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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 2836 – 2840 www.elsevier.com/locate/procedia
Advanced in Control Engineeringand Information Science
Stability Criterion for BAM Neural Networks of NeutralType with Interval Time-Varying Delays Guoquan Liu a*, Simon X. Yanga,b a
College of Automation,Chongqing University, Chongqing 400044,China School of Engineering ,University of Guelph, Guelph, Ontario, Canada
b
Abstract In this paper, the asymptotic stability for bidirectional associative memory (BAM) neural networks of neutral-type with interval time-varying delays is investigated. The discrete delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. By employing the Lyapunov-Krasovskii functional method and using the linear matrix inequality (LMI) technique, a new delay-range-dependent stability criterion is established in terms of LMI. In addition, the proposed LMI based results can be easily checked by LMI control toolbox in Matlab.
© 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords:Asymptotic stability; Bidirectional associative memory neural networks; Neutral-type; Linear matrix inequality; Interval time-varying delays;
1. Introduction It is well known that bidirectional associative memory (BAM) neural network is a type of recurrent neural network. BAM neural network was introduced by [1]-[2]. During the past years, the dynamics of BAM neural networks have been widely studied due to their extensive applications in many areas such as associative memory, pattern recognition, optimization and automatic control. In practice, time delays are likely to be present due to the finite switching speed of amplifiers and occur in signal transmission among * Corresponding author. Address: College of Automation, Chongqing University, Chongqing 400044, China. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.534
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neurons in the electronic implementation of neural networks, In addition, time delay is often a source of oscillations, chaos and instability in various types of neural networks. Thus, the study of the stability problem of BAM neural networks with time delays has received great attention in recent years and a number of results have been reported [3]-[7]. On the other hand, many dynamical neural networks are described with neutral functional differential equations that include neutral delay differential equations. These neural networks are called neutral neural networks or neural networks of neural-type. Recently, a few results about the global asymptotic or exponential stability for BAM neural networks of neural-type have been derived in the literatures [8][10]. In [8], a delay-dependent global asymptotic stability criterion is presented for BAM neural networks of neutral-type by using the Lyapunov method. In [9], Liu and zhang furthermore investigated asymptotic stability for BAM neural networks of neutral-type, a novel delay-dependent stability conditions were established. In [10], by utilizing the Lyapunov-krasoviskii functional and combining with the LMI approach; three sufficient conditions were given ensuring the global exponential stability for BAM neural networks of neutral-type with time-varying delays. Up to now, the asymptotic stability problem has not been touched for BAM neural networks of neutral-type with interval time-varying delays, which is still open problem. Based the aforementioned discussions, a class of BAM neural networks of neutral-type with interval time-varying delays is considered in this paper. Based on the Lyapunov-krasoviskii stability method and the LMI technique, a new asymptotic stability criterion is presented in terms of LMI. Notations: The notations are quite standard. ℜ n and ℜ n×n denote the n-dimensional Euclidean space and the set of all n × n real matrices, respectively; For a real symmetric matrix X , the notation X ≥ 0 (respectively, X > 0 ), means that X is positive semi-definite (respectively, positive definite); The superscripts "T " and "-1" stand for matrix transposition and matrix inverse, respectively; The mathematical expectation operator with respect to the given probability measure Ｐ is denoted by E {} ⋅ . diag {⋅⋅⋅} denotes a block diagonal matrix; ∗ denotes the elements below the main diagonal of a symmetric block matrix. 2. Problem Formulation Consider the following BAM neural networks of neutral-type with interval time-varying delays: u& (t ) = − Au (t ) + W1 f (v(t )) + W2 f (v(t − τ (t ))) + W3v&(t − h1 (t )) + I , v&(t ) = − Bv(t ) + V1 g (u (t )) + V2 g (u (t − σ (t ))) + V3u& (t − h2 (t )) + J ,
(1)
where u = [u1 , u2 ,..., un ] and v = [ v1 , v2 ,..., vn ] are the neuron state vectors. A = diag {a1 , a2 ,..., an } > 0, T
T
B = diag {b1 , b2 ,..., bm } > 0, W1 ,W2 ,W3 ,V1 ,V2 and V3 are known constant matrices with appropriate dimension, f
and g denote the neuron activations, I and J denote the constant external inputs. τ (t ) and σ (t ) represent the discrete transmission delays with 0 ≤ τ 1 ≤ τ (t ) ≤ τ 2 ,τ&(t ) ≤ τ d < 1, 0 ≤ σ 1 ≤ σ (t ) ≤ σ 2 ,τ&(t ) ≤ σ d < 1, (2) where τ 1 ,τ 2 ,τ d , σ 1 , σ 2 and σ d are constants. h1 (t ) and h2 (t ) represent the neutral delays with 0 ≤ h1 (t ) ≤ h1 , h&1 (t ) ≤ h1d < 1,0 ≤ h2 (t ) ≤ h2 , h&2 (t ) ≤ h2 d < 1.
Assume that the neuron activation functions f and g satisfy the following hypotheses (A1) f and g are bounded functions. (A2) f and g are Lipschitz continuous, i.e., there exist real scalars l j > 0 and ki > 0 , such that
(3)
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f (ς 1 ) − f (ς 2 ) ≤ l j ς 1 − ς 2 , g (ς 1 ) − g (ς 2 ) ≤ ki ς 1 − ς 2 , for all ς 1 , ς 2 ∈ ℜ and ς 1 ≠ 2 for any i = 1, 2,..., n, j = 1, 2,..., m. It
is clear that under the assumptions (A1) and (A2), system (1) has at least one equipment point. Suppose T T that u ∗ = ( u1∗ , u2∗ ,..., um∗ ) and v∗ = ( v1∗ , v2∗ ,..., vn∗ ) be one equilibrium point of system (1). For convenience, we shift u ∗ , v∗ to the origin by taken the following transformation:
(4)
x(⋅) = u (⋅) − u ∗ , f (⋅) = f (u (⋅)) − f (u ∗ (⋅)), y(⋅) = v(⋅) − v∗ , g (v(⋅)) = g (v(⋅)) − g (u ∗ (⋅)).
Then, system (1) can be written as
x& (t ) = − Ax(t ) + W1 f ( y (t )) + W2 f ( y (t − τ (t ))) + W3 y& (t − h1 (t )), y& (t ) = − By (t ) + V1 g ( x(t )) + V2 g ( x (t − τ (t ))) + V3 x& (t − h2 (t )).
(5)
From (A1) and (A2), we can derive that activation f and g satisfy (G1) f and g are bounded functions. (G2) f and g are Lipschitz continuous, i.e., there exist real scalars l j > 0 and ki > 0 , such that f (ς 1 ) − f (ς 2 ) ≤ l j ς 1 − ς 2 , g (ς 1 ) − g (ς 2 ) ≤ ki ς 1 − ς 2 , for all ς 1 , ς 2 ∈ ℜ and ς 1 ≠ ς 2 for any i = 1, 2,..., n, j = 1, 2,..., m
with f (0)) = 0, g (0)) = 0. In order to obtain the main result, a basic lemma is always made throughout this paper. Lemma 1. For any constant matrix M > 0 , any scalars a and b with a < b , and a vector function x(t ) :[ a, b] → ℜn such that the integrals concerned as well defined, the following holds: T
⎡ b x( s )ds ⎤ M ⎡ b x( s )ds ⎤ ≤ (b − a ) b xT ( s ) Mx( s )ds. ∫a ⎢⎣ ∫a ⎥⎦ ⎢⎣ ∫a ⎥⎦
(6)
3. Stability Analysis
In this section, we propose a new stability criterion for BAM neural networks of neutral-type with interval time-varying delays described in (5). Theorem 1. Under assumptions (G1) and (G2) hold. The equilibrium point of system (5) is asymptotically stable, if there exist positive definite matrices P1 , P2 , Qi , i = 1, 2,...,6 , Ri , i = 1, 2,3, 4, Z1 , Z 2 , and diagonal positive definite matrices M 1 , M 2 such that the following LMI holds:
where
0 ⎡Ω1 ⎢ ∗ −Ω 3 Ω=⎢ ⎢∗ ∗ ⎢ ∗ ⎣∗
Ω2 0 Ω4 ∗
⎡Ξ1 0 ⎢∗ Ξ 4 Ω1 = ⎢ ⎢∗ ∗ ⎢ ⎣∗ ∗
Ξ2 Ξ5 Ξ7 ∗
⎡ Ξ 24 ⎢ ∗ Ω4 = ⎢ ⎢ ∗ ⎢ ⎣ ∗
with
0 Ξ 26 ∗ ∗
0 ⎤ 0 ⎥⎥ < 0, 0 ⎥ ⎥ −Ω5 ⎦
Ξ3 ⎤ ⎡Ξ10 0 ⎢ 0 Ξ Ξ 6 ⎥⎥ 12 , Ω2 = ⎢ ⎢Ξ14 Ξ15 Ξ8 ⎥ ⎥ ⎢ Ξ9 ⎦ ⎣ Ξ18 Ξ19 Ξ 25 0 ⎤ 0 Ξ 27 ⎥⎥ , Ω = diag {Z1 Ξ 28 0 ⎥ 5 ⎥ ∗ Ξ 29 ⎦
(7)
Ξ11 0 Ξ16 Ξ 20 Z2} ,
0 ⎤ Ξ13 ⎥⎥ , Ω = diag {Ξ 22 Ξ17 ⎥ 3 ⎥ Ξ 21 ⎦
Ξ 23
R1
R2
R3
R4 } ,
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Guoquan Liu and Simon Yang / Procedia Engineering 15 (2011) 2836 – 2840 Guoquan Liu etX.al/ Procedia Engineering 00 (2011) 000–000
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T T Ξ1 = − P1 A − AP1T + Q1 + AT Q6 A + R1 + R2 + (τ 2 − τ 1 ) Z1 , Ξ 2 = PW 1 1 − A Q6W1 , Ξ 3 = − AM 2 , 2
T T T T Ξ 4 = PW 1 2 − A Q6W2 , Ξ 5 = PW 1 3 − A Q6W3 , Ξ 6 = Q3 − P2 B − BP2 + B Q5 B + R3 + R4 + (σ 2 − σ 1 ) Z 2 , 2
T T T T Ξ 7 = − BM 1T , Ξ8 = PV 1 1 − BM 2 − B Q5V1 , Ξ 9 = PV 2 2 − B Q5V2 , Ξ10 = PV 2 3 − B Q5V3 ,
Ξ11 = Q2 + M 1W1 + W1T M 1T + W1T Q6W1 , Ξ12 = M 1V1 + W1T M 2T , Ξ13 = W1T Q6W2 , Ξ14 = M 1V2 , Ξ15 = W1T Q6W3 , Ξ16 = M 1V3 + W1T Q6W3 , Ξ17 = Q4 + M 2V1 + V1T M 2T + V1T Q5V1 , Ξ18 = M 2W2 + V1T Q5V2 , Ξ19 = V1T Q5V2 , Ξ 20 = M 2W3 + V1T Q5V3 , Ξ 21 = V1T Q5V3 , Ξ 22 = (1 − τ d )Q1 , Ξ 23 = (1 − σ d )Q3 , Ξ 24 = −(1 − τ d )Q2 + W2T Q6W2 , Ξ 25 = W2T Q6W3 , Ξ 26 = −(1 − σ d )Q4 + V2T Q6V2 , Ξ 27 = V2T Q5V3 , Ξ 28 = −(1 − h1d )Q5 + W3T Q6W3 , Ξ 29 = −(1 − h2 d )Q6 + V3T Q5V3 .
Proof. Construct a lyapunov-krasoskill functional for system (5) as follows V (t ) = V1 (t ) + V2 (t ) + V3 (t ) + V4 (t ),
(8)
where n
V1 (t ) = xT (t ) P1 x(t ) + yT (t ) P2 y(t ) + 2∑ m1 j ∫ j =1
V2 (t ) = ∫
t
n
f j ( s )ds + 2∑ m2i ∫ i =1
xi ( t )
0
gi ( s ) ds,
t
t −τ ( t )
+∫
T T T T ⎣⎡ x ( s)Q1 x( s ) + f ( y ( s ))Q2 f ( y ( s)) ⎦⎤ ds + ∫t −σ (t ) ⎡⎣ y ( s)Q3 y ( s) + g ( x( s))Q4 g ( x( s)) ⎤⎦ ds
t
t −h1 ( t )
V3 (t ) = ∫
y j (t )
0
t
t −τ1
y& T ( s )Q5 y& ( s )ds + ∫
t
t −h2 ( t )
xT ( s ) R1 x( s )ds + ∫
t
t −τ 2
V4 (t ) = (τ 2 − τ 1 ) ∫
−τ1
−τ 2
∫
t
t +δ
x& T ( s)Q6 x& ( s)ds,
xT ( s ) R2 x( s ) ds + ∫
t
yT ( s) R3 y ( s) ds + ∫
t −σ 1
t
t −σ 2
xT ( s ) Z1 x( s )dsdδ + (σ 2 − σ 1 ) ∫
−σ 1
−σ 2
∫
t
t +δ
yT ( s) R4 y ( s) ds,
yT ( s ) Z 2 y ( s )dsdδ .
Calculating the derivative of V (t ) along the trajectory of system (6) is (9)
V& (t ) = V&1 (t ) + V&2 (t ) + V&3 (t ) + V&4 (t ),
where
V&1 (t ) = 2 xT (t ) P1 [ − Ax(t ) + W1 f ( y (t )) + W2 f ( y (t − τ (t ))) +W3 y& (t − h1 (t ))]
+ 2 yT (t ) P2 [ − By (t ) + V1 g ( x(t )) +V2 g ( x(t − σ (t ))) + V3 x& (t − h2 (t )) ] + 2M 1 f T ( y (t )) × [ − By (t ) + V1 g ( x(t )) + V2 g ( x (t − τ (t ))) + V3 x& (t − h2 (t )) ] + 2M 2 g T ( x(t )) [ − Ax (t )
(10)
+W1 f ( y (t )) + W2 f ( y (t − τ (t ))) + W3 y& (t − h1 (t )) ] ,
V&2 (t ) = xT (t )Q1 x(t ) − (1 − τ&(t )) xT (t − τ (t ))Q1 x (t − τ (t )) + f T ( y (t ))Q2 f ( y (t )) − (1 − τ&(t )) f T ( y (t − τ (t ))) × Q2 f T ( y (t − τ (t ))) + y T (t )Q3 y (t ) − (1 − σ& (t )) yT (t − τ (t ))Q3 y (t − τ (t )) + g T ( x(t ))Q4 g ( x(t )) − (1 − σ& (t )) g T ( x(t − σ (t )))Q4 g ( x(t − σ (t ))) + [ − By (t ) + V1 g ( x(t )) + V2 g ( x(t − τ (t )))
+V3 x& (t − h2 (t ))] Q5 [ − By (t ) + V1 g ( x(t )) + V2 g ( x(t − τ (t ))) + V3 x& (t − h2 (t )) ] − (1 − h& (t )) y& T (t − h (t ))Q y& (t − h (t )) + [ − Ax (t ) + W f ( y (t )) + W f ( y (t − τ (t ))) T
1
1
5
1
1
(11)
2
+W3 y& (t − h1 (t ))] Q6 [ − Ax(t ) + W1 f ( y (t )) + W2 f ( y (t − τ (t ))) + W3 y& (t − h1 (t )) ] − (1 − h& (t )) x&T (t − h (t ))Q x& (t − h (t )), T
2
2
6
2
V&3 (t ) = xT (t ) R1x(t ) − xT (t − τ1 ) R1x(t − τ1 ) + xT (t ) R2 x(t ) − xT (t − τ 2 ) R2 x(t − τ 2 ) + yT (t ) R3 y (t ) − yT (t − σ1 ) R3 y(t − σ1 ) + yT (t ) R4 y(t ) − yT (t − σ 2 ) R4 y(t − σ 2 ), t −τ1 V&4 (t ) = (τ 2 − τ 1 ) 2 xT (t ) Z1 x(t ) − (τ 2 − τ 1 ) ∫ xT ( s ) Z1 x( s ) ds + (σ 2 − σ 1 ) 2 yT (t ) Z 2 y (t ) t −τ 2
− (σ 2 − σ 1 ) ∫
t −σ 1
t −σ 2
T
y ( s ) Z 2 y ( s )ds.
(12) (13)
By Lemma 1, we known that −(τ 2 − τ 1 ) ∫
t −τ1
t −τ 2
T
t −τ1 t −τ1 xT ( s ) Z1 x( s )ds ≤ − ⎡ ∫ x( s)ds ⎤ Z1 ⎡ ∫ x( s )ds ⎤ , ⎢⎣ t −τ 2 ⎥⎦ ⎣⎢ t −τ 2 ⎦⎥
(14)
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Guoquan Liu and Simon X. Yang / Engineering Procedia Engineering (2011) 2836 – 2840 Guoquan Liu et al / Procedia 00 (2011) 15 000–000
−(σ 2 − σ 1 ) ∫
t −σ1
t −σ 2
T
t −σ1 t −σ 1 yT ( s ) Z 2 y ( s )ds ≤ − ⎡ ∫ y ( s )ds ⎤ Z 2 ⎡ ∫ y ( s )ds ⎤ . ⎢⎣ t −σ 2 ⎣⎢ t −σ 2 ⎦⎥ ⎦⎥
(15)
By utilizing relationships (9)-(15), we have
(16)
dV (t ) ≤ ξ T (t )Ωξ (t ) dt ,
where Ω is defined in (9) and ξ T (t ) = ⎡⎣ xT (t ), yT (t ), f T ( y (t )), g T ( x(t )), xT (t − τ (t )), yT (t − τ (t )), xT (t − τ 1 ), xT (t − τ 2 ), yT (t − σ 1 ), yT (t − σ 2 ), f T ( y (t − τ (t ))), g T ( x(t − σ (t ))), y& (t − h1 (t )), x& (t − h2 (t )),
5
(∫
t −τ1
t −τ 2
) (∫ T
x( s )ds ,
t −σ1
t −σ 2
)
T ⎤ y ( s )ds ⎥ . ⎦
It is obvious that for Ω < 0 , which indicates from the Lyapunov stability theory that the system (5) is asymptotic stable. This completes the proof. 4. Conclusion
This paper addresses the problem of the asymptotic stability for BAM neural networks of neutral-type with interval time-varying delays. A new delay-range-dependent asymptotic stability condition for the considered systems is proposed in terms of LMI based on the Lyapunov-Krasovskii function method and the LMI technique. Acknowledgment. This work was supported by the Fundamental Research Funds for the Central Universities (No. CDJXS11172237). References [1] Kosko B, Adaptive bidirectional associative memories. Applied Optics 1987; 26: 4947-4960. [2] Kosko B, Bidirectional associative memories. IEEE Trans. Syst. Man, Cyber 1988; 18: 49-60. [3] Liao X F, Yu J B, Chen GR, Novel stability criteria for bidirectional associative memory neural networks with time delays. International Journal of Circuit Theory and Applications 2002; 30: 519-546. [4] Chen, AP, Cao, J, Huang, LH. Exponential stability of BAM neural networks with transmission delays. Neurocomputing 2004; 57: 435-454. [5] Li, YT, Yang, CB. Global exponential stability analysis on impulsive BAM neural networks with distributed delays. Journal of Mathematical Analysis and Applications 2006;324: 1125-1139. [6] Huang, ZT, Luo, XS, Yang, QG. Global asymptotic stability analysis of bidirectional associative memory neural networks with distributed delays and impulse. Chaos Solitons & Fractals 2007; 34: 878-885. [7] Hu L, Liu H, Zhao Y B. New stability criteria for BAM neural networks with time-varying delays. Neurocomputing 2009;72: 3245-3252. [8] Park J H, Park CH, Kwon O M, Lee S M. A new stability criterion for bidirectional associative memory neural networks of neutral-type. Applied Mathematics and Computation 2008;199: 716-722. [9] Liu J, Zong, G D. New delay-dependent asymptotic stability conditions concerning BAM neural networks of neutral type. Neurocomputing 2009; 72: 2549-2555. [10] Balasubramaniam P, Rakkiyappan R. Global exponential stability for neutral-type BAM neural networks with time-varying delays. International Journal of Computer Mathematics 2010; 87: 2064-2075.
