Time and Frequency Domain Analysis of the Linear Fractional-order Systems

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1 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology Te ad Frequecy Doa Aaly of he Lear Fracoal-order Sye Maha K.Bhole Irueao Dep. Bhara Vdyapeeh College of Egg. Nav Muba, Ida Mueh D. Pal Reearch Scholar, IDP Sye & Corol Egg. Ida Iue of Techology Bobay, Muba, Ida Vhweh A. Vyawahare Reearch Scholar, IDP Sye & Corol Egg. Ida Iue of Techology Bobay, Muba, Ida Abrac Rece year have ee a reedou upurge he area relaed o he ue of Fracoal-order (FO) dffereal equao odelg ad corol. FO dffereal equao are foud o provde a ore realc, fahful, ad copac repreeao of ay real world, aural ad aade ye. FO coroller, o he oher had, have bee able o acheve a beer cloed-loop perforace ad robue, ha her eger-order couerpar. I h paper, we provde a yeac ad rgorou e ad frequecy doa aaly of lear FO ye. Varou cocep le ably, ep repoe, frequecy repoe are dcued deal for a varey of lear FO ye. We alo gve he ae pace repreeao for hee ye ad coe o he corollably ad obervably. The exerce preeed here covey he fac ha he e ad frequecy doa aaly of FO lear ye are very lar o ha of he eger-order lear ye. Keyword- Fracoal-order ye, fracoal calculu, ably aaly. I. INTRODUCTION The aheacal odelg of FO ye ad procee, baed o he decrpo of her propere er of Frac- oal Dervave (FD, lead o dffereal equao of - volvg FD ha u be aalyzed. Thee are geerally ered a Fracoal Dffereal Equao (FDE. The advaage of fracoal calculu have bee decrbed ad poed ou he la few decade by ay auhor [], [], [3], [8], [9], [4]. The lae ad very exhauve leraure urvey abou he FC ad FO ye gve [7]. I ha bee how ha he FO odel of real ye (epecally drbued paraeer ype ad eory ype) are ore adequae ha he uually ued I e g erorder (IO) odel. Fracoal Dervave (FD provde a excelle rue for he decrpo of eory ad heredary propere of varou aeral ad procee. Th he a advaage over he IO odel, whch poe led eory. The advaage of FD becoe appare applcao cludg odelg of dapg behavour of vco-elac aeral, cell dffuo procee [8], rao of gal hrough rog agec feld, odelg echacal ad elecrcal propere of real aeral, a well a he decrpo of rheologcal propere of roc, ad ay oher feld [5]. I feedbac corol, by roducg proporoal, egral ad dervave corol aco of he for α, / α, αr +, we ca acheve ore robu ad flexble deg ehod o afy he corolled ye pecfcao. Sude have how ha a FO coroller ca provde beer perforace ha eger order (IO) coroller. The paper orgaed a follow : Seco II ad III gve pecal fuco ad defo of fracoal calculu heory repecvely. Seco IV defe lear FO ye geeral. Seco V decrbe he ably aaly of fracoal-order ye, Seco VI expla he repreeao of fracoal-order ye ad Seco VII aalycal reul of FO ye are gve wh he cocluo Seco VIII. II. SPECIAL FUNCTIONS OF FR ACTION AL CALCULUS (FC) Soe pecal fuco eed o be ued Fracoal Calculu (FC). Thee fuco play pora role he heory of FC ad he heory of fracoal dffereal equao (FDE. A. Gaa Fuco Oe of he o bac fuco of FC Euler gaa fuco Γ(z), whch geeralze he facoral fuco z! ad allow z o ae alo o-eger ad eve coplex value []. The gaa fuco (Γ(z)) gve by he followg expreo, u z ( z) e u du. () P a g e

2 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology Noe ha whe zz + we have Γ(z + ) = z! B. Mag-Leffler Fuco The expoeal fuco e z play a very pora role he heory of eger order dffereal equao. I paraeer geeralzao fuco for a coplex uber z gve by [], E (z) z ( ), () The paraeer fuco of he ML fuco, whch alo pora FC defed a, E, ( z) ( ) z, (, ). (3) Th bac defo very ueful dervg he repoe of a FO ye o ay forcg fuco, for exaple, ep repoe, rap repoe. III. DEFINITIONS FOR FR AC TIO N AL-DIFFERINTEGRALS The hree equvale defo [6],[] o frequely ued for he geeral fracoal dervave (FD) are he Gruwald-Leov (GL) defo, he Rea-Louvlle ad he Capuo defo []. I all he defo below, he fuco f ( aued o be uffcely ooh ad locally egrable. ) The Gruwald-Leov defo of fracoal-order ug Podluby led eory prcple [4] gve by l h [( a) / h] j a D f ( h ( ) C j f ( jh), (4) j where [.] ea he eger par ad coeffce. C j he boal ) The Rea-Louvlle defo obaed ug he Rea-Louvlle egral gve a, d f ( ) D f ( d, ( ) d ( ) (5) a a for ( < α < ) ad Γ(.) he Gaa fuco. 3) The Capuo defo ca be wre a, a D a f ( ) f ( d, ( ) ( ) for ( < r < ), where f ( (6) he h order dervave of he fuco f (. Sce we deal wh caual ye he corol heory, he lower l fxed a a ad for he brevy wll o be how h paper. We ee ha he Capuo defo ore rercve ha he RL. Neverhele, preferred by egeer ad phyc becaue he FDE wh Capuo dervave have he ae al codo a ha for he eger-order dffereal equao. Noe ha he FD calculaed ug hee hree defo cocde for a ally relaxed fuco ( f ( ). IV. LINEAR FRACTIONAL-ORDER SYSTEMS A geeral lear FO ye gve by he FO rafer fuco a : Y ( G( U ( b ( ) a ( ), (7) where a,, Y( ad U( are he Laplace rafor of he oupu y( ad pu u( repecvely. I ca be repreeed by he bloc dagra a how Fg.(). Fgure. Bloc dagra repreeao of lear FO ye. Fg.() repree he geeral bloc dagra of a cloedloop FO ye wh Y( ad U( are he Laplace rafor of he oupu y( ad pu u( repecvely, he ga, G ( he ye rafer fuco, ad H ( he feedbac copoe. Y( ad U( are o uual polyoal bu are peudo-polyoal wh fracoal-order. I h wor we have ae uy feedbac for all exaple. Fgure. Bloc dagra repreeao of cloed-loop lear FO ye. V. STABILITY OF FRACTIONAL-ORDER SYSTEMS The ably aaly pora corol heory. Recely, here ha bee oe advace corol heory of fracoal dffereal ye for ably. I he FO ye he delay dffereal equao order o-eger whch P a g e

3 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology ae dffcul o evaluae he ably by fdg roo or by ug oher algebrac ehod. The ably of FO ye ug polyoal crera (e.g Rouh or Jury ype) o poble due o he fracoal power. A geeralzao of he Rouh-Hurwz crero ued for ably aaly for fracoal-order ye preeed []. However, h ehod very coplcaed. T h e g e o e r c e h o d u c h a Nyqu ype ca be ued for he ably chec he BIBO ee (bouded-pu bouded-oupu. Roo locu aoher geoerc ehod ha ca be ued for aaly for FO ye [], [4]. Alo, for lear fracoal dffereal ye of fe deo ae-pace for, ably ca be vegaed. The ably of a lear fracoal dffereal equao eher by raforg he -plae o he F -plae ( F ) v he w -plae ( w ) or o, explaed [3]. The robu ably aaly of a Fracoal-order Ierval Polyoal (FOIP) faly preeed [5] ad [6]. A. Sably ug Rea urface The udy of he ably of FO ye ca be carred ou by obag he oluo of he dffereal equao ha characerze he. To carry ou h udy eceary o reeber ha a fuco of he ype a a... a, (8) wh R,,, a ul-valued fuco of he coplex varable whoe doa ca be ee a a Rea urface of a uber of hee. The prcpal hee defed by arg( ). I he cae of Q, ha, / v, v beg a pove eger, he v hee of he Rea urface are deered by,,,..., v. j e, ( ) ( 3), (9) where he prcpal R e a h e e. T hee hee are rafored o aoher plae called w -plae wh he relao w. The rego of hee hee o he w -plae ca be defed by : j w we, ( ) ( 3), () Thu, a equao of he ype (8) whch geeral o a polyoal, wll have a fe uber of roo, aog whch oly a fe uber of roo wll be o he prcpal hee of he Rea urface. The roo whch are he ecodary hee of he Rea urface are o-dapg ad oly he roo ha are he prcpal hee of he Rea urface are repoble for a dffere dyac: daped ocllao, ocllao of coa aplude, ocllao of creag aplude. For he cae of ye, whoe characerc equao a polyoal of he coplex varable w he ably codo expreed a [6], arg( w ), () where w are he roo of he characerc polyoal w. For he parcular cae of he well ow ably codo for lear e-vara ye of egerorder recovered: arg( w ). () B. Frequecy Repoe - Bode Plo I geeral, he frequecy repoe ha o be obaed by he evaluao of he rraoal-order rafer fuco of he FO ye alog he agary ax for j, (, ) [6]. The frequecy repoe ca be obaed by he addo of he dvdual corbuo of he er of order reulg, P( G( Q( ( ( z ) ), (3) where z ad are he zero ad pole repecvely. For each of hee er he agude plo wll have a lope whch ar a zero ad ed o db/decade, ad he phae plo wll go fro o /. VI. REPRESENTATION OF FRACTIONAL-ORDER SYSTEMS A. Laplace Trafor I ye heory, he aaly of dyacal behavor of- e ade by ea of rafer fuco. Hece roduco of he Laplace rafor (LT) of fracoal-order dervave eceary for he udy. Foruaely, LT for eger-order ye ca be very ealy appled a a effecve ool eve for fracoal ye []. Ivere Laplace raforao (ILT) alo ueful for e doa repreeao of ye for whch oly he frequecy repoe ow. The o geeral forula aug zero al codo he followg: d f ( L d L f (. (4) Th very ueful order o calculae he vere Laplace rafor of eleeary rafer fuco, uch a o eger order egraor /. 3 P a g e

4 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology B. Sae-pace Repreeao For lear fracoal dffereal ye of fe deo ae-pace for, ably vegaed [6]. Coder he coeurae-order TF defed by (7), aocaed wh h TF, caocal ae-pace repreeao ca be propoed, whch are lar o he clacal oe developed for IO dffereal equao ye. Corollable Caocal For : Defg he fr ae er of Laplace rafor a, X ( U (, (5) a ( ) ad he reag elee of he ae vecor a recurve way fro h oe a x D x,,,..,, (6) he ae repreeao, expreed he corollable caocal for, gve by [6], A a y b where a D b a x Ax Bu, D x D x D x., D x D x, B, b b a where (7) a a x x x b, f or, C B AB A B A B (8) o Corollably crero ha he ye corollable f ad oly f arx C defed by (8), whch called a corollable arx full-ra. Rearragg he above FO ae equao, he obervable caocal for ca be obaed wh he arce A, B ad C arce. The obervably codo alo ae a for eger-order LTI ye. VII. ANALYTICAL RESULTS Soe FO ye are aalyzed h eco. Ther ably, ep repoe, frequecy repoe, ad he SS repreeao dcued. The aaly doe ug MATLAB []. The adard coercally avalable ulao ofware cao be ued for evaluag he ep, rap, frequecy repoe of he FO ye. Recely, MATLAB wo oolboxe dedcaed o FO ye are avalable. They are CRONE [9] ad NINTEGER oolbox [8]. A. Exaple Coder he FO egraor ye wh TF of he for, G(. (9) For he FO egraor f. 5, he coder.5 w, hece G ~ ( w) w The ye wh he above fuco ha oe ope-loop pole a org. The Rea urface of he fuco v w ha wo Rea hee. Now f. 5 ~ G( w) 3 w, ad coder.5 w, he The ye wh he above TF ha hree ope-loop pole a org. Sep Repoe: The ye rafer fuco, Coder ep pu, Y( U( U(,, () Y (. () Tag vere Laplace rafor of he equao we ge (. ( ) y () 4 P a g e

5 The Fg.(3) how he ep repoe of he ye for α =.,.5,.8, ad.5. j Frequecy Repoe: Pu he ye fuco gve by (9) we ca plo he agude ad phae plo. The agude ad phae plo of he ye for α =.,.5,.8, ad ploed a how he Fg.(4). Fro he above repoe we ca coclude ha: ) The agude ha a coa lope of α db/decade. ) The phae plo a horzoal le a απ/. IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology B. Exaple Coder he coeurae ye gve by he followg rafer fuco [6] [7] G (. (3) Fgure 4. Frequecy repoe of Exaple () for dffere value of α Phycal gfca roo are he fr Rea hee, whch expreed by relao / v / v, where arg(w ). I h exaple coplex cojugae roo fr Rea hee are w ,, j arg( w ).66, whch afy codo, arg( w ) / v / a how Pole-zero, plo how Fg.(5). Fgure 3. Sep repoe of Exaple () TABLE I. OPEN LOOP POLES AND CORRESPONDING ARGUMENTS OF EXAMPLE () Argue rada The ye gve he equao ca be wre a G (. (4) Coder w he ye ha Rea hee. ~ G ( w). (5) 9.8w.5w The ope-loop pole ad her approprae argue of he ye are how able I. Fg.(5) gve he pole-zero plo of he ope-loop ye. w, =.997 ± j.8 arg(w, ) = 3.3 w 3,4 =.997 ± j.444 arg(w 3,4 ) =.698 w 5,6 =.7465 ± j.64 arg(w 5,6 ) =.43 w 7,8 =.566 ± j.8633 arg(w 7,8 ) =.5 w 9, =.59 ± j.965 arg(w 9, ) =.834 w, =.54 ± j. arg(w, ) =.595 w 3,4 =.38 ± j.977 arg(w, ) =.65 w 5,6 =.543 ± j.8359 arg(w 5,6 ) =. w 7,8 =.7793 ± j.6795 arg(w 7,8 ) =.77 w 9, =.984 ± j.396 arg(w 9, ) =.4 w, =.45 ± j.684 arg(w, ) =.66 5 P a g e

6 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology Fgure 5. Ope loop pole-zero plo of Exaple () The roo fr Rea hee afy he ably crera, hece he ye able. Oher roo of he ye le ecodary Rea hee. The fr Rea hee rafored fro plae o w - plae a follow: / arg( w ) /, ad arg( w ). ( 6) Therefore fro h coderao agle obaed arg( arg( w). (7) The cloed loop pole are gve able II ad are ploed Fg.(6). TABLE II. CLOSED LOOP POLES AND CORRESPONDING ARGUMENTS OF EXAMPLE () Argue rada w, = -.98± j.3 arg(w, ) = 3.5 w 3,4 =.9557 ± j.4483 arg(w 3,4 ) =.73 w 5,6 =.7764 ± j.6694 arg(w 5,6 ) =.43 w 7,8 =.5776 ± j.8863 arg(w 7,8 ) =.48 w 9, =.768 ± j.9956 arg(w 9, ) =.84 w, =.73 ± j.43 arg(w, ) =.587 w 3,4 =.399 ± j.55 arg(w, ) =.7 w 5,6 =.5488 ± j.8676 arg(w 5,6 ) =.6 w 7,8 =.348 ± j.653 arg(w 7,8 ) =.584 w 9, =.94 ± j.47 arg(w 9, ) =.47 w, =.7989 ± j.6953 arg(w, ) =.76 Fgure 6. Cloed loop pole-zero plo of Exaple () Sep Repoe: The ye TF, Y( U(.8 For ep repoe of he ye, U(.. (8) Calculag he redue ad pole by paral fraco are how able III. TABLE III. RESIDUES AND CORRESPONDING POLES Redue.64 ± j ±j ± j ± j ± j ± j ± j ± j ± j ± j ± j ± j..467 ± j..997 ± j.8.7 ± j ± j ± j ± j ± j ± j ± j ± j.64 Ug vere Laplace rafor [6], L r ( p ) r E, ( p ), (9) 6 P a g e

IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology where E, (.) he Mag Leffler (ML) fuco a defed Seco II, r are he redue ad p are he correpodg pole for o.

7 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology where E, (.) he Mag Leffler (ML) fuco a defed Seco II, r are he redue ad p are he correpodg pole for o. To plo ep repoe we have ued he MATLAB ubroue lf() developed by Podluby []. The ep repoe plo ploed a how Fg.(7). The ep repoe how a uderdaped ye. Th obvou a he wo able pole he prcpal Rea hee are very cloe o he agary ax he -plae. See Fg.(6) for he correpodg w -plae uao. ye obaed a, ((. ) Y( X (.8 (.).5 (.)9. (3). 9.65( ).5) Y(.5X (. (3) Coder pu u( ad ag vere Laplace rafor we ge, D..9 y(.65d y(.5y(.5u(, (3) Cae : Le ( x ( y ad. D x( x( (33) I geeral we have. x D x,,,..,.. D x(.5x (.65x(.5u(, (34) The corollable caocal for herefore gve by, Fgure 7. Sep repoe of Exaple () Frequecy Repoe: Pu j he gve ye fuco. The agude plo ad phae plo of he ye ug MATLAB ploed a how he Fg.(8). The ga arg ad he phae arg abou 77. D D D... x ( x ( x(.5 u(.5.65 x ( x( x( y ( u (. (35) Cae : Le ( x (.9 y ad x x ( D (. (36) The corollable caocal for herefore gve by, D D.9.3 x ( x ( ).5 x(.65 x( Fgure 8. Frequecy repoe of Exaple () Sae-pace Repreeao: The caocal for of he (.5 u. (37) The corollable arx of h ye full ra ad hece he ye corollable. I alo how ha here ca be o uque ae pace repreeao for a fracoal-order ye. I he aaly of h 7 P a g e

8 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology coeurae FO ye we coclude ha he ye able, corollable ad obervable. C. Exaple 3 Coder he coeurae ye gve by he followg rafer fuco [6] : G (. (38).5.5 The ye gve he equao ca be wre a G (. (39) Coder w, he ye ha wo Rea hee. Traforg he ye oo w - p l a e w e g e, ~ G ( w). (4) w w.5 The ope-loop pole ad her approprae argue of he ye are how able I V. TABLE IV. OPEN LOOP POLES AND CORRESPONDING ARGUMENTS OF EXAMPLE 3 Argue rada Fgure. Cloed-loop pole-zero plo of Exaple (3) Sep Repoe: The ep repoe obaed ug vlap ubroue [3] for he cloed-loop ye wh uy ga a how Fg.(). I oberved ha he ML fuco calculao e coug ad ay o gve proper reul all he cae. I uch cae hey ca alo be ploed ug vlap. ubroue (uercal ILT) [], [3]. w, =. ± j.5 arg(w, ) =.4636 The ope-loop pole-zero plo how he Fg.(9). The pole le he uable rego / 4 arg( w ) / 4, ad he fr Rea hee / arg( w ) /. Fgure. Sep repoe of Exaple (3) Fgure 9. Ope-loop pole-zero plo of Exaple (3) The cloed-loop pole ad her approprae argue of he ye are how able V. TABLE V. CLOSED LOOP POLES AND CORRESPONDING ARGUMENTS OF EXAMPLE 3 Argue rada w, =. ± j.8 arg(w, ) =.84 The cloed-loop pole-zero plo how he Fg.(). The pole are he able rego, whch ple ha he cloed-loop ye able. Fgure. Frequecy repoe of Exaple (3) Frequecy Repoe: Pu j he gve ye fuco. The agude plo ad phae plo of he ye ug MATLAB ploed a how he Fg.(). The ga arg ad he phae arg abou 93. Th how ha he ye able wh a wde rage of ga ad phae arg. Sae-pace Repreeao: The caocal for of he ye obaed a, 8 P a g e

9 Y( X (, (4).5.5 Ug he procedure a gve Seco VI, we ge,.5 x ( x ( D u( (4) x(.5 x( y ( u (. (43) IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology Where A,.5 B, C The ye foud o be corollable ad obervable. Fro he ope-loop ad cloed-loop pole-zero plo, ad he ga arg ad phae arg ca be cocluded ha he ye able he cloed-loop cofgurao. D. Exaple 4 Coder he coeurae ye gve by he followg ope loop rafer fuco []..5 G (. (44) The ye gve he equao ca be wre a G (. (45) Coder w, he ye ha wo Rea hee. Traforg he ye oo w - p l a e w e g e, Fgure 3. Ope-loop pole-zero plo of Exaple (4) The cloed-loop pole, zero ad her approprae argue of he ye are how able V II. TABLE VII. CLOSED LOOP POLES AND CORRESPONDING ARGUMENTS OF EXAMPLE 4 Argue rada w =.867 arg(w ) =. w =.83 arg(w ) =. w 3,4 =.995 ± j.99 arg(w 3,4 ) =.3985 Zero Argue rada w 5 =. arg(w 5 ) =. The pole-zero plo of he cloed-loop ye he w - plae a how he Fg.(4). There are pole ad zero he uable rego ad pole he able rego o he ecod Rea hee whch lar o he cae of opeloop ye. ~ w G ( w). (46) 4 3 w 3w w w The ope-loop pole, zero ad her approprae argue of he ye are how able V I. TABLE VI. OPEN LOOP POLES AND CORRESPONDING ARGUMENTS OF EXAMPLE 4 Argue rada w = 3. arg(w ) =. w =. arg(w ) =. w 3,4 =. ± j. arg(w 3,4 ) =.356 Zero Argue rada Fgure 4. Cloed-loop pole-zero plo of Exaple (4) Sep Repoe: The cloed-loop ep repoe obaed ug vlap ubroue [3] for uy ga a how Fg.(5). I how ha he ye uable. w 5 =. arg(w 5 ) =. The ope-loop pole-zero plo of he ye he w - plae a how he Fg.(3). I how he uable rego / 4 arg( w ) / 4, ad he fr Rea hee / arg( w ) /. Alo here are pole ad zero he uable rego ad pole he able rego o he ecod Rea hee. Fgure 5. Sep repoe of Exaple (4) 9 P a g e

10 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology Frequecy Repoe: Pu j he gve ye fuco. The agude plo ad phae plo of he ye ug MATLAB ploed a how he Fg.(6). The ga arg abou 35dB ad phae arg. VIII. CONCLUSION The fracoal-order odel of real ye are ore adequae ha he uually ued eger order odel. A he ae e fracoal-order coroller provde beer perforace coparo o eger order coroller. The o pora feaure uch a ably, corollably, obervably, ably arg of lear fracoal-order ye are uded durg he wor. They are dcued ug Bode dagra, e repoe, ae pace repreeao. The e ad frequecy doa aaly of fracoal-order ye foud o be lar o ha of eger order ye. ACKNOWLEDGMENT Our ha o all hoe who have drecly or drecly helped u copleo of h wor. Specal ha o our faly eber for her uppor ad ecouragee. Fgure 6. Frequecy repoe of Exaple (4) Sae-pace Repreeao: The caocal for of he ye obaed a, Y( X ( , (47) Ug he procedure a gve Seco VI, we ge,.5 D x ( x( x 3( x4( x ( x( x ( ) 3 3 x4( u( ( 48) x ( ( ) ( ) x y u ( x 3( (49) x4( Where A, 3 C, D B, The ye foud o be corollable ad obervable. Fro he ope-loop ad cloed loop pole-zero plo, ep repoe we coclude ha he ye uable. REFERENCES [] T.F. Noeacher, W.G. Glocle, A Fracoal Model for echacal re relaxao, 3 rd ed. Phl Magle: 64():89-93, 99 [] S. Weerlud ad L. Ea, Capacor heory, 3rd.IEEE Tra Delecrc ad Iulao; (5), 86-39, 994. [3] R.L. Bagley, R.A. Calco, Fracoal order ae equao for he corol of vcoelac rucure, 3rd ed. J Gud Corol Dy; 4():34-, 99. [4] I. Podluby, Fracoal Dffereal Equao, Maheac Scece ad Egeerg, 3rd ed. Sa Dego: Acadec Pre, 999. [5] W. LePage, Coplex Varable ad he Laplace Trafor for Egeer, 3rd ed. McGraw-Hll, New Yor, NY, USA, ere: Ieraoal Sere Pue ad Appled Maheac 96. [6] C. Moje, Y. Che, B. Vagre, D. Xue ad V. Felu, Fracoal-order Sye ad Corol Fudaeal ad Applcao, Sprger Lodo Dordrech Hedelberg New Yor,. [7] I. Podluby, Fracoal-order ye ad P I α D β -coroller, IEEE Traaco o Auoac Corol, vol. 44, o., pp. 8-4, 999. [8] R. Hlfer, Ed., Applcao of Fracoal Calculu Phyc, World Scefc, Rver Edge, NJ, USA,. 9] K. B. Oldha ad J. Spaer, The Fracoal Calculu: Theory ad Applcao of Dffereao ad Iegrao o Arbrary Order, Acadec Pre, New Yor, NY, USA, 974. [] R. Capoeo, G. Dogola, L.Forua ad I.Pera, Fracoal Order Sye Modelg ad Corol Applcao, World Scefc Publhg Co. Pe. Ld. vol.7, ere A. [] F. Merrh-Baya ad M. Afhar, Exedg he Roo-Locu Mehod o Fracoal-Order Sye, Joural of Appled Maheac, Hdaw Publhg Corporao vol.8. [] M. Ieda ad S. Taahah, Geeralzao of Rouh algorh ad ably crero for o-eger egral ye, Elecroc ad Coucao Japa vol., o., pp. 4-5, 977. [3] A.G. Radwa, A.M. Sola, A.S. Elwal ad A. Sedee, O he ably of lear ye wh fracoal-order elee, Chao, Solo ad Fracal 4, 3738, 9. [4] T. Machado, J. A., Roo Locu of Fracoal Lear Sye, Coucao Nolear Scece ad Nuercal Sulao () do.6/j.c... P a g e

11 IJACSA Specal Iue o Seleced Paper fro Ieraoal Coferece & Worhop O Eergg Tred I Techology [5] N. Ta, O. F. Ozguve, M. M. Ozye, Robu ably aaly of fracoal order erval polyoal, ISA Traaco 48 (9) [6] H. Kag, H. S. Lee, J. W. Bae, Robu Sably Aaly of Coeurae Fracoal Order Ierval Polyoal, ISECS Ieraoal Colloquu o Copug, Coucao, Corol, ad Maagee,9. [7] J. Machado, V. Kryaova, F. Maard, Rece hory of fracoal calculu, I: Cou Nolear Sc Nuer Sula, Elver,. [8] D. Valro ad S da Coa, Neger: A o-eger corol oolbox for MaLab, I: Fracoal dervave ad applcao, Bordeaux, 4. [9] P. Melchor, B. Oro, O. Lavalle, A. Oualoup, The CRONE oolbox for Malab: fracoal pah plag deg roboc, Laboraore dauoaque e de Producque (LAP),. [] The MahWor Ic. MATLAB Corol Sye Toolbox, Uer Gude,. [] I. Podluby, Mag-Leffler fuco, olehp:// 5. [] H. Shega, Y. Lb, ad Y. Che, Applcao of Nuercal Ivere Laplace Trafor Algorh Fracoa Calculu, Proceedg of FDA. The 4h IFAC Worhop Fracoal Dffereao ad Applcao. Badajoz, Spa, Ocober 8-,. [3] K. Hollebec, Ivlap. : A Malab fuco for uercal vero of Laplace rafor by he de hoog algorh, hp:// [4] R.L. Mag, Fracoal Calculu Boegeerg,Begell Houe, 6. [5] R. Goreflo, F. Maard, A. Carper, Fracoal calculu: Iegral ad dffereal equao of fracoal order, Fracal ad Fracoal Calculu Couu Mechac. Sprger Verlag, 997. P a g e

The Signal, Variable System, and Transformation: A Personal Perspective

The Signal, Variable System, and Transformation: A Personal Perspective The Sgal Varable Syem ad Traformao: A Peroal Perpecve Sherv Erfa 35 Eex Hall Faculy of Egeerg Oule Of he Talk Iroduco Mahemacal Repreeao of yem Operaor Calculu Traformao Obervao O Laplace Traform SSB A