This brief provides improved conditions for the existence of a unique equilibrium point and its global asymptotic stability of cellular neural networks with time delay. Both delay-dependent and delay-independent conditions are obtained by using more general Lyapunov-Krasovskii functionals. These conditions are expressed in terms of linear matrix inequalities, which can be checked easily by recently developed standard algorithms. Examples are provided to demonstrate the reduced conservatism of the proposed criteria by numerically comparing with those reported recently in the literature. © 2005 IEEE.published_or_final_versio

Daniel W. C. Ho

James Lam

Senior Member

Shengyuan Xu

Yun Zou

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

Ho, DWC

Xu, S

Lam, J

Zou, Y

HKU Scholars Hub

Title Novel global asymptotic stability criteria for delayed cellularneural networksAuthor(s) Xu, S; Lam, J; Ho, DWC; Zou, YCitation Ieee Transactions On Circuits And Systems Ii: Express Briefs,2005, v. 52 n. 6, p. 349-353Issued Date 2005URL http://hdl.handle.net/10722/43015Rights©2005 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.IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 6, JUNE 2005 349Novel Global Asymptotic Stability Criteria forDelayed Cellular Neural NetworksShengyuan Xu, James Lam, Senior Member, IEEE, Daniel W. C. Ho, Member, IEEE, and Yun ZouAbstract—This brief provides improved conditions for the exis-tence of a unique equilibrium point and its global asymptotic sta-bility of cellular neural networks with time delay. Both delay-de-pendent and delay-independent conditions are obtained by usingmore general Lyapunov–Krasovskii functionals. These conditionsare expressed in terms of linear matrix inequalities, which can bechecked easily by recently developed standard algorithms. Exam-ples are provided to demonstrate the reduced conservatism of theproposed criteria by numerically comparing with those reportedrecently in the literature.Index Terms—Cellular neural networks (CNNs), global asymp-totic stability, linear matrix inequality (LMI), time-delay systems.I. INTRODUCTIONCELLULAR NEURAL networks (CNNs), which were in-troduced in [8], have been a subject of intense researchactivities in the literature over the past years and have foundextensive applications in image processing, pattern recognitionand classification, solving nonlinear algebraic equations, andother areas [7], [11]. It has been shown that such applicationsrely on the analysis of the dynamical behavior of CNNs. Sincestability is one of the most important issues related to such be-havior, the study of the stability problem of CNNs has receivedmuch attention in recent years and a great number of results onthis issue have been reported; see, e.g., [10], [14], and the refer-ences therein.On the other hand, time delays are often encountered invarious practical systems such as chemical processes, longtransmission lines in pneumatic systems, biological and neuralnetworks [3], [12]. The existence of time delays may lead tooscillation, divergence or instability in neural networks dueto finite speed of information processing. Therefore, manyresearchers have investigated the problem of stability analysisfor neural networks with time delays. For example, the globalasymptotic stability of delayed CNNs with bounded and non-monotonic activation functions were studied in [2] and [5],respectively, where several sufficient conditions were given viaManuscript received November 19, 2003; revised November 4, 2004.This work was supported by the Research Grants Council under GrantHKU7127/02P, by the Foundation for the Author of National Excellent DoctoralDissertation, China, under Grant 200240, by the National Natural ScienceFoundation, China, under Grant 60304001 and Grant 60074007, and by theFok Ying Tung Education Foundation under Grant 91061. This paper wasrecommended by Associate Editor M. Gilli.S. Xu and Y. Zou are with the Department of Automation, Nanjing Universityof Science and Technology, Nanjing 210094, China.J. Lam is with the Department of Mechanical Engineering, University ofHong Kong, Hong Kong (e-mail: james.lam@hku.hk).D. W. C. Ho is with the Department of Mathematics, City University of HongKong, Hong Kong.Digital Object Identifier 10.1109/TCSII.2005.849000different approaches. These results were extended to a moregeneral class of CNNs with delays in [6], [18]. Recently, alinear matrix inequality (LMI) approach to the stability anal-ysis of delayed CNNs was introduced in [13], in which somesufficient conditions were presented. The results in [13] werefurther improved in [1] by using different Lyapunov–Krasovskiifunctionals. It is noted that all the above mentioned asymptoticstability results are delay-independent; that is, they do notinclude any information on the size of delays. It is known thatdelay-dependent stability conditions, which employ the infor-mation on the size of delays, are generally less conservativethan delay-independent ones especially when the size of thedelay is small [12], [16]. Considering this, a delay-dependentasymptotic stability condition for symmetric delayed CNNswas proposed in [9]. When the CNN is not symmetric, thedelay-dependent results can be found in [17].In this paper, attention is focused on the derivation ofimproved conditions for the existence of a unique equilib-rium point and its global asymptotic stability of CNNs withtime-invariant delay. By constructing more general Lya-punov–Krasovskii functionals, less conservative stabilityconditions are established. These conditions are expressed interms of LMIs. It is worth pointing out that the LMI conditioncan be checked numerically very efficiently by resorting torecently developed interior-point methods, and no tuning ofparameters will be involved [4]. Both delay-dependent anddelay-independent results are given. Examples are provided todemonstrate the less conservatism of the proposed criteria.Notation: Throughout this brief, for real symmetric matricesand , the notation (respectively, ) meansthat the matrix is positive semi-definite (respectively,positive definite). The superscript “ ” represents the transpose.The notation refers to the Euclidean vector norm. We useto denote the minimum eigenvalue of a real symmetricmatrix. Matrices, if not explicitly stated, are assumed to havecompatible dimensions.II. MAIN RESULTSConsider a continuous-time CNN with a constant delay de-scribed by the following nonlinear retarded functional differen-tial state equation:(1)where1057-7130/$20.00 © 2005 IEEE350 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 6, JUNE 2005is the state vectoris the neuron activation function with ,, and are the interconnectionmatrices representing the weighting coefficients of the neurons,is a constant vector representing the bias.The scalar is a constant delay of the system.The following assumption will be made throughout the paper.Assumption 1: [6] The activation function is nonde-creasing, bounded and globally Lipschitz; that is(2)Then, by [6], it can be seen that there exists an equilibriumfor (1). For the sake of simplicity in the stability analysis of (1),we make the following transformation to (1):(3)whereis an equilibrium point of system (1). Under the transformationof (3), it is easy to see that (1) becomes(4)whereis the state vector of the transformed system, andwithand , for . It is noted that satisfies(2); that is(5)Before presenting the main results, we give the followinglemma, which will be used in the sequel.Lemma 1: [15] 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 (4), which is independent of the size of thedelay.Theorem 1: The origin of the delayed CNN in (4) is theunique equilibrium point and it is globally asymptotically stablefor all delay if there exist matrices , ,and two diagonal matrices and such that the fol-lowing LMI holds:(6)where(7)(8)Proof: We prove this theorem in two steps. Firstly, weshow the uniqueness of the equilibrium point by contradiction.Let be the equilibrium point of the delayed CNN in (4). Then,we have(9)Suppose . By (9), it is easy to see that(10)and(11)Note that(12)whereThen, it follows from (10)–(12) that(13)Now, applying Lemma 1, we havewhereThis together with (13) implies(14)On the other hand, pre- and post-multiplying (6) byXU et al.: NOVEL GLOBAL ASYMPTOTIC STABILITY CRITERIA 351and its transpose, respectively, we obtainwhich, by the Schur complement formula, results inThis contradicts with (14). Therefore, we have that at the equi-librium point , which, by (9), implies . That is,the origin of the delayed CNN in (4) is the unique solution to(9), and hence, (4) has a unique equilibrium point.Next, we show that the unique equilibrium point of (4) isglobally asymptotically stable. To this end, we denoteandDefine a Lyapunov–Krasovskii functional candidate for system(4) aswhere(15)(16)(17)Then, the time derivative of along the solution of (4) gives(18)Considering the relationship in (5) and noting that andare diagonal matrices, we can deduce(19)(20)Noting and using Lemma 1, we have(21)LetThen, by (18)–(21), it can be shown that(22)From the inequality in (6), it is easy to see thatTherefore, by using Lemma 1 again, we obtainThis together with (22) giveswhereNow, applying the Schur complement formula to (6) gives that. Therefore(23)whereFinally, by [12], it follows from (23) that the time-delay systemis asymptotically stable. This completes the proof.Remark 1: The Lyapunov–Krasovskii functional used inTheorem 1 is more general than that in [1], which will resultin the fact that Theorem 1 will be less conservative than thecondition derived in [1]. It is also worth pointing out thatTheorem 1 is obtained under the assumption that the activationfunctions are nonstrict, while [13, Theorem 2] is obtainedunder the assumption that the activation functions are strict.Therefore, Theorem 1 in this paper is applicable to a widerclass of delayed CNNs than [13, Theorem 2].Next, we provide an example to illustrate the reduced con-servatism of Theorem 1 by comparing it with recently reporteddelay-independent asymptotic stability results in the literature.Example 1: Consider a delayed continuous-time CNN in (4)with parametersWe assume thatFor this delayed CNN, it can be checked that the matrix inequal-ities in both [1, Theorem 1] and [13, Theorem 2] have no solu-tions. Also, it is found that the asymptotic stability conditions in[6] and [18] are not satisfied. Therefore, Theorem 1 in [1], The-orem 2 in [13], and the results in [6] and [18] fail to conclude352 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 6, JUNE 2005whether this delayed CNN is asymptotically stable or not. How-ever, by resorting to the Matlab LMI Control Toolbox, we findthat the LMI in Theorem 1 in this paper is feasible. Hence, byTheorem 1, it can be seen that the delayed CNN is asymptoticstable. This shows that the condition given in Theorem 1 is lessconservative than those in [1], [6], [13], and [18].Considering delay-dependent asymptotic stability conditionsare generally less conservative than delay-independent ones, wenow provide such a condition for the delayed CNN (4) in thefollowing theorem.Theorem 2: The origin of the delayed CNN in (4) is theunique equilibrium point and it is globally asymptotically stablefor any delay if there exist matrices , ,and three diagonal matrices , andsuch that the following LMI holds:(24)where is given in (7), andProof: Under the condition of the theorem, the unique-ness of the equilibrium point can be established by following asimilar line as in the proof of Theorem 1, and thus is omitted.Now, we prove that the unique equilibrium point of (4) isglobally asymptotically stable. To this end, we define a Lya-punov–Krasovskii functional candidate for (4) aswhere , and are defined in (15)–(17), re-spectively, andThen, the time-derivative of along the solution of (4)provides(25)Noting that is a diagonal matrix, we then have(26)Considering (18), (19), (25) and (26), it can be deduced that(27)Now, by Lemma 1, it can be shown that(28)Then, it follows from (20), (27), and (28) that(29)whereBy (24), it is easy to have thatThen, by Lemma 1, it can be deduced thatThis together with (29) gives(30)XU et al.: NOVEL GLOBAL ASYMPTOTIC STABILITY CRITERIA 353whereApplying the Schur complement formula to (24), we have thatfor allwhich, by the Schur complement formula implies that. ThenwhereTherefore, we have that (4) is asymptotically stable for any delay.The following examples are given to show the reduced con-servatism of the delay-dependent asymptotic stability result inTheorem 2.Example 2: Consider a delayed continuous-time CNN in (4)with parametersIn this example, we suppose thatFor this delayed CNN, it can be checked that the asymptoticstability conditions in [6] and [18], Theorem 1 in [1], Theorem2 in [13] as well as Theorem 1 in this paper are not satisfied.Therefore, they fail to conclude whether this delay system isasymptotically stable or not. Now, we resort to Theorem 2 inthis paper to check the asymptotic stability of the delay system.It is found that the LMI in (24) is feasible for all delay. Therefore, by Theorem 2, we conclude that thedelay system is delay-dependent asymptotically stable. If we useTheorem 1 in [17], we can calculate the largest allowed valueof , which is 75.27% smaller than that obtained byour method. This shows that the condition given in Theorem 2 isless conservative than that in [17] when checking the asymptoticstability of a given delay neural network.III. CONCLUSIONThis paper has provided improved criteria for the existence ofa unique equilibrium point and its global asymptotic stability ofcontinuous delayed CNNs. Both delay-dependent and delay-in-dependent stability conditions have been proposed in terms ofLMIs, which can be checked easily by using recently developedstandard algorithms. Examples via numerical comparisons withthe recently reported results have demonstrated the lower con-servatism of the proposed criteria in this paper.REFERENCES[1] S. Arik, “Global asymptotic stability of a larger class of neural networkswith contant time delay,” Phys. Lett. A, vol. 311, pp. 504–511, 2003.[2] S. Arik and V. Tavsanoglu, “Equilibrium analysis of delayed CNNs,”IEEE Trans Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 1, pp.168–171, Jan. 1998.[3] P. Baldi and A. 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Novel global asymptotic stability criteria for delayed cellular neural networks

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Novel global asymptotic stability criteria for delayed cellular neural networks

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