The Cost Function Minimization for Predictive Control by Newton-Raphson Method

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1 Proceed of e Ieraoal lcoferece of Eeer ad Coper Sce 8 Vol II IECS 8, 9- arc, 8, Ho Ko e Co Fco zao for Predcve Corol by ewo-rapo eod B. Drş, H. eraş,. Yşa, F. eraş, R. Kaza Abrac-e ewo-rapo eod oe of e o wdely ed eod for zao. I ca be ealy eeralzed for olv o-lear dffereal eqao ye. I dy, Geeralzed Predcve Coroller GPC wa appled o a 6R robo aplaor baed o o corol. ewo-rapo -R eod wa ed o ze e co fco ex e GPC a repree error bewee referece raecory ad acal raecory e corol of robo. e ewo-rapo eod reqre le erao ber for coverece ad redce e calclao. dy pree a dealed dervao of e Geeralzed Predcve Corol alor w ewo-rapo zao eod. e rel of alar pa ad poo error belo o o were exaed ad copared w Recrve ea Sqare RS pleeed Geeralzed Predcve Corol. e lao rel owed a ewo-rapo eod proved corol perforace of e GPC. Keyword- Predcve corol, eeralzed predcve corol, co fco zao. I. IRODCIO Predcve corol alor wa developed fro e oparaeer odel predcve corol alor cld e odel Alor Corol AC, e Dyac arx Corol DC ad o o, o e odel Predcve Corol PC alor o exaple of wc e Geeralzed Predcve Corol GPC alor [-4]. Becae e GPC alor paraeer odel orer a e o-paraeer odel, redced e GPC alor calclao e, ad eaced perforace of e corol ye. e GPC ed e corol of o- pae pla, ope-loop able pla ad pla w varable or ow dead e. I alo rob w repec o odel error, over ad der paraeerzao, ad eor oe [5-6]. e copaoal perforace of a GPC pleeao acrp receved oveber, 7. wor wa ppored by cefc reearc fd very of Saarya, ra o B.Drş w Saarya very, Depare of Elecrc - Elecroc Eeer, 5487 Adapazar, RKEY correpod aor poe: ; fax: ; e-al: bdr@ aarya.ed.r. H. eraş Dlpıar very, Depare of Elecrc - Elecroc Eeer, 447 Kaya, RKEY e-al: era@dlpar.ed.r.. Yşa Saarya very, Depare of Coper Eeer, 5487 Adapazar, RKEY e-al: ya@aarya.ed.r. F. eraş Saarya very, Depare of Coper Eeer, 5487 Adapazar, RKEY e-al: era@aarya.ed.r. R. Kaza Saarya very, Depare of ecacal Eeer, 5487 Adapazar, RKEY e-al: aza@aarya.ed.r. larely baed o e zao alor coe for CF bloc. ere are everal zao alor a ave bee pleeed GPC c a o-rade [7], Splex, ad Scceve Qadrac Prora [8,9]. e eleco of a zao eod ca be baed o everal crera c a; ber of erao o a olo, copaoal co ad accracy of e olo. I eeral ee approace are erao eve a reale corol dffcl. Very few paper addre real-e pleeao or e ey ed pla w lare e coa [8,9]. o prove e ably, a faer opzao alor eeded. e ewo-rapo eod oe of e o wdely ed eod for zao. I a qadrac alor cover beer a oer. I reqre le erao ber for coverece ad redce e calclao. I dy, Geeralzed Predcve Corol GPC wa appled o 6R robo aplaor for o corol. ewo- Rapo -R eod wa ed o ze e co fco ex GPC a repree error bewee referece raecory ad acal raecory e corol of robo. e rel of alar pa ad alar velocy belo o o were exaed ad copared w rel obaed fro e Recrve ea Sqare pleeed Geeralzed Predcve Corol. Alo, proce e of bo alor were ow. II. GEERAIZED PREDICIVE CORO e Geeralzed Predcve Corol GPC rodced by Clare a eeralzao of e odel-baed corol ad able for coroll of e procee w varable dead e ad a pla wc laeoly o-pae ad ope loop able. ay reearc owed e effecvee of corol alor []. e GPC ye for e roboc aplaor ve Fre. I co of ree copoe, e roboc aplaor or laor, coroller ad paraeer eaor. Were, orqe,, e corol p o e aplaor ye, e raecory, y, e op, ad y r e referece op. F. Bloc dara of e GPC ye for roboc aplaor e ear of a odel-baed predcve coroller e pla odel. e Corolled Aorereve Ieraed ISB: IECS 8

2 Proceed of e Ieraoal lcoferece of Eeer ad Coper Sce 8 Vol II IECS 8, 9- arc, 8, Ho Ko ov Averae CARIA odel cooly ed e GPC, a applcable o ay le-p leop pla: A q y B q ξ / Were,, y are e pla p ad op. A ad B are polyoal e bacward f operaor q - : a A q Ι a q a q - a a q a, B b q b q b b q - b b b q Were, ξ, a correlaed rado eqece, ad e e of e operaor q ere a eral corol law. Eqvalely o e procedre of Clare [] a opal - ep forward predcor ve by []: y F q y E q G q free repoe forced repoe e fr er of eqao called free repoe, a repree e pla predced op y, we ere o fre corol aco. e ecod er called forced repoe, a repree e op predco de o e ypoecal fre corol aco -,. o elec a ood corol eqece, we wold w a al rac error e y - y r over a cera op orzo. Frerore, o preve ay exploo of e corol aco, a ecod er eerally added, o a e fal perforace dex a a for lar o e follow:,, yˆ yr λ 4 bec o: for Were, ad are e ad e ax co orzo, e corol orzo ad λ a we facor for e corol cree eqece o be calclaed. Frerore, e cora a e corol cree are forced o zero provde a beer coverece of e op o e e po. If we defe e wo follow vecor fored w polyoal olo of eqao : [ f... f ] f, e vecor of e free repoe Addoally, f we deoe: y y... y 5 [ ] [,... ] ~ 6 Ad e arx fored w e coeffce of e G polyoal, wc fac correpod o e ep repoe vale : G e op predco a e follow for: y G ~ f 8 ow, eqao 4 ca be rewre a arx for: [ G ~ f y ] [ G ~ r f y r ] λ ~ ~ 9 Ad e opal corol law coe fro ~. ~ G G Λ G y f r I dy, frly, we ed Recrve ea Sqare RS for zao of co fco e GPC. I eod A q ad B q paraeer wc are copo G ad f are recalclaed eac of corol ep. We defe follow fored w e CARIA odel olo of eqao for ed Recrve ea Sqare []. y ay a y a b b b ξ / a b eqao plfed; ˆ y Θ Φ e If a b ; ˆ Θ, Φ ad e are polyoal arx. We deoe follow: Θ ˆ a,, a a, b,, b b Φ y,, y a,,, b 4 e ξ / 5 ˆ Θ paraeer wc ex A q ad B q paraeer pdaed for eac of a corol ep a follow: 7. e eqao of a: P Φ K 6 µ Φ P Φ e a calclaed ed K. µ e are facor µ. 95. P paraeer arx. P Ι / δ for fr ep corol. Ι arx, δ a coa wc vale -6. e calclao of P for oer ep corol e follow eqao. P P Ι K Φ 7 µ ISB: IECS 8

3 . e eqao of error: y e Φ Θ 8 Error calclaed by ed eqao.. ˆ Θ calclaed e follow eqao: K e ˆ ˆ Θ Θ 9 q A ad q B paraeer are recalclaed fro ˆ Θ e eqao 9. Coeqely, e paraeer of GPC are pdaed by proce. III. COS FCIO IIIZAIO by EWO- RAPHSO EHOD e obecve of e CF alor o ze eqao 4 w repec o [,,, ], deoed. accopled by e e eqao 4 o zero ad olv for. W ewo-rapo ed a e CF alor, zed eravely o deere e be. A erave proce yeld eredae vale for deoed. For eac erao of a eredae corol p vecor alo eeraed ad deoed a erao #,,, e ewo-rapo pdae rle for : were e acoba deoed a: e Hea a O Solv eqao drecly reqre e vere of e Hea arx. ee procee cold be copaoally expeve. Oe ecqe o avod e e of a arx vere o e decopoo [] o olve for e corol p vecor. accopled by rewr Eqao e for of a ye of lear eqao, Axb. rel A b, ad x I for Eqao ca be olved w wo roe ppled [] e ower/pper ralar decopoo roe ldcp, ad e ye of lear eqao olver lbb. Afer x calclaed, olved by evala x. procedre repeaed l e perce cae eac elee of le a oe -7. We olv for x, calclao of eac elee of e acoba ad Hea eeded o eac of ewo- Rapo erao. e elee of e acoba r y y y λ ax,,. e, elee of e Hea : r y y y y y λ ax,. ad,. e alway evalae o zero. e la copoe eeded o evalae e calclao of e op of e pla, y, ad dervae. IV. DYAIC ODE OF ROBO AIPAOR A pror forao eeded for aplaor corol aaly ad aplaor de a e of cloed for dffereal eqao decrb e dyac beavor of e aplaor. Varo approace are avalable o forlae e robo ar dyac, c a arae-eler, ewo-eler ad Recrve arae [,]. e cofrao of e x o roboc aplaor odel ad Deav-Hareber paraeer ca be ee Fre ad able I repecvely [4]. I dy, arae-eler ed for dyac odel of e x o roboc aplaor. arae-eler eqao of e oo, d d τ θ θ Proceed of e Ieraoal lcoferece of Eeer ad Coper Sce 8 Vol II IECS 8, 9- arc, 8, Ho Ko ISB: IECS 8

4 Proceed of e Ieraoal lcoferece of Eeer ad Coper Sce 8 Vol II IECS 8, 9- arc, 8, Ho Ko were τ eeralzed orqe appled o e ye fro o, araa fco K P, K : oal ec eery of e aplaor, P : oal poeal eery of e aplaor, θ e alar poo of e o, ad. θ e fr order dervave of e θ. Eqao wc were ed for e calclao of e oal ec eery of e aplaor are ve 7, 8, ad 9. K 7 K dk K 8 dk x& y& z& d 9 Eqao wc were ed for e calclao of e oal poeal eery of e aplaor are ve,,, ad. V. SIAIO RESS I paper, wa deed 6R x-dof roboc aplaor corol ewo-rapo -R pleeed Geeralzed Predcve Coroller GPC alor baed o o corol. I wa copared w RS Recrve ea Sqare pleeed GPC accord o e lao rel. oal lao e ecod ad oal ep ber. I addoally, robo aplaor carre 5 load a e ed-effecer. able II. Soe rel of lao robo aplaor by -R ad RS pleeed GPC o Alor Alar Ial Dered Acal Pa Ale Ale Ale Error rad rad rad rad o -R RS o -R RS o -R RS o -R RS o -R RS o -R RS F. e odel of 6R Robo aplaor able I. Deav-Hareber Paraeer of PA 56 Robo Ar α a θ d deree eer eer -9 θ θ θ θ θ 5 6 θ 6.56 P P r P For exaple, e rel of alar pa were ve Fre. Alar veloce ad alar poo error were ve belo o robo ar o, ad 5 Fre 4, 5 ad 6 repecvely. Addoally, al- dered-acal ale, alar pa error were ve able II ad alo qare of alar velocy error vale were ve able III. A ee able II, ale pa error are aller e - R a oe e RS. O e oer ad, e qare of alar velocy error are proporoal o e er. e le alar velocy error a e o lead o le er. W -R coroller appeed fewer ol a e o. A ee F., -R coroller raced o dered raecory ooer ad cloer a RS coroller. I rac perforace, bo alor were fod o be afacory, accord o lao rel of alar velocy ve F 4-6. Alo, e dfferece of alar pa error are ee ae F. able III. Soe rel of alar velocy error rad/ec o o o o4 o5 o6 -R RS able IV. Poo error of robo ar ad proce e CP e of coroller r A r Poo error Proce e r x, y, z, Were e a of e lb, e ravy vecor, A e rao arx /. -R RS e poo error of robo ar ad proce e belo o corol alor were ve able IV. e ISB: IECS 8

5 Proceed of e Ieraoal lcoferece of Eeer ad Coper Sce 8 Vol II IECS 8, 9- arc, 8, Ho Ko poo error of e -R coroller are leer a oe by RS. For proce e, lao of e RS ad ae 469, werea e -R coroller a ae 65, for e ve raecory F 4-6. ewo-rapo eod redced e e of co fco zao ad alo redced e proce e. a. -R b. RS F. Alar Pa of o by -R ad RS pleeed GPC a. -R b. RS F. 4 Alar Velocy ad Alar Poo Error for o by -R ad RS pleeed GPC a. -R b. RS F. 5 Alar Velocy ad Alar Poo Error for o by -R ad RS pleeed GPC ISB: IECS 8

6 Proceed of e Ieraoal lcoferece of Eeer ad Coper Sce 8 Vol II IECS 8, 9- arc, 8, Ho Ko a. -R b. RS F. 6 Alar Velocy ad Alar Poo Error for o 5 by -R ad RS pleeed GPC I paper, copaoally effce of co fco zao e GPC alor wa exaed. ere over copao e zao of e co fco. ewo-rapo pdae eod reqre le erao ber for coverece ad redce e calclao I applcao, Recrve ea Sqare RS pleeed GPC bewee ewo-rapo pleeed GPC coroller were appled o e 6R robo ar aplaor baed o o corol. e rel of alar pa ad alar velocy belo o o, poo error of ed-effecer ad alo proce e of coroller were copared. Accord o e lao rel, ewo- Rapo pleeed GPC redced poo error ad alo proce e for robo aplaor corol. ea a e ewo-rapo proved corol perforace of e GPC. [] F. era, H. era,. Ya, C. Oz, Effec of e raecory Pla o e odel Baed Predcve Roboc aplaor Corol, ec. oe Cop. Sc., 869, pp [] W.H. Pre, B.P. Flaery, S.A. eoly, ad W.. Veerl, ercal Recpe C:e Ar of Scefc Cop, Cabrde very Pre 988. [] W.. Slver, O e Eqvalece of araa ad ewo-eler Dyac for aplaor, e Ieraoal oral of Roboc Reearc, Vol [] C. S. G. ee, Robo Ar Keac- Dyac, ad Corol, Coper, Vol [4] ar.., Beczy A.K, Y X., Coordaed Corol of wo Robo Ar, CH8-/86//9/. copy IEEE, 986. REFERECES [] Clare D W. Applcao of eeralzed predcve corol o dral procee. IEEE corol Sy. aa, 988, 8, pp [] Rcale, odel predcve erc corol: Applcao o dral procee, Aoaca, 978, 45, pp [] Roa R., era R K. odel alorc corol AC, Bac eorecal propere, Aoaca,98, 84, pp [4] C R Cler, B Raaer, Dyac arx corol: A coper corol alor, Proceed of o Aoaca Corol Coferece, Sa Fracco, 98. [5] Köer R., De ad perforace of a elle predcve coroller for a x-deree-of-freedo robo e Ela ewor, Iforao Scece, Vole 76, Ie, pp , e 6. [6] D. Solaway, P. Haley, eral eeralzed predcve cool: a ewo- Rapo pleeao, Proceed of e IEEE Ieraoal Sypo o Ielle Corol, pp. 77-8, 996. [7] G. A. oae,.. Wll,.. a ad A.. orr, Arfcal eral ewor Baed Corol, Ieraoal Coferece o Corol 99, Vol., pp [8] eo So ad Swo Par. eral odel-predcve Corol for olear Cecal Procee, oral of Cecal Eeer of apa, 99, V6, 4, pp [9] D.C. Pcoo ad.h. ar, olear Ieral odel ad odel Predcve Corol eral ewor, 5 IEEE Ieraoal Sypo o Ielle Corol 99, pp ISB: IECS 8