International Journal of Innovative Computing, Information and Control Volume 7, Number 1, January 2011

c ICIC International ⃝2011 ISSN 1349-4198 pp. 433–444

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS WITH MULTIPLE TIME-VARYING DELAYS Ruey-Shyan Gau1 , Chang-Hua Lien2,∗ and Jer-Guang Hsieh3 1

Department of Computer Science Municipal Kaohsiung Senior Vocational Industrial High School Kaohsiung 807, Taiwan 2

Department of Marine Engineering National Kaohsiung Marine University Kaohsiung 811, Taiwan 3

Department of Electrical Engineering I-Shou University Kaohsiung 840, Taiwan ∗ Corresponding author: [email protected]

Received September 2009; revised February 2010 Abstract. In this paper, the global exponential stability and global asymptotic stability for a class of interval delayed neural networks (IDNNs) with multiple time-varying delays are considered. Delay-dependent and delay-independent criteria are proposed to guarantee the robust stability of IDNNs via linear matrix inequality (LMI) approach. Some numerical examples are illustrated to show the eﬀectiveness of our results. From the illustrative examples, significant improvement over the recent results can be demonstrated. Keywords: Interval delayed neural network, Global exponential stability, Delay-dependent criterion, Delay-independent criterion, Linear matrix inequality

1. Introduction. In recent years, neural networks have been used in many mathematical and practical applications, such as optimization, recognition, prediction, diagnosis, decision, association, approximation, and generalization. Various neural networks have been investigated, such as bidirectional associative memory neural networks [1], cellular neural networks [2], Cohen-Grossberg neural networks [3], and Hopfield neural networks [4]. The existence of time delays is often a source of oscillation and instability of a neural networks. Hence the stability of delayed neural networks (DNNs) are important and significant in practical applications. The DNNs may be applied in many areas including the moving images processing, pattern classification, and automatic control engineering [1-4]. For many applications, artificial neural networks are usually implemented by integrated circuits [1-4]. In the implementation of artificial neural networks, time delay is produced from finite switching and finite propagation speed of electronic signals. During the implementation on very large scale integrated chips, parameter perturbations and transmitting time delays will destory the stability of DNNs [4-13]. In the analysis for uncertain DNNs, it is reasonable to assume that the parameters are varying in some prescribed intervals. Such DNNs are called the interval delayed neural networks (IDNNs) [4-7,9,13]. Hence some less conservative stability conditions for IDNNs will be proposed in this paper. Depending on whether the stability criterion itself contains the size of delay, criteria for IDNN can be classified into two categories, namely delay-independent criteria [5,7,9] and delay-dependent criteria [4,6-8,10-12]. Usually the latter is less conservative when the value of delay is small. In the Lyapunov-based delay-dependent results, the slow-varying 433

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constraints τ˙i (t) < 1, i ∈ n, are usually imposed on the time-varying delays [4,6,7,13]. These constraints will be relaxed and delay-dependent results will be proposed in this paper. In [5,9], 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 dealing with many control problems. The LMI problem can be solved quite eﬃciently by using the toolbox of Matlab [14]. In [4,6-8,10-13], stability criteria for IDNNs were proposed via LMI approach. In this paper, LMI-based delay-dependent and delay-independent criteria are proposed by using a new Lyapunov functional. In general, our approach is useful and is easy to be generalized to other forms of uncertain DNNs. When a practical neural network is designed for some specific purposes, all the results proposed in this paper can be used to guarantee the stability of these practical neural networks. By the proposed decomposition for interval matrices and approaches, our results are shown to be less conservative than other recent literatures for our illustrative numerical examples. The notation used throughout this paper is as follows. For a matrix A, we denote the transpose by AT , spectral norm by ∥A∥, symmetric positive (negative) definite by A > 0 (A < 0), maximal (minimal) eigenvalue of symmetric matrix by λmax (A) (λmin (A)). A ≤ B denotes that the matrix B − A is symmetric positive semi-definite. For a vector n x, the Euclidean norm is denoted as √ ∥x∥. ℜ denotes the n-dimensional real space,

n := {1, 2, . . . , n}, ∥xt ∥s :=

∥x (t + s)∥2 + ∥x˙ (t + s)∥2 , I denotes the identity −τM ≤s≤0 [ ] matrix, diag [ai ] denotes diagonal matrix with the diagonal elements ai , i ∈ n. V A, A { ( ) denotes A = (aij ) ∈ ℜn×n | A ≤ A ≤ A, i.e., aij ≤ aij ≤ a ¯ij , i, j ∈ n} with A = aij and A = (aij ). sup

2. Problem Statement and Preliminaries. Consider the following IDNN with multiple time-varying delays: x˙ (t) = −CI x (t) + AI y (x (t)) + B I y (x (t − τ (t))) + J,

where x (t) = x (t − τ (t)) =

[ [

(1a)

t ≥ 0,

(1b)

t ∈ [−τM , 0] ,

(1c)

y (x (t)) = f (x (t)) , x (t) = ϕ (t) ,

t ≥ 0,

(x1 (t)) (x2 (t)) · · ·

(xn (t))

]T

x1 (t − τ1 (t)) x2 (t − τ2 (t)) · · ·

, xn (t − τn (t))

]T

,

n ≥ 2 is the number of neurons in the network, 0 ≤ τi (t) ≤ τM , τ˙i (t) ≤ τD , i ∈ n, y (x (t)) ] [ ]T [ is the output, J = J1 J2 · · · Jn is the external bias vector, CI ∈ V C I , C I is ] [ a positive diagonal matrix with C I = diag [ci ], C¯I = diag [¯ ci ], ci > 0, AI ∈ V AI , AI ] [ is the feedback matrix with AI = [ai ], AI = [ai ], BI ∈ V B I , B I is the delay feedback [ ] matrix with B I = [bi ], B I = bi , and ϕ ∈ C1 is the initial function, where C1 is the set of diﬀerentiable functions from [−H, 0] to [−H, 0]. The activation functions of IDNN (1) given by [ ]T , f (x (t)) = f1 (x1 (t)) f2 (x2 (t)) · · · fn (xn (t)) [ ]T , f (x (t − τ (t))) = f1 (x1 (t − τ1 (t))) f2 (x2 (t − τ2 (t))) · · · fn (xn (t − τn (t))) are bounded monotonically nondecreasing and satisfy |fi (ξ1 ) − fi (ξ2 )| ≤ Li · |ξ1 − ξ2 | ,

ξ1 , ξ2 ∈ ℜ,

i ∈ n,

(2)

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS

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where Li > 0, i ∈ n are some positive constants. [ ]T Assume x˜ = x˜1 x˜2 · · · x˜n ∈ ℜn is an equilibrium point of system (1), then we can obtain the following system: z˙ (t) = −CI z (t) + AI g (z (t)) + BI g (z (t − τ (t))) , where z (t) =

[ [

z1 (t) z2 (t) · · · zn (t)

(3)

]T

= x (t) − x˜, ]T g1 (z1 (t)) g2 (z2 (t)) · · · gn (zn (t)) ,

g (z (t)) = [ ]T , (4a) z (t − τ (t)) = z1 (t − τ1 (t)) z2 (t − τ2 (t)) · · · zn (t − τn (t)) [ ]T g (z (t − τ (t))) = g1 (z1 (t − τ1 (t))) g2 (z2 (t − τ2 (t))) · · · gn (zn (t − τn (t))) , (4b) gi (zi (t)) = fi (xi (t)) − fi (˜ xi ) = fi (zi (t) + x˜i ) − fi (˜ xi ) ,

gi (0) = 0,

gi (zi (t − τi (t))) = fi (xi (t − τi (t))) − fi (˜ xi ) = fi (zi (t − τi (t)) + x˜i ) − fi (˜ xi ) .

(4c) (4d)

Let W = diag [wi ] and Y = diag [yi ] be two diagonal matrices with wi , yi > 0. From (2) and (4c)-(4d), we have |gi (zi (t))| = |fi (zi (t) + x˜i ) − fi (˜ xi )| ≤ Li |zi (t)| , g T (z (t)) W g (z (t)) =

n ∑

wi |gi (zi (t))|2 ≤

i=1

n ∑

(4e)

i=1

g (z (t − τ (t))) Y g (z (t − τ (t))) = T

wi L2i |zi (t)|2 =z T (t) ΓW Γz (t) , n ∑

yi |gi (zi (t − τi (t)))|2

i=1

≤

n ∑

yi L2i |zi (t − τi (t))|2 = z T (t − τ (t)) ΓY Γz (t − τ (t)) ,

(4f)

i=1

where Γ = diag[Li ]. Remark 2.1. The forms fi (xi ) = 0.5 (|xi + 1| − |xi − 1|) and fi (xi ) = tanh (xi ) are two general activation functions satisfying (2). From interval matrix assumptions, matrices C I , AI and BI can be rewritten as ) 1( CI = C + ∆C, C = C I + C I , ∆C = diag [∆ci ] , 2 1 |∆ci | ≤ cˆi , cˆi = (¯ ci − ci ) > 0, i ∈ n, 2 ) 1( AI = A + ∆A, A = AI + AI , ∆A = [∆aij ], 2 ) 1( a ¯ij − aij > 0, i, j ∈ n, |∆aij | ≤ a ˆij , a ˆij = 2 ) 1( BI = B + ∆B, B = B I + B I , ∆B = [∆bij ], 2 ) ( 1 ¯bij − b > 0, i, j ∈ n. |∆bij | ≤ ˆbij , ˆbij = ij 2 Uncertain matrices ∆C, ∆A and ∆B are assumed to take the form ∆C = MC FC (t)NC ,

∆A = MA FA (t) NA ,

∆B = MB FB (t)NB ,

(5a)

(5b)

(5c)

(5d)

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where n n [ ] z√ }| √ { [√ ] ∆ci 2 MC = NC = diag cˆi , FC (t) = diag , MA = diag a ˆi1 · · · a ˆin ∈ ℜn×n , cˆi i=1

[ √ ] √ diag a ˆ11 · · · a ˆ1n n2 ×n NA = , ] ∈ℜ [ √ ... √ diag a ˆn1 · · · a ˆnn

n n z√ }| √ { ˆbi1 · · · ˆbin MB = diag

2

∈ ℜn×n ,

i=1

[ √ ] √ ˆb11 · · · ˆb1n ∈ ℜn2 ×n , · · · NB = [ ] √ √ ˆbn1 · · · ˆbnn diag diag

] ∆a11 / a ˆ11 · · · ∆a1n / a ˆ1n n2 ×n2 ··· FA (t) = , [ ] ∈ℜ ˆn1 · · · ∆ann / a ˆnn diag ∆an1 / a diag

[

/ / ] ∆b11 ˆb11 · · · ∆b1n ˆb1n n2 ×n2 · · · FB (t) = . [ / / ] ∈ℜ diag ∆bn1 ˆbn1 · · · ∆bnn ˆbnn [

diag

From (5a)-(5c), we have FCT (t) FC (t) ≤ I, FAT (t) FA (t) ≤ I, FBT (t) FB (t) ≤ I. Definition 2.1. [5] 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−αt for all t ≥ 0 . Lemma 2.1. [15] Let U , V , W and M be real matrices of appropriate dimensions with M satisfying M = M T , then M + U V W + W T V T U T < 0 for all V T V ≤ I, if and only if there exists a positive scalar ε > 0 such that M + ε−1 U U T + εW T W < 0. Lemma 2.2. (Schur complement [14]) For a given matrix S = T T , the following conditions are equivalent: , S22 = S22 S11 (1) S < 0, −1 T (2) S22 < 0, S11 − S12 S22 S12 < 0.

[

S11 S12 ∗ S22

] with S11 =

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS

437

3. Global Exponential Stability Analysis. In this section, we present a delay-dependent criterion for the global exponential stability of system (1). Theorem 3.1. The equilibrium point x˜ of system (1) with τD ≤ 1 (resp., τD > 1 or unknown, is unique and globally exponentially stable (GES) with convergence rate α > 0, if there exist an n × n positive definite symmetric matrix P, some n × n positive diagonal matrices R, Q1 , Q2 (resp., Q1 = 0, Q2 = 0), W , Y , a matrix U ∈ ℜn×n , and three positive constants εA , εB , εC , such that the following LMI condition is satisfied: Σ=

Σ11 + Σ∆11 Σ12 Σ13 Σ14 Σ15 Σ16 Σ17 Σ18 ∗ Σ22 0 0 0 0 0 0 ∗ ∗ Σ33 Σ34 Σ35 Σ36 Σ37 Σ38 ∗ ∗ ∗ Σ44 + Σ∆44 0 0 0 0 ∗ ∗ ∗ ∗ Σ55 + Σ∆55 0 0 0 ∗ ∗ ∗ ∗ ∗ Σ66 0 0 ∗ ∗ ∗ ∗ ∗ ∗ Σ77 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ88

0 such that V˙ (zt ) ≤ −ρ · e2at · ∥zt ∥2 . From the condition V˙ (zt ) ≤ 0, we have V (zt ) ≤ V (z0 ) ,

440

R.-S. GAU, C.-H. LIEN AND J.-G. HSIEH

where T

V (z0 ) = z (0) P z (0) + + τM ·

n ∫ ∑ i=1

n ∫ 0 ∑

−τM

i=1

0

−τi (0)

e2αs · (s + τM ) · ri · z˙i2 (s) ds

∫

0

T

= z (0) P z (0) + ∫

−τi (0)

e2αs · z T (s) Q1 z (s) ds ∫

0 2αs

+ [

[ ] e2αs · q1i zi2 (s) + q2i gi2 (zi (s)) ds

e −τi (0)

· g (z (s)) Q2 g (z (s)) ds + τM ·

0

T

−τM

e2αs · (s + τM ) z˙ T (s) Rz˙ (s) ds

] 3 ≤ λmax (P ) + τM · λmax (Q1 ) + τM · λmax (Q2 ) + τM · λmax (R) · ∥z0 ∥2s = δ1 · ∥z0 ∥2s , and 3 δ1 = λmax (P ) + τM · λmax (Q1 ) + τM · λmax (Q2 ) + τM · λmax (R) .

On the other hand, we have V (zt ) ≥ e2αt · z T (t) P z (t) ≥ λmin (P ) · e2αt · ∥z (t)∥2 . Consequently, we obtain

√

∥z (t)∥ ≤

δ1 · ∥z0 ∥s · e−αt , λmim (P )

t ≥ 0.

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˜, [ ]T i.e., the equilibrium point z˜ = 0 · · · 0 of (3). Assume z˜ is an equilibrium point of the system (3). Then we have −CI z˜ + AI g (˜ z ) + BI g (˜ z ) = 0. Multiplying both sides of preceding equation by 2˜ z T P , we have ( ) z˜T −P CI − CIT P − 2e−2ατM · R + 2e−2ατM · R z˜ + 2˜ z T P AI g (˜ z ) + 2˜ z T P BI g (˜ z ) = 0. This implies [ ] z˜T −P CI − CIT P − 2e−2ατM · R + 2e−2ατM · R + 2α · P + Q1 − e−2ατM · (1 − τD ) · Q1 z˜ [ ] +2˜ z T P AI g (˜ z ) + 2˜ z T P BI g (˜ z ) + g T (˜ z ) Q2 − e−2ατM · (1 − τD ) · Q2 g (˜ z ) ≥ 0. From (9), we have z˜T [ΓW Γ + ΓY Γ] z˜ − g T (˜ z ) [W + Y ] g (˜ z ) ≥ 0, z˜ z˜ [ T T ] ≥ 0, 0 z˜ z˜ 0 g T (˜ z ) g T (˜ z ) Σ1 g (˜ z) g (˜ z) where Σ1 is defined in (11). Note that the condition Σ < 0 in (6) is equivalent to Σ1 < 0 in (11), implying 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 IDNN (1). This completes the proof of Theorem 3.1.

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS

441

Remark 3.1. By setting α = 0 in Theorems 3.1, we can obtain the global asymptotic stability result of system (1). By setting α = 0 and R = U = 0 in Theorem 3.1, we may obtain the following delayindependent asymptotic stability condition (independent of τM ) for the system (1). Corollary 3.1. The equilibrium point x˜ of system (1) with τD ≤ 1 is unique and globally asymptotically stable (GAS), if there exist an n × n positive definite symmetric matrix P , some n × n positive diagonal matrices Q1 , Q2 , W , Y , and three positive constants εA , εB , εC , such that the following LMI condition is satisfied: ˆ 11 + Σ ˆ ∆11 0 ˆ 13 ˆ 14 ˆ 15 Σ ˆ 16 Σ ˆ 17 Σ Σ Σ Σ ˆ 22 0 Σ 0 0 0 0 0 T ˆ ˆ ˆ 0 Σ33 + Σ∆33 0 0 0 0 Σ13 ˆ = ˆT ˆ 44 + Σ ˆ ∆44 0 Σ Σ 0 0 Σ 0 0

c ICIC International ⃝2011 ISSN 1349-4198 pp. 433–444

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS WITH MULTIPLE TIME-VARYING DELAYS Ruey-Shyan Gau1 , Chang-Hua Lien2,∗ and Jer-Guang Hsieh3 1

Department of Computer Science Municipal Kaohsiung Senior Vocational Industrial High School Kaohsiung 807, Taiwan 2

Department of Marine Engineering National Kaohsiung Marine University Kaohsiung 811, Taiwan 3

Department of Electrical Engineering I-Shou University Kaohsiung 840, Taiwan ∗ Corresponding author: [email protected]

Received September 2009; revised February 2010 Abstract. In this paper, the global exponential stability and global asymptotic stability for a class of interval delayed neural networks (IDNNs) with multiple time-varying delays are considered. Delay-dependent and delay-independent criteria are proposed to guarantee the robust stability of IDNNs via linear matrix inequality (LMI) approach. Some numerical examples are illustrated to show the eﬀectiveness of our results. From the illustrative examples, significant improvement over the recent results can be demonstrated. Keywords: Interval delayed neural network, Global exponential stability, Delay-dependent criterion, Delay-independent criterion, Linear matrix inequality

1. Introduction. In recent years, neural networks have been used in many mathematical and practical applications, such as optimization, recognition, prediction, diagnosis, decision, association, approximation, and generalization. Various neural networks have been investigated, such as bidirectional associative memory neural networks [1], cellular neural networks [2], Cohen-Grossberg neural networks [3], and Hopfield neural networks [4]. The existence of time delays is often a source of oscillation and instability of a neural networks. Hence the stability of delayed neural networks (DNNs) are important and significant in practical applications. The DNNs may be applied in many areas including the moving images processing, pattern classification, and automatic control engineering [1-4]. For many applications, artificial neural networks are usually implemented by integrated circuits [1-4]. In the implementation of artificial neural networks, time delay is produced from finite switching and finite propagation speed of electronic signals. During the implementation on very large scale integrated chips, parameter perturbations and transmitting time delays will destory the stability of DNNs [4-13]. In the analysis for uncertain DNNs, it is reasonable to assume that the parameters are varying in some prescribed intervals. Such DNNs are called the interval delayed neural networks (IDNNs) [4-7,9,13]. Hence some less conservative stability conditions for IDNNs will be proposed in this paper. Depending on whether the stability criterion itself contains the size of delay, criteria for IDNN can be classified into two categories, namely delay-independent criteria [5,7,9] and delay-dependent criteria [4,6-8,10-12]. Usually the latter is less conservative when the value of delay is small. In the Lyapunov-based delay-dependent results, the slow-varying 433

434

R.-S. GAU, C.-H. LIEN AND J.-G. HSIEH

constraints τ˙i (t) < 1, i ∈ n, are usually imposed on the time-varying delays [4,6,7,13]. These constraints will be relaxed and delay-dependent results will be proposed in this paper. In [5,9], 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 dealing with many control problems. The LMI problem can be solved quite eﬃciently by using the toolbox of Matlab [14]. In [4,6-8,10-13], stability criteria for IDNNs were proposed via LMI approach. In this paper, LMI-based delay-dependent and delay-independent criteria are proposed by using a new Lyapunov functional. In general, our approach is useful and is easy to be generalized to other forms of uncertain DNNs. When a practical neural network is designed for some specific purposes, all the results proposed in this paper can be used to guarantee the stability of these practical neural networks. By the proposed decomposition for interval matrices and approaches, our results are shown to be less conservative than other recent literatures for our illustrative numerical examples. The notation used throughout this paper is as follows. For a matrix A, we denote the transpose by AT , spectral norm by ∥A∥, symmetric positive (negative) definite by A > 0 (A < 0), maximal (minimal) eigenvalue of symmetric matrix by λmax (A) (λmin (A)). A ≤ B denotes that the matrix B − A is symmetric positive semi-definite. For a vector n x, the Euclidean norm is denoted as √ ∥x∥. ℜ denotes the n-dimensional real space,

n := {1, 2, . . . , n}, ∥xt ∥s :=

∥x (t + s)∥2 + ∥x˙ (t + s)∥2 , I denotes the identity −τM ≤s≤0 [ ] matrix, diag [ai ] denotes diagonal matrix with the diagonal elements ai , i ∈ n. V A, A { ( ) denotes A = (aij ) ∈ ℜn×n | A ≤ A ≤ A, i.e., aij ≤ aij ≤ a ¯ij , i, j ∈ n} with A = aij and A = (aij ). sup

2. Problem Statement and Preliminaries. Consider the following IDNN with multiple time-varying delays: x˙ (t) = −CI x (t) + AI y (x (t)) + B I y (x (t − τ (t))) + J,

where x (t) = x (t − τ (t)) =

[ [

(1a)

t ≥ 0,

(1b)

t ∈ [−τM , 0] ,

(1c)

y (x (t)) = f (x (t)) , x (t) = ϕ (t) ,

t ≥ 0,

(x1 (t)) (x2 (t)) · · ·

(xn (t))

]T

x1 (t − τ1 (t)) x2 (t − τ2 (t)) · · ·

, xn (t − τn (t))

]T

,

n ≥ 2 is the number of neurons in the network, 0 ≤ τi (t) ≤ τM , τ˙i (t) ≤ τD , i ∈ n, y (x (t)) ] [ ]T [ is the output, J = J1 J2 · · · Jn is the external bias vector, CI ∈ V C I , C I is ] [ a positive diagonal matrix with C I = diag [ci ], C¯I = diag [¯ ci ], ci > 0, AI ∈ V AI , AI ] [ is the feedback matrix with AI = [ai ], AI = [ai ], BI ∈ V B I , B I is the delay feedback [ ] matrix with B I = [bi ], B I = bi , and ϕ ∈ C1 is the initial function, where C1 is the set of diﬀerentiable functions from [−H, 0] to [−H, 0]. The activation functions of IDNN (1) given by [ ]T , f (x (t)) = f1 (x1 (t)) f2 (x2 (t)) · · · fn (xn (t)) [ ]T , f (x (t − τ (t))) = f1 (x1 (t − τ1 (t))) f2 (x2 (t − τ2 (t))) · · · fn (xn (t − τn (t))) are bounded monotonically nondecreasing and satisfy |fi (ξ1 ) − fi (ξ2 )| ≤ Li · |ξ1 − ξ2 | ,

ξ1 , ξ2 ∈ ℜ,

i ∈ n,

(2)

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS

435

where Li > 0, i ∈ n are some positive constants. [ ]T Assume x˜ = x˜1 x˜2 · · · x˜n ∈ ℜn is an equilibrium point of system (1), then we can obtain the following system: z˙ (t) = −CI z (t) + AI g (z (t)) + BI g (z (t − τ (t))) , where z (t) =

[ [

z1 (t) z2 (t) · · · zn (t)

(3)

]T

= x (t) − x˜, ]T g1 (z1 (t)) g2 (z2 (t)) · · · gn (zn (t)) ,

g (z (t)) = [ ]T , (4a) z (t − τ (t)) = z1 (t − τ1 (t)) z2 (t − τ2 (t)) · · · zn (t − τn (t)) [ ]T g (z (t − τ (t))) = g1 (z1 (t − τ1 (t))) g2 (z2 (t − τ2 (t))) · · · gn (zn (t − τn (t))) , (4b) gi (zi (t)) = fi (xi (t)) − fi (˜ xi ) = fi (zi (t) + x˜i ) − fi (˜ xi ) ,

gi (0) = 0,

gi (zi (t − τi (t))) = fi (xi (t − τi (t))) − fi (˜ xi ) = fi (zi (t − τi (t)) + x˜i ) − fi (˜ xi ) .

(4c) (4d)

Let W = diag [wi ] and Y = diag [yi ] be two diagonal matrices with wi , yi > 0. From (2) and (4c)-(4d), we have |gi (zi (t))| = |fi (zi (t) + x˜i ) − fi (˜ xi )| ≤ Li |zi (t)| , g T (z (t)) W g (z (t)) =

n ∑

wi |gi (zi (t))|2 ≤

i=1

n ∑

(4e)

i=1

g (z (t − τ (t))) Y g (z (t − τ (t))) = T

wi L2i |zi (t)|2 =z T (t) ΓW Γz (t) , n ∑

yi |gi (zi (t − τi (t)))|2

i=1

≤

n ∑

yi L2i |zi (t − τi (t))|2 = z T (t − τ (t)) ΓY Γz (t − τ (t)) ,

(4f)

i=1

where Γ = diag[Li ]. Remark 2.1. The forms fi (xi ) = 0.5 (|xi + 1| − |xi − 1|) and fi (xi ) = tanh (xi ) are two general activation functions satisfying (2). From interval matrix assumptions, matrices C I , AI and BI can be rewritten as ) 1( CI = C + ∆C, C = C I + C I , ∆C = diag [∆ci ] , 2 1 |∆ci | ≤ cˆi , cˆi = (¯ ci − ci ) > 0, i ∈ n, 2 ) 1( AI = A + ∆A, A = AI + AI , ∆A = [∆aij ], 2 ) 1( a ¯ij − aij > 0, i, j ∈ n, |∆aij | ≤ a ˆij , a ˆij = 2 ) 1( BI = B + ∆B, B = B I + B I , ∆B = [∆bij ], 2 ) ( 1 ¯bij − b > 0, i, j ∈ n. |∆bij | ≤ ˆbij , ˆbij = ij 2 Uncertain matrices ∆C, ∆A and ∆B are assumed to take the form ∆C = MC FC (t)NC ,

∆A = MA FA (t) NA ,

∆B = MB FB (t)NB ,

(5a)

(5b)

(5c)

(5d)

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R.-S. GAU, C.-H. LIEN AND J.-G. HSIEH

where n n [ ] z√ }| √ { [√ ] ∆ci 2 MC = NC = diag cˆi , FC (t) = diag , MA = diag a ˆi1 · · · a ˆin ∈ ℜn×n , cˆi i=1

[ √ ] √ diag a ˆ11 · · · a ˆ1n n2 ×n NA = , ] ∈ℜ [ √ ... √ diag a ˆn1 · · · a ˆnn

n n z√ }| √ { ˆbi1 · · · ˆbin MB = diag

2

∈ ℜn×n ,

i=1

[ √ ] √ ˆb11 · · · ˆb1n ∈ ℜn2 ×n , · · · NB = [ ] √ √ ˆbn1 · · · ˆbnn diag diag

] ∆a11 / a ˆ11 · · · ∆a1n / a ˆ1n n2 ×n2 ··· FA (t) = , [ ] ∈ℜ ˆn1 · · · ∆ann / a ˆnn diag ∆an1 / a diag

[

/ / ] ∆b11 ˆb11 · · · ∆b1n ˆb1n n2 ×n2 · · · FB (t) = . [ / / ] ∈ℜ diag ∆bn1 ˆbn1 · · · ∆bnn ˆbnn [

diag

From (5a)-(5c), we have FCT (t) FC (t) ≤ I, FAT (t) FA (t) ≤ I, FBT (t) FB (t) ≤ I. Definition 2.1. [5] 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−αt for all t ≥ 0 . Lemma 2.1. [15] Let U , V , W and M be real matrices of appropriate dimensions with M satisfying M = M T , then M + U V W + W T V T U T < 0 for all V T V ≤ I, if and only if there exists a positive scalar ε > 0 such that M + ε−1 U U T + εW T W < 0. Lemma 2.2. (Schur complement [14]) For a given matrix S = T T , the following conditions are equivalent: , S22 = S22 S11 (1) S < 0, −1 T (2) S22 < 0, S11 − S12 S22 S12 < 0.

[

S11 S12 ∗ S22

] with S11 =

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS

437

3. Global Exponential Stability Analysis. In this section, we present a delay-dependent criterion for the global exponential stability of system (1). Theorem 3.1. The equilibrium point x˜ of system (1) with τD ≤ 1 (resp., τD > 1 or unknown, is unique and globally exponentially stable (GES) with convergence rate α > 0, if there exist an n × n positive definite symmetric matrix P, some n × n positive diagonal matrices R, Q1 , Q2 (resp., Q1 = 0, Q2 = 0), W , Y , a matrix U ∈ ℜn×n , and three positive constants εA , εB , εC , such that the following LMI condition is satisfied: Σ=

Σ11 + Σ∆11 Σ12 Σ13 Σ14 Σ15 Σ16 Σ17 Σ18 ∗ Σ22 0 0 0 0 0 0 ∗ ∗ Σ33 Σ34 Σ35 Σ36 Σ37 Σ38 ∗ ∗ ∗ Σ44 + Σ∆44 0 0 0 0 ∗ ∗ ∗ ∗ Σ55 + Σ∆55 0 0 0 ∗ ∗ ∗ ∗ ∗ Σ66 0 0 ∗ ∗ ∗ ∗ ∗ ∗ Σ77 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ88

0 such that V˙ (zt ) ≤ −ρ · e2at · ∥zt ∥2 . From the condition V˙ (zt ) ≤ 0, we have V (zt ) ≤ V (z0 ) ,

440

R.-S. GAU, C.-H. LIEN AND J.-G. HSIEH

where T

V (z0 ) = z (0) P z (0) + + τM ·

n ∫ ∑ i=1

n ∫ 0 ∑

−τM

i=1

0

−τi (0)

e2αs · (s + τM ) · ri · z˙i2 (s) ds

∫

0

T

= z (0) P z (0) + ∫

−τi (0)

e2αs · z T (s) Q1 z (s) ds ∫

0 2αs

+ [

[ ] e2αs · q1i zi2 (s) + q2i gi2 (zi (s)) ds

e −τi (0)

· g (z (s)) Q2 g (z (s)) ds + τM ·

0

T

−τM

e2αs · (s + τM ) z˙ T (s) Rz˙ (s) ds

] 3 ≤ λmax (P ) + τM · λmax (Q1 ) + τM · λmax (Q2 ) + τM · λmax (R) · ∥z0 ∥2s = δ1 · ∥z0 ∥2s , and 3 δ1 = λmax (P ) + τM · λmax (Q1 ) + τM · λmax (Q2 ) + τM · λmax (R) .

On the other hand, we have V (zt ) ≥ e2αt · z T (t) P z (t) ≥ λmin (P ) · e2αt · ∥z (t)∥2 . Consequently, we obtain

√

∥z (t)∥ ≤

δ1 · ∥z0 ∥s · e−αt , λmim (P )

t ≥ 0.

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˜, [ ]T i.e., the equilibrium point z˜ = 0 · · · 0 of (3). Assume z˜ is an equilibrium point of the system (3). Then we have −CI z˜ + AI g (˜ z ) + BI g (˜ z ) = 0. Multiplying both sides of preceding equation by 2˜ z T P , we have ( ) z˜T −P CI − CIT P − 2e−2ατM · R + 2e−2ατM · R z˜ + 2˜ z T P AI g (˜ z ) + 2˜ z T P BI g (˜ z ) = 0. This implies [ ] z˜T −P CI − CIT P − 2e−2ατM · R + 2e−2ατM · R + 2α · P + Q1 − e−2ατM · (1 − τD ) · Q1 z˜ [ ] +2˜ z T P AI g (˜ z ) + 2˜ z T P BI g (˜ z ) + g T (˜ z ) Q2 − e−2ατM · (1 − τD ) · Q2 g (˜ z ) ≥ 0. From (9), we have z˜T [ΓW Γ + ΓY Γ] z˜ − g T (˜ z ) [W + Y ] g (˜ z ) ≥ 0, z˜ z˜ [ T T ] ≥ 0, 0 z˜ z˜ 0 g T (˜ z ) g T (˜ z ) Σ1 g (˜ z) g (˜ z) where Σ1 is defined in (11). Note that the condition Σ < 0 in (6) is equivalent to Σ1 < 0 in (11), implying 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 IDNN (1). This completes the proof of Theorem 3.1.

NOVEL STABILITY CONDITIONS FOR INTERVAL DELAYED NEURAL NETWORKS

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Remark 3.1. By setting α = 0 in Theorems 3.1, we can obtain the global asymptotic stability result of system (1). By setting α = 0 and R = U = 0 in Theorem 3.1, we may obtain the following delayindependent asymptotic stability condition (independent of τM ) for the system (1). Corollary 3.1. The equilibrium point x˜ of system (1) with τD ≤ 1 is unique and globally asymptotically stable (GAS), if there exist an n × n positive definite symmetric matrix P , some n × n positive diagonal matrices Q1 , Q2 , W , Y , and three positive constants εA , εB , εC , such that the following LMI condition is satisfied: ˆ 11 + Σ ˆ ∆11 0 ˆ 13 ˆ 14 ˆ 15 Σ ˆ 16 Σ ˆ 17 Σ Σ Σ Σ ˆ 22 0 Σ 0 0 0 0 0 T ˆ ˆ ˆ 0 Σ33 + Σ∆33 0 0 0 0 Σ13 ˆ = ˆT ˆ 44 + Σ ˆ ∆44 0 Σ Σ 0 0 Σ 0 0