ISSN 746-7659 England UK Journal of Informaon and Compung Scence Vol. No. 8 pp.- Sably Analyss of Fuzzy Hopfeld Neural Neworks w mevaryng Delays Qfeng Xun Cagen Zou Scool of Informaon Engneerng Yanceng eacers Unversy 4 Yanceng Cna (Receved June 7 8 acceped Augus 8) Absrac. In s paper e problem of asympoc sably for akag-sugeno (-S) fuzzy Hopfeld neural neworks w me-varyng delays s suded. Based on e Lyapunov funconal meod consderng e sysem w unceranes or wou unceranes new delay-dependen sably crera are derved n erms of Lnear Marx Inequales (LMIs) a can be calculaed easly by e LMI oolbox n MALAB. e proposed approac does no nvolve free wegng marces and can provde less conservave resuls an some exsng ones. Besdes numercal examples are gven o sow e effecveness of e proposed approac. Keywords: asympoc sably; -S fuzzy model; Hopfeld neural neworks; me-varyng delay. Inroducon Hopfeld neural neworks (HNNs) were frs nroduced by Hopfeld []. e dynamc beavor of HNNs as been wdely suded due o er poenal applcaons n sgnal processng combnaoral opmzaon and paern recognon [-4]. ese applcaons are mosly dependen on e sably of e equlbrum of neural neworks. us e sably analyss s a necessary sep for e desgn and applcaons of neural neworks. Somemes neural neworks ave o be desgned suc a ere s only global sable equlbrum. For example wen a neural nework s appled o solve e opmzaon problem mus ave unque equlbrum wc s globally sable. Bo n bologcal and arfcal neural neworks e neracons beween neurons are generally asyncronous wc nevably resul n me delays. me-delay s ofen e man facor of nsably and poor performance of neural nework sysems [5]. erefore los of effors ave been made on sably analyss of neural neworks w me-varyng delays n recen years [6-9]. e free-wegng marx meod was proposed o nvesgae e delay-dependen sably [] and some less conservave delay-dependen sably crera for sysems w me-varyng delay were presened [-6]. However Researcers ave realzed a oo many slack varables nroduced wll make e sysem syness complcaed lead o a sgnfcan ncrease n e compuaonal burden and canno resul n less conservave resuls ndeed [7-9]. In praccal sysems ere always are some unceran elemens and ese unceranes may come from unknown nernal or exernal nose envronmenal nfluence and so on. Hence as been e focus of nensve researc n recen years [] [] []. I s well-known a e -S fuzzy models ave been very mporan n academc researc and praccal applcaons and e fuzzy logc eory as sown o be an effcen meod o dealng w e analyss and syness ssues for complex nonlnear sysems [-4]. Very recenly some resuls ave been produced n e sudy of sably analyss of -S fuzzy Hopfeld neural neworks sysems w me-varyng delays [5-7] o e bes of our knowledge e robus sably problem for unceran fuzzy HNNs w me-varyng nerval delays as no been fully nvesgaed wc remans as an open and callengng ssue. In s paper e problem of sably analyss for -S fuzzy HNNs w me-varyng delays s consdered. Based on Jensen negral nequaly and some mporan Lemma new suffcen condons are derved n erms of LMIs. By consrucng a Lyapunov-Krasovsk funcon wou free-wegng marces approac e proposed crera n s paper are muc less conservave an some exsng resuls. Numercal examples are gven o sow e applcably of e obaned resuls. e res of s paper s arranged as follows. Secon gves problem saemen and some prelmnares used n laer secons. Secon presens our man resuls. Secon 4 provdes e numercal examples and Secon 5 concludes e paper.. Problem Saemen and Prelmnares Publsed by World Academc Press World Academc Unon
Journal of Informaon and Compung Scence Vol. (8) No. pp - In s bref we wll consder e followng HNNs w unceranes represened by a -S fuzzy model and e rule of e -S fuzzy model s of e followng form: Plan rule : z ( ) s M and z ( ) s M and z ( ) s M IF n n HEN x( ) ( A A ( )) x( ) ( B B ( )) f ( x( )) ( C C ( )) f ( x( d( ))) () x( ) ( ) [ ] q M j ( j n) s e fuzzy se z( ) [ z( ) z( ) z ( )] s e premse varable vecor n x () R s e sysem sae varable e me delay d( ) s e me-varyng delay w an upper bound of d () and q s e number of IF-HEN rules. n A () B () and C () unknown marces a represen e me-varyng parameer unceranes and are assumed o be admssble f e followng assumpon s sasfed. Assumpon [] : H a sasfy E E and E [ A ( ) B ( ) C ( )] H ( )[ E E E ] () are gven real consan marces. e class of paramerc unceranes are () ( ) I F ( ) J F ( ) () s sad o be admssble J s also a known marx sasfyng I JJ (4) and denoes unknown me-varyng marx funcons. I s assumed a all elemens are Lebesgue F () measurable sasfyng F () F ( ) F ( ) I R (5) o oban our man resuls we nroduce e followng lemmas. Lemma [8] : Le M P Q be e gven marces suc a Q en P M P M Q M M Q Lemma [7] mm : For any consan marx M R M vecor funcon en e followng nequaly olds: ( ( s) ds) M ( ( s) ds) ( s) M( s) ds d () M s a scalar : R m R s a Lemma [8] For any scalars W W s a connuous funcon and sasfes d() en W W mn W W W W d( ) d( ) Lemma 4 [9] Assume a () s gven by ()-(5). Gven marces M and E of approprae dmensons e nequaly M ( ) N N ( ) M (6) olds for all F () sasfes F ( ) F( ) I. en e followng nequaly M F ( ) N ( M F ( ) N ) (7) olds f and only f ere exss a scalar sasfyng JIC emal for subscrpon: publsng@wau.org.uk
4 Qfeng Xun e al.:sably Analyss of Fuzzy Hopfeld Neural Neworks w me-varyng Delays M N * I J * * I Usng a sandard fuzzy nference meod e sysem () s nferred as follows: q. (8) x( ) ( z( ))[ ( A A ( )) x( ) ( B B ( )) f ( x( )) ( C C( )) f ( x( d( )))] (9) from e fuzzy ses eory we ave ( z ( )). Man Resuls w ( z( )) ( z( )) w ( z( )) M ( z( )) n q j j wj ( z( )) j q ( z ( ))... me-varyng delay sysems wou unceranes eorem []. For gven scalars sysem (9) s asympocally sable f ere exs marces LMIs old: P Q ( ) R and R () w approprae dmensons suc a e followng j j * R q j 4. () * * R PB PC * Q * * Q * * * ( ) Q * * * * * * * * * A P PA Q Q Q () () A R B R C R A R B R C R (4) JIC emal for conrbuon: edor@jc.org.uk
Journal of Informaon and Compung Scence Vol. (8) No. pp - 5 R R R R R R R R * * * 4R 4R * * * * * * * * * * * * R R R R R R * R R * * 4R 4R * * * * * * * * * * * * R R R R R R * R R * * 4R 4R * * * * * * * * * * * * R R R R R R * R R 4 * * 4R 4R * * * * * * * * * * * * (5) (6) (7) (8) Proof. Coose a Lyapunov-Krasovsk funconal canddae as follows: JIC emal for subscrpon: publsng@wau.org.uk
6 Qfeng Xun e al.:sably Analyss of Fuzzy Hopfeld Neural Neworks w me-varyng Delays V( x ) V ( x ) V ( x ) V ( x ) V ( x ) 4 ( ) V ( ) ( ) x x Px V ( x ) x ( s ) Q x ( s ) ds x ( s ) Q x ( s ) ds V( x ) x ( s) Q () x( s) ds d 4 x V ( x ) x ( s ) R x ( s ) dsd ( s) R x( s) dsd en e me dervave of ( ) V x along e rajecory of sysem (9) yelds ( ) V ( ) ( ) x x Px (9) V ( x ) x ( )( Q Q ) x( ) x ( ) Q x( ) x ( ) Qx( ) () V( x ) x ( ) Qx( ) ( ) x ( d( )) Qx( d( )) () V4 ( x ) x ( )( R R ) x( ) ( ) ( ) ( ) x s R x ( ) () By usng Lemma and Lemma we ave x ( s ) R x ( s ) ds = x d () ( s) R x( s) ds d () d ( ) d ( ) and = x ( s) R x( s) ds {[( x ( s ) ds ) R x ( s ) ds ] / d ( ) d ( ) d ( ) [( x( s) ds) R x( s) ds] / ( d( ))} x ( s ) R x ( s ) ds W W d () d () x R R * R x ( s) R x( s) ds x ( s) R x( s) ds {[ x( s) ds) R ( ) x( s) ds] / ( d( ) ( ) ) d d d ( ) d ( ) W W W W max{ } () ( ) x( ) x( d( )) x( d( )) x( d( )) R R x( d( )) x( ) * R x( ) [ x( s) ds) R x( s) ds] / ( d( ))} R R x( ) x( ) W x( d( )) * R x( d( )) W I can be sown from ()(9)-(4) and Lemma a W W W W 4 4 max{ } (4) x( d( )) R R x( d( )) 4 x( ) * R x( ) JIC emal for conrbuon: edor@jc.org.uk
Journal of Informaon and Compung Scence Vol. (8) No. pp - 7 V ( x ) V ( x ) V ( x ) V ( x ) V ( x ) () 4 q ( z( )) ( ) ( ) j j (5) 4. = [ x( ) x( ) x( ) x( d( ) f ( x( )) f ( x( d( )))] Hence sysem (9) s asympocally sable. s complees e proof. Wen ere s no fuzzy and no unceranes n (9) e sysem s reduced o x( ) Ax( ) Bf ( x( )) Cf ( x( d( ))) (6) Corollary For gven scalars sysem (6) s asympocally sable f ere exs marces LMIs old: P Q ( ) R and R j j * R j 4. * * R PB PC * Q * * Q * * * * * * * * * defned as n eorem. w approprae dmensons suc a e followng A P PA Q Q Q and * * * ( ) Q.. me-varyng delay sysems w unceranes Now we sall dscuss e feasble robus sably crera for me-varyng delay sysems w uncerany. eorem. For gven scalars ( ) and e sysem(9) s robus sably f ere exs marces P Q ( ) R R and of approprae dmensons and scalar suc a e followng LMIs old: j M N * I J * * I q j 4 (8) j s defned n () and M PH H R H R N E E E Proof. Assume a nequales (8) old from Lemma and Lemma 4 j M F ( ) N [ M F ( ) N ] ( q; j 4) old. From (6) can be verfed a q ( z( )) ( ) M ( ) N [ M ( ) N ] ( ). j Hence sysem (9) s robus sably from eorem. Wen ere s no fuzzy n (9) e sysem s reduced o x( ) ( A A) x( ) ( B B) f ( x( )) ( C C) f ( x( d( ))) (9) Corollary For gven scalars and sysem (9) s asympocally sable f ere exs marces P Q ( ) R R and J w approprae dmensons and scalar J j (7) are JIC emal for subscrpon: publsng@wau.org.uk
8 Qfeng Xun e al.:sably Analyss of Fuzzy Hopfeld Neural Neworks w me-varyng Delays suc a e followng LMIs old: j M N * I J * * I s defned n (7) and j j 4 () Remark : wen M PH H R H R N E E E wll reduced o e sysem n []. J 4. Numercal Examples In s secon ree numercal examples are gven o llusrae e effecveness of e proposed meods. Example In s example we consder e DNNs (9) w A.4..5. B.4.6 C..5 A.9.4.7.6 B.5.7 C.. A B C e me-varyng delays are aken as d (). sn and e acvaon funcon s descrbed by x x e e f( x) e membersp funcon s x x ( z( )) sn x ( z( )) cos x usng MALAB LMI e e oolbox o solve e LMIs n eorem some posve defne feasble marces are gven as follows P.777 -.747 -.747.997 Q.95 -.5 -.5.987 Q.77 -.7 -.7.8 Q.6 -.8 -.8.4.58 -.66.96 -.48 R -.66.558 R -.48.476 and e sae rajecores of e sysems w dfferen nal condons are sowed as follows (Fgs. -) Fgs.- sow a e sae rajecores of e sysems are convergng o zero w dfferen nal sae a s o say sysem (9) s asympocally sable wen eorem olds. Example In s example we consder e DNNs (7) and corollary w A.7.8 B...5. C... e e acvaon funcon s descrbed by f( x) e w gven s sowed n able. x x e e x x e maxmum allowable upper bound of able : Maxmum allowable upper bound of w gven....5. Muralsankar e al. [5] <6.7 <6.7 <6.7 <6.7 6.64 Wu e al.[] <.8 <.8 <.8 <.8.77 Corollary 6.4 5.884 4.9. 9.4548 JIC emal for conrbuon: edor@jc.org.uk
Journal of Informaon and Compung Scence Vol. (8) No. pp - 9 Fg. : e sae rajecores w x() Fg. : e sae rajecores w x () 4 Fg. : e sae rajecores w x() 4 Accordng o e able s example sows a our resuls are beer an ose resuls dscussed n [5] wen s small enoug aloug free-wegng marx approac s adoped n [5]. Example In s example we consder e DNNs () and eorem w.7.8.9.4.. A A B.5. B.5.7 C....6 sn C..4 F () cos. H I E E E... x x e e e acvaon funcon s descrbed by f( x) e membersp funcon s x x ( z( )) sn x e e ( z( )) cos x and we coose e parameer J w dfferen values e maxmum allowable upper bound of w gven s sowed n able. able : Maxmum allowable upper bound of w gven.5..5..5. J.55.66.77.9.5 -.I.64.667.475.89.548 -.5I.4.9.55 - - - e me-varyng delays are aken as d (). +.5sn and J. I usng MALAB LMI oolbox o solve e LMIs n eorem some posve defne feasble marces are gven as follows: P.56.65.65.76 Q..8.8.4 Q..8.8.4 Q.7...6 JIC emal for subscrpon: publsng@wau.org.uk
Qfeng Xun e al.:sably Analyss of Fuzzy Hopfeld Neural Neworks w me-varyng Delays.598.6 R.6.77.56.54 R.54.95 and e sae rajecores of e sysems w dfferen nal condons are sowed as follows(fgs. 4-6) Fg. 4 e sae rajecores w x() 4 Fg. 5 e sae rajecores w x () Fg. 6 e sae rajecores w x() 4 able sows e maxmum allowable upper bound of me-delay w gven e allowable lower bound. From Fgs.4-6 can be seen a e sae rajecores of e sysems are convergng o zero w dfferen nal sae a s o say sysem (9) s robus sable wen eorem olds. Example 4 In s example we consder e DNNs () and corollary w A.7.8 B...5. C. sn.. E E E. F () cos H I x x e e e acvaon funcon s descrbed by f( x) and we coose e parameer J w dfferen values x x e e e maxmum allowable upper bound of w gven s sowed n able. ab. : Maxmum allowable upper bound of J w gven.5..5....75.4.55.68.94 -.I.54.59.55.5 - -.5I.4.4.54 - - - e me-varyng delays are aken as d ().5.sn and J. I usng MALAB LMI oolbox o solve e LMIs n corollary some posve defne feasble marces are gven as follows: P....8 Q.46.5.5.54 Q.46.5.5.54 Q.5...57 JIC emal for conrbuon: edor@jc.org.uk
Journal of Informaon and Compung Scence Vol. (8) No. pp -.69.46 R.46.78 R.479.5.5.479 and e sae rajecores of e sysems w dfferen nal condons are sowed as follows(fgs. 7-9) Fg. 7 e sae rajecores w x() 4 Fg. 8 e sae rajecores w x () Fg. 9 e sae rajecores w x() 4 able sows e maxmum allowable upper bound of me-delay w gven e allowable lower bound. From Fgs.7-9 can be seen a e sae rajecores of e sysems are convergng o zero w dfferen nal sae a s o say sysem () s robus sable wen corollary olds. 5. Conclusons We presen mproved crera of robus sably for HNNs w me-varyng delay and unceranes n s paper. e obaned sably condons are expressed w LMIs. By comparng e expermenal resuls from numercal examples s demonsraed e mprovemen of our proposed crera over some exsng ones. Acknowledgemens s work s parally suppored by e nnovaon eam of Group Inellgence Collaborave Compung n Yanceng eacers Unversy of Cna. 6. References J. J. Hopfeld Neural neworks and pyscal sysems w emergen collec compuaonal ables Proc. Na. Acad. Sc. USA 79() pp. 554-558 98. W. J. L and. Lee Hopfeld neural neworks for affne nvaran macng IEEE rans. Neural Neworks (6)() pp. 4-4. G. Joya M. A. Aenca and F. Sandoval Hopfeld neural neworks for opmzaon: Sudy of e dfferen dynamcs Neurocomp. 4() pp. 9-7. S.S. Young P.D. Sco and N.M. Nasrabad Objec recognon usng mullayer Hopfeld neural nework IEEE rans. Image Process. 6()(997) pp. 57-7. C.M. Marcus and R.M. Weservel Sably of analog neural neworks w delays Pys. Rev. A 9()(989) pp. 47-59. Y. He G.P. Lu and D. Rees New delay-dependen sably crera for neural neworks w me-varyng delay IEEE rans. Neural New. 8(7) pp. -4. O.M. Kwon J.H. Park and S.M. Lee On robus sably for unceran neural neworks w nerval me-varyng delays IE Conrol eory Appl. (8) pp. 65-64. JIC emal for subscrpon: publsng@wau.org.uk
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