616 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: … · 2020. 10. 9. · 616 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 3, MARCH 2009 Global Asymptotic

616 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 3, MARCH 2009

Global Asymptotic Stability and Robust Stability ofa Class of Cohen–Grossberg Neural Networks

With Mixed DelaysHuaguang Zhang, Senior Member, IEEE, Zhanshan Wang, and Derong Liu, Fellow, IEEE

Abstract—This paper is concerned with the global asymptoticstability of a class of Cohen–Grossberg neural networks with bothmultiple time-varying delays and continuously distributed delays.Two classes of amplification functions are considered, and somesufficient stability criteria are established to ensure the globalasymptotic stability of the concerned neural networks, which canbe expressed in the form of linear matrix inequality and are easy tocheck. Furthermore, some sufficient conditions guaranteeing theglobal robust stability are also established in the case of parameteruncertainties.

Index Terms—Cohen–Grossberg neural networks, distributeddelays, global asymptotic stability, linear matrix inequality (LMI),multiple time-varying delays, nonnegative equilibrium points,robust stability.

I. INTRODUCTION

C OHEN–GROSSBERG [11] proposed in 1983 a neuralnetwork model described by the following system:

(1)where is the state of neuron, is an amplificationfunction, is a well-defined function to guarantee theexistence of solution of system (1), is an activationfunction describing the effects of input on the output of neuron,

is the connection weight coefficient of the neural network,is an external constant input, and . System

(1) includes a number of models from neurobiology, population

Manuscript received June 23, 2007; revised December 02, 2007 and June 12,2008. First published August 04, 2008; current version published March 11,2009. This work was supported in part by the National Natural Science Foun-dation of China under Grants 60534010, 60572070, 60728307, 60774048, and60774093, by the 111 Project of the Ministry of Education of China under GrantB08015, by the Natural Science Foundation of Liaoning Province under Grant20072025, by the Program for Cheung Kong Scholars and Innovative ResearchGroups of China under Grant 60521003, by the National High Technology Re-search and Development Program of China under Grant 2006AA04Z183, and bythe Postdoctor Foundation of Northeastern University under Grant 20080314.This paper was recommended by Associate Editor M. Di Marco.

H. Zhang and Z. Wang are with the School of Information Science and En-gineering, Northeastern University, Shenyang, Liaoning 110004, China, andalso with the Key Laboratory of Integrated Automation of Process Industry(Northeastern University), Ministry of Education of China, Shenyang, Liaoning110004, China (e-mail: [email protected]; [email protected]).

D. Liu is with the Department of Electrical and Computer Engineering, Uni-versity of Illinois at Chicago, Chicago, IL 60607 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCSI.2008.2002556

biology, and evolution theory, as well as the Hopfield neural net-work model [20] as a special case. In electronic implementationof analog neural networks, delays always exist due to the trans-mission of signal and the finite switching speed of amplifiers.On the other hand, it is desirable to introduce delays into neuralnetworks when dealing with problems associated with motions.Therefore, model (1) and its delayed version have attracted theattention of many researchers (see, e.g., [1], [5]–[7], [10], [19],[26], [27], [29], [33], [35], [36], [40], [42], [45]). Among them,[42] introduced constant delays into (1), which yields the fol-lowing form:

(2)where are bounded constant delays, are the con-nection weight coefficients, and other notations are the same asthose in system (1), i.e., and .

We remark that one can consider several types of delays in(2), [15]–[17], [31], and [32]. References [17] and [18] first in-vestigated the stability problem of asymmetric Hopfield neuralnetworks with continuously distributed delays. The character-istic of this kind of continuously distributed delays is that thedelays range over the infinitely long duration. Dynamics of dif-ferent kinds of neural networks with this kind of continuouslydistributed delays are widely studied [6], [8], [9], [25], [34]. Al-though the results of [6], [8], [9], [25], and [34] are generallyeasy to verify, all the results in [6], [8], [9], [25], and [34] takethe absolute value operation on the connection weight coeffi-cients. Therefore, the sign difference of entries in connectionmatrix is ignored, which leads to the ignorance of the neuron’sexcitatory and inhibitory effects on the neural network. How-ever, some kinds of continuously distributed delays in a prac-tical system often range over a finite duration. For example,one application of this kind of continuously distributed delaysystems can be found in the modeling of feeding systems andcombustion chambers in a liquid monopropellant rocket motorwith pressure feeding [12], [13]. This kind of continuously dis-tributed delay systems have been investigated in [22], [23], [38],and [41], but all the results in [22], [23], [38], and [41] are onlysuitable for the case of linear systems with constant delay. Itis well known that neural networks are large-scale and com-plex nonlinear dynamic systems. Different types of delays mayoccur, and it is useful to investigate the dynamics of neural net-works with different types of delays.

Motivated by the aforementioned discussion, the purpose ofthis paper is to establish sufficient conditions for the globalasymptotic stability of the following system with both multiple

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time-varying delays and a kind of continuously distributed de-lays ranging over a finite duration:

(3)

where and are connection weight coefficients of theneural network, time delays and areall bounded, is an external constant input, and

are activation functions, respectively, and other no-tations are the same as those in system (1), i.e., ,

, and .The neural network model studied here is more general and

includes a large class of existing neural networks of the fol-lowing two cases.Case 1) for all and .Case 2) for all and ,

.For example, in Case 1), model (3) contains the models

studied in [3], [4], [7], [10], [19]–[21], [24], [26], [27], [33],[37], [42], [45]–[47], [49], and [50]. In Case 2), model (3)contains the classical Cohen–Grossberg neural model [11],[28], [30] and the famous Lotka–Volterra neural model [43],[44]. Instead of using the Jensen inequality approach, as usedin [3] and [46], we construct a suitable Lyapunov–Krasovskiifunctional in this paper to deal with the continuously distributeddelays, and some sufficient conditions are derived to ensurethe global asymptotic stability of system (3) for Case 1) or2), respectively, which can be expressed in the form of linearmatrix inequality (LMI) and are independent of the values oftime-varying delays and amplification functions. Corollariesare also given for some special cases of system (3). Moreover,a global robust stability criterion is also established for system(3) with parameter uncertainties.

Throughout this paper, let , , , , anddenote the transpose, the inverse, the

smallest eigenvalue, the largest eigenvalue, and the Euclideannorm of a square matrix , respectively. Letdenote a positive (negative) definite symmetric matrix. Letand 0 denote the identity matrix and the zero matrix with com-patible dimensions, respectively. Time delays andare all bounded, i.e., , ,

, , , , ,, , and .

II. GLOBAL ASYMPTOTIC STABILITY RESULTS

In this section, we need the following assumptions andlemma.

Assumption 2.1: There exist constants such that thefunction satisfies

(4)

for , , and .Assumption 2.2: The bounded activation functions ,

, and satisfy the following conditions:

(5)

(6)

(7)

for , , and and for , , and.

Let , ,, and .

Assumption 2.3: The amplification function is positiveand continuous, i.e., for all .

Lemma 2.1 (See [37]): Let and be two real vectors withappropriate dimensions, and let and be two matrices withappropriate dimensions, where . Then, for any two posi-tive constants and , the following inequality holds:

(8)

According to [35], for every external constant input , neuralnetwork (3) has an equilibrium point if

, , , , and satisfy the aforementionedconditions . Letting , thenmodel (3) is transformed into the following form:

(9)

or in a vector matrix form by the method in [48]

(10)

where ,,

, ,,



,, ,

,,

, ,,

, ,, , , and

. The initial condition of (10) is of the formfor , and its supremum bound is

.We now state and prove the results of this paper as follows.Theorem 2.1: Suppose that Assumptions 2.1–2.3 hold,

, and . If there exist positivediagonal matrices , , , , and and positive definitesymmetric matrices , , , , , , , and such thatthe following conditions hold:

(11)

(12)

(13)

(14)

(15)

(16)

then the equilibrium point of model (10) is globally asymp-totically stable, where and( , , and ).

Proof: We will prove the theorem in two steps. First, wewill show that (11)–(16) are sufficient conditions guaranteeingthe uniqueness of the equilibrium point of system (10). Consider(10) at equilibrium point , i.e.,

(17)If , , and , then it is easy tosee that . Now, suppose that . Then, ,

, and . Multiplying , ,and on both sides of (17) yields

(18)

where we have used Lemma 2.1.However, for , , and , from

(12)–(14), we have

(19)Obviously, (18) contradicts with (19), which, in turn, impliesthat, at the equilibrium point , , , and

. This means that the origin of (10) is a unique equi-librium point.

Second, we will show that conditions (11)–(16) are also suf-ficient conditions guaranteeing the global asymptotic stabilityof the equilibrium point of system (10). Consider the followingLyapunov–Krasovskii functional

(20)

where

where , , , , , ,, and and .



The derivative of along the trajectories of (10) is asfollows:

(21)

By Lemma 2.1, for and positive constants ,, we have

Then, from (21) it yields

(22)

Similarly, the derivative of along the trajectories of(10) is as follows:

(23)

Thus, we have

(24)

The derivative of along the trajectories of (10)is given by (25), shown at the bottom of the page.where , ,

, , and.

(25)



By Assumptions 2.1 and 2.2 and Lemma 2.1, we have (26),shown at the bottom of the page. Therefore, see (27), shown atthe bottom of the next page. Now, we choose such that

(28)Then

(29)

Meanwhile, see the second equation at the bottom of the nextpage, where is defined in (13). If we choose such that

(30)

then we have for .Similarly, for sufficiently large number , the condition

in (31), shown at the bottom of page 622, hold for ,where is defined in (12).

(26)



Moreover, by condition (15) in Theorem 2.1, we have. Then, there exists a sufficiently large constant

such that

(32)

Therefore, by (14), (28), (30), (31), and (32), and from (27), wehave for , , ,and . if and only if

, , , and . Ac-cording to Lyapunov stability theory, the origin of (10) is glob-ally asymptotically stable.

If we take another Lyapunov functional, we can obtain thefollowing result.

Theorem 2.2: Suppose that Assumptions 2.1–2.3 hold,, and . If there exist positive

diagonal matrices , , , , , , , , andsuch that the following LMI holds:

(33)

then the equilibrium point of model (10) is globally asymptoti-cally stable, where , ,

(27)



, ,, ,

, ,, ,

, ,, ,

, ,, denotes the corre-

sponding symmetric part in a matrix, , and( , , and ).

Proof: Consider the following Lyapunov–Krasovskiifunctional:

(34)

where and are defined in (20), except that, and

where , , , , and (and ).

The derivatives of along the trajectoriesof (10) is as (35), shown at the bottom of the page,where , ,

, , and( ).

By Assumption 2.2, the following inequalities hold for pos-itive diagonal matrices , , and ( and

)

(36)

Combining (24), (35), and (36), we have

(37)

where

.

Obviously, if , then for . ByLyapunov stability theory, the origin of system (10) is globallyasymptotically stable.

We have the following results, which can be derived in a sim-ilar manner to the proof of Theorem 2.2.


(31)

(35)



diagonal matrices , , , , and such that the followingLMI holds:

(38)

then the equilibrium point of model (10) is globally asymp-totically stable, where

,, , ,

, ,, ,

, denotes the corre-sponding symmetric part in a matrix, , and

( , , and ).Proof: Consider the following Lyapunov–Krasovskii

functional

(39)

where and are the same as those defined in(20), except that , and

In a similar manner to the proof of Theorem 2.2, Theorem 2.3can easily be derived.


diagonal matrices , , , and such that the followingLMI holds:

(40)

then the equilibrium point of model (10) is globally asymp-totically stable, where

, ,, ,

, ,, denotes the corresponding sym-

metric part in a matrix, , and( , , and ).

When and , i.e., ,, system (10) becomes

(41)

We have the following results, which can be derived in a similarmanner to the proof of Theorem 2.2.

Theorem 2.5: Suppose that Assumptions 2.1–2.3 hold and. If there exist positive diagonal matrices ,

, and such that the following condition holds:

(42)

then system (41) is globally asymptotically stable, where( , , and ).

Corollary 2.1: Suppose that Assumptions 2.1–2.3 hold and. If there exist positive diagonal matrices ,


(43)

then system (41) is globally asymptotically stable, where( , , and ).

For neural network (41) with ( and), we have the following result, which can be ob-

tained directly from Theorem 2.5.Corollary 2.2: Suppose that Assumptions 2.1–2.3 hold and

. If there exist positive diagonal matricesand such that the following condition holds:

(44)then the equilibrium point of model (41) with is glob-ally asymptotically stable, where (

, , and ).For the following Cohen–Grossberg neural networks:

(45)

where , , , , and are the sameas those in (3) ( ). We have the following result.

Corollary 2.3: Suppose that Assumptions 2.1–2.3 hold and. If there exist positive diagonal matrices

and such that the following condition holds:

(46)then the equilibrium point of model (45) is globally asymptot-ically stable for a given , where , isa square matrix whose th row is composed by the th row ofsquare matrix , and the other rows are all zero( ).



III. GLOBAL ROBUST STABILITY RESULT

Consider the Cohen–Grossberg neural network (9) with un-certainties in the form of

(47)

or in a vector matrix form

(48)

where , , and denote the unknown con-nection weight coefficients representing time-varying parameteruncertainties, and other notations are the same as those in (9).For convenience of description, we let ,

, ,, , and (

, , and ).Assumption 3.1: Parameter uncertainties satisfy

, , and ,respectively, where , , , , , and are the knownstructural matrices and is an unknown time-varying func-tion matrix satisfying ( and

).Lemma 3.1 (See [39]): If , , and are real matrices

of appropriate dimensions with satisfying , thenfor all , if and only

if there exist a positive constant , such that.

Theorem 3.1: Suppose that Assumptions 2.1–2.3 and 3.1hold, , and . If there existpositive diagonal matrices , , , , , , , , and

, positive constants , , , and

, , , , such that thefollowing LMI holds:

(49)then system (47) is globally robustly stable, where

, ,,

, ,, ,

, ,,,,

, denotes the correspondingsymmetric part in a matrix, and others are defined in Theorem2.2.

Proof: Consider the same Lyapunov–Krasovskii func-tional defined in (20). In a similar manner to the proof ofTheorem 2.2, the derivative of (20) along the trajectories of(48) is of the form

(50)

with being defined in (37)

(51)

where , ,, ,

, , ,, ,

, ,, and others are defined in The-

orem 2.2.According to Assumption 3.1, Lemma 3.1, and the Schur

complement, in a similar routine to the proof in [46], (51) isequivalent to (49). The details are omitted.

IV. GLOBAL ASYMPTOTIC STABILITY OF COHEN–GROSSBERGNEURAL NETWORKS WITH NONNEGATIVE

EQUILIBRIUM POINTS

In the original paper of Cohen–Grossberg [11], theCohen–Grossberg neural network model was proposed asa kind of competitive-cooperation dynamical system for de-cision rules, pattern formation, and parallel memory storage.Therefore, each state of the neuron might be the popula-tion size, activity, or concentration of the th species in the



system, which is always nonnegative. It is clear that this subsetof Cohen–Grossberg neural networks includes the famousLotka–Volterra recurrent neural networks [11], [43], [44].

When systems (1) and (3) are applied to biology, initial con-ditions will be of the following type:

(52)

where each is a continuous function defined on .In applications, the activation functions of model (3) may

not be bounded, for example, in the Lotka–Volterra model [43],[44]. Therefore, in this section, we will employ the followingassumptions and lemmas.

Assumption 4.1: satisfies (4) and ,.

Assumption 4.2: Activation functions , , andsatisfy conditions (5)–(7), and , , and

, .Assumption 4.3: The amplification function for all

and , and for any ,holds for all .

Lemma 4.1: Assume that satisfies Assumption 4.3.Then, the solution of the system (3) is positive with the initialcondition (52).

If Assumption 4.3 holds, then the nonnegative equilibriumpoint of system (3) is a solution of the following:

(53)

where

Although (53) might possess multiple solutions, in a similar rou-tine to [30Proof of Proposition 1], we can show that if is anasymptotically stable nonnegative equilibrium point of system(3), then it must be a solution of the following problem:

(54)

where .Lemma 4.2 (See [30]): Equation (54) has a unique solution

for every if and only if is norm coercive, i.e.,

and locally univalent, where ,, ,

, if and if, if and if , .

Lemma 4.3 (See [14], [30]): Let be an matrix,, and . If there exists

a positive definite diagonal matrix suchthat

holds, then, for any positive definite diagonal matrix andnonnegative definite diagonal matrix satisfying ,we have , i.e., is nonsingular.

Theorem 4.1: Suppose that Assumptions 4.1–4.3 hold,, and ( and ).

Then, system (3) with initial conditions (52) has a unique non-negative equilibrium point, which is globally asymptoticallystable if there exist positive definite diagonal matrices , ,


(55)

where denotes the symmetric parts in a matrix, and its vari-ables are described

Proof: We prove Theorem 4.1 in two steps. First, we willshow that (55) ensures the existence and uniqueness of a non-negative equilibrium point of system (3).

From (55), we know that , which also means that

is positive definite, where we have used Lemma 2.1 and the factsthat and

According to Lemma 4.2, we only need to prove thatis norm coercive and locally univalent. We first prove thatis locally univalent. For any , withoutloss of generality, by some rearrangement of , we can assumethat if ; if ;and if for some integers .Moreover, if is sufficiently close to , without lossof generality, we can also assume that if ;

if ; if ;



if ; and iffor some integers . It can be seen that

(56)

(57)

where , ,, ,

otherwise

otherwise

otherwise

otherwise.

Then, , , , and ().

If , then we have

(58)By (56), without loss of generality, we can assume that

where and for some integer . Therefore,(58) can be written as

According to Lemma 4.3 and from ,is nonsingular. By matrix de-

composition, we can easily obtain and , respec-tively. Therefore, and , i.e., , whichimplies that is locally univalent.

We next prove that is norm coercive. Suppose thatthere exists a sequence suchthat . Then, there exists some indexsuch that , which implies that

in the case of the bounded activation functions, , and . In the case of ,

, or , we willprove that .

By simple algebraic manipulations, it leads to

(59)

Let ,, ,

, and by Lemma 2.1, (59) be-comes



(60)

where . Therefore,, which implies that

is norm coercive. In combination with Lemma 4.2, (55)ensures the existence and uniqueness of the nonnegative equi-librium point of system (3).

Second, we will prove that (55) is also a sufficient conditionto guarantee the global asymptotical stability of the nonnegativeequilibrium point of system (3). It suffices to show thatensures the global asymptotic stability of nonnegative equilib-rium point.

Let be the nonnegative equilibrium pointof system (3) and . Then, model (3) is trans-formed into the following form:

(61)

with

where, the initial condition

satisfies (52), and others are the same as those defined in (10).Since is the nonnegative equilibrium point of system (3),

then, from (54), we know that holds for all .This implies that , , and

hold for all and .Consider the same Lyapunov functional used in the proof of

Theorem 2.3 with , and following the same routine as theproof of Theorem 2.3, we can show that guarantees theglobal asymptotic stability of the nonnegative equilibrium pointof system (3). Details are omitted.

Remark 4.1: If , is a constantdelay in system (61), , and under the same assump-tions of Theorem 4.1, we can prove that conditions (42) in The-orem 2.5 and (43) in Corollary 2.1 ensure the global asymptoticstability of the nonnegative equilibrium point of system (61).

Remark 4.2: In a similar routine to the proof of Theorem 3.1,we can easily derive the robust stability criterion from Theorem4.1 in the case of uncertainties satisfying Assumption 3.1.

V. ILLUSTRATIVE EXAMPLES

In this section, we will use two examples to show the effec-tiveness of the obtained results.

Example 5.1: Consider the following Cohen–Grossbergneural networks with two neurons:

(62)

where and areany bounded constant delays and are some kinds ofamplification functions satisfying Assumption 2.3, whose exactvalues of lower and upper bounds may be unknown ( ).Obviously, the results in [10] and [19] cannot be applied to thisexample. Pertaining to this example, the result in [27] and [7,Th. 2] are not satisfied. Applying Corollary 2.3 of this paper, wehave , ,and . Therefore, the neural net-work (62) is globally asymptotically stable. Moreover, if we let

in (62), , [24, Ths. 1 and 2], [47, Th. 1],and [4, Th. 3] are not satisfied either.

Example 5.2: Consider the following two Cohen–Grossbergneural networks with two neuron:

(63)

(64)

where , ,

, and .

Pertaining to this example, [28, Th. 4] cannot judge the sta-bility of system (63). Applying Corollary 2.2 and Theorem 2.5,we can show that systems (63) and (64) are both globally asymp-totically stable, respectively.

In the following, we set initial conditions andfor , respectively.

When , the equilibrium points of system (63)are , , , and .Among them, is the nonnegative equilibriumpoint. is the unique equilibrium point ofsystem (64). Obviously, different dynamics between systems(63) and (64) are caused by different amplification functions.



VI. CONCLUSION

Under different assumptions of amplification function, somesufficient criteria are derived for the global asymptotic stabilityof a class of Cohen–Grossberg neural networks with both mul-tiple time-varying delays and continuously distributed delays,which are independent of the size of the time-varying delays andamplification functions. The results can be applied to some spe-cial cases of Cohen–Grossberg neural networks, for example,Hopfield neural networks and Lotka–Volterra recurrent neuralnetworks. Two numerical examples are employed to demon-strate the effectiveness of the obtained results.

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Huaguang Zhang (SM’04) was born in Jilin, China,in 1959. He received the B.S. and M.S. degrees incontrol engineering from Northeastern Dianli Uni-versity of China, Jilin City, China, in 1982 and 1985,respectively, and the Ph.D. degree in thermal powerengineering and automation from Southeast Univer-sity, Nanjing, China, in 1991.

He was with the Department of Automatic Control,Northeastern University, Shenyang, China, in 1992,as a Postdoctoral Fellow for two years. Since 1994,he has been a Professor and the Head of the Insti-

tute of Electric Automation, School of Information Science and Engineering,Northeastern University. His main research interests include fuzzy control, sto-chastic system control, neural-network-based control, nonlinear control, andtheir applications.

Dr. Zhang is an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS,MAN, AND CYBERNETICS—PART B and Automatica. He was the recipient ofthe Outstanding Youth Science Foundation Award from the National NaturalScience Foundation of China in 2003. He was named the Cheung Kong Scholarby the Education Ministry of China in 2005.

Zhanshan Wang was born in Liaoning, China, in1971. He received the M.S. degree in control theoryand control engineering from Fushun Petroleum In-stitute, Fushun, China, in 2001 and the Ph.D. degreein control theory and control engineering from North-eastern University, Shenyang, China, in 2006.

He is currently an Associate Professor with theSchool of Information Science and Engineering,Northeastern University. His research interestsinclude stability analysis of recurrent neuralnetworks, fault diagnosis, fault-tolerant control, and

nonlinear control.

Derong Liu (S’91–M’94–SM’96–F’05) received thePh.D. degree in electrical engineering from the Uni-versity of Notre Dame, Notre Dame, IN, in 1994.

From 1993 to 1995, he was a Staff Fellow withthe General Motors Research and DevelopmentCenter, Warren, MI. From 1995 to 1999, he was anAssistant Professor with the Department of Elec-trical and Computer Engineering, Stevens Instituteof Technology, Hoboken, NJ. He has been with theUniversity of Illinois, Chicago, since 1999, wherehe is currently a Full Professor of electrical and

computer engineering and computer science and has also been the Director ofGraduate Studies with the Department of Electrical and Computer Engineeringsince 2005. He has published eight books (four research monographs and fouredited volumes).

Dr. Liu is an Associate Editor of Automatica. He is currently the Editor of theIEEE Computational Intelligence Society’s Electronic Letter and an AssociateEditor for the IEEE TRANSACTIONS ON NEURAL NETWORKS, the IEEE Compu-tational Intelligence Magazine, and the IEEE Circuits and Systems Magazine.He is the General Chair for the 2008 IEEE International Conference on Net-working, Sensing and Control (Sanya, China). He was the Program Chair forthe 2008 International Joint Conference on Neural Networks, the 2007 IEEEInternational Symposium on Approximate Dynamic Programming and Rein-forcement Learning, the 21st IEEE International Symposium on Intelligent Con-trol (2006), and the 2006 International Conference on Networking, Sensing andControl. He is an elected AdCom Member of the IEEE Computational Intel-ligence Society (2006–2008), the Chair of the Chicago Chapter of the IEEEComputational Intelligence Society, the Chair of the Technical Committee onNeural Networks of the IEEE Computational Intelligence Society, and the PastChair of the Technical Committee on Neural Systems and Applications of theIEEE Circuits and Systems Society. He was the recipient of the Michael J. BirckFellowship from the University of Notre Dame (1990), the Harvey N. Davis Dis-tinguished Teaching Award from Stevens Institute of Technology (1997), theFaculty Early Career Development (CAREER) Award from the National Sci-ence Foundation (1999), and the University Scholar Award from the Universityof Illinois (2006–2009). He is a member of Eta Kappa Nu.


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