Proceedings of the World Congress on Engineering nd Computer Science 2007 WCECS 2007, October 24-26, 2007, Sn Frncisco, USA obustness Anlysis of Pulse Width Modultion Control of Motor Speed Wei Zhn Abstrct A first principle model for DC permnent mgnetic motor is used in the robustness nlysis of Pulse Width Modultion (PWM) motor speed control. A Simulink model is developed for simultion nd nlysis. Bsed on the simultion result, the min fctors tht contributed to the verge speed vrition re identified using Design of Experiment (DOE). A robust solution is derived to improve the verge speed control ccurcy using esponse Surfce Method (SM). The robustness of the new design is verified using the simultion model. Index Terms Design of Experiment, Monte Crlo Anlysis, Pulse Width Modultion, esponse Surfce Method. I. INTODUCTION Pulse Width Modultion (see [7]) is commonly used in industry. For instnce, in mny utomotive pplictions, bttery/lterntor is often used s power source. The bttery/lterntor hs constnt voltge of bout 2 Volts. When voltge different from this vlue is needed to control n ctutor, one cn either use hrdwre voltge regultor or PWM control. The hrdwre solution is usully not desirble due to the high cost nd pckging issues ssocited with it. The ide of using PWM control is very simple: the power to the ctutor is effectively reduced by n mount tht cn be djusted with PWM duty cycles. A 50% PWM control commnd is shown in Fig., where the PWM frequency is defined to be /Ts, nd the PWM duty cycle is defined to be On/Ts * 00%. When the commnd is high, the constnt voltge source is connected to the ctutor. When the commnd is low, the constnt voltge source is disconnected from the ctutor. A rule of thumb for PWM duty cycle (see [4]) selection is DesiredVoltge DutyCycle = min 00% () No lvoltge This is very ttrctive solution. However, it ws found in [9] tht the results re usully not ccurte. In rel world ppliction one gets lrge vrition in the result. The cuses for the vrition cn be from the nonlinerity in the system, the vrition in the system prmeters nd the voltge source etc. Mnuscript received July 5, 2007. Wei Zhn is with the Deprtment of Engineering Technology nd Industril Distribution, Texs A&M University, College Sttion, TX 77843-3367 USA (Tel.: 979-862-4342; Fx: 979-847-3367; e-mil: wei.zhn@ tmu.edu). u(t) On T S Fig.. A typicl PMW control commnd Using the Six Sigm (see []) pproch, the robustness of the PWM control of motor speed ws discussed in [9]. Detils nd references in controlling DC permnent mgnetic motors cn be found in [8]. In this pper, we will try to generlize the result in [9] by dding the temperture fctor in the robustness nlysis. First, the model for the motor developed in [3] is modified so tht the temperture is tken into considertion. The bseline performnce of PWM control is estblished using Monte Crlo nlysis (see [2] nd [2]). DOE nlysis (see [5] nd [6]) is conducted to find the min fctors contributing to the speed vrition. A very useful tool, esponse Surfce Method (SM) (see [0] nd []), is used to derive robust design. The new design is verified nd compred to the bseline using Monte Crlo nlysis. II. MODEING A DC permnent mgnetic motor cn be modeled by the following equtions: where di = e i dt T = K i m e = K b 2 d θ = T 2 dt J i dθ b dt m T J eb K b = K i (This cn be esily derived bsed on the fct tht energy goes in is equl to energy comes out) ; K i is the torque constnt, in N-m/A; K b is the bck-emf constnt, in V/rd/sec; i ()is t the rmture current, in A; t (2) ISBN:978-988-9867-6-4 WCECS 2007
Proceedings of the World Congress on Engineering nd Computer Science 2007 WCECS 2007, October 24-26, 2007, Sn Frncisco, USA voltge / _inv -K- s lod Gin3 Integrtor I s I Integrtor -Ktorque gin 9.5493 Gin4 speed speed bck emf voltge -K- rpm gin is the rmture resistnce, in Ω ; eb ()is t the bck emf, in V; T ()is t the lod torque, in N-m; Tm () t is the motor torque, in N-m; θ is the rotor displcement, in rd; is the rmture inductnce, in H; e (t) is the pplied motor voltge, in V; Fig. 2. Simulink model for DC permnent mgnetic motor minly determined by the PWM duty cycle. In this pper, the focus will be on the verge speed. Thus, without loss of generlity, we ssume tht the PWM frequency is 40 Hz. Using the model, it cn be esily shown tht with the following motor/control prmeter set: K i = 0.02 Nm/A, = 0. Ω, =.0e-4 H, J =9.0e-5 kgm 2, V = 2 V, DC = 20%, T lod = 0.3 Nm, T=20 o C, 28.5% duty cycle would give n verge speed of 3000 rpm. The rule of thumb defined in eqution () would give duty cycle of 64.83% which would result in n verge speed of 4570 rpm. J is the rotor inerti, in kg- m 2. The coil resistnce nd torque/bck emf gin re dependent on the temperture: ( T ) = [ + 0.0039( T 20)] (3) K i ( T ) = K[ 0.002( T 20)] (4) where K is the nominl torque gin nd is the nominl resistnce of the coil, both t 20 o C. The first eqution in (2) is derived using the Kirchoff Voltge w: the sum of the voltge drops cross the resistor, the inductor, nd the bck emf is equl to the pplied voltge. The second eqution in (2) is simply from the liner pproximtion of the torque-current curve. The third eqution in (2) is from the fct tht the bck emf voltge is proportionl to the ngulr velocity of the rotor. The fourth eqution in (2) is derived using Newton s Second w. Bsed on (2), we build Simulink model (Fig. 2). The vlues for the motor model prmeters such s K i,,, J, the temperture, nd the control prmeter (Duty Cycle) re ssigned in Mtlb script file. (3) nd (4) re lso evluted in the sme script file for simplicity. This llows us to esily simulte thousnds of different prmeter vlues for sttistic nlysis. It ws discovered in [9] tht pek-to-pek speed is minly determined by the PWM frequency nd the verge speed is III. BASEINE ESUT The verge error defined by the following formul verge error = verge speed trget speed (5) will be used s metric for performnce. To estblish the bseline performnce, we use the Simulink model with the following ssumption on the model prmeters: Nominl resistnce: norml distribution, men vlue.0e- Ω, σ = 5.0e-3 Ω ; Inductnce: norml distribution, men vlue.0*e-4 H, σ = 5.0e-6 H; Inerti: norml distribution, men vlue 9.0e-5 kg m^2, σ = 4.5e-6 kg m^2; Nominl torque gin: norml distribution, men vlue 2.0e-2 Nm/A, σ =.0e-3 Nm/A; Bck emf gin: norml distribution, men vlue 2.0e-2 v/rd/s, σ =.0e-3 v/rd/s; Voltge: norml distribution, men vlue 2 v, σ =.5v; Temperture: uniform distribution, rnge: -0 o C ~ 60 o C. od: T = 0.3 Nm; where σ is the stndrd devition of the norml distributions. The rndom vlues for model prmeters/input were generted using Minitb. One thousnd set of vlues were generted in the rndom mnner. The trget verge speed is 3000 rpm. From previous section, the duty cycle for PWM control should be set to 28.5%. A Mtlb script ws written to red ech set of the prmeter vlues, run the model nd then record the output. ISBN:978-988-9867-6-4 WCECS 2007
Proceedings of the World Congress on Engineering nd Computer Science 2007 WCECS 2007, October 24-26, 2007, Sn Frncisco, USA Summry for bseline verge error (rpm) A nderson-drling Normlity Test A-Squred 0.36 P-Vlue 0.454 Men -37.32 StDev 765.26 V rince 58562.34 Skew ness -0.004050 Kurtosis -0.0878949 N 000-2400 -600-800 -0 800 600 2400 Minimum -2780.00 st Q urtile -667.75 Medin -33.00 3rd Q urtile 40.00 Mximum 2490.00 95% C onfidence Interv l for Men -84.80-89.83 95% C onfidence Interv l for Medin -209.47-8.72 Men 95% Confidence Intervls 95% C onfidence Interv l for StDev 733.3 800.36 Medin -200-80 -60-40 -20-00 -80 The simultion result is shown in Fig. 3. The error distribution is pproximtely norml with men of -37 rpm nd stndrd devition of 765 rpm. IV. DETEMINING THE MAIN FACTOS To understnd how ech prmeter in the model ffects the result, we conduct DOE for motor PWM control. The 2-level full fctoril DOE with the following vribles is chosen: esistnce: 0.085 Ω, 0.5 Ω ; Inductnce: 8.5e-5 H,.5e-4 H; Inerti: 7.65e-5 kg m^2,.04e-4 kg m^2; Torque/bck emf gin: 0.07 Nm/A, 0.023 Nm/A; Voltge: 8 v, 6 v; Temperture: -0 o C, 60 o C; Duty cycle: 25%, 35%. Term 2.0 E F DG DF DE AG EG AD EF AE D AF FGB AG C CD BE CE BG CG CF BD AC AB BF BC 0 20 Preto Chrt of the Stndrdized Effects (response is verge error (rpm), Alph =.05) 40 60 80 Stndrdized Effect 00 Fig. 3. Bseline performnce 20 Fig. 4. Preto chrt of effects for motor speed Fctor A B C D E F G Nme J K V T D The totl number of test is therefore 2 7 = 28. The result is shown in Fig. 4. It cn be seen tht bttery voltge, temperture nd PWM duty cycle re the min fctors tht cuse the speed control vrition. Since temperture is usully not monitored, voltge is usully monitored, nd duty cycle is control vrible, we focus on the voltge nd duty cycle. This will be justified in Section V. V. ESPONSE SUFACE METHOD To further investigte the impct of PWM duty cycle nd bttery voltge on the verge motor speed, we do the following to find the response surfce (i.e., the verge motor speed s function of PWM duty cycle nd bttery voltge): Assume the nominl vlue for ll the motor prmeters; Vry the bttery voltge from 8 v to 6 v by incrementl of 0.5 v; Vry the PWM duty cycle from 5% to 65% by incrementl of 0.3%; Simulte the model nd record the verge speed. The simultion result is plotted in Fig. 5. A horizontl plne pssing through trget speed of 3000 rpm is lso plotted in the sme figure. It cn be seen from the response surfce tht there is lrge vrition to the verge speed s the voltge vries. ISBN:978-988-9867-6-4 WCECS 2007
Proceedings of the World Congress on Engineering nd Computer Science 2007 WCECS 2007, October 24-26, 2007, Sn Frncisco, USA Fig. 5. Averge motor speed s function of voltge nd duty cycle The intersection of these two surfces defines reltionship between Voltge nd PWM duty cycle DC = f(v) (6) Since the curve specified by (6) lies on plne, we cn plot it in two dimensionl plne, s shown in Fig. 6. The function in (6) cn be estimted by using numericl curve fitting method s follows DC V V V V 4 3 2 = 0.029.6 + 33.6 39.5 + 85.5 (7) If we djust the duty cycle using (7), then we will hve constnt verge motor speed of 3000 rpm. This leds us to new PWM control design: The duty cycle of PWM is djusted ccording to the vlue of the voltge being pplied to the motor. Of course the ctul result will hve vrition since we still hve the prt-to-prt vrition nd temperture vrition. To compre the new design to the bseline design, we conduct the sme Monte Crlo nlysis with the new design. The result is shown in Fig. 7. The error distribution is pproximtely norml with men of -57 rpm nd stndrd devition of 340 rpm. The men vlue is similr to tht of the bseline. The stndrd devition is reduced by 55%. It is worth mentioning tht the result shown in Fig. 7 is chieved without tightening the prt to prt vrition for the motor. Also, the temperture fctor is tken into considertion in the Monte Crlo nlysis, even though we decided in section IV to leve the temperture fctor lone nd focus on the voltge nd duty cycle. The nlysis shows tht it ws resonble ssumption to ignore the temperture fctor. If we cn dd temperture sensor, then we cn hopefully further reduce the vrince. This is still under investigtion. Fig. 6. Duty cycle s function of Voltge ISBN:978-988-9867-6-4 WCECS 2007
Proceedings of the World Congress on Engineering nd Computer Science 2007 WCECS 2007, October 24-26, 2007, Sn Frncisco, USA Summry for verge error (rpm) A nderson-drling Normlity Test A-Squred.03 P-V lue 0.00 Men -57.04 StDev 340.67 V rince 6056.0 Skew ness -0.68904 Kurtosis -0.276665 N 000-800 -400 0 400 800 Minimum -090.00 st Q urtile -290.50 Medin -44.5 3rd Q urtile 93.50 Mximum 060.00 95% C onfidence Interv l for M en -78.8-35.90 95% C onfidence Interv l for M edin -7.65-5.00 Men 95% Confidence Intervls 95% C onfidence Interv l for StDev 326.37 356.30 Medin -80-70 -60-50 -40-30 -20 Fig. 7. Performnce of new design (verge speed error) VI. CONCUSIONS A first principle model, which tkes temperture into considertion, is used to simulte the PWM control of DC permnent mgnetic motors. Using DOE, the min cuse of the verge speed vrition is found. A new robust PWM control design is derived using SM. The verge speed error vrition for the new design is reduced by 55% compred to the bseline PWM control. This is chieved without tightening the tolernce bnds for the motor prmeters. In other words, the robustness of the PWM control is chieved without dditionl cost. As one cn esily see tht the methodology used in this pper is not limited to motor speed control. The bsic robustness nlysis concept cn be used in mny other res. VII. EFEENCES [] M. Hrry nd. Schroeder, Six Sigm: The Brekthrough Mngement Strtegy evolutionizing the World s Top Corportions, st edition, Doubledy, New York, 2000. [5] G. Tguchi, Tguchi on obust Technology Development, ASME Press, New York, 993. [6] G. Tguchi, System of Experimentl Design, Unipub/Krus/Americn Supplier Institute, Derborn, MI, 987. [7] A.K. Gelig nd A. N. Churilov, Stbility nd Oscilltions of Nonliner Pulse-Modulted Systems, Birkhuser, Boston, 998. [8]. Vlentine, Motor Control Electronics Hndbook, McGrw-Hill, 998. [9] W. Zhn, obust Design For Pulse Width Modultion Control Using Six Sigm Tools, The 2th Annul Interntionl Conference on Industry, Engineering & Mngement Systems, Florid, Mrch 2007. [0].H. Myers nd D.C. Montgomery, esponse Surfce Methodology, John Wiley & Sons, Inc., 995. [] E. del Cstillo, Process Optimiztion: A Sttisticl Approch, Interntionl Series in Opertions eserch & Mngement Science, Vol. 05, Americn Institute Physics, 2007. [2] J. S. iu, Monte Crlo Strtegies in Scientific Computing, Springer, New York, June 200. [2] G. Csell, Monte Crlo Sttisticl Methods, Springer, September 2004. [3] W. Zhn, Sensorless Speed Control For DC Permnent Mgnetic Motors, The Ninth IASTED Interntionl Conference on Control nd Applictions, Montrel, My 2007. [4] M. Brr, "Pulse Width Modultion," Embedded Systems Progrmming, September 200, pp. 03-04. ISBN:978-988-9867-6-4 WCECS 2007