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Global robust exponential stability of delayed neural networks: An LMI approach
Chaos, Solitons and Fractals 32 (2007) 1742–1748 www.elsevier.com/locate/chaos
Global robust exponential stability of delayed neural networks: An LMI approach Ou Ou College of Information Engineering, Chengdu University of Technology, Chengdu 610059, PR China Accepted 1 December 2005
Abstract In this paper, the problems of determining the robust exponential stability and estimating the exponential convergence rate for neural networks with parametric uncertainties and time delay are studied. Based on Lyapunov– Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique, some delay-dependent criteria are derived to guarantee global robust exponential stability. The exponential convergence rate can be easily estimated via these criteria. Ó 2006 Published by Elsevier Ltd.
1. Introduction Neural networks have many applications in pattern recognition, image processing, association, optimal computation, etc. Some of these applications require that the equilibrium points of the designed network be stable. So, it is important to study the stability of neural networks. In biological and artificial neural networks, time delays often arise in the processing of information storage and transmission. In recent years, the stability of delayed neural networks (DNN) has been investigated by many researchers (e.g. [1–10]). The exponential stability property is particularly important, when the exponential convergence rate is used to determine the speed of neural computations. The exponential stability property guarantees that, whatever transformation occurs, the network’s ability to store rapidly the activity pattern is left invariant by self-organization. Thus, it is not only theoretically interesting but also practically important to determine the exponential stability and to estimate the exponential convergence rate for delayed neural networks in general. Delay-dependent exponential stability of neural networks with constant or time-varying delays is considered in [11]. Global exponential stability for neural networks with time-varying delays has been studied in [12]. In practice, the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties. It is important and interesting to investigate the robust stability of neural networks with parametric uncertainties. In [13– 16], the authors studied the robust stability of delayed neural networks. In [17], the authors studied the robust global exponential stability of Cohen–Grossberg neural networks with time delays, but the criteria are based on the intervalised network parameters. In this paper, we further extend the results of [11,12] to the general delayed neural networks with norm-bounded parametric uncertainties
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.12.026
O. Ou / Chaos, Solitons and Fractals 32 (2007) 1742–1748
1743
duðtÞ ¼ AuðtÞ þ ðW þ DW Þf ðuðtÞÞ þ ðW 1 þ DW 1 Þf ðuðt sðtÞÞÞ þ U; dt
ð1Þ
where u(t) = [u1(t), u2(t), . . . , un(t)]T is the neuron state vector, A = diag(a1, a2, . . . , an) is a positive diagonal matrix and 0 < ai 6 ai 6 ai ; W and W1 are interconnection weight matrices, 0 < s(t) 6 s0 is the differentiable time delay, and it is assumed that s_ ðtÞ 6 d < 1; d P 0. U is a constant input vector, DW, DW1 are parametric uncertainties, and f(u) = [f1(u1), f2(u2), . . . , fn(un)]T denotes the neuron activation function. As in many papers, we assume that each activation function in (1) satisfies the following sector condition: there is a real constant k 2 R+ such that jfj ðxÞ fj ðyÞj 6 kjx yj;
8x; y 2 R; j ¼ 1; 2; ; n
The uncertainties DW, DW1 are defined by DW ¼ HFE;
DW 1 ¼ H 1 F 1 E1 ;
ð2Þ
where H, H1, E, E1 are known constant matrices of appropriate dimensions, and F, F1 are unknown matrices representing the parameter uncertainties, which satisfy F T F 6 I;
F T1 F 1 6 I
ð3Þ
in which I is the identity matrix of appropriate dimension. The uncertainty model of (2) and (3) has been widely adopted in robust control and filtering for uncertain systems. The matrices H(H1) and E(E1) characterize how the uncertain parameters in F(F1) enter DW(DW1). Note that F(F1) can always be restricted as in (3) by appropriately choosing H(H1) and E(E1). Some notations to be used are defined here. We use WT to denote the transpose of a square matrix W, W1 to denote the inverse of a square matrix W, kM(m)(W) to denote the operation of taking the maximum (minimum) eigenvalues of a square matrix W, I to denote identity matrix, and k Æ k to denote Euclidean norm of a vector. We use W > 0(W < 0) to denote a symmetric positive (negative) definite matrix W. We use xt to represent a segment of x(h) on [t s(t), t] with kxtk = supts(t)6h6tkx(h)k. Definition 1. If there is a unique equilibrium point u ¼ ½u1 ; u2 ; . . . ; un T of system (1) and there exist > 0 and c() > 0 such that kuðtÞ u k 6 cðÞet
sup
kuðhÞ u k;
8t > 0
ð4Þ
sð0Þ6h60
then system (1) is said to be globally robustly exponentially stable, where is the degree of exponential stability. In the following, we always shift the equilibrium u* of (1) to the origin. If we make a transform x(t) = u(t) u*, then it transforms model (1) to the following: dxðtÞ ¼ AxðtÞ þ ðW þ DW ÞgðxðtÞÞ þ ðW 1 þ DW 1 Þgðxðt sðtÞÞÞ; dt
ð5Þ
where gj ðxj ðtÞÞ ¼ fj ðxj ðtÞ þ uj Þ fj ðuj Þ; j ¼ 1; 2; . . . ; n. Note that gj also satisfies a sector condition in the form of jgj ðxj Þj 6 kjxj j;
j ¼ 1; 2; . . . ; n.
ð6Þ
Correspondingly, the initial condition of system (5) is x(h) = u(h) u*, h 2 [s(0), 0]. We denote it as u(h) throughout this paper. Clearly, an equilibrium u* of system (1) is globally robustly exponentially stable if and only if the origin of (5) is globally robustly exponentially stable.
2. Global robust exponential stability of neural networks with parametric uncertainties and time delay In this section, two criteria are obtained for the robust exponential stability of neural networks with parametric uncertainties and time delay via Lyapunov stability theorem for functional differential equations [18] and linear matrix inequality (LMI) technique [19]. Before stating the main result, the following lemmas are needed. Lemma 1. Given any a, b 2 Rn, any matrix Q > 0 of appropriate dimension and any scalar s > 0, we have 1 2aT b 6 saT Qa þ bT Q1 b. s
ð7Þ
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Lemma 2 [19, Schur complement]. The LMI QðxÞ SðxÞ > 0; SðxÞT RðxÞ
ð8Þ
where Q(x) = Q(x)T, R(x) = R(x)T, and S(x) depend affinely on x, is equivalent to RðxÞ > 0;
QðxÞ SðxÞRðxÞ1 SðxÞT > 0.
ð9Þ
Theorem 1. If there exist a symmetric positive matrix P, positive matrices Q0 and Q, and scalars s0 > 0, s > 0 such that the following LMI hold 3 2 PH 1 ð1; 1Þ PW PH PW 1 7 6 W T P Q 0 0 0 7 6 0 7 6 T 7<0 H P 0 s I 0 0 ð10Þ M ¼6 0 7 6 7 6 T 2s0 W P 0 0 ð1 dÞe Q 0 5 4 1 0 0 0 ð1 dÞse2s0 I H T1 P with ð1; 1Þ ¼ 2P ðAT P þ P AÞ þ k 2 Q0 þ s0 k 2 ET E þ k 2 Q þ sk 2 ET1 E1 ; A ¼ diagða1 ; a2 ; . . . ; an Þ, then the system (1) has the unique equilibrium point and the unique equilibrium is globally robustly exponentially stable for all time delay 0 < s(t) 6 s0 and s_ ðtÞ 6 d < 1. Moreover sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2sð0Þ kM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þ 1e2 kuðtÞ u k 6 ð11Þ kuket . km ðP Þ Proof. Clearly, it suffices to prove the global robust exponential stability of the origin of the system described by (5). We will first prove that the origin is the unique equilibrium point of (5). Consider the equilibrium equation of (5) Ax þ ðW þ DW Þgðx Þ þ ðW 1 þ DW 1 Þgðx Þ ¼ 0;
ð12Þ
where x* is the equilibrium point of (5). Clearly, the origin x* = [0, 0, . . . , 0]T is the equilibrium point of (5). We assume that the solution x* 5 [0, 0, . . . , 0]T is also a equilibrium point. Multiplying both sides of (12) by 2(x*)TP, we obtain 2ðx ÞT PAx þ 2ðx ÞT P ðW þ HFEÞgðx Þ þ 2ðx ÞT P ðW 1 þ H 1 F 1 E1 Þgðx Þ ¼ 0.
ð13Þ
By Lemma 1 and inequalities (3), (6), we have 2 T T 2ðx ÞT PWgðx Þ 6 ðx ÞT PWQ1 0 W P ðx Þ þ k ðx Þ Q0 ðx Þ;
1 2ðx ÞT PFEgðx Þ 6 ðx ÞT PHH T P ðx Þ þ s0 k 2 ðx ÞT ET Eðx Þ; s0 e2s0 T ðx Þ PW 1 Q1 W T1 P ðx Þ þ ð1 dÞe2s0 k 2 ðx ÞT Qðx Þ 2ðx ÞT PW 1 gðx Þ 6 1d e2s0 T ðx Þ PW 1 Q1 W T1 P ðx Þ þ k 2 ðx ÞT Qðx Þ; 6 1d e2s0 ðx ÞT PH 1 Q1 H T1 P ðx Þ þ ð1 dÞse2s0 k 2 ðx ÞT ET1 E1 ðx Þ 2ðx ÞT PH 1 F 1 E1 gðx Þ 6 ð1 dÞs e2s0 6 ðx ÞT PH 1 Q1 H T1 P ðx Þ þ sk 2 ðx ÞT ET1 E1 ðx Þ. ð1 dÞs Therefore, we get 1 e2s0 2 T ðx ÞT ðAT P þ AP Þ þ PWQ1 PHH T P þ s0 k 2 ET E þ PW 1 Q1 W T1 P þ k 2 Q 0 W P þ k Q0 þ s0 1d e2s0 2 T 1 T PH 1 Q H 1 P þ sk E1 E1 ðx Þ P 0. þ ð1 dÞs
ð14Þ ð15Þ
ð16Þ
ð17Þ
ð18Þ
O. Ou / Chaos, Solitons and Fractals 32 (2007) 1742–1748
1745
Thus 1 e2s0 2 T PW 1 Q1 W T1 P þ k 2 Q PHH T P þ s0 k 2 ET E þ ðx Þ ðAT P þ AP Þ þ PWQ1 0 W P þ k Q0 þ s0 1d e2s0 ðx Þ P 0. PH 1 Q1 H T1 P þ sk 2 ET1 E1 þ ð1 dÞs T
ð19Þ
Using the well-know Schur complement, (10) can be expressed as 2 T 2P ðAT P þ AP Þ þ PWQ1 0 W P þ k Q0 þ
þ
1 e2s0 PHH T P þ s0 k 2 EE þ PW 1 Q1 W T1 P þ k 2 Q 1d s0
e2s0 PH 1 Q1 H T1 P þ sk 2 ET1 E1 < 0. ð1 dÞs
ð20Þ
The contradiction in (19) and (20) means that the assumption (x*) 5 [0, 0, . . . , 0]T is violated. So, x* = [0, 0, . . . , 0]T is the unique equilibrium point of (5). Now, we will show that the conditions given in Theorem 1 also imply the global robust exponential stability of the origin of (5). To this end, we choose a Lyapunov–Krasovskii functional as Z t V ðxðtÞÞ ¼ e2t xT ðtÞPxðtÞ þ e2l gT ðxðlÞÞðQ þ sET1 E1 ÞgðxðlÞÞ dl. ð21Þ tsðtÞ
The derivative of V(x(t)) along the trajectory of (5) is V_ ðxðtÞÞ ¼ 2e2t xT ðtÞPxðtÞ þ e2t ½_xT ðtÞPxðtÞ þ xT ðtÞP x_ ðtÞ þ e2t gT ðxðtÞÞðQ þ sET1 E1 ÞgðxðtÞÞ ð1 s_ ðtÞÞe2ðtsðtÞÞ gT ðxðt sðtÞÞÞðQ þ sET1 E1 Þgðxðt sðtÞÞÞ 6 e2t 2xT ðtÞPxðtÞ xT ðtÞðAT P þ PAÞxðtÞ þ 2xT ðtÞPWgðxðtÞÞ þ 2xT ðtÞPHFEgT ðxðtÞÞ þ 2xT ðtÞPW 1 gðxðt sðtÞÞÞ þ 2xT ðtÞPH 1 F 1 E1 gðxðt sðtÞÞÞ þ gT ðxðtÞÞQgðxðtÞÞ
þ gT ðxðtÞÞsET1 E1 gðxðtÞÞ ð1 dÞe2s0 gT ðxðt sðtÞÞÞðQ þ sET1 E1 Þgðxðt sðtÞÞÞ .
ð22Þ
By Lemma 1, we have the following inequalities: 2xT ðtÞPW 1 gðxðt sðtÞÞÞ 6
e2s0 T x ðtÞPW 1 Q1 W T1 PxðtÞ 1d þ ð1 dÞe2s0 gT ðxðt sðtÞÞÞQgðxðt sðtÞÞÞ;
ð23Þ
e2s0 2xT ðtÞPH 1 F 1 E1 gðxðt sðtÞÞÞ 6 xT ðtÞPH 1 Q1 H T1 PxðtÞ ð1 dÞs þ ð1 dÞse2s0 gT ðxðt sðtÞÞÞET1 E1 gðxðt sðtÞÞÞ. Combined with (14), (15), (23) and (24), we get _V ðxðtÞÞ 6 e2t xT ðtÞ 2P ðAT P þ P AÞ þ PWQ1 W T P þ k 2 Q0 þ 1 PHH T P þ s0 k 2 ET E 0 s0 2s0 e2s0 e PW 1 Q1 W T1 P þ PH 1 Q1 H T1 P þ k 2 Q þ sk 2 ET1 E1 xðtÞ ¼ e2t xT ðtÞXxðtÞ; þ 1d ð1 dÞs
ð24Þ
ð25Þ
where 2 T X ¼ 2P þ ðAT P þ P AÞ þ PWQ1 0 W P k Q0
1 e2s PW 1 Q1 W T1 P PHH T P s0 k 2 EE 1d s0
e2s0 PH 1 Q1 H T1 P k 2 Q sk 2 ET1 E1 . ð1 dÞs
Obviously, by the Schur complement, if (10) holds, X > 0 and then V_ ðxðtÞÞ < 0. So, we have V ðxðtÞÞ < V ðxð0ÞÞ.
ð26Þ
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Noting that V ðxð0ÞÞ ¼ xT ð0ÞPxð0Þ þ
Z
0
e2l gT ðxðlÞÞðQ þ sET1 E1 ÞgðxðlÞÞ dl
sð0Þ
6 kM ðP Þkuk2 þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þkuk2
Z
0
e2l dl
sð0Þ
1 e2sð0Þ kuk2 6 kM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þ 2 and V ðxðtÞÞ P e2t km ðP ÞkxðtÞk2 . We have
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2sð0Þ kM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þ 1e2 kxðtÞk 6 kuket . km ðP Þ
Thus, the origin of system (5) is globally robustly exponentially stable with degree , namely, the equilibrium point u ¼ ½u1 ; u2 ; . . . ; un T of system (1) is globally robustly exponentially stable with degree . The proof of Theorem 1 is completed. h Remark 1. The exponential stability criterion (10) gives a sufficient condition for the global robust exponential stability in terms of an LMI. It can be easily solved by using some existing software packages, for example, the MATLAB LMI toolbox. Remark 2. Uncertain neural networks with multiple time delays can also be studied similarly. Remark 3. Note that the stability criterion (10) implies uniformly exponential stability with degree for all admissible DW, DW1. Under the condition of (3), the exponential stability depends on the structure of uncertainties, i.e., matrices H, E, H1 and E1. If LMI (10) is satisfied, the lower bound of exponential convergence rate is guaranteed for all neural networks described by the uncertain model (1)–(3). Remark 4. Based on criterion (10) and given the time delay conditions: 0 6 s(t) 6 s0 and s_ ðtÞ 6 d < 1, one can know not only the solution will converge the unique equilibrium u* but in what speed. On the other hand, if some converge rate should be guaranteed, one could know the allowable time delay conditions: s0 and d. The lower bound of exponential convergence rate or the allowable time delay conditions can be determined by solving the following three optimization problems: Case 1: to estimate the lower bound of exponential convergence rate > 0. max Op1 : s.t. condition (10) is satisfied, s0 and d fixed. Case 2: to estimate the allowable maximum time delay s0. max s0 Op2 : s.t. condition (10) is satisfied, > 0 and d fixed . Case 3: to estimate the allowable maximum change rate of time delay d. max d Op3 : s.t. condition (10) is satisfied, > 0 and s0 fixed.
ð27Þ
ð28Þ
ð29Þ
If the change rate of time delay is equal to 0, i.e., s(t) s0, then the systems (1) reduces to the neural networks with constant delay du ¼ AuðtÞ þ ðW þ DW Þf ðuðtÞÞ þ ðW 1 þ DW 1 Þf ðuðt s0 ÞÞ þ U dt and consequently, Theorem 1 reduces to the following Corollary.
ð30Þ
O. Ou / Chaos, Solitons and Fractals 32 (2007) 1742–1748
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Corollary 1. If there exist a symmetric positive matrix P, positive matrices Q0 and Q, and scalars s0 > 0, s > 0 such that the following LMI hold 2 3 PH 1 ð1; 1Þ PW PH PW 1 6 W T P Q 7 0 0 0 6 7 0 6 T 7 6 7<0 0 s0 I 0 0 M ¼6H P ð31Þ 7 6 T 7 2s0 0 0 e Q 0 4 W 1P 5 H T1 P 0 0 0 se2s0 I with ð1; 1Þ ¼ 2P ðAT P þ P AÞ þ k 2 Q0 þ s0 k 2 ET E þ k 2 Q þ sk 2 ET1 E1 ; A ¼ diagða1 ; a2 ; . . . ; an Þ then the system (1) has the unique equilibrium point and is globally robustly exponentially stable for time delay s(t) s0. Moreover vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u ukM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET E1 Þ 1e2s0 1 t 2 kuðtÞ u k 6 ð32Þ kuket . km ðP Þ The proof follows the same ideas in the proof of Theorem 1, and is omitted here. The corresponding Lyapunov–Krasovskii functional has the following form: V ðxðtÞÞ ¼ e2t xT ðtÞPxðtÞ þ
Z
t
e2l gT ðxðlÞÞðQ þ sET1 E1 ÞgðxðlÞÞ dl.
ð33Þ
ts0
The lower bound of exponential convergence rate or the allowable time delay conditions can be determined by solving the following two optimization problems: Case 4: to estimate the lower bound of exponential convergence rate > 0. max Op4 : s.t. condition (10) is satisfied, s0 fixed. Case 5: to estimate the allowable maximum time delay s0. max s0 Op5 : s.t. condition (10) is satisfied, > 0 fixed.
ð34Þ
ð35Þ
Remark 5. All the above optimization problems (Op1 Op5) can be solved by MATLAB LMI toolbox. Especially, Op1 and Op4 can estimate the lower bound of global exponential convergence rate , which means that the exponential convergence rate of any neural network included in (1)–(3) is at least equal to . It is useful in real-time optimal computation. 3. Conclusions In this paper, the problems of exponential stability and exponential convergence rate for uncertain neural networks with time delay have been studied. Some global exponential stability criteria, which depend on time delay, are derived via the approach of the Lyapunov–Krasovskii functional. The stability criteria are given in terms of linear matrix inequalities (LMIs) and hence are computationally efficient.
References [1] Marcus CM, Westervelt RM. Stability of analog neural network with delay. Phys Rev A 1989;39(1):347–59. [2] Ye H, Michel AN, Wang K. Global stability and local stability of Hopfield neural networks with delays. Phys Rev E 1994;50(5):4206–13. [3] Arik S. IEEE Trans. Stability analysis of delayed neural networks. Circ Syst I 2000;47(7):1089–92. [4] Liao X, Yu J. Qualitative analysis of bi-directional associative memory networks with time delays. Int J Circ Theory Appl 1998;26(4):219–29. [5] Liao TL, Wang FC. Global stability condition for cellular neural networks with delay. Electron Lett 1999;35(16):1347–9. [6] Cao J. IEEE Trans. A set of stability criteria for delayed cellular neural networks. Circ Syst I 2001;48(4):494–8.
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[7] Zhang Y, Heng PA, Leung KS. Convergence analysis of cellular neural networks with unbounded delay. IEEE Trans Circ Syst I 2001;48(6):680–7. [8] Liao X, Chen G, Sanchez EN. LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans Circ Syst I 2002;49(7):1033–9. [9] Li CD, Liao XF, Zhang R, Prasad A. Global robust exponential stability analysis for interval neural networks with time-varying delays. Chaos, Solitons & Fractals 2005;25(3):751–7. [10] Liao X, Li C, Chen G, Sanchez EN. Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Networks 2002;15(7):855–66. [11] Zhang HB, Li CG, Liao XF. A note on the robust stability of neural networks with time delay. Chaos, Solitons & Fractals 2005;25(2):357–60. [12] Zeng Z, Wang J, Liao X. Global exponential stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans Circ Syst I 2003;50(10):1353–8. [13] Liao X, Yu J. Robust stability for interval Hopfield neural networks with time delay. IEEE Trans Neural Networks 1998;9(5):1042–6. [14] Arik S. Global robust stability analysis of neural networks with discrete time delays. Chaos, Solitons & Fractals 2005;26(5):1407–14. [15] Arik S. Global robust stability of delayed neural networks. IEEE Trans Circ Syst I 2003;50(1):156–60. [16] Singh V. Robust stability of cellular neural networks with delay: linear matrix inequality approach. IEE Proc—Contr Theor Appl 2004;151(1):125–9. [17] Chen T, Rong L. Robust global exponential stability of Cohen–Grossberg neural networks with time delays. IEEE Trans Neural Networks 2004;15(1):203–6. [18] Hale JK, Lunel SMV. Introduction to the theory of functional differential equations. New York: Springer; 1991. [19] Boyd S, Ghaoui LEI, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM; 1994.
Global robust exponential stability of delayed neural networks: An LMI approach Ou Ou College of Information Engineering, Chengdu University of Technology, Chengdu 610059, PR China Accepted 1 December 2005
Abstract In this paper, the problems of determining the robust exponential stability and estimating the exponential convergence rate for neural networks with parametric uncertainties and time delay are studied. Based on Lyapunov– Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique, some delay-dependent criteria are derived to guarantee global robust exponential stability. The exponential convergence rate can be easily estimated via these criteria. Ó 2006 Published by Elsevier Ltd.
1. Introduction Neural networks have many applications in pattern recognition, image processing, association, optimal computation, etc. Some of these applications require that the equilibrium points of the designed network be stable. So, it is important to study the stability of neural networks. In biological and artificial neural networks, time delays often arise in the processing of information storage and transmission. In recent years, the stability of delayed neural networks (DNN) has been investigated by many researchers (e.g. [1–10]). The exponential stability property is particularly important, when the exponential convergence rate is used to determine the speed of neural computations. The exponential stability property guarantees that, whatever transformation occurs, the network’s ability to store rapidly the activity pattern is left invariant by self-organization. Thus, it is not only theoretically interesting but also practically important to determine the exponential stability and to estimate the exponential convergence rate for delayed neural networks in general. Delay-dependent exponential stability of neural networks with constant or time-varying delays is considered in [11]. Global exponential stability for neural networks with time-varying delays has been studied in [12]. In practice, the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties. It is important and interesting to investigate the robust stability of neural networks with parametric uncertainties. In [13– 16], the authors studied the robust stability of delayed neural networks. In [17], the authors studied the robust global exponential stability of Cohen–Grossberg neural networks with time delays, but the criteria are based on the intervalised network parameters. In this paper, we further extend the results of [11,12] to the general delayed neural networks with norm-bounded parametric uncertainties
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.12.026
O. Ou / Chaos, Solitons and Fractals 32 (2007) 1742–1748
1743
duðtÞ ¼ AuðtÞ þ ðW þ DW Þf ðuðtÞÞ þ ðW 1 þ DW 1 Þf ðuðt sðtÞÞÞ þ U; dt
ð1Þ
where u(t) = [u1(t), u2(t), . . . , un(t)]T is the neuron state vector, A = diag(a1, a2, . . . , an) is a positive diagonal matrix and 0 < ai 6 ai 6 ai ; W and W1 are interconnection weight matrices, 0 < s(t) 6 s0 is the differentiable time delay, and it is assumed that s_ ðtÞ 6 d < 1; d P 0. U is a constant input vector, DW, DW1 are parametric uncertainties, and f(u) = [f1(u1), f2(u2), . . . , fn(un)]T denotes the neuron activation function. As in many papers, we assume that each activation function in (1) satisfies the following sector condition: there is a real constant k 2 R+ such that jfj ðxÞ fj ðyÞj 6 kjx yj;
8x; y 2 R; j ¼ 1; 2; ; n
The uncertainties DW, DW1 are defined by DW ¼ HFE;
DW 1 ¼ H 1 F 1 E1 ;
ð2Þ
where H, H1, E, E1 are known constant matrices of appropriate dimensions, and F, F1 are unknown matrices representing the parameter uncertainties, which satisfy F T F 6 I;
F T1 F 1 6 I
ð3Þ
in which I is the identity matrix of appropriate dimension. The uncertainty model of (2) and (3) has been widely adopted in robust control and filtering for uncertain systems. The matrices H(H1) and E(E1) characterize how the uncertain parameters in F(F1) enter DW(DW1). Note that F(F1) can always be restricted as in (3) by appropriately choosing H(H1) and E(E1). Some notations to be used are defined here. We use WT to denote the transpose of a square matrix W, W1 to denote the inverse of a square matrix W, kM(m)(W) to denote the operation of taking the maximum (minimum) eigenvalues of a square matrix W, I to denote identity matrix, and k Æ k to denote Euclidean norm of a vector. We use W > 0(W < 0) to denote a symmetric positive (negative) definite matrix W. We use xt to represent a segment of x(h) on [t s(t), t] with kxtk = supts(t)6h6tkx(h)k. Definition 1. If there is a unique equilibrium point u ¼ ½u1 ; u2 ; . . . ; un T of system (1) and there exist > 0 and c() > 0 such that kuðtÞ u k 6 cðÞet
sup
kuðhÞ u k;
8t > 0
ð4Þ
sð0Þ6h60
then system (1) is said to be globally robustly exponentially stable, where is the degree of exponential stability. In the following, we always shift the equilibrium u* of (1) to the origin. If we make a transform x(t) = u(t) u*, then it transforms model (1) to the following: dxðtÞ ¼ AxðtÞ þ ðW þ DW ÞgðxðtÞÞ þ ðW 1 þ DW 1 Þgðxðt sðtÞÞÞ; dt
ð5Þ
where gj ðxj ðtÞÞ ¼ fj ðxj ðtÞ þ uj Þ fj ðuj Þ; j ¼ 1; 2; . . . ; n. Note that gj also satisfies a sector condition in the form of jgj ðxj Þj 6 kjxj j;
j ¼ 1; 2; . . . ; n.
ð6Þ
Correspondingly, the initial condition of system (5) is x(h) = u(h) u*, h 2 [s(0), 0]. We denote it as u(h) throughout this paper. Clearly, an equilibrium u* of system (1) is globally robustly exponentially stable if and only if the origin of (5) is globally robustly exponentially stable.
2. Global robust exponential stability of neural networks with parametric uncertainties and time delay In this section, two criteria are obtained for the robust exponential stability of neural networks with parametric uncertainties and time delay via Lyapunov stability theorem for functional differential equations [18] and linear matrix inequality (LMI) technique [19]. Before stating the main result, the following lemmas are needed. Lemma 1. Given any a, b 2 Rn, any matrix Q > 0 of appropriate dimension and any scalar s > 0, we have 1 2aT b 6 saT Qa þ bT Q1 b. s
ð7Þ
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Lemma 2 [19, Schur complement]. The LMI QðxÞ SðxÞ > 0; SðxÞT RðxÞ
ð8Þ
where Q(x) = Q(x)T, R(x) = R(x)T, and S(x) depend affinely on x, is equivalent to RðxÞ > 0;
QðxÞ SðxÞRðxÞ1 SðxÞT > 0.
ð9Þ
Theorem 1. If there exist a symmetric positive matrix P, positive matrices Q0 and Q, and scalars s0 > 0, s > 0 such that the following LMI hold 3 2 PH 1 ð1; 1Þ PW PH PW 1 7 6 W T P Q 0 0 0 7 6 0 7 6 T 7<0 H P 0 s I 0 0 ð10Þ M ¼6 0 7 6 7 6 T 2s0 W P 0 0 ð1 dÞe Q 0 5 4 1 0 0 0 ð1 dÞse2s0 I H T1 P with ð1; 1Þ ¼ 2P ðAT P þ P AÞ þ k 2 Q0 þ s0 k 2 ET E þ k 2 Q þ sk 2 ET1 E1 ; A ¼ diagða1 ; a2 ; . . . ; an Þ, then the system (1) has the unique equilibrium point and the unique equilibrium is globally robustly exponentially stable for all time delay 0 < s(t) 6 s0 and s_ ðtÞ 6 d < 1. Moreover sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2sð0Þ kM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þ 1e2 kuðtÞ u k 6 ð11Þ kuket . km ðP Þ Proof. Clearly, it suffices to prove the global robust exponential stability of the origin of the system described by (5). We will first prove that the origin is the unique equilibrium point of (5). Consider the equilibrium equation of (5) Ax þ ðW þ DW Þgðx Þ þ ðW 1 þ DW 1 Þgðx Þ ¼ 0;
ð12Þ
where x* is the equilibrium point of (5). Clearly, the origin x* = [0, 0, . . . , 0]T is the equilibrium point of (5). We assume that the solution x* 5 [0, 0, . . . , 0]T is also a equilibrium point. Multiplying both sides of (12) by 2(x*)TP, we obtain 2ðx ÞT PAx þ 2ðx ÞT P ðW þ HFEÞgðx Þ þ 2ðx ÞT P ðW 1 þ H 1 F 1 E1 Þgðx Þ ¼ 0.
ð13Þ
By Lemma 1 and inequalities (3), (6), we have 2 T T 2ðx ÞT PWgðx Þ 6 ðx ÞT PWQ1 0 W P ðx Þ þ k ðx Þ Q0 ðx Þ;
1 2ðx ÞT PFEgðx Þ 6 ðx ÞT PHH T P ðx Þ þ s0 k 2 ðx ÞT ET Eðx Þ; s0 e2s0 T ðx Þ PW 1 Q1 W T1 P ðx Þ þ ð1 dÞe2s0 k 2 ðx ÞT Qðx Þ 2ðx ÞT PW 1 gðx Þ 6 1d e2s0 T ðx Þ PW 1 Q1 W T1 P ðx Þ þ k 2 ðx ÞT Qðx Þ; 6 1d e2s0 ðx ÞT PH 1 Q1 H T1 P ðx Þ þ ð1 dÞse2s0 k 2 ðx ÞT ET1 E1 ðx Þ 2ðx ÞT PH 1 F 1 E1 gðx Þ 6 ð1 dÞs e2s0 6 ðx ÞT PH 1 Q1 H T1 P ðx Þ þ sk 2 ðx ÞT ET1 E1 ðx Þ. ð1 dÞs Therefore, we get 1 e2s0 2 T ðx ÞT ðAT P þ AP Þ þ PWQ1 PHH T P þ s0 k 2 ET E þ PW 1 Q1 W T1 P þ k 2 Q 0 W P þ k Q0 þ s0 1d e2s0 2 T 1 T PH 1 Q H 1 P þ sk E1 E1 ðx Þ P 0. þ ð1 dÞs
ð14Þ ð15Þ
ð16Þ
ð17Þ
ð18Þ
O. Ou / Chaos, Solitons and Fractals 32 (2007) 1742–1748
1745
Thus 1 e2s0 2 T PW 1 Q1 W T1 P þ k 2 Q PHH T P þ s0 k 2 ET E þ ðx Þ ðAT P þ AP Þ þ PWQ1 0 W P þ k Q0 þ s0 1d e2s0 ðx Þ P 0. PH 1 Q1 H T1 P þ sk 2 ET1 E1 þ ð1 dÞs T
ð19Þ
Using the well-know Schur complement, (10) can be expressed as 2 T 2P ðAT P þ AP Þ þ PWQ1 0 W P þ k Q0 þ
þ
1 e2s0 PHH T P þ s0 k 2 EE þ PW 1 Q1 W T1 P þ k 2 Q 1d s0
e2s0 PH 1 Q1 H T1 P þ sk 2 ET1 E1 < 0. ð1 dÞs
ð20Þ
The contradiction in (19) and (20) means that the assumption (x*) 5 [0, 0, . . . , 0]T is violated. So, x* = [0, 0, . . . , 0]T is the unique equilibrium point of (5). Now, we will show that the conditions given in Theorem 1 also imply the global robust exponential stability of the origin of (5). To this end, we choose a Lyapunov–Krasovskii functional as Z t V ðxðtÞÞ ¼ e2t xT ðtÞPxðtÞ þ e2l gT ðxðlÞÞðQ þ sET1 E1 ÞgðxðlÞÞ dl. ð21Þ tsðtÞ
The derivative of V(x(t)) along the trajectory of (5) is V_ ðxðtÞÞ ¼ 2e2t xT ðtÞPxðtÞ þ e2t ½_xT ðtÞPxðtÞ þ xT ðtÞP x_ ðtÞ þ e2t gT ðxðtÞÞðQ þ sET1 E1 ÞgðxðtÞÞ ð1 s_ ðtÞÞe2ðtsðtÞÞ gT ðxðt sðtÞÞÞðQ þ sET1 E1 Þgðxðt sðtÞÞÞ 6 e2t 2xT ðtÞPxðtÞ xT ðtÞðAT P þ PAÞxðtÞ þ 2xT ðtÞPWgðxðtÞÞ þ 2xT ðtÞPHFEgT ðxðtÞÞ þ 2xT ðtÞPW 1 gðxðt sðtÞÞÞ þ 2xT ðtÞPH 1 F 1 E1 gðxðt sðtÞÞÞ þ gT ðxðtÞÞQgðxðtÞÞ
þ gT ðxðtÞÞsET1 E1 gðxðtÞÞ ð1 dÞe2s0 gT ðxðt sðtÞÞÞðQ þ sET1 E1 Þgðxðt sðtÞÞÞ .
ð22Þ
By Lemma 1, we have the following inequalities: 2xT ðtÞPW 1 gðxðt sðtÞÞÞ 6
e2s0 T x ðtÞPW 1 Q1 W T1 PxðtÞ 1d þ ð1 dÞe2s0 gT ðxðt sðtÞÞÞQgðxðt sðtÞÞÞ;
ð23Þ
e2s0 2xT ðtÞPH 1 F 1 E1 gðxðt sðtÞÞÞ 6 xT ðtÞPH 1 Q1 H T1 PxðtÞ ð1 dÞs þ ð1 dÞse2s0 gT ðxðt sðtÞÞÞET1 E1 gðxðt sðtÞÞÞ. Combined with (14), (15), (23) and (24), we get _V ðxðtÞÞ 6 e2t xT ðtÞ 2P ðAT P þ P AÞ þ PWQ1 W T P þ k 2 Q0 þ 1 PHH T P þ s0 k 2 ET E 0 s0 2s0 e2s0 e PW 1 Q1 W T1 P þ PH 1 Q1 H T1 P þ k 2 Q þ sk 2 ET1 E1 xðtÞ ¼ e2t xT ðtÞXxðtÞ; þ 1d ð1 dÞs
ð24Þ
ð25Þ
where 2 T X ¼ 2P þ ðAT P þ P AÞ þ PWQ1 0 W P k Q0
1 e2s PW 1 Q1 W T1 P PHH T P s0 k 2 EE 1d s0
e2s0 PH 1 Q1 H T1 P k 2 Q sk 2 ET1 E1 . ð1 dÞs
Obviously, by the Schur complement, if (10) holds, X > 0 and then V_ ðxðtÞÞ < 0. So, we have V ðxðtÞÞ < V ðxð0ÞÞ.
ð26Þ
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Noting that V ðxð0ÞÞ ¼ xT ð0ÞPxð0Þ þ
Z
0
e2l gT ðxðlÞÞðQ þ sET1 E1 ÞgðxðlÞÞ dl
sð0Þ
6 kM ðP Þkuk2 þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þkuk2
Z
0
e2l dl
sð0Þ
1 e2sð0Þ kuk2 6 kM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þ 2 and V ðxðtÞÞ P e2t km ðP ÞkxðtÞk2 . We have
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2sð0Þ kM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET1 E1 Þ 1e2 kxðtÞk 6 kuket . km ðP Þ
Thus, the origin of system (5) is globally robustly exponentially stable with degree , namely, the equilibrium point u ¼ ½u1 ; u2 ; . . . ; un T of system (1) is globally robustly exponentially stable with degree . The proof of Theorem 1 is completed. h Remark 1. The exponential stability criterion (10) gives a sufficient condition for the global robust exponential stability in terms of an LMI. It can be easily solved by using some existing software packages, for example, the MATLAB LMI toolbox. Remark 2. Uncertain neural networks with multiple time delays can also be studied similarly. Remark 3. Note that the stability criterion (10) implies uniformly exponential stability with degree for all admissible DW, DW1. Under the condition of (3), the exponential stability depends on the structure of uncertainties, i.e., matrices H, E, H1 and E1. If LMI (10) is satisfied, the lower bound of exponential convergence rate is guaranteed for all neural networks described by the uncertain model (1)–(3). Remark 4. Based on criterion (10) and given the time delay conditions: 0 6 s(t) 6 s0 and s_ ðtÞ 6 d < 1, one can know not only the solution will converge the unique equilibrium u* but in what speed. On the other hand, if some converge rate should be guaranteed, one could know the allowable time delay conditions: s0 and d. The lower bound of exponential convergence rate or the allowable time delay conditions can be determined by solving the following three optimization problems: Case 1: to estimate the lower bound of exponential convergence rate > 0. max Op1 : s.t. condition (10) is satisfied, s0 and d fixed. Case 2: to estimate the allowable maximum time delay s0. max s0 Op2 : s.t. condition (10) is satisfied, > 0 and d fixed . Case 3: to estimate the allowable maximum change rate of time delay d. max d Op3 : s.t. condition (10) is satisfied, > 0 and s0 fixed.
ð27Þ
ð28Þ
ð29Þ
If the change rate of time delay is equal to 0, i.e., s(t) s0, then the systems (1) reduces to the neural networks with constant delay du ¼ AuðtÞ þ ðW þ DW Þf ðuðtÞÞ þ ðW 1 þ DW 1 Þf ðuðt s0 ÞÞ þ U dt and consequently, Theorem 1 reduces to the following Corollary.
ð30Þ
O. Ou / Chaos, Solitons and Fractals 32 (2007) 1742–1748
1747
Corollary 1. If there exist a symmetric positive matrix P, positive matrices Q0 and Q, and scalars s0 > 0, s > 0 such that the following LMI hold 2 3 PH 1 ð1; 1Þ PW PH PW 1 6 W T P Q 7 0 0 0 6 7 0 6 T 7 6 7<0 0 s0 I 0 0 M ¼6H P ð31Þ 7 6 T 7 2s0 0 0 e Q 0 4 W 1P 5 H T1 P 0 0 0 se2s0 I with ð1; 1Þ ¼ 2P ðAT P þ P AÞ þ k 2 Q0 þ s0 k 2 ET E þ k 2 Q þ sk 2 ET1 E1 ; A ¼ diagða1 ; a2 ; . . . ; an Þ then the system (1) has the unique equilibrium point and is globally robustly exponentially stable for time delay s(t) s0. Moreover vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u ukM ðP Þ þ ðk 2 kM ðQÞ þ sk 2 ET E1 Þ 1e2s0 1 t 2 kuðtÞ u k 6 ð32Þ kuket . km ðP Þ The proof follows the same ideas in the proof of Theorem 1, and is omitted here. The corresponding Lyapunov–Krasovskii functional has the following form: V ðxðtÞÞ ¼ e2t xT ðtÞPxðtÞ þ
Z
t
e2l gT ðxðlÞÞðQ þ sET1 E1 ÞgðxðlÞÞ dl.
ð33Þ
ts0
The lower bound of exponential convergence rate or the allowable time delay conditions can be determined by solving the following two optimization problems: Case 4: to estimate the lower bound of exponential convergence rate > 0. max Op4 : s.t. condition (10) is satisfied, s0 fixed. Case 5: to estimate the allowable maximum time delay s0. max s0 Op5 : s.t. condition (10) is satisfied, > 0 fixed.
ð34Þ
ð35Þ
Remark 5. All the above optimization problems (Op1 Op5) can be solved by MATLAB LMI toolbox. Especially, Op1 and Op4 can estimate the lower bound of global exponential convergence rate , which means that the exponential convergence rate of any neural network included in (1)–(3) is at least equal to . It is useful in real-time optimal computation. 3. Conclusions In this paper, the problems of exponential stability and exponential convergence rate for uncertain neural networks with time delay have been studied. Some global exponential stability criteria, which depend on time delay, are derived via the approach of the Lyapunov–Krasovskii functional. The stability criteria are given in terms of linear matrix inequalities (LMIs) and hence are computationally efficient.
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