Control Strategy of Buck Converter Driven Dc Motor: a Comparative Assessment

Transcription

1 Autralian Journal of Baic and Applied Science, 4(10): , 2010 ISSN Control Strategy of Buck Converter Driven Dc Motor: a Comparative Aement M.A. Ahmad, R.M.. Raja Imail, M.S. Ramli Control and Intrumentation Reearch Group Faculty of Electrical and Electronic Engineering, Univeriti Malayia Pahang, Malayia. Abtract: hi paper preent the detailed account on the control deign for input tracking of a buck converter driven dc motor. Proportional-Integral (PI), Proportional-Integral-type Fuzzy Logic controller (PI-type FLC) and Linear Quadratic Regulator (LQR) are the technique propoed in thi invetigation to control the peed of a dc motor. he dynamic ytem compoed from buck-converter/dc motor i conidered in thi invetigation and derived in the tate-pace and tranfer function form. Complete analye of imulation reult for PI, PI-type FLC and LQR technique are preented in frequency domain and time domain. Performance of the controller are examined in term of input tracking capability, duty cycle input energy and armature current. Finally, a comparative aement of the impact of each controller on the ytem performance i preented and dicued. Key word: Buck-converter driven dc motor, PI controller, PI-type FLC, LQR controller. INRODUCION Dc motor ha good peed control repondence, wide peed control range. It i widely ued in peed control ytem which need high control requirement, uch a rolling mill, double-hulled tanker, and high preciion digital tool. When it need control the peed teple and moothne, the motly ued way i to adjut the armature voltage of motor. One of the mot common method to drive a dc motor i by uing PWM ignal with repect to the motor input voltage (Wu, H., 2008). However, the underlying hard witching trategy caue unatifactory dynamic behavior. he reulting trajectorie exhibit a very noiy hape. hi caue large force acting on the motor mechanic and alo large current which detrimentally tre the electronic component of the motor a well a of the power upply (Antritter, F., 2006). Since it i uually neceary to add a power upply component, anyway, thi contribution hall preent a control for the entire ytem of buck-converter/dc motor. he combination of dc to dc power converter with dc motor ha been reported in (Boldea, I. and S.A. Naar, 1999). In particular, the compoition of a buck converter with a dc motor ha been propoed in (Linare-Flore, J. and H. Sira-Ramirez, 2004; Linare-Flore, J. and H. Sira-Ramirez, 2004). he buck type witched dc to dc converter i well known in power-electronic. Due to the fact that the converter contain two energy toring element, a coil and a capacitor, mooth dc output voltage and current with very mall current ripple can be generated. he control iue of the converter/motor i to deign the controller o that the dc motor can track a precribed trajectory velocity preciely with minimum error. In order to achieve thee objective, variou method uing different technique have been propoed. DC machine are extenively ued in many indutrial application uch a ervo control and traction tak due to their effectivene, robutne and the traditional relative eae in the deviing of appropriate feedback control cheme, epecially thoe of the PI and PID type. he increaing availability of feedback controller deign technique and the rapid development of circuit imulation program, uch a PSpice, offer much wider poibilitie to analyze, and redeign, currently ued dc motor drive ytem (Linare-Flore, J. and H. Sira-Ramirez, 2004). he implementation of PID controller for power converter and motor control ha been reported in (Liping, G., 2005; Kadwane, S.G., 2006; Liping, G., 2007; Zhou, H., 2008). Moreover, there have been everal publication which apply fuzzy logic controller to control power electronic converter (Ayob, S.M., 2006; Ayob, S.M., 2007; Viwanathan, K., 2005). he mooth trajectory input tracking uing dynamic feedback controller for buck-converter driven dc motor ha been reported in (Linare-Flore, J. and H. Sira-Ramirez, 2004; Linare-Flore, J. and H. Sira- Ramirez, 2004). Correponding Author: Mohd Ahraf Ahmad, Faculty of Electrical and Electronic Engineering, Univeriti Malayia Pahang Pekan, 26600, Pahang, Malayia. el: Fax: mahraf@ump.edu.my 4893

2 hi paper preent invetigation of peed control of a dc motor baed on mooth trajectory tracking. A imulation environment i developed within Simulink and Matlab for evaluation of the control trategie. In thi work, the dynamic model of the buck converter driven dc motor i derived in the tranfer function and tate-pace form. hree feedback control trategie which are conventional PI, PI-type Fuzzy Logic and LQR are developed in thi imulation work. Performance of each controller are examined in term of angular velocity input tracking, duty cycle input energy and armature current. Finally, a comparative aement of the impact of each controller on the ytem performance i preented and dicued. 2. Buck-converter Driven Dc Motor Sytem: he implified model of the overall ytem buck-converter driven dc motor i hown in Figure 1. he witching device have been replaced by an ideally witched voltage ource. hi i indicated by the multiplication of U e with the witching variable u {0,1}. An additional reitance R L ha been added to the model in order to take into account the ohmic reitance of the coil winding. he motor ha been modeled by an inductance L M with ohmic reitance R M and electromagnetic voltage ource K E. An input voltage U e ha been ued which value i equal to the maximum voltage of the dc motor. In thi tudy, the buck converter circuit with coil inductance, L, coil reitance, R L and capacitance, C i conidered. 3. Modelling of Buck-converter Dc Motor Sytem: hi ection provide a brief decription on the modelling of the buck-converter driven dc motor, a a bai of a imulation environment for development and aement of the propoed control technique. he dynamic ytem compoed from converter/motor i conidered in thi invetigation and derived in the tranfer function and tatepace form. Conidering the dynamic ytem of the converter/motor, the ytem can be modelled a uu R i L di u dt L e L L C du dt C il C ia dia uc RMia LM KE dt d J KMia dt (1) (2) (3) (4) where J i the moment of inertia and K M i the tacho generator gain of the motor. In (1) and (3), the very low meaurement amplifier reitance R 4 and R 6 have been neglected. From the equation (1) to (4), the tate pace modeling complete with mechanical equation that decribe the dynamic of the motor haft, a linear fourth order ytem i obtained a x AxBu y Cx with the vector x i u i L C a and the matrice A, B and C are given by (5) 4894

3 RL L L C C A 1 RM K 0 L L L KM J Ue B L M M M, 0 0 0, C E (6) According to (Ortega, R., 1998), the dicrete input u {0,1} can be replaced with the duty ratio {0,1} when uing a PWM-trategy to generate from an analog input ignal. Hence, for the given etup we may refer to a o-called averaged dynamic model, given by x AxB where the new input i the duty cycle. Note that thi linear ytem decription i valid only a long a it can be enured that no aturation effect occur in the coil. Otherwie the inductance L would depend nonlinearly on the current i L. In thi tudy, the value of the parameter are defined a L M = 8.9 mh, R M = 6, K E = V-/rad, K M = N-m/A, J = kg-m 2, U e = 24 V, L = 1.33 mh, R L = 0.2, and C = 470 μf. For the converter, a witching frequency of f = 45 khz ha been ued. 4. Control Algorithm: Proportional-Integral (PI) Control Scheme: o demontrate the performance of the PI controller in controlling the motor angular velocity, the angular velocity () of the dc motor i fed back and compared to the deired angular velocity d (7) () a hown in Figure 2. he angular velocity error i regulated through the proportional and integral gain and applied to the buck converter driven dc motor in term of control duty cycle, (), where it can be obtained a () G ()[ () ()] c d Hence, the cloed loop tranfer function i obtained a () Gc () G() () 1 G () G() d c G () K K / G () ()/ () where and. c p i Before deigning the PI controller, the tranfer function of the buck converter driven dc motor hould be obtained. From (5) and (6), the output-to-control mall ignal tranfer function of Figure 1 can be obtained a 4895

4 G () Ue JLLM C JC ( R LR L ) K M 4 3 M L M KM J KELC ( RLRMCLLM) K M J K R C ( R R ) K K M E L L M E 2 (8) Fig. 1: Decription of the buck-converter driven DC motor ytem. From (8), the output-to-control tranfer function of the buck converter driven dc motor at the nominal operating point i obtain a G () (9) he tranfer function in (9) ha complex conjugate pole at j which caue a 180 phae delay at the approximate frequency of 2490 rad/. he bode plot for the ytem i hown in Figure 3. he bode plot how that the gain margin i 31.1 db and the phae margin i 159 which indicate the intability of the ytem. A PI controller wa deigned for the control of the buck converter driven dc motor at teady tate to reduce teady tate ocillation. One pole wa placed at the origin and one zero wa placed at 57.5 rad/. he dc gain of the controller wa adjuted to obtain ufficient phae margin and high croover frequency. he tranfer function of the PI controller i given by G ( ) c he bode plot for the PI compenated ytem i hown in Figure 4. he bode plot how that the gain margin i 12.4 db, the phae margin i 71 and the bandwidth i 2680 rad/. Proportional-Integral-type Fuzzy Logic Control (PI-type FLC): Fuzzy logic can be defined a a theory of vaguene and uncertaintie. hi theory provide an approximate yet effective, mean of decribing the behaviour of the ytem, which are too complex and ill defined to permit precie 4896

5 Fig. 2: PI control tructure. Fig. 3: Bode plot of buck-converter driven DC motor. mathematical analyi. ypically, Fuzzy Logic Controller (FLC) conit of three tage, namely fuzzification, rule deciion making and defuzzification. Fuzzification i a proce to tranform the non-fuzzy value from the phyical meaurement into a fuzzy linguitic range, i.e., poitive big, poitive mall, negative mall and o on. he aignment of the crip input into fuzzy form i realized by memberhip function. he crip input are firt been normalized o that the input cover all the memberhip function range. hi can be realized by uing input caling factor that act a forward gain. Preently, there i no generalized tandard procedure on how to elect the appropriate hape of memberhip function for pecific application. Memberhip function hape can be either trapezoid, triangular, ingletone or bell-hape. However, triangular with 50% overlap between the adjacent memberhip function i more preferred hape ince it contribute to a le computational proce time. rule deciion making conit of two component. hey are rule table and rule evaluator. he rule tored in rule table actually relate the input-output relationhip. For intance, for two input with equal five fuzzy et, there will be 25 rule that relate the input-output relationhip. he rule evaluator will decide which rule hould be fired with a help of linguitic rule IF... HEN.... For n input, there will be a maximum of 2n rule fired. he lat tage i the defuzzifiction tage. Defuzzification i a tage where the fuzzy form i tranform to phyical value. wo method uually carried out to perform the tak are centroid method and mean maximum method (Ayob, S.M., 2006; Ayob, S.M., 2007). In general, FLC can be claified into three type of controller, namely PI-type FLC, PD-type FLC and PIDtype FLC. Each name reflect their identical performance to their conventional PI, PD and PID control performance but with tuning adjutment feature. PID-type FLC need three input, namely error, change of error and um of error. hi factor ignificantly expand the rule table and make the deign more complicated. Compared to PID- 4897

6 Fig. 4: Bode plot of PI compenator buck-converter driven DC motor. type FLC, PI-type FLC and PD-type FLC are much impler and more applicable. It i known, the PI-type FLC i more practical than PD-type FLC becaue PD-type FLC uually produce ytem teady tate error becaue of lack integral function in it control nature (Petrov, M., 2002). he hybrid fuzzy control ytem propoed in thi work i hown in Figure 5, where d () and () are the deired angular velocity and angular velocity of the dc motor, wherea k 1, k 2 and k 3 are caling factor for two input and one output of the fuzzy logic controller ued with the normalied univere of dicoure for the fuzzy memberhip function. In thi tudy, triangular memberhip function are choen for angular velocity error, integral of angular velocity error, and duty cycle with 50% overlap. Normalized univere of dicoure are ued for both angular velocity error and it integral and duty cycle. Scaling factor k 1 and k 2 are choen in uch a way a to convert the two input within the univere of dicoure and activate the rule bae effectively, wherea k 3 i elected uch that it activate the ytem to generate the deired output. Initially all thee caling factor are choen baed on trial and error. o contruct a rule bae, the angular velocity error, angular velocity error integral, and duty cycle are partitioned into five primary fuzzy et a Angular velocity error E = {NM NS ZE PS PM}, Angular velocity error integral V = {NM NS ZE PS PM}, Duty Cycle U = {NM NS ZE PS PM}, where E, V, and U are the univere of dicoure for angular velocity, angular velocity integral and duty cycle, repectively. A PI-type fuzzy logic controller wa deigned with 11 rule a a cloed loop component of the control trategy for maintaining the peed of dc motor. he rule bae wa extracted baed on underdamped ytem repone and i hown in able 1. he three caling factor, k 1, k 2 and k 3 were choen heuritically to achieve a atifactory et of time domain parameter. hee value were recorded a k 1 = , k 2 = and k 3 = 0.6. Linear Quadratic Regulator (LQR) Control Scheme: A more common approach in the control of buck converter with DC motor ytem involve the utilization linear quadratic regulator (LQR) deign (Ogata, K., 1999). Such an approach i adopted at thi tage of the invetigation here. Figure 6 illutrate the LQR control tructure. In order to deign the LQR controller a linear tatepace model of the buck converter with motor wa obtained by lineariing the equation of motion of the ytem. For a linear time invariant (LI) ytem 4898

7 Fig. 5: PI-type FLC tructure. Fig. 6: LQR control tructure. able 1: Linguitic rule of FLC No. Rule 1. If (e i NM) and (e/ i ZE) then (u i PM) 2. If (e i NS) and (e/ i ZE) then (u i PS) 3. If (e i NS) and (e/ i PS) then (u i ZE) 4. If (e i ZE) and (e/ i NM) then (u i PM) 5. If (e i ZE) and (e/ i NS) then (u i PS) 6. If (e i ZE) and (e/ i ZE) then (u i ZE) 7. If (e i ZE) and (e/ i PS) then (u i NS) 8. If (e i ZE) and (e/ i PM) then (u i NM) 9. If (e i PS) and (e/ i NS) then (u i ZE) 10. If (e i PS) and (e/ i ZE) then (u i NS) 11. If (e i PM) and (e/ i ZE) then (u i NM) x AxBu the technique involve chooing a control law while minimizing the quadratic cot function c J [ xt ( ) Qxt ( ) ut ( ) Rut ( )] dt 0 u ( x) which tabilize the origin (i.e., regulate x to zero) (10) QQ 0 R R 0 where and. he term linear-quadratic refer to the linear ytem dynamic and the quadratic cot function. he matrice Q and R in (10) are called the tate and control penalty matrice, repectively. If the component of Q are choen large relative to thoe of R, then deviation of x from zero will be penalized heavily relative to deviation of u from zero. On the other hand, if the component of R are large relative to thoe of Q, then control effort will be more cotly and the tate will not converge to zero a quickly. A famou and omewhat urpriing reult due to Kalman i that the control law which minimize J c alway take the form u ( x) Kx. he optimal regulator for a LI ytem with repect to the quadratic cot function above i alway a linear control law. With thi obervation in mind, the cloed-loop ytem take the form x ( ABK) x and the cot function J take the form 4899

8 J [ xt ( ) Qxt ( ) ( Kxt ( )) R( Kxt ( ))] dt c 0 0 xt () ( QK RK) xt () dt In thi invetigation, the tracking performance of the LQR applied to the buck converter with motor wa invetigated by etting the value of vector K and N which determine the feedback control law and for elimination of teady tate error capability repectively. For the buck converter with motor decribed by the tate-pace model given by Equation (5), the LQR gain matrix for I Q I 22 and R 1 wa calculated uing Matlab and wa found to be K N and. 5. Implementation and Reult: In thi ection, the propoed control cheme are implemented and teted within the imulation environment of the buck converter driven dc motor and the correponding reult are preented. he control trategie were deigned by undertaking a computer imulation uing the fourth-order Runge-Kutta integration method at a ampling frequency of 1 khz. he angular velocity of the dc motor i required to follow a mooth trajectory within the range from 0 to 150 rad/ a hown in Figure 7. he ytem repone namely angular velocity, duty cycle input energy and armature current are oberved. he performance of the control cheme are aeed in term of input tracking capability, time repone pecification, duty cycle repone and armature current repone. Finally, a comparative aement of the performance of the control cheme i preented and dicued. In Figure 8, the angular velocity repone i hown whereby it reache the required velocity from 0 to 150 rad/. It demontrate that the propoed control cheme are capable in tracking the deired input. Performance of the controller in term of time repone pecification and integral quare error (ISE) are ummarized in able 2. It i oberved that, while the ISE of LQR i five time lower a compared to PI, the ISE of PI-type FLC i extremely lower a compared to both PI and LQR controller. With lower ISE, the actuator repone required le time to achieve to deired velocity. he comparion of the pecification of the angular velocity repone for all control cheme are ummarized in Figure 9. It i hown that, the rie time for all controller are almot equal. However, in term of ettling time, PI-type FLC reult in fatet input tracking repone, followed by LQR and PI. Figure 10 how the duty cycle input energy repone of the buck converter driven DC motor uing PI, PI-type FLC and LQR. It i noted that, the duty cycle for PI controller and PI-type FLC ettled down at 150 m, while the duty cycle for LQR controller ettled down at 200 m, with the final average duty cycle value of for all controller. However, the PI controller and PI-type FLC reult in mooth input energy a compared to the LQR which reult in a full duty cycle from 50 m to 160 m. It i alo noted that, the duty cycle repone of PI-type FLC reache the teady tate value fater than the cae of PI and LQR controller. It how that, by combining PI with FLC, the peed of the duty cycle can be improved. It alo how that, the LQR controller utilize high energy conumption due to a lot of witching in the control ignal during the tranaction between 0 to 150 rad/ when the ytem i tracking the deired repone. hi i the main diadvantage of time optimal control and it i the reaon why the time optimal controller i not ued in practice. he armature current repone i illutrated in Figure 11. It how that, all controller produce high amplitude of armature current before ettled down at 200 m. It i noted that PI-type FLC produce the highet maximum magnitude of armature current with the value of A, followed by LQR controller and PI controller with the value of A and A, repectively. On top of that, for all controller, the buck-converter upplie the highet amount of power to the motor during the tranition period. Similar to the angular velocity and duty cycle repone, the armature current repone uing PI-type FLC ettled down to zero value fater than the LQR and PI controller. 4900

9 Fig. 7: he trajectory reference input. Fig. 8: Angular velocity repone. Fig. 9: Comparative aement of time repone pecification of angular velocity. 4901

10 Fig. 10: Duty cycle repone. Fig. 11: Armature current repone. able 2: Specification of the angular velocity repone of the DC motor Controller Specification of angular velocity repone Rie time () Settling time () ISE PI PI-FLC LQR Concluion: Invetigation into angular velocity control of buck-converter driven dc motor with PI controller, PI-type FLC and LQR controller have been preented. Performance of the controller are examined in term of angular velocity, duty cycle input energy and armature current repone. Acceptable input tracking capability ha been achieved with all control trategie. A comparion of the reult ha demontrated that the PI-type FLC i capable in tracking the mooth trajectory angular velocity with very minimum delay, a compared to the PI and LQR controller. he reult demontrated that the trajectory tracking of angular velocity of buck-converter driven dc motor can uccefully be handled by all PI controller, PI-type FLC and LQR controller. In term of peed of the angular velocity repone, the PI-type FLC provide fatet input tracking repone a compared to other, which i proven by the mallet value of ettling time. However, PI-type FLC reult in a lightly higher input energy of duty cycle a compared to the PI controller. On the other hand, LQR controller conume a high input energy which indicate the diadvantage of time optimal control. 4902

11 REFERENCES Antritter, F., P. Maurer and J. Reger, Flatne Baed Control of a Buck-converter Driven DC Motor. Proceeding of 4 th IFAC Sympoium on Mechatronic Sytem, 4(1). Ayob, S.M., Z. Salam and N.A. Azli, Simple PI Fuzzy Logic Controller Applied in DC-AC Converter. Proceeding of 33rd Annual Conference of the IEEE Indutrial Electronic Society, pp: Ayob, S.M., Z. Salam and N.A. Azli, A Dicrete Single Input PI Fuzzy Controller for Inverter Application. Proceeding of 33rd Annual Conference of the IEEE Indutrial Electronic Society, pp: Boldea, I. and S.A. Naar, Electric Drive. Boca Raton, FL: CRC Pre LLC. Kadwane, S.G., S.P. Vepa, B.M. Karan and. Ghoe, Converter Baed DC Motor Speed Control Uing MS320LF2407A DSK. Proceeding of 1t IEEE Conference on Indutrial Electronic and Application, pp: 1-5. Linare-Flore, J. and H. Sira-Ramirez, A Smooth Starter for a DC Machine: A Flatne Baed Approach. Proceeding of 1t International Conference on Electrical and Electronic Engineering and X Conference on Electrical Engineering, pp: Linare-Flore, J. and H. Sira-Ramirez, DC Motor Velocity Control through a DC-to-DC Power Converter. Proceeding of 43rd Conference on Deciion and Control, pp: Liping, G., J.Y. Hung and R.M. Nelm, Comparative Evaluation of Linear PID and Fuzzy Logic for a Boot Converter. Proceeding of 31t Annual Conference of IEEE Indutrial Electronic Society, pp: Liping, G., Implementation of Digital PID Controller for DC-DC Converter Uing Digital Signal Proceor. Proceeding of IEEE International Conference on Electro/Information echnology, pp: Ogata, K., Modern Control Engineering, New Jerey: Prentice-Hall. Ortega, R., A. Loria, P.J. Nicklaon and H. Sira-Ramirez, Paivity-baed Control of Euler-Lagrange Sytem. London: Spinger. Petrov, M., A. Ganchev and A. aneva, Fuzzy PID Control of Nonlinear Plant. Proceeding of 1t International IEEE Sympoium, 1: Wu, H., X. Chen and L. Hu, Embedded Sytem of DC Motor Speed Control Baed on ARM. Proceeding of International Colloquium on Computing, Communication, Control, and Management, pp: Viwanathan, K., R. Oruganti and D. Srinivaan, Nonlinear Function Controller: A Simple Alternative to Fuzzy Logic Controller for a Power Electronic Converter. IEEE ranaction on Indutrial Electronic, 52: Zhou, H., DC Servo Motor PID Control in Mobile Robot with Embedded DSP. Proceeding of International Conference on Intelligent Computation echnology and Automation, pp: