A Comparison of Fractional Smith Predictors

Transcription

1 ) with it gralizd itgral ad drivativ actior bojour, 3 Europa Cotrol Corc ECC) July 7-9, 3, Zürich, Switzrlad A Copario o Fractioal Sith Prdictor N Maari, M outit ad JC rigaou Abtract : A w cla o ractioal Sith prdictor i propod i thi papr, aociatig Itral Modl Cotrol or th dig o ti dlay cotrollr to th CRONE tplat i ordr to iprov robut to tatic gai variatio Nurical iulatio illutrat itriic ad tructural robut prorac o th ractioal PI ad PID prdictor Copario o th two cotrollr how that th Sith PID prt th gratt ability to rjct gai variatio ad yt tructur ucrtaity Kyword : Fractioal PID, i dlay yt, Itral odl cotrol, Sith prdictor, Robut prorac I INRODUCION PID cotrol ad th tuig o PID cotrollr rai a activ rarch ild Fractioal calculu [,,4] ha crtaily cotributd to thi it activity, thak to th ractioal PID cotrollr itroducd by Podluby [3] with it gralizd itgral ad drivativ actio Sic thi iitial papr, a lot o publicatio hav b dvotd to th tuig o th P I α D β cotrollr [7,8,] i dlay yt ca b cotrolld by claical PID rgulator, but it i wll kow that or icrad prorac, a pciic cotrollr i rquird, which i calld th Sith prdictor [6] Bcau th ti dlay caot b caclld, it ha to b icludd i th clod loop rc odl, accordig to Itral Modl Cotrol IMC) thory [9] So, th quivalt dback cotrollr corrpod to th Sith prdictor, which ca b itrprtd a a gralizd PID cotrollr or ti dlay yt Quit aturally, cobiig thi claical approach to ractioal itgral ad drivativ actio, th ractioal Sith prdictor ha b itroducd [,7] Sic th ial work o Bod [], Maab [5] ad Outaloup [], it ha b dotratd that th ai itrt o ractioal cotrollr i to iprov robut o th clod loop to plat ucrtaiti Robut to tatic gai variatio wa th ai atur o th arly ractioal cotrollr Bad o th CRONE op loop rc odl, a w cla o ractioal PI ad PID cotrollr ha b did [4] Uig IMC pricipl [9], it i aily Maucript rcivd Octobr 5, N Maari, Maîtr d Coérc, i with LIAS Laboratory, Uivrity o Poitir, Frac -ail: zhaaari@uiv-poitirr) JC rigaou, oorary Proor, i with IMS Laboratory, Uivrity o Bordaux I, Frac -ail: jaclaudtrigaou@ i-bordauxr) M outit, PhD tudt, i with LIAS Laboratory, Uivrity o Poitir, Frac -ail: aartoutit@uiv-poitirr) gralizd to ractioal PI ad PID Sith cotrollr I thi papr, w propo to copar th prorac o th two cotrollr i tr o robut prorac wo typ o robut ar ivtigatd: robut to tatic gai variatio ad aalyi o it liitatio caud by th ti dlay, ad robut to th ral tructur o th plat, which i grally or coplx tha that o th odl ud or paratr tuig h papr i orgaizd a ollow Sctio II i a ridr o ractioal PI ad PID cotrollr bad o th CRONE odl Uig IMC thory, ractioal Sith cotrollr ar did i ctio III Fially, bad o urical iulatio, a copario o th robut prorac o th cotrollr i prtd i ctio IV II FRACIONAL PI AND PID CONROLLERS A Itroductio h objctiv i to dig a cotrollr C ), accordig to th covtioal dback loop o igur, whr ) i th plat Our dig objctiv i that th clod loop yt bhav a a giv rc odl ) : ) ) ) ) ) ) ) CL Figur : Covtioal dback cotrol loop B Fractioal PI cotrollr Outaloup ha did a idal op loop CRONE tplat []: OL ) with < < ) + i ordr to gt rid o tatic gai variatio h rc clod loop trar uctio corrpodig to thi CRONE tplat i: ) 3) / 3 EUCA 399

2 Coidr th irt ordr yt: ) 4) who tatic gai i ubjct to variatio Bcau our dig objctiv i CL ) ), it i traightorward to gt th corrpodig cotrollr: ) 5) which w ca writ a: C ) + K p) i 6) with i ad K p 7) C ) i a claical PI cotrollr aociatd to a ractioal itgrator / W di it a th PI) cotrollr: it i th gralizatio o th PI cotrollr to th ractioal ca Rark: hak to thi ractioal cotrollr, th op loop trar uctio C ) ) atch prctly th CRONE tplat ), i th curv rprtig C j) j) i th Nichol chart i xactly vrtical []), o thr i prct rjctio o tatic gai variatio C Fractioal PID cotrollr Lt ) a + a 8) b th proc to b cotrolld, whr i ubjct to variatio Robut to tatic gai variatio i providd by th ollowig op loop CRONE tplat [4]: ) OL 9) + ) whr < < provid robut ad ad ar tuig paratr h corrpodig clod loop rc odl i: ) ) ) i a gralizd cod ordr rc odl, whr, ad charactriz th ti rpo ad th dapig o th clod loop yt A ractioal PID cotrollr PID ) corrpodig to th trar uctio [4]: d ) + K p + ) ) i applid to ) iur xactly th op loop tplat OL ) ad th clod loop rc odl ) h tuig o th paratr corrpod to: i K p a i d a K p i ) Lt u rid that ay PID cotrollr ha to iclud a iltrig o th drivativ actio d i th dig o th cotrollr hi iltrig actio i cary or a ralitic ipltatio o th cotrollr ad to rduc oi itivity o th cotrol iput u t) O th othr had, th prc o / i th op loop trar uctio 9) prvt th curv OL j) i th Nichol chart) to b xactly vrtical, o thr i o logr prct rjctio o tatic gai variatio hi rark appli alo to th ractioal Sith prdictor III FRACIONAL SMI PREDICORS A Itral Modl Cotrol IMC [9] r to th clod loop tructur illutratd o igur Figur : Itral Modl Cotrol loop ) i th yt, ) i th odl o th yt: it rprt th ur kowldg cocrig th tru yt K ) i th IMC cotrollr: Lt ) b th rc odl o th clod loop For th ythi o K ), w coidr th oial ca, i w au that ) rprt xactly ) I orovr w au that thr i o diturbac i th yt, w gt: y t) y t) thu w t), i th yt oprat i op loop So w ca writ: ) ) ) ) ) 3) or quivaltly: ) ) ) which i th rlatio diig K ) 4) Bcau w t), th yt oprat i op loop, o otic that thr i o tability probl i th oial ca, providd that ) ad K ) ar tabl trar uctio 399

3 Idd, thr i grally a iatch btw ) ad ), o th clod loop trar uctio i: ) ) ) CL ) ) )) 5) which a that th odlig rror ) ) i th cau o clod loop itability hr i xact quivalc btw IMC ad covtioal dback [9], corrpodig to th quivalc quatio: ) K ) ) 6) ) ) ) hi quivalc allow ipltatio o IMC uig a covtioal dback loop; thi approach i ud by th Sith prdictor Lt B PI Sith cotrollr ), ) 7) b th proc to b cotrolld It i wll kow that it i ipoibl to cacl th ti dlay : o it rai carily i th clod loop rpo W hav to iclud thi ti dlay i th clod loop rc odl accordig to th IMC pricipl [9] Morovr, bcau our objctiv i to gt rid o tatic gai variatio, w hav to u th clod loop CRONE odl 3) So th rultig rc IMC odl i: ) 8) h, uig th quivalc btw IMC ad covtioal dback, it i traightorward to dotrat that th cotrollr ) corrpod to: ) 9) A traoratio o thi quatio, ipird by th Sith prdictor tchiqu [6], how that ) corrpod to th tructur rprtd igur 3: Notic that i w ipo, th prviou quatio di th claical itgr ordr Sith PI prdictor [6] Lt C PID Sith cotrollr ), ) a + a ) b th proc to b cotrolld Uig th a approach a prviouly, w hav to u th gralizd CRONE clod loop rc odl, o: ) ) + h, uig IMC thodology, it i traightorward to dotrat that th cotrollr ) corrpod to: a + a ) + ) + ) ) hi cotrollr corrpod to th tructur rprtd igur 4: Figur 4: Fractioal PID Sith cotrollr C ) i copod o a ractioal PID cotrollr, who paratr tuig ar giv by ), aociatd to a clod loop tr, icludig th ti dlay ad a cod ractioal itgrator Obviouly, thi clod loop tr icra th coplxity o th Sith cotrollr O th othr had, it guarat ucoditioal tability o th clod loop i th oial ca, i i ) ) ), IV A COMPARAIVE ANALYSIS OF SMI PREDICORS Figur 3: Fractioal PI Sith cotrollr hi cotrollr i copod o a ractioal PI cotrollr who paratr tuig ar giv by 7), aociatd to a clod loop tr, aalogou to that o th Sith prdictor, icludig th ti dlay itgrator ad a cod ractioal A Probl Statt Robut o ractioal cotrollr dpd o vral actor Firt o all, bcau thy hav b digd with th CRONE tplat, it i cary to tt thir ability to rjct tatic gai variatio, i gai variatio aociatd to chag o th plat opratig poit Scodly, bcau th yt odl ud or th dig i a approxiatio o th tru plat odl, it i cary to ivtigat th iluc o thi iatch o robut So, w propo two typ o tt 3993

4 h irt o cocr itriic robut, i robut to gai variatio wh thr i o iatch btw th odl ) ad th tru yt ) h cod o cocr tructural robut, i robut to gai variatio wh thr i a iatch btw th yt ad it odl Au that th tru yt i charactrizd by: ) ) 3) whr i a tru ti dlay ad ) rprt th dyaical part o th yt Practically, w hav to coidr two odl: ) or th PI Sith cotrollr,, ) or th PID Sith cotrollr hr xit ay tchiqu to driv th odl: thr ar ltary graphical) tchiqu lik Strjc, Broïda, ) ad tchiqu bad o yt idtiicatio I our approach, w hav drivd th two odl uig output rror idtiicatio [3,5] which i bad o o liar optiizatio [6] Notic that oly i ) ) Bcau o iatch ) ) ), w gt : + 4) whr ) i a quivalt ti dlay corrpodig to B Itriic Robut B PI Sith prdictor W coidr ) ), ) 5) with o ad 5 vari i th itrval : o < < o 6) hi tt itrval ha b ud or all th othr cotrollr Bcau w xpct a cotat ovrhoot o th clod loop rpo) wh vari, w charactriz robut with th valu D % o th achivd ovrhoot Each cotrollr ha b digd with 5 ad 4 h rult ar prtd i abl or th thr valu o th ti dlay utabl abl : Fractioal PI cotrollr, ) ), A xpctd, robut i corrctly achivd oly or all ti dlay i ) ad thr i a dgradatio or larg ti dlay otic itability or ) B PID Sith prdictor W coidr : ) ) ) + ) + ), with o 5 ; 5 ; 7) A prviouly, robut dpd o th ti dlay, but alo o th clod loop ti rpo So, w hav ud two valu or thi ti rpo, rrd a low ad at hy ar obtaid with th ollowig tuig paratr: Slow : rd / Fat : rd / h rpctiv valu o th clod loop ovrhoot ar prtd i tabl ad abl : Fractioal PID cotrollr, ) ), low rpo, abl 3: Fractioal PID cotrollr, ) ), at rpo, Bcau th claical Sith prdictor corrpod to, w hav alo copard th ractioal prdictor to it itgr ordr quivalt or th two rpo: Slow : 5 45rd / Fat : 5 rd / h corrpodig valu o th clod loop ovrhoot ar prtd i tabl 4 ad abl 4: Covtioal Sith PID cotrollr, ) ), low rpo, 3994

5 abl 5: Covtioal Sith PID cotrollr, ) ), at rpo, Firtly, copario with covtioal Sith prdictor how that th ractioal cotrollr xhibit a bttr robut i all ituatio, thak to th CRONE tplat owvr, a prviouly, thi robut i xcllt oly or all valu o ad a low rpo; it i dgradd or larg valu o ad atr rpo C Structural robut C PI Sith cotrollr W coidr ad, ) ) + ) + with ) ) o 5 ; 5 ; 8) 9) ) ha b tiatd with output rror tchiqu Etiat ar litd i tabl 6 J abl 6 Output rror tiatio o ), W ca otic that ) i a rough approxiatio o ), a idicatd by th larg valu o quadratic critrio J Notic that +, accordig to 4), with 65 h, w hav ttd tabl 7) th clod loop robut to variatio or th thr valu o, uig th prviou low ca Notic that it ha b cary to odiy or ach valu o, with 3 hu: 5 4 ; 5 ; abl 7: PI Sith prdictor applid to th ti dlayd cod ordr yt Idd, w ca copar th rult to tho o th PID Sith prdictor, low ca W otic idiatly that th Sith PID xhibit bttr robut prorac tha th Sith PI So, w ca coclud: though th two Sith prdictor hav good robut itriic robut, th PI Sith prdictor i ot abl to ur th a prorac lvl a th PID prdictor i thr i a tructural iatch btw ) ad ) Idd, thi i xplaid by th prc o th drivativ actio o th PID cotrollr ad by th iportat iatch idicatd by th quadratic critrio C PID Sith cotrollr W coidr ow : ad ), a + a ) ) + ) + ) + with: ) ) 3 o 3 3 ; ; 5 3) 3) A prviouly, ) ha b tiatd with th output rror tchiqu, tabl 8, with th a xcitatio a a J abl 8: Output rror tiatio o ), Notic ow th low valu o quadratic critrio J, idicatig a good atch btw ) ad ) Agai, w otic that +, with 65 h, w hav ttd robut to gai variatio, or two valu o th ti rpo: Slow : rd / Fat : rd / W otic good robut to variatio or th dirt valu o abl 9 ad, igur 9 ad ) with th low rpo, whra thr i a iportat dgradatio ad v itability) or 5 with th at rpo abl 9: PID Sith prdictor applid to th ti dlayd third ordr yt, low rpo 3995

6 utabl abl : PID Sith prdictor applid to th ti dlayd third ordr yt, at rpo owvr, th PID Sith cotrollr xhibit a bttr ability tha th PI o to rjct tructural iatch bcau it prt or tuig paratr ad orovr bcau ) odl ha or lxibility tha ) to rduc, tructural iatch D Cocluio, Fractioal ad itgr ordr Sith prdictor xhibit quivalt prorac i th oial ca o ) ad with o iatch btw th yt ad it odl: thi a that thy ar thortically abl to provid prct cotrol with larg ti dlay Practically, du to thir high itivity to gai ucrtaity or larg ti dlay, th ral itrt o covtioal Sith prdictor i liitd by th rik o clod loop itability A xhibitd by prviou rult, ractioal cotrollr provid or curity to th ur thak to thir tatic gai robut, i thir ability to rjct tatic gai ucrtaity Robut to tructural ucrtaity i a tial atur or all yt, but particularly or ti dlay yt h ractioal PID Sith cotrollr i crtaily th o which xhibit th gratt ability to rjct tatic gai ad tructural ucrtaiti V CONCLUSION A w cla o ractioal Sith prdictor ha b propod i thi papr: it aociat Itral Modl Cotrol pricipl or th dig o ti dlay cotrollr to th CRONE tplat i ordr to iprov robut to tatic gai variatio A particular atur o th cotrollr i thir tuig iplicity: i act, it rli tially o yt idtiicatio uig ) or ) odl Nurical,, iulatio hav illutratd itriic ad tructural robut prorac o th ractioal PI ad PID prdictor owvr, copario hav how that th Sith PID cotrollr prt th gratt ability to rjct gai variatio ad yt tructur ucrtaity So, a itrtig applicatio o th PID prdictor i th robut cotrol o a larg cla o ucrtai plat i aociatio with yt idtiicatio [8,9] REFERENCES [] Bod, W 945) Ntwork aalyi ad dback apliir dig, Va Notrad Nw York, 945 [] Fliu-Battl, V, Prz, R, arcia, F, Rodriguz L «Sith prdictor bad robut ractioal ordr cotrol: applicatio to watr ditributio i a ai irrigatio pool, Joural o Proc Cotrol, Vol 9, pp 56-59, 9 [3] Ljug, L Syt idtiicatio: thory or th ur Prtic all, Nw Jry, USA987 [4] Maari, N, rigaou, JC, outit M O ractioal PI ad PID cotrollr, 4 th IFAC Workhop o Fractioal calculu ad it applicatio, FDA, Badajoz, Spai, [5] Maab, S h o itgr itgral ad it applicatio to cotrol yt, Joural o th Ititut o Elctrical Egir o Japa, pp , 96 [6] Marquardt, D A algorith or lat quar tiatio o o liar yt J Soc Idutr Appl Math Vol ), pp 43-44, 963 [7] Moj, CA, Viagr, B, Ch, YQ, Fliu-Battl, V, Lau, P, Sabatir, J «Optial tuig or ractioal ractioal PID cotrollr, Fractioal Dirtiatio ad it applicatio, 5, pp [8] Moj, CA, Ch, YQ, Viagr, B, Xu, D, Fliu-Battl V, Fractioal ordr Syt ad Cotrol: Fudatal ad Applicatio Sprigr-Vrlag Lodo [9] Morari, M, Zairiou, S Robut proc cotro l, Prtic all, 989 [] Outaloup A ad Mathiu, B La coad CRONE, du calair au ultivariabl», rè, Pari, 995 [] Outaloup A, La dérivatio o tièr : théori, ythè t applicatio rè, Pari, 995 [] Ptra I, Fractioal ordr o liar yt: odllig, aalyi ad iulatio, Sprigr Vrlag, [3] Podluby I Fractioal ordr yt ad P I α D β cotrollr IEEE raactio o Autoatic Cotrol, Vol 44, pp 8-3, 999 [4] Podluby I Fractioal dirtial quatio, Acadic Pr, Sa Digo 999 [5] Richalt, J «Pratiqu d l idtiicatio» rè, Pari, 99 [6] Sith O A cotrollr to ovrco dad ti ISA Joural, Vol 6, pp 8-33, 959 [7] outit M, Maari N ad rigaou, JC, A ti dlay ractioal cotrollr or robut cotrol o ucrtai plat Sypoiu FDA, Najig, Chia, [8] outit M, Maari N ad rigaou, JC «Sythè d corrctur ractioair robut avc pri copt d icrtitud» Corc CIFA, robl, Frac, [9] outit M, Maari N ad rigaou JC, A idtiicatio procdur or th tuig o a robut ractioal cotrollr, FDA 3, robl, Frac, 3 [] Valrio D ad JS Da Cota, uig rul or ractioal PID, i hortical Dvlopt ad Applicatio i Phyic ad Egirig, Sprigr,