Stability Analysis of Fuzzy Hopfield Neural Networks with Timevarying

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1 ISSN 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

2 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

3 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

4 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

5 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

6 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

7 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 B.4.6 C..5 A 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 Q Q Q R R 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 Muralsankar e al. [5] <6.7 <6.7 <6.7 < Wu e al.[] <.8 <.8 <.8 <.8.77 Corollary JIC emal for conrbuon: edor@jc.org.uk

8 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 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 J I I 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 Q Q Q JIC emal for subscrpon: publsng@wau.org.uk

9 Qfeng Xun e al.:sably Analyss of Fuzzy Hopfeld Neural Neworks w me-varyng Delays R R 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 I I 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 Q Q JIC emal for conrbuon: edor@jc.org.uk

10 Journal of Informaon and Compung Scence Vol. (8) No. pp R R 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 W. J. L and. Lee Hopfeld neural neworks for affne nvaran macng IEEE rans. Neural Neworks (6)() pp G. Joya M. A. Aenca and F. Sandoval Hopfeld neural neworks for opmzaon: Sudy of e dfferen dynamcs Neurocomp. 4() pp S.S. Young P.D. Sco and N.M. Nasrabad Objec recognon usng mullayer Hopfeld neural nework IEEE rans. Image Process. 6()(997) pp C.M. Marcus and R.M. Weservel Sably of analog neural neworks w delays Pys. Rev. A 9()(989) pp 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 JIC emal for subscrpon: publsng@wau.org.uk

11 Qfeng Xun e al.:sably Analyss of Fuzzy Hopfeld Neural Neworks w me-varyng Delays Y. Cen and Y. Wu Novel delay-dependen sably crera of neural neworks w me-varyng delay Neurocompung 7(9) pp J. Qu H. Yang J. Zang and Z. Gao New robus sably crera for unceran neural neworks w nerval mevaryng delays Caos Solons Fracals 9(9) pp Y. He M. Wu J.H. Se and G.P. Lu Parameer-dependen Lyapunov funconal for sably of me-delay sysems w polyypc-ype unceranes IEEE rans. on Auomac Conrol49(5)(4) pp Y. He Q.G. Wang and.h. Lee Furer mprovemen of free-wegng marces ecnque for sysems w mevaryng delay IEEE rans. on Auomac Conrol 5()(7) pp M. Syed Al and P. Balasubramanam Sably analyss of unceran fuzzy Hopfeld neural neworks w medelay Commun Nonlnear Sc. Numer. Smula. 4(9) pp J.K. an and S.M. Zong Improved delay-dependen sably creron for neural neworks w me-varyng delay Appl. Ma. Compu. 7(4)() pp O. M. Kwon S.M.Lee J.H.Park and E.J.Ca New appoaces on sably crera for neural neworks w nerval me-varyng delays Appl. Ma. Compu. 8(9)() pp S. Muralsankar A. Manvannan and N. Gopalakrsnan Asympoc sably crera for -S fuzzy neural neworks w dscree nerval and dsrbued me-varyng delays Neural Compu. & Applc. () pp.s57-s67.. L. Wang A.G. Song and S.M. Fe Combned convex ecnque on delay-dependen sably for delayed neural neworks IEEE rans. on Neural Neworks and Learnng Sysems 4(9)() pp S.Y. Xu and J. Lam On equvalence and effcency of ceran sably crera for me-delay sysems IEEE rans. on Auomac Conrol 5()(7) pp.95-. J. Yu Furer resuls on delay-dsrbuon-dependen robus sably crera for delayed sysems Inernaonal Journal of Auomaon and Compung 8()() pp.-8. J. Yu J. an H. Jang and H. Lu Dynamc oupu feedback conrol for markovan jump sysems w me-varyng delays IE Conrol eory Appl. 6(6)() pp.8-8. H. Wu W. Feng and X Lang New sably crera for unceran neural neworks w nerval me-varyng delays Cogn. Neurodyn. (8) pp akag and M. Sugeno Fuzzy denfcaon of sysems and s applcaons o modelng and conrol IEEE rans. Sys. Man Cybern. 5(985) pp.6-. Y.Y. Cao P.M. Frank Sably analyss and syness of nonlnear me-delay sysems va lnear akag-sugeno fuzzy models IEEE rans. Fuzzy Sys. 4() pp.-9. F. Lu M. Wu Y. He and R. Yokoyama New delay-dependen sably crera for -S fuzzy sysems w mevaryng delay Fuzzy Ses Sys. 6() pp.-4. C.G. Zou L.P. Zang H.B. Jang and J.J. Yu Robus Fuzzy Conrol of Fuzzy Impulsve Sngularly Perurbed Sysems w Unceranes Adv. Sc. Le. () pp H.Y. L B. Cen Q. Zou and W.Y. Qan Robus sably for unceran delayed fuzzy Hopfeld neural neworks w markovan jumpng parameers IEEE rans. on Sysems Man and Cybernecs Par B: Cybernecs 9() 9. C.J. Zu and S.P. Wen Socasc sably of fuzzy Hopfeld neural neworks w me-varyng delays Inernaonal Conference on Informaon Scence and ecnology (ICIS) pp.4-7. S. Boyd L.E. Gou E. Feron and V. Balakrsnan Lnear marx nequales n sysem and conrol eory SIAM Pladelpa PA 994. K. Gu Inegral nequaly n e sably problem of me-delay sysems n: proceedngs of 9 IEEE CDC Sydney Ausrala pp L L. Guo and C. Ln A new creron of delay-dependen sably for unceran me-delay sysem IE Conrol eory and Applcaons ()(7) pp J. J. Yu Novel Delay-Dependen Sably Crera for Socasc Sysems w me-varyng Inerval Delay Inernaonal Journal of Conrol Auomaon and Sysems ()() pp C.G. Zou X. Q. Zeng and J. J. Yu Novel Sably Crera of -S Fuzzy Hopfeld Neural Neworks w mevaryng Delays and Unceranes Inernaonal jon Conference on Neural Neworks(IJCNN) July 6- Bejng Cna4. JIC emal for conrbuon: edor@jc.org.uk