Dynamic stability of Time-Delayed Feedback Control System by FFT based IHB Method

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1 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee ynm ly of Tme-elye eek onol Syem y T e IHB eo R ITRA epmen of enl Engneeng Nonl Ine of Tenology gp eeme Unvey e Bengl Pn-739 INIA kmme@gmlom p://wwwngpn/fly_elpp?= A BANI epmen of vl Engneeng Nonl Ine of Tenology gp eeme Unvey e Bengl Pn-739 INIA knk@gmlom p://wwwngpn/fly_elpp?= S HATTERJEE epmen of enl Engneeng Bengl Engneeng n Sene Unvey Sp e Bengl Pn-73 INIA y@gmlom p://wwwen/oyml-eee-me-menem A: - Te foe ffng ollo nvege y nenonl me-elye plemen feek y f oe nfom e nemenl mon lne meo long w onnon ene T-IHB T-IHB n effenly evelop feeny epone ve w ll le n nle olon n olon ne Appele eon n pek vle of epone n gl eon n e kew-ne n feeny epone ve oeve w e noon of gn n ely e feeny epone ve w ll le olon n e eve w ppope oe of gn n ely n e pmy n eony ly zone of lne ly nly Te el one y meo e ompe w nmel negon meo n ey m pefely ey-o: - oe nfom; nemenl mon lne meo; e leng onnon; nenonl me-elye feek; loe ly Inoon onollng eonn von of flele mne omponen n l meme lwy een n mpon e of ee fo engnee In e p vo meo nlyl em-nlyl n nmel meo wee vlle fo onollng eonn von Tog ve von onol peo o pve onol ene peene of nvole me ely n e feek eoly lm e pefomne of n ve onol yem n n e wo e e yem epone my even eome none T eemely ffl o mee e ee oeve of von onol yem ne e peene of nonollle me-ely Te ommon meml meo vlle fo e nly of e foe l of yem e meo of lple Tme Sle TS n g fow mon lne HB nly B meo wok well only fo wekly non-lne yem n fo me-ely mlle ompe o e nl me peo of von of e yem A ompeenve vey of e een ee on e fel vlle n Olg e l 3 ve evelope n ve von oe e on lne me-elye E-ISSN: Ie 4 Volme 8 Ooe 3

2 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee e feek w ey ve eme e elye eono H e l 4 ve epoe meelye e feek onol of e pmy n /3 -mon eonne of foe ffng ollo Uw e l 5 o 7 ve nvege e pplon of me-elye veloy feek fo onollng e von of l yem ne em long 8 n Ay 9 ve e von onol of elf-ee yem Te pmy eonne of nleve em ne e me-elye feek onol een e y eee ve e me-elye feek onol of vo fon-ne nle Von onol y eve me-elye eleon feek lo een e y eee Rm e l 3 one e egenvle gnmen polem fo lne voy yem ng e feek onol n e peene of me-ely El-Bony n Elkoly 4 peen nlyl n nmel e on e effe of me-elye feek of nonlne SO yem ne eenl n pme eon A e ele peene of nvole me ely n e feek eoly lm e pefomne of n ve onol yem y elzng g onol gn In oe o mven nele on me-ely nenonlly noe n e feek p wee e gn n e me-ely e o onollle e lee vey evel e on e effe of nenonl me-elye feek fo l of eln nonlne yem e e o e e of e o knowlege n nfomon Te peen y eefoe move y e nee fo ee em-nlyl peon of omple peo v f oe nfom e nemenl mon lne meo long w onnon ene IHB-T Sne e yem o e feek onol lw w wll e evelope n noe n e feek p ongly non-lne e effeny of e meo o y e epone n ly of yem empe Te ly of e peo olon emne y loe eoy Te el ow e epone ve n e ppee o e ee level w ppope oe of ely n gn pmee oe Tnfom Be Inemenl Hmon Blne eo fo Tme-elye eek Syem oe ezon one e e of non-lne ony efeenl eon fo ml egee of feeom ynml yem w me-ely of e followng genel fom f e peo onon n In veo eon nly e nknown epone of e non-lne yem o n genel e epenen vle veo e me-elye fnon e non-menonl eon feeny e me-ely n f e eenl mon eon mple Ove o enoe evve w epe o e nonmenonl me n e nege oe of e -mon epone eng onee Te f ep of e IHB meo e Newon Rpon eve poee To on peo olon of E one nee o ge olon e egnnng of e poee w my e ken e olon of lne yem A negong olon n e epee y ng e oeponng nemen ymolze y o em follow 3 n f f f T e lef e vle of E 3 n e ege e negong e wle on e g e e m of known e n e nemen Sng E 3 no E one on e followng nemenl eon: 4 f f Now epnng E 4 y Tylo ee o e nl e p o e f oe e lneze nemenl eon e one 5 f f Hee e ee o oeve em w wll vn wen e olon e Now eple 6 n f ee E 5 ke e fom E-ISSN: Ie 4 Volme 8 Ooe 3

3 f 7 Te em n n n genel e me vyng n e lle e evlen nemenl m mpng ffne ely feeny n eon epevely E 5 o 7 epeen e of lne eon oe vle oeffen ony ffeenl eon OE Te eon ep of e IHB meo o epn e genelze oone n e oeponng nemen no oe ee fo peo epone n o 8 A Y YA 8 wee enoe ny pove nege n o o o Y n n n 8 O Y 8 wee o n n A 8e n A T 8f Hee e ppe p ymol T enoe e npoe of m Tee olon nle ll mon p o en oe n n ve mon oe om now on me n ppomely known olon n o e fon new olon Teefoe ll fnon of me n 6 e known fo gven n ey n e oele y epnng ng oe ee follow: n o 9 n o 9 n n o 9 Now e oe oeffen of ee fnon n e lle mo effenly y pplyng T f oe nfom lgom oneng e me-elye plemen of e fom e oe ee of n e epee : - ω - ω n o Sng E 8 9 n no E 5 we on e followng lne m eon fo e nknown nemen } { A } { } { } { A Y Φ } { } { } { f ee / T n / T Le efne J e Jon gen o ngenl m w epe o } { A n w e me n e efne y A Y A Y A Y n A Y SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee E-ISSN: Ie 4 Volme 8 Ooe 3

4 Evlon of Jon m Te elemen of e Jon m n E e effevely lle ng T meo Hee we wll evle only n el Te el fo n e pple ely fom Leng n 995 5Te elemen of n em of oe omponen of e epee 3 om E n o n o 4 wee n e e oe oeffen of Eng f e oeffen of on o e of E 4 we on 5 T mple n 4 Ne evle e oeffen of on o e of E 4 fo en n o 6 o n 6 Ne we pply Glekn' meo o e mon lne meo o evle e oe oeffen lplyng o e of e ove eon y n epevely n negng ove me fom zeo o n en ee o zeo we ve epevely o n 7 n / o / n 8 Ue of e followng genel gonome elon n oogonly elon eween ne n one fnon e elpfl o mplfy E 7 n 8 Hee n e zeo o nege n ll fo 9 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee E-ISSN: Ie 4 Volme 8 Ooe 3

5 fo fo 9e fo fo fo π 9f Teefoe fo e one em we ve o π / n A ml epeon n e one fo ne em n no own o mnn evy nlly fom E 7 n 8 we on fo n o o fo gn o gn o ll fo n Ne we ee e oeffen of on o e of E 4 fo en ωω ωω n o o n Applyng e mon lne meo ove we on n o 3 gn o n 3 n o - - gn n 3 Te elemen of n em of oe omponen of e epee 4 wee fo gn 5 5e gn 5f n fo 5g Te elemen of n em of oe omponen of e epee SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee E-ISSN: Ie 4 Volme 8 Ooe 3

6 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee 6 wee 7 fo 7 gn fo 7 - fo ll 7 7e 7f n gn fo 7g Te elemen of n em of oe omponen of e epee 8 ee 4 gn fo gn - n fo Ne {} {} n {Φ} n E e lle ng T lgom n olve fo e nknown oeffen veo A nlly e m E olve e me ep ng e Newon-Rpon eve poe 3 P followng n pme onnon Te T-IHB meo w vle pmee elly e o pme onnon fo onng e epone gm of nonlne yem Afe onng e olon fo e pl vle of pmee e olon fo new pmee lgly pee fom e ol one n e one y eon ng e pevo olon n ppomon Te mn m of e p followng n pme onnon o effevely e e fon eene pmee of e yem ve In y n leng poee 6 ope fo e pme onnon Inong e p pmee e gmenng eon fo genel yem n e wen : X 9 T wee{ X} { A} T A goo oe of e T fnon {g X} g X { X} {X} oneng e nemen n {A} n e nemenl eon one { A} 3 T A Togee w we on e gmene nemene nemenl eon { Y } { A} { } { Φ} { } f { Φ} { } 3 J T A Φ J X { X} A wee J efoe n J X e Jon m w mofe w epe o {X} oneng e poon of e elm p of e olon n e gmenng eon n e wen T X { X' } { X X} 33 Te f peon of e new pon {X n } of e olon long e elm p gven n em of e wo pevo pon {X } n {X } follow: { Xn} { X} Δγ{ X} n 34 { X} { X X}/ X X wee n y ep leng ken n e ompon y epeene g A poon of e elm p E-ISSN: Ie 4 Volme 8 Ooe 3

7 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee 4 Sly nly of peo olon en e ey-e olon fo me-ely yem ompe y ng e T-IHB meo e ly of e peo o lmo peo olon eke y men of loe eoy fo wo mpon eon le ne n e nge fom nle one n fon pon n e loe y monong egenvle of e monoomy m In ppe loe eoy mofe o nlye e ly of e peo olon of me-elye plemen feek yem T one y peng e e vle o e ey-e olon w el n yem of lneze eon w peolly vyng oeffen Te pee eon of moon lwy onomo en e olon pee y { } e nemenl lne m ony ffeenl eon { } { } { } { } 35 In e pe fom E 3 n e wen { } { } 36 { } { } { z } B { z} 37 ee e non m B peo w ne peo T e B=B+T Te ly of E 3 eke y evlng e egenvle of e monoomy m Z w nfom e e veo {z n } =nt o {z n+ } =n + T If e ole mgne of e egenvle e le n ny e olon le If le one of e egenvle mgne gee n one en e peo olon nle Te wy e egenvle leve e n le eemne e ne of fon Te epl fom of Z n e wen 7 n Z ep B 38 wee T / N n N epeen e nme of von e o ve one peo T Te effen nmel evlon Z eve y mkng e of e efnon of e m eponenl n B B ep B I B 39! n! wee I e eny m o mll me nevl e ee n eon onvege ply n e vle of e m eponenl n e ely ppome y fne nme of em 3 ffng Ollo w Tme- elye eek Te ffng ollo ne mono-mon eon nlye w me-elye plemen feek onol Te eon of moon of e yem wen 3 f g 4 wee e non-menonl plemen epone fnon of non-menonl me Hee non-menonlze w epe o e nl me peo of e ollo o enoe evve w epe o ω e nonmenonl eon feeny Te fong fnon ken f o 4 δ n β enoe epevely e mpng non-lne ffne n pmee e onol gnl n memlly epee 4 g n e oeponng gn n ely pmee g Tme-elye feek onol eme of SO menl ollo 4 onollng Lne Von 4 eml moel Te non-menonl eon of moon of e ove nmpe ollo w e popoe onol epee f g 43 Te nfe fnon T of e lne yem govene y E38 gven y X T 44 ge ee lle omple feeny vle I ol e noe fo g= E43 ee o e nfe fnon of e nonolle yem 4 Sly nly Te ly of e vl elm of e yem ene y e oo of e e eon gven elow: E-ISSN: Ie 4 Volme 8 Ooe 3

8 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee ge 45 Sng n ny el nme n E45 n epng e mgny n el p yel g o 46 n g n 46 Sne g n' e zeo n 47 Te genel olon of E46 n n 48 Sng E48 no 46 yel e followng eon fo e l ly lne: n g o n 49 A e nonolle yem mgnlly le g lo l ly lne nlly e le egon n e g v plne e one fom e gn of e oo eneny efne elow R gn Re 5 Inee n e eee n e R nme of nle nme of nle oo oo + - g3 ep e egon of ly n e g v plne Bee e ly egon n e lowe nge of ely of pl mpone egon ene fo lle e pmy ly zone n e gy e egon e lle eony ly zone In een nmel eploon pmee vle e oen fom e pmy well eony ly zone Te negve gn of e onol gn n e eony ly zone ele e f e onol yem le ne negve feek onol g 3 Sly egon ne me-elye plemen feek Se egon e le Blk egon e pmy ly zone Gy egon e eony ly zone Sol lne epeen e l ly lne fo ffeen n on w ome of e e oo e pely mgny 5 Nmel on Te olon one y T-IHB meo fo ffng ollo ve een peene n ompe g 4 ow e ompon of olon one y T-IHB pe nemen PI 8 n nmel negon NI meo Te oe oeffen of mpe ffng ollo e oe fom Leng n ng Te oe oeffen one ome pef feene y PI meo epoe n Tle- Te oe oeffen one y T-IHB meo fo ome feeny e le n Tle- I ol e noe T-IHB ple of nopong lge nme of oe oeffen fo onng moe e olon f neey In e le only f oeffen e gven Te mple ve feeny plo of e foe ffng ollo y T-IHB NI n PI meo e gven n g 4 By PI meo e feeny mple plo fo foe ffng ollo een one n e nge of 9-3 / Te feeny epone plo fo e me ollo ng T-IHB n NI meo n e peen y one n e nge of 3- / I oeve fom e plo e T-IHB meo e ll ne of pole le n nle olon A le n ne / n onne p o n mple vle of 67 m feeny / om pon le peo olon mp o pek epone vle of 94 m E-ISSN: Ie 4 Volme 8 Ooe 3

9 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee og n nle n of peo olon T mp penomenon ypl e of non-lne yem Te mple of epone en ee ply n ome own o e mple of m feeny 9 / I oeve e omplee eonne ve w le-nlele ne evelope effenly y T- IHB meo I een fom e fge e ee pon one y PI meo n e nge of 9 o 3 / me well w e eonne ve one y e peen meo T-IHB n NI A e ele NI meo n pove nle olon Alo n Ref 3 e feeny epone ve evelope only fo p of e feeny nge w no nfomon o e ly of e olon one Te foe ffng ollo lo nvege fo von onol e o ffeen vle of feek gn n me-ely y o T-IHB n NI meo g 5 epeen feeny epone gm fo ffeen vle of gn n ely n e pmy ly zone g 3 g 5 epeen feeny epone ve of e nonolle yem e g= n = g 5 o 5 ow e feeny epone ve fo onn gn of n vyng ely I oeve ee onno eon n pek vle of mple e ely nee Te pek epone vle een ee fom 7 m o 34 m fo e vle of gn n ely w popoone eon of le n nle ne of olon I my e oeve e o e noon of gn n ely e feeny epone ve f o e lef nng n lenon n e nl feeny of e yem n n ll e e feeny epone ve e oe w e mp penomen In g 5e e gn vle een nee o 5 w ely vle of 5 A el e pek epone ppee ppely o vle of 37 m w ll le olon n e mp penomenon vne olly Ne n g 5f o 5 gn vle fe 3 n ely vle e glly nee Hee lo ppele eon n pek epone een eve Te nle egon e lo ee A fe nee of gn o vle of 5 n ely of n g 5 ppee e pek epone lo o vle of 7m w ll le ne of olon Ne n g 6 e negve gn -5 n g vle of ely 475 vle e gnfnly elee fom e eony ly egon of g 3 I een e of gn n ely el n epone ve w ll le olon o ll e feeny epone ve fo ffeen gn n ely vle ee gl eon n pek vle of epone n kewne of e ve In e of plemen feek Seleon of gn n ely vle fom e oe wo egon n e eony ly zone e olon eome none w mo nle olon So n e of plemen feek pope eleon of gn n ely n e pmy n eony zone n ppe pek epone o ny ee vle Tle oe oeffen one y Pe Inemen meo ω Tle oe oeffen one y T-IHB meo ω E-ISSN: Ie 4 Volme 8 Ooe 3

10 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee g 4 eeny epone ve of nonolle yem δ=4 β=5 = g= = E-ISSN: Ie 4 Volme 8 Ooe 3

11 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee g 6 eeny epone gm fo gn negve n ely n e eony ly zone δ= β =5 = 6 onlon g 5 eeny epone gm fo ffeen gn pove n ely n e eony ly zone δ= β=5 = Te ffng ollo ne monomon eon nvege fo ee ppeon of pek epone y nenonl me-elye plemen feek Te followng onlon e wn: Te ompon of Jon m n ene peo olon gly effen n fe n ompon o mple IHB meo A lge nme of mon n e nopoe n T-IHB f neey; wee mple IHB meo enone ffly w lge nme of mon Te olon one y NI me pefely w le olon one y T-IHB meo T-IHB n onvenenly nle ny ype of nonlney wee nonlney fe neey nfomon menle o IHB meo n only e nle y mple IHB meo Te omplee feeny epone ve w ll pole le n nle olon n olon ne n e vey effenly evelope y T-IHB meo n mp e n e oeve Te noon of gn ely n e foe ffng ollo el n ppele eon n e pek vle of epone E-ISSN: Ie 4 Volme 8 Ooe 3

12 SEAS TRANSATIONS on APPLIE n THEORETIAL EHANIS R A Bnk S eee Refeene: o g=3 n =4 ll olon eome le n mp penomen no longe oeve I oeve ppope eleon of gn n ely pmee n nenonl me-elye feek gnfnly nge e eonne ve n ly of olon n omeme n ee ppeon of von G n S Nle Svey on Reen Rel n e Sly n onol of Tmeely Syem ASE Jonl of ynm Syem eemen n onol N Olg BT Holm-Hnen A novel ve von opon ene: elye eono Jonl of Son n Von N Jll N Olg lple enl elyeeono von oe fo ml-egee-of feeom menl e Jonl of ynm Syem eemen n onol ASE H H EH owell LN Vgn Reonne of monlly foe ffng ollo w me-ely e feek Nonlne ynm E Uw H von Bemen R m Hoen Tme-elye onol of e Eke Engneeng n Sl ynm E Uw H von Bemen P Poom Tme-elye onol egn fo ve onol of e: pnple n pplon Sl onol Hel onong P Poom E Uw H von Bemen Tme-elye pove veloy feek onol egn fo ve onol of e Jonl of Engneeng en ASE A Von onol fo e pmy eonne of nleve em y me-ely e feek Jonl of Son n Von S eee Tme-elye feek onol of fon ne nle Inenonl Jonl of Non-Lne en S eee Von onol y eve me-elye eleon feek Jonl of Son n Von Y Rm A Sng n J E oee Se feek onol w me-ely enl Syem n Sgnl Poeng A El-Bony n S El-koly Reonne of non-lne SO yem w me-ely n lne feek onol Py Sp 8 9 no 5 A Y T Leng n S Non-lne von of ople ffng ollo y n mpove nemenl mon lne meo Jonl of on n von A Bnk n T Sly Anly of Two-Pon oong Syem n Sge Ollon ASE Jonl of omponl n Nonlne ynm Vol P emnn E Hmmon n T H oo Effen Nmel Temen of Peo Syem Applon o Sly Polem Inenonl Jonl of Nmel eo n Engneeng A Y T Leng n T ng Pe nemen nly of mpe ffng ollo Inenl Jonl of nmel meo n engneeng Vol A Von onol fo pmelly ee Le n yem Inenonl Jonl of Non-Lne en Ay Vn e Pol ollo ne elye feek Jonl of Son n Von E-ISSN: Ie 4 Volme 8 Ooe 3