CHAPTER 3. FRACTIONAL-ORDCEHRACPOTENRTR3O. LFLREARCSTIONAL-ORDE1R3 CONTROLLERS 13 CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 13
CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 13
Figure 3.2: PFIiDgurcone 3troller:.2: P IDfromconptroller:oints tofromplanpeo. iFnitgsutroe p3l.a2n: eP. ID controller: from points to plane.
n δ n δ Figure 3.2: P IDFconor troller:a wide cfromlFaossr poaofwicnoitdsnettrocollaplelsadsnooef.bcjeocnttsrowleledrecoobmjemctesnFwdoretharewcforidamecmtcioelansadsl toPhfIecnofDrnaδtcrtoilolnedaloPbIjecDts we recommend the fractional P I D CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 13 controller, wconhichtroller,is a partiwhicularch is caaspartie of cularP IλDcδasconeconotroller,ftroller,P IλDwδhwconerehictroller,hλ i=s an,partiwnherecularλ =casn,enof P IλDδ controller, where λ = n, n ∈ ∈ ∈ For a wide class of contNrollanded obδjectsNRw.eandrIenctoegerδmmeordernRd. thIenitnfegerrtaegratorctioordernal PisIiinmpDtegratorδortanN andt ifors δimpsteadyortanR. -sItnatetforegeresrrorteadyorder-stiatentegratorerror is important for steady-state error ∈ ∈ ∈ controller, which is a particancellaticular caseonofcancellatibutP IλonDδthecononotroller,therbut onhandwtheheretheoλtherfracti= nhand,onaln theicancellatintegralfractiionals onalsobutinitmpegralonortantheis taolsforthero imphandortanthet forfractional integral is also important for ∈ N and δ R. Integer orderobtaintinegratorg a Bobotidsaei’insmpinidgortaneaal Blotodforpe’tsrsateadyindsefaelr-lsfotuoatenpcttireoarrornnsresfeorpbfonstuaninecitniwogintharesBconsopdonsetan’setidwphaseiathl leoconsop ttanrantsphasfer fuenction response with constant phase ∈ Bode’s ideal loop transfer function cancellation but on the othermargihandn ftheor desfmargiractiiredonalnfreqforinutdesencyegraliredrangeis afreqlso (uimpe.g.encyortan[1,range3t,for23(,margie.g.51]).[1,n 3for, 2des3, 51]).ired frequency range (e.g. [1, 3, 23, 51]). obtaining a Bode’s ideal loop transfer function response with constant phase Figure 3.2: P ID controller: from points to plane. margin for desired frequency range (e.g. [1, 3H, .2W3., B51]).ode, Network Analysis and Feedback Amplifier Design. 3Fo.r2a .w2ide classBof co3ondt.ro2elle.d’2Dsob. jVeicandts B weNostrandearoelcodmlmeoe’nods Compantpheidfrtacertioanyanl,l IncPslIfon.,De oNeδ rpwf uYtork,nr3ac. n1945.t2is.o2fner fBuondceti’osnideal loop transfer function controller, which is a particular case of P IλDδ controller, where λ = n, n ∈ Fractional order N and δ R. Integer order integrator is important for steady-state error Bode∈ suggesBoteddeansuideggealstsehdapane oifdetalhe slohapopetransof thefer lofuncop titranson inferhifsuncworktiononin his work on 3.2.2 Bode’s ideacancellatil looonpbuttonrathenostherfehandr ftheunfractictonaliointegral is also important for Bode suggested an ideal shape of the loop transfer function in his work on obdetainsiniggna Boofde’sfeideadbacdel loospitgnrkanBode’samfeofr fuplifencteiofiedbacnsres rsidealponsikne amw1945.i thlooppliconsfietanIdet rsphastransfailenlo1945.opertransde function:Idesignfearl offlouncofepetidbactranson haskfeamrformfuncpli:fietionrs ihasn 1945.form:Ideal loop transfer function has form: Bode suggested an ideal margishapn fore desofiredthefrequloencyoprangetrans(e.g. fe[1,r3, f2unc3, 51]).tion in his work on systems and controllers α α design of feedback amplifiers in 1945. Ideal loop transfer function hassform: s s α 3.2.2 Bode’s ideal loop transfer functioLn(s) = L(s,) = , (3.6) L(s) =(3.6) , (3.6) α ωgc Bode suggested an ideal shape of the loop transfer function in his work on ωgc ω s ! " ! " ! gc " desHendrikign ofLfe(e dbacWs)adek=am Bodeplifiers in 1945., Ideal loop transfer function has form: (3.6) where ωgc is desired crosα sover frequency and α is slope of the ideal cut-off wωhgcere ωgcs is desired crossover frequency and α is slope of the ideal cut-off (1905-1982)! "L(s) = , (3.6) where ωgc is desired crossover frequency and α is slope of the ideal cut-off ω characteristic. ! gc " where ωgc is desired crossover frequenccyhaarnadctαeriistisclo. pe of the ideal cut-off characteristic. where ω is desired crossover frequency and α is slope of the ideal cut-off Pgc hase margin is Φm = π(1 + α/2) for all values of the gain. The amplitude characteristic. characteristic. Phase margin is Φm = π(1 + α/2) for all values of the gain. The amplitude oPhaose margoin is Φm = π(1 + α/2) for all values of the gain. The amplitude mPhaasregmianrgiAn ismΦmis= iπn(1fi+nα/i2)tyfor. aTll hvaeluecs oofnthestgaiann.tTphehamapslietudemargin 60 , 45 and 30 ocorreso pond o o o o Phase margin is Φ = π(1 + α/2) formaallrgvianlueAs ofistheinfigaio niotn.y. TThoheeacmopnlistudetant phase margin 60 , 45 and 30 correspond m margin Am is infinity. The constant phasme margin 60 , 45 and 30 correspond margin Am is infinity. The constant phase margin 60 , 45 and 30 correspond to tothe sthelopes αslop= 1e.3s3, α1.5=and 11.636.3,o 1.5o and o1.66. margin Am is infinity. The constant −phatos−e mthea−rg−silopn 6e0s−,α45=and1.3−30, cor1r.5espanondd to1.6the6. slopes α = 1.33, 1.5 and 1.66. The NTyqhueist NcuryveqfuoriisdetalcBuodrevtreanfsfoerrfuindcteioanlis−Bsimopdlyea sttr−aaignhtslifneer fun−ction is simply a stra−ight lin−e − to the slopes α = 1.33, thro1u.g5h tahenodrigin w1it.h6a6r.g(LT(jωh))e=Nαπy/2q(sueeise.tg. c[1u, 3r9v]).e for ideal Bode traTnshferNfuynquctisiotncuisrvsiemfoprlyidaesatlrBaiogdhet ltirnaensfer function is simply a straight line − −tBhordeo’sutgrahnsf−etrhfuencotiornig(3i.n6) cwanitbhe usaedrgas(La r(ejfeωren)c)e s=ysteαmπin/t2he (see e.g. [1, 39]). 1 The Nyquist curve forfolloidweiangl form:Bode tratnhsrfoerugfuhnctthieonorisigsiinmwpliytha astrrga(iLgh(tjωlin))e = απ/2 (see e.g. [14, 39]). K through the origin with arg(L(jω)) = απ/2 (see e.g. [1, 39]). through the origin with arg(L(Bjωod))e’=s αtGrπca(s/n)2=sf(esreefuen.gc.(t0i[ Fractional-orFractional order sysdertems asystemsnd controllers Bode’s ideal loopC HAtransfPTER 3. FRACTerIONAL -OfunctionRDER CONTROLLERS 14 CHAPTER 3. FRACCTIHONAAPL-OTREDRER 3C.ONTFRROLALCERTS IONAL-OR14DER CONTROLLERS 13 W(s) Y(s) K G(s) = ! W(s) Y(s) s K G(s) = s! W(s)+ E(s) U(s) Y(s) CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 14 Figure 3.3: Bode’s ideal loop. Gc(s) Gs(s) W(s) Y(s) – K (a) Open loop: G(s) =Figures! 3.3: Bode’s ideal loop. Magnitude: constant slope of α20dB/dec.; • − (a) Open loop: Crossover frequency: a function of K; • General characteristics of the Bode’Phase: hidealorizontal ltransfine of α πer; function: FiMgureagni3.3:tudeBo:dec’sonsidetanal tlosop.lope of α20dB•/dec.; − 2 1 • − Nyquist: straight line at argument α π . Gs (s) = Crossover frequency: a function of K;• − 2 βn βn 1 β1 β0 (a) Open loop: a s a s − a s a s Start • (b) Closed loop: n + n−1 +!+ 1 + 0 Phase: horizontal line of α π ; Magnitude: •constant slope of α20dB/dec.−; 2 !! "" Gain margin: Am = infinite; • Nyquist: strai−ght line at argumeFntigαu• πr.e 3.2: P ID controller: from points to plane. β β β Crossover fre•quency: a function of K; − 2 α n n−1 0 ! " • Phase margin: constant : Φm = π 1 2 ; a D y(t) + a D y(t) +!+ a D y(t) = u(t) π • − n n−1 0 Phas(eb:)hCorliozsoendtaloloinpe: of α ; α α 63 / 90 • − 2 Step response: y(t) = Kt Eα,α+1 !( Kt ") , π • − Nyquist: straight line at argument α . where Ea,b(z) is Mittag-Leffler function of two parameters [43]. n δ Back • Gain margin: A F−=ori2nfianitwe;ide class of controlled objects we recommend the fractional P I D • m Full screen (b) Closed loop: α λ δ Phase margconin: cotroller,nstant : Φmw=hπic1h is ;a particular case of P I D controller, where λ = n, n • 3.2.3 −Il2lustrative example Close Gain margin: Am = infinite; α α ∈ • Step responNse: andy(t) = δKt Eα,αR+1.!( KInt "t)eger, order integrator is important for steady-state error Phase margi•n: constant : Φ = π 1 α ; We can−illustrate the fractional order control properties by an example. As- End • where Ea,b(zm) is Mit−ta2g-Leffl∈er function of two parameters [43]. α α suming that the transfer function of the DC motor is Step response: y(t) = Kt cancellatiEα,α+1 !( Kt ") , on but on the other hand the fractional integral is also important for • − 2 where E (z) is Mittag-Leffler function of two parameters [43]. Km 5 a,b obtaining a Bode’s ideal loopG(ts)r=ansfer ,function respons(3.9)e with constant phase 3.2.3 Illustrative example Js(s + 1) margin for deswithiredJ beinfgreqthe puayencyload inerrangetia. Our s(pe.g.ecificati[1,on is 3th,e 2co3ns,ta51]).nt phase 3.2.3 IlluWsteracantiviellusextrateampthele fractional order control properties by an example. As- margin independent of the payload changes. suming that the transfer function of the DC motor is We can illustrate the fractional order control properties by an Aexssample.ume thaAst w- e would like to have a closed loop system that is insensitive suming that the transfer function of the DC motor is to gain variations with a constant phase margin of 60o. Bode’s ideal loop Km G(s) = transfer, function that gives this phase(3margi.9) n is 3K.2.2 BJs(os +d1e) ’s ideal loop transfer function G(s) = m , (3.9) Js(s + 1) 1 G (s) = . (3.10) with J being the payBoloaddeinerstiua.ggeOurstsepedcifiancationideis althe scohnapstaneto pohafsset√3hes loop transfer function in his work on with J beingmthaergipnayilndeoad pinendeertian.t Oofuther spepcaifiycloadationchisangethe sc.onstant phase margin independeAntssofumthee tphaaytloadwecwhangeodeulds.liignke toofhavfee aecdbaclosed lokopamsysteplim tfiehatrsis ininsen1945.sitive Ideal loop transfer function has form: Assume that we would like to have a closed loop system that is insensitive to gain variations with a constant phase margin of 60o. Bode’s ideal loop to gain variations with a constant phase margin of 60o. Bode’s ideal loop α transfer function that gives this phase margin is s transfer function that gives this phase margin is L(s) = , (3.6) 1 1 G (s) = . Go(s) = . (3.10) (3.10) ωgc o s√3 s s√3 s ! " where ωgc is desired crossover frequency and α is slope of the ideal cut-off characteristic. Phase margin is Φm = π(1 + α/2) for all values of the gain. The amplitude o o o margin Am is infinity. The constant phase margin 60 , 45 and 30 correspond ! µ to the slopes α = 1.33, 1.5 and 1.66. Bode’s ideal loop transf− er− function− PI D controllers The Nyquist curve for ideal Bode transfer function is simply a straight line through the origin with arg(L(jω)) = απ/2 (see e.g. [1, 39]). Can be used as a Brefodere’sencetrans fsystemer functio inn (the3.6) fcorm:an be used as a reference system in the CHAPTER 3. FRACTIONAL-ORDER CfolloONTROwLLiEngRS form: 14 W(s) Y(s) K K G(s) = ! G (s) = (0 < α < 2) (3.7) s c sα + K K Figure 3.3: Bode’s ideal loop. Go(s) = α (0 < α < 2), (3.8) CCHHAAPPTTEERR 33.. FFRRAACCTTIIOONNAALL--OORRDDEERR CCOONNTTRROOLLLLEERRSS 1155 s (a) Open loop: 2020 log log10 |G(j |G(j!!)|)| 10 where Gc is transfer function of closed loop and Go(s) is transfer function in Magnitude: constant slope of α20dB/dec.; • − -- 20 20 "" dB/dec dB/dec Crossover frequency: a function opof Ke; n loop. General characteristics of Bode’s ideal transfer function are: • Phase: horizontal line of α π ; • − 2 Nyquist: straight line at argument α π . • − 2 (b) Closed loop: loglog !! argarg ( (G(jG(j!!)))) # Gain marg#in: Am = infinite; • 22 α Phase margin: constant : Φm = π 1 2 ; • # − " # α α Step resp"onse: y(t) = Kt Eα,α+1 !( Kt ") , • 22 − where Ea,b(z) is Mittag-Leffler function of two parameters [43]. ## loglog !! 3 3.2.3 Illustrative example 6 FiFiguregure3.4:3.4: BoBodedeplotsplotsoofftranstransferferffunctiunctionon(3.8).(3.8). We can illustrate the fractional order control properties by an example. As- SsSiuinnmicceengGGoothat((ss)) ==theCC(trans(ss))GG((sfers),), fwwunctiee cancanonfindfindof thethetheDccConontrollermtrollerotor transitranss ferfer ffunctiunctionon iinn tthehe follofollowwiingng formform Km J G(s) =1 , 1 (3.9) J 22//33 1Js(s + 1) 22//33 1 CC((ss))== ss ++ 1/3 ==KK ss ++ 1/3 ,, ((33..1111)) KKm ss1/3 ss1/3 m !! "" !! "" with J being the payload inertia. Our specification iλs tδhe constant phase wwhhiicchhiissaapartiparticularcularccasaseeoofftthehetranstransferferffunctiunctiononofofthethePPIIλDDδ ccoonnttrroolllleerr((33..11)),, margin independent of the payload changes. wwhheerree KK ==JJ//KKmm iissttheheconcontrollertrollercconsonsttanant.t. Assume that we would like to have a closed loop system that is insensitive TThhee phasphasee margimarginn ooff tthehe concontrolledtrolled ssyysstemtem wwiithth aa fforworwaardrd loloopop concontrollertroller to gain variations with a constant phase margin of 60o. Bode’s ideal loop GGcc((ss))iiss transfer function that gives this phase margin is ΦΦmm ==aarrgg[[CC((jjωω00))GG((jjωω00))]]++ππ,, ((33..1122)) wwhheerree ωω00 iisstthheeccrroossssoovveerr ffrreeqquueennccyy.. 1 Go(s) = . (3.10) TThheeoobbttaaiinneeddpphhaassee mmaarrggiinniiss s√3 s 11 44ππ ππ ΦΦm ==aarrgg[[CC((jjωω))GG((jjωω))]]++ππ==aarrgg ++ππ==ππ == .. ((33..1133)) m ((jjωω))44//33 −−3322 33 ## $$ TThhee cconsonsttanantt phasphasee mmaargirginn iiss nnotot dedeppeendendenntt ofof thethe ppaayyloadload cchhangeangess aandnd thethe ssyysstteemm ggaaiinnKK andandphasphaseecurcurvvee iissaa horhoriizonzonttalallliineneatat 22ππ//33.. −− SStteepprreessppoonnssee ooffcclloosseedd ccoonnttrroollllooooppccaannbbeeeexxpprreesssseedd aass:: 1 11 1 11++11//33 11++11//33 yy((tt))==LL−− ==tt EE1+1/3, 2+1/3 tt ,, ((33..1144)) s (s11++11//33+ 1) 1+1/3, 2+1/3 − % s (s + 1)& − % & ' ( wwhheerree sstteepp rreessppoonnssee iissiinnddeeppeennddeenntt oofftthheeppaayyllooaaddiinneerrtt'iiaa aannddαα(==44//33.. CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 14 W(s) Y(s) K G(s) = s! Figure 3.3: Bode’s ideal loop. (a) OpenCHlAooPp:TER 3. FRACTIONAL-ORDER CONTROLLERS 14 W(s) Y(s) Magnitude: constant slope of Kα20dB/dec.; • G(s) = −s! Crossover frequency: a function of K; • Phase: horizontal line of α π ; • − 2 Nyquist: straighFit gureline3.3:at Boargumde’s ideenaltloop.α π . • − 2 (b) Close(a)d lOopoepn:loop: Magnitude: constant slope of α20dB/dec.; • − Gain mCarorsgsoivne:r fAreqmue=ncyi:nafifnunitceti;on of K; • • Phase: horizontal line of α π ; α Phas•e margin: constant :−Φ2m = π 1 2 ; • Nyquist: straight line at argument α π . − • α − 2 α Step response: y(t) = Kt Eα,α+1 !( Kt ") , • (b) Closed loop: − where Ea,b(z) is Mittag-Leffler function of two parameters [43]. Gain margin: A = infinite; • m Phase margin: constant : Φ = π 1 α ; • m − 2 α α Step response: y(t) = Kt Eα,α+1 !( Kt ") , 3.2.3 Illus•trative example − where Ea,b(z) is Mittag-Leffler function of two parameters [43]. We can illustrate the fractional order control properties by an example. As- suming that the transfer function of the DC motor is 3.2.3Bode’Illusst ridealative eloopxamp transfle er function: example PI!Dµ controllers We can illustrate the fractional order controlK properties by an example. As- G(s) = m , (3.9) suming thatThethe transftransfererfuncti functionon of the JofsD (aCs DCm+ot1o )motorr is is Fractional-order PID controllers: K G(s) = m , J is payload iner(3tia.9) from points to plane with J being the payload inertia. JOsu(sr+s1p) ecification is the constant phase CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 13 margin independent of the payload changes. with J beiAssumeng the pa ythatload iwneert iwa.ouldOur likspec itofica thaionveis at hclosede consta nloopt pha se margin independent of the payload changes. Assume that wsysteme wou lthatd lik ise tinsensitivo have aec tolos gained l ovariationsop system withtha ta is insensitive Assume that we would like to have a closed loop system thatois insensitive to gain variations with a constant phase moargin of 60 . Bode’s ideal loop to gain varconstantiations wit hphasea con smargintant pha sofe m 60argi.n Bode’of 60o.s idealBode’ sloopideal loop transfer ftransferunctionfunctiCtransfHthatAPonTEthatgierR v3 .efunctiongisFveRtshiAtChisTsphaseIOphase thatNAL- margiOgivmargiRDesEnR thisisCnOiN sphaseTROLL EmarginRS is 15 20 log |G(j !)| 1 G (10s) = 1. (3.10) G o(s) =s√3 s . (3.10) CHAPTER 3. FRACTIOoNAL-ORDsE√3R sCONTROLLERS 15 CHAPTER 3. FRACTIONAL-ORDER- 20C "O dB/decNTROLLERS 15 20 log |G(j !)| 10 Figure 3.2: P ID controller: from points to plane. 20 log 10 |G(j !)| - 20 " dB/dec 7 10 CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 15 For a wide class of controlled objects we recommend the fractional P InDδ λ δ - 20 " dB/declog ! controller, which is a particular case of P I D controller, where λ = n, n 20 log 10 |G(j !)| arg ( G(j )) ∈ ! N and δ R. Integer order integrator is important for steady-state error # ∈ - 20 " dB/dec cancellation but on the other hand the fractional integral is also important for 2 log ! arg ( G(j!)) obtaining a Bode’s ideal loop transfer function response with constant phase # " # margin for desired frequency range (e.g. [1, 3, 23, 51]). 22 log ! arg ( G(j!)) log ## ! arg" ( G(j# !)) 2 3.2.2 Bode’s ideal loop transfer function 2 log ! # 2 # Bode suggested an ideal shape of the loop transfer function in his work on Figure 3.4:# Bode plots of transfer function (3.8). " log ! # 2 design of feedback amplifiers in 1945. Ideal loop transfer function has form: " 2 α Since Go(s) = FiC(gures)G(3.4:s), wBoedecanplotsfindofthetransconfertrollerfunctiontrans(3.8).fer function in the # s following form # L(s) = , (3.6) log ! ωgc log ! ! " Since Go(s) = C(s)G(s),J we can find1 the controller trans1fer function in the ! µ Bode’s idealC (sloop) = transfs2/3 + er =function:K s2/3 + example, (3.11) following form K s1/3 s1/3 where ωgc is desPIired crDossover fcontrrequency andollersα is slope of the ideal cut-off FigureFigure3.4: Bo3.4:mde!plotsBodeofplotstrans"ferof ftransuncti! onfer(3.8).functi"on (3.8). which is a particular caseJof the2/trans3 fer1 function of2/3the P1IλDδ controller (3.1), characteristic. C(s) = s + 1/3 = K s + 1/3 , (3.11) Km s s SwinhcerGe K(s)==JC/K(sm)Gi(sst),heweconcantrollerfind theconsctonantrollert. transfer function in the Phase margin is Φm = π(1 + α/2) for all values of the gain. The amplitude SinceSince o G o ( s ) = C ( s ) G ( s ), !, w thee can contrfind" theoller!con transftroller "ertrans functionfer functi ison in the followwhingch iforms a particular case of the transfer function of the P IλDδ controller (3.1), margin A is infinity. The constant phase margin 60o, 45o and 30o correspond folloThweingphasforme margin of the controlled system with a forward loop controller m Comparison of unit step responses Gwc(hse)reisK = J/Km is tJhe controller1 constant. 1 to the slopes α = 1.33, 1.5 and 1.66.of the fractional-order “reality” and C(s) = s2/3 + = K s2/3 + , (3.11) The phase margin of theJ controlled12//33 sy1stem with 1a/3forw2/3 ard lo1op controller − − − its integer-order “model” Km Φm = arsg[C(jω0)G(jω0)] +s π, (3.12) C(s) =! s "+ 1/!3 = K s" + 1/3 , (3.11) The Nyquist curve for ideal Bode transfer function is simply a straight line Gc(s) is Km s λ δ s wwhihcehries ωa 0partiis tcularhe crocassseovoefrthefretransqu!enfercy.function of"the P I!D controller"(3.1), through the origin with arg(L(jω)) = απ/2 (see e.g. [1, 39]). Φm =thisarg[ Cis( jaω 0par)G(ticularjω0)] + πcase, of fractionalλ δ ( 3PID.12) whwerheTicKhhe=iosbJat/apartiKinmediscularpthheasconecmastrolleraergoifntciheonss transtant. fer function of the P I D controller (3.1), Bode’s transfer function (3.6) can be used as a reference system in the wThheerephasω0 eismargithe cnroossfotvheer confreqtrolleduency.system with a forward loop controller where K = J/Km is the controller cons1tano t. 4 π π GThe(s)ΦiT sobtainedh=e oabrtga[iCne(djωp )phasehGas(ejωm)a] r+ gmarginiπn =is arg is 60 :+ π = π = . (3.13) following form: c Tmhe phase margin of the controlled(jω)4s/y3stem with−a 3forw2 ard3 loop controller K Φm = arg[C(jω0)G#(jω10)] +$π, 4 π π(3.12) G (s) = (0 < α < 2) (3.7) Gc(s) is c α The cΦonsm =tanartg[phasC(jωe)Gm(ajrgiω)n] +isπn=otadergpende4n/t3 of+theπ =pπayload c=hange. s (a3nd.13the) s + K where ω0 is the crossover frequency. (jω) − 3 2 3 system gain K and phase curΦmve=isaarghor[#C(izonjω0t)alG$ (lijneω0at)] +2ππ,/3. (3.12) TThhee coonsbtatiannedt phasphase marrgiginn iiss not dependent of the pay−load changes and the K StepSte prresponse:esponse of closed control loop can be expressed as: Go(s) = (0 < α < 2), (3.8) wsyhsetreemωg0aiins Kthande crophasssoveecurr frveqius eanhorc1y.izontal line at 4 2ππ/3.π sα Φm = arg[C(jω)G(jω)] + π = arg + π = π − = . (3.13) TSthepe orebstpaoinns1eedofpchloasee1dmcoanrtgrionl l(iosjoωp1)+4c/1a3/n3 be expre−sse3d2as:1+31/3 y(t) = L− = t E t , (3.14) where Gc is transfer function of closed loop and Go(s) is transfer function in s (s1+1/3 + 1) # $ 1+1/3, 2+1/3 − The constant phase1%margin i1s not de&pende1+nt1/of3 the1payload ch1ange+1/3s and4 πthe π open loop. General characteristics of Bode’s ideal transfer function are: Φ y(=t) =argL[−C(jω)G(jω)] + π = targ E1+1/3, 2+1/3+ 'πt= π ,( =(3.14.) (3.13) sywstheemregsmatienpKreandsponphassesi(sesicur1n+d1/ve3ep+eisn1da)ehornt iozonf tthale plineaylato4a/d3 2inπe/r3−t.ia and α = 4/3. % & (jω) − − 3 2 3 Step response of closed control loop can be#expressed$as': ( 8 Twhhereconssteptanretspphasonse eis minadrgiepenndiesnnt otof tdeheppeandeyloandt iofnerthetia apnadyαload= 4/c3h.anges and the 1 1 1+1/3 1+1/3 systye(mt) =gaLin− K and phase cur=vte is aE1hor+1/3i,zon2+1/t3al ltine at, 2π/(33..14) s (s1+1/3 + 1) − − Step respo%nse of closed&control loop can be expressed as: where step response is independent of the payload inert'ia and α( = 4/3. 1 1 1+1/3 1+1/3 y(t) = L− = t E t , (3.14) s (s1+1/3 + 1) 1+1/3, 2+1/3 − % & where step response is independent of the payload inert'ia and α( = 4/3. PI!Dµ controllers PI!Dµ controllers Control of the “reality” and the “model” using a classical PD controller, which is optimal for the “model” 9 CHAPTER 3. FRACTIONAL-ORDER CONTROLLERS 16 3.3 Design of controller parameters The tuning of P IλDδ controller parameters is determined according to the given requirements. These requirements are, for example, the damping ratio, the steady-state error (ess), dynamical properties, etc. One of the methods being developed is the method of dominant roots [37], based on the given stability measure and the damping ratio of the closed control loop. Assuming that, the desired dominant roots are a pair of complex conjugate root as follows: s = σ jω , 1,2 − ± d designed for the damping ratio ζ and natural frequency ωn. The damping constant (stability measure) is σ = ζωn and the damped natural frequency of oscillation ω = ω 1 ζ2. The design of parameters: K , T , λ, T and δ d n − p i d can be computed numerically from characteristic equation. More specifically, ! µ ! µ! PI D controllers for simpPIle planDt mode l contrP (s), this canollers:be done by so lvdesigning min C(s)P (s) + 1 s= σ jωd . Kp,Ti,λ,Td,δ || || − ± The design of PI!D" controllers can be based on gain Control of the “reality” and the Another possible way to obtain the controller parameters is using the tuning “model” using a classical PD and phase margin specifications: controller, which is optimal formula, based on gain and phase margins specifications [52]: for the “model”, after detuning (aging) of the [C (jω )] [P (jω )] [C (jω )] [P (jω )] = 1 , p p p p Am controller (when TD = 1). " " − # # − [C (jω )] [P (jω )] + [C (jω )] [P (jω )] = 0, " p # p # p " p [C (jω )] [P (jω )] [C (jω )] [P (jω )] = cos Φ , " g " g − # g # g − m [C (jω )] [P (jω )] + [C (jω )] [P (jω )] = sinΦ , " g # g # g " g − m where Φm is a phase margin, Am is a gain margin, ωp and ωg is 0dB (crossover) frequency. Last but not least we should mention the optimization algorithm based on the integral absolute error (IAE) minimization [44]: t t IAE (t) = e(t) dt = w(t) y(t) dt, | | | − | &0 &0 where w(t) is the desired value of closed control loop and y(t) is the real value of closed control loop. This method does not insure the desired stability measure of the closed control loop. Measure of stability has to be checked out additionally. We can use a frequency method described in [39]. PI!Dµ controllers PI!Dµ controllers: design IV.2 A Design Approach • The plant model is assumed to be Control of the „reality“ using the PD controller, which is optimal for the „model“, and using the " PD controller. • and the fractional order PID controller is • It is expected that the gain and phase margin of the compensated systems are and • Question: how to choose parameters FOC tutorial III 27 ©Dingyu Xue, NEU, PR China PI!Dµ controllers: design PI!Dµ controllers: design Requiring Design of fractional-order PID controller it can be found that parameters is determined by given requirements. These requirements can be, for example: • the damping ratio, • the steady-state error, • dynamical properties, • etc. • For • It is found that PI!Dµ controllers: design PI!Dµ control: MATLAB I. General Description of Linear • Four equations, with 7 variables We have four equations with seven variables: Fractional Order Systems I.1 The normal form FOC tutorial III 28 ©Dingyu Xue, NEU, PR China The rest of the variables can be determined by minimizing the ISE criterion • Thus compared with IO LTI’s, information on orders are also used • A FOTF model class/object can be defined in MATLAB to describe the system model FOC tutorial III Courtesy: Dingyü3 Xue, YangQuan Chen ©Dingyu Xue, NEU, PR China PI!Dµ controllers: design PI!Dµ control: MATLAB II. MATLAB Class Design Sample plots for various combinations of ! and µ: • For different µ, ! combinations II.1 Create a @fotf directory • The fotf class can be defined as FOC tutorial III 4 ©Dingyu Xue, NEU, PR China FOC tutorial III 30 ©Dingyu Xue, NEU, PR China Courtesy: Dingyü Xue, YangQuan Chen PI!Dµ controllers: design PI!Dµ control: MATLAB • A fractional order PID controller can be II.2 Other Overload Functions The designedoptimal PIsuch!D µthat controller: • Function call • A display function FOC tutorial III 31 ©Dingyu Xue, NEU, PR China FOC tutorial III 5 ©Dingyu Xue, NEU, PR China Courtesy: Dingyü Xue, YangQuan Chen PI!Dµ control: MATLAB PI!Dµ control: MATLAB II.3 Define a fotf Object in MATLAB • Feedback function • Example • MATLAB command • Display • These functions are suitable for interconnections of fractional order systems FOC tutorial III 6 ©Dingyu Xue, NEU, PR China Courtesy: Dingyü Xue, YangQuan Chen FOC tutorial III Courtesy: Dingyü9 Xue, YangQuan Chen ©Dingyu Xue, NEU, PR China PI!Dµ control: MATLAB PI!Dµ control: MATLAB II.4 Overload functions for block • A common function unique • Plus function for parallel connections FOC tutorial III 7 ©Dingyu Xue, NEU, PR China FOC tutorial III 10 ©Dingyu Xue, NEU, PR China Courtesy: Dingyü Xue, YangQuan Chen Courtesy: Dingyü Xue, YangQuan Chen PI!Dµ control: MATLAB PI!Dµ control: MATLAB • Multiplication for series connection ExampleII.5 of An inter Exampleconnection: for interconnection • Unity negative feedback system with • Minus The closed-loop system can be • The closed-loop system can be established • Uminus obtained as • The closed-loop system • Inv FOC tutorial III 11 ©Dingyu Xue, NEU, PR China FOC tutorial III 8 ©Dingyu Xue, NEU, PR China Courtesy: Dingyü Xue, YangQuan Chen • The closed-loop systems with even more complicated structures can easily be obtained with the overloaded functions FOC tutorial III 12 ©Dingyu Xue, NEU, PR China CHAPTER 4. CONTINUED FRACTION EXPANSION (CFE) 22 + + + + + + 1 - - - + + + Y2n* (s) Z2n -1(s) Y2n -2 (s) Z2n-3(s) Y2 (s) (s) CHAPContinTER 4. CONTuedINUED F RfractionsACTION EXPANSI O(CFE)N (CFE) 18 CFEs and nested multipleZ1 loops Figure 4.6: Nested multiple-loop control system of the first type. can be expressed in the form: expansion, which is identical with the equation (6.1): 1 Z(s) = Z (s) + b (s) 1 1 1 Y (s) + G(s) a (s) + 2 1 0 b (s) CHAPTER 4. CONTINUED FRACTION EXPANSION (CFE) 22 Z (s) + ! 2 3 1 a1(s) + b3(s) Y4(s) + a2(s)+ a (s)+... + + + + + + ...... 3 1 CHAPTER 4. CONTINUED FRACTION EXPANSION (CFE) 19 - - - 1 + + + 1 b1(s) b2(s) b3(s) * Y2n 2(s) + Y2n(s) − 1 = a0(s) + . . . (4.1) Z2n 1(s) + a1(s+)+ a2(s)+ a3(s)+ − Y2n(s) G(s) Z2n -1(s) - Similarly to the above considerations, we can obtain a continued fraction Y (s) where a s and b s are rational functions of the variable s, or are constants. 2n -2 i i H(s) expansion of the transfer function of the other interesting type of a nested Z (s) multiple-loop control system, depicted in Fig. 4.7: The application of theCHAmPTeEtRho4.d CyOiNelTdsINUaEDratiFRonalACTIOfuncN EXtiPAon,NSIOGN((sC),FEw) hich is19an 2n-3 1 approximation of the irrational function G(s). Z(s) = Nested loop of type I (4.9) Figure 4.1: A control l+oop with a negative feedback. Y2 (s) 1 G(s) Z1(s) + - CHAPTER 4. CONTINUED FRACTION EX1PANSION (CFE) 22 On the other hand, for interpolation purposes, rational functi! ons are some- Y2(s) + Z1 (s) 1 Z3(s) + times superior toThpeorlaytnioonmaliafulns.ctiTonhiRs(sis),shouldroughbleyH(s)wsriptteneakiinngt,heduforme tooftha ecionr tianbuilousity + + + ...... +...... +...... +...... Figure 4.6: Nested multiple-loop control system of the first type. 1 - - 1 fraction: - + 1+ + to model functions with poles. (As it can be see1n later, branch points can be Y2n 2(s) + *− 1 FigRu(rse)4.=1: A control loop with a negative feedback. (4.3) expansion, which is identical with the equation (6.1): Y2n(s) 1 Z2n 1(s) + considered as accumulations of interlaced poles and zeros). These techniques 1 − Y2n(s) b1s + Z(s) = Z (s) + 1 1 1 Y (s) + Z2n -1(s) are based on the approTxhime rattiioonsnal fuonfctaion Rirrati(s) bs2houldsonal+ befuncwrittention,in theG(forms), bofya aconrtiantuioousnal 2 1 Z (s) + fraction: ...... 3 1 Y4(s) + (s) function defined by the quotient of two polynomial1s in the variable s: ...... Y2n -2 R(s) = (4.3) 1 1 1 1 b1s + bn 1s + Y2n 2(s) + Z (s) 1 − 1 2n-3 − bns Z2n 1(s) + b2s + Pµ(s) − Y2n(s) G(s) Ri(i+1)...(i+m) = ...... If bk > 0, k = 1, . . . , n, then the system is stable. If some bk is negative, ! Qν(s) Similarly to the above considerations, we can obtain a continued fraction Y2 (s) then the system is unstable. µ 1 expansion of the transfer function of the other interesting type of a nested p + p s + . . . + pbn s1s + 0 1 µ− multiple-loop control system, depicted in Fig. 4.7: Considering the continued fraction (4.3) as a tobolnsfor designing a corre- Z1 (s) = ν (4.2) If b > 0, k = 1q, . . .+, n,qthsen+th.e. s.y+stemq iss stable. If some b is negative, 1 sponding LC ckircuit, we can0 conc1lude that stabνility of a linear sysktem is equiv- Z(s) = (4.9) 1 then the system is unstable. Z (s) + Figure 4.6: Nested multiple-loop control system of the first type. 1 1 alent to remaliz+ab1ility=of iµts+tesνt +fun1ction R(s) with the help of only passive Y (s) + Considering the continued fraction (4.3) as a tool for designing a corre- 2 1 electric components. Z3(s) + sponding LC circuit, we can conclude that stability of a linear system is equiv- ...... e.x.p. .a.n.s. i.o. n. .,. .w. h. .i.c.h. is identical with the equation (6.1): 1 alent to realizability of its test function R(s) with the help of only passive 1 1 passing through the points (si, G(si)), . . . , (si+m, G(si+m)). Y2n 2(s) + Z(s) = Z1(s) + − 1 1 electric components. Z2n 1(s) + Y2(s) + − Y (s) 1 4.4 CFE and nested multiple-loop 2n Z (s) + 3 1 Y4(s) + ...... 4c.o4ntrCoFlEsyasntdemnessted multiple-loop 1 4.3 CFE and stability of linear systems 1 Y2n 2(s) + CHAPTER 4. CONTINUED FRACTION EXPANSION (CFE) 20 − 1 Let us now estacbolisnhtarnoilntseyresstienmg nsew relationship between continued frac- Z2n 1(s) + − Y2n(s) It is also knowtionnsthatand nceosntetdinmuuousltiplef-raclooptionconteroxlpanssysteimonss. can be used for investi- Let us now establish an interesting new+relationship between continued frac- 1 Similarly to the above considerations, we can obtain a continued fraction gatingCFEsstability of Wandelifirnesart recals ynestedslttehemksno. Fwonrfactthi tsmhat, thetultiplehectharacrans- fertefunctris tloopsiiconpRoly(s)noomf theialconQtrol(s) CFEs and nested multiple loops CHAPTER 4. CONTINUED FRACTIOtiNonEsXaPnAdNnSeIsOteNd(mCFulEt)iple-loop c1o9ntrol systems. expansion of the transfer function of the other interesting type of a nested loop with aWneegfirasttivreecalfeletdhebakcnok wshnofactwn tihatn Ftighe. 4tr.ans1 isfergifvuncten ibony R[8(]s) of the control of the differential equation of the system should be dYiv2n* (s)ided in two parts, the multiple-loop control system, depicted in Fig. 4.7: + loop with a negative feedback shown in Fig. 4.1 is given by [8] G(s) 1 “even” part (con- taining even powers of s) and Gth(se)“odd” part (containing odd Z(s) = (4.9) FiRgure(s)4.2:= Nested multiple.-loop control system – level(14..4) 1 G(s) Z (s) + powers of s): H(s) R(s1) =+ G(s)H(s) . (4.4) 1 1 CHAPTER 4. CONTINUED FRACTION EXPANSION (CFE)Y (s) + 23 1 + G+(s)H(s) + 2 1 From (4.4) it immediQ(s) ately= m(follos) +wsnthat(s).the trans1 fer function of the circuit Z3(s) + Figure 4.1: A control loop withFaromnega(4.4)tive feietdibmmediack. ately follows that the- transfer functi+ on of the circuit + + + + + ...... Y2n(s) 1 CHAPTER 4. CONTINUEDsFhRoAwCnTIOinN FEXigP.A4N.S2IOiNs (CFE) 20 * - - - + + CHAPTER 4.TChOeNnTINtUhEiDs FtRwAoCTIpOaNsrhEtoXswPnAoNifnSItFOhiNge.(C4cF.2hE)aisracter2i0stic polynomYia2nl(s)are used for creating 1 Y2n 2(s) + The rational function R(s) should be written in the form of a continuous Z2n -1(s) − 1 + 1 1 1 1 Z2n 1(s) + fraction: its test functi+on in1 the form of Pa2nf(racs) =tion, in which=the hi,ghest power of(4s.5i)s − Y2n(s) - 1 1 P2n(s) = =Z2n -1(s) , (4.5) R(s) = - 1 +(41.13)+Y12∗n(Ys2∗)n(s) Y2nY(2ns()s) Y2n -2 (s) 1 · contained in the dY*e(s)nominator: · b1s + Y*2n(s) 2n where Y12n(s) = Y ∗ (sFi) gure+ 1. 4.3: Nested multiple-loop control system – level 2. Z (s) wherbe2sY+2n(s) = Y2∗n(s) +2n1. 2n-3 Figure 4.2: Nested multiple-loop c.o.n.trol. . . .sy.ms.te. .m(. s–)level 1. n(s) Figure 4.2: Nested multiple-loop control system – level 1. R(s) = or R(s) = . Y2n-4 (s) + + 1 Using the equations (4.4) and (4.5) we obtain the transfer function of the + 1 bn+1s + n(s) m(s) - 1 − bsnysstem shown in Fig. 4.3: Nested loop of type II - + " # + Z1 (s) If bk > 0, k = 1, . . . , n, then theYs*y(s)stem is stable. If some bk is negative, Y*2n(s) 1 then the system is unstable. 2n Q2n 1(s) = Z2n 1(s) + P2n(s) = Z2n 1(s) + . (4.6) − − − Y2n(s) Figure 4.7: Nested multiple-loop control system of the second type. Considering the continued fractiZon2n -1(s)(4.3) as a tool for designing a corre- Z (s) sponding LC circuit, we can conclude2nt -1hat stability of a Clinoemarbisniysngtemtheis eeqquuivati- ons (4.4) and (4.5) we find the transfer function of Both types of nested multiple-loop systems, presented in this section, can alent to reaFilizgureabili4.3:ty oNef isttsedtemstultifuplenc-tloioopn cRon(strol) wsiytshtethemthe–nlehesevtelepld2o.mf oultinlyplepa-slosiopve system shown in Fig. 4.4: Figure 4.3: Nested multiple-loop control system – level 2. be used for simulations and realizations of arbitrary transcendental transfer electric components. Using the equations (4.4) and (4.5) we obtain theCtransHAPferTfEuncRti4on. ofCOtheNTINUED FQR2nAC1(TsI)ON EXPANSION (C1 FE) 20 functions. For this, the transfer function should be developed in a continued Using the equations (4.4) and (4.5) we obtain the transfer funcPti2onn 2of(sthe) = − = fraction, which after truncation can be represented by a nested multiple-loop system shown in Fig. 4.3: − 1 + Q2n 1(s)Y2n 2(s) 1 system shown in Fig. 4.3: − − Y2n 2(s) + system shown in Fig. 4.6 or Fig. 4.7. 4.4 CFE and nested multiple-loo1p + − Q2n 1(s) Q2n 1(s) = Z2n 1(s) + P2n(s) = Z2n 1(s) + 1 . (4.6) 1 − Q2n −1(s) = Z2n −1(s) + P2n(s) = Z2n −1(s) + Y2n(s). (4.6) - 1 − − − Y2n(s) = (4.7) control systems CHAPTER 4. CONTINUED FRACTION EXPA1NSION (CFE) 20 Combining the equations (4.4) and (4.5) we find the transfer function of * 4.5 CFE and rational and irrational numbers Combining the equations (4.4) and (4.5) we find the transfer function of Y2n 2(s) + Y2n(s) Lettheusnnesotewdemstultiablpleish-loaopn isnytsetreemstsinhogwneiwn Freig.lat4.4:ionship between continued frac- − 1 the nested multiple-loop system shown in Fig. 4.4: + Z2n 1(s) + Every rational and irrational number may be written in CFE form. Basically tions and nested multiple-loop control systems. 1 − Y2n(s) Q2n 1(s) 1 Figure 4.2: Nested m-ultiple-loop control system – level 1. the process of finding a continued fraction development consists of two steps: P2n 2(s) = Q2n −1(s) = 1 We firstPr2ecaln −2(sl)the=kno1 w+nQfact2n 1−(tshat)Y2nthe2(s)tr=ansfer functiTonheR1t(rsa)nosffetrhefuconnctroltion of the system shown in Fig. 4.5 is then given by the if the fraction m/n is greater than 1, then divide. Otherwise, write the fraction − 1 + Q2n −1(s)Y2n −2(s) Y2n 2(s) + 1 Y2n* (s) loop with a negative feedback −shown i−n Fig. 4Y.21n i−s2(gs)iv+enQb2yn [18(]s) + + m/n as 1/(n/m) and proceed with the first step. Continue until a numerator − relatiQ2onsn −1(hsi)p 1 − 1 = 1 (4.7) - + CHAPTER 5o.f 1 iAs oPbtPainRed.OCXonItMinueAd TfraIctOionNs arOe oFftenFuRsedAtoCgTet IgOoodNraAtioLnal OPERATOR 25 = G1(s) Figure(4.2:4.7) Nested multiple-loop control system – level 1. CHAPTER 5. APPROXIMATION OF FRACTIONAL OPERATOR 25 Y2n R2((ss))+= 1 . Q2n 3(s)(4.=4) Z2n 3(s) + P* 2n 2(s) approximations for real numbers. Let us consider the numbers written in the Y2n −2(s) + 1 + G(s)H(1s) − − Y2n(s)− CHAPTER 5. APPROXIMATION OF FRACTIONAL OPERATOR 25 CHAPTER 4. CONTINUE−D FRAZC2Tn IO1(Ns) +EXP1ANSION (CFE) 21 + + 1 form: a0 + b1/(a1 + b2/(a2 + b3/(a3 + . . . ))). In ”simple” continued fractions, Z2n −1(s) + Y2n(s) 1 − = Z2n 3(s) + (4.8) Y2n(s) - + all the bi, i, are 1 and the number can be re-written as [a0; a1, a2, a3, . . . ]. From (4.4) it immediately follows that the transfer function of the circuit − 1 no∀ minator polynomials. PrCoHbaAPblTEy,Rt5hi. sAfaPcPtROwXaIsMnoATtIeOdNfOoFr tFheRACfiTrsItONtiAmLeOiPnE[R1A7T],OR 25 The transfer function of the system shown in Fig. 4.5 is then given by the Z2nY -1(s) (s) + nominator polynomials. Probably, this fact was noted for the first time in [17], * 2n 2 shownTihne Ftriagn.s4fe.r2fiusnction of the system shown in Fig. 4.5 is then given by the Y2n(s) − 1 Example 4.5.1. relationship nominator polynomials. Probably, this fact was noteCdHAfoPrTEtheR 5.firAstPPtRimOXeIMinAT[1IO7]N, OF FRACTIONAL OPERATOR 25 relationship Z2n 1(s) + where the following idea appeared: a dense interlacing of simple poles and − Y2n(s) where the followThienCgFE ifodreπa, whiacph pgiveesatrheed”b:est”aapprodnoexmnimisnaaetitonoripnofotlayengroivlmeaniacolsirde.nPrg,roisbaoblf y,stihims fpacltewapsonolteesd foar nthedfirst time in [17], 1 1 Figure 4.3: Nested multiZple(s)-loop control system – level 2. where the following idea appeared: a dense interlacing of simple poles and Q2n 3(s) P=2nZ(s2)n =3(s) + P2n 2(s) = , (4.5) 2n -1 [3, 7, 15, z1e, r2o92s, 1a,l1o,n1g, 2a, 1,li3n, e1, i1n4, 2th, 1e, 1s,w2ph, el2ar,en2t,eh2e,is.f.o,.]l.lionTwhinsegovmeirdyealawragpaepye,aredq:uiavadelennset intoteralacbinrganofchsimcpulet;poles and Q − (s) = Z − (s) + P − (s) 2n 3 2n 3 1 + 12n Y2 ∗ (s) Y2n(Cso)ntinuing this process, we obtain the transfer function of the nested multiple- nominator polynomials. Probably, this fact was noted for the first time in [17], − − ·− 2n 1 zeros azleornosgaalotenlrmign292ea liminneansethinteαhattshtheepfolloslapnwliaengniecosn,ivse,rigenintsizsoeoramomgos eaeoldownapprogaayyl,xi,nimeeqaitiqunonuitvhiea[13].vlsaenpleltantetoist,aoinbasroamnbecrwhaanyc,cuehqtu;ivcaulent;t to a branch cut; = Z2n 3(s) + 1 (4.8) and s , 0 < α < 1, viewed as an operatowrh,erheatsheafobllroawnincghidceuatapaploeanregd:thaednenesgeaitnitveerlacing of simple poles and = Z2n −3(s) + lo1oUsp icngonttheFirogurel esqyu(4.3:s4atit.8e)monsNessht(eo4.4)dwmnuandltiinpleF(-i4.5)log.op4.cw6oentroliobtain tshyensteftmoherm–transleovefl a2fe.rcofuncntintiuoned offrathection α α where Y2n(s) = Y2∗n(s) ++1. −+ Y2n 2(s) ++ 1 α 355 and s , 0 < α < 1, viewed as an operator, has a branch cut along the negative Y1 − (s) + 1 and s and, 0 as belonging to this group. The starting po(iHn(ts)of)1/αthe(Gm(set)) ho= 0d; is tHhe(s)s=tat(Gem(s))enα t (5.3) − of the following relationships: (H(s))1/α (G(s)) = 0; H(s) = (G(s))α (5.3) − CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 32 CHAPTER 6. G6E.N2ERDAoLmAinPoPlRadOdAeCr HcirTcOuitFRACTANCE DEVICES 32 General Approach to Fractances 6.2GeneralDomino ApprladZd1 eroachcZ 3ircuit toZ 2n -3 FractancesZ 2n -1 CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 32 Y Y4 Y2n -2 Y2n 6.2 DoZ(s)mino ladd2 er circuit Z1 Z 3 Z 2n -3 Z 2n -1 A devices or a circuit exhibiting fractional-order Z1 Z 3 Z 2n -3 Z 2n -1 behaviour is called a fractance. Figure 6.2: Finite ladder circuit. Z(s) DominYo 2ladder lattice neYtw4orks can approximate fractYion2nal -2operator moYre 2n Z(s) Y2 Y4 Y2n -2 Y2n effectively than the lumped networks [10, 30]. Let us consider the circuit depicted in Fig. 6.2, where Z2k 1(s) and Y2k(s), domino ladder circuit network, − • k = 1, . . . , n, are given impedances of the circuit elements. The resulting a tree structure of electrical elements, impedance Z(s) of the entire circuit can be found easily, if we consider it in • the right-to-left directiFigureon: 6.2: Finite ladder circuit. 1 • transmission line circuit Z(s) =FiZ (gures)+ 6.2: Finite ladder circuit. (6.1) Domino lad1 der lattice networks can approxim1ate fractional operator more Y (s) + 2 1 effectively than the lumpedZ n(es)tw+orks [10, 30]. 3 1 Let us consider the circuit depiYct(esd) +in Fig. 6.2, where Z2k 1(s) and Y2k(s), Domino ladder lattice networks can4 approximate frac−tional operator more k = 1, . . . , n, are given impedances of th. .e. .c.i.r.c.u. .it. . .e.le. .m. .e.n.t.s... . .T. .h. .e. .r.e.sulting 1 Design of fractances can be using a truncated CFE, effectively thanimptehdeanlcue mZ(ps)eodf tnheetenwtiorerkcisrcu[i1t0ca,n3b0e].found easily, if1we consider it in Y2n 2(s) + the right-to-left direction: − 1 Let us consider the circuit depicted in Fig. 6.2,Zw2n h1(esr) e+ Z2k 1(s) and Y2k(s), − Y2n(s)− which gives a rational approximation. 1 k = 1, . . . , n, Za(rse) =gZiv(se)n+ impedances of the circuit elements. T(6h.1e) resulting 1 1 The relatioYns(hsi)p+between the finite domino ladder network, shown in Fig. 6.2, impedance Z(s) of the en2tire circuit can be foun1d easily, if we consider it in and the continued fracZ (tis)on+(6.1) provides an easy method for designing a circuit 3 1 the right-to-left diwrectiith a gon:iven impedance Z(sY).4(sF)o+r this one has to obtain a continued fraction ...... expansion for Z(s). Then the obtained particular expressions for Z2k 1(s) and 1 − Y2k(s), k = 1, . . . , n, will give the types 1of necessary comp1onents of the circuit Y2n 2(s) + Z(s) = Z1(s)+ and their nominal values. − 1 (6.1) 1 Z2n 1(s) + Y (s) + − Y2n(s) 2 Example 6.2.1. 1 To deZsig(nsa)c+ircuit with the impedance The relationshi3p between the finite domino ladder network1, shown in Fig. 6.2, and the continued fractionY4(6.1)(s) p+rovides 4a+n e4ass2y+m1ethod for designing a circuit Z(s) .=...... , ...... (.6..2.). . . with a given impedance Z(s). For this one 3has to obtain a continued fraction s + s 1 expansion for Z(s). Then the obtained particular expressions for Z2k 1(s) and 1 − Y2k(s), k = 1, . . . , n, will give the typeYs2onf n2ec(ess)sa+ry components of the circuit and their nominal values. − 1 Z2n 1(s) + − Example 6.2.1. Y2n(s) To design a circuit with the impedance The relationship between the finite domsi4n+o4sl2a+d1der network, shown in Fig. 6.2, Z(s) = , (6.2) and the continued fraction (6.1) provides asn3 +eass y method for designing a circuit witCh HaAgPivTenERim6p.edGanEcNe EZR(sA).L FAoPr PthiROs AoneCHhasTOtoFoRbtaiACnTAaNconCEtinDuEedVIfractiCESon 33 CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 31 expansion for Z(s). Then the obtained particular expressions for Z2k 1(s) and − 6.1.2 Negative impedance converters Y2kw(se),hkav=e 1to, .d.e.v, enl,owp iZll(gsi)veinthceonttyipnuesedoffrnaeccteisosnary components of the circuit and their nominal values. GeneralIt can be shown Apprthat the useoachof CFE for atonalog ueFractancesrealization of arbitrary GeneralCHAPTER 6. GE ApprNERAsL4 +AP4oachPs2R+OA1CH TtoO FR AFractancesCT1ANCE DEVICES 33 transfer functions may lead to the appearance of negative impedances. This Z(s) = = s + . (6.3) Example 6.2.1. s3 + s 1 1 observation is not unknown. For example, in the paper [10] S. C. Dutta Roy Towedehsaivgentao cdiervceuloitpwZi(tsh) tihnecoimntpineudeadncfreaction s + 1/2 Example 1: design a domino ladder3 cir9cuit 1with recalls Khovanskii’s continued fraction expansion for x found 1/2in [19] and CHAPTER 6. GENERAL APPROACH TO FRAsC+TANCE DEVICES 33 S. C. Dutta Roy on Khovanskii’s CFE for x : CHAPTER 64. GEN2 ERAL APPROACH T2O FRA2CTANCE DEVICES 33 makes a remark that s + 4s s+4 1+ 4s2 + 1 1 s Z(s) = 3 = s + 3. (6.3) Z(ss) =+ s 3 1 , 1 (6.2) “. . . if x is replaced by the complex frequency variable s, then we have to dweveehloavpe Zto(sd)evienlocpoZn(tsin) suinedc+onfsrtaincuteisdon+fraction 3 9 1 the realization would require a negative resistance. Thus, the [Kho- From this expansion it follows that s + 4 2 s4 + 4s2 + 1 2 2 1 vanskii’s] CFEs do not seem to be useful for realization of fractional s Z+(s4)s= + 1 = s + 1 s . (6.3) Find a CFE: Z(s) = 9 3= s + 1 1 1. 2 (6.3) inductor or capacitor.” s3 + s s + s 1 s 3+1 CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 31 Z1(s) = s; Z3(s) = s; Y2(s) =s +s;3 9Y4(s)1= s. 2 3 9 1s + 3 From this expansion it follows that 3 2 2 Then he describes a method for circumventing this difficulty, which gives s + s 2 2 3 6.1.2 Na ecognatintuievdefraimctiopneexdpaanscioen wciothnpvoesirtitve crosefficients. Therefore, for the analogue realization in the form ofs the first Cauer’s 9 1 3 2 HowHoevwereve, r,thethe ppossibilityossibility of reali ofzati ronealizationof negative im ofped ancnegatives in eleectri c canonic LCZ1(csir)c=uist;F[r2o0m] tZwhi3es(seh)xpa=avnse iostno; itchfoolYlo2ws(ess)tthah=et fso;llowinYg4(sv)a=luess.of coils and It can be shown that the use of CFE for analogue realization of arbitrary 2 3 3 impedancescircuits has bee ninp oelectricinted out by Bcirodecuits[3, Ch ahaspter I Xbeen]. Later ,pointedin 1970s, op eoutra- capacitoFrsro: m this expansion it follows that 9 1 2 transfer functions may lead to the appearance of negative impedances. This TherTherefeforeor,efor, the analogueZ1(s) = s; realiZz3(asti)on= isn; theY2form(s) = ofs; the Yfi4rs(st) =Caues. r’s tional amplifiers appeared, which significantly simplified creation of circuits ex- 2 3 3 canonic LC circuit [20] we ha9ve to9choose the 1follo1wing values o2f c2oils and observatibyon hH.ibisit niWnotg .n unkeBodegatnoivewrne sin.istF aor1945.nceesx, amnegpleativ,eincaptheacitpapanceesr, a[n10d]nSeg.aCtiv. eDinuttaductaRoncyes. Z1(s) = s; Z3(s) = s; Y2(s) = s; Y4(s) = s. L1 = 1 [H];TherefLore3,=for the[Hanalogue]; C2rea=lizati[onF ]i;n the Cform4 =of the[F ]fi.rst Cauer’s Such circuits are called negative-impedance converte1r/s2[9]. capacitors: 2 3 3 recalls Khovanskii’s continued fraction expansion for x found in [19] and For the firstcan oCauer’nic LC circsu2 itcanonic[20] we hav eLCto c h3ciroosecuitsthe fo llwowieng mv3austlues o ftakcoilseand The simplest scheme of a negative impedance converter (or current inverter) makes a remark that Thereforec,apforacittohers: analogue9 realization 1in the form of 2the first Cauer’s is shown in Fig. 6.1. The circuit consists of an operational amplifier, two ExamcanpleonLic16=L.2.2.C1 [cHirc];uit [2L0]3 =we h[aHve]; to chCo2o=se th[eF ]f;ollowiCng4 =valu[eFs ]o. f coils and “.re.s.isitforxs iosf reeqpualal credsisbtayncteheR,coamndpalecxomfrpeoqnueenntcwyitvhartihaebilmepse,dtahnecen Z. The 2 9 3 1 3 2 Tchaepafucintcotriso: n Z(s) gLiv1 e=n1b[Hy ]e; quatiL3on= (6.2)[H]; canCb2 e= wri[Ftten]; aClso4 =in [tFhe]. form entire circuit, considered as a single element, has negative impedance Z. 2 3 3 the realization would require a negative resistance. Thus, the [Kho- − Example 6.2.2. This means that I = V /( Z)). 4 29 1 2 vanskii’s] CFEs do noint seemin to−be useful for realization of fractional L = Exam1 [H]ple; s6.2.2.+L 4=s +[H1 ]; 1 C = [F1]; C = [F ]. For example, taking a resistor of resistance R instead of the element Z, we The funct1ionZZ((ss))=given b3y equation= (6.2)+ 2can be written a.4lso in the form(6.4) inductor or capacitor.” Z The functiosn3 Z+(ss2) given bysequati1on3(6.2) can1 be written3 also in the form obtain a circuit, which behaves like a negative resistance RZ . The negative 4 2 + − s + 4s + 1 4 1 2 3s 19 1 resistance means that if such an element of negative resistance, for instance, s + 4s + 1 1 1 Then he describes a method for circumventing this difficulty, which gives Example 6.2.2.Z(s) = 3 Z(s) = = + = + + . . (6.4) (6.4) s + s s3 1 s 2s11 2 1 10 kΩ is connected in series with a classical 20 kΩ resistor, then the resistance The function Z(s) given by equatison+ s(6.2)+can be +written also in the form a continued−fraction expansion with positive coefficients. 9 3s 1 9 1 of the resulting connection is 10 kΩ. 3s 3s + + 2 However, the possibility of realization of negative impedances in electric s4 + 4s2 + 1 1 2s 1 2 2s From this expansion it follows that 3s circuits has been pointed out by Bode [3, Chapter IX]. Later, in 1970s, opera- Z(s) = 3 = + 3s . (6.4) R s + s s 1 1 Iin From this expansion it follows that + tional amplifiers appeared, which significantly +simplified creation of circuits ex- From this exp1ansion it follows9that 3s 1 9 1 2 Z1(s) = ; Z3(s) = ; Y2(s) = ; + Y4(s) = . hibiting negative resistances, negativVeincapacita_nces, and negative inductances. s 1 2s 9 3s 2s 1 2 3s 2 1 Z1(s) = ; 9 Z3(s) = ; 1Y2(s) = ; Y42(s) = . Such circuits are called negative-impedance converters [9]. Z (s) = ; Z (s) =s ; Y (2ss) = ; Y3s(3ss) = . 3s Z Therefore1 , for the analogue3 realization2 in the form of4 the second Cauer’s The simplest scheme of a negative impedance converter (or current inverter) s Therefore, for the2sanalogue realizati3ons in the form of the3s second Cauer’s R From this expansion it follows that is shown in Fig. 6.1. The circuit consists of an operational amplifier, two canonTicheLreCforeci,rcforucaitntohe[n2ic0analogue]LCwecirhcuaivter[e2at0lio] zwacehtiohonaovseeintotthhecehofoformosellothweofinfogthellovwaisnleugceovsandluoefCs caoofueiclsor’silas nadnd capacitors: capac1itors: 9 1 2 resistors of equal resistancFeigRu,rea6n.d1: aNcegoamtipveo-nimenptedwanitche ctohneveimrtepre. dance Z. The canonic LCZci(rsc)ui=t [2;0] weZha(vse) =to ch;oose Yth(es)fo=llowi;ng vaYlue(s)o=f coil.s and 1 s 3 2s 2 3s 4 3s entire circuit, considered as a single element, has negative impedance Z. capacitors: 2 2 3 3 − C1 = 1 [F ]; C3 = [F ]; L2 = 3 [H]; L4 = [H]. This means that I = V /( Z)). TCh1e=ref1ore[F, ]for; theCanalogue3 = [Fr];ealizati9Lon2 =in3t[heH]form; ofL4the= sec[Ho2nd]. Cauer’s in in − 29 3 2 FoGeneralr example, taking a rAppresistor of reoachsistance R itonstead Fractancesof the element Z, we GeneralC1 = 1 [F ]Appr; C3 = oach[F ]; L 2to= 3 [ HFractances]; L4 = [H]. Z canonic LC cExamircuitple[206].2.3.we9have to choose the following va2lues of coils and obtain a circuit, which behaves like a negative resistance R . The negative Examcappleacito6r.2.3.s: To design a circuit with the impedance − Z CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 34 resistance means that if such an element of negative resistance, for instance, ExamTo deplesign6a.2.3.circuit with the impedance s4 + 3s2 + 8 Example II: design a domino2 ladZ(s)der= circuit, with 3 (6.5) 10 kΩ is connected in series with a classical 20 kΩ resistor, then the resistance Toneo dehsasigCton1aobtai=ci1rcn[uFiat];conwitihnCutoush3e=ifmractip[eFdon]a;nrepresce Len2 tati=2s33on+[Hof4]s;the functiL4 =on Z[(Hs),]. − Negative impedance converters are available: CHAPTER 6. GENERAL APP9ROsA4C+H 3TsO2 +FR8ACTANCE DEVIC2ES 34 of the resulting connection is 10 kΩ. CHAPTER 6. GENEs4R+AL3sA2P+P8ROAC1 H TO FRACT1ANCE DEVICES 34 Z(s) = Z(s) ==s4 +s +3s2 + 8 , (6.6) (6.5) Example 6.2.3. 2s3 + 4s 22s + 4s 1 one has to obtain a continuZous(s)fracti= on repres3 2s +entati, on of the function Z(s)(, 6.5) one has to obtain a continuous fraction2srepres+ 4ens tation1 of the1function Z(s), R To design a circuit4with 2the impedance s + I s 4+ 3s 2+ 8 1 1 in Z(s) = s + 3s + 8 = 1 s + −112 3 (6.6) + We have: Z(s) = 3 = s + 1 s (6.6) 2s 3+ 4s 2 4 2 1 −2 2s + 4s s +22s3s+s++ 8 Vin _ From this expansion it folloZw(sst)ha=t 1 1, 1 1 (6.5) 2s3 + 4s s +s + 1 1 −−1212 3 3 3 Z (s) = s; Z (s) = s; Y (s) = 2s; Ys(ss) = s. Z 1 2 3 −12 2 −2−4 2 −2 From this expansion it follows that R FTrohmeretfhiores ,exforpanstheionanalogueit followsretahaliztation in the form of the first Cauer’s 1 1 3 canonic ZL1C(s)ci1=rcuist; [20] Zw3e(s)h=ave t1osc;hoosYe2(tsh)e=fo2lslo; wingY4v(asl)u=es of sc.o3ils and Z1(s) = s2; Z3(s) = −12s; Y2(s) = 2s; Y4(s)−=2 s. 2 −12 −2 Figure 6.1: Negative-impedance converter. capaciTtohresr:efore, for the analogue realization in the form of the first Cauer’s Therefore, for the analogue realization in the form of the first Cauer’s canonic L1C circuit [20] we ha1ve to choose the following values of 3coils and L1 = [H]; L3 = [H]; C2 = 2 [F ]; C4 = [F ]. canocnaipcacLitCors2c:ircuit [20] we −ha1v2e to choose the following valu−es2of coils and capaHcietorersw: e se1e negative induct1ances and capacitance. Such elem3 ents cannot L = [H]; L = [H]; C = 2 [F ]; C = [F ]. be realized1 1us2ing passive 3elec−tr1ic2 component2s. However, th4ey −ca2n b3e realized NoticeL negativ= [H]; e valuesL = - [canH]; beC done= 2 [F ] ; usingC =OpAmps[F ]. with tH1hereh2ewlep soeef naecgtiavteiv3ecoinm−dpu1oc2tnaennctess, annadmcealp2yacoitpaenrcaet.inSgucahmeple4lmifieenr−tss.2caWnneotcan reaHbliezrereetawhliezsesbdeyeusnnienegggaapttiavisveseivineinpedeleudcacttnraiccneccoeomsnpavonenrdetencrtas.wpahHciociwthaenwvecares,. tdhSeeusyccrhciabeneldebmeinerenpatrlsiezvceidaonunsot bseurbwesaeitlcihtzieotdhne.uhsienlpg opfaasscitvivee ecloemctpriocnecnotms, pnoanmeenltys.opHeroawtienvgera,mtphleifiyercsa. nWbe craenalized realize this by negative inpedance converter which was described in previous with the help of active components, namely operating amplifiers. We can subsection. re6al.i3ze thiTs rbyanesgmativiessinipoendanlcienceosnvecritrercwuhiitch was described in previous subsection. Let6u.s3consTiderratnhescmirciusitsidoepnictleidninesFigc.i6r.c3,uwihtere Z2k 1(s) and Y2k(s), k = − 1, . . . , n, are given impedances of the circuit elements. This structure is known Let us consider the circuit depicted in Fig. 6.3, where Z2k 1(s) and Y2k(s), k = 6.3 Transmission lines circuit − as a1, t.r.a. ,nnsm, airsesigoinvenlinime poerdanscyems oeftrthiceaclirdcoumitienloemlaedntdse.rTlhaitstiscteruncteutwreoirskknaoswwnell. TheasreasutrlatninsgmismsiponedlaineceorZa(ss)ymofetrhiecaelndtoirmeinciorcluaditdecranlabtteicexnpertewsoserkd ass waeCll.FE Let us consider the circuit depicted in Fig. 6.3, where Z2k 1(s) and Y2k(s), k = desTcriheberdesublytin(6.1)g impweidthanscyemetriZ(s) calof thdeisetrintbutiire conircuofit elemencan be tsex−,pZre2sksed1(ass) a=CZF2Ek(s) 1, . . . , n, are given impedances of the circuit elements. This str−ucture is known anddesY2ckrib1e(ds)b=y (Y6.1)2k(sw),ithk =symetri1, . . .cal, n.distribution of elements, Z2k 1(s) = Z2k(s) as a tran−smission line or a symetrical domino ladder lattice−network as well. andIn tYh2ek c1a(se) =if Yw2ke(sa)s,skum=e1,t.h.e. ,sna.me elements in transmision line in series The resulti−ng impedance Z(s) of the entire circuit can be expressed as a CFE (Za) aInndtthheecsaasme eif iwneshaussnutm(eZtbh),ewsaemgetelaemsternutcstuinretrdaenpsimctiseidoninlinFeigi.n6s.e4r.ies described by (6.1) with symetrical distribution of elements, Z2k 1(s) = Z2k(s) (IZma)peadnadntchee osaf msuecihn kshinudnto(fZtb)r,awnsemgiestioanstlrinucet,uwrehdicehpictaend bine Fuisg−e.d6.4a.lso as a and Y2kIm1p(sed)a=ncYe 2okf(ss)u,chk k=in1d,o.f. .tr,ann.smision line, which can be used also as a model −of real cable line (solved by O. Heaviside in 1887), can be expressed as Imnotdehel ocfarseealifcablewe lianessu(smolve etdhebysaOm. eHeealveimsideentisn i1887),n trancansmbiseioenxprelinsesedinaseries Z(s) = Za.Zb. (6.7) (Za) and the same in shunt (ZbZ),(sw)e=getZaa.sZtbr.ucture depicted in Fig. (66..47.) If Iwmepseudbasnticteutoefasurecshistkaincde oZf atr=anRsma!insdiona clianpea,ciwtahniche Zcab n=b1e/suCse,dthaelnsothaes a If we substitute a resistance Z = R a!nd a capacitance Z = 1/sC, then the impedance is a b modeiml poefdraenacle cisable line (solved by O. Heaviside in 1887), can be expressed as R 1/2 R 1/2 jπ/4 Z(s) = RsZ−(1s/2)== RZaω.Z−1b/.2 e−jπ/4 s=jω. (6(.86).7) Z(s) = s− = ω− e− . (6.8) "CC "C |s|=jω If we substitute a resistanc"e Za = R a"!nd a capacitance Zb = 1/sC, then the impedance is R 1/2 R 1/2 jπ/4 Z(s) = s− = ω− e− s=jω. (6.8) "C "C | CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 34 one has to obtain a continuous fraction representation of the function Z(s), s4 + 3s2 + 8 1 1 Z(s) = = s + (6.6) 3 2 1 2s + 4s 2s + 1 1 s + 12 3 − s CHAPTECRH6A. PGTEENRER6.ALGAEPNPERROAALCHAPTPORFORAACCHTATNOCEFRDAEVCITCAENSCE34DEVICES 34 CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 34 −2 From this expansion it follows that one has tooneobtaihasn atoconobtaitinunousa confractitinonuousrepresfractientationonrepresof theenftatiunctiononofZthe(s), function Z(s), one has to obtain a continuous fracti1 on representation1of the function Z(s), 3 Z1(s) = s; Z3(s) =s4 + 3ss2;+ 8 4 Y12(s)2 = 2s; 1 Y4(s) = s. 4 2 2 −12 s + 3s + 8 1 −12 s + 3s + 8 1 Z(s) = 1 3 = s + (6.6) Z(s) = = s + 2Zs (+s)4=s 23 (6=.6) s +1 (6.6) There3fore, for t2he analogue re1alization2s in+ 4t2hess +form2 of the first C1auer’s 2s + 4s 2s + 1 2s1+ canonic LC circuit [20] we hav1e to ch1oose the followings +values of1coils a1nd CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVsI+CES 34 −12 3 s + 12 3 s −12 3 capacitors: − s −2 s −2 one has to obtain a continuous fraction repres1 Frenomtatithions eofxptheansiofunctin1it foonllo−wZ2s(stha), t 3 From this expansionLi1t =follow[Hs t]ha; t FL1ro3m=this ex[pHans]; io1n iCt 2fo=llo2ws[Ftha]; t C4 = [F3]. s4 + 3s2 + 8 1 2 Z (s) =1 s; −Z12(s) = s; Y (s) = 2s; Y (s−) 2= s. Z(s) = 1 = s + 1 1 2 3 1 −1(26.6) 3 2 1 4 −2 3 Z1(s2)s=3 +s4;s HeZr32e(sw) e=see nse;gativ1YeZ2(i1sn()ds)=u=c2tsa;nsc;es Ya4n(Zds)3(c=sa)p=acsit.ancse;. SucYh2(esl)em=e2nst;s canYn4o(ts) = s. 2 2−Ts h+1e2refore, for the analogue2 realizati−on2 −in12the form of the first Cauer’s −2 be realized using pa1ssive el1ectric components. However, they can be realized Therefore, for the analoguecanonriecaliLzCaticsTonir+hcueirintef[tore2he0], wformfore hatofvhee thetanalogueo chfiorsotseCtauehreearfli’sozllaotiwoning ivnaltuhees oformf coilsofanthed first Cauer’s −12 3 canonic LC circuwitit[h20t]hwee hchaeaplvpaeciotofrsac:hacotnioovsneictchoLemCsfpolcolionrwceuninittgs,[v2an0lau]mewseeolyfhacoovpielesrtaontidcnhgooasme pthliefiefrosll.owWineg cvanlues of coils and −2 capacitors: realize this by negcaat1pivaeciitnoprse:dance co1nverter which was described in3previous From this expansion it follows that L1 = [H]; L3 = [H]; C2 = 2 [F ]; C4 = [F ]. 1 subsection. 1 2 1 −12 3 1 −2 3 1 L1 = [H]; 1 L3 = [H]; C2 = 2 [F ]; 3 C4 = [F ]. 2 −1H2 ere we see Ln1eg=ative[Hind];uctanLce3s−=an2d cap[aHcit];ance. CSu2c=h e2le[mFe];nts canCn4o=t [F ]. Z1(s) = s; Z3(s) = s; Y2(s) = 2s; Y4(s2) = s. −12 −2 2 Here we see n−eg1a2tive inductances and capacitance.−S2uch elements cannot be realized usHinegrepawsesivseeelnecetgraicticvoemipnodnuecnttasn. cHesowaenvderc, atphaeycitcancbee. rSeuaclihzedelements cannot CHAPTER 7. REALIZATION OF FRACTIONAL CONTROLLERS 40 Therefore, forbe rtehealizanalogueed using6p.rae3sasilivzeaetiTleonwcrtitrahicnntchtosheemmhpeformolnipsenostfsiof.aocHtntheiovwe elcfivioersnmr,tpetoChsneaueeynctcsira,’srnncbamue erileytaliozpeedrating amplifiers. We can be realized using passive electric components. However, they can be realized canonic LC cirwcuitiht [t2h0e] hweelphoafveacttoivechcooomrsepaoltinhzeenttfhso,ilslonbwayminnegelygvaaotiplvueersaintpoinfegdcaoanimclseplaciofinnedvrse.rteWr we hciachn was described in previous 1k! with the help of active components, namely operating amplifiers. We can 1k! capacitors: realize this by neLgeattiuves icnopnessduiadbnseecrcetichooenn.cvierrctueritwdheipchicwteads idnesFcriigb.e6d.3in, wprheevrieouZs2k 1(s) and Y2k(s), k = 1µF − R 1k! 1 subsection. 1, . . . , n, are given rimeapliezedatnhcisesboyfnt3ehgeactivrceuiintpeeldemanecnetsc.onTvheristesrtrwuhcticuhrewiasskdneoswcrnibed in previous L = [H]; L = [H]; C = 2 [F ]; C = [F ]. 1µF 1k! 1 2 3 −1a2s a transmi2ssion sliunbeseocrtiaon4s.ym−et2rical domino ladder lattice network as well. 6.3 Transmission lines circuit 1µF Here we see negative inductances and capacitance. Such elements cannot The resulting impedance Z(s) of the entire circuit can be expressed as a CFE 1k 6.3 Transmission lines circuit 1k! ! be realized using passive electric componeLnetsu. sHcoonwseidveerr,thteheciyrccuaitndbepeicrteeadlizinedFig. 6.3, where Z (s) and Y (s), k = described by (6.1) with symetrical distribution of elements2,k Z12k 1(s) =2k Z2k(s) 1µF 1 F 6.3 Transmission lines cir−cu−it µ with the help Lofetaucsticvoensciodmerptohneecnirtcs,uitnadme1p,e.ilc.yt.e,donp,inearrFaetigin.ivg6e.n3a,immwpphleifidreaenrZsc2.eks oW1(fset)hcaeancndircYu2ikt(se)le,mke=nts. This structure is known and Y2k 1(s) = Y2k(s), k = 1, . . . , n.− 1µF 1k! − ZF(s) realize this by negative inpedance convertaesrawthraicnhsmwisassiodnelsicnreiboerdainsympreetvricoaulsdomino ladder lattice network as well. C 1, . . . , n, are given imIpnedtahnececsaosfethife wcLierctuauistsseuclomenmeseindtthesre. tTshahemicseisrtceruuliectmtudereneptisctkeindowitnrnaFnisgm. 6is.3io,nwhlienree Zin2kse1r(ise)sand Y2k(s), k = General Approach to Fractances − General Approach 1toµF Fractances subsection. as a transmission line or a sTyhmeertersiucalt1li,nd.go.m.im,inpo,edalarnedcdgeeivrZel(nast)itmiocfpetehndeatewnncoterirskeoacfsirtcwhueiltlc.ciracnubiteeelxepmresnstesd. aTshaisCsFtrEucture is known (Za) and the same in shunt (Zb), we get a structure depicted in Fig. 6.4. The resulting impedance Z(dess) ocrif tbheed ebnyti(r6.1)e cirwcuithit csaynmetribe ecalxprdeissstriedbutias ona CofFEelements, Z2k 1(s) = Z2k(s) Impedance of sauscha tkrianndsmoifsstiroannslmineisioornalisnyem, ewtrhicicahl dcoamn ibnoe ulasd−ederallsaottaicse anetwork as well. described by (6.1) with symetriandcalY2kdi1s(tris)buti= Yon2k(sof), elemenk = 1, ts. .,. ,Zn. (s) = Z (s) Sample implementations:Figure 7.4: RC binary tree circuit. CHAPTER 6. G−ETNhEe RreAsuLltAinPg PimRpOeAdaCnHc2ek TZ1O(s)FoRfAt2hCk eTeAnNtirCeEciDrcEuiVt IcCanESbe ex3p4ressed as a CFE 6.3 Transmission mliondeelsofcreiarlcIncuabletihte clianese i(sf wolve eadssubmyeOt.heHseaam−veisiedelemiennt1887),s in tracnansmibsieonexliprene sinsesderaiess and Y2k 1(s) = Y2k(s), k = 1, . . . , n. !5 CHAPTER 6. GENER−AL APPROACH TO FRACTANCE DEVdesICEScri3b5ed by (6.1) with symetrical distribution of elements, Z2k 1(s) = Z2k(s) In the case if we assume(Ztah)easnadmteheelseammeentins sinhutnrtan(Zsmb)i,swioengleint ea isntrsuecrtiuerse depicted in Fig. 6.4. − !10 Z(s) = Za.Zb. (6.7) !15 Z Z oneZ has to obtaiZ n aandconYt2iknu1ous(s) =fractiY2k(ons), represk = 1,en. .tati. , non. of the function Z(s), Let us consider the ci1rcuit d3epicted5 in Fig. 6.32n-1Im, wphedearencZe2okf 1s(usc)haknidndY2okf(tsr)a,nksm=ision line, which can be used also as a (Za) and the same in shunt (Zb), we get a st−r−ucture depicted in Fig. 6.4. !20 In4the ca2se if we assume the same elements in transmision line in series !25 1, . . . , n, are given impedances oIf twhe csiurbcusmtitioteudelteelmoaefnrrteesa.sliTscstablehain+scseli3tnersZuca(s+tu=olv8reReids abk1!nyndoOwa.nHceaapvaiscideitani1nce1887),Zb =can1/bseCe,xtprehesnsetdhaes Impedance of such kind of transmision line, which can be used also as a Magnitude [dB] Z(s) Y Y Y Y Y Y Y Y !30 as a transmission l1 ine o2 r 3a sym4 e5trica6l dom2n-1inoZ2nla(dsd()Ze=ra)laatntidc3ethneetswamorek=ianssswh+uelnl.t (Zb), we get a structure depic(t6e.d6)in Fig. 6.4. Z Z impZ edance isZ 1 !35 model2 of real 4 cable li6ne (solved 2nby O. Heavis2isde +iTnak41887),s e Zcan2(s)b=e expreZa.sZsbe.d as (6.7) !40 2s + 2 3 4 The resulting impedance Z(s) of the entire circuit canImbepedxapnrecseseodf asuscahCkFinEd of transmision line, which can be used also as a 10 10 10 1 1 Frequency [rad/sec] If wZe(su) b=stituZtea.Za br.eRsistan1c/e2 Za = R a!nd1/a2 ca(p6jaπ.c7/i4)tance Zb = 1/sC, then the Figure 6.3: General structure of transmission line. model of real −cable line (solv−ed by− O.sH+eaviside in 1887), can be expressed as described by (6.1) with symetrical distribution ofZelemen(s) =ts, Z2ks 1(s) = Z2k(sω) e s=jω.3 (6.8) !30 − −12 Za Za Za impZaedance is C C | s and Y (s) =IYf w(ess)u, bkst=itu1t,e. .a. ,rens.istance Za = R a!nd a c"apacitance Zb "= 1/sC, then the !40 2k 1 2k CHAPTER 6. GENERAL APPROACH TZO(sF)R=ACTZA−aN.2ZCbE. DEVICES 34 (6.7) − CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 34 !50 In the case imif pweedaanscseumis e the same elements in transmision linRe in 1s/e2ries R 1/2 jπ/4 Z(s) From this expansion itZf(os)ll=ows thas−t = ω− e− s=jω. (6.8) !60 Zb Zb Zb Zb Zb Zb Zb Zb If we substitute aC resistance CZa = R a!nd|a capacitance Zb = 1/sC, then the (Za) and the same inZa shuntZa(Zb), wZa e get aRonestZrahuasctutoreobtaidRepinctaedconintiFn"uigous. 6.f4racti. on"representation of the function Z(s), Phase [deg] !70 one has to obtain a continuous fracti11/2 on represen1/2tatiThenjonπ/41of the function Z(s), 3 Z(s) = s− =impedωa−nce eis− s=jω. (6.8) !80 Impedance of such kind of transmZi1s(iosCn) =line,s;whichCZc3a(ns)b=esu4 s+ed3| ssa2;l+so8asYa12(s) = 2s; 1 Y4(s) = s. 4 " 2 2 " 12 2 !90 − − 2 3 4 Figure 6.4: Transmission line circuit compossed o+f tw3osinp+edanc8 e Za1and Zb.(s) = 1 3 = s + (6.6) 10 10 10 model of real cable line (solveZd(sb)y=O. Heaviside i=n 1887),s + can be2esxpre+ 4ssed as2 R 1/(26.6) R1 1/2 jπ/4 Frequency [rad/sec] There3fore, for t2he analogue re1aZli(zsa)ti=on in st−2hes +form= ofωthe− fie−rst Csaue=jωr.’s (6.8) 2s + 4s 2s + C 1 C 1 | 6.4 Tree structure ciZrc(usi)t= Z .Z . 1 1 (6.7") "s + canonic LCa cibrcuit [20] we have to choose the followi1n2g valu3es of coils and Figure 7.5: Bode plots of the I1/2 controller where half order integral was s + − s Let us consider the total impedance ofctheapfaracctianctoercsirc: uit as shown in Fig. 6.5, −12 3 approximated by transmission line depicted in Fig. 7.2. If we suwbhsicthithuastaereacurseivseisttraucntucre wZitha t=he cRombai!nnatdionaofctawopaimcpietdaancceseZZb = 1/sC, thens the −2 a −2 impedanandceZib.sImpedance is derived as geometrical mean of Z1a andFroZmb, this expansion1it follows that 3 From this expansionLi1t =follow[Hs t]ha; t L1 3 = [H]; 1 C2 = 2 [F ]; C4 = [F3]. Z(s) = Za.Zb. 2 Z (s) = s; −Z12(s) = s; Y (s) = 2s; Y (s−) 2= s. R1 1/2 R 1/121 jπ/4 2 3 −12 3 2 4 −2 Z(sZ)1=(s) = s−; ! H=eZr3e(sw) e=ωs−ee nsee;−gativYse=2(jisnω).d=uc2tsa;nces Ya4n((ds6).8c=)apacsit.ance. Such elements cannot C2 C −Th1e2refore, |for the analogue realizati−on2 in the form of the first Cauer’s If we substitute a resista"nce Za =beR arnedaal"iczapeadcitauncseinZbg=p1/assCs, itvhen electric components. However, they can be realized the impedance shTowhsetrhefforeracta,ncfore chartahecterisanaloguetics asc(asene oalnsroiec(a6.liL8)zC)aticonircuiint [t2he0] wforme haofve theto chfiorsotseCtauehe rf’sollowing values of coils and canonic LC circuwitit[h20t]hwee hhealvpe otof achcotiovse tchoemfpololonwenintgs,vanlaumeseolyf coopilesrantidng amplifiers. We can R 1/2 R 1/2 jπc/4apacitors: Z(s) = s− = ω− e− s=jω, capacitor"s:C re"aClize this | by negat1ive inpedance co1nverter which was described in3previous which is a fractional order in1tegral with absolute value o1f imLpe1danc=e prop[Hor-]; L3 = [H]; 3 C2 = 2 [F ]; C4 = [F ]. CHAPTER 6. 1/G2 ENERAL APPROsuACbHseTctOioFnR.ACTANCE D2EVICES 35 −12 −2 tional to ω− and phasL1e =angle is[Hcons];tant πL/43, in=dependent[oHf th]e; frequenCcy2. = 2 [F ]; C4 = [F ]. − However, it is impossible t2o construct such an infin−ity1sH2truecrteurewaes ssheowe inegative inductances−an2d capacitance. Such elements cannot Z Z Z Z Fig. 6.5. TherefoHre1,eirteseewmes tsoe3bee inmepgoratatnivt5 eto isntuddbyuecthtreaeenafflecicezt2n-1seodfathnuedsniuncmga-ppaacsistiavnecel.ecSturicchceolmempoentesntcsa.nHnotwever, they can be realized ber of stage in the recursive self infinity (binary) structure on the impedance be realized using6p.a3ssive eTlewcrtitrahicntchosemmhpeolnipsenostfsi.aocHtniovwe elcvioenmr,petohsneeynctcsia,rnncbamue erileytaliozpeedrating amplifiers. We can characteristics [31]. CHAPTER 6. GENERAL APPROACH TO FRACTANCE DEVICES 35 Z(s) Y Y Y Y Y Y Y Y A similawr ti1rteehstrtuhcte2ureh3aeslpa froacft4alamc5todiveleofcr6oumgrhepainoltinezr2n-1efanctethsb,iestwn2nebaeynmtnweoelygaotipveraintpinegdaanmceplciofinevrse.rteWr we hciachn was described in previous materials of very different conductivities, e.g. an electrode and an electrolyte Z2 Z4 Z6 Z2n realize this by neLgeattiuves icnopnessduiadbnseecrcetichooenn.cvierrctueriZt1wdheipchicwtZea3ds idnesFcriigZb5.e6d.3in, wprheevrieZo2n-1uZs2k 1(s) and Y2k(s), k = was studied in [21]. There was described a slightly different structure of tree − subsection. 1, . . . , n, are given impedances of the circuit elements. This structure is known Figure 6.3: General structure of transmissionZ(s)line. as a transmission lineY1or a Ys2 yYm3 etriYc4alY5domYin6 o ladY2n-1der lYa2nttice network as well. 6.3 TransZ missiZon linZ es circuZit 6.3Za TraZansTmheiZrsaessuioltninglimnZpeaesdacnicrec2Zu(si)tof 4the entir6 e circuit ca2nn be expressed as a CFE described Lbeyt (u6.1)s conwsidither tshyemetricircuicalt dedpiisctritedbutiin Fonig. 6of.3,elemenwhere tsZ2,k Z12(ks) 1a(nsd)Y=2k(Zs2)k, (ks=) Z(s) Figure 6.3: General structure of transmission line. − − LZbet us ZbconZbsider ZbtheZbcircuiZbt de1p, .ic.t.Zbe,dn,inarFZbeig.iv6e.n3,imwphedreanZc2eks o1(fst)haencdircYu2ikt(se)le,mke=nts. This structure is known and Y2k 1(s) = Y2k(s), k = 1, . . . , n.− Za Za Za − Za Za Za Za Za CHAPTER 7. REALIZATION OF FRACTIONAL CONTROLLERS 41 1, . . . , n, are given imIpnedtahnecaecsaoasfettrhiafenswcmiericsuasistosneulmelimneeentothrse.aTssayhmisesttreriulceacmtluderonemtisinkionolawtdrnadenrsmlatisticoennleitnweorikn asserwielsl. CHAPTER 7. REALIZATION OF FRACTIONAL CONTROLLERS 42 Generalas a transm iApprssion line oroacha sTyhmeertersiucal tlintodgomiminp oeFractancesdlandcdeerZl(ast)tiocfethnetewnotrirke acsircwueiltl.can be expressed as a CFE General Approach to Fractances Figure 6.4: Transmission line circui(Zt caom) aponsdedtohf etwsoaimnpediancn eshZua nandt (Zbb.), we get a structure depicted in Fig. 6.4. describeZ(s)d by (Zb6.1) wZbithZbsymetriZb calZb disZbtributionZb of elemenZb ts, Z2k 1(s) = Z2k(s) The resulting impedance Z(s) of the entire circuit can be expressed as a CFE − Impedance of such kZina d of trZa nsmisiZoan line, whiZcah can be used also as a Sample implementations -- transmission lines -- responses: described by (6.1) with symetriandcalY2kdi1s(tris)buti= Yon2k(sof), elemenk = 1, ts. .,. ,Zn2.k 1(s) = Z2k(s) 6.4 Tree structure circuit − − and Y (s) = Ymo(sde), lko=f r1e,a. l.I.nc,ablenth. e clianese i(sf wolve eadssubmyeOt.heHseaamveisiedelemiennt1887),s in tracnansmibsieonexliprene sinsesderaiess CHAPTER 6. GE2Nk ER1 AL APPRO2kACH TO FRACTANCE DEVICES 36 0.6 0.4 − Figure 6.4: Transmission line circuit composed of two inpedance Za and Zb. 0.4 Let us consider the total impedance of the fractanc(Ze cai)rcauintdastshhoe wsanmineFiing. 6.5,shunt (Zb), we get a structure depicted in Fig. 6.4. 0.2 In the case if we assume the same elements inZt(rsa)ns=misioZnal.iZnbe. in series (6.7) 0.2 0 0 which has a recursive structure with the comZabinatioInmopfetdwaonicmepoedf asnuccehs Zkaind of transmision line, which can be used also as a !0.2 (Za) and the samZae in shunt (Zb), we get a structure depicted in Fig. 6.4. !0.4 Output signal: U [V] !0.2 and Z . Impedance is derived as geometrical mea6n.o4f Z TandreZe, structure circuit Output signal: U [V] b If we suZbbsmtiotudetel aoaf rreeaslbisctableancelineZa(s=olvRed ab!nydOa. Hceaapvaiscideitanince1887),Zb =can1/bseCe,xtprehesnsetdhaes !0.6 !0.4 ImpZaedance of such kind of transmision line, which can be used also as a !0.8 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.005 0.01 0.015 0.02 0.025 0.03 CHAPTER 7. REALIZATION OFTimeFR [sec]ACTIONAL CONTROLLERS 44 CHAPTER 7. REALIZATION OFTimeF R[sec]ACTIONAL CONTROLLERS 43 impedaZanceLeitsus consider the total impedance of the fractance circuit as shown in Fig. 6.5, 1 model of realZc(ables) Zb= lineZa.(sZbolv. ed by O. Heaviside iTnak1887),e Zcan(s)b=e expreZa.sZsbe.d as (6.7) 1 0.5 Zb which has a recursive structure with the combination of two impedances Za 0.5 ! R R 0 Iandf wZZe(bs. u)Ibm=spteidtuaZntceae.Zaisbr.deesriisvteadna1cs/e2geZoame=triRcalam!neda1n/a2ocf aZ(pa6jaπ.andc7/i4)taZnbc, e Zb = 1/sC, then the 0 Za Z(s) = s− = ω− e− s=jω. (6.8) Input signal: E [V] !0.5 Za Input signal: E [V] !0.5 If we substitute a resistance Za = R and a cimappaceidtaanncceeZibs= 1/sC,Cthen C | sine !1 unit step 0 0.005 0.01 0.015 0.02 0.025 0.03 If we substitute a resistance Za = R a!nd a c"apacitance Zb "= 1/sC, then the !1 Zb Z(s) = Za.Zb. 0 0.005 0.01 0.015 0.02 0.025 0.03 Time [sec] the impedance shows thZbe fractance characteristics as (see also (6.8)) Time [sec] impedance is R R 0.6 1/2 1/2 jπ/4 1/2 Za 0.4 Figure 7.7: Time response of the I controller to sin input where half order Zb Z(s) = s−! = ω− e− . (6.8) 1/2 s=jω Figure 7.6: Time response of the I controller to unit-step input where half integral0.4was approximated by transmission line depicted in Fig. 7.2. R 1/2 R 1/2 jπ/4 C C | order inte0.2gral was approximated by transmission line depicted in Fig. 7.2. Z(s) = s− = ωZb− Re− s=jω, R " " 0.2 If 1w/e2 substitute a 1r/e2sistajnπc/e4Za = R and a capacitance Zb = 1/sC, then 0 Then 0 Z(s) "C Z("s)C= s− | = ω− e− s=jω. (6.8) Output signal: U [V] !0.2 "Cthe impedan"ce sChows the fractan|ce characteristics as (see also (6.8)) Output signal: U [V] !0.2 !0.4 which is a fractional order integral with absolute value of impedance propor- !0.4 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.005 0.01 0.015 0.02 0.025 0.03 Time [sec] Time [sec] Figur1e/26.5: A self-similar tree circuit composed of two inpedance Za and Zb. tional to ω− and phase angle is constant π/4, independent of the frequencyR. R 1 1/2 1/2 jπ/4 1 − Z(s) = s− = ω− e− s=jω, 0.5 However, it is impossible to construct such an infinity structure as show iCn C | 0.5 circuit, where capacitance C was the same at every stage and resistance R " " 0 0 Fig. 6.5i.ncTreahseerdebfoyrteh,eirtatsieoeamats etvoerbyestageimpoforbranctanthting.o stTuhedyrestuhltie ngeffiemcptedoancf thee num- which is a fractional order integral with absolute value of impedance propor- Input signal: E [V] !0.5 Input signal: E [V] !0.5 ber of sZta(sg)ehiand thheeforremcuorfsaivceonsteinluf eidnfifrnacitiyon(bexipnaanrsyio)n.structure o1/n2 the impedance saw ramp !1 tional to ω− and phase angle is constant π/4, independent of the frequency. !1 0 0.005 0.01 0.015 0.02 0.025 0.03 phase angle is constant !"/4, independent of the frequency 0 0.005 0.01 0.015 0.02 0.025 0.03 Time [sec] characteristics [31]. − Time [sec] However, it is impossible to construct such an infinity structure as show in A similar tree structure as a fractal model of rough interface between two 1/2 Fig. 6.5. Therefore, it seems to be important to study the effect of the num- Figure 7.8: Time response of the I1/2 controller to saw input where half order Figure 7.9: Time response of the I controller to ramp input where half order integral was approximated by transmission line depicted in Fig. 7.2. integral was approximated by transmission line depicted in Fig. 7.2. materials of very different conductivities, e.g. anbelrecotfrsotdaegeanind tahne erleeccutrsoilvyeteself infinity (binary) structure on the impedance was studied in [21]. There was described a slightclhyardaicffteerreisntticsstr[u31ct].ure of tree A similar tree structure as a fractal model of rough interface between two materials of very different conductivities, e.g. an electrode and an electrolyte was studied in [21]. There was described a slightly different structure of tree III.1 Commensurate Order Systems General Approach to Fractances Stability of fractional order systems • What is a commensurate order system? • If one selects the greatest common divisorAn ! of Example Sample implementations: Commensuratethe orders, order such systems: that CHAPTER 7. REALIZATION OF FRACTIONfractanceAL CONTROLLERS 38 where p and q are the orders of the rational approximation, P and Q are polynomials of degree p and q, respectively. Block diagram of the analogue • A fractional order system R 2 An Example ZF Ri _ R 1 An Example _ E + + + U • A fractional order system CHAPTER 7. REALIZATION OF FRACTIONAL CONTROLLERS 40 Figure 7.1: Analogue fractional-order integrator. CHAPTER 7. REALIZATION OF FRACTIONAL CONTROLLERS 39 CHAPTER 7. REALIZATION OF FRACTIONAL CONTROLLERS 39 •• AIf !fractional can be found, order the system commensurate order 1k realization of fractional-order operator is shown in Fig. 7.1. 1k! ! 0.47µ F 0.47µ F 0.47µ F 0.47µ F 0.47µ F 0.47µ F • It is obvious that !=1/6, thus the original order 0.47µ F 0.47µ F 0.47µ F 0.47µ F 0.47µ F 0.47µ F For example: 1 F model can easily be written R 1k! µ λ 7.2 Realization of fractional I controller 1µF 1k! ZF(s) 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! 22k ! Z (s) 22k 22k 22k 22k 22k 22k 22k 22k 22k 22k 22k 22k system can be written as F ! ! ! ! ! ! ! ! ! ! ! ! 7.2.1 Description of circuit 1µF FOC tutorial III 14 ©Dingyu Xue, NEU, PR China 0.47µ F 0.47µ F 0.47µ F 0.47µ F 0.47µ F 0.47µ F λ 1k 0.47µ F 0.47µ F 0.47µ F For e0.47µxpeFriment0.47µal mF easure0.47µmeFnt we built a fractional-order I controller which 1k! ! • It is obvious that !=1/6, thus the original order λ µ Figure 7.2: RCistransa pamirtiscsuionlarlicneasecircuiof tt.he P I D controller, (if Kp = 0 and Td =10µ)F. The con- 1µF Figure 7.2: RC transmitrsosilonlerliwneascirrcuiealit.zed in three forms, namely by: the symetrical domino ladder or It is obvious that =1/6, thus the original order 1k 1k 1k 1k 1k 1k can be written as ! ! ! ! ! ! Z (s) 1µF 1k! • ! transmission line (see Fig. 7.2) for n=6, the finite doFmino ladder (see Fig. 7.3) system can be written as 1k! 1k! 1k! 1k! 1k! 1k! C for n = 6 and the tree structure (see Fig. 7.4) for n = 4, and Fractances with 1µF 1µF 1µF 1µF 1µF 1µF 1µF 1µF 1µF 1µF 1µF im1µpFedance1ZµFF were1cµoFnnecte1dµFto feedback in operational amplifier (Fig. 7.1). Z (s) system can be written as Z (s) F It should be noted that the described methods work for arbitrary orders, but F where the circuit elements with computed values are not usually available. Because Figure 7.4: RC binary tree circuit. of this, in our experiment we proposed and realized the integrator with order Figure 7.3: RC domino ladder circuit. Figure 7.3: RC dominλo=lad0d.5e.r circuit. !5 It should be mentioned that this simple case of th!e10 controller order can FOC tutorial III 15 we have Ti = 1.4374. The transfer function of the realized analogue fractional- !15 we have Ti = 1.4374. The transfer functiobneorfetahleizreedaliazlesdoaunsailnogutehferamcteiotnhaold- s described in [31, 33, 38, 54], which do not order Iλ controller is: !20 where ©Dingyu Xue, NEU, PR China order Iλ controller is: involve expli0c.5it rational approximations. !25 C(s) = 10..45374 s− . (7.2) C(s) = 1.4374 s−In .the case, if we will us(e7.i2d)entical resistors (R-seMagnitude [dB] r!30ies) and identical ca- λ Adjustment of the integration constant Ti of the fractionalλ-order I con- !35 Adjustment of the integration constanpacit Ttorsi of th(Ce -frsahcutniotn)ali-nortdhere Ifraccotna-nces, then the behaviour of the circuit will where troller (or half order integrator) depicted in Fig. 7.1 was done by resistor Ri. !40 2 3 4 troller (or half order integrator) depictedbien aFsig.a 7.ha1lfw-asorddoneer inbtyegrresatisotror/dRiffi.erentiator. We used the resisto10r values R = 10 10 FOC tutorial III 15 If we change the resistor Ri, the integration constant changes the value in the Frequency [rad/sec] If we change the resistor Ri, the integrati1onkΩcons, (Rtan=t cRhanges, j = t1he, . .v.a,luen)inandthethe capacitor values C = 1µF , (C = C, j = required interval. j j ©Dingyu Xue, NEU, PR China required interval. !30 For measuremets we used freq1u, .en. .cy, n1)0.0FHozrabnetd taermpmliteasudeurem10enVt. results we used two operational amplifiers For measuremets we used frequency 100 Hz and amplitude 10 V. ± !40 TL081CN in inverting± connection. FOC tutorial III 15 !50 7.2.2 Experimental resultTshe resistors R1 and R2 are R1 = R2 = 22kΩ. The integration constant Ti ©Dingyu Xue, NEU, PR China 7.2.2 Experimental results 2 !60 can be computed from relationship Ti = 1/ R/(Ri C), and for Ri = 22kΩ 0.5 ∗ Phase [deg] Transmission line approximation of0.5the s− !70 Transmission line approximation of the s− ! !80 In Fig. 7.5 and Fig. 7.6 - 7.9 the measured characteristics of realized ana- In Fig. 7.5 and Fig. 7.6 - 7.9 thλ e measured characteristics of realized ana- !90 logue fractional-order I controller, where half order integral was approxi- 2 3 4 logue fractional-order Iλ controller, where half order integral was approxi- 10 10 10 mated by transmission line depicted in Fig. 7.2, are presented. It can be seen Frequency [rad/sec] mated by transmission line depicted in Fig. 7.2, are presented. It can be seen from Fig. 7.5 that the realized analogue fractional-order Iλ controller provides from Fig. 7.5 that the realized analogue fractional-order Iλ controller provides 2 2 1/2 a good approximation in the frequency r2ange [10 rad/s2ec, 5 10 rad/sec]. (For Figure 7.5: Bode plots of the I controller where half order integral was a good approximation in the frequency range [10 rad/sec, 5 10 rad/·sec]. (For comparison see expected Bode plots shown in Fig. ·3.2.2.) comparison see expected Bode plots shown in Fig. 3.2.2.) approximated by transmission line depicted in Fig. 7.2. Stability of fractional order systems III.2 Stability of Fractional Order Systems • The ! curve – if , then the stable condition for the system is Unstable region with four poles in the stable region FOC tutorial III 17 ©Dingyu Xue, NEU, PR China> 1), and Hl(s) 2n (1 + sT )α ! " theHapl1(psro)xim=ation f1or+low frequencies (ω << 1). (5.2) TThheettrraannsferr ffuunncctitoinonofotfhtehseysstyemstesmhowshnoiwn nFiign. 4F.5igi.s 4th.5enisgitvheennbygitvheen by the 5.2 General Cα FwEheremHshe(st) his tohedapproximation for high frequencies (ωT >> 1), and Hl(s) Figure 4.4: Nested multiple-loop control system – level 3. Hh(s) = 1 ! " (5.1) relatirelationsonshhiipp H (s) = (Exam11++plesT )5.2.1.α the approximation for low freq(u5en.2ci)es (ω << 1). In gl eneral [10], aPrearformitionangl athepprCoFEximofathetionfunctiof theon (5.1),functwioithn GT (=s)1,=α s= 0α.,5,0w>3 1), and2 Hl(s) * Y2n 2(s) + Example: s s + 1.333s + 0.478s4 + 0.064s3 + 0.0028442 Y2n(s) − − 11 0.3513s + 1.405s + 0.8433s + 0.1574s + 0.008995 Y2n 2(sZ) + (s) + Example 5.2.1. ! " H1(s) = − 2n 1 1 the approximation for low frequencies (ω << 1). 4 3 2 − Y2n(s) 1 s + 1.333s + 0.478s + 0.064s + 0.002844 Z2n 1(s) + Example 5.2.2. Z2n -1(s) − Y (s) Performing the CFE of the function H(5.1),h(s) w=ith T = 1, α = 0.5, we obtain: (5.1) Continuing this process, we obtain the transfer function of the nested2mn ultiple- where H (s) is the approximation for highPefrformirequngethenciCeFEs of(ωtheTfuncti(>1 +>onsT1),(5.2),)α awndith TH= 1(,sα)= 0.5, we obtain: Examh ple 5.2.1. Example 5.2.2. l loop control system shown in Fig. 4.6 in the form of a continued fraction 4 3 2 α Y2n -2 (s) Continuing this process, we obtain the transfer function of the nested multiple- 4 3 1 2 the approPxeimrformiationgn ftheor lCoFEw fofreqtheuen0functi.c3i5e1s3ons(ω+(5.1),<1.<4051ws)i.th+ 0T.H8=P4(e3srformi13),ssα=+ng=40sthe.01+.55,+C72FE.4ws eof+obtaithe00.4.04f8uncti0s8n:+990on.502(5.2),56 with T =(15,.2α)= 0.5, we obtain: loop control system shown in Fig. 4.6 in the form of a continued fraction H (s) = H2(s)l = 1 4 3 2 4 3 s 2 s + 1.333s + 0.478s +9s 0+.01624s s++4.302.s00+4280.457436s + 0.02256 + + Z2n-3(s) + + ! " s + 4s + 2.4s + 0.448s + 0.0256 1 4 3 2 H (s) = - - Example 5.2.1. 2 4 3 2 + + 0.3513s +w1h.e4r0e5Hs (+s)0is.8t4h3e3aspp+rox0i.m1a5t7io4ns +for0h.0ig0h8f9r9eq5u9sen+cie12ss(ω+T4.3>2s> +1),0.5a7nd6s +H0.(0s25)6 H1(s) = h l Figure 4.5: Nested multipleY* (s)-loop control system – level 4. Example4 5.2.2. 3 2 2n Performing the CFE ofs the+ 1t.3hfuncti3e3asppr+oonx0im.4(5.1),5a7t.8i3osn f+oCrw0al.oi0rthw6ls4frosTenq+u’=se0n.mc10ie0,es2αt8(hω4=o4