Input-Output Stability Of Neural Network With Hysteresis

Transcription

1 Iero Jour of Comuer Iformo Sysems d Idusr geme Acos ISSN Voume 6 (014) IR Lbs wwwmrbse/csm/dehm Iu-Ouu Sby Of Neur Nework Wh Hyseress GPdmvh 1 JNeem 1 PVSv Kumr GVDeek 1 1 CRRo Advced Isue of hemcs Sscs d Comuer Scece Uversy of Hyderbd cmushyderbd Id dmvh@crromscsres VNRVg Jyoh Isue of Egeerg & Techoogy Bchuy Nzme (SO)Hyderbd Id svkumr_v@vrve Absrc I hs er yss of hyseress eur ework owrds sby re roosed I he rese reserch esece symoc sby d Iu-ouu sby of euos s mode for hyserec euros re dscussed These eur eworks c so be emoyed for mge erco ose erferg ches We esbsh suffce coos for vrous sby yss of hs css of eur eworks The resu mroves he erer ubcos due o he Iu ouu yss of he ework wh eur deys Keywords- Hyseress Neur Neworks Asymoc sby Iu ouu sby I INTRODUCTION Hyseress c be observed my egeerg sysems such s coro sysems eecroc crcus d so observed ms such s frogs [15] d cryfsh [9]: I ezoeecrc cuor; hyseress mes h for cer u; here s o uue ouu d he ouu deeds o he u hsory [7] hemc modes descrbg he dymc ercos of hyseress eur ework hve bee dscussed ([][4-8]) I hs er we cosder he css of couous me hyseress eur ework s mode descrbed by he foowg form of eur dey dffere euos z m z z z b f (1) 1 z dz () 0 Here z 1 1 The fucos ssoced wh he sysem (1) re gve by z( s) ( s)for s[ 0] for ech 1 here m{ } C([ 0] R) for ech 1 Le z ( ) m z ( ) z ( ) The euo (1) c be wre s 1 z z b f for 0 Dffereg wr () d usg sysem (3) d [ ( )] z b f z 1 z b f z 1 d z b f z 1 he sysem (1) c be wre he foowg form whch s mhemcy covee o work wh d b f 1 1 b f b f I I 1 For y s [ 0] 1 () (3) (4) ( s) ( s) ( s ) ( s ) ( s) From C([ 0] R) we hve C([ 0] R) Dymc Pubshers Ic USA

2 Iu-Ouu Sby Of Neur Nework Wh Hyseress 39 Therefore ( ) ( s)for s[ 0] re he coos ssoced wh he ework () The sysem (1) my be vewed s frs order dffere euos of eur ye wh vryg us From ework (1) we c observe h fuco f deeds o oy o he ouu of sysem bu so hsory of he re of chge of s ouu Iu-ouu rereseo d se vrbe rereseo re wo dffere behvors of ookg he sme [0] The wo yes of rereseos re used s ech of hem gve dffere kd of roch o how he sysem works There ess very cose reosh bewee he yes of sby resus Hece oe c fd dog hese wo roches The er roch s med he deermo of ouu bouds gve he chrcerscs of he feedbck sysem d s u Boh he u d he ouu bouds re defed some ormed sces Thus he ssue of u-ouu sby s referred o s L sby yss L sby heory hs bee eesvey suded he erure ([1] [16][17][0][3]) O he oher hd he echues of fuco yss oeered by Sdberg [16-17] d Zmes [3] hve deveoed euy rdy d geered rge umber of resus cocerg he u-ouu roeres of oer feedbck sysems The L sby of er feedbck sysems wh sge me-vryg secor-bouded eeme s suded [13] The subec of feedbck sysems sby hs bee eesvey dwe uo he erure [0] I he rese vesgo we esbsh resus o deg wh he crcumsces uder whch coos of (4) s L - sbe Defo 1The souo 0 s L - sbe for (4) f s sbe d for ( 0 u0) D where D / 0 0 Le F( 0 u 0) be y souo of (4) for F( u ) u for 0 0 here ess whch ( ) 0 such h f u F( 0 u0) 0 II he EXISTENCE AND UNIQUENESS I s esy o see h he eubrum of he sysem (4) s souo of he foowg sysem of euos For 1 1 b 1 f I (5) Throughou hs dscusso we ssume h he fucos f ssfy he foowg coos: For 1 here es osve ues L such h f u f v L u v (6) for u v R d 0 Now our frs resu s cocered wh he esece of uue eubrum 1 for he sysem (4) for Theorem 1 Assume h coo (6) ssfed I do ssume h he decy res he syc weghs b d he rmeers L ssfy he foowg euy 1 1 L b 1 (7) The uder hese coos here es uue eubrum o for he sysem (4) ( 1 ) T 1 Proof If b f 1 Defe mg s eubrum o wh ssfes he foowg for ech = 1 H { H 1 ; 1 } Where H b 1 f 1 I [14] ocy verbe 0 for ech R C m H : R (8) (9) R s R oo sef f s roer h s homeomorhsm of H 1 ( K) s comc for y comc se K R So f we verfy h H s roer we c hve h H s becve mg d hece H() = 0 hs uue souo Therefore (4) hs oher se of uue eubrum souo for ech R Frs we rove h f (4) hve eubrum he s uue Suose h here re wo eubrum os d The we c ob

3 393 Pdmvh e 1 b f f 1 1 (10) Thus we hve m H( ) Hece H ( ) 0 hs uue 1 souo d (4) hs uue eubrum o L b III STABILITY ANALYSIS Ths mes we hve Gob symoc sby of eubrum mes h he 1 1 L b we rec h he eubrum ssoced wh (11) If we ssume 1 fucos (6) ssfes m ( ) L b 1 1 (1) Lemm 1 [1] Le f c If f () We hve 0 Therefore Hece f (4) f hve eubrum hs uue eubrum To show he s bouded he f 0 s esece of he eubrum of (4) s eough o show h H s homeomorhsm of R oo sef From he Theorem Assume h he foowg eues re uueess of eubrum roof we hve h f ssfed he we hve H( ) H ( ) hece H s for ech =1 L b 1 oe-o-oe Therefore H s ocy verbe C 0 mg 1 To rove roer suffce o rove h h H s Furher here es 0 such h H( ) for 1 b L Cosder 1 H ( ) H ( ) H (0) 1 b f f 1 Thus we hve 0 1 ( ) 0 H b f f b f f L b Le m 1 L b 1 1 From (10) we hve 0 d hece H( ) whch mes we hve m H( ) So we hve rec s erfec he sese o hs or guesses re eeded Now u I s goby symocy sbe deede of deys f every souo of (4) corresodg o rbrry choce of 1 L b b d 1 b L L b c (13) The he uue eubrum souo symocy sbe of (4) s goby Proof From (4) d (5) we hve d ( ) b ( f f ) 1 ( ) b ( f f ) 1 b ( f f ) (14) 1 Cosder

4 Iu-Ouu Sby Of Neur Nework Wh Hyseress 394 E ( ) b ( f f ) 1 1 L b L b b L b b s ds 1 1 L b s 1 s ds 1 L b b (15) 1 1 Dffereg wh resec o d Usg (14) (6) d 1 usg he euy b b for y 0 for re b we hve de ( 1 1 b L 1 1 b L 1 1 b L 1 L b b ) L b b L b L b b L b b L b L b b (16) Rerrgg he erms we hve

5 395 Pdmvh e de 1 ( 1 b L b L 1 1 L b b L b L L b L b b L b b L b b L b b L b L b b L b b 1 1 ( ) b ( f f ) 1 1 E L b L b b So ( ) E 0 (17) Ths er eds o b ( f f ) de( ) 1 (18) 1 Usg (6) we hve Where 1 b L 1 1 b L 1 L b L b b Thus we hve de 0 0 Sce de Thus we hve 1 E s ds E 1 0 Sce E 0 we hve 1 0 s ds E 0 s ds (19) 1 0 Now we rove h Sce 1 0 ' re bouded s ds 0 From (19) we hve E E 0 Thus for ech 1 E Ths mes we hve 0 1 L b su ( s ) E 0 s [ 0] su s L b s [ 0] 1 0

6 Iu-Ouu Sby Of Neur Nework Wh Hyseress 396 Thus we hve 0 E su ( s ) s [ 0] Therefore ' ( ) s 1 L b 1 (0) s s uform bouded Thus s so bouded Whou oss of geery we c ssume h I 0 euo (4) d f(0) 0 1 o mke comuoy rcbe d o rovde deh yss The foowg emms c be used he dervo of sby L - Lemm : Le 1 The for b oegve we hve () ( b) ( b ) () ( b) b 1 1 hece b ( b ) Lemm 3: If 0 u for rbrry come umbers ( ) 1 m{1 } he Defo : The orm of eeme y ( y1 y y ) of Eucde -sce E s gve by y y 1 Lemm 4 [1] Le V be Luov fuco o E such h V u c u E o D / 0 0 for some c0 0 The u 0 s P L - sbe for (A) [Where u 1 f u (A)] Lemm 4[1]: If here essv u Luov fuco he u 0 s sbe for A Theorem 3: Assume h he coos for 1 b L b L L b L b b The he 0 s Proof 1 L b 1 b L b L b 1 L b 1 1 (1) P L - sbe for he sysem (4) Le E E where 1 1 E b f 0 () Cery E s osve defe 0 d E s couous o D h E s ocy Lschz o D ( )( ) D D ( 0) / 0 Cosder E E 1 b f 1 1 b f 1 E 0 0 Now we verfy For y b f f 1 sg he euy for y 1 r r r r P r r r r we hve P P P1 P U

7 397 Pdmvh e E E 1 b f 1 1 b f 1 b f f 1 Usg emm 3 d euy (6) we hve E E b L 1 b L b L 1 where m{1 } From he defo of D we hve E E b L b L b L L b 1 where b L 1 Thus we hve E s ocy Lschz o D Dfferee euo () wr d usg euo (4) d emm 3 we hve de de b f 1 d b f [ b f ] [ b f b f ] 1 1 Smfyg we hve [ where m{1 } 1 1 b f b f 1 b f b f 1 b f 1 Thus we hve de b f 1 b f 1 1 b f b f b f 1 1 b f b f 1 1

8 Iu-Ouu Sby Of Neur Nework Wh Hyseress 398 From (6) d rerrgg he erms we hve de 1 b L 1 1 b L b L 1 L b b L b b 1 1 Usg he euy 1 for y o- egve b we hve b ( b ) 1 1 de b L b L 1 1 b L b L 1 L b b 1 1 b L 1 1 L b 1 1 L b b L b b L b b 1 1 Rerrgg erms we hve de 1 b L 1 b L b L 1 L b b L b b 1 1 (3) Smfyg we hve de b L 1 b L b L 1 L b b L b 1 b L b 1 (4) 1 1 L b 1 L b 1 1

9 399 Pdmvh e ( ) Le F F 1 where 1 L b 1 b 1 1 F s ds 1 1 L b L b 1 L b s ds 1 1 Cery F s osve defe 0 d s couous o D F s ocy Lschz o (5) F 0 0 Now we verfy h y ( )( ) D D ( 0) / 0 F F D For cosder L b L b L b L b b s s ds s s ds Usg he euy for y 1 r r r r r r r r P P P 1 P By yg he smr rocedure for Lschz o h F D E s ocy s verfed bove c esy verfy s so ocy Lschz o D Dfferee euo (5) wr 1 L 1 b b L b 1 df 1 1 L b 1L b 1 1 Defe V E F E F (6) (7) Cery V s osve defe 0 0 d s couous o D Now we verfy h V ocy Lschz o Lschz o (4) d (6) dv D D V 0 s V s ocy Dffereg (7) wr d usg b L 1 b L 1 L b L b b L b 1 b L b L b 1 L b 1 1

10 Iu-Ouu Sby Of Neur Nework Wh Hyseress L b 1 b L b L b 1 L b 1 1 Smfyg dv b L 1 b L 1 L b L b b L b 1 b L b L b 1 L b 1 1 Thus we hve dv b L 1 b L 1 L b L b b L b 1 b L b L b 1L b 1 1 where b L 1 b L L b 1 1 L b b L b 1 b L b L b 1 L b 1 1 From (14) 0 he we hve (8) dv 0 0 V s Luov fuco for (4)Now we Hece verfy h 1 Sce P V o D for some 0 dv From emm 4 0 s P L - sbe for (4) Eme 1 Cosder he ework descrbed by he sysem (4) wh = L1 L b Furher choose m{1 } f s foows for =1 f1 h L1 1 f f h L 1 These rmeers of he ework ssfes coos of Theorem 1

11 401 Pdmvh e L L b b 1 If 1/ b L b1 L 1 1 K d 1 1 m c K1 d 1 d 1 1 m b L b1 L 1 1 K d 1 1 m c K1 d 1 d 1 1 m Thus coos of heorem re ssfed coos of Theorem 3 re If 1 A b L L b b 1 1 B b L L b 1 L b 1 b L b L b 1 L b 1 1 A B1=1050 A B =1050 Therefore coos of Theorem d 3 re ssfed hus he eubrum of he ework s symocy sbe d L-sbe If 5 coos of heorem re o ssfed d f 15 The coos of heorem3 re o ssfed So ys mor roe whe ssfyg he coos of heorems d 3 resecvey Cosder he mode (1) wh vryg us he he mode become d I () b f 1 1 b f ( ) ( ) ( ) (9) b f I where I 1 Due o he fc h he u fucos re me vryg (o oger coss) he mode (9) co hve re secfed eubrum ers I he cos of eur eworks wh omzo robems se d ouu covergece of he ework s bsc cosr some of he resos o cosder se d ouu covergece of he NNs wh me-vryg us re dscussed ( [5][18]) os of he reserch o hese modes (9) s focused o he ouu covergece yss d s sed h sudyg he se covergece of NNs wh me vryg us s mode (9) geer s dffcu robem ([11][18][4][5]) I our vesgo we sudy se covergece d rovde yss resrced segs (A1-A3) We ssume h (A1) he fucos f 1 re goby Lschz couous moooe oe decresg cvo fucos d h s here es L 0 such h f ( ) ( ) 0 f L (30) for y R d Ad so ssume for some 0 f( ) R 1 (A) I ( ) re ocy Lschz couous h s f for every u R here es eghborhoods U such h I u resrced o U u resecvey re Lschz couous (A3) I () ssfes he coos m I ( ) I (31) where I re some coss Th s m I( ) I Theorem 4 Assume h (A1) (A) d (A3) re ssfed d here ess cos vecor R such h (1 ) B f ( ) I 0 (3)

12 Iu-Ouu Sby Of Neur Nework Wh Hyseress 40 The gve y 0 R he sysem (9) hs uue ( ; ) defed o [0 ) souo 0 Proof of Theorem 4 s dscussed [5] Now we ob suffce coos for he eubrum er o be goby eoey sbe The eubrum er of (9) s sd o be goby eoey sbe f here es coss 0 d 1 such h () e for y 0 We deoe su ( ) where 0 eubrum of he sysem (9) he uue Theorem 5 Assume h he coos (A1) (A) d (A3) re ssfed d furher suose h here es osve cos such h 1m L 1 b 1 1 (33) 1 L b 1 The he eubrum w of sysem (9) s goby eoey sbe Proof of Theorem 5 s dscussed [5] IV CONCLUSION AND REARKS I he rese vesgo he uhors hve cosdered css of couous- me hyserec euro mode Sby yss s much desred for hese sysems from he o of vew of he re word ure We hve obed suffce coos for Iu-ouu sby of uue eubrum We hve obed symoc sby of he souos of hs sysem The resus re ec he sese h he crer obed re esy verfbe s hey re eressed erms of he rmeers of he sysem These modes c be ed o vrey of re me cos such s he hgher order hrdwre coro sysems c be reced by hs eur ework for reducg he comey hese eur eworks c be desred d red o fer ou vryg eves of ose erferece he che d rovde ecee d secury d hese eur eworks c so be emoyed for mge erco vryg ose erferg ches I order o rovde d secury messge (usuy referred o s he e) w be rsformed by he seder o rdom ookg messge (usuy referred o s he chere) by usg reversbe mg d rsmed o he recever However durg he rsmsso of he chere osy che he chere ges ered dsowg he egme recever o correcy ge bck he e To ddress hs robem he cheres(ose-free) c be sored s sbe ses of our ework so h wheever osy chere s u o he ework coverges fer fe umber of eros o oe of he sbe ses(he oe o whch s Hmmg dsce s he mmum) whch w resu he correc e fer decryo The messge o be rsmed w be sored s bry mge [3] Ths mge w he be ecryed usg CDA sredg echue where PN-seueces w be geered usg LFSR whose coeco oyom s rmve Nose of cer eve w be dded o he ecryed mge d rsmed o he recever The recever woud he u hs osy er o he ework (The ework woud sore he ecryed mges of he messges h w be eveuy rsmed) The er ouu by he ework w he be decryed usg CDA desredg echue ACKNOWLEDGENT The uhors ckowedge he fc suor of hs work from he DST WOS-A of Id (Gr Sco No100/ (IFD)/139/01-13 ded ) Pry suored by CS-DSTGOI roec LrNo SR/S4/S: 516/07 ded104008ad so hks o NVNredr SRrhm d he oymous revewers for her vube suggesos REFERENCES [1] Aro Sruss Luov Fucos d L Souos of Dffere Euos Source: Trscos of he Amerc hemc Socey Vo 119 No 1 (Ju 1965) Pubshed by: Amerc hemc Socey [] CFeg d RPmodo Sby yss of bdreco ssocve memory eworks wh me deys IEEE Trscos o Neur Neworks November 003 Voume- 14 Number [3] D Gds P Thgve & A K Ngr (011): Nose remov usg hyserec Hofed ueg ework messge rsmsso sysems Iero Jour of Comuer hemcs 88: [4] GPdmvh Kumr PVS d Bhu S (013) Gob eoe sby resu for come vued hyserec euro mode I J che Iegece d Sesory Sg Processg Vo 1 No [5] GPdmvhPVSv kumreoe Sby of Hyseress Neur Neworks wh Vryg Ius /1/$3100_c 01 IEEE [6] GW Hoffm A eur ework mode bsed o he ogy wh he mmue sysem JTheore Bo 1 (1986) [7] HY YSwd Assocve memory ework comosed of euros wh hyserec roery Neur Neworks 3(1990) 3-8 [8] Hrsch W (1989) Coverge cvo dymcs couous me eworks Neur Neworks [9] JPSegudo d ODrez Dymc d sc hyseress cryfsh receors BoCybervo

13 403 Pdmvh e [10] KGosmy d Pgzhou Lu Dymcs of hyserec euro oer yss 8(007) [11] LuDHuS& WgJ (004) Gob ouu covergece of css of couous-me recurre eur eworks wh me-vryg hreshods IEEE Trscos o Crcus d Sysems-II: Eress Brefs [1] ossheb S The crce crero d he LP sby of feedbck sysems SIA J Cor Om Vo 0 No [13] P Gurf Quve L sby yss of css of er me vryg feedbck sysems I J A h Comu Sc 003 Vo 13 No [14] Qg Zhg Xoeg we d J Xu A ove gob eoe sby resu for dscree-me ceur eur eworks wh vrbe deys Iero our of eur sysemsvo16o6(006) [15] RLDddy"A mode of vsuomoor mechsms he frog oc ecum hbosc vo [16] Sdberg IW A freuecy dom coo for he sby of feedbck sysems cog sge me vryg oer eeme Be Sys Tech J 1964 Vo [17] Sdberg IW Some resus o he heory of hysc sysems govered by oer fuco euos Be Sys Tech J 1965 Vo 44 No [51] [18] Sug Hu LuD o he gob ouu covergece of css of recurre eur eworks wh me-vryg us Neur Neworks 18 (005) [19] Su Bhrkv d Jerry ede The hyserec hofed eur ework IEEE Trsco s o eur eworks Vo 11 o 4 Juy 000 [0] Vdysgr Noer Sysems Ayss d Ed1993 New Jersey: Uer Sdde Rver [1] YPSgh The symoc behvor of souos of er hrd order dffere euos ProcAmerhSoc 0(1969) [] YTkefu d KCLee A hyseress bry euro: A mode suressg he oscory behvor of eur dymcs BoCyber vo [3] Zmes G (1990): O u-ouu sby of me-vryg oer feedbck sysems-pr II: Coos vovg crces he freuecy e d secor oeres IEEE Trs Auom Cor Vo AC 11 No [7] [4] Zhg Y J Cheg Lv d Le Zhg Ouu covergece yss for css of deyed recurre eur eworks wh me-vryg us IEEE Trscos o Sysems d Cyberecs-PrB: Cyberecs Vo 36 No1 Februry 006 [5] Zheyu Guo Lhog Hug Gob ouu covergece of css of recurre deyed eur eworks wh dscouous euro cvos Neur Process Le (009) 30:13-7 DOI /s z