi (ξ1 ) − fi (ξ 2 ) ≤ Li , ξ1 , ξ 2 ∈ ℜ, i ∈ n, ξ1 − ξ 2
(2)
where Li > 0, i ∈ n , are some positive constants. Assume x = [ x1 x2 xn ]T ∈ ℜn is an equilibrium point of system (1), then we can obtain the following system:
with A = ( a ij ) and A = (aij ) .
II. PROBLEM FORMULATION Consider the following uncertain DNN with interval time-varying delay:
z (t ) = −[C + ∆C ]z (t ) + [ A + ∆A] g ( z (t )) + [ B + ∆B ] g ( z (t − τ (t ))),
x (t ) = −[C + ∆C ]x(t ) + [ A + ∆A] y (t ) + [ B + ∆B ] y (t − τ (t )) + J ,
where
t ≥ 0,
(3)
(1a)
y (t ) = f ( x(t )), t ≥ 0 ,
(1b)
x(t ) = φ (t ),
(1c)
t ∈ [−τ M , 0],
z (t ) = [ z1 (t ) z2 (t ) zn (t )]T = x(t ) − x, g ( z (t )) = [ g1 ( z1 (t ))
g 2 ( z2 (t )) g n ( zn (t ))]T ,
gi ( zi (t )) = fi ( xi (t )) − fi ( xi ) = fi ( zi (t ) + xi ) − fi ( xi ),
where x(t ) = [ x1 (t ) x2 (t ) xn (t )] , n ≥ 2 is the number of neurons in the network, 0 ≤ τ (t ) ≤ τ M , τ(t ) ≤ τ D , y(t) is T
the output, J = [ J1 J 2 J n ]T is the external bias vector, C is a positive diagonal matrix, A is the feedback matrix, B is the delay feedback matrix, and φ is the initial continuous function. The linear fractional perturbation matrices ∆C, ∆A, and ∆B are assumed to satisfy the following conditions: ∆C = M C ∆ C (t ) N C , ∆A = M A ∆ A (t ) N A , ∆B = M B ∆ B (t ) N B , (1d)
gi (0) = 0.
(4a)
Let W j = diag[ w ji ] and Y j = diag[ y ji ] , j = 1, 2, be two diagonal matrices with w ji , y ji > 0. From (2) and (4a), we have 0≤
gi ( zi (t )) ≤ Li , 0 ≤ gi ( zi (t )) zi (t ) ≤ Li ⋅ zi2 (t ), zi (t )
0 ≤ g i2 ( zi (t )) ≤ Li ⋅ g i ( zi (t )) zi (t ) ≤ L2i ⋅ zi2 (t ),
(4b) (4c)
where −1
∆ C (t ) = [ I − FC (t )ΘC ] FC (t ), ΘC Θ < I , T C
g T ( z (t ))W1 g ( z (t )) ≤ g T ( z (t ))ΓW1 z (t ), (1e)
g T ( z (t ))ΓW2 z (t ) ≤ z T (t )ΓW2 Γz (t ), −1
∆ A (t ) = [ I − FA (t )Θ A ] FA (t ), Θ AΘ < I , T A
(4d)
(1f) g T ( z (t − τ (t )))Y1 g ( z (t − τ (t ))) ≤ g T ( z (t − τ (t )))ΓY1 z (t − τ (t )),
∆ B (t ) = [ I − FB (t )Θ B ]−1 FB (t ), Θ B ΘTB < I ,
(1g)
where MC, MA, MB, NC, NA, NB, ΘC, ΘA, and ΘB are some given constant matrices with appropriate dimensions. FC(t), FA(t), FB(t) are some unknown matrices representing the parameter
g T ( z (t − τ (t )))ΓY2 z (t − τ (t )) ≤ z T (t − τ (t ))ΓY2 Γz (t − τ (t )),
(4e) where Γ = diag[ Li ] .
Journal of Marine Science and Technology, Vol. 22, No. 2 (2014 )
148
Remark 1. The activation function fi (xi) = 0.5(xi + 1 − xi – 1) is a general form satisfying (2) with Γ = I. Definition 1 [2]. The equilibrium point x of system (1) is said to be the globally exponentially stable (GES) with convergence rate α, if there are two positive constants α and Ψ such that x(t ) − x ≤ Ψ ⋅ e −α t for all t ≥ 0.
Lemma 1 [10, 18]. Suppose ∆ (t ) = [ I − F (t )Θ]−1 F (t ) with unknown matrix F(t) satisfying F T (t ) F (t ) ≤ I , Θ is a given constant matrix and satisfies ΘΘT < I , then for real matrices H, E and X with X = X T, the following statements are equivalent:
(I) The inequality is satisfied X + H ∆ (t ) E + E T ∆T (t ) H T < 0,
(II) There exists a scalar ε > 0, such that X * *
H −ε ⋅ I *
ε ⋅ ΘT < 0.
(5)
−ε ⋅ I
Σ1 + Σ Σ 2 S12 < 0, > 0, R1 > S22 , Σ = * S22 Σ3
0 Σ34
* *
Σ 44 *
Σ16 0 Σ 2 = Σ36 0 0
Σ17 0 Σ37 0 0
Σ18 0 Σ38 0 0
Σ19 0 0 0 0
0 0 0 Σ 410 0
0 0 0 , 0 Σ511
Σ66 * * Σ3 = * * *
0 Σ77 * *
0 0 Σ88 *
Σ69 0 0 Σ99
0 Σ 710 0 0
* *
* *
* *
Σ1010 *
0 0 Σ811 , 0 0 Σ1111
Σ12 = e−2ατ M ⋅ R2 , Σ13 = −C T U T , Σ14 = PA + Γ(W1 − W2 ),
−I
0 0 0]T S12T },
(6)
Σ 25 = Γ(Y1 − Y2 ), Σ33 = −U T − U + τ M ⋅ ( R1 + τ M ⋅ R2 ), Σ34 = UA + V , Σ35 = UB, Σ36 = UM C , Σ37 = UM A , Σ38 = UM B , Σ44 = −2W1 + Q2 , Σ 410 = ε ⋅ N TA , Σ55 = −2Y1 − (1 − τ D ) ⋅ e −2ατ M ⋅ Q2 , Σ511 = ε ⋅ N BT , Σ69 = ε ⋅ ΘTC ,
Σ710 = ε ⋅ ΘTA , Σ811 = ε ⋅ ΘTB , Σ66 = Σ77 = Σ88 = Σ99 = Σ1010 = Σ1111 = −ε ⋅ I . Proof. The Lyapunov functional candidate of the system (1) and (2) is given by V0 ( zt ) = e 2α t ⋅ z T (t ) Pz (t ) + V1 ( zt ) + V2 ( zt ), V1 ( zt ) = ∫
where * is the symmetrical form of matrix,
−I
0 Σ33
Σ15 Σ 25 Σ35 , 0 Σ55
Σ19 = −ε ⋅ NCT , Σ 22 = −e−2ατ M ⋅ [(1 − τ D ) ⋅ Q1 + R2 ] + 2ΓY2Γ,
Theorem 1. The equilibrium point x of system (1) with (2) and τ D ≤ 1 (resp., τ D > 1 or unknown) is unique and globally exponentially stable (GES) with convergence rate α > 0, if there exist some n × n positive definite symmetric matrices P, Q1, Q2 (resp., Q1 = 0, Q2 = 0), R1, R2, S22, a 5n × 5n positive definite symmetric matrix S11, some n × n positive diagonal matrices V, W1, W2, Y1, Y2, some matrices U ∈ ℜn×n, S12 ∈ℜ5n×n, and a positive constant ε, such that the following LMI conditions are satisfied:
+ [I
Σ14
Σ15 = PB, Σ16 = PM C , Σ17 = PM A , Σ18 = PM B ,
In this section, we present a delay-dependent criterion for the global exponential stability of system (1) with (2).
Σ = e−2ατ M ⋅ {τ M ⋅ S11 + S12 ⋅ [ I
Σ13
Σ11 = − PC − C T P + 2α ⋅ P + Q1 + 2α ⋅ΓV + 2ΓW2 Γ − e−2ατ M ⋅ R2 ,
ε ⋅ ET
III. GLOBAL EXPONENTIAL STABILITY ANALYSIS
S11 S= *
Σ11 Σ12 * Σ 22 * Σ1 = * * * * *
0 0 0]
t t −τ ( t )
+∫
(7a)
e 2α s ⋅ [ z T ( s )Q1 z ( s ) + g T ( z ( s ))Q2 g ( z ( s ))]ds
t t −τ M
e 2α s ⋅ ( s − (t − τ M )) ⋅ zT ( s )( R1 + τ M ⋅ R2 ) z ( s )ds,
(7b)
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
n
V2 ( zt ) = 2e 2α t ⋅ ∑ ∫ i =1
zi ( t ) 0
(7c)
vi gi ( s ) ds,
where V = diag[v1 vn ] and the integral term
∫
zi ( t ) 0
From condition in (4c), the time derivative of V2 ( zt ) is bounded by n
vi gi ( s )ds
V2 ( zt ) = 4α ⋅ e2α t ∑ ∫ i =1
is nonnegative in view of (4b). The time derivatives of V0 ( zt ) in (7) along the trajectories of system (3) with (4) satisfy V0 ( zt ) = e
2α t
+e =e
⋅ z (t ) ⋅ 2α Pz (t ) + e T
2α t
2α t
2α t
149
zi ( t ) 0
n
≤ 4α ⋅ e2α t ∑ ∫ i =1
⋅ z (t ) Pz (t )
n
vi gi ( s )ds + 2e 2α t ∑ vi gi ( zi (t )) zi (t ) i =1
zi ( t ) 0
n
vi Li sds + 2e2α t ∑ vi gi ( zi (t )) zi (t ) i =1
T
= e 2α t ⋅ [2α ⋅ z T (t )ΓVz (t ) + 2 g T ( z (t ))Vz (t )].
⋅ z (t ) Pz (t ) + V1 ( zt ) + V2 ( zt )
(8c)
T
Define
⋅ [ z (t )(− PC − C P + 2α ⋅ P ) z (t ) T
T
Z T (t ) = [ z T (t ) z T (t − τ (t )) zT (t ) g T ( z (t )) g T ( z (t − τ (t )))].
+ 2 z T (t ) PAg ( z (t )) + 2 z T (t ) PBg ( z (t − τ (t )))] + V1 ( zt ) + V2 ( zt ),
By Leibniz-Newton formula and LMI (6), the following additional nonnegative inequality can be introduced: (8a) T
where C = C + ∆C , A = A + ∆A, B = B + ∆B. With τ D ≤ 1 , Q1 > 0, Q2 > 0, or τ D > 1, Q1 = 0, Q2 = 0, the time derivative of V1 ( zt ) is bounded by V1 ( zt ) = e2α t [ zT (t )Q1 z (t ) + g T ( z (t ))Q2 g ( z (t ))
0≤e
−2ατ M
S12 Z (t ) ds S22 z ( s )
= e −2ατ M ⋅ {τ (t ) ⋅ Z T (t ) S11Z (t ) + 2Z T (t ) S12 [ z (t ) − z (t − τ (t ))] +∫
− (1 − τ(t )) ⋅ e−2ατ (t ) ⋅ zT (t − τ (t ))Q1 z (t − τ (t ))
Z (t ) S11 ⋅∫ t −τ ( t ) z ( s) * t
t t −τ ( t )
zT ( s ) S22 z ( s )ds}
≤ e −2ατ M ⋅ {τ M ⋅ Z T (t ) S11Z (t ) + 2Z T (t ) S12 [ z (t ) − z (t − τ (t ))]
− (1 − τ(t )) ⋅ e−2ατ (t ) ⋅ g T ( z (t − τ (t )))Q2 g ( z (t − τ (t )))
+∫
+ τ M ⋅ z (t )( R1 + τ M ⋅ R2 ) z (t ) T
t t −τ ( t )
zT ( s ) S 22 z ( s )ds}.
(8d)
From (3), we have −∫
t t −τ M
−τM ⋅ ∫
e
2α ( s −t )
t t −τ M
⋅ z ( s )( R1 + τ M ⋅ R2 ) z ( s )ds T
− zT (t )(U T + U ) z (t )
e 2α ( s −t ) ⋅ z ( s )T R2 z ( s )ds ]
+ zT (t )U [−Cz (t ) + Ag ( z (t )) + Bg ( z (t − τ (t )))] + [Cz (t ) + Ag ( z (t )) + Bg ( z (t − τ (t )))]T U T z (t ) = 0.
≤ e −2α t [ z T (t )Q1 z (t ) + g T ( z (t ))Q2 g ( z (t ))
By the inequality in [4], we have
− (1 − τ D ) ⋅ e −2ατ M ⋅ z T (t − τ (t ))Q1 z (t − τ (t ))
− (1 − τ D ) ⋅ e
−2ατ M
−τ M ⋅ e −2ατ M ⋅ ∫
⋅ g ( z (t − τ (t )))Q2 g ( z (t − τ (t ))) T
t t −τ ( t )
− τ M ⋅ e −2ατ M ⋅ ∫
t
z ( s )T R2 z ( s )ds ].
z ( s )T R2 z ( s )ds
t t −τ ( t )
z ( s )T R2 z ( s )ds T
z ( s )T R1 z ( s )ds
t −τ ( t )
t t −τ ( t )
≤ −τ (t ) ⋅ e −2ατ M ⋅ ∫
+ τ M ⋅ zT (t )( R1 + τ M ⋅ R2 ) z (t )
−e −2ατ M ⋅ ∫
(8e)
t t z ( s )ds R2 ∫ z ( s )ds ≤ −e−2ατ M ⋅ ∫ t −τ (t ) t −τ (t )
(8b)
= −e −2ατ M ⋅ ( z (t ) − z (t − τ (t )))T R2 ( z (t ) − z (t − τ (t ))).
(8f)
Journal of Marine Science and Technology, Vol. 22, No. 2 (2014 )
150
From (4d) and (4e), we have + EC
g T ( z (t ))ΓW1 z (t ) − g T ( z (t ))W1 g ( z (t )) ≥ 0,
z T (t )ΓW2 Γz (t ) − g T ( z (t ))ΓW2 z (t ) ≥ 0,
(9a)
g T ( z (t −τ (t )))ΓY1 z (t −τ (t )) − g T ( z (t −τ (t )))Y1 g ( z (t −τ (t ))) ≥ 0, z T (t − τ (t ))ΓY2 Γz (t − τ (t )) − g T ( z (t − τ (t )))ΓY2 z (t − τ (t )) ≥ 0, (9b)
HC + H A H B
EA
T
0 0 HC ∆ C (t ) EB 0 ∆ A (t ) 0 H A 0 0 ∆ B (t ) H B T
0 0 ∆ C (t ) 0 E ∆ t ( ) 0 A C 0 ∆ B (t ) 0
EA
T
EB , (12)
where
From the inequality R1 > S 22 in (6) and conditions (8)-(9), we have
EC = M CT P 0 M CT U T
0 0 , H C = − N C 0 0 0 0,
V0 ( zt ) + 2e2α t ⋅ [ g T ( z (t ))ΓW1 z (t ) − g T ( z (t ))W1 g ( z (t ))
E A = M AT P 0 M AT U T
0 0 , H A = [ 0 0 0 N A
EB = M BT P 0 M BT U T
0 0 , H B = [ 0 0 0 0 N B ].
+ z T (t )ΓW2 Γz (t ) − g T ( z (t ))ΓW2 z (t )] + 2e 2α t ⋅ [ g T ( z (t − τ (t )))ΓY1 z (t − τ (t ))
− g T ( z (t − τ (t )))ΓY2 z (t − τ (t ))] t −τ ( t )
I − F (t )Θ −1 F (t ) C C C = 0 0
zT ( s )( S 22 − R1 ) z ( s ) ds
≤ e 2α t ⋅ Z T ⋅ Σ1 ⋅ Z ,
(10)
where Σ11 0 * Σ 22 Σ1 = * * * * * *
−C T U
PA + ΓW
0
0
Σ33
UA + V
*
Σ 44
*
*
PB + ΓY 0 UB + Σ , 0 Σ55
(11)
Σ11 = − PC − C T P + 2α ⋅ P + Q1 + 2α ⋅ΓV + 2ΓW2 Γ − e−2ατ M ⋅ R2 , Σ 22 , Σ33 , Σ 44 , Σ55 , and Σ have been defined in (6). From (5), the matrix in (11) can be rearranged as
Σ11 0 * Σ 22 Σ1 = * * * * * *
Σ13
Σ14
0
0
Σ33
Σ34
*
Σ 44
*
*
Σ15 0 Σ35 + Σ 0 Σ55
0] ,
T
0 0 ∆ C (t ) 0 ∆ A (t ) 0 ∆ B (t ) 0 0
+ 2e2α t ⋅ [ z T (t − τ (t ))ΓY2 Γz (t − τ (t ))
t
T
From conditions (1e)-(1g), we have
− g T ( z (t − τ (t )))Y1 g ( z (t − τ (t )))]
≤ e 2α t ⋅ Z T ⋅ Σ1 ⋅ Z + ∫
T
0
[ I − FA (t )Θ A ] 0
FC (t ) 0 0 ΘC FA (t ) 0 0 = I − 0 0 0 FB (t ) 0
FA (t ) 0 −1 [ I − FB (t )Θ B ] FB (t ) 0
−1
0 ΘA 0
0 0 Θ B
−1
0 0 FC (t ) FA (t ) ⋅ 0 0 , 0 0 FB (t ) where 0 0 FC (t ) 0 FA (t ) 0 0 0 FB (t )
T
0 0 FC (t ) 0 FA (t ) 0 ≤ I . 0 0 FB (t )
By using Lemma 1, LMI condition Σ < 0 in (6) will imply Σ1 < 0 in (10). By the S-procedure of [6] with conditions (8)-(10) and Σ1 < 0 , there exists a positive constant ρ > 0 such that 2 V0 ( zt ) ≤ − ρ ⋅ e2α t ⋅ z (t ) .
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
151
From (9) and Σ in (6), we have
From the condition V ( zt ) ≤ 0 , we have
zT [− PC − C T P + 2α ⋅ P + 2α ⋅ ΓV + Q1 − e−2ατ M ⋅ (1 − τ D ) ⋅ Q1
V0 ( zt ) ≤ V0 ( z0 ) , where
+ 2Γ(W2 + Y2 )Γ]z + 2 zT [ PA + PB ]g ( z )
V0 ( z0 ) = z T (0) Pz (0)
+ 2 zT [Γ(W1 − W2 ) + Γ(Y1 − Y2 )]g ( z ) + g T ( z )[−2W1 − 2Y1 ]g ( z )
0
+∫
−τ ( t )
+∫
−τ M
0
e 2α s ⋅ [ z T ( s )Q1 z ( s ) + g T ( z ( s ))Q2 g ( z ( s ))]ds
+ g T ( z )[Q2 − e −2ατ M ⋅ (1 − τ D ) ⋅ Q2 ]g ( z ) ≥ 0,
e 2α s ⋅ ( s + τ M ) ⋅ zT ( s )( R1 + τ M ⋅ R2 ) z ( s )ds
[ zT zT 0 g T ( z ) g T ( z )] Σ [ zT zT 0 g T ( z ) g T ( z )]T ≥ 0.
n
+ 2∑ ∫ i =1
zi (0) 0
This implies
vi gi ( s )ds
[ zT zT 0 g T ( z ) g T ( z )] Σ1 [ zT zT 0 g T ( z ) g T ( z )]T ≥ 0,
≤ [λ max ( P ) + τ M ⋅ λ max (Q1 ) + τ M ⋅ λ max (ΓQ2 Γ)
+ τ M2 ⋅ λ max ( R1 + τ M ⋅ R2 ) + λ max (ΓV ) ⋅ z0
where Σ1 is defined in (11). Note that the condition Σ < 0 in
2 s
(6) is equivalent to Σ1 < 0 in (11), this will imply the result
2
= δ1 ⋅ z 0 s ,
z = g ( z ) = [0 0]T .
Hence the equilibrium point z =
[0 0]T is unique i.e., x is the unique equilibrium point □ of uncertain DNN (1). This completes the proof.
with
δ1 = λ max ( P) + τ M ⋅ λ max (Q1 ) + τ M ⋅ λ max (ΓQ2 Γ) + τ M2 ⋅ λ max ( R1 + τ M ⋅ R2 ) + λ max (ΓV ).
On the other hand, we have
V0 ( zt ) ≥ e2α t ⋅ z T (t ) Pz (t ) ≥ λ min ( P) ⋅ e2α t ⋅ z (t ) . 2
Consequently, we have z (t ) = x(t ) − x ≤
δ1 λ mim ( P)
⋅ z0 s ⋅ e −α t , t ≥ 0.
From Definition 1, this implies that the equilibrium point x of system (1) is globally exponentially stable with convergence rate α. Next we will prove the uniqueness of the equilibrium point x , i.e., the equilibrium point z = [0 … 0]T of (3). Assume z is an equilibrium point of the system (3). Then we have
− Cz + Ag ( z ) + Bg ( z ) = 0. Multiplying both sides of preceding equation by 2 zT P , we have zT (− PC − C T P) z + 2 zT PAg ( z ) + 2 zT PBg ( z ) = 0.
By setting α = 0 and R1 = R2 = U = S = 0 in Theorem 1, we may obtain the following delay-independent asymptotic stability condition (independent of τM) for system (1) with (2). Corollary 1. The equilibrium point x of system (1) with (2) and τD ≤ 1 (resp., τD > 1 or unknown) is unique and globally asymptotically stable (GAS), if there exist some n × n positive definite symmetric matrices P, Q1, Q2 (resp., Q1 = 0, Q2 = 0), some n × n positive diagonal matrices W1, W2, Y1, Y2, and a positive constant ε, such that the following LMI conditions are satisfied: Σˆ 11 0 * Σˆ 22 * * * * * * Σˆ = * * * * * * * * * *
Σˆ 13
Σˆ 14
Σˆ 15
Σˆ 16
Σˆ 17
Σˆ 18
0
0 ˆΣ 33
0
0
0
0
0
0
0
0
0
*
0 ˆΣ 44
0 ˆΣ 39
0
0
*
*
0 ˆΣ 55
0
*
*
*
0 ˆΣ 66
0 ˆΣ 58
*
*
*
*
0 ˆΣ 77
*
*
*
*
*
0 ˆΣ 88
*
*
*
*
*
*
0 ˆΣ 99
*
*
*
*
*
*
*
0
0 0 ˆΣ 69 0
0 0 0 Σˆ 410 0 < 0, 0 Σˆ 710 0 0 Σˆ 1010
(13) where
Journal of Marine Science and Technology, Vol. 22, No. 2 (2014 )
152
Σˆ 11 = − PC − C T P + Q1 + 2ΓW2 Γ, Σˆ 13 = PA + Γ(W1 − W2 ),
Table 1. Some comparisons for system (1) with (2) and (14).
Σˆ 14 = PB, Σˆ 15 = PM C , Σˆ 16 = PM A , Σˆ 17 = PM B , Σˆ 18 = −ε ⋅ N CT , Σˆ 22 = −[(1 − τ D ) ⋅ Q1 ] + 2ΓY2 Γ, Σˆ 24 = Γ(Y1 − Y2 ),
Some upper bounds of the time delay for the stability of system (1) with (2) and (14) Results [10] Our results
τD = 0 (Constant) τD = 0.5
τM < ∞
τM < ∞ (GAS)
τM = 2.9653
τM < ∞ (GAS)
Σˆ 410 = ε ⋅ N BT , Σˆ 58 = ε ⋅ ΘTC , Σˆ 69 = ε ⋅ ΘTA , Σˆ 710 = ε ⋅ ΘTB ,
τD = 0.9
τM = 0.8629
τM < ∞ (GAS)
Σˆ 55 = Σˆ 66 = Σˆ 77 = Σˆ 88 = Σˆ 99 = Σˆ 1010 = −ε ⋅ I .
τD = 1 or unknown
Not provided
Σˆ 33 = −2W1 + Q2 , Σˆ 39 = ε ⋅ N , Σˆ 44 = −2Y2 − (1 − τ D ) ⋅ Q2 , T A
Remark 2. In Corollary 1 with α = 0 and R1 = R2 = U = S = 0, the obtained result is delay-independent of τM. Hence the result “τM < ∞” can be guaranteed when the LMI (13) is feasible.
α = 0, τM = 5000000 (GAS) α = 0.1, τM = 215 (GES)
IV. NUMERICAL EXAMPLES
With α = 0.1, τD = 1, τM = 215, LMI conditions in Theorem 1 still have a feasible solution. The system (1) with (2), (14), τD = 1, and τM = 215, is exponentially stable with convergence rate α = 0.1. In order to show the improvement, we summarize some comparisons in Table 1.
Example 1. Consider the UDNNs in (1) with (2) and the following parameters: (Example 2 of [10])
Example 2. Consider the UDNNs in (1) with (2) and the following parameters: (Example 1 of [14])
1 0 −1 0.5 −2 0.5 C= , A = 0.5 −1.5 , B = 0.5 −2 , 0 1
0 0 0 1.2769 0 0.6231 0 0 C= , 0 0 0.923 0 0 0 0.448 0
0 0 MC = M A = M B = , −0.1 −0.1 1 0 0.5 0 0.1 0.1 NC = , NA = , NB = , 0 0 0.5 0 1 0
0.4 0 Γ= , ΘC = Θ A = Θ B = 0.3. 0 0.8
(14)
With τD = 0.2, LMI conditions in Corollary 1 have a feasible solution: 0.8112 0.1937 0.3581 0.0991 P= , Q1 = , 0.1937 0.9452 0.0991 0.2138 0 2.9551 −0.6585 0.8847 Q2 = , V = , 0.8847 −0.6585 3.1723 0
−0.0373 −1.6033 A= 0.3394 −0.1311
0.4852 0.5988 −0.086 0.3253
−0.3351 0.2336 −0.3224 1.2352 , −0.3824 −0.5785 −0.9534 −0.5015
0.8674 −1.2405 −0.5325 0.022 0.0474 −0.9164 0.036 0.9816 B= , 1.8495 2.6117 −0.3788 0.8428 −2.0413 0.5179 1.1734 −0.2775 0 0 0 L1 0 0 0 0.1137 0 0.1279 0 0 0 L2 0 0 , Γ= = 0 0 0.7994 0 0 0 L3 0 0 0 0.2368 0 0 0 L4 0
0 0 2.3825 0.1643 W1 = , W2 = , 1.9513 0.0368 0 0
M C = M A = M B = N C = N A = N B = 0, ΘC = Θ A = Θ B = 0. (15)
0 0 0.6111 0.3178 Y1 = , Y2 = , ε = 0.2989. 0.165 0.0491 0 0
2.
The system (1) with (2), (14), and τD = 0.2, is asymptotically stable and the equilibrium point x is unique. With τD = 1, LMI conditions in Corollary 1 are not feasible. Hence Theorem 1 should be used to show delay-dependent results.
50, J = 0, x(t ) = [10 −10 5 −5]T , t ∈ [−50, 0] , the system state trajectories are shown in Fig. 1, where 0 is the unique equilibrium point.
Some comparisons of proposed results are shown in Table With fi ( xi ) = Li × 0.5 ( xi + 1 − xi − 1 ) , i = 1, 2, 3, 4, τ (t) =
C.-H. Lien and K.-W. Yu: Exponential Stability for Uncertain Delayed NNs
153
Table 2. Some comparisons for system (1) with (2) and (15).
REFERENCES
Some upper bounds of the time delay for the stability of system (1) with (2) and (15) Results [7] [14] Our results τD = 0 1 ≤ τ (t) ≤ 3.5841 1 ≤ τ (t) ≤ 3.8363 τM < ∞ (GAS) τD = 0.5 1 ≤ τ (t) ≤ 2.5802 1 ≤ τ (t) ≤ 2.7299 τM < ∞ (GAS) τD = 0.9 1 ≤ τ (t) ≤ 2.2736 1 ≤ τ (t) ≤ 2.3811 τM < ∞ (GAS) α = 0, τM = 12016155 τD = 1 or 1 ≤ τ (t) ≤ 2.2393 1 ≤ τ (t) ≤ 2.3114 (GAS) unknown α = 0.1, τM = 229 (GAS)
1. Boyd, S. P., Ghaoui, El, Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System And Control Theory, SIAM, Philadelphia, USA (1994). 2. Chen, A., Cao, J., and Huang, L., “Global robust stability of interval cellular neural networks with time-varying delays,” Chaos, Solitons & Fractals, Vol. 23, pp. 787-799 (2005). 3. Chua, L. O. and Roska, T., Cellular Neural Networks and Visual Computting, Cambridge University Press, Cambridge, U.K. (2002). 4. Gau, R. S., Lien, C. H., and Hsieh, J. G., “Global exponential stability for uncertain cellular neural networks with multiple time-varying delays via LMI approach,” Chaos, Solitons & Fractals, Vol. 32, pp. 1258-1267 (2007). 5. Gau, R. S., Lien, C. H., and Hsieh, J. G., “Novel stability conditions for interval delayed neural networks with multiple time-varying delays,” International Journal of Innovative Computing, Information and Control, Vol. 7, pp. 433-444 (2011). 6. Gu, K., Kharitonov, V. L., and Chen, J., Stability of Time-Delay Systems, Birkhauser, Boston, Massachusetts, USA (2003). 7. He, Y., Liu, G. P., and Ress, D., “New delay-dependent stability criteria for neural networks with time-varying delay,” IEEE Transactions on Neural Networks, Vol. 18, pp. 310-314 (2007). 8. Li, C., Liao, X., and Zhang, R., “Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach,” Physics Letters A, Vol. 328, pp. 452-462 (2004). 9. Li, C., Liao, X., Zhang, R., and Prasad, A., “Global robust exponential stability analysis for interval neural networks with time-varying delays,” Chaos, Solitons & Fractals, Vol. 25, pp. 751-757 (2005). 10. Li, T., Guo, L., and Sun, C., “Robust stability for neural networks with time-varying delays and linear fractional uncertainties,” Neurocomputing, Vol. 71, pp. 421-427 (2007). 11. Liao, X. and Wang, J., “Global and robust stability of interval Hopfield neural networks with time-varying delays,” International Journal of Neural Systems, Vol. 13, pp. 177-182 (2003). 12. Liao, X., Wong, K. K., Wu, Z., and Chen, G., “Novel robust stability criteria for interval-delayed Hopfield neural networks,” IEEE Transactions on Circuits and Systems I, Vol. 48, pp. 1355-1359 (2001). 13. Lien, C. H., Yu, K. W., Lin, Y. F., Chang, H. C., and Chung, Y. J., “Stability analysis for Cohen-Grossberg neural networks with time-varying delays via LMI approach,” Expert Systems with Applications, Vol. 38, pp. 6360-6367 (2011). 14. Tian, J. and Zhou, X., “Improved asymptotic stability criteria for neural networks with interval time-varying delay,” Expert Systems with Applications, Vol. 37, pp. 7521-7525 (2010). 15. Xu, S., Lam, J., and Ho, D. W. C., “Novel global robust stability criteria for interval neural networks with multiple time-varying delays,” Physics Letters A, Vol. 342, pp. 322-330 (2005). 16. Yang, X., Li, C., Liao, X., Evans, D. J., and Megson, G. M., “Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays,” Applied Mathematics and Computation, Vol. 171, pp. 108-121 (2005). 17. Yu, K. W., “Global exponential stability criteria for switched systems with interval time-varying delay,” Journal of Marine Science and Technology, Vol. 18, pp. 298-307 (2010). 18. Zheng, C. D., Jing, X. T., Wang, Z. S., and Feng, J., “Further results for robust stability of cellular neural networks with linear fractional uncertainty,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, pp. 3046-3057 (2010).
10
x1 x2 x3 x4
8 6 4 2 0 -2 -4 -6 -8 -10
0
50
100 150 200 250 300 350 400 450 500 t
Fig. 1. The system state trajectories of DNN (1) with (15).
V. CONCLUSIONS In this paper, the global stability and uniqueness of equilibrium point for a class of uncertain delayed neural networks with time-varying delay and linear fractional perturbations has been investigated. Based on the LMI approach and some proposed additional nonnegative inequalities, some delaydependent and delay-independent criteria have been proposed to guarantee the exponential stability and asymptotic stability of UDNN, respectively. Some numerical examples by using the proposed results have shown great improvement over recent published results.
ACKNOWLEDGMENTS The research reported here was supported by the National Science Council of Taiwan, R.O.C. under grant no. NSC 99-2221-E-022-003 and NSC 101-2514-S-022-001. The authors would like to thank the editor and anonymous reviewers for their helpful comments.
