sensorless vector control of a permanent magnet synchronuous motor with fuzzy logic observer

7/28/2019 Sensorless Vector Control of a Permanent Magnet Synchronuous Motor With Fuzzy Logic Observer

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O R IGINA L PA PE R

Bilal Gu mu s Mehmet O zdemir

Sensorless vector control of a Permanent magnet synchronuous

motor with fuzzy logic observer

Received: 24 December 2004 / Accepted: 12 March 2005 / Published online: 15 June 2005 Springer-Verlag 2005

Abstract In this paper, a new method for sensorlessvector control of a permanent magnet synchronousmotor (PMSM) using a fuzzy logic observer is developed.This method is based on determination of rotor position

and thereby speeds by estimating back-emf componentswhich result from fuzzy logic observers. The rotor posi-tion angle and rotating speed are estimated by evaluatingthe instantaneous values of stator voltages and currents.The estimators are two fuzzy logic observers. They havetwo inputs: the estimated stator currents and the differ-ence between the measured and estimated stator cur-rents. In addition, the outputs of the fuzzy logicobservers are the back-emf components in an ab refer-ence frame. The proposed method was implementedusing a MATLAB/Simulink software package program.The obtained results are within acceptable error limits fora wide speed range, from 40 rad/s up to 500 rad/s.

Keywords Permanent magnet synchronous motor Sensorless control Fuzzy logic observer

1 Introduction

Permanent magnet synchronous motors (PMSM) arefrequently used in industrial applications. Especiallytheir compact design, high efficiency, high power/weightand torque/inertia ratios can be shown as the mostimportant advantages of PMSMs. On the other hand,the high cost and their time-varying magnetic charac-teristics are the disadvantages of PMSMs [18]. PMSMs

can be divided into two main groups with respect to theplacement of their magnets on the rotor. PMSMs areclassified as surface- mounted PMSM (SPMSM) andinterior PMSM (IPMSM) if their magnets are mounted

on the surface of the rotor and inside the rotor,respectively. In this work, SPMSM has been used.

The vector control technique is one of the mostimportant closed loop techniques for AC machines invariable speed applications. In vector control applica-tions, speed and rotor position can be obtained by usingconventional electromechanical sensors such as tacho-generators and encoders mounted on the rotor. How-ever, this increases the dimensions and cost of the driversystem. [14, 6, 79]. The need for using sensors can beeliminated if the speed and position of the rotor areestimated. This is called sensorless control technique.

The main aim of sensorless control technique is to

estimate the rotor position and hence the speed usingmachine parameters and measured voltages and cur-rents. In PMSMs employing a back-emf-based sensor-less scheme, the information on the rotor position isextracted from the estimated back-emf waveforms bymeans of inverse trigonometric functions [4, 7]. Theestimated position and speed of the motor are used inclose loop control applications. There are variousdeveloped estimating methods such as sliding estimators[2] or voltage injection through the machine [9].

Fuzzy systems have been used extensively and suc-cessfully in control systems over the past few decades[10]. Fuzzy logic is an intelligent control method which

has proved to be convenient in some applications andhas successful results when compared with classic con-trol methods [10]. Fuzzy logic processing can be sug-gested to achieve a better approximation of the absoluterotor position when measurement uncertainties arepresent and there is a lack of a precise law of variationfor the current differences in the PMSM control.

In this study, a new method using a fuzzy logic ob-server for sensorless vector control of the PMSM ispresented for estimating the position with acceptableerror values in a wide speed range.

B. Gu mu s (&)Department of Electrical & Electronics Engineering,Dicle University, Engineering and Architecture Faculty,21280 Diyarbakr,E-mail: [email protected]

M. O zdemirDepartment of Electrical & Electronics Engineering,Engineering Faculty, Frat University, ElazgE-mail: [email protected]

Electrical Engineering (2006) 88: 395402DOI 10.1007/s00202-005-0301-7


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2 Mathematical model of PMSM

A one-phase electrical equation of the machine can begiven as [5]:

v Zi Ri dW

dt Ri

d

dtLi Wmh; 1

where Wm corresponds to the amplitude of the naturalmagnetic flux of the permanent magnets. The termddtWmh corresponds to the back-emf (induced voltage)

and can also be writtendWmh

dh xe; wherexe corresponds to

the electrical speed. Supposing that the machine is sinu-soidal, the induced voltage has the following form:

~E xe ~Wm

sinhesin he

2p3

sin he

4p3

24

35 xe~Wm Khe : 2

In the PMSM case, the torque is expressed by:

~Te p ~Is t ~Wm K he ; 3where Is is stator currents matrix and p is the number ofpole pairs. The torque equation can be rewritten as

~Te p~Wm~IaKah ~IbKbh ~IcKch 4

If the sinusoidal stator currents are replaced in Eq. 4then the torque expression becomes [4]:

~Te 3

2p ~Wm ~Is 5

If the mechanical equation of the machine is:

j

dxm

dt Kd xm Tl Te 6

Here, Kd is the friction coefficient, Tl represents loadtorque and Te the induced torque.

Fig. 1 Phasor diagram of SPMSM

Fig. 2 Sensorless vector control scheme (of PMSM)

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The mathematical model of a PMSM can be definedin the dq frame by the following equations [4]:

vdvq

!

Rs pLd xLqxeLd R pLq

!idiq

!

0

KExe;

!7

where td;tq and id, iq determine dq axis voltages and

currents, respectively. Rs is the stator resistance, xe is theelectrical angular speed, KEis the back-emf constant, p isthe differential operator and Ld and Lqcorrespond to thedq axis inductances. In a SPMSM,

Ld Lq L: 8

The general representation of Eq. 7 constitutes themathematical model of an interior mounted PMSM.Consequently Eq. 7 can be used for a surface mountedPMSM by replacing L with Ld and Lq.

The equation in the ab reference frame for SPMSMis obtained by considering Eq. 8 as:

vavb

! Rs pL 00 Rs pL

! iaib

!KExe sin hecos he !; 9where, ta and tb represent ab axis voltages, respec-tively. This equation can be rewritten as

vavb

! R

iaib

!|fflfflffl{zfflfflffl}

A

pLiaib

!|fflfflfflffl{zfflfflfflffl}

B

KExe sin he

cos he

!|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

C

; 10

where, the portion Cof the equation represents the backemf [4]. The phasor diagram for SPMSM is illustrated inFig. 1.

Fig. 3 Input, output functions of the proposed fuzzy logic observer

Fig. 5 Scaled membership functions of ia iainput function

Fig. 6 Scaled membership functions of the ea output function

Table 1 Rule base of proposedfuzzy logic observer

NL Negative large, NM Nega-tive medium, NS Negative sm-all, Z Zero, PS Positive small,PM Positive medium, PL Posi-tive large

INPUT 1 INPUT 2

NL NM NS Z PS PM PL

NL X NL NM Z PM PL PLNM NL NM NS Z PS PM PLNKS NM NM NS Z PS PM PMZ NM NS NS Z PS PS PMPS NM NM NS Z PS PM PMPM NL NM NS Z PS PM PLPL NL NL NM Z PM PL X

Fig. 4 Scaled Leave single space scaled membership functions of iainput function

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Fig. 7 The general structureof the proposed positionestimator

Fig. 8 Simulation results of sensorless vector control. a Rotor speed, b Stator currents, c Induced torque, d id and iq currents

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3 Sensorless vector control of SPMSM

The block representation of a sensorless vector controlstructure for PMSM is shown in Fig. 2. Here, the maindifference between sensorless control and the classicalvector control scheme [8] is that in the sensorless control,the speed sensor is removed and replaced by a positionestimator. The vector control method shown in Fig. 2

imposes a rotor flux vector angle he which is aligned tothe d-axis. In the case of PMSM, he is equal to theinstantaneous flux vector angle since there is no slipfrequency and reference isd is taken to be zero.

A current-controlled voltage source inverter was usedas a driver, since the power value of the motor is low.Here, a hysteresis band current controller with fixedswitching frequency is chosen and the ideal switches areused in the simulation. The switches frequency is takento be 10 kHz with a hysteresis current band of 0.02 A.

3.1 Fuzzy logic observer

Two fuzzy logic observers are proposed for this opera-tion. Each obtains one of the components of back emfs.The fuzzy logic observer has two inputs and one output.One of the inputs is the estimated current componentand the other is the difference between the estimated andmeasured currents. The output function is induced backemf. Seven membership functions were used for the twoinputs and output (Fig. 3). Limits of the membershipfunctions were determined by considering the parame-ters of PMSM. The general structure of the fuzzy logicobserver is given in Fig. 4.

Membership functions for ia current is shown inFig. 4 where triangular membership functions have beenchosen and scaled in the range of 3.6 A. These are themaximum values for PMSM currents. Similarly, mem-bership functions for the other input ia ia are given inFig 5. Here, the membership function was scaled in the

Fig. 9 Simulation results for 40 rad/s reference speed with fuzzy logic observer: a Estimated and measured positions, b error betweenmeasured and estimated positions, c a component of back-emf output from fuzzy logic observer, d b component of back-emf output fromfuzzy logic observer

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range of 7.2 A for the worst case. Functions for theb-axis currents were formed in the same way. Inaddition, eaand eb output functions of the fuzzy logicobserver were formed from seven triangular membershipfunctions scaled in the range of 18 V as shown inFig. 6.

The rule base for the fuzzy logic observer was

constructed by evaluating the truth degrees of theestimated and measured currents. The rule base forboth observers is the same and is shown in Table 1.Rules were extracted using the Mamdani method. Inthe table, two rules are ignored because they make nosense.

In the observer, the centroid method is used as thedefuzzification method. The position estimator showssimilarities with the sliding mode estimations. But in theproposed method, back emfs are obtained from thefuzzy logic observer instead of the machine model.

3.2 General structure of the position estimator

It is known that back-emf terms contain the rotorposition information as given in Eq. 9. Hence, it isessential to obtain these back emfs in the ab referenceframe to calculate the rotor position. The block diagramof the proposed method is given in Fig. 7.

The relationships between va, vb and the estimatedcurrents, ia; ib; and voltages, ea; eb;can be given as

dia

dtva

L

Rs

Lia

ea

L;

dib

dtvb

L

Rs

Lib

eb

L:

11

Here, back-emf components, ea; eb are obtained fromthe output of fuzzy logic observers, then estimated cur-rents ia; and ib are calculated using Eq. 10. In the close-loop operation, the currents ia; ib and voltages ea; eb are

Fig. 10 Simulation results for 300 rad/s reference speed with fuzzy logic observer: a Estimated and measured positions, b error betweenmeasured and estimated positions, c a component of back-emf output from fuzzy logic observer, d b component of back-emf output fromfuzzy logic observer

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obtained for every sampling period. The estimated po-sition information, he can be calculated as

^he tan1

ea

eb

12

So the determination of position was done by the fuzzylogic observer that works to decrease the error with

feedback.The estimated position information varies between

p/2 and p/2. A Matlab function was written for sum-ming the position values by considering the startingpoint as reference in order to avoid instability. The samefunction is used for reversal speed operation by modi-fying the sign of position value. Velocity estimation isperformed by the help of variation in position infor-mation obtained by this function. A longer samplingperiod can be used in speed estimation because of thefact that speed variation is much slower than position

variation. The proposed system eliminates analyticsolution of the higher order system, error determinationand correction methods. Error correction methods arequite complex [14, 6, 79]. The purpose of this study isto overcome such difficulties by using the proposedmethod.

4 Simulation results

The general structure of PMSM, including the sensorlessvector control, is implemented in the Matlab/ Simulinksoftware package program. Therefore, the proposedsensorless control technique has been simulated forvarious speed references to demonstrate the perfor-mance of the system.

Figure 8 shows the simulation results obtained forsensorless vector control using the fuzzy logic ob-server. A sinusoidal reference speed of 16.6 Hz with a

Fig. 11 Simulation results for 500 rad/s reference speed with fuzzy logic observer: a estimated and measured positions, b error betweenmeasured and estimated positions, c a component of back-emf output from fuzzy logic observer, d b component of back-emf output fromfuzzy logic observer

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magnitude of 500 rad/s was given to demonstrate theperformance of the system in four-quadrant regions.The initial position estimation in Fig. 8 was taken aszero. The motor speed follows the reference speed asshown in Fig. 8a. Three-phase stator currents areillustrated in Fig. 8b. The reference torque shown inFig. 8c is not saturated due to the sinusoidal change inthe reference speed. Figure 8d shows dq currentswhere the torque component current follows the ref-erence torque.

The back-emf components, ea, eb, which are esti-mated by the fuzzy logic observer, the position infor-mation calculated from these back emfs, and the errorbetween the measured and estimated position values forvarious speeds are illustrated in Fig. 9 and Fig. 10.

Figure 9 shows the measured and estimated positionsfor 40 rad/s which can be accepted as a low rotatingspeed for PMSMs. It can be seen that the maximumerror between the measured and estimated positions isnot more than 20. The proposed control algorithmmakes position estimation acceptable at low speed ran-ges where sensitive position control is not required.

In Fig. 10, it can be seen that the maximum electricalposition error is approximately 10. The proposed fuzzylogic observer makes the estimation at high accuracy atmedium and high-speed ranges. Therefore, this error re-sults in negligible speed variations. Similar results are gi-ven in Fig. 11 for a speed range of 500 rad/s. As discussedabove, the error rate of the position information deter-mined by the proposed method does not exceed 20 for awide speed bandwidth [40500 rad/s]. Mechanical rotorposition error in [9] is more than 20. In this work, themaximum electrical position error was approximately 20which corresponds to 10 of mechanical rotor positionerror. Hence, these error values are acceptable as bad

speed conditions for most of the PMSM applications.

5 Conclusions

Methods depending on the complex and some constantparameters are used in order to decrease errors thatoccur in the position estimation method used in thesensorless drive systems. It is necessary to determine thecorrection coefficients in these methods for each system.It was seen that improper choosing of the coefficientsleads to some errors in the system. However, they wereeliminated without the necessity for correction coeffi-

cients by the proposed method based on fuzzy logic.Moreover, the disadvantages of using different estima-tion methods for lower and higher speeds are eliminated.In the proposed method, the differences between theestimated currents, which are determined by means ofback emfs, and the measured currents are used as input

parameters. So the error that occurs in the system isdecreased by this feedback. In this work, the positionestimation has been done with acceptable error in thewide speed range from 40 rad/s to 500 rad/s. In thisspeed interval, the maximum electrical position errorwas approximately 20 and the error did not exceed 5for most values of the position. If the number of thepoles of the machine is taken into consideration, itwould be seen that the mechanical position error be-comes smaller since the electrical position is twice that ofthe mechanical position. It can be observed that theproposed method can make estimations with a highaccuracy ratio for a wide speed interval.

Appendix

References

1. Batzel TD, Lee KY (2000) Slotless permanent magnet syn-chronous motor operation without high resolution rotor anglesensor. IEEE Trans Indust App 15(4):366376

2. Consoli A, Scarcella G, Testa A (2001) Industry application ofzero-speed sensorless control techniques for PM synchronousmotors. IEEE Trans Indust App 37(2):513519

3. Corley MJ, Lorenz RD (1998) Rotor position and velocityestimation for a salient- pole permanent magnet synchronousmachine at standstill and high speeds. IEEE Trans Indust App37(4):784789

4. Chen Z, Tomita MD, Doki S, Okuma S (2003) An extendedelectromotive force model for sensorless control of interiorpermanent-magnet synchronous motors. IEEE Trans IndustApp 50(2):288288

5. Simon T (1999) Implementation of a speed field oriented con-trol of 3-Phase PMSM motor using TMS320F240. TI Appli-cation report SPRA588, Texas Instruments

6. Tursini M, Petrella R, Parasitili F (2003) Initial rotor positionestimation method for PM motors. IEEE Trans Indust App39(6):16301640

7. Kojabadi HM, Ahrabian G (2000) Similation, analysis of theinterior permanent magnet synchronous motor as a brushlessAC drive. Simulation Practice and Theory 7:691707

8. Su nter S, Altun H (2005) Control of a permanent magnet

synchronous motor fed by a direct ACAC converter. ElectEng 87(2):8392

9. Tursini M, Petrella R, Parasitili F (2003) Initial rotor positionestimation method for pm motors. IEEE Trans Indust App39(6):16301640

10. Simon D (2000) Design and rule base reduction of a fuzzy filterfor the estimation of motor currents. Int J Approx Reasoning 25

Table 2 Motor parameters

P

(W)

W

(rpm)

RS(X)

Ld(H)

J

(kgm2)

Kd(Nms)

p Un(V)

Tm(Rated

torque)

(Nm)

Ke(Wb).

75 8,00 0 5.25 0 .46103 9.9107 0 2 19.1 0.029 0.0246

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sensorless vector control of a permanent magnet synchronuous motor with fuzzy logic observer

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