A New Criterion for Stability of Delayed Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks

Transcription

1 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR) ISSN , Volue 7, Nube 3 (Jue 018), pp MEACSE Publicatios A New Citeio fo Stability of Delayed Takagi-Sugeo Fuzzy Cohe-Gossbeg Neual Netwoks Neyi Ozca Depatet of Electical ad Electoics Egieeig, Uludag Uivesity, Busa, Tukey Abstact I this pape, we deive a ew citeio fo the global asyptotic stability of Takagi-Sugeo(T-S) fuzzy Cohe-Gossbeg eual etwoks with ultiple tie delays. By eployig a oe geeal type of fuzzy Lyapuov fuctioal, we establish a easily veifiable delay idepedet stability coditio that establishes a elatioship betwee the syste paaetes of the delayed Takagi-Sugeo (T-S) fuzzy Cohe-Gossbeg eual etwoks with espect to the odeceasig ad slope-bouded activatio fuctios. Keywods: Cohe-Gossbeg Neual Netwoks, Tie delays, Lyapuov Stability Theoes, T-S Fuzzy Systes. Itoductio I the ecet yeas, ay papes [1]-[6] have studied equilibiu ad stability popeties of Cohe- Gossbeg eual etwoks (CGNNs) itoduced i [7] as this class of eual etwoks has foud ipotat applicatios i the aeas of patte ecogitio, iage ad sigal pocessig, paallel coputatio ad cotol systes. I ost of the applicatios of eual etwoks, the key poit is to desig a eual etwok possessig a globally asyptotically stable equilibiu poit. Theefoe, i the ecet liteatue, stability of Cohe-Gossbeg eual etwoks with o without delay paaetes has eceived a geat deal of attetio ad ay diffeet sufficiet coditios fo global asyptotic stability of the equilibiu poit fo delayed Cohe-Gossbeg eual etwoks have bee peseted [1]-[7]. O the othe had, it has bee show i [8] that fuzzy logic theoy ay help to ipove the desied behavio of coplex dyaical systes ad itoduced the Takagi-Sugeo (T-S) fuzzy odel. I [8], it has also bee poved that T-S fuzzy systes ca be used to tasfo a oliea syste ito a set of TS liea odels. A T-S fuzzy syste is essetially a oliea syste descibed by a set of IF-THEN ules. Soe cetai oliea coplex systes ca be appoxiated by the oveall fuzzy liea T-S odels i ode to coduct a ivestigatio ito the stability aalysisof coplex oliea systes. I a ecet pape [9], by usig Lyapuov stability theoes, soesufficiet coditios fo the stability the T-S fuzzy systes have bee peseted. The esults obtaied i [9] have led to ay eseaches studyig the T-S fuzzy odels to dive futhe stability esults fo vaious classes of fuzzy eual etwoks with tie delays [10]-[19]. I this pape, wewilldealwiththestability of poble of Takagi-SugeofuzzyCohe-Gossbegeualetwokswithultiple tie delaysadobtai a ewalteativeciteiofothe global asyptoticstability of theclass of delayedfuzzycohe- Gossbegeualetwoks. 43

2 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 Stability Aalysis of Takagi-Sugeo Fuzzy Cohe-Gossbeg Neual Netwoks Coside the followig Cohe-Gossbeg eual etwoks with ultiple tie delay x i t = d i(x i t ) c i x i t + j =1 a ij f j x j t + j =1 b ij f j x j t τ ij + u i (1) whee is the ube of the euos i the eual syste, x i deotes the state of the ith euo, d i (x i ) epeset the aplificatio fuctios, ad c i (x i ) epeset the behaved fuctios. The costats a ij ae the itecoectio paaetes of the euos withi the eual syste, the costats b ij ae itecoectio paaetes of the euos with tie delay paaetes τ ij.the f i (. ) deote the activatio fuctios of euos. The costats u i ae soe exteal iputs. I syste (1), τ ij 0 ae costat tie delays with τ = ax τ ij, 1 i, j. Accopayig the eual syste (1) is a iitial coditio of the fo: x i t = φ i t C τ, 0, R), whee C τ, 0, R) deotes the set of all cotiuous fuctios fo τ, 0 to R. The assuptios o the fuctios d i (x), c i (x) ad f i (x) i (1) ae defied to be as follows: H 1 : The fuctios d i (x), (i = 1,,, ) satisfy the coditios 0 < μ i d i x ρ i, x R whee μ i ad ρ i ae soe positive costats. H : The fuctios c i (x), (i = 1,,, ) satisfy the coditios c i x c i (y) x y = c i x c i(y) x y γ i > 0, i = 1,,,, x, y R, x y whee γ i ae soe positive costats. H 3 : The fuctios f i (x), (i = 1,,, ) satisfy the coditios f i x f i y l i x y, i = 1,,,, x, y R, x y whee l i ae soe positive costats. I ode to siplify the poofs, the equilibiu poit x of Cohe-Gossbeg eual etwok odel (1) ca be tasfoed to the oigi. Usig the tasfoatio z t = x t x, we ca tasfosyste (1) ito a ew syste of the fo : z i t = α i (z i t ) β i z i t + j =1 () j =1 a ij g j z j t + b ij g j z j t τ ij whee the followig popeties hold: α i z i t = d i z i t + x i, i = 1,,, β i z i t = c i z i t + x i c i (x i ), i = 1,,, g i z i t = f i z i t + x i f i (x i ), i = 1,,, Sice x(t) x asz(t) 0, establishig stability of the oigi of syste () will be the aiobjective. I [9], the T-S fuzzy Cohe-Gossbeg eual etwok with ultiple tie delays is defied by the followig atheatical odel : Plat Rule : IF θ 1 t is M 1 ad ad θ p t is M p THEN 44

3 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 z i t = α () i (z i t ) β () i z i t + a () j =1 ij g j z j t + b () ij g j z j t τ ij j =1 (3) wheeθ l t (l = 1,,, p)ae the peise vaiables. M l ( 1,,,, l 1,,, p ae the fuzzy sets ad is the ube of IF-THENules. By ifeig fo the fuzzy odels, (3) ae stated as follows [19] : z i t = (θ t ) α () i (z i t ) β i zi t + j =1 a ij gj z j t + b () ij g j z j t τ ij whee θ t = θ 1 t, θ t,, θ p t T p, ω θ t = l=1 M l (θ l t ) ad θ t = ω (θ t ) j =1 (4) =1 ω (θ t ) deote the weight ad aveaged weight of each fuzzy ule, espectively. The te ω l θ l t is the gade ebeship of θ l t i ω l. It is assued that ω θ t 0, 1,,,.Theefoe, it follows that (θ t ) = 1 fo all t 0. Fo the odel of T-S fuzzy eual syste (4), the assuptios H 1, H ad H 3 ae ow espectively foulated as follows: 0 < α i zi t ρ i, i = 1,,, z i t β i zi t γ i zi t 0, i = 1,,, g i (z i t ) k i z i t, z i t g i z i t 0, i = 1,,, Stability of Delayed Takagi-Sugeo Fuzzy Cohe-Gossbeg Neual Netwoks Ithissectio, weobtaithefollowigstabilityesult : Theoe 1:Ude AssuptiosH 1, H ad H 3, the oigi of the delayed T-S fuzzy Cohe-Gossbeg eual etwoks defied by (4) is globally asyptotically stable if the followig coditio holds: δ = μγ ρ K 1 T ( A + A ) B 1 B I > 0 wheek = diag k 1, k,, k μ = i μ i, ρ = ax ρ i 1,,,, = 1,,,, A = (a ij ) x, A = ( a ij ) x ad B = (b ij ) x. Poof:Coside the Lyapuov fuctioal: V z t = z(t) 0 z(t) ad γ = ax γ i,l = ax l i, i = sds + σ g i (s)ds + ε g j z j (ζ )dζ + σρ ξ 0 t t τ ij b ij gj (z j ζ )dζ whee the ξ, ε ad σ ae soe positive costats to be deteiated late. Calculatig the tie deivative of V(x t ) alog the tajectoies of syste (4) yields V z t = z i t z i t + σ g i (z i t )z i t σρ ξ g j (z j t τ ij ) t t τ ij + ε g j z j (t τ ij ) + σρ ξ g j (z j t ) 45

4 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 = θ t z i t α i zi t β i zi t + (θ t ) α i (zi t )a ij zi t g j (z j t ) + (θ t ) α i (zi t )b ij zi t g j (z j t τ ij ) σ θ t α i zi t β i zi t g i z i t +σ (θ t ) α i (zi t )a ij gi (z i t )g j (z j t ) +σ (θ t ) α i (zi t )b ij gi (z i t )g j (z j t τ ij ) +σρ +ε g j z j (t ) ε g j z j (t τ ij ) ξ b ij gj (z j t ) σρ ξ b ij gj z j t τ ij (5) Ude the Assuptios H 1, H ad H 3, i [9], the followig iequalities have bee show to be held: θ t z i t α i zi t β i zi t μγ z i t (θ t ) α i (zi t )a ij zi t g j (z j t ) μγ z i t + υ g i z i t (6) (7) (θ t ) α i (zi t )b ij zi t g j (z j t τ ij ) μγ z i t + ε g j z j (t τ ij ) (8) σ θ t α i zi t β i zi t g i z i t σμγ g T z t + K 1 g z t (9) T σ (θ t ) α i (zi t )a ij gi (z i t )g j (z j t ) σρ A + A g z t (10) whee υ = a ρ μγ, ε = b ρ, μγ 46

5 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 () ad a ad b ae soe positive costat such that a ij = 1,,, Weotethefollowigiequality () a ad b ij b fo all, i, j = 1,,, ad σ (θ t ) α i (zi t )b ij gi (z i t )g j (z j t τ ij ) σ θ t α i zi t b ij g i z i t g j z j t τ ij σ θ t ρ b ij g i z i t g j z j t τ ij σ (θ t )ρ 1 b ξ ij g i (z i t ) + ξ b ij g j (z j t τ ij ) σρ 1 b ξ ij g i z i t + ξ b ij g j z j t τ ij (11) Usig (6)-(11)V (z t ) yields Let V z t υ g i z i t + ε g i z i t σμγ g T z t K 1 g z t +σρ j=1 j=1 +σρ g T T z t A + A g z t 1 b ξ ij g i z i t + σρ ξ b ij g i z i t υ + ε g T z t g z t σμγ g T z t K 1 g z t +σρ +σρ g T T z t A + A g z t 1 B ξ g T z t g z t + σρ ξ B 1 g T z t g z t ξ = B B 1, = 1,,, The, we have V z t υ + ε g T z t g z t σμγ g T z t K 1 g z t +σρ g T T z t A + A g z t 47

6 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 +σρ B 1 B g T z t g z t σρ g T z t = υ + ε g T z t g z t ( μγ ρ K 1 A + A T = υ + ε g T z t g z t σρ g T z t Ω g z t The fact that Ω is a positive defiite atix iplies that B 1 B ) g z t V z t υ + ε g z t σρλi (Ω) g z t = ( υ + ε σρλ i Ω ) g z t the fo: Fo the choice σ > υ+ε, V z t < 0 fo all g z t 0. Let g z t = 0. The, V z t is of ρλ i Ω V z t = θ t z i t α i zi t β i z i t + (θ t ) α i (zi t )b ij zi t g j (z j t τ ij ) ε g j z j (t τ ij ) + σρ ξ b ij gj z j t τ ij (1) which, whe cobied with (6) ad (8) i (1), esults i V z t = μγ z i t + μγ z i t + ε g j z j (t τ ij ) j=1 ε g j z j (t τ ij ) σρ ξ b ij gj z j t τ ij = μγ z i t σρ ξ j=1 b ij gj z j t τ ij μγ z i t It is easy to see thatv z t < 0 fo all z(t) 0. Now let g z t = 0 ad z t = 0. The V z t = σρ ξ j=1 b ij gj z j t τ ij Note that if g j z j t τ ij 0 fo ay pais of i ad j, the V z t < 0. Hece, it follows that V z t = 0if ad oly if g z t = 0, z t = 0 ad g j z j t τ ij = 0 fo all i ad j ad V z t < 0 i all othe cases. It is easy to veify that V z t is adially ubouded sicev z t as z(t). Theefoe, we ca diectly coclude that the oigi of the T-S fuzzy Cohe-Gossbeg eual etwok odel (4) is globally asyptotically stable. 48

7 Coclusios Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 This pape has peseted a ew delay idepedet sufficiet coditio fo the global asyptotic stability of delayed Takagi-Sugeo (T-S) fuzzy Cohe-Gossbeg eual etwoks by usig a class of fuzzy Lyapuov fuctioal with espect to the odeceasig ad slope-bouded activatio fuctios. This stability coditio ca be easily checked as it is copletely expessed i tes of the etwok paaetes of the eual syste. It is also show that the stability citeio obtaied i this pape fo delayed Takagi Sugeo (T-S) fuzzy Cohe-Gossbeg eual etwoks ipoves ad geealizes a ecetly published coespodig stability esult. Refeeces [1] Li, L. ad Jia J., Lagage p-stability ad expoetial p-covegece fo stochastic Cohe-Gossbeg eual etwoks with tie-vayig delays, Neual Pocessig Lettes, vol. 43, 016, [] Li, R.,Cao, J.,Alsaedi, A. ad Ahad, B. Passivity aalysis of delayed eactio-diffusio Cohe Gossbeg eual etwoks via Hady-Poica iequality, Joual of the Fakli Istitute,vol. 354, 017, [3] Li, B. ad Sog, Q., Soe ew esults o peiodic solutio of Cohe-Gossbeg eual etwok with ipulses, Neuocoputig, vol. 177, 016, [4] Nie, X., Zheg, W. X. ad Cao, J., Multistability of eistive Cohe-Gossbeg eual etwoks with o-ootoic piecewise liea activatio fuctios ad tie-vayig delays, NeualNetwoks, vol. 71, 015, [5] Du, Y., Zhog S. ad Zhou, N., Global asyptotic stability of Makovia jupig stochastic Cohe Gossbeg BAM eual etwoks with discete ad distibuted tie-vayig delays, Applied Matheatics ad Coputatio, vol. 43,014, [6] Aik, S. adoa, Z., Global stability aalysis of Cohe-Gossbeg eual etwoks with tievayig delays, Physics Lettes A, vol. 341, 005, [7] Cohe, M.A. ad Gossbeg, S., Absolute stability ad global patte foatio ad paallel eoy stoage by copetitive eual etwoks, IEEE Tasactios o Systes, Ma adcybeetics, vol. 13, 1983, [8] Takagi, T. adsugeo, M., Fuzzy idetificatio of systes ad its applicatios to odelig ad cotol, IEEE Tasactios o Systes, Ma ad Cybeetics, vol. 15, 1983, [9] Hou, Y.Y.,Liao, T.L. ad Ya, J.J., Stability aalysis of Takagi-Sugeo fuzzy cellula eual etwoks with tie-vayig delays, IEEE Tasactios o Systes, Ma ad Cybeetics, vol. 37, 007, [10] Yaaoto, H. ad Fuuhashi, T., A ew sufficiet coditio fo stable fuzzy cotol syste ad its desig ethod, IEEE Tasactios o Fuzzy Systes, vol. 9, 001, [11] Huag, H., Ho, D.W.C. ad La, J., Stochastic stability aalysis of fuzzy Hopfield eual etwoks with tie-vayig delays, IEEE Tasactios o Cicuits Systes-I, Fudaetal Theoy ad Applicatios, vol. 5, 005, [1] Tseg, K.H,Tsai, J. S. ad Lu, C. Y., Desig of delay-depedet expoetial estiato fo T-S Fuzzy Neual etwoks with ixed tie-vayig iteval delays usig hybid Taguchi-Geetic algoith, Neual Pocessig Lettes, vol. 36, 01, [13] Xie, W. ad Zhu, Q., Mea squae expoetial stability of stochastic fuzzy delayed Cohe-Gossbeg eual etwoks with expectatios i the coefficiets, Neuocoputig, vol. 166, 015, [14] Jia, J. ad Jiag, W., Lagage expoetial stability fo fuzzy Cohe-Gossbeg eual etwokswith tie-vayig delays, Fuzzy Sets ad Systes, vol. 77, 017, [15] Mathiyalaga, K., Pak, J. H., Sakthivel, R., Athoi, S. M., Delay factioig appoach to obust expoetial stability of fuzzy Cohe-Gossbeg eual etwoks, Applied Matheaticsad Coputatio, vol. 30, 014, [16] Bao, G., We, S. ad Zeg, Z., Robust stability aalysis of iteval fuzzy Cohe-Gossbeg eual etwoks with piecewise costat aguet of geealized type, Neual Netwoks, vol. 33, 01,

8 Iteatioal Joual of Coputig Acadeic Reseach (IJCAR), Volue 7, Nube 3, Jue 018 [17] Li, C.,Li, Y. ad Ye, Y., Expoetial stability of fuzzy Cohe-Gossbeg eual etwokswith tie delays ad ipulsive effects, Couicatios i Noliea Sciece ad NueicalSiulatio, vol. 15, 010, [18] He, D. ad Xu, D., Attactig ad ivaiat sets of fuzzy Cohe-Gossbeg eual etwoks with tievayig delays, Physics Lettes A, vol. 37, 008, [19] Sea, S., A aalysis of global stability of Takagi-Sugeo fuzzy Cohe-Gossbeg eual etwoks with tie delays, Neual Pocessig Lettes, x. 50