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A novel global robust stability criterion for neural networks with delay
Physics Letters A 337 (2005) 369–373 www.elsevier.com/locate/pla
A novel global robust stability criterion for neural networks with delay Vimal Singh Department of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey Received 17 August 2004; accepted 2 February 2005
Communicated by A.P. Fordy
Abstract A criterion based on the intervalised network parameters for the global robust stability of Hopfield-type neural networks with delay is presented. The criterion is compared with an earlier criterion. 2005 Elsevier B.V. All rights reserved. Keywords: Dynamical interval neural networks; Equilibrium analysis; Global robust stability; Hopfield neural networks; Neural networks
1. Introduction A number of applications of neural networks in areas such as pattern recognition, image processing, associative memory, optimisation problems, etc., have emerged. In some of these applications, it is required that the equilibrium point of the designed network be unique and globally asymptotically stable. In electronic implementation of neural networks, a time delay will occur in the interaction between the neurons, which may affect the stability of the network creating oscillatory and unstable characteristics. The stability of neural networks with delay has received a considerable attention (see [1–31] and the references cited therein). In practice, the weight coefficients of the neurons depend on certain resistance and capacitance values which are subject to uncertainties. Ensuring the global asymptotic stability and uniqueness of the equilibrium point of the designed network in the presence of such parametric uncertainties is an important problem. The global robust stability of Hopfield-type delayed neural networks (DNNs) based on the intervalised network parameters has been studied in [5,16,24]. A criterion for the global robust stability of delayed cellular neural networks (DCNNs) based on norm-bounded uncertainties has been given in [31]. In this Letter, a novel criterion for the global robust stability of Hopfield-type DNNs based on the interE-mail address: [email protected] (V. Singh). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.004
370
V. Singh / Physics Letters A 337 (2005) 369–373
valised parameters is presented. The criterion is easily testable and compares well with a criterion due to Arik [24].
2. System description The DNN model [5,16,24] to be considered herein is described by n n dxi (t) = −di xi (t) + aij yj xj (t) + aijτ yj xj (t − τj ) + ui , dt j =1
i = 1, . . . , n
j =1
⇒ x˙ = −Dx + Ay x(t) + Aτ y x(t − τ ) + u,
(1)
where x = [x1 . . . xn ]T denotes the state vector, D = (di ) the positive diagonal matrix, A = (aij ) the feedback matrix, Aτ = (aijτ ) the delayed feedback matrix, τj the transmission delays, ui the constant external inputs and yj the activation functions and the superscript T to any vector (or matrix) denotes the transpose of that vector (or matrix). The activation functions are assumed to satisfy [16] |yj (ξ )| Mj , ∀ξ ∈ R and yj (ξ1 ) − yj (ξ2 ) Lj for each ξ1 , ξ2 ∈ R, ξ1 = ξ2 , ξ1 − ξ2 where Mj and Lj are positive constants. The quantities di , aij and aijτ may be considered as intervalised [5,16,24]: D I := D = diag(di ): D D D, i.e., d i di d i , i = 1, . . . , n, ∀D ∈ D I , AI := A = (aij )n×n : A A A, i.e., a ij aij a ij , i, j = 1, . . . , n, ∀A ∈ AI , AτI := Aτ = (aijτ )n×n : Aτ Aτ Aτ , i.e., a τij aijτ a τij , i, j = 1, . . . , n, ∀Aτ ∈ AτI . 0
The system (1) with these parameter ranges will be referred [5,16,24] to as globally robust stable if there is a unique equilibrium point x ∗ = [x1∗ . . . xn∗ ]T of the system, which is globally asymptotically stable for all D ∈ D I , A ∈ AI , Aτ ∈ AτI . By shifting the equilibrium point x ∗ = [x1∗ . . . xn∗ ]T of (1) to the origin and using the relation z(·) = x(·) − x ∗ , (1) is transformed to n n dzi (t) = −di zi (t) + aij gj zj (t) + aijτ gj zj (t − τj ) , dt j =1
i = 1, . . . , n
j =1
⇒ z˙ = −Dz + Ag z(t) + Aτ g z(t − τ ) ,
(2)
where z = [z1 . . . zn ]T is the state vector of the transformed system, T T g z(t) = g1 z1 (t) . . . gn zn (t) , g z(t − τ ) = g1 z1 (t − τ1 ) . . . gn zn (t − τn ) , gj zj (·) = yj zj (·) + xj∗ − yj xj∗ . The functions gj satisfy 0 (gj (zj ))/zj Lj and gj (0) = 0. Thus, it suffices to prove the global robust stability of the null solution of (2). In the following, B > 0 ( 0) implies that the matrix B is symmetric positive definite (positive semi-definite), I denotes the identity matrix, τ ∗ A = max Aτ , Aτ , LM = max{Li }, dm = min{d i }, 2 2 2 and S = {sij } is a symmetric matrix defined by −2a ii , if i = j, sij = −a ∗ , if i = j, aij∗ = max |a ij + a j i |, |a ij + a j i | . ij
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3. Main result In this section, we present a sufficient condition for the global robust stability of system (1). This result is given in the following: Theorem 1. Suppose the condition Aτ 2 (Aτ )∗ 2 holds. Then system (1) is globally robust stable if
∗ 2 dm S+ 2 − 1 − Aτ 2 I > 0. LM
(3)
Proof. Choose the positive-definite Lyapunov functional n V z(t) = αzT (t)z(t) + 2
z i (t)
gi (s) ds +
i=1 0
n t
gi2 zi (ς) dζ,
(4)
i=1 t−τi
where α is a positive constant. Along the trajectories of (2)
n z(t) T T gi (zi (t)) T V˙ z(t) = − z (t) g z(t) g z(t − τ ) M −2 di gi zi (t) zi (t) 1 − g(z(t)) Li zi (t) g(z(t − τ )) i=1 n
di dm −2 gi2 zi (t) , − Li LM
(5)
i=1
where
−αA
2αD
T M = −αA −α(Aτ )T
2
dm I − A − AT − I LM −(Aτ )T
−αAτ
−Aτ .
(6)
I
The second and third terms in (5) are non-positive, implying that V˙ (z(t)) is negative-definite if M > 0. Thus, M > 0 is a sufficient condition for the global asymptotic stability of the origin of (2). Using the well-known Schur’s complements, the condition M > 0 can be expressed as ˆ > 0, ˆ T D −1 A Q = N − (1/2)α A ˆ = [A Aτ ] and where A
dm 2 I − A − AT − I N= LM −(Aτ )T
−Aτ
> 0.
(7)
I
Clearly, the condition Q > 0 is satisfied by letting α < (λ1 /λ2 ) where λ1 denotes the minimum eigenvalue of N ˆ In the light of Schur’s complements, (7) becomes ˆ T D −1 A. and λ2 the maximum eigenvalue of (1/2)A 2
dm I − A − AT − I − Aτ (Aτ )T > 0. LM
(8)
Thus, the origin of (2) is globally asymptotically stable if (8) holds. This global asymptotic stability result automatically implies that the origin of (2) is the unique equilibrium point if (8) holds. Finally, (8) is satisfied if (3) holds. Note that (3) is the worst case of (8). This completes the proof of Theorem 1. 2
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V. Singh / Physics Letters A 337 (2005) 369–373
4. Comparison with an earlier criterion The following is the criterion given in [24] for the global robust stability of system (1):
∗ 2 dm S > 0 and 2 − 1 − Aτ 2 I 0. LM
(9)
A perusal of the proof given in [24] reveals that, for the applicability of (9), the assumption Aτ 2 (Aτ )∗ 2 is required (though this was not explicitly stated in [24]). Whereas (3) represents a single condition, two conditions are to be satisfied separately when dealing with (9). Moreover, the first of (9) requires the restriction a ii < 0. A closer examination of (3) and (9) reveals that adding the two conditions in (9) yields (3). This means that (3) is less restrictive than (9). Note that, in situations where both of (9) hold, (3) will be satisfied. On the other hand, it is not always required to satisfy both of (9) for (3) to hold, i.e., if one of (9) is violated, then (3) may possibly still be satisfied. As an illustration of this, consider a second-order DNN characterised by 0.1 0.1 0.4 0.4 1 0 τ τ A=A= , A =A = , D=D= , L1 = L2 = 1. (10) 0.1 −1 0.4 0.4 0 1 In this example, the first of (9) fails while (3) is satisfied. As another example, consider −1 0.5 −2 0.5 0.7 0.7 τ τ A= , A= , A =A = , 0.5 −1.5 0.5 −2 0.7 0.7 1 0 D=D= , L1 = L2 = 1. 0 1
(11)
Here, the second of (9) is violated while (3) holds.
5. Conclusion A criterion based on the intervalised network parameters for the global robust stability of Hopfield-type neural networks with delay is presented. The criterion turns out to be an improvement over the criterion reported in [24].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
T. Roska, C.W. Wu, M. Balsi, L.O. Chua, IEEE Trans. Circuits Systems I 39 (1992) 487. T. Roska, C.W. Wu, L.O. Chua, IEEE Trans. Circuits Systems I 40 (1993) 270. P.P. Civalleri, M. Gilli, L. Pandolfi, IEEE Trans. Circuits Systems I 40 (1993) 157. S. Arik, V. Tavsanoglu, IEEE Trans. Circuits Systems I 45 (1998) 168. X. Liao, J. Yu, IEEE Trans. Neural Networks 9 (1998) 1042. J. Cao, D. Zhou, Neural Networks 11 (1998) 1601. T.L. Liao, F.C. Wang, Electron. Lett. 35 (1999) 1347. J. Cao, Phys. Rev. E 59 (1999) 5940. S. Arik, V. Tavsanoglu, IEEE Trans. Circuits Systems I 47 (2000) 571. N.N. Takahashi, IEEE Trans. Circuits Systems I 47 (2000) 793. T.L. Liao, F.C. Wang, IEEE Trans. Neural Networks 11 (2000) 1481. Q. Zhang, R.N. Ma, J. Xu, Electron. Lett. 37 (2001) 575. J. Cao, IEEE Trans. Circuits Systems I 48 (2001) 494. Z. Yi, A. Heng, K.S. Leung, IEEE Trans. Circuits Systems I 48 (2001) 680. J. Cao, IEEE Trans. Circuits Systems I 48 (2001) 1330. X. Liao, K. Wong, Z. Wu, G. Chen, IEEE Trans. Circuits Systems I 48 (2001) 1355. T. Chen, Neural Networks 14 (2001) 977.
V. Singh / Physics Letters A 337 (2005) 369–373
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
S. Arik, IEEE Trans. Neural Networks 13 (2002) 1239. X. Liao, G. Chen, E.N. Sanchez, IEEE Trans. Circuits Systems I 49 (2002) 1033. X. Liao, G. Chen, E.N. Sanchez, Neural Networks 15 (2002) 855. S. Arik, IEEE Trans. Circuits Systems I 49 (2002) 1211. C.Y. Sun, K.J. Zhang, S.M. Fei, C.B. Feng, Phys. Lett. A 298 (2002) 122. J. Cao, J. Wang, IEEE Trans. Circuits Systems I 50 (2003) 34. S. Arik, IEEE Trans. Circuits Systems I 50 (2003) 156. S. Arik, Phys. Lett. A 311 (2003) 504. W. Lu, L. Rong, T. Chen, Int. J. Neural Systems 13 (2003) 193. L. Xiaofeng, K. Wong, Z. Wu, IEEE Trans. Neural Networks 14 (2003) 222. C.Y. Sun, C.B. Feng, Neural Process. Lett. 17 (2003) 107. H. Qi, L. Qi, IEEE Trans. Neural Networks 15 (2004) 99. V. Singh, IEEE Trans. Neural Networks 15 (2004) 223. V. Singh, IEE Proc. Control Theory Appl. 151 (2004) 125.
373
A novel global robust stability criterion for neural networks with delay Vimal Singh Department of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey Received 17 August 2004; accepted 2 February 2005
Communicated by A.P. Fordy
Abstract A criterion based on the intervalised network parameters for the global robust stability of Hopfield-type neural networks with delay is presented. The criterion is compared with an earlier criterion. 2005 Elsevier B.V. All rights reserved. Keywords: Dynamical interval neural networks; Equilibrium analysis; Global robust stability; Hopfield neural networks; Neural networks
1. Introduction A number of applications of neural networks in areas such as pattern recognition, image processing, associative memory, optimisation problems, etc., have emerged. In some of these applications, it is required that the equilibrium point of the designed network be unique and globally asymptotically stable. In electronic implementation of neural networks, a time delay will occur in the interaction between the neurons, which may affect the stability of the network creating oscillatory and unstable characteristics. The stability of neural networks with delay has received a considerable attention (see [1–31] and the references cited therein). In practice, the weight coefficients of the neurons depend on certain resistance and capacitance values which are subject to uncertainties. Ensuring the global asymptotic stability and uniqueness of the equilibrium point of the designed network in the presence of such parametric uncertainties is an important problem. The global robust stability of Hopfield-type delayed neural networks (DNNs) based on the intervalised network parameters has been studied in [5,16,24]. A criterion for the global robust stability of delayed cellular neural networks (DCNNs) based on norm-bounded uncertainties has been given in [31]. In this Letter, a novel criterion for the global robust stability of Hopfield-type DNNs based on the interE-mail address: [email protected] (V. Singh). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.004
370
V. Singh / Physics Letters A 337 (2005) 369–373
valised parameters is presented. The criterion is easily testable and compares well with a criterion due to Arik [24].
2. System description The DNN model [5,16,24] to be considered herein is described by n n dxi (t) = −di xi (t) + aij yj xj (t) + aijτ yj xj (t − τj ) + ui , dt j =1
i = 1, . . . , n
j =1
⇒ x˙ = −Dx + Ay x(t) + Aτ y x(t − τ ) + u,
(1)
where x = [x1 . . . xn ]T denotes the state vector, D = (di ) the positive diagonal matrix, A = (aij ) the feedback matrix, Aτ = (aijτ ) the delayed feedback matrix, τj the transmission delays, ui the constant external inputs and yj the activation functions and the superscript T to any vector (or matrix) denotes the transpose of that vector (or matrix). The activation functions are assumed to satisfy [16] |yj (ξ )| Mj , ∀ξ ∈ R and yj (ξ1 ) − yj (ξ2 ) Lj for each ξ1 , ξ2 ∈ R, ξ1 = ξ2 , ξ1 − ξ2 where Mj and Lj are positive constants. The quantities di , aij and aijτ may be considered as intervalised [5,16,24]: D I := D = diag(di ): D D D, i.e., d i di d i , i = 1, . . . , n, ∀D ∈ D I , AI := A = (aij )n×n : A A A, i.e., a ij aij a ij , i, j = 1, . . . , n, ∀A ∈ AI , AτI := Aτ = (aijτ )n×n : Aτ Aτ Aτ , i.e., a τij aijτ a τij , i, j = 1, . . . , n, ∀Aτ ∈ AτI . 0
The system (1) with these parameter ranges will be referred [5,16,24] to as globally robust stable if there is a unique equilibrium point x ∗ = [x1∗ . . . xn∗ ]T of the system, which is globally asymptotically stable for all D ∈ D I , A ∈ AI , Aτ ∈ AτI . By shifting the equilibrium point x ∗ = [x1∗ . . . xn∗ ]T of (1) to the origin and using the relation z(·) = x(·) − x ∗ , (1) is transformed to n n dzi (t) = −di zi (t) + aij gj zj (t) + aijτ gj zj (t − τj ) , dt j =1
i = 1, . . . , n
j =1
⇒ z˙ = −Dz + Ag z(t) + Aτ g z(t − τ ) ,
(2)
where z = [z1 . . . zn ]T is the state vector of the transformed system, T T g z(t) = g1 z1 (t) . . . gn zn (t) , g z(t − τ ) = g1 z1 (t − τ1 ) . . . gn zn (t − τn ) , gj zj (·) = yj zj (·) + xj∗ − yj xj∗ . The functions gj satisfy 0 (gj (zj ))/zj Lj and gj (0) = 0. Thus, it suffices to prove the global robust stability of the null solution of (2). In the following, B > 0 ( 0) implies that the matrix B is symmetric positive definite (positive semi-definite), I denotes the identity matrix, τ ∗ A = max Aτ , Aτ , LM = max{Li }, dm = min{d i }, 2 2 2 and S = {sij } is a symmetric matrix defined by −2a ii , if i = j, sij = −a ∗ , if i = j, aij∗ = max |a ij + a j i |, |a ij + a j i | . ij
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371
3. Main result In this section, we present a sufficient condition for the global robust stability of system (1). This result is given in the following: Theorem 1. Suppose the condition Aτ 2 (Aτ )∗ 2 holds. Then system (1) is globally robust stable if
∗ 2 dm S+ 2 − 1 − Aτ 2 I > 0. LM
(3)
Proof. Choose the positive-definite Lyapunov functional n V z(t) = αzT (t)z(t) + 2
z i (t)
gi (s) ds +
i=1 0
n t
gi2 zi (ς) dζ,
(4)
i=1 t−τi
where α is a positive constant. Along the trajectories of (2)
n z(t) T T gi (zi (t)) T V˙ z(t) = − z (t) g z(t) g z(t − τ ) M −2 di gi zi (t) zi (t) 1 − g(z(t)) Li zi (t) g(z(t − τ )) i=1 n
di dm −2 gi2 zi (t) , − Li LM
(5)
i=1
where
−αA
2αD
T M = −αA −α(Aτ )T
2
dm I − A − AT − I LM −(Aτ )T
−αAτ
−Aτ .
(6)
I
The second and third terms in (5) are non-positive, implying that V˙ (z(t)) is negative-definite if M > 0. Thus, M > 0 is a sufficient condition for the global asymptotic stability of the origin of (2). Using the well-known Schur’s complements, the condition M > 0 can be expressed as ˆ > 0, ˆ T D −1 A Q = N − (1/2)α A ˆ = [A Aτ ] and where A
dm 2 I − A − AT − I N= LM −(Aτ )T
−Aτ
> 0.
(7)
I
Clearly, the condition Q > 0 is satisfied by letting α < (λ1 /λ2 ) where λ1 denotes the minimum eigenvalue of N ˆ In the light of Schur’s complements, (7) becomes ˆ T D −1 A. and λ2 the maximum eigenvalue of (1/2)A 2
dm I − A − AT − I − Aτ (Aτ )T > 0. LM
(8)
Thus, the origin of (2) is globally asymptotically stable if (8) holds. This global asymptotic stability result automatically implies that the origin of (2) is the unique equilibrium point if (8) holds. Finally, (8) is satisfied if (3) holds. Note that (3) is the worst case of (8). This completes the proof of Theorem 1. 2
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V. Singh / Physics Letters A 337 (2005) 369–373
4. Comparison with an earlier criterion The following is the criterion given in [24] for the global robust stability of system (1):
∗ 2 dm S > 0 and 2 − 1 − Aτ 2 I 0. LM
(9)
A perusal of the proof given in [24] reveals that, for the applicability of (9), the assumption Aτ 2 (Aτ )∗ 2 is required (though this was not explicitly stated in [24]). Whereas (3) represents a single condition, two conditions are to be satisfied separately when dealing with (9). Moreover, the first of (9) requires the restriction a ii < 0. A closer examination of (3) and (9) reveals that adding the two conditions in (9) yields (3). This means that (3) is less restrictive than (9). Note that, in situations where both of (9) hold, (3) will be satisfied. On the other hand, it is not always required to satisfy both of (9) for (3) to hold, i.e., if one of (9) is violated, then (3) may possibly still be satisfied. As an illustration of this, consider a second-order DNN characterised by 0.1 0.1 0.4 0.4 1 0 τ τ A=A= , A =A = , D=D= , L1 = L2 = 1. (10) 0.1 −1 0.4 0.4 0 1 In this example, the first of (9) fails while (3) is satisfied. As another example, consider −1 0.5 −2 0.5 0.7 0.7 τ τ A= , A= , A =A = , 0.5 −1.5 0.5 −2 0.7 0.7 1 0 D=D= , L1 = L2 = 1. 0 1
(11)
Here, the second of (9) is violated while (3) holds.
5. Conclusion A criterion based on the intervalised network parameters for the global robust stability of Hopfield-type neural networks with delay is presented. The criterion turns out to be an improvement over the criterion reported in [24].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
T. Roska, C.W. Wu, M. Balsi, L.O. Chua, IEEE Trans. Circuits Systems I 39 (1992) 487. T. Roska, C.W. Wu, L.O. Chua, IEEE Trans. Circuits Systems I 40 (1993) 270. P.P. Civalleri, M. Gilli, L. Pandolfi, IEEE Trans. Circuits Systems I 40 (1993) 157. S. Arik, V. Tavsanoglu, IEEE Trans. Circuits Systems I 45 (1998) 168. X. Liao, J. Yu, IEEE Trans. Neural Networks 9 (1998) 1042. J. Cao, D. Zhou, Neural Networks 11 (1998) 1601. T.L. Liao, F.C. Wang, Electron. Lett. 35 (1999) 1347. J. Cao, Phys. Rev. E 59 (1999) 5940. S. Arik, V. Tavsanoglu, IEEE Trans. Circuits Systems I 47 (2000) 571. N.N. Takahashi, IEEE Trans. Circuits Systems I 47 (2000) 793. T.L. Liao, F.C. Wang, IEEE Trans. Neural Networks 11 (2000) 1481. Q. Zhang, R.N. Ma, J. Xu, Electron. Lett. 37 (2001) 575. J. Cao, IEEE Trans. Circuits Systems I 48 (2001) 494. Z. Yi, A. Heng, K.S. Leung, IEEE Trans. Circuits Systems I 48 (2001) 680. J. Cao, IEEE Trans. Circuits Systems I 48 (2001) 1330. X. Liao, K. Wong, Z. Wu, G. Chen, IEEE Trans. Circuits Systems I 48 (2001) 1355. T. Chen, Neural Networks 14 (2001) 977.
V. Singh / Physics Letters A 337 (2005) 369–373
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
S. Arik, IEEE Trans. Neural Networks 13 (2002) 1239. X. Liao, G. Chen, E.N. Sanchez, IEEE Trans. Circuits Systems I 49 (2002) 1033. X. Liao, G. Chen, E.N. Sanchez, Neural Networks 15 (2002) 855. S. Arik, IEEE Trans. Circuits Systems I 49 (2002) 1211. C.Y. Sun, K.J. Zhang, S.M. Fei, C.B. Feng, Phys. Lett. A 298 (2002) 122. J. Cao, J. Wang, IEEE Trans. Circuits Systems I 50 (2003) 34. S. Arik, IEEE Trans. Circuits Systems I 50 (2003) 156. S. Arik, Phys. Lett. A 311 (2003) 504. W. Lu, L. Rong, T. Chen, Int. J. Neural Systems 13 (2003) 193. L. Xiaofeng, K. Wong, Z. Wu, IEEE Trans. Neural Networks 14 (2003) 222. C.Y. Sun, C.B. Feng, Neural Process. Lett. 17 (2003) 107. H. Qi, L. Qi, IEEE Trans. Neural Networks 15 (2004) 99. V. Singh, IEEE Trans. Neural Networks 15 (2004) 223. V. Singh, IEE Proc. Control Theory Appl. 151 (2004) 125.
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