PEDS 2007

Adaptation of Motor Parameters in Sensorless PMSM Drives

Antti Piippo, Marko Hinkkanen, and Jorma LuomiPower Electronics Laboratory

Helsinki University of TechnologyP.O. Box 3000, FI-02015 TKK, Finland

Abstract-The paper proposes an on-line method for theestimation of the stator resistance and the permanent magnetflux in sensorless permanent magnet synchronous motor drives.An adaptive observer augmented with a high-frequency signalinjection technique is used for sensorless control. The observercontains excess information that is not used for the speed andposition estimation. This information is used for the adaptationof the motor parameters. At low speeds, the stator resistance isestimated, whereas at medium and high speeds, the permanentmagnet flux is estimated. Steady-state analysis and small-signalanalysis are carried out to investigate the parameter estimation,and adaptation mechanisms are defined for the parameters. Theconvergence of the parameter estimates is shown by simulationsand laboratory experiments. The stator resistance adaptationworks down to zero speed in sensorless control.

I. INTRODUCTION

Permanent magnet synchronous machines (PMSMs) areused in many high-performance applications. For vector con-trol of PMSMs, information on the rotor position is required.In sensorless control, the methods for estimating the rotorspeed and position can be classified into two categories:fundamental-excitation methods [1], [2] and signal injectionmethods [3], [4]. The methods can also be combined bychanging the estimation method as the rotor speed varies [5],[6].The fundamental-excitation methods used for sensorless

control are based on models of the PMSM. Hence, theelectrical parameters are needed for the speed and positionestimation [7]. The errors in the stator resistance estimateresult in an incorrect back-emf estimate and, consequently,impaired position estimation accuracy. The operation can alsobecome unstable at low speeds in a loaded condition. Thedetuned estimate of the permanent magnet (PM) flux resultsin incorrectly estimated electromagnetic torque [8], and alsoimpairs the position estimation accuracy. Errors in the d-and q-axis inductances of a salient PMSM also affect theestimation and the torque production, and can degrade thecurrent control performance.The stator resistance and the PM flux depend on the motor

temperature, and thus change rather slowly. On the other hand,magnetic saturation decreases the inductances, which thusdepend on the load condition. The inductances can be modeledas functions of the stator flux or the stator current, but anestimation scheme is required for the stator resistance and thePM flux. The rotor back-emf is proportional to the PM flux andthe resistive voltage drop to the stator resistance. At mediumand high speeds, the effect of the PM flux estimation error ismore significant than that of the stator resistance estimation

error. On the other hand, the back-emf is small at low speeds,and the stator resistance estimate plays an important role inthe estimation.

Several methods have been proposed to improve the per-formance of a PMSM drive by estimating the electricalparameters. In [7], an MRAS scheme is used for the on-line estimation of the stator resistance and the PM flux withposition measurement. Reactive power feedback is used forestimating the PM flux in [9]. The stator current estimationerror and a neural network can be used for estimating both thePM flux and the stator resistance [10]. The stator inductancesand the PM flux are estimated using the steady-state voltageequations and the flux harmonics, respectively in [11]. A DC-current signal is injected to detect the resistive voltage dropfor the resistance estimation in [12], and the PM flux linkageis estimated by taking it as an additional state of an extendedKalman filter in [13].Some parameter estimation schemes have also been devel-

oped for sensorless control methods. In [7], an MRAS schemeis applied for the stator resistance estimation. A parameterestimator is added to two position estimation methods forestimating the stator resistance and the PM flux in [14]. In[15], these parameters are estimated using both the steady-state motor equations and the response to an alternatingcurrent signal. In [14], [15], the convergence of the estimatedparameters to their actual values is not shown. [16] proposes amethod where the resistance and the inductances of a salientPMSM are extracted from an extended EMF model. Threeelectrical parameters are estimated simultaneously, but thebehavior of the stator resistance estimate is not convincing.

This paper proposes a method for the online estimation ofthe stator resistance and the PM flux in sensorless control.The method is based on a speed-adaptive observer that isaugmented with a high-frequency (HF) signal injection tech-nique at low speeds [17]. The excess information availablein the observer is used for the adaptation of the parameters.At medium and high speeds, the PM flux is estimated fromthe d-axis current estimation error. At low speeds, the statorresistance is estimated from a speed correction term producedby the signal injection method. The sensitivity of the d-axiscurrent estimation error and the speed correction term to theparameter errors are investigated by means of steady-state andsmall-signal analyses, and adaptation laws are designed for theestimation of the parameters. The stability and the convergenceof the parameter estimators are investigated by means of sim-ulations and laboratory experiments. The resistance adaptationis shown to work down to zero speed in sensorless control.

1-4244-0645-5/07/$20.00©2007 IEEE 175

i/ F£

Us,ref Ad- + / m

==>justable --A1t

model js

AL is~~~~1

where estimated quantities are marked by ^ and Us,ref is thestator voltage reference. The estimate of the stator current andthe estimation error of the stator current are

(5)

(6)is = L-'(,Os -,pm)is = il - is*1

Fig. 1. Block diagram of the adaptive observer.

II. PMSM MODELThe PMSM is modeled in the d-q reference frame fixed to

the rotor. The d axis is oriented along the PM flux, whoseangle in the stator reference frame is 0m in electrical radians.The stator voltage equation is

Us Rsis + s +WmJ+5 (1)

where us [ud Uq ]T is the stator voltage, is [id iq ]T thestator current, sb= [bd /q ]T the stator flux, Rs the statorresistance, Wm =m the electrical angular speed of the rotor,and

J K0 -1

The stator flux isI

respectively, where iV is the measured stator current expressedin the estimated rotor reference frame. The feedback gainmatrix A is proportional to the rotor speed up to the nominalspeed [17].The speed adaptation is based on an error term

Fe = Clis (7)

where Ci = [O Lq], i.e., the current error in the estimated qdirection is used for adaptation. The estimate of the electricalangular speed of the rotor is obtained using a PI speedadaptation mechanism

Wm =-kpF -kiJ Fedt (8)

where kp and ki are nonnegative gains. The gain selection isdescribed in [17]. The estimate Om for the rotor position isevaluated by integrating Win

sb= Lis+ bpm (2)where 14pm = [/pm O]T is the PM flux and

L Ld fq]L Lq

is the inductance matrix, Ld and Lq being the direct- andquadrature-axis inductances, respectively. The electromagnetictorque is given by

Te = 3p2TJTis (3)

where p is the number of pole pairs.

III. SPEED AND POSITION ESTIMATION

A. Adaptive ObserverAn adaptive observer [17] is used for the estimation of the

stator current, rotor speed, and rotor position. The speed andposition estimation is based on the estimation error betweentwo different models; the actual motor can be considered as areference model and the observer including the rotor speedestimate Wm as an adjustable model. The error term usedin a speed adaptation mechanism is based on the estimationerror of the stator current. The estimated rotor speed, obtainedusing the adaptation mechanism, is fed back to the adjustablemodel.The adaptive observer is formulated in the estimated rotor

reference frame. The block diagram of the adaptive observeris shown in Fig. 1. The adjustable model is based on (1) and(2), and defined by

,bs = Us,ref -Rsis- mJQs + Ais (4)

B. High-Frequency Signal Injection

The adaptive observer described above is augmented with aHF signal injection method to stabilize the speed and positionestimation at low speeds [17]. By using the HF signal injectionmethod with an alternating voltage u, as a carrier excitationsignal [18], an error signal E 2K Om proportional to theposition estimation error 0m = m 0-m is obtained, KSbeing the signal injection gain. The error signal is used forcorrecting the estimated position by influencing the directionof the stator flux estimate of the adjustable model. For thecombined observer, the adjustable model (4) is modified to

Qs = Us,ref -Rss-(m - )JQs + Als

where

SE: = apE + -aj+ dt

(9)

(10)

is the speed correction term, -yp and tYj being the gains of thePI mechanism driving the error signal E to zero. In accordancewith [6], these gains are selected as

2Ki/yP 2K '

2

6KE (1 1)

where ai is the approximate bandwidth of the PI mechanism.At low speeds, the combined observer relies both on the

signal injection method and on the adaptive observer. Theinfluence of the HF signal injection is decreased linearly as thespeed increases by decreasing both the HF excitation voltageand the bandwidth ai. At speeds higher than a threshold speedWoA, the estimation is based only on the adaptive observer.

176

IV. PARAMETER ADAPTATION

The current estimation error is of the adaptive observercontains information that can be used for the parameteradaptation. In [7], [10], the components of is are used forthe adaptation of two parameters in a PMSM drive equippedwith a motion sensor. Solving the parameters from the steady-state voltage equations as in [11] would require filtering toprevent incorrect operation in transients. It is to be noted thatthe PMSM dynamics offer only two degrees of freedom. Sincethe component iq is used for the speed estimation in this paper,one parameter can thus be adjusted using id

In low-speed operation, the HF signal injection methodprovides additional information through the speed correctionterm w . Instead of solving the parameters directly from theresponse to the injected signal [12], wo is used here for theparameter adaptation. Hence, if wo differs from zero in steadystate or if the current estimation error id is nonzero, the motorparameter estimates are inaccurate and the variables wo and 'dcan be driven to zero by adjusting the parameter estimates.

A. Steady-State Analysis Stator Resistance Adaptation

At low speeds, the stator resistance estimate plays an im-portant role in the speed and position estimation, particularlyin loaded conditions. To extract the sensitivity of the d-axiscurrent estimation error id and the speed correction term w,to the parameter errors, the combined observer is investigatedin steady state. The position estimation error is assumed zerobecause the HF signal injection method is in use.

In steady state, the equations of the PMSM and the adaptiveobserver can be written as

Us = Rsis + WmJ(Lis + /pm) (12)

Us = Rsis + (Win - o)J(Lis + 'pm)- Ais (13)respectively. The estimated quantities are expressed in termsof their actual values and estimation errors, i.e. x = x.The stator voltage is eliminated by combining (12) and (13),yielding

Rsis + WmJ(Lis + ,pm) (14)= (Rs -Rs)(is--I's)+ (Wmi-n )J[L(is-i-) +(spm 14pm)] -Ais

When it is assumed that the estimation errors and the speedcorrection term wo are small, their products can be omittedand the equation reduces to

(RsI + WmJL + A)is (15)-Ris- WmJ/npm - J(Lis + v/pm)

The q component of the current estimation error is zeroin steady state because it is driven to zero by the adaptationmechanism (8). The observer gain A is proportional to the rotorspeed and is also omitted since low speeds are investigated.In addition, the d component of the stator current is ignored

TABLE IMOTOR DATA

Nominal voltage UN 370 VNominal current IN 4.3 ANominal frequency fN 75 HzNominal torque TN 14.0 NmStator resistance RS 3.59 QDirect-axis inductance Ld 36.0 mHQuadrature-axis inductance Lq 51.0 mHPM flux <bpm 0.545 VsTotal moment of inertia 0.015 kgm2

because it is controlled to a considerably smaller value thanthe q component. In the component form, the result is

Rs -Lqiq10'I 0 ° RLgmELd opm j ej L- q - Om opm

(16)For investigating the sensitivity of id and wo to the pa-

rameter errors, the variables id and wo are solved from (16).Assuming Rs<pm »> WmoLdLqiq the result is

L 2LqM[.td Rl k ;q iq R,Wmiq l

Rs (17)L e j fpm L q Som

-L opm

It can be seen that both variables on the left-hand side dependboth on Rs and <pm. The correction term w, is selected forparameter adaptation because iq has more effect on id than onw . When using relative parameter errors, the speed correctionterm is

1 ( Rs 4bpmW = jRRsi + opmbmYpm s mpm

(18)

The dependence between the stator resistance error and wois proportional to the resistive voltage drop, whereas thedependence between the PM flux error and wo is proportionalto the back-emf.The effect of the parameter errors on w, was calculated

numerically, using the parameters given in Table I. The effectof the stator resistance error Rs/R, =-0.1 is illustrated inFig. 2(a), and the effect of the PM flux error <pm/<)pm -0.1in Fig. 2(b). The results obtained from the approximate equa-tion (18) are shown. In addition, the figures show the resultsobtained using (15) and the results of steady-state simulations.The speed values used for the calculation and simulation inFig. 2(a) are -0.067 p.u., 0 p.u., and 0.067 p.u, and the torquevalues used in Fig. 2(b) are -TN, 0, and TN. According toFig. 2, the assumptions used in (18) result in only a small error.At low speeds and large values of load torque, the resistivevoltage drop dominates, and the stator resistance is thus areasonable selection for the adaptation.

For the stator resistance adaptation, the closed-oop systemshown in Fig. 3 is investigated. The gain

FR = -tq/pm (19)corresponds to (17) with bpm 0= . The stator resistance isestimated by integration from w , the transfer function of theadaptation mechanism thus being

GR(S) -kR/S (20)

177

50

- 0

2-50-1

50

0-

72 -50

0Te/TN

(a)

1

TC

-0.05 0Wm (p.u.)

(b)

tpm~~~~~~~~<pqJpm+~ F, (s) -G s0V'pm

Fig. 4. Closed-oop system of the PM flux adaptation.

The estimates are replaced with the actual quantities and theirerrors, i.e. x = x, and the estimation errors are assumedsmall so that their products can be omitted. In addition, (23b)is substituted for the current error in (23a), the result being

Q = (-RsL- -WmJ- AL-1) s

A0.05

Fig. 2. Effects of parameter estimation errors on speed correction term wE: (a)effect of resistance error R/R=R - 0.1 as a function of torque for variousrotor speed values and (b) effect of PM flux error <-pm/<-pm =-0.1 as afunction of rotor speed for various electromagnetic torque values. Solid linesshow the calculated values from (15) without additional assumptions, dashedline shows calculated values from (18), and the circles are simulated values.

RsRS+ \_ R,

FR GR(S)

Fig. 3. Closed-oop system of the stator resistance adaptation.

where kR is the stator resistance adaptation gain. The closed-loop system has a bandwidth

aR = iqkR/4pm (21)which can be affected by properly selecting the gain kR.

B. Small-Signal Analysis PM Flux AdaptationAt medium and high speeds, the speed correction term w,

is not available since the HF signal injection is not used. Theparameter adaptation can be based on 'd, which is obtainedfrom the adaptive observer. Small-signal analysis is used forobtaining the sensitivity of td to the parameter errors.

For the analysis, the rotor speed is assumed constant, and theobserver is written assuming zero orientation error, yielding

,bs = u5 Rsis-WmJQs + Ais (22a)

is = L l (,Os- Opm) (22b)where only the stator resistance and the PM flux errors aretaken into account. The stator flux error Qs and the statorcurrent error is are solved by subtracting the estimates in (22)from the actual variables solved from (1) and (2). The resultis

II -

,Os =Rsis + Rsis-S,J+bsis = L- (,Os -,pm)

Ais (23a)(23b)

+ (R5L-' + AL-1) /pm

BC Di, =L 0s L1 14pmC D

Rsis (24a)

(24b)

The stator resistance error can be detected from is onlywhen the motor is loaded, and when the stator current is varies,the gain from Rs to is changes. Because the stator resistanceis estimated at low speeds using the HF signal injection, onlythe PM flux is selected for the adaptation at medium andhigh speeds. It is also to be noted that only two quantities,the current error components 'd and tq, can be used for theadaptation and only two quantities can thus be estimated. Sincethe rotor speed is estimated using the q component of thecurrent estimation error, the d component is selected for thePM flux adaptation.

For investigating the behavior of the PM flux adaptation, theclosed-oop system of Fig. 4 is studied. The transfer functionfrom the scalar-valued flux error pnpm to the current error idis obtained from

F. (s)= [1 0]{C(sI-A) 1B+D}[1 O]T (25)

Integration is used for the adaptation of <pm, and the transferfunction corresponding to the adaptation is

G1,p(s) =-kp-Ld/S (26)

where kp is the adaptation gain. The closed-oop transferfunction , \ , \

G, (s) 1 +Ff(s)GG(s)IX- + Fp, (s)Gp (s)

(27)

from Opm to <pm can be evaluated in any operating point.The closed-oop transfer function (27) was analyzed nu-

merically in different operating points. The parameter valuesgiven in Table I were used for the calculations. Fig. 5 showspoles and Fig. 6 unit step responses obtained with a constantadaptation gain k= O.2WB, where WB is the base angularspeed. The speed varies from 0 to 1 p.u. in Fig. 5. Thestator current and the rotational direction have no effect on theresults. Due to symmetry, only the upper half-plane is shownin the pole plot. The step responses in Fig. 6 were obtainedat speeds Wm = 0.25 p.u., 0.5 p.u., 0.75 p.u., and 1.0 p.u.

178

1.5 11

--: I -

H1.

0.5-

Wm (p.U.)

0.2

0 10.5 0

-8.2 0

-1 -0.5Re{s} (p.u.)

0.5 -1-v) WA 0 Wm L

WO'm

Fig. 7. Functions f (wm,) (solid) and g(wm ) (dashed) as functions ofestimated rotor speed.

0 .

0

Fig. 5. Pole plot of the closed-oop system of PM flux adaptation. Thevicinity of the origin is also shown as a magnification.

0 /00 0.025 0.05 0.075 0.1

t (s)Fig. 6. Step response of the closed-oop system of PM flux adaptation forvarious rotor speeds.

According to the results, the small constant gain gives arelatively slow response. However, the actual PM flux changesslowly since it depends on the temperature. Although thereare complex poles, the dominant pole is real-valued and it isalmost constant as the speed increases above 0.5 p.u. Hence,the response time to a changing PM flux does not varysignificantly at higher speeds, which can also be seen fromthe step responses in Fig. 6.

C. Gain SchedulingThe adaptation of the stator resistance is in use only at low

speeds in which the HF signal injection method is used. ThePM flux adaptation is not used simultaneously with the statorresistance adaptation, and is enabled when the rotor speed ishigher than WA. The adaptation gains kR and kp are changedwith the varying rotor speed and stator current.

In order to have a nonnegative bandwidth (21) for the statorresistance adaptation, the gain kR and the current iq musthave the same sign. A constant bandwidth aR is infeasible,since iq = 0 would imply infinite adaptation gain. A signumfunction in the gain kR could cause chattering near zero tq.Therefore, the gain kR is changed proportionally to iq, i.e.

kR = OqROJTn)Pm I2 (28)B

which results in the adaptation bandwidth*2

aR = aROf(Wrn)]y (29)B

The parameter aRO is a constant corresponding to the adap-tation bandwidth at zero speed and at approximately nominal

Fig. 8. Block diagram of the control system.

load, IB is the base value of the current, and f(Jm) is aspeed-dependent function shown in Fig. 7.The gain kp, for the PM flux adaptation is varied according

tokV, = ko g9(Wm) (30)

where kpo is a positive constant and g(Wm ) is a speed-dependent function shown in Fig. 7. At speeds higher thanw1,p, the gain kp is thus kept constant.

V. SIMULATION RESULTS

The proposed method was investigated by means of simu-lations and laboratory experiments. Fig. 8 shows the blockdiagram of the control system comprising cascaded speedand current control loops. P14ype speed control with activedamping is used. The data of the six-pole interior-magnetPMSM (2.2 kW, 1500 rpm) are given in Table I. The basevalues for voltage, current, and angular speed are defined asUB = 2/3UN, IB = W2N, and WB = 27fN, respectively.The electromagnetic torque is limited to 22 Nm, which is1.57 times the nominal torque TN. The high-frequency carrierexcitation signal has a frequency of 833 Hz and an amplitudeof 40 V. The threshold speeds WA = 0.13 p.u. and wX = 0.2p.u., the resistance adaptation bandwidth aRo = 0.01 p.U.,and the constant kpo = 0.2WB. The current and speed controlbandwidths are 5.33 p.u. and 0.067 p.u., respectively, and thebandwidth ai = 0.067 p.u.The MATLAB/Simulink environment was used for the sim-

ulations. Fig. 9 shows results obtained at zero speed reference.Except the stator resistance, the parameter values used in thecontroller were equal to those of the motor model. In the

179

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0.1

7 -0.11.5

0O-0.520

0

2 -20

0.1

0.05 =vD~

C O0

0.6

7 -0.22F

1 2 3t (s)

Fig. 9. Simulation results showing stator resistance adaptation. First subplotshows electrical angular speed of the rotor (solid), its estimate (dashed), and itsreference (dotted). Second subplot shows the load torque reference (dotted),the electromagnetic torque (solid), and its estimate (dashed). Third subplotshows the estimation error of the rotor position. Last sublot shows the statorresistance (dashed) and its estimate (solid).

beginning of the simulation, the stator resistance estimate is15 % smaller than the actual stator resistance. A nominal loadtorque step is applied at t = 1 s, and the stator resistanceestimate starts converging to the actual resistance immediately.At t = 2 s, a 15 % step increase occurs in the stator resistance,and its estimate again follows the actual stator resistance.The stator resistance estimate converges close to the actualresistance in less than 1 s.

Results showing the behavior of the estimated PM flux aredepicted in Fig. 10. The estimated flux is 15 % bigger thanits actual value in the beginning of the simulations, and otherparameter estimates are equal to the actual values in the motormodel. The speed reference is changed from zero to 0.5 p.u.at t = 0.5 s, and a nominal load torque step is applied att = 1 s. In Fig. 10(a), the parameter adaptation is not inuse, whereas in Fig. 10(b), the adaptation is used. Fig. 10(a)shows that the erroneous PM flux results in an error in therotor position estimate both at no load and when a load isapplied. In addition, the electromagnetic torque is smaller thanthe estimated torque. According to Fig. 10(b), the adaptationpractically removes the PM flux error in less than 0.2 s afterthe motor is started, and the errors in the rotor position andthe torque are avoided.

VI. EXPERIMENTAL RESULTS

The experimental setup is illustrated in Fig. 11. The PMSMis fed by a frequency converter that is controlled by a dSPACEDS1 103 PPC/DSP board. Mechanical load is provided bya PMSM servo drive. An incremental encoder is used formonitoring the actual rotor speed and position. The nominalDC4ink voltage is 540 V, and the switching frequency andthe sampling frequency are both 5 kHz. The dc-ink voltageof the converter is measured, and a simple current feedforward

00L-0.520

0

2 -20

1.2

0.6

000

0.6

S 072 -0.2

2

0O-0.520

0

t -20

1.2

0.6

000

0.5

0.5

t (s)(a)

1

1

1.5

1.5t (s)(b)

Fig. I 0. Simulation results showing PM flux adaptation: (a) withoutparameter adaptation (b) with parameter adaptation. Explanations of the curvesare as in Fig. 9 except that the last subplot shows the PM flux (dashed) andits estimate (solid).

compensation for dead times and power device voltage dropsis applied.

Results obtained in low-speed operation are depicted inFigs. 12 and 13, showing the behavior of the stator resistanceadaptation. Additional 1-Q resistors were added between thefrequency converter and the PMSM as shown in Fig. 11.The resistance was changed stepwise by opening or closinga manually-operated three-phase switch connected in parallelwith the resistors. The experiment in Fig. 12 corresponds tothe simulation in Fig. 9. The error in the stator resistance isdecreased after the load torque step at t = 1 s. The estimatedstator resistance follows well the actual resistance after thestepwise increase at t = 2 s.

In the experiment of Fig. 13, the drive was operating atvery low speed (W -0.05 p.u.) in the regenerating mode.The load torque was at the positive nominal value. The stator

180

Computer with DS1103

Fig. 11. Experimental setup. Mechanical load is provided by a servo drive.

0.1

S0

1-0.5l 5rF Ar

0.1

S 07 -0.1

1.5

E-- 0-0.520

0

2 -20

0.1

0.0 1

Fig. 13. Experimental results showing stator resistance adaptation. Explana-tions of the curves are as in Fig. 12.

0.6

a 07 -0.2

2F

2 3t (s)

Fig. 12. Experimental results showing stator resistance adaptation. Firstsubplot shows the electrical angular speed of the rotor (solid), its estimate(dashed), and its reference (dotted). Second subplot shows the load torquereference (dashed) and the electromagnetic torque estimate (solid). Thirdsubplot shows the estimation error of the rotor position. Last sublot showsthe stator resistance (dashed) and its estimate (solid).

resistance estimate was forced to an incorrect value at t 0.6s. When the resistance adaptation is activated again at t 1 s,the estimated resistance returns close to the actual resistance inabout 1 second. After the stepwise decrease in the resistanceat t - 2.5 s, the estimated resistance settles close to the newvalue. It is to be noted that in the experiments in Figs. 12and 13, the inverter unidealities contribute to the resistanceseen by the controller. Therefore, the estimated resistance isnot precisely the stator resistance shown in the figures.

Figs. 14 and 15 show results in the medium-speed operation,where the PM flux adaptation is in use. The experiment inFig. 14 corresponds to the simulation in Fig. 10. The resultsare comparable to that of the simulation, and the initial 15% PM flux error is rapidly removed after the accelerationat t = 0.5 s. The PM flux estimate also stays close tothe actual flux when the load torque step is applied. Fig.15 shows results in constant-speed operation, the rotor speedbeing 0.5 p.u. In Fig. 15(a), the load torque was at the positivenominal value, while in Fig. 15(b) the load torque was atthe negative nominal value. The drive operated thus in themotoring and in the regenerating modes in Figs. 15(a) and

0

0L-0.520

0

2 -20

1.2

0.6

000

t (s)

Fig. 14. Experimental results showing PM flux adaptation. Explanations ofthe curves are as in Fig. 12 except that the last subplot shows the PM flux(dashed) and its estimate (solid).

15(b), respectively. The PM flux estimate was forced to an

erroneous value at t - 0.5 s, and the adaptation was enabledagain at t r-1 s. The erroneous PM flux estimate causes an

error in the electromagnetic torque estimate, and the positionestimation error also impairs the performance of the drive.After t -

1 s, the estimated PM flux quickly converges closeto its actual value, leading to a reduced position estimationerror and improved torque estimation accuracy.

VII. CONCLUSIONS

This paper proposed a method for the estimation of thestator resistance and the PM flux in a sensorless PMSM drive.The adaptive observer augmented with an HF signal injectiontechnique at low speeds was used for the adaptation of theparameters in addition to the speed and position estimation.The system was investigated by means of steady-state and

181

0-°t-0.520

0

t -20

0.1

0.05 =

0&,

2t (s)

3 4

1

0.5 1 1.5

ACKNOWLEDGEMENTThe authors gratefully acknowledge the financial support

given by ABB Oy, Walter Ahlstrom foundation, and KAUTEfoundation.

OL-0.5

20

0O2 -20

1.2

0.6

0 0.5t (s)(a)

1.5

0.6

0.5

0.40.50

-1.5'

20

-20

1.2

0.6

0 0.5 1 1.5t (s)

(b)

Fig. 15. Experimental results showing PM flux adaptation. Explanations ofthe curves are as in Fig. 14.

small-signal analyses in order to develop reasonable adaptationalgorithms. The simulation and experimental results show thatthe stator resistance adaptation reduces the resistance error

significantly. Even though the high-frequency signal injectionremoves the position estimation error in steady state even

without any resistance adaptation, the good accuracy of theresistance estimate reduces estimation errors when the signalinjection is not in use. The decreased speed correction term

due to the resistance adaptation also improves the dynamicperformance of the combined observer. The PM flux adapta-tion reduces the position estimation error at medium and highspeeds and improves the electromagnetic torque estimationaccuracy. The parameter estimates converge rapidly close to

the actual parameters, and the the sensitivity to the parameter

variations is reduced.

REFERENCES

[1] R. Wu and G. R. Slemon, "A permanent magnet motor drive without a

shaft sensor," IEEE Trans. Ind. Applicat., vol. 27, no. 5, pp. 1005-1011,Sept./Oct. 1991.

[2] R. B. Sepe and J. H. Lang, "Real-ime observer-based (adaptive) controlof a permanent-magnet synchronous motor without mechanical sensors,"IEEE Trans. Ind. Applicat., vol. 28, no. 6, pp. 1345-1352, Nov./Dec1992.

[3] M. Schroedl, "Sensorless control of AC machines at low speed andstandstill based on the INFORM method," in Conf Rec. IEEE-IAS Annu.Meeting, vol. 1, San Diego, CA, Oct. 1996, pp. 270-277.

[4] P. L. Jansen and R. D. Lorenz, "Transducerless position and velocityestimation in induction and salient AC machines," IEEE Trans. Ind.Applicat., vol. 31, no. 2, pp. 240-247, March/April 1995.

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