Enginring, Tchnology & Applid Scinc Rsarch Vol. 7, No. 5, 217, 1976-1981 1976 Ral Tim Spd Control of a DC Motor Basd on its Intgr and Non-Intgr Modls Using PWM Signal Abdul Wahid Nasir Elctrical & Elctronics Eng. Dpt. NIT Jamshdpur Jamshdpur, India 214rs3@nitjsr.ac.in Idamakanti Kasirddy Elctrical & Elctronics Eng. Dpt. NIT Jamshdpur Jamshdpur, India 215rs2@nitjsr.ac.in Arun Kumar Singh Elctrical & Elctronics Eng. Dpt. NIT Jamshdpur Jamshdpur, India aksingh.@nitjsr.ac.in Abstract This papr xploits th advantag of non-intgr ordr modling of a procss ovr intgr ordr, in thos cass whr th procss modl is rquird for control purpos. Th prsnt cas dals with spd control of a DC motor. Basd on th ral tim opn loop rspons, DC motor is bing modld as intgr and non-intgr ordr first ordr plus dlay systm. Both ths modls ar thn sparatly usd for dtrmining two sts of Proportional-Intgral-Drivativ (PID) controllr paramtrs through Ziglr Nichols (ZN) closd loop tuning mthod. In addition to this, a modl basd control tchniqu i.. Intrnal Modl Control (IMC) is also implmntd using both intgr and non-intgr modl rspctivly. For carrying out th ral tim spd control of DC motor, LabVIEW platform has bn usd. Aftr going through th rsults, it is obsrvd that th controllr prformanc considrably improvs, if non-intgr ordr modl is usd for controllr dsign rathr than intgr ordr modl. Kywords-DC motor spd; fractional ordr systm; PID; intrnal modl control; LabVIEW I. INTRODUCTION DC motor is having a prominnt rol in almost all industrial and robotic applications. Its simplicity, rliability, conomic fasibility and th as with which such typs of DC motor spd can b controlld, maks it omniprsnt. With th rcnt advancmnts in powr lctronics, various switching control tchniqus mrgd. Puls Width Modulation (PWM) tchniqu is on of thm, which has bn usd hr for th spd control of DC motor by manipulating its duty cycl. Hr ZN-PID & IMC control tchniqu is usd to mt th control objctiv. Sinc both ths aforsaid control tchniqus rquir modl of th procss for controllr dsign, thrfor intgr and non-intgr (fractional) ordr modl is dtrmind from opn loop stp rspons of th DC motor. Hr, th concpt of fractional calculus has bn introducd in procss modlling. Th branch of calculus which is volving du to th gnralization of ordr of convntional intgro-diffrntial oprator from intgr to non-intgr, is known as fractional calculus. Du to incrasing intrst in this among rsarchrs in rcnt dcads [1-3], th solutions of various problms incorporating non-intgr ordr intgro-diffrntial oprator ar now availabl making its application fasibl in various filds of scinc and tchnology. Ths non-intgr ordr fundamntal oprators, whr α is th oprator ordr and a & t giv th limits of th oprator is dfind as follows: d, dt D 1, (1) a t t a d, Diffrnt dfinitions of this fractional intgro-diffrntial ar givn in [4]. Th trms fractional ordr and non-intgr ordr ar usd intrchangably. For carrying out all ral tim xprimntal work, LabVIEW platform has bn usd. For intrfacing th DC motor dynamics with LabVIEW platform through computr, a LabJack U3-HV data acquisition card (DAC) was mployd. For th RPM masurmnt of th motor, an infrard (IR) proximity snsor was usd. In Sction II stp-wis outlin of th work carrid out is givn. Exprimntal stup is dscribd in sction III. Sction IV givs th modlling of a DC motor basd on its opn loop data. Th control schms implmntd for th spd control of DC motor is discussd brifly in sction V. Exprimntal rsults ar prsntd and discussd in sction VI. Basd on th various rsults obtaind, conclusion ar drawn and scop for futur work is givn in Sction VII. II. EXPERIMENTAL SETUP Th xprimntal stup for th DC motor spd control systm is shown in Figur 1. It includs on 12 volts / 1-amp DC motor having maximum RPM of 2. To th shaft of this motor is attachd on circular disc, whos on half of th circular ara is colord in silvr and anothr half in black. An infrard (IR) proximity snsor having on IR transmittr and on IR rcivr is ngagd to sns th RPM of motor. Th IR transmittr continuously transmits th IR ray on th rotating disc which is rflctd back to th IR rcivr to giv positiv output whn falls on silvr ara and zro output whn falls on black ara, as th black ara absorbs th IR ray rathr than rflcting it. Thus dpnding on th rotational spd of th motor a pulsating signal of spcific frquncy is gnratd at
Rotation of motor (RPM) Enginring, Tchnology & Applid Scinc Rsarch Vol. 7, No. 5, 217, 1976-1981 1977 th output of IR transmittr. At th intrval of vry.1 scond, th output frquncy is masurd and updatd to th control algorithm mployd in LabVIEW nvironmnt through a LabJack mak U3-UV, data acquisition card. Basd on currnt RPM information, th controllr gnrats control signal in th form of PWM signal through th sam data acquisition card. This signal is thn fd to L298N DC motor drivr circuit, through which th spd of motor is rgulatd to dsird point. This is how a ral tim closd loop spd control systm prototyping is implmntd. DC MOTOR III. LabVIEW ENVIRONMENT Fig. 1. COMPUTER IR PROXIMITY SENSOR L298N MOTOR DRIVER DC motor spd control systm LabJack U3-HV DAC INTEER & NON-INTEER ORDER MODELLIN For th control systm undr study i.. spd control of a DC motor, first of all diffrnt opn loop stp rsponss ar notd to analyz th systm bhavior. Sinc PWM is th input manipulating variabl and RPM of th motor is output, controlld variabl, thrfor th duty cycl (in prcntag) of PWM signal is varid to obtain th stady stat RPM corrsponding to ach input signal. This input-output data is thn plottd as shown in Figur 2. 16 14 12 8 6 4 25 3 35 4 45 5 55 6 65 7 75 Duty cycl of PWM signal (%) Fig. 2. Input output stady stat charactristics From th natur of th plot, it can b concludd that within th spcific input rang of duty cycl i.. 25% to 75%, DC motor spd rspons almost xhibits linar charactristic. So, th controllr dsignd for any oprating condition within this rang should work wll throughout th RPM rang of 55 to 16. Th opn loop transfr function rlating input i.. armatur voltag and output i.. rotational spd is givn by (2) [5]: DC _ motor whr () s Kt V () s Ls R Js b K K t (2) θ : rotational spd of motor (RPM) V: armatur voltag (V) J: momnt of inrtia of rotor (Kg.m2) b: motor viscous friction constant (N.m.s) K : lctromotiv forc constant (V/rad/sc) K t: motor torqu constant (N.m/Amp) R: lctric rsistanc (ohm) L: lctric inductanc (H) In many cass, whr R>>L, th transfr function givn by (2) rducs to first ordr systm givn by (3). DC _ motor Kt R Js b K K Now th nxt stp is to stimat th DC motor modl paramtr basd on th stp rspons. A stp chang in duty cycl of PWM signal is mad from 5% to 75%, and RPM rspons data is obtaind. Sinc this rspons vry much matchs th charactristics of stp rspons of first ordr plus dlay systm. Thrfor, it is dducd that th currnt systm can b modlld as first ordr plus dlay systm. Making us of th flxibl fatur [6-7] of fractional ordr calculus in modlling, th prsnt systm is not only modlld as intgr but also as non-intgr ordr systm as givn by (4) and (5) rspctivly. intgr nonintgr K s 1 Ls K s 1 Ls Hr, ntic Algorithm (A) is usd to stimat th valus of paramtrs for intgr modl i.. gain (K), tim Constant (τ) and dlay (L) to minimiz th valu of Intgral of Squard Error (ISE), th objctiv function considrd [8-9]. Similarly, th unknown paramtrs of non-intgr modl ar obtaind i.. α, th ordr of th filtr, along with K, τ & L. A is stochastic sarch computational tool basd on th thory of natural volution, basically involving th procss of slction, crossovr and mutation. In 1975, Holland [1], proposd th lmntary principls of A. Basd on this novl topic, latr som litraturs also surfacd [11]. To bgin with, a random st of paramtrs is prsumd, which mimics th gns of chromosoms, to b th possibl solution of th sarch problm. An objctiv function is dfind that rflcts th fitnss of ths chromosoms, which whn subjctd to minimization or maximization undr crtain constraints yild th optimal rsult. Th objctiv function considrd for stimation of modl paramtrs is Intgral of Squard Error (ISE) subjct to minimization is givn by (6). Diffrnt sts of t (3) (4) (5)
Enginring, Tchnology & Applid Scinc Rsarch Vol. 7, No. 5, 217, 1976-1981 1978 modl paramtrs ar chosn from th sarch spac basd on A at vry simulation run, and th stp rspons data of that particular modl is obtaind. This obtaind modl data is thn compard to alrady xisting systm data to gnrat rror which is ultimatly usd to calculat ISE. This ISE valu is again passd to A as fdback information, and this procss continus until optimal minimum is rachd. Figur 3 rprsnts th implmntation of A for modl paramtr stimation. 2 2 sp (6) ISE { y ( t) - y( t)} dt { ( t)} dt Th discussd A is implmntd using Matlab for currnt problm of finding modl paramtrs. For carrying out all typs of simulation work for modl paramtr stimation rlatd to non-intgr ordr plants, FOMCON toolbox [12], has bn usd in Matlab. Tabl I shows th A charactristics considrd in th prsnt cas. Tabl II givs th intgr and non-intgr ordr transfr function modl for DC motor spd with thir corrsponding ISE valus and Figur 4 rprsnts th stp rspons for th DC motor spd systm along with its intgr and non-intgr modl. In th inst of Figur 4, zoomd rsponss for spcific tim duration ar givn, so that th diffrnc among all th thr rsponss can b idntifid asily. STEP INPUT Fig. 3. SYSTEM DC MOTOR MODEL A Systm data Modl data Es () ISE Modl paramtrs stimation using A Natur of Modl Intgr Ordr Non- Intgr Ordr TABLE II. IV. MODEL PARAMETERS Transfr Function 22.58.235s 1 23.358.91.25s 1.347s.364s CONTROLLER DESIN In spit of advancmnts in diffrnt schms of controllr in th rcnt past, PID still ruls th application domain du to its simpl architctur and its ability to control various typs of linar and non-linar procsss. Most of th tuning of controllr paramtrs for PID rquirs modl of th procss. Anothr vry simpl control tchniqu basd on modl of th procss is Intrnal Modl Control (IMC). Both ths control tchniqus i.. PID & IMC ar implmntd for th prsnt PID & IMC ar implmntd for th prsnt cas of DC motor spd control, and ar discussd in nutshll undr th following subsction rspctivly. A. PID Control Transfr function rprsntation of PID controllr rlating rror, E(s) and control signal, U(s) is givn by (7). U( s) 1 Kp 1 T s d E() s T s i whr K p, T i & T d ar th controllr tuning paramtrs. Th transfr function of PI controllr can b obtaind by substituting T d=, in (7). Similarly, for obtaining P-typ controllr transfr function, st T i= and T d=. Th basic PID controllr tuning ruls wr proposd in [13], and wr basd on mpirical data obtaind from xtnsiv xprimnts. Authors in [14], furthr modifid th tuning ruls. Sinc thn various rsarchrs ar proposing svral othr mthods for PID controllr tuning. At prsnt, Ziglr Nichols closd loop mthod/cycling mthod is usd. Although it is xprimntal basd control dsign, but sinc th modl of th procss to b controlld is availabl, thrfor, without disturbing th actual procss, th controllr paramtrs can b obtaind. First of all, ultimat gain, K u and ultimat priod, P u is dtrmind by making T d= and T i= and varying proportional gain, K p such that th closd loop rspons bcoms oscillatory having constant amplitud. Th valu of K p at which sustaind oscillation occurs is th ultimat gain, K u and th priod of th oscillation is th ultimat priod, P u. Aftr that th P, PI & PID paramtr can b obtaind using Tabl III. ISE 44 154 (7) Fig. 4. TABLE I. Opn loop stp rsponss A CHARACTERISTICS Population siz 5 Fitnss Scaling Function Rank Crossovr Function.8 Crossovr Fraction Scattrd Migration Fraction.2 Ending Critrion Function tolranc of 1-4 TABLE III. ZN CLOSED LOOP PID TUNIN Controllr Typ Kp T i T d P.5K u -- -- PI.45K u P u/1.2 -- PID.6K u P u/2 P u/8 B. Opn Loop IMC Intrnal modl control (IMC) is a modl basd control tchniqu, whr a procss is mbddd within th
duty cycl of PWM (%) Spd of Motor (RPM) Enginring, Tchnology & Applid Scinc Rsarch Vol. 7, No. 5, 217, 1976-1981 1979 controllr [15]. Th opn loop IMC structur is givn in Figur 5, whr q(s) is th modl basd controllr and (s) is th procss. rs () qs () us () () s ys () Fig. 5. Modl basd Controllr Procss Opn loop intrnal modl control Considr a first ordr procss (s), givn by (8), whos controllr transfr function q(s) is givn by (9). K s 1 (8) 1 q( s) f ( s) ( s) (9) whr -1 (s) is th invrtibl part of (s), with right hand sid zros and dlay bing factord out if thy ar prsnt, to mak q(s) stabl. f(s) is th filtr dfind by (1). 1 f() s ( s 1) (1) Th filtr, f(s) is incorporatd in th controllr transfr function q(s) to mak it propr i.. physically ralizabl. Th opn loop control rspons y(s) is, y( s) q( s) ( s) r( s) f ( s) r( s) (11) As sn from (11), th output rspons dpnds on filtr tim constant λ, hnc it is considrd as controllr paramtr. In th prsnt cas, IMC tchniqu will also us non-intgr modl, othr than intgr ordr modl. Thrfor, bfor implmnting non-intgr modl in ral tim control in LabVIEW platform, it is firstly convrtd to th quivalnt highr intgr ordr modl using mthod dscribd in [16], bcaus th dynamics of non-intgr ordr systm can b wll rprsntd with th hlp of highr intgr ordr modl and vic-vrsa. intgr & non-intgr modl, which hav bn tabulatd as follows in Tabl V. TABLE V. P, PI AND PID CONTROLLER PARAMETERS Controllr Typ K p T i(s) T d(s) P IO modl.16 -- -- FO modl.14 -- -- PI IO modl.95 1.528 -- FO modl.126 1.86 -- PID IO modl.127.917.229 FO modl.168.652.163 Th P, PI & PID obtaind from intgr ordr modl is tstd against th stp chang in RPM. Similarly, th controllr prformanc is also valuatd for controllrs obtaind from non-intgr ordr modl. All th ral tim closd loop controlld rspons data is collctd and plottd to hav th comparativ study. Figurs 6-8, rprsnt th srvo rspons for P, PI & PID control rspctivly. Tabl VI givs th various prformanc indics for th diffrnt controllrs undr tst. Spd of Motor (RPM) duty cycl of PWM (%) 15 5 St-Point controllr paramtr drivd from IO modl controllr paramtr drivd from FO modl 1 2 3 4 5 6 7 8 9 1 Tim (s) 1 8 6 4 2 1 2 3 4 5 6 7 8 9 1 Tim (s) Fig. 6. P control output from FO modl P control output from IO modl Srvo Rspons of P controlld systm V. RESULTS AND DISCUSSIONS Th PID controllr paramtrs wr tund using Ziglr Nichol s closd loop tchniqu. As discussd in subsction IV.A, ultimat gain K u and ultimat priod, P u is dtrmind. Sinc two typs of modl ar availabl for th procss. i.. intgr and non-intgr ordr modl, thrfor for ach natur of modl, K u and P u ar drivd rspctivly as shown in Tabl IV. 15 5 1 2 3 4 5 6 7 8 Tim (s) 1 St-Point controllr paramtr drivd from IO modl controllr paramtr drivd from FO modl TABLE IV. KU AND PU VALUES Paramtr IO modl FO modl K u.2125.28 P u(sc) 1.8338 1.335 8 6 4 2 PI control output from FO modl PI control output from IO modl Basd on ths valus of K u and P u, th controllr paramtrs for P-typ, PI-typ and PID-typ control ar dtrmind using ZN formula as givn in Tabl IV for both th 1 2 3 4 5 6 7 8 Tim (s) Fig. 7. Srvo Rspons of PI controlld systm
duty cycl of PWM (%) Spd of Motor (RPM) Enginring, Tchnology & Applid Scinc Rsarch Vol. 7, No. 5, 217, 1976-1981 198 Aftr analyzing th closd loop stp rsponss in Figurs 6 8 and th data indicating various prformanc indics in Tabl VI, it can b asily dducd that prformanc for th class of controllrs obtaind using non-intgr ordr modl is suprior in vry aspct than thos obtaind from intgr ordr modl for all P, PI & PID control schms. Thr is on insignificant xcption in P-typ controllr whos maximum ovrshoot is slightly mor for th controllr dsign using non-intgr ordr modl rathr than intgr ordr modl. Othr prformanc indx paramtrs such as ris tim (t r) and pak tim (t p) hav not bn includd in Tabl VI, as ths paramtrs ar almost idntical for both cass, i.. for controllr obtaind from intgr and non-intgr modl. Evn, Figurs 6-8, th closd loop stp rspons, convys th sam mssag. Moving to othr control tchniqu mployd for ral tim spd control of DC motor, i.. IMC, whr th filtr tim constant, λ is controllr paramtr. Th valu of λ dtrmins th natur of control prformanc. Th aggrssivnss of controllr is invrsly proportional to λ valu, whras robustnss is dirctly proportional to valu of λ. Hnc th valu of λ is obtaind through xtnsiv simulations and is found to b λ=.1. Finally, IMC opn loop control is implmntd in LabVIEW nvironmnt using th sam valu of λ Hr only opn loop control has bn implmntd in ordr to just validat th fact that, sinc non-intgr modl capturs th ral tim systm dynamics bttr than intgr ordr modl, thrfor th modl basd control tchniqu incorporating non-intgr modl givs improvd rsults rathr than intgr modl. Figur 9 shows th opn loop control rsponss for IMC obtaind from both intgr and non-intgr ordr modl for th stp chang mad at t=.4 scond, from initial valu of to 6 RPM. Tabl VII givs th transfr functions of diffrnt IMC along with prformanc indics data. Figur 9 and Tabl VII data clarly dpict th suprmacy of IMC basd on non-intgr ordr modl ovr intgr ordr modl. TABLE VI. PERFORMANCE INDICES FOR P, PI & PID CONTROLLED SYSTEM P-typ controllr dsign using PI-typ controllr dsign using PID-typ controllr dsign using Prformanc Intgr Non-intgr Intgr Non-intgr Intgr Non-intgr Indics modl modl modl modl modl Modl Sttling tim, ±5% (s) 1.26 1.13 3.94 2.96 3.48 2.56 Sttling tim, ±2% (s) 1.43 1.28 5.2 3.72 3.91 2.86 Ovrshoot (%) 38 41 45 41 41 37.5 Undrshoot (%) -- -- 4 33 36.5 33.4 Offst 34 274 Nil Nil Nil Nil ISE 1.16+7 8.12+6 3.24+6 2.52+6 3.5+6 2.35+6 IAE 3.29+4 2.72+4 9.49+3 7.19+3 8.71+3 6.44+3 VI. CONCLUSION 8 In this papr, two diffrnt control tchniqus, PID and IMC hav bn implmntd for th ral tim spd control of a DC motor. Th aim of th prsnt papr is to highlight th fact that non-intgr ordr modl is mor fficint in capturing th dynamics of ral tim systms as compard to intgr ordr modl, and also th controllrs basd on non-intgr modls yild bttr prformanc rathr than thos basd on th intgr modl. Th conclusion is wll validatd by th rsults obtaind from xprimntal stup. This work can b xtndd to control any systm, whrvr a modl is rquird for controllr dsign Spd of Motor (RPM) 6 4 2.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Tim (s) 3 St-Point IMC basd on IO modl IMC basd on FO modl 15 5 St-Point controllr paramtr drivd from IO modl controllr paramtr drivd from FO modl 1 2 3 4 5 6 7 8 9 1 Tim (s) duty cycl of PWM (%) 2 1 control output for IMC basd FO modl control output for IMC basd IO modl.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Tim (s) Fig. 9. IMC opn loop control rsponss 1 8 6 4 2 PI control output from FO modl PI control output from IO modl 1 2 3 4 5 6 7 8 9 1 Tim (s) Fig. 8. Srvo Rspons of PID controlld systm IMC basd on intgr ordr modl non-intgr ordr modl. TABLE VII. Controllr TF, q(s).235s 1 22.58(.1s 1).91.25s 1 23.358(.1s 1) IMC PERFORMANCE INDICES offst ISE IAE 3 3.22+5 1.51+3 ngligi bl 2.79+5 1.12+3
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