Updtd: Tudy, Octor 8, EE 434 - Cotrol Sytm LECTUE Copyright FL Lwi 999 All right rrvd BEEFTS OF FEEBACK Fdc i uivrl cocpt tht ppr i turl ytm, itrctio of pci, d iologicl ytm icludig th ic cll d mucl cotrol ytm i th ody Chrl rwi howd tht fdc ovr log tim priod i rpoil for turl lctio of pci Voltrr howd tht fdc i rpoil for th itrctio of prdtor/pry pci i clod coytm Adm Smith howd tht fdc occur i th lc of th itrtiol coomi of tio Fdc i hum gird ytm h my fit Uig fdc, o c mov th clod-loop pol o tht th ytm prformc improv Prformc c improvd i trm of ft trit rpo, prct ovrhoot, d ocilltio frqucy O c oti improvd prformc i th prc of diturc, uch wid gut i ircrft cotrol ytm O c lo lct uitl comptor to oti good tdy-tt trcig y tudyig th tdy-tt rror th ot w dicu: Pol Plcmt iturc jctio Stdy-Stt Error POLE PLACEMET US FEEBACK A importt proprty of fdc i tht it mov th ytm pol Th xt xmpl how how to u thi for fdc dig Exmpl - Movig th Pol Uig Fdc A op-loop plt h trfr fuctio
Thi ytm h utl complx pol pir t 3 j rt K ut yt t i dird to dd fdc comptor K how i th figur, whr w wt to u th proportiol-drivtiv P fdc K P with th drivtiv gi d P th proportiol gi W wt to lct th gi to oti th prformc pcifictio: clod-loop POV of 4% d ttlig tim of c A POV of 4% corrpod to dmpig rtio of / Sic t 5, ttlig tim of c m / 5 4 c Thi giv rl prt of th pol of 5 Th rgio of th -pl pcifid y thi t prformc pc i how i th figur Th clod-loop pol mut o th 45 o li how d to th lft of 5 j 45 o -5 W c tify th pc y lctig 5 77, 5, 7678 Th, th dird clod-loop chrctritic polyomil i d 5 35 Th clod-loop trfr fuctio i
K T K P P O tht th pol my plcd ritrrily dird y lctig th two P gi To oti th dird clod-loop chrctritic polyomil, t 5 35 d olv to gt th P gi 6, 5 P STUBACE EJECTO P Fdc c ud to rduc th ffct of diturc o th output dt ut yt th op-loop ytm how, could rprt plt g ircrft d could rprt th ctutor g ircrft lvtor Sigl ut might th ur iput i pilot iput, d dt diturc tht ct dirctly o th plt without hvig to go through th ctutor g wid gut o th lvtor cotrol urfc Th iput/output trfr rltio i giv y U To rduc th ffct of th diturc dt o th output yt d lo produc dirl trfr from th cotrol to yt, o could dd fdc comptor H d fdforwrd comptor how dt rt ut yt H 3
Thi i ow two-dgr-of-frdom rgultor coitig it do of fdc d fdforwrd compot W hll tht th -OF rgultor llow to to lct oth th clod-loop trfr fuctio from rt to yt wll th clod-loop trfr fuctio from dt to yt Th w commd rt i ow th rfrc iput t fuctio my igifictly diffrt from th origil fuctio of ut Thi of drivig cr, whr ytm iput ut i th cclrtor pdl dprio Aftr turig o th crui cotrol, howvr, th drivr' rfrc iput rt i th pd commd to th crui cotrol Th cotrol ytm utomticlly comput th ut rquird to mtch thi commdd vlu or Accordig to Mo' formul, th clod-loop trfr rltio i giv y H H H H, whr H H H H W would li to lct th comptor H d o tht H i mll ot tht th clod-loop trfr fuctio hv th rtur diffrc i th domitor Exmpl - Comptor ig Exmpl Op-Loop Sytm Th plt i ircrft with lightly dmpd mod giv y d th ctutor i motor tht i modld Prformc Spcifictio 4
ig -OF rgultor o tht, i clod loop: Th diturc di wy i pproximtly 5 cod Th trfr fuctio from rt to yt h dmpig rtio of / d turl frqucy of c Th figur giv rpo tht my pilot li, i trm of good compromi tw ovrhoot d pd of rpo c Comptor ig To ccomplih thi, firt fid th dird clod-loop trfr fuctio follow Sic th dird ttlig tim i rpo to diturc i t 5, o hould lct tim cott i th diturc chl of c, or rl prt of th pol of t lt / Slct thrfor dird clod-loop trfr fuctio i th diturc chl of H d Thi mt th ttlig tim prformc pc To mt th pc i th commd chl, hvig i mid th coicl form for th chrctritic polyomil, lct th dird trfr fuctio i th commd chl of H d Though o c chg th umrtor uig -OF rgultor, w hv dcidd to lv it th m i op loop ow, o imply plug i d olv for d H Thu, H H Sutitutig for, llow o to olv for H Uig thi ow i H H d utitutig for, llow o to olv for Th, filly o oti H 5
ctul dig, o mut t cr o tht oth d H hv rltiv dgr of t lt o, othrwi th comptor cot implmtd Thi i ot difficult to do Thi dig tchiqu i formlizd i wht i ow th Polyomil Equtio ig Approch to fdc ytm Stdy-Stt Error Fdc c improv th trcig cpiliti of plt y mig th tdy-tt rror mllr ut yt A op-loop ytm i how fi th trcig rror t u t y t For good trcig, th rror hould mll, for th th ctul output yt follow th dird commd ut Th trcig rror i foud E U U U U To fid th tdy-tt rror i rpo to uit tp commd o t u th Fil Vlu Thorm to oti E U d Thrfor, th tdy-tt rror i qul to zro oly if th C gi i xctly qul to Thi i difficult rquirmt to mt i prctic ic, for itc th C gi my dpd o om tritor prmtr which my vry with tmprtur, tc figur To oti mor rout trcig ytm dig, dd fdc how i th rt t K ut yt Th clod-loop trfr fuctio i giv y 6
K T K Howvr, to fid th tdy-tt rror, o dtrmi th trfr fuctio from rt to th rror t Uig Mo' formul, thi i giv y E K To fid th tdy-tt rror i rpo to uit tp commd, lct d u th Fil Vlu Thorm to oti E K K Thrfor, th tdy-tt rror i mll if th C loop gi K i imply lrg ough O c gurt thi y imply lctig th comptor K=, lrg vlu of cott proportiol gi Exmpl 3- Stdy-Stt Error Alyi dt rt t K yt thi clod-loop ytm, th plt i d th cotrollr i th itgrl comptor K with th itgrl gi Th trcig rror i giv y t r t y t Thi xmpl mphiz tht o mut oft fid multipl trfr fuctio i th m digrm to lyz it fct, i thi xmpl w hll hv to fid four trfr fuctio 7
8 Fid th Stdy-Stt Error i po to Uit Stp iturc dt To fid th rqutd rror, t = d = / Th trfr fuctio from dt to t i giv y K E E Uig ow th Fil Vlu Thorm yild E Th ytm i of typ, o th rror wrt uit tp diturc i qul to zro ot tht thi m th output yt go to zro, ic th commd i rt= Fid th Stdy-Stt Error i po to Uit Stp Commd rt To fid th rqutd rror, t =/ d = Th trfr fuctio from rt to t i giv y K E E Uig ow th Fil Vlu Thorm yild E Th ytm i of typ, o th rror wrt uit tp commd i qul to zro ot tht thi m th output yt go to o, ic th commd i rt=u - t c Fid th POV For Uit Stp Commd Th clod-loop trfr fuctio from th commd rt to th output yt i giv y K K
9 Th clod-loop chrctritic polyomil i, d comprig thi to th tdrd form o tht O c thrfor lct th itgrl gi to oti y dird vlu of dmpig rtio, d hc of POV which i giv y % / POV d Fid th Output yt if, t u t d t u t r t t Th trfr rltio tw th two iput d th output i giv y K K K or Sttig ow, o oti For th giv vlu of o my ow prform PFE d dtrmi yt
Stdy-Stt Error d Sytm Typ Stdy-tt rror dpd o th choic of th comptor Th comptor c lctd to m tdy-tt rror qul to zro A clod-loop trcig ytm i how i th figur A trcr i mt to ur tht th ctul plt output yt follow or trc th dird rfrc commd rt Trcig ytm r digd y tudyig th tdy-tt rror proprti of th clod-loop ytm Th trcig rror i dfid t y t r t Lt th plt giv i th form d m m Th comptor i giv K, which i -itgrtor with gi of ot tht th gi h diffrt uit dpdig o th vlu for i i dimiol, whil h uit c -, d o o Th loop gi i d K m m o for rltiv dgr rquirmt w d m W um th plt do ot hv y pol t th origi = A ytm with pol t th origi i clld typ ytm Th trfr fuctio from rfrc commd to rror i giv uig Mo' formul m m d d K E Accordig to th Fil Vlu Thorm, th tdy-tt rror i giv y d d K E K ut yt rt t
Thi dpd o th commd rt wll th ytm typ vlutig thi, ot tht du to th pcil form w lctd for, o h d=, = po to Uit Stp Poitio Commd Lt th rfrc commd th uit tp, which corrpod to poitio commd Th, f =, corrpodig to typ zro ytm, th th tdy-tt rror to poitio commd i qul to O c m thi mll y lctig lrg gi o tht i much grtr th ot tht o mut ow th ig of, ic th cotrol gi mut hv th m ig yt rt t ozro tdy-tt rror to tp commd f =, corrpodig to typ o ytm, th th tdy-tt rror to poitio commd i qul to d d fct, for y o h zro tdy-tt rror Thrfor, for typ ytm, th tdy-tt rror i rpo to uit tp c md mll dird y icrig th gi For typ or lrgr ytm, th tdy-tt rror i rpo to uit tp i qul to zro Thi m tht trcig ytm c follow poitio commd prfctly log thr i t lt o itgrtor i th fdforwrd pth po to Uit mp Vlocity Commd
Lt th rfrc commd th uit rmp, which corrpod to vlocity commd Th, f =, corrpodig to typ zro ytm, th th tdy-tt rror i rpo to uit rmp commd i qul to f =, corrpodig to typ o ytm, th th tdy-tt rror i qul to d d O c m thi mll y lctig lrg gi o tht i much grtr th ot tht o mut ow th ig of, ic th cotrol gi mut hv th m ig yt rt t ozro tdy-tt rror to vlocity commd f =, corrpodig to typ two ytm, th th tdy-tt rror i qul to d d fct, th tdy-tt rror i zro for y Thrfor, for typ ytm, th tdy-tt rror i rpo to uit rmp i ifiity Such ytm cot follow vlocity commd For typ ytm, th tdy-tt rror i rpo to uit rmp c md mll dird y icrig th gi For typ or lrgr ytm, th tdy-tt rror i rpo to
uit tp i qul to zro Thi m tht trcig ytm c follow poitio commd prfctly log thr r t lt two itgrtor i th fdforwrd pth trl Modl Pricipl Th itrl modl pricipl i vry importt i fdc dig t tt tht trcig ytm c follow prfctly commd rt log th fdforwrd pth coti modl of th commd Thu, poitio followr d to hv t lt o itgrtor i th fdforwrd pth, vlocity followr d t lt two itgrtor, tc otdrd Trcig Sytm ot ll fdc ytm r i th tdrd form w hv dicud o fr Exmpl 4- otdrd Trcig Sytm th xmpl w hv o fr, thr i uity gi outr trcig loop o tht th rror t ppr xplicitly i th loc digrm Thi m it vry y to u Mo' Formul to dtrmi th trfr fuctio dd for tdy-tt rror lyi dt rt yt K Th figur how otdrd trcig ytm with o uity gi outr loop Th trcig rror i dfid t r t y t, ut it cot drw i th figur Suppo tht th plt i d th cotrollr i th itgrl comptor K 3
4 with th itgrl gi To fid th tdy-tt rror i rpo to uit tp commd, t =/ d = Th trfr fuctio from rt to yt i giv y K ow, comput E Thi i th rror o which o prform th Fil Vlu Thorm fct, E Thu, thi i vry d dig t i ot typ ytm, ic th itgrtor do ot ppr i th fdforwrd pth