In this paper, a new delay-dependent asymptotic stability condition for delayed Hopfield neural networks is given in terms of a linear matrix inequality, which is less conservative than existing ones in the literature. This condition guarantees the existence of a unique equilibrium point and its global asymptotic stability of a given delayed Hopfield neural network. Examples are provided to show the reduced conservatism of the proposed condition. © 2006 IEEE.published_or_final_versio

Ho, DWC

Lam, J

Xu, S

IEEE Transactions on Circuits and Systems II Analog and Digital Signal Processing

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Title A new LMI condition for delay-dependent asymptotic stability ofdelayed Hopfield neural networksAuthor(s) Xu, S; Lam, J; Ho, DWCCitation IEEE Transactions On Circuits And Systems Ii: Express Briefs,2006, v. 53 n. 3, p. 230-234Issued Date 2006URL http://hdl.handle.net/10722/44940Rights©2006 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.230 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 3, MARCH 2006A New LMI Condition for Delay-DependentAsymptotic Stability of DelayedHopfield Neural NetworksShengyuan Xu, James Lam, Senior Member, IEEE, and Daniel W. C. Ho, Senior Member, IEEEAbstract—In this paper, a new delay-dependent asymptotic sta-bility condition for delayed Hopfield neural networks is given interms of a linear matrix inequality, which is less conservative thanexisting ones in the literature. This condition guarantees the exis-tence of a unique equilibrium point and its global asymptotic sta-bility of a given delayed Hopfield neural network. Examples areprovided to show the reduced conservatism of the proposed condi-tion.Index Terms—Global asymptotic stability, Hopfield neural net-works, linear matrix inequality, time delays.I. INTRODUCTIONRECENTLY, there has been increasing interest in thestudy of Hopfield neural networks since Hopfield neuralnetworks have found extensive applications in solving someoptimization problems, associative memory, classification ofpatterns, reconstruction of moving images, and other areas[8]–[10]. It is now well known that applications of neuralnetworks heavily depend on their dynamic behavior. Sincestability is one of the most important issues related to suchbehavior, the problem of stability analysis of Hopfield neuralnetworks has attracted considerable attention in recent years.Based on different assumptions on the network parameters,a great number of results on global asymptotic stability havebeen proposed; see, e.g., [1], [15], and the references therein.In the implementation of artificial neural networks, however,time delays are unavoidable due to the finite switching speedof amplifiers. It has been shown that the existence of time de-lays in a neural network may lead to oscillation, divergenceor instability. Therefore, the stability issue of Hopfield neuralnetworks with time delays has been studied. Via different ap-proaches, a great number of stability conditions for delayedHopfield neural networks have been reported in the literatureManuscript received March 9, 2005; revised June 13, 2005 and July 27,2005. This work was supported in part by Hong Kong University under CRCG10205878, the Program for New Century Excellent Talents in University by theMinistry of Education of China (NCET-04-0508), the National Natural ScienceFoundation of China under Grant 60304001, the Fok Ying Tung EducationFoundation under Grant 91061, and the Foundation for the Author of NationalExcellent Doctoral Dissertation of China under Grant 200240. This paper wasrecommended by Associate Editor J. L. Moiola.S. Xu is with the Department of Automation, Nanjing University of Scienceand Technology, Nanjing 210094, China.J. Lam is with the Department of Mechanical Engineering, University ofHong Kong, Hong Kong.D. W. C. Ho is with the Department of Mathematics, City University of HongKong, Hong Kong.Digital Object Identifier 10.1109/TCSII.2005.857764[4]–[7], [12]–[14], [16], [20]. These stability results can be clas-sified into two types; that is, delay-independent stability anddelay-dependent stability; the former does not include any in-formation on the size of delay while the latter employs such in-formation. It is known that delay-dependent stability conditionsare generally less conservative than delay-independent ones es-pecially when the size of the delay is small. Although delay-de-pendent stability results for delayed Hopfield neural networkswere proposed in [7], [18], [19], [21], and [22], they are suffi-cient conditions and have conservatism to some extent, whichleaves open room for further improvement.In this paper, we develop an improved delay-dependentasymptotic stability condition for Hopfield neural networkswith time delays by utilizing Lyapunov functionals. Under theproposed condition, both the existence of a unique equilibriumpoint and the global asymptotic stability of a given delayedHopfield neural network are guaranteed. The derived conditionis expressed in terms of a linear matrix inequality (LMI), whichcan be checked numerically very efficiently by resorting torecently developed standard algorithms such as interior-pointmethods, and no tuning of parameters will be involved [3]. Ex-amples are provided to demonstrate the reduced conservatismof the proposed condition.Notation: Throughout this paper, for real symmetric ma-trices and , the notation (respectively, )means that the matrix is positive semi-definite (re-spectively, positive definite). The superscript “ ” representsthe transpose. The notation refers to the Euclidean vectornorm. We use to denote the minimum eigenvalue ofa real symmetric matrix. Matrices, if their dimensions are notexplicitly stated, are assumed to have compatible dimensions.II. MAIN RESULTSConsider a continuous time-delayed Hopfield neural networkwhich is described by the following nonlinear retarded func-tional differential equations:(1)or equivalently(2)where1057-7130/$20.00 © 2006 IEEEXU et al.: NEW LMI CONDITION FOR DELAY-DEPENDENT ASYMPTOTIC STABILITY 231is the state vectorIn (2), , are the de-layed interconnection matrix representing the weighting coeffi-cients of the neurons, is a constant inputvector. The scalar is a constant delay of the system.Throughout the paper, we make the following assumption.Assumption 1: [2], [21]: The activation function isboundless and satisfies(3)for any , and .Now, let be an equilibrium point of(2), and set(4)Then, it is easy to see that (2) can be transformed to(5)whereis the state vector of the transformed system, andwithand , for . It is noted that satisfies(3); that is(6)for any , and .It can be seen that the global asymptotic stability of of(2) is equivalent to that of the trivial solution of (5). Therefore,in the next, attention will be focused on the study of the globalasymptotic stability of (5).Before presenting the main results, we give the followinglemma, which will be used in the sequel.Lemma 1: [17]: Let , and be real matrices of appro-priate dimensions with . Then for any vectors , withappropriate dimensionsNow, we are in a position to present a new asymptotic stabilitycondition for system (5), which is dependent on the size of thedelay.Theorem 1: The origin of the delayed Hopfield neural net-work in (5) is the unique equilibrium point and it is globallyasymptotically stable for any delay if there exist ma-trices , , , , , and two diagonal matricesand such that the LMI shown in (7) at the bottomof the page, holds, where(8)(9)(10)(11)Proof: We will first prove the uniqueness of the equilib-rium point by contradiction. Let be the equilibrium point ofthe delayed Hopfield neural network in (5). Then, we have(12)Suppose . Then, from (12), we have(13)which can be re-written as(14)where(15)(16)By (6), we have that forThat is(17)(7)232 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 3, MARCH 2006Therefore, using Lemma 1, we obtainThis together with (14) gives(18)On the other hand, considering (7), it can be shown thatThat is, we get the inequality shown at the bottom of the page.Pre- and post-multiplying this inequality byand its transpose, respectively, we obtainwhich contradicts with (18). Therefore, we have . That is,(5) has a unique equilibrium point.Next, we show that the unique equilibrium point of (5) isglobally asymptotically stable. To this end, we define a Lya-punov–Krasovskii functional candidate aswhereand(19)(20)(21)Noting (15)–(17) and applying Lemma 1, we have that the time-derivative of , , 2, 3, along the solution of (5) gives(22)(23)(24)Furthermore(25)XU et al.: NEW LMI CONDITION FOR DELAY-DEPENDENT ASYMPTOTIC STABILITY 233Now, by the Leibniz–Newton formula, it is easy to show thatThis together with (22)–(25) gives(26)wherewhere , and are given in (9)–(11), respectively.Now, applying the Schur complement formula to the LMI in (7)results inThen, it is easy to show that for all , we havewhich, by the Schur complement formula, implies .This together with (26) gives(27)where . Finally, by [11], it followsfrom (27) that the delayed Hopfield neural network in (5) isasymptotically stable for any delay . This completesthe proof.Remark 1: Theorem 1 provides a new delay-dependentglobal asymptotic stability condition for delayed Hopfieldneural networks in terms of an LMI; this is derived by choosingan appropriate Lyapunov functional different from those in [7],[19], [21], which leads to a less conservative stability conditionin (7).III. EXAMPLESIn this section, we provide examples to illustrate the reducedconservatism of Theorem 1 by comparing it with recently re-ported delay-dependent asymptotic stability results in the liter-ature.Example 1: [21]: Consider a delayed Hopfield network de-scribed bywhere the activation function is given as234 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 3, MARCH 2006Then, it is easy to see thatBy the methods in [7], [19] and [21], the maximum delay thatguarantees the delayed Hopfield network to be asymptoticallystable are calculated to be 0.3636, 2.2361 and 0.4926, respec-tively. However, by Theorem 1 in this paper, it is found that LMI(7) is feasible for any arbitrarily large (as long as the numericalcomputation is reliable). Therefore, it can be seen that Theorem1 provides a less conservative condition for this example.Example 2: Consider a delayed Hopfield network in (5) withparametersFor this example, both of the delay-dependent conditions in [7],[21], [22] cannot be satisfied for any . Therefore, theycannot provide any results on the maximum allowed delay .However, by the method in [18] and [19], we obtain as 1.7484and 0.4121, respectively, while by Theorem 1 in this paper, wehave , which also shows that the condition given inTheorem 1 is less conservative than those in [7], [18], [19], [21],and [22].IV. CONCLUSIONAn improved delay-dependent asymptotic stability conditionfor Hopfield neural networks with time delays has been devel-oped in this paper. This condition is expressed in terms of anLMI, which can be checked easily. Examples have been pro-vided to demonstrate the less conservatism of the proposed re-sults.REFERENCES[1] S. Arik, “Global asymptotic stability of a class of dynamical neural net-works,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no.4, pp. 568–571, Apr. 2000.[2] , “Global asymptotic stability of a larger class of neural networkswith constant time delay,” Phys. Lett. A, vol. 311, pp. 504–511, 2003.[3] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994.[4] J. Cao, “Global stability conditions for delayed CNNs,” IEEE Trans.Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 11, pp. 1330–1333,Nov. 2001.[5] J. Cao and D. W. C. Ho, “A general framework for global asymptoticstability analysis of delayed neural networks based on LMI approach,”Chaos, Solitons, Fractals, vol. 24, pp. 1317–1329, 2005.[6] J. Cao and J. 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A new LMI condition for delay-dependent asymptotic stability of delayed Hopfield neural networks

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A new LMI condition for delay-dependent asymptotic stability of delayed Hopfield neural networks

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