DC Motor with Load Coupled by Gears Speed Control using Modified Ziegler-Nichols Based PID Tunings

Transcription

1 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, 014 Abstrct DC Motor with od Coupled by Gers Speed Cotrol usig Modified Ziegler-Nichols Bsed PID uigs Aborisde, Dvid O. Deprtet of Electroic/Electricl Egieerig, doke Akitol Uiversity of echology, Ogbooso. Oyo-stte, Nigeri. el: E-il: doborisde@yhoo.co he used of DC otor i vrious pplictios hs bee icresed due to the ese with which it speed c be cotrolled to give the desired perforce chrcteristics uder vrious coditio. PID cotrollers re widely used i DC otor speed cotrol due to its siple structure d robustess to the odelig error, however their effectiveess is ofte liited due to the poor selectio (or tuig) of its preters. o fcilitte the deteritio of the pproprite vlues of the preters of the PID cotroller for the cotrol of DC otor t y set poit therefore required usig pproprite tuig ethod. his pper provides better uderstdig of how PID cotroller is tued usig Ziegler-Nichols Step Respose, Cohe-Coo Method d Chie Hroes Reswick (CHR) ethod. Experietl results of PID cotrol of DC otor with lod coupled by ger shows tht CHR-PID tuigs gives uch iproved perforce over Ziegler-Nichols Step Respose d Cohe-Coo PID-tuig with settlig tie of 355sec., 11sec., d 111 sec. respectively t ech set poit Keywords: DC otor, Ziegler-Nichols Step Respose, Cohe-Coo ethod d CHR ethod 1 Itroductio he used of DC otor i vrious pplictios such s steel rollig ills, electric tris, electric vehicles, electric cres d robotic ipultors hs bee icresed i the recet yers due to the ese with which it speed c be cotrolled to give the desired perforce chrcteristics uder vrious coditio (Wleed & Kheri 01). he purpose of DC otor speed cotroller is to geerte sigl represetig the deded speed, d to drive the otor t tht speed (Pl &.Mudi 01). Although, proportiol-itegrl-derivtive (PID) cotrollers re widely used i DC otor speed cotrol due to its siple structure d robustess to the odelig error, their effectiveess is ofte liited due to the poor selectio (or tuig) of its preters required to stisfy the desig requireets s best s possible (Åströ & Hägglud 1995). o fcilitte the deteritio of the pproprite vlues of the preters of the PID cotroller for the DC otor uder cotrol t y set poit therefore required usig pproprite tuig ethod. Cotroller tuig ethods provide the cotroller preters i the for of forule or lgoriths. hey esure tht the obtied cotrol syste would be stble d would eet give objectives. here re severl estblished techiques for tuig the PID cotroller preters (Aborisde & Adewuyi 014). Ziegler d Nichols proposed the well-kow Process Rectio Curve techique d the Closed-oop Cyclig ethod (Ziegler & Nichols 194). hese tuig ethods re very siple, but cot gurtee to be lwys effective. My vrits of the trditiol Ziegler-Nichols PID tuig ethods hve bee proposed for giig better d ore cceptble cotrol syste respose bsed o the desirble cotrol objectives such s percet of overshoot, itegrl of bsolute vlue of the error (IAE), settlig tie, ipulted vrible behvior d etc. Soe of these tuig ethods hve cosidered oly oe of these objectives s criterio for their tuig lgorith d soe of the hve developed their lgorith by cosiderig ore th oe of the etioed criterio. Eve though PID tuig ethod usig Cohe-Coo (Seborg et l. 1989) d Chie Hroes Reswick (Neil 00) techiques hve bee extesively studied by y reserchers, the detils o how the techiques re ipleeted i DC otor speed cotrol re still vgue. his pper is iteded to provide better uderstdig of how PID cotroller is tued usig Cohe-Coo Method d CHR pproch. he pper lso copres the trsiet perforce of the DC otor usig Cohe-Coo Method d CHR techiques with Ziegler-Nichols ethod. his pper is orgized s follows. Sectio gives brief itroductio of the DC otor d its odel cocepts if exterl lod is ito it through ger tri. Sectio 3 describes the ipleettio of Cohe-Coo, CHR d Ziegler-Nichols techiques to tue PID cotroller. he siultio results usig MAAB re preseted i detil i Sectio 4 d Sectio 5 cocludes the pper.. Model of DC Motor with od Coupled by Gers DC otor drives re used for y speed d positio cotrol systes where their excellet perforce, ese of cotrol d high efficiecy re desirble chrcteristics. Essetilly, DC otor cosists of sttor, 58

2 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, 014 rotor d couttor. he sttor is the housig of the otor d cotis get, berig d etc. he rotor is the rottig prt of the otor, which cotis coil of wire through which curret flows. he coil of wire i the rotor coects to the couttor d receives curret through brushes. he couttor esures tht the curret flows i the proper directio while the rotor turs. rditiolly, DC otors ws cosidered to be vrible speed otor d re curretly used i y idustries becuse of its torque speed chrcteristics tht c be vried over wide rge while retiig high efficiecy. Oe of the jor chlleges of DC otors is the cotrol of its speed. DC otor cotrol is geerlly relized by djustig the teril voltge pplied to the rture but other ethods such s djustig the field resistce, isertig resistor i series with the rture circuit re lso vilble (Chp 005). A coo ctutor tht provides rotry otio i cotrol systes is the DC otor (Ogt 1990, Kuo 1991). he electric circuit of the DC otor with exterl lod coupled to the otor spidle through ger tri is show i Figure 1. he getic flux ϕ betwee the sttor d the rotor is give by the lier reltio ϕ k = φ (1) = f I f where k f is costt d I f reltio: kiφ where k is the sttor curret. he torque developed by the otor is give by the = () is the rture coil costt d i is costt rture curret. Substitutig (1) ito (), the torque ow hs the for = k i (3) where k = k k I. f f Figure 1: Equivlet odel of DC otor with exterl lod coupled by gers (Cludio & Joh 011) he Kirchhoff s lw of voltges for the rotor etwork is di v = + Ri + vb (4) dt he voltge vb is proportiol to the otor speed, i.e., v dθ b = kb = k ω b (5) dt where k is the bck ef costt, θ b is the gulr positio or displceet, d ω is the gulr velocity of the otor. Applyig the plce trsfor to (4) with zero iitil coditios, the rture curret is V Vb V kbsθ I = = (6) s + R s + R Sice, the shft torque is used for drivig lod gist the iertil d frictiol torque, therefore, θ d dθ = J + B = ki (7) dt dt where J is the torque iertil d B is the coefficiet of frictio. Apply the plce trsfor to (7) to yield 59

3 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, 014 J s Θ + BsΘ I = (8) k where ll iitil coditios re ssued zero. After soe lgebric ipultios the followig trsfer fuctio is derived Θ k = (9) V [( s + R )( J s + B ) + kkb ] s Cosiderig the effect of the lod coupled by gers to the otor spidle, the block digr show i Figure is drw: Figure : Equivlet block digr of DC otor with exterl lod coupled by ger (Cludio & Joh 011) he trsfer fuctio tht reltes the uloded otor torque O with the lod positio Θ is: Θ ( J s + B ) s = O 1 + s ( J s + B ) ( J s + B ) s = [( J + J ) s + ( B + B )] s where 1 = (11) 1 J B J : Nuber of teeth of the iput (otor) : Nuber of teeth of the output ger (lod) : Iertil oetu of the lod : Viscous frictio of the lod : Iertil oetu of the lod referred to the otor spidle B : Viscous frictio of the lod referred to the otor spidle J B O O = J = B where + J + B J O is the totl iertil oetu of the otor spidle while BO (10) (1) is the totl viscous frictio of the otor spidle. By puttig (1) i (10), the ew trsfer fuctio reltig the uloded otor torque O with the lod positio Θ is coputed s Θ = (13) O ( J Os + BO ) s he coplete equivlet block digr of the syste is show i Figure 3 60

4 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, 014 Figure 3: Equivlet block digr of DC otor with exterl lod (Cludio & Joh 011) he ssocited trsfer fuctio of the equivlet block digr of the syste show i Figure 3 is coputed s Θ k G( s) = = (14) V [( s + R )( J s + B ) + k k ] s O O b With the preters of the DC otor show i ble 1, the trsfer fuctio describig the dyic behvior of the DC otor becoe Θ 0.56 G( s) = = (15) V s s s ble 1: Preters of DC Motor with exterl lod coupled by gers Preter Vlue Rted otor speed, N 1750 rp Rted teril voltge 40 V R 11. Ω J B J B K K 1 b 0.11H 0. kg N - - sec/rd 0.0kg N - - sec/rd 1.8 N - /p 1.5 V - sec/rd 3. PID uigs echique he Proportiol-plus-Itegrl-plus-Derivtive (PID) cotrollers hve foud wide cceptce d pplictios i the idustries for the pst few decdes. It hs siple cotrol structure which ws uderstood by plt opertors d which they foud reltively esy to tue. I spite of the siple structures, PID cotrollers re prove to be sufficiet for y prcticl cotrol probles d hece re prticulrly ppelig to prcticig egieers. A typicl structure of PID cotrol syste is show i Figure 4, where it c be see tht i PID cotroller, the error sigl e(t) is used to geerte the proportiol, itegrl, d derivtive ctios, with the resultig sigls weighted d sued to for the cotrol sigl u(t) pplied to the plt odel

5 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, s s s Figure 4: PID cotrol structure A theticl descriptio of the PID cotroller is t 1 de( t) u( t) = k p e ( t) + e( τ ) dτ + d (16) dt i 0 where u(t) is the iput sigl pplied to the otor odel to obti the output sigl y (t), the error sigl e(t) is defied s e( t) = r( t) y( t), d r(t) is the referece iput sigl. A budt out of reserch work hs bee reported i the pst o the tuig of PID cotrollers. I this study the followig ethods for tuig re used. 3.1 he Ziegler Nichol s Step Respose A siple ethod of coputig the preters of PID cotroller developed by Ziegler d Nichols d published i 194 is kow s Ziegler-Nichols step respose ethod. his ethod is pplied to the ope loop step respose of the lod coupled DC otor displyed i Figure 4. his respose is pproxited by the trsfer fuctio give by first-order plus ded tie (FOPD) give by (Ngrth & Gopl 00): G b ( s) = e s (17) 1+ s where is the ppret tie dely d is the ppret tie costt, which re deteried by drwig tget lie t the iflectio poit of the curve d fidig the itersectios of the tget lie with the tie xis d the stedy-stte level lie. he PID preters of the Ziegler-Nichols step respose ethod is deterie s give i ble where is coputed s (Åströ & Witterk 1995). = b / (18) ble : Ziegler-Nichols step respose ethod Cotroller Preter k p i d PID Cotroller 1. / / 6

6 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, 014 Figure 4: Ope loop step respose of the lod coupled DC otor 3. Cohe-Coo Method Cohe-Coo ethod is other versio of the Ziegler-Nichols PID tuig ethod. his ethod is lso bsed o delyed first order rise, d the ethod of tuig the PID gis to chieve good respose ore sesitive th the Ziegler-Nichols ethod. Referrig to the FOPD odel (17), deote, the cotroller preters with this ethod is desiged by the direct use of ble 3. ble 3: Cohe-Coo ethod Cotroller Preter k p i PID Cotroller τ.5 τ τ τ = k / d τ = /( + ) d τ τ 3.3 Chie Hroes Reswick (CHR) Method Chie-Hroes-Reswick (CHR) ethod is lso other odified versio of the Ziegler-Nichols ethod (Neil 00). his ethod, developed i 195 by Chie-Hroes-Reswick provides better wy to select copestor for process cotrol pplictios. Copred with the trditiol Ziegler-Nichols tuig forul, the CHR ethod uses the tie costt of the plt explicitly. he CHR-PID cotroller tuig foruls re surized i ble 4 for set-poit regultio. ble 4: CHR tuig forule for set-poit regultio PID Cotroller with 0% overshoot with 0% overshoot k p i d 0.6/.5 k p i d 0.95/ Experietl Results he siulik odel of the PID cotrol of the lod coupled DC otor is show i Figure 5. I the experiet to evlute the perforce of PID cotrol tuig techique, the preters of Ziegler-Nichols Step Respose, Cohe-Coo d CHR ethods re clculted. o be ble to defie PID preters of the three lyzed tuig ethods, ope loop step respose of the lod coupled DC otor ws exied. Figure 4 shows the siultio result of the ope loop step respose of the lod coupled DC otor through which d 63

7 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, 014 vlues used i clcultig the PID preters for the three tuig ethods re geerted. he three tuig ethods re ipleeted usig Mtlb-SIMUINK. Experietl results of PID cotrol of DC otor with lod coupled by ger usig Ziegler-Nichols Step Respose, Cohe-Coo d CHR tuig ethods t three differet set poits re give i Figure 6, 7, 8 d 9 respectively. First set poit is 1600 rp, secod set poit is 1750 rp d the third set poit is 1900 rp. he secod set poit chge occurs t 650 th secod d the third set poit chge occurs t 5000 th secod. Coprisos of the percetge overshoot d settlig tie for ech tuig ethod t vrious set poits re preseted i ble 5. As see i the figure d tble it is cler tht CHR-PID-tuigs gives uch iproved perforce with sllest settlig tie of 355sec., 11sec., d 111 sec respectively t ech set poit th Ziegler-Nichols Step Respose d Cohe-Coo PID-tuig. 5. Coclusio I this pper the speed of DC otor with lod coupled by gers is cotrolled usig PID cotroller. Regrdig to the tuig of PID cotroller odified Ziegler-Nichols ethods hs bee preseted. hese ethods re ipleeted usig MAAB/SIMUINK siultio. Fro the obtied results show i Figure 9 d ble 5 it is cocluded tht CHR bsed tuig of PID cotroller outperfored the Ziegler-Nichols d Cohe- Coo tuig ethods t ech set-poit. he perforce of CHR-PID bsed tuig is t the cost of its less settlig-tie to rech desired vlue. Overll rsfer Fuctio siout Step4 e u PID Cotroller Sturtio s s s y o Workspce Ziegler-Nichols Scope1 siout1 Step1 e u PID Cotroller1 Sturtio s s s y o Workspce1 Cohe-Coo Scope Step e u PID Cotroller Sturtio s s s y siout o Workspce Scope Chie-Hroes-Reswick (0% overshoot) Scope3 Figure 5: Ope loop step respose of the lod coupled DC otor ble 5: Copriso of results t vrious set poits Set Poit Mxiu Overshoot (%) Rise ie (sec.) Settlig ie (sec.) (rp) (rp) (rp) Ziegler-Nichols Cohe-Coo CHR Method

8 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, Ziegler-Nichols 000 Speed (rp) ie (sec) Figure 6: Experietl result of PID cotrol for Ziegler-Nichols Step Respose 500 Cohe-Coo 000 Speed (rp) ie (sec) Figure 7: Experietl result of PID cotrol for Cohe-Coo tuig ethod 65

9 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, Chie-Hroes-Reswick 000 Speed (rp) ie (sec) Figure 8: Experietl result of PID cotrol for CHR tuig ethod Speed (rp) 1500 Refereces ie (sec) 66 Ziegler Nichol's Cohe-Coo Chie-Hroes-Reswick Figure 9: Copriso of PID cotrol of ll uig Methods Åströ K.J. d. Hägglud (1995), PID Cotrollers: heory, Desig d uig, Reserch rigle Prk, North Croli: Istruet Society of Aeric. Aborisde, D. O d Adewuyi, P. A (014), Evlutio of PID uig Methods o Direct Gs-Fired Ove, It. Jourl of Egieerig Reserch d Applictios, Vol. 4, Issue 3(Versio 1), pp Åströ, K., J., Witterk, B. (1995), Adptive Cotrol Addiso-Wesley Publishig Copy Ic., USA. Cludio Urre, d Joh Ker (011), A New Model for Alog Servo Motors: Siultios d Experietl Results, Cdi Jourl o Autotio, Cotrol d Itelliget Systes, Vol., No., pp. 9-

10 Cotrol heory d Ifortics ISSN (Pper) ISSN (Olie) Vol.4, No.5, Mrch 011. Chp, S.J. (005), Electric Mchiery Fudetls, 4th editio, New York: McGrw Hill.. Kuo B.C. (1991), Autotic Cotrol Systes, 6th ed., Pretice Hll, NJ. Neil Kuyvehove (00), PID uig Methods Autotic PID uig Study with MthCd, Clvi College ENGR Ogt K. (1990), Moder Cotrol Egieerig, d ed., Pretice Hll, NJ. Ngrth J. d M. Gopl (00), Cotrol Syste Egieerig, New Age Itertiol Publictios, 3rd Editio. Pl A.K. d R.K.Mudi (01), Speed Cotrol of DC Motor usig Rely Feedbck ued PI, Fuzzy PI d Self-ued Fuzzy PI Cotroller, Cotrol heory d Ifortics. Vol, No.1. Seborg, D. E.,.F. Edgr, D.A.Mellichp (1989), Process Dyics d Cotrol, Joh Wiley & sos. Wleed I. H. d Kheri A. M. (01), Speed Cotrol Of Seprtely Excited DC Motor Usig Fuzzy Neurl Model Referece Cotroller, Itertiol Jourl of Istruettio d Cotrol Systes (IJICS) Vol., No.4. Ziegler J.G. d N. B. Nichols (194), Optiu Settigs for Autotic Cotrollers, rs. ASME, Vol. 64, pp

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