Optimal Speed Control for Direct Current Motors Using Linear Quadratic Regulator

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1 Optiml Speed Control for Direct Current Motors Using Liner Qudrtic Regultor Amir Hshim Obeid Ahmed School of Electricl & Nucler Engineering, Sudn University of Science nd Technology Received: Accepted: Abstrct: Direct Current (DC) motors hve been extensively used in mny industril pplictions. Therefore, the control of the speed of DC motor is n importnt issue nd hs been studied since the erly decdes in the lst century. This pper presents comprison of time response specifiction between conventionl Proportionl-Integrl-Derivtives (PID) controller nd Liner Qudrtic Regultor (LQR) for speed control of seprtely excited DC motor. The gol is to determine which control strtegy delivers better performnce with respect to DC motor s speed. Performnce of these controllers hs been verified through simultion using MATLAB/SIMULINK softwre pckge. According to the simultion results, liner qudrtic regultor method gives the better performnce, such s settling time, stedy stte error nd overshoot compred to conventionl PID controller. This shows the superiority of liner qudrtic regultor method over conventionl PID controller. Keywords: Optiml Control, Liner Qudrtic Regultor, Proportionl-Integrl-Derivtive Controller, Direct Current Motors, Speed Control. ا: ت ار ا (DC) م %ق وا"! ا ات ا. +* (ن ا'& " ت ار ا 12 / درا"/ + ا.د ا,و ن ا 34. ھ+ه ا.ر 2 م ر% 7.ا 56 ت ا;": 7 ا> 7 ا ا"- ا'-ا 45 (PID) و ا=& ا 7 ا (LQR) '& " ك ار ا ذو ا<+ ا 5. ا/ف ذ* إ"ا: ا'& ا 3 أداء أ D1 C 7'& " ك ار ا. و 2 & ا C أداء ھ+ه ا'ت HIل اة 7 "ام ح> 7%L.MATLAB/SIMULINK و LO اة %: أن ط ا=& ا 7 ا أداء أ DM D1 ز ا" S ا Q ح ا;"ار و:وز ا/ف ر%! ا ا.PID وھ+ا ل 5.ق ط ا=& ا 7 ا.PID ا ا Introduction Electricl derives involving vrious types of DC motors turn the wheel of industry. The min reson for their populrity is the bility to control their torque nd flux esily nd independently. Therefore, DC motors re comprehensively used in vrious industril pplictions such s electricl equipment, computer peripherls, robotic mnipultors, ctutors, steel rolling mills, electricl vehicles, nd home pplinces. Its pplictions spred from low horse power to the multimeg wtt due to its wide power, torque, speed rnges, high efficiency, fst response, nd simple nd continuous control chrcteristics [1-4]. Controlling the speed of DC motor is pivotl issue. The speed of DC motor cn be chnged by controlling the rmture nd field voltges. In this pper, the controller is designed to control the rmture voltge while the field voltge is fixed s constnt. Over the pst decdes, mny techniques hve been developed for the DC motor control. Some of these methods were bsed on clssicl nd lso intelligent pproches [5-1]. For DC motors, fctors such s unknown lod chrcteristic nd prmeter vrition influence seriously the controlling effect of speed controller. The most commonly used controller for the speed control of DC motors is conventionl PID controller. Trditionl PID controllers hve been successfully used in control pplictions since 194s nd re the most often used industril controller tody. Conventionl PID controllers hve severl importnt fetures. The reson is tht the conventionl PID controller is esy to implement either by hrdwre or by softwre. No deep mthemticl theory is necessry to understnd how the conventionl PID controller works, so everybody is ble to 32

2 imgine wht is hppening inside the controller during the control process. Furthermore, it hs the bility to eliminte stedy stte offset trough integrl ction nd it cn nticipte the chnges through derivtive ction. In ddition to this, trditionl PID controllers hve very simple control structure nd inexpensive cost. In spite of the mjor fetures of the fixed PID controller, it hs some disdvntges such s the high strting overshoot in speed, the sensitivity to controller gins nd the sluggish response due to sudden chnge in lod torque disturbnce. Therefore, gret del of ttention hs been focused on dptive or self-tuning of conventionl PID controller gins. Tuning PID controller prmeters is very difficult, poor robustness; therefore, it's difficult to chieve the optiml stte under field conditions in the ctul production. In order to overcome some problems tht fced by conventionl PID controller nd chieve ccurte control performnce of speed control of DC motor, the other type of control methods cn be developed such s liner qudrtic regultor [11-16]. Liner qudrtic regultor design technique is well known in modern optiml control theory nd hs been widely used in mny pplictions. It hs very nice robustness property. This ttrctive property ppels to the prcticing engineers. Thus, the liner qudrtic regultor theory hs received considerble ttention since 195s. The liner qudrtic regultor technique seeks to find the optiml controller tht minimizes given cost function (performnce index). This cost function is prmeterized by two mtrices, Q nd R, tht weight the stte vector nd the system input respectively. These weighting mtrices regulte the penlties on the excursion of stte vribles nd control signl. One prcticl method is to Q nd R to be digonl mtrix. The vlue of the elements in Q nd R is relted to its contribution to the cost function. To find the control lw, Algebric Riccti Eqution (ARE) is first solved, nd n optiml feedbck gin mtrix, which will led to optiml results evluting from the defined cost function is obtined [17-2]. In this pper, to chieve ccurte control performnce of speed control of DC motor, optiml liner qudrtic regultor technique is presented. The reminder of the pper is orgnized s follows: t first the dynmic model of the seprtely excited DC motor is briefly reviewed for the purpose of speed control. The next section the bsic concept nd design of liner qudrtic regultor controller is briefly reviewed. Then the simultion results re presented. Finlly, the lst section sttes the min conclusion. Dynmic Model of DC Motor Direct current motors re widely used for vrious industril nd domestic pplictions. Exmples re s robotic nd ctutor for utomtion process, mechnicl motion, nd others. Accurte speed control of the DC motor is the bsic requirement in such pplictions. There re two min wys of controlling DC motor: The first one nmed rmture control consists of mintining the sttor mgnetic flux constnt, nd vrying the rmture current. Its min dvntge is good torque t high speeds nd its disdvntge is high energy losses. Figure 1. A seprtely excited DC motor model The second wy is clled field control, nd hs constnt voltge to set up the rmture current, while vrible voltge pplied to the sttor induces vrible mgnetic flux. Its dvntges re energy efficiency, inexpensive controllers nd its disdvntges re torque tht decreses t high speeds. In this pper, the seprtely excited DC motor model is chosen ccording to its good electricl nd mechnicl performnces more thn other DC motor models. The electric circuit of the seprtely 33

3 excited DC motor is shown in figure 1. Objective is to control the speed of the seprtely excited DC motor by rmture voltge control [1-4]. Assuming constnt field excittion the rmture circuit electricl eqution is written s: di V R i L = + + E b (1) di V R i L = + + K b ω Where V is the input terminl voltge (rmture voltge) in volt, E b is the bck emf in volt, R is the rmture resistnce in ohm, L is the rmture inductnce in H. K b is the bck emf constnt in Vs/rd, ω is represents ngulr speed in rd/s, nd i is the rmture current in A. The dynmics of the mechnicl system is given by the following torque blnce eqution: T d ω K i = J + B ω T d t = (2) Where J is the moment of inerti of the motor in kgm 2 /s 2, T is the motor torque in Nm, B is the viscous friction coefficient in Nms, nd K T is the torque fctor constnt in Nm/A. Eqution (1) nd eqution (2) re rerrnged to obtin: d i R K b V = i ω + (3) d t L L L d ω K B = i ω (4) d t JT J To design desired controller using the liner qudrtic regultor technique, the system must first be expressed in the stte spce form. In the stte spce model of seprtely excited DC motor, the eqution (3) nd eqution (4) cn be expressed by choosing the ngulr speed (ω) nd rmture current (i ) s stte vribles nd the rmture voltge (V ) s n input. The output is chosen to be the ngulr speed [1-4]. The physicl nd functionl prmeters of the seprtely excited DC motor used for simultion testing re given in Tble1. Design of the LQR Controller Liner qudrtic regultor design technique is well known in modern optiml control theory nd hs been widely used in mny pplictions. The stndrd theory of the optiml control is presented in [17-2]. R K b di L L i = + 1 V dω K T B ω L (5) J J i y = 1 ω Tble 1. Prmeters of the seprtely excited DC motor Prmeters Vlues Armture Resistnce, R 1Ω Armture Inductnce, L.5H Moment of Inerti, J.1kgm 2 /s 2 Viscous Friction Coefficient, B. 3Nms The Bck EMF Constnt, K b The Torque Fctor Constnt, K T.23Vs/rd.23Nm/A Under the ssumption tht ll stte vribles re vilble for feedbck, the LQR controller design method strts with defined set of sttes which re to be controlled. In generl, the system model cn be written in stte spce eqution s follows: x& = Ax + Bu (6) n m Where x R nd u R denote the stte vrible, nd control input vector, respectively. A is the stte mtrix of order n n; B is the control mtrix of order n m. Also, the pir (A, B) is ssumed to be such tht the system is controllble. The liner qudrtic regultor controller design is method of reducing the performnce index to minimize vlue. The minimiztion of it is just the mens to the end of chieving cceptble performnce of the system. For the design of liner qudrtic regultor controller, the performnce index (J) is given by: T T J = ( x Qx + u Ru) (7) Where Q is symmetric positive semi-definite ( ) stte weighting mtrix of order n n, nd R is symmetric positive definite ( > ) control 34

4 weighting mtrix of order m m. The choice of the element Q nd R llows the reltive weighting of individul stte vribles nd individul control inputs s well s reltive weighting stte vector nd control vector ginst ech other. The weighting mtrices Q nd R re importnt components of n LQR optimiztion process. The compositions of Q nd R elements hve gret influences of system performnce. The designer is free to choose the mtrices Q nd R, but the selection of mtrices Q nd R is normlly bsed on n itertive procedure using experience nd physicl understnding of the problems involved. Commonly, tril nd error method hs been used to construct the mtrices Q nd R elements. This method is very simple nd very fmilir in liner qudrtic regultor ppliction. However, it tkes long time to choose the best vlues for mtrices Q nd R. The number of mtrices Q nd R elements re dependent on the number of stte vrible (n) nd the number of input vrible (m), respectively. The digonl-off elements of these mtrices re zero for simplicity. If digonl mtrices re selected, the qudrtic performnce index is simply weighted integrl of the squred error of the sttes nd inputs. The term in the brckets in eqution (7) bove re clled qudrtic forms nd re quite common in mtrix lgebr. Also, the performnce index will lwys be sclr quntity, whtever the size of Q nd R mtrices [21-25]. The conventionl liner qudrtic regultor problem is to find the * optiml control input lw u tht minimizes the performnce index under the constrints of Q nd R mtrices. The closed loop optiml control lw is defined s: * u = Kx (8) Where K is the optiml feedbck gin mtrix, nd determines the proper plcement of closed loop poles to minimize the performnce index in eqution (7). The feedbck gin mtrix K depends on the mtrices A, B, Q, nd R. There re two min equtions which hve to be clculted to chieve the feedbck gin mtrix K. Where P is symmetric nd positive definite mtrix obtined by solution of the ARE is defined s: T 1 T A P+ PA PBR B P+ Q= (9) Then the feedbck gin mtrix K is given by: 1 T K = R B P (1) Substituting the bove eqution (8) into eqution (6) gives: x& = Ax BKx = ( A BK) x (11) If the eigenvlues of the mtrix (A-BK) hve negtive rel prts, such positive definite solution P lwys exits. Simultion Results In order to verify the vlidity of the liner qudrtic regultor controller, severl simultion tests re crried out using MATLAB/SIMULINK softwre pckge. The performnce of liner qudrtic regultor controller hs been investigted nd compred with the conventionl PID controller. Simultion tests re bsed on the fcts tht whether the liner qudrtic regultor controller is better nd more robust thn the trditionl PID controller or not. For the comprison, simultion tests of the speed response were performed ccording to the nominl condition, moment of inerti vrition, nd rmture inductnce vrition of the seprtely excited DC motor. To determine the feedbck gin mtrix K, the elements of the weighting mtrices Q nd R re chosen s: Q = [.2 ;.28] nd R =.2, respectively. Speed (rd/sec) Time (sec) Figure 2. Comprison of output speed responses mong LQR nd conventionl PID controllers After solving the ARE nd substituting into eqution (1), the optiml vlues of control LQR PID 35

5 feedbck gin mtrix K re obtined s K= [ ]. Figure 2 shows the step responses of speed control of the seprtely excited DC motor t nominl condition by two controllers. According to the simultion results, liner qudrtic regultor method give the better performnce compred to trditionl PID controller. The time response specifictions of the conventionl PID controller nd liner qudrtic regultor technique obtined from the simultion of the seprtely excited DC motor speed control is shown in Tble 2. Bsed on the Tble 2, liner qudrtic regultor technique hs the fstest settling time of 2s while trditionl PID controller hs the slowest settling time of 4.5s. For the percent overshoot, liner qudrtic regultor technique does not hve overshoot nd conventionl PID controller hs the gretest vlue of percent overshoot of 17%. Furthermore, there is no stedy stte error using liner qudrtic regultor controller. However, the rise time for trditionl PID controller is smllest vlue thn for liner qudrtic regultor controller. Tble 2. Performnces metrics for LQR nd PID controllers Time Response Specifictions LQR PID Settling Time (T s ) 2s 4.5s Rise Time (T r ) 2.5s 1.1s Overshoot % 17 Stedy Stte Error (e ss ).3 For high performnce pplictions the proposed liner qudrtic regultor scheme should be robust to prmeter vritions. Chnges in the moment of inerti nd the rmture inductnce re investigted through simultions. The simultion studies re undertken by chnging one prmeter t time while keeping other prmeters unchnged. The seprtely excited DC motor is commnded to ccelerte from rest to reference speed under no torque lod. Figure 3 shows the seprtely excited DC motor responses of optiml liner qudrtic regultor pproch nd conventionl PID controller when the moment of inerti is incresed by 1% of its originl vlue, whilst Figure 4 depicts the speed response when the rmture inductnce incresed by 1% of its originl vlue. Speed (rd/sec) Time (sec) Figure 3. Responses of the DC motor using two controllers with vrition in the moment of inerti Speed (rd/sec) Time (sec) Figure 4. Responses of the DC motor using two controllers with vrition in the rmture inductnce From Figure 3, it cn be seen tht the increment of the moment of inerti does not impose ny significnt effect on the performnce of the liner qudrtic regultor technique but only ffects the rise time. A comprison is illustrted in Tble 3 between LQR nd PID controller quntittively. Tble 3. Performnces of two controllers under incresed J Time Response Specifictions LQR PID Settling Time (T s ) 3.6s 7.8s Rise Time (T r ) 5.1s 2.1s Overshoot % 4 Stedy Stte Error (e ss ).3 LQR PID LQR PID 36

6 It is very much cler from Figure 4 tht the proposed liner qudrtic regultor controller is less sensitive to prmetric vritions nd robust trcking performnce is chieved in presence of the uncertin prmeters. Furthermore, it cn be noted tht the increse in rmture inductnce cuses gretest vlue of percent overshoot, settling time, nd stedy stte error in clssicl PID controller thn optiml liner qudrtic regultor controller which is ffected only by slowest rise time. The time response prmeters percent overshoot; settling time, rise time, nd stedy stte error for LQR nd PID controller re presented in Tble 4. Tble 4. Performnces of two controllers under incresed L Time Response Specifictions LQR PID Settling Time (T s ) 3.3s 1.3s Rise Time (T r ) 2.6s 1.4s Overshoot % 2 31 Stedy Stte Error (e ss ).3 Conclusions Optiml LQR strtegy nd conventionl PID controller hve been considered in this pper for controlling the speed of seprtely excited DC motor. The performnce of the two controllers is vlidted through simultions. A number of simultion results re presented for comprison. Bsed on the comprtive simultion results, one cn conclude tht the liner qudrtic regultor controller relizes good dynmic behvior of the seprtely excited DC motor with rpid settling time, no overshoot, nd zero stedy stte error compred to conventionl PID controller under nominl condition. But the comprison between the speed control of the seprtely excited DC motor by liner qudrtic regultor technique nd conventionl PID controller shows clerly tht the liner qudrtic regultor technique gives better performnces thn conventionl PID controller ginst prmeter vritions. Furthermore, the simultion results so obtined show tht the conventionl PID controller gives gretest vlue of percent overshoot nd longer settling time. References [1] Weiyo L. nd Qi Z. (29), Speed Control of DC Motor using Composite Nonliner Feedbck Control, 29 IEEE Interntionl Conference on Control nd Automtion Christchurch, New Zelnd. [2] Moleykutty G. (28), Speed Control of Seprtely Excited DC Motor, Americn Journl of Applied Sciences, Vol. 5, No. 3, pp [3] Hung J., Kuo C. (1999), Robust position control of DC servomechnism with output mesurement noise, Electr. Eng., Vol. 88, pp , 26. [4] Chpmn J., Electric Mchinery Fundmentls, The McGrw-Hill Compnies. [5] Tossporn C., Piyoros J. nd Thn R. (21), Sliding Mode Control with PID Tuning Technique: An Appliction to DC Servo Motor Position Trcking Control, Energy Reserch Journl 1 (2), pp [6] Igor K., Goce S. (25), Comprison of Sliding Mode nd Proportionl Integrl Control for Brushless DC Motor, Electronics Sozopol, Bulgri, September, pp [7] Um M., kishore Y. nd K. Amresh, Sliding Mode Speed Control of DC Motor, Interntionl Conference on Communiction Systems nd Network Technologies, 211. [8] Rhul M., Tejbeer K. (211), DC Motor Control using Fuzzy Logic Controller, Interntionl Journl of Advnced Engineering Sciences nd Technologies, Vol.8, No.2, pp [9] Arpit G., Ankit Un., Anurg B. (212), Performnce Comprison of PID nd Fuzzy Logic Controller using Different Defuzzifiction Techniques for Positioning Control of DC Motors, Journl of Informtion Systems nd Communiction, Vol.3, No.1, pp [1] Bsil H., Moyed A. (211), Fuzzy PID Controllers using FPGA for Rel Time DC Motor Speed Control, Intelligent Control nd Automtion, Vol. 2, pp [11] Ang K., Chong G., Li Y. (25), PID control system nlysis, design, nd technology, IEEE Trns. Control System Technology, Vol. 13, pp [12] Yniv O., Ngurk M. (23), Robust, PI controller design stisfying sensitivity nd uncertinty specifictions, IEEE Trns. Automtion Control, Vol. 48, pp [13] Kim H., Mrut I., Sugie T. (28), Robust PID controller tuning bsed on the constrined prticle swrm optimiztion, Automtic, Vol. 44, Iss. 4, pp [14] Meshrm P. nd Rohit G. (212), Tuning of PID Controller using Ziegler-Nichols Method for Speed Control of DC Motor, IEEE- Interntionl Conference On Advnces In Engineering, Science And Mngement (ICAESM -212), pp

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