THE SPEED CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR USING FUZZY LOGIC AND SELF TUNING FUZZY PI CONTROLLER Abulhakim KARAKAYA Ercüment KARAKAŞ e-mail: akarakaya@kou.eu.tr e-mail: karakas@kou.eu.tr Department of Electrical Eucation, University of Kocaeli, Imit 4380, Kocaeli, Turkey. Keywors: Permanent Magnet Synchronous Motor, F an STFLPI. ABSTRACT This paper obtains a nonlinear mathematical moel of PMSM, an realies simulation of obtaine moel in Matlab/Simulink program. Spee control of motor moel is mae with Fuy Logic (FL) an Self Tuning FLPI (STFLPI) controllers. Controller performances are compare from the spee graphs obtaine. I. INTRODUCTION In high performance applications, the Permanent Magnet Synchronous Motors (PMSMs) are becoming popular as compare to other types of ac motor ue to some of their avantageous features incluing high torue, high power, high efficiency an low noise. Insensitivity to parameter variation an, reaching of the spee to a reference value at shortest time ue to any isturbances, are some of the important criteria of the high performance rive systems use for rive PMSMs in robotics, rolling mills, machine tolls etc. The conventional proportional integral (PI) an proportional integral erivative (PID) controllers have been wiely utilie as spee controllers in PMSM rives. However in orer to obtain the best results from the controls, the - axis reactance parameters of the PMSM must be known exactly. This is rather ifficult an conventional fixe gain PI an PID controllers are very sensitive to step change of comman spee, parameter variations an loa isturbance []. Therefore, a special controller of PMSM is neee to make spee control in high performance rive systems [2]. In the literature on PMSM, it is seen that; Güney et al [3] examine ynamic behaviour moel of permanent magnet synchronous motor using PWM inverter an fuy logic controller for stator phase current, flux an torue control of PMSM. Ohm et al [4], establishe a mathematical moel of PMSM an obtaine parameters of PMSM experimentally. Singh et al [5] examine current, voltage, spee an torue variation graphs an realie performance analysis with FL controller of PMSM river. Uin an Rahman [2] compare simulation results with responses obtaine from experiments an mae FL funamental spee control of Interior PMSM. Senjyu et al [6] worke on measurement of real parameters for high spee PMSMs an mae comparison between calculate values an measure values. This paper obtains mathematical moel of PMSM, realies simulation of the moel obtaine in Matlab/Simulink program. The parameters use in simulation, are the real measure values from PMSM of 260.75 W power, an from motor spee graphs obtaine with these parameters, rise time, settling time, overshoot an steay-state error analyses are mae. In the spee control block in Fig. Proportional Integral (PI), Fuy Logic (FL), Fuy Logic PI (FLPI) an Self Tuning FLPI (STFLPI) controllers are use an performances of controllers are compare. Necessary parameters are shown in Table 9. Wr θr Fig.. Spee control block iagram of PMSM. II. MATHEMATICAL MODEL OF PMSM. Fig. shows spee control block iagram of PMSM. The PMSM is fe by a current-controlle pulse with moulate (PWM) inverter. The motor currents are ecompose into i an i components which are respectively flux an torue components in the rotorbase - coorinates system [7]. Motor moel is constitute with following euations: [. i + ( L L ). i i ] 3 P T e = λ m. () 2 2 ( i ) v r s. i + wr. L. i = (2) t L
( i ) v r. i w.( L. i + λ ) t = s r L m (3) sets: Negative Big (NB), Negative Meium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Meium (PM) an Positive Big (PB). ( w ) t P w. 2 Te TL B wrm = (4) J rm. = (5) r w rm Where T L is the loa torue, B is the viscous friction, J is the moment of inertia, V an V represent the - axes stator voltages, i an i are the - axis stator currents. L an L are the - axis inuctances, r s is the per phase stator resistance, w r shows the electrical velocity of the rotor. λ m is expression of the flux linkage ue to the rotor magnets linking the stator, T e is the motor prouce torue an w rm is the mechanical velocity of the rotor. III. FUZZY LOGIC CONTROLLER If structure of FL controller is investigate as shown in Fig. 2 (a), controller has two input variables; spee error e(k) an change of spee error ce(k) [9]. At the same time, change in reference phase current i (k) is output i (k). (a) (b) Fig. 2. (a) Structure of FL controller. (b) FL controller internal structure. e(k) an ce(k) are calculate as in euations (24) an (25) for every sampling time: e(k) = w (k)-w r (k) (6) ce(k) = e(k)-e(k-) (7) Where w (k) is reference spee an w r (k) is actual spee value. In the first stage, the crisp variables e(k) an ce(k) are converte into fuy variables e an ce using the triangular membership functions shown in Fig. 3. The universes of iscourse of the input variables e an ce are respectively (-0, 0) ra/s an (-0.89, 0.89) ra/s. The universe of iscourse of the output variable i is (-, ) A. Each universe of iscourse is ivie into seven fuy Fig. 3. Membership functions of the fuy variables e, ce an i. In the secon stage, the FL controller executes the 49 control rules shown in Table taking the fuy variables e an ce as inputs an the output uantity i is processe in the efuification unit. The rules are formulate using the knowlege of the PM synchronous motor behavior an the experience of control engineers. Table. Fuy control rules for spee controller. Error "e" NB NS NS Z PS PM PB NB NB NB NB NB NM NS Z Change of error "ce" NM NB NB NB NM NS Z PS NS NB NB NM NS Z PS PM Z NB NM NS Z PS PM PB PS NM NS Z PS PM PB PB PM NS Z PS PM PB PB PB PB Z PS PM PB PB PB PB As shown in Fig. 2 (b), the inference engine output variable i is converte into a crisp value i (k) in the efuification unit. Various efuification algorithms have been propose in the literature [8]. Here, the centroi efuification algorithm is use in which the crisp value is calculate as the center of gravity of the membership function of i as in euation (8): i = n i= ( i ) µ [( i ) ] i i n µ [( i ) ] i= i The reference current i (k) for the vector control system is obtaine by integrating i (k) as in euation (9): i i ( k ) + i (8) = (9)
IV. SELF TUNING FUZZY PI CONTROLLER Block iagram of STFLPI controller is shown in Fig. 4. Output of FL controller is improve by self tuning mechanism. The necessary backgroun for this mechanism is given in the following subsection. Fig. 4. Block iagram of STFLPI controller. A. Membership functions Input membership functions e(k) an ce(k) are (-, ) ra/s an output membership function i (k) is (-, ) A. At the same time, the scaling factor for self tuning mechanism inputs (E(k), CE(k)) an α are use as (0, ). For input an output variables, necessary rule bases are shown in Table an membership functions in Fig. 5. Membership functions are shown for self tuning mechanism block in Fig. 6. For etermination of gain upating factor α (7x7) control rules (Zero (Z), Very Small (VS), Small (S), Small Big (SB), Meium Big (MB), Big (B), Very Big (VB)) as shown in Table 2 an triangle membership functions shown in Fig. 0, are use. Change of error "ce" Table 2. Fuy rules for calculation of α. Error "e" VB B MB SB S VS Z VB VB VB VB B SB S Z B VB VB B B MB S VS MB VB MB B VB VS S VS SB S SB MB Z MB SB S S VS S VS VB B MB VB VS VS S MB B B VB VB Z Z S SB B VB VB VB V. COMPARATIVE STUDY OF SPEED RESPONSES A. No-loa conition System is run, while motor shaft is uner no loa conition. While reference spee is 00 ra/s, graphs obtaine are shown in Fig. 7. In Table 3, t ro is the rise time of angular spee, t so is settling time of angular spee, O s is the overshoot an e ss is the steay-state error an shown result to obtaine from controllers. 0 00 90 80 70 Fig. 5. Memberships functions for E, CE an I Wr (ra/s) 60 50 40 w B. Scaling factors The relationships between the scaling factors (G e, G ce an G I ) of input an output variables of the STFLPI are as in euations (0), () an (2): E e. Ge ce. Gce i =. I G CE = (0) = () α. (2) I In euation (2), α is the gain upating factor. Unlike FLPI controller (which uses only G I ) the actual output ( i (k)) for STFLPI controller is obtaine using the effective scaling factor (α.g I ) as shown in Fig. 4. Suitable values for G e, G ce an G I are respectively etermine to be 0.009, an 0.5. C. The rule-bases Rules of FLPI controller is shown in Table. The gain upating factor (α) is calculate using fuy rules. Rule base in Table 2 is use for calculation of α. 30 20 0 0 0 0.5.5 2 2.5 3 3.5 4 4.5 5 x 0-3 Fig. 7. Spee responses of PMSM uner no loa obtaine with FL an STFLPI controllers. Table 3. While PMSM is uner no loa, spee performance analyses of FL an STFLPI controllers. Controller t ro (s) t so (s) O s (%) e ss FL 0.002832 0.0036 0.03 0.0028 STFLPI 0.000990 0.003 0.0487 0.0055 Table 4. While PMSM is uner no loa, comparison of Controllers t ro (%) Controllers t so (%) STFLPI -FL 48 STFLPI -FL 46 Controllers O s (%) Controllers e ss (%) FL- STFLPI 58 FL- STFLPI 32 In Table 4, controllers are compare among themselves in percentages. It is seen that in rise time of angular spee an in settling time of the angular spee the STFLPI, but
in overshoot an in steay-state error the FL, exhibit the best performance. B. Loa conition System is run, while motor shaft is uner loa conition. Graphs obtaine are shown in Fig. 8 while reference spee is 00 ra/s. In Table 5, t rl is the rise time of the angular spee an t sl is the settling time of the angular spee both uner loa conition. Also shown are results obtaine from controllers. 00 Wr (ra/s) 03 02 0 00 99 98 97 96 95 w Wr (ra/s) 90 80 70 60 50 40 30 20 0 w 0 0 2 3 4 5 6 7 8 x 0-3 Fig. 8. Spee responses of PMSM uner loa obtaine with FL an STFLPI controllers. Table 5. While PMSM is uner loa, spee performance analyses of FL an STFLPI controllers. Controller t rl (s) t sl (s) O s (%) e ss FL 0.002907 0.0036 0.078 0.0027 STFLPI 0.00070 0.004 0.0893 0.0054 Table 6. While PMSM is uner loa, comparison of Controllers t rl (%) Controllers t sl (%) STFLPI -FL 46 STFLPI -FL 44 Controllers O s (%) Controllers e ss (%) FL- STFLPI 66 FL- STFLPI 34 In Table 6, controllers are compare among themselves in percentages. It is seen that in rise an settling times of angular spee STFLPI an in overshoot an steay-state error the FL controllers exhibit the best performance. C. Step loa torue application After motor makes no-loa eparture, step loa torue of nominal loa (0.83 Nm) is applie to the system at 0.04 s. While reference spee is 00 ra/s, graphs obtaine are shown in Fig. 9. In Table 7, t i is the settling time, w i is the angular spee change, O si is the overshoot, an e ssi is the steay-state error of the motor, all of which are etermine uner a step loa of nominal value. 94 0.039 0.0395 0.04 0.0405 0.04 0.045 0.042 0.0425 0.043 Fig. 9. Spee response to step loa torue application with FL an STFLPI controllers. Table 7. While PMSM is uner step loa, spee performance analyses of FL an STFLPI controllers. Controller t i (s) w i (%) O si (%) e ssi FL 0.0060 3.65570 0.0290 0.0028 STFLPI 0.0006 2.230 0.022 0.0053 Table 8. While PMSM is uner step loa, comparison of Controllers t i (s) Controllers w i (%) STFLPI -FL 82 STFLPI -FL 26 Controllers O si (%) Controllers e ssi (%) STFLPI -FL 40 FL- STFLPI 30 In Table 8, controllers are compare among themselves in percentages. It is seen that in settling time an angular spee change, in overshoot the STFLPI, an in steaystate error the FL controllers exhibit the best performance. VI. CONCLUSION In this paper, ifferent controllers for PMSM are use an the following results are obtaine in the spee control; STFLPI controller gives the best performance in settling time uner no loa, loa an step loa conitions. Inspection of Tables 4, 6 an 8 reveals that uner no loa conition, in settling time STFLPI controller is 46% better than FL controller. Uner loa conition in settling time, STFLPI controller is 44% better than FL controller. Uner step loa, STFLPI controller is 82% better than FL controller. From an observation of these percentages one can see that superiority of STFLPI in settling time is most marke uner step loa conition. In general it can be conclue that in practices with step loa application use of STFLPI controller, an in those with small steay-state error reuirement use of FL controller is best for the given system.
Table 9. Parameters of PMSM. V (V) 530 f (H) 50 P 6 r s (Ω) 5.25 L ( mh).83 L (mh) 3.33 λ m (Wb) 0.09653 B (Nm/(ra/s)) 0.0004324 J (kgm 2 ) 0.000054 T L (Nm) 0.83 KAYNAKLAR. Rahman, M. A., Houe, M. A., 997. On-line self-tuning ANN base spee control of a PM DC motor. IEEE/ASME Trans. On Mechatronics 2 (3), pp.69-78. 2. Uin, M. N., Rahman M. A., 2000. Fuy logic base spee control of an IPM synchronous motor rive. Journal of Avance Computational Intelligence 4 (2), pp. 22-29. 3. Güney, İ., Yüksel, O., Serteller, F., 200. Dynamic behaviour moel of permanent magnet synchronous motor fe by PWM inverter an fuy logic controller for stator phase current, flux an torue control of PMSM. International Electric Machines an Drive Conference, pp. 479-485. 4. Ohm, D.Y., Brown, J. W., Chava, V.B., 995. Moeling an parameter characteriation of permanent magnet synchronous motor. Proceeing of the 24th Annual Symposium of Incremental Motion Control Systems an Devices, Sn Jose, pp. 8-86. 5. Singh, B., Putta Swamy, C.L., Singh, B.P., Chanra, A., Al-Haa, K., 995. Performance analysis of fuy logic controlle permanent magnet synchronous motor rive. Proceeings of IEEE-ICON l (), 399-405. 6. Senjyu, T., Kuwae Y., Urasaki N., Ueato K., 200. Accurate parameter measurement for high spee permanent magnet synchronous motors. Power Electronics Specialists Conference 2, pp. 772-777. 7. Krause, P.C., 987. Analysis of electric machinery. McGraw-Hill, New York. 8. Sousa, G.C.D., Bose, B.K., 99. A fuy set theory base control of a phase-controlle converter c machine rive. Inustry Applications Society Annual Meeting Conference Recor, pp. 854-86. 9. Yager, R. R., Filev, D. P., Essentials of fuy moeling an control, John Wiley & Sons Inc., Canaa, 994.