Speed sensorless control of a DC-motorvia adaptive filters

M. Hilairet and F. Auger

Abstract: In the motor control industry, digital signal processor (DSP) systems offer majorimprovements compared to analogue designs, notably enabling to replace speed or positionsensors by the implementation of sensorless control algorithms. A new viable method that estimatesthe rotor velocity of a permanent magnet DC-motor from the ‘rotor slot harmonics’ present in the DCcurrent signal is proposed. This approach is based on both two adaptive digital filters and a frequencyestimator, which is fitted to this particular application and compared to a classical technique. Finally,an algorithm that selects the best speed estimation to reach high-speed transient behaviour and toobtain speed estimation error equal to zero at steady state is proposed.

1 Introduction

Process industries use more and more motor drives becauseof their higher control parameter, higher flexibility andlower power consumption. One way to control the speedand the torque of these motors is to include tachometersor position encoders into the feedback loop. But thesesensors, their conditioning and their connections cost a lotand are a significant source of failure that reducesreliability. When they are mounted on the driving shaft ofa single-shaft motor, they are also located at a place thatshould preferably be assigned to the load. Therefore theirelimination is an attractive prospect, which can be achievedby estimating speed from the stator terminal currentmeasurements. This is what is commonly called the sensor-less control of electrical machines [1].

The absence of a speed transducer is an important factordue to cost requirements as an important application fieldis the automobile industry. Some applications are automaticwindow lifters, electric sunroofs, sliding doors, enginecooling fans, air conditioning, cockpit heating and so on.However, sensorless control is not only suited for verysmall motors. In fact, several problems afflict electricalmotor drives [2] and so far, redundant or conservativedesign has been used in every application where continuityof operations is a key feature, including home and civilappliances, such as heat pumps and electric vehicles. Thisis especially important in high impact automotive appli-cations, such as an electrical vehicle (EV) or a hybridone, where even limp-back operation is preferred over nooperation [3].

Fault-tolerance has become an increasingly interestingtopic in the last decade where automation has becomemore complex. A trend towards more autonomic controlsystems also drives the interest for fault-tolerance. In

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-epa:20060149

Paper first received 13th April and in revised form 22nd September 2006

M. Hilairet is with the LGEP/SPEE Labs, CNRS UMR 8705, Supelec,Universites Paris VI et XI, Gif sur Yvette 91192, France

F. Auger with the IREENA, EA 1770, Universites de Nantes, Saint-Nazaire44602, France

E-mail: mickael.hilairet@lgep.supelec.fr

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mass produced industrial systems, unit cost is a paramountissue. Hence, cheap fault-tolerant control has become animportant industrial research area. The objective is to givesolutions that provide fault-tolerance to the most frequentfaults and thereby reduce the costs of handling the faults.The fault tolerant controller also increases the reliabilityof the process.

At least two classes of solutions exist to estimate the rotorspeed of DC-motors without mechanical sensors. The firstone, which can be described by a dynamic model [4],depends on the back electro magnetic field (EMF). Thislinear model can be used to design an observer, which esti-mates both the current and the rotor velocity [5–13]. Butthis observer is inherently parameter-dependent, makingthe estimation sensitive to parameter uncertainties and toparameter variations caused by internal heating. These par-ameters may also be tracked [14], but the resulting non-linear model increases the computational cost.

The second kind of solution, to which our approachbelongs, is based on a passive monitoring of the rotor slotharmonics as in the case of AC machines [15–19].Mechanical and magnetic saliencies in the rotor (eccentrici-ties, variations of the air-gap permeance) generate spectrallines in the DC-current signals whose frequencies andamplitudes do not depend on the motor parameters.However, this approach is complicated by the presence ofmany spectral lines arising from both the voltage sourceinverter and the motor itself. To illustrate this remark,Fig. 1 shows a spectral analysis of the armature current ofa 0.5 kW DC-motor at 10 Hz (600 rpm). This spectrumincludes spectral lines not only at multiples of the funda-mental chopping frequency (solid lines, three harmonicsat 100, 200 and 300 Hz), but also at frequencies, which beexpressed as fsh ¼ nrðNr=pÞfm (dashed lines, three harmo-nics at 16 � 10; 32 � 10 and 48 � 10 Hz), where nr is asigned integer, and fm;Nr and p are, respectively, the mech-anical rotor frequency, the total number of rotor slots andthe number of stator pole pairs. In our case, the DC-motoris composed of 48 conductors with three conductors perrotor slot. So, the number of rotor slots Nr is equal to 16.This is confirmed by the spectral analysis.

Fig. 2 represents a time frequency analysis of the arma-ture current during a speed transient. As we can see, thecurrent is mainly composed of a continuous component

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(at zero frequency), three major harmonic components,function of the rotor velocity, and a constant frequencyequal to 1000 Hz, which may be due to mechanicalfrequency oscillation.

So as to estimate the rotor frequency fm from these spectrallines, non-parametric (FFT-based) approaches cannot beused, because of the trade-off between high-frequency resol-ution (obtained with long-data records) and quick time inresponse to speed variations (obtained with short-datarecords). Therefore we propose a new approach that tries tocomply with the most of the practical requirements.

2 Proposed estimator

The proposed frequency estimator is composed of threesuccessive blocks computed at each sample time Ts

(Fig. 3) [20].

1. The first block is a band-pass filter with a centre fre-quency l0 nearly the normalised frequency 16 fmTs. Theobjective of this filter is to remove as much as possible irre-levant harmonic, so as to keep only sinusoidal one. Such afilter can be realised with a second-order (IIR) [21]

HpbðzÞ ¼Y1ðzÞ

XðzÞ¼

1

2ð1 �HptðzÞÞ ð1Þ

HptðzÞ ¼r2

� ð1 þ r2Þbz�1

þ z�2

1 � ð1 þ r2Þbz�1 þ r2z�2ð2Þ

with b ¼ cosð2pl0Þ and

r2¼ ð1 � tanðpDlÞ=ð1 þ tanðpDlÞÞ

0 500 1000 1500 2000 2500−90

−80

−70

−60

−50

−40

−30

−20

−10

0

frequency (Hz)

PS

D (

dB)

Fs = 5000 Hz, NFFT = 2048, 80% overlap, Hamming window

Fig. 1 Estimated power spectral density of the current in steadystate

The rotor velocity can be recovered from the dashed spectral lines

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The bandwidth Dl needs to be chosen in order to satisfy thecompromise between two antagonistic characteristics: thetracking capability and the estimator resolution. Fig. 4ashows the frequency response of the band-pass filter forl ¼ 0:05 and Dl ¼ 0:001. This filter strongly attenuatesthe frequency out side the bandwidth. Moreover,Hpbð0Þ ¼ Hpbð0:5Þ ¼ 0, as (1) and (2).2. The signal y1½n� obtained at the output of the band-passfilter excites a fifth-order (FIR) filter whose frequencyresponse nearly l0 is slightly different from an idealdouble derivative filter GðlÞ ¼ �ð |2plÞ2 ¼ 4p2l2.

For low-frequency estimation in relation to the samplingfrequency, we can search the coefficients of this FIR filter,which has the same behaviour at the origin as an ideal

filter GðlÞ ¼ 4p2l2. For an N-order filter (so of length

N þ 1) with a transfer function HðzÞ ¼PN

k¼0hkz�k and

with a frequency response HðlÞ ¼ Hðz ¼ e|2plÞ, the simi-larity of behaviour is obtained by imposing a contactpoint of order N at l ¼ 0 between the wished frequencyresponse GðlÞ and the obtained one HðlÞ

dnH

dlnð0Þ ¼

dnG

dlnð0Þ for n [ ½0;N �

with

dnG

dlnð0Þ ¼

8p2 for n ¼ 2

0 else

�

−0.02

0

0.02

0.04

0.06

Rea

l par

t

Signal in time

Log. scale [dB]

Ene

rgy

spec

tral

den

sity

SP, Lh=100, Nf=128, log. scale, imagesc, Threshold=0.001%

Time [ms]

Fre

quen

cy [k

Hz]

0 200 400 600 800 1000 1200 14000

0.5

1

1.5

2

Fig. 2 Time frequency analysis of the current during a transientoperation

Fig. 3 Schematic diagram of the proposed frequency estimator

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Fig. 4 Frequency response of the band-pass filter and double derivative filter

a Band-pass filter with l ¼ 0.05 and Dl ¼ 0.001b Double derivative filter with l ¼ 0 (i), l ¼ 0.25 (ii), and l ¼ 0.175 (iii)

These relations lead to a linear system of dimension N þ 1,which is

XNk¼0

knhk ¼�2 for n ¼ 2

0 else

�

The coefficients hk are the solution of a Vandermodesystem. For N ¼ 5, the solution is

h0 ¼ �45

12h1 ¼

154

12h2 ¼ �

214

12

h3 ¼156

12h4 ¼ �

61

12h5 ¼

10

12

The frequency response is close to that of an ideal doublederivative filter (Fig. 4b (i)) for jlj , 0:12, so forFs . 10 f . The use of higher-order filters does not seemto be justified because their behaviours are not significantlybetter. On the other hand, for jlj . 0:12, it is necessary touse an adjustable filter parametrised by the normalised fre-quency l where its behaviour at l is near an ideal doublederivative filter. For a fifth order FIR filter, the transferfunction and frequency response are

HðzÞ ¼ h0 þ h1z�1

þ � � � þ h5z�5

HðlÞ ¼ h0 þ h1e�|2pl

þ � � � þ h5e�|10pl

This principle leads to impose the followings relations asfollows

Hðl0Þ ¼ Hð�l0Þ ¼ Gðl0Þ ¼ 4p2l20

H0ðl0Þ ¼ �H

0ð�l0Þ ¼ G

0ðl0Þ ¼ 8p2l0

H00ðl0Þ ¼ H

00ð�l0Þ ¼ G

00ðl0Þ ¼ 8p2

which form a linear system of dimension 6. With u ¼ 2pl0,the solution is

h5 ¼ð1 þ sinðuÞ2Þu� sinðuÞ cosðuÞ

2 sinðuÞ3

h4 ¼ �uþ sinðuÞð11 � 12 sinðuÞ2Þh5

2 sinðuÞ cosðuÞ

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h3 ¼u

sinðuÞ� 4 cosðuÞh4 � 2ð5 � 6 sinðuÞ2Þh5

h2 ¼ �4 cosðuÞh3 � ð10 � 12 sinðuÞ2Þh4

� 4 cosðuÞð5 � 8 sinðuÞ2Þh5

h1 ¼ �2 cosðuÞh2 � ð3 � 4 sinðuÞ2Þh3

� 4 cosðuÞð1 � 2 sinðuÞ2Þh4

� ð5 � 20 sinðuÞ2 þ 16 sinðuÞ4Þh5

h0 ¼ u2� cosðuÞh1 � cosð2uÞh2 � cosð3uÞh3

� cosð4uÞh4 � cosð5uÞh5

Figs. 4b (ii) and (iii) show the frequency response forl ¼ 0:25 and 0.175, respectively. The principal operationsfor the coefficient computation is the evaluation of

b ¼ cosðuÞ; g ¼ sinðuÞ; g2;g3; g4, and three divisions. Noother more trigonometric computation is necessarybecause of the following relations

cosð2uÞ ¼ 1 � 2 sinðuÞ2

cosð3uÞ ¼ cosðuÞð1 � 4 sinðuÞ2Þ

cosð4uÞ ¼ 1 � 8 sinðuÞ2 þ 8 sinðuÞ4

cosð5uÞ ¼ cosðuÞð1 � 12 sinðuÞ2 þ 16 sinðuÞ4Þ

3. If y1½n� ¼ A cosð2plnþ wÞ, the signal obtained at theoutput of the double derivative filter is y2½n� ¼ 4p2l2y1½n�.Because the two signals y1½n� and y2½n� are colinear, the esti-mation of l could be considered by the operation

l½n� ¼1

2p

ffiffiffiffiffiffiffiffiffiffiy2½n�

y1½n�

s¼

1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy1½n� y2½n�

y1½n�2

s

To avoid division by zero, the frequency is calculated by

l½n� ¼1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPþ1k¼0 a

ky1½n� k� y2½n� k�Pþ1k¼0 a

ky1½n� k�2

s

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09 at 04:23 from IEEE Xplore. Restrictions apply.

with 0 � a � 1, this corresponds to the minimisationof the quadratic criterior J ðl½n�Þ ¼

Pþ1k¼0 a

kðy2½n�k�

�4p2l½n�2y1½n�k�Þ2. Obviously, this expression is computed

in a recursive form

l½n� ¼1

2p

ffiffiffiffiffiffiffiffiffiN ½n�

D½n�

s

with N ½n� ¼ aN ½n� 1� þ y1½n� y2½n�

D½n� ¼ aD½n� 1� þ y1½n�2

The parameter a has to satisfy the compromise between hightracking capabilities for fast frequency variations and theminimisation of the variance of the frequency estimationerror. Experimental results give satisfactory behaviourwith a ¼ e�2pDl [20].

Finally, to track time-varying frequencies, the frequencyl, estimated by the frequency estimator, is used at the nexttime sample to update the two filter coefficients.

3 Conventional speed estimator

A conventional method for speed estimation is the compu-tation of the discrete time state space model of the system inreal-time [5, 6, 8, 9, 11, 12]. The continuous state spacemodel of a permanent magnet DC-motor, with state spaceX ¼ ½iv�T is

dX

dt¼

�R=L �K=LK=J �f =J

� �X þ

1=L0

� �U þ

0

�1=J

� �Cr

where i;v; u and Cr are, respectively, the armature current,the rotor velocity, the armature voltage and load torque.

One solution for speed estimation is to assume the rotorvelocity nearly constant between two sample times. In thiscase the speed estimator has a low cost but it is possibleto obtain instability phenomenon in the closed-loop. Toavoid this problem, we opt for the estimation of the loadtorque. The new state space is X ¼ ½i v Cr�

T¼

½x1 x2 x3�T with transistion matrix Ac, command matrix

Bc and measurement matrix C

Ac ¼

�R=L �K=L 0

K=J �f =J 1=J

0 0 0

264

375

Bc ¼

1=L

0

0

264

375 C ¼ 1 0 0

� �

Among all the estimation techniques (Luenberger observer,Kalman filter, adaptive observer) [5, 8, 9, 11–13], wechoose the suboptimal Kalman filter (SKF) [22, 23]because of two main reasons.

† Reduced computational time, because the system islinear and stationnary. The Kalman gains are constant.† The tuning easiness of the different degrees of freedom.

A second-order series expansion of the exponentialmatrix eAcTs is sufficient for discretising the continuousstate space model with a sampling period Ts equal to

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200ms. This leads to

Ad ¼

a11 a12 a13

a21 a22 a23

a31 a32 a33

24

35 Bd ¼

b1

b2

b3

24

35

with

a11 ¼ 1 � RTe=Lþ ðR2=L2� K2=ðLJ ÞÞT 2

e =2

a12 ¼ �KTe=Lþ ðRK=L2þ Kf =ðLJ ÞÞT2

e =2

a13 ¼ KT2e =ð2LJ Þ

a21 ¼ KTe=J � ðRK=ðLJ Þ þ fK=J ÞT 2e =2

a22 ¼ 1 � f Te=J þ ð f 2=J2� K2=ðLJ ÞÞT 2

e =2

a23 ¼ Te=J þ f T 2e =ð2J

2Þ

a31 ¼ a32 ¼ 0

a33 ¼ 1

b1 ¼ 1=L� RTe=ð2L2Þ

b2 ¼ KTe=ð2LJ Þ

b3 ¼ 0

Finally, the SKF is

x1p½k� ¼ a11x1½k � 1� þ a12x2½k � 1�

þ a13x3½k � 1� þ b1U ½k � 1�

x2p½k� ¼ a21x1½k � 1� þ a22x2½k � 1�

þ a23x3½k � 1� þ b2U ½k � 1�

e½k� ¼ y½k� � x1p½k�

x1½k� ¼ x1p½k� þ k1e½k�

x2½k� ¼ x2p½k� þ k2e½k�

x3½k� ¼ x3½k� þ k3e½k�

The three gains k1; k2 and k3 are the final values of the gainK defined by

Pp½k� ¼ AdP½k � 1�Atd þ Q

K½k� ¼ Pp½k�CtðCPp½k�C

tþ RÞ�1

P½k� ¼ Pp½k� � K½k�CPp½k�

limk!þ1K ½k� ¼

k1

k2

k3

0B@

1CA

with

R ¼ sicQ ¼

si 0 0

0 sv 0

0 0 sCr

24

35

Because R is a scalar, only the ratio Q/R is important. So,we assign to R the value 1 and we adjust the dynamics ofthe filter via the three degrees of freedom noted ða1;a2;a3Þ

R ¼ 1 Q ¼

a1 0 0

0 a2 0

0 0 a3

24

35

The degrees of freedom are adjusted so that the instan-taneous current and speed estimation are similar to the real

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measure values. After trial and error loops, the selection is

a1 ¼ 10; a2 ¼ 100; a3 ¼ 0:01

4 Experimental results

An experimental validation of the proposed algorithm wascarried out on a 0.5 kW permanent magnet DC-motor,whose shaft is coupled to an induction machine (Table 1).The motor is fed by a transistor bridge chopper operatingin differential mode (þU, 2U), which received thecontrol signals from two IP regulators in cascade. This con-troller is composed of a integral action in relation to theerror, a proportional action relatively to the measuredsignal and an anti-windup action. The goal of the anti-windup action is to counteract the integration of the control-ler that produces oscillations when the control signalsaturates. This can be performed by feeding back theoutput of the saturation to the controller. Figs. 5 and 6shows the approach used for the anti-windup action.

For information, the execution time of the measurement,estimation and control is given in Table 2. All the algor-ithms are developed with the C language and implementedin a TMS320C31 Digital Signal Processor of TexasInstruments with a clock frequency of 40 MHz (Fig. 7).The execution time of the frequency estimator is nearlyequal to the time required by the controller (current andspeed control). The proposed algorithm has a medium com-plexity and offers some advantages compared to state obser-vers. In fact, this speed estimator is not parameter dependentand so gives zero speed estimation error at steady state.

Fig. 8 shows a speed estimation for an operating pointat 15 Hz (900 rpm) for two different bandwidths of theIIR filter. In both cases, the average speed estimationerror is nearly equal to zero, and the variance depends onthe parameter Dl. For Dl ¼ 0:001, the speed estimationis nearly equal to the estimation value obtained from a

Table 1: Model parameters

Rated output power Pn ¼ 0.5 kW

Rated torque Cn ¼ 1.75 Nm

Rated speed N ¼ 3000 rpm

Rated voltage Vn ¼ 180 V

Rated current In ¼ 3.50 A

Resistance R ¼ 6.72 V

Inductance L ¼ 357 mH

Torque constant K ¼ 0.59 Nm/A

Inertia load J ¼ 5.4 � 1023 kgm2

Viscous coefficient f ¼ 1.6 � 1023 Nm/rad/s

Number of pole pairs p ¼ 1

Fig. 5 Speed controller with anti-windup action

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classical speed estimation algorithm [24, 25] such as

v½n� ¼um½n� � um½n� 12�

12Ts

Fig. 9 shows a speed estimation with a speed initialisation at180 rpm (þ20%) below the real value. Good estimation isreached approximately after 0.3 and 0.2 s, respectively.

Fig. 10 represents the output signals of the band-passfilter and double derivative filter, respectively, for arotational speed of 600 rpm. The harmonic at 16 times therotational frequency of the machine has an amplitude of�15 mA. The FIR filter tends to increase the noise of theoutput signal and thus deteriorates the quality of thefrequency estimation (speed).

Fig. 11 shows a linear speed variation from 2000 to400 rpm, which confirms the tracking capability of thealgorithm in a real situation. For Dl ¼ 0:002, the speedestimation error remains low. It is this value that is optedfor the feedback speed control.

Because the SKF gives a better estimation of the speedduring high transient and at lower speed, than the frequencyestimator, we opt to use the later during medium and highspeed and also during low-transient operating mode,and the SKF otherwise. A selection algorithm (Fig. 12)selects speed estimation according to the operating mode.Because the SKF gives a reliable speed estimation comparedto the frequency estimator, we use the SKF information asa base speed for the selection algorithm condition (seecondition 3 of Fig. 13). Table 3 briefly summarises theselected speed estimation according to the operating modeof the machine. This is illustrated by the algorithm moredetailed in Fig. 13 and described in what follows.

Test 1 informs about the operating mode of the machine:steady-state or transient operating mode. If the steady stateis well established, the logical variable RP2 is equal to 1.Then if the estimated speed by the frequency estimator is

Fig. 6 Current controller with anti-windup action and back EMFcompensation

Table 2: Execution time of the algorithms

Operations Execution

time, ms

current measurement 5.6

speed measurement 7.6

frequency estimator 20 or 54

stationnary Kalman filter 5

controller (current, speed and

selection algorithm)

24.5

605

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Fig. 7 Digital signal processor board (Texas Instruments TMS320C31) with its interface hardware

superior to 50 rpm in absolute value, and the distancebetween the value of the reference speed and the value ofthe speed estimated by the frequency estimator is inferiorto 200 rpm in absolute value (low-transient operatingmode), this means that the speed estimated by the frequencyestimator is used in the speed control loop. Otherwise, if test1 is not valid, then the variables N, RP1 and RP2 reduce to

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zero, which means a transition from steady state towards atransient operating mode.

During a transient operating mode, if the distancebetween the value of the reference speed and the value ofthe speed estimated by the SKF is lower than 60 rpm inabsolute value, and the speed estimated by the frequencyestimator is greater than 50 rpm in absolute value, test 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

DC current

A

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6850

900

950measure and estimate speed

rpm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−20

−10

0

10

20speed estimation error

time (s)

a

rpm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

DC current

A

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6850

900

950measure and estimate speed

rpm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−20

−10

0

10

20speed estimation error

time (s)

b

rpm

Fig. 8 Experimental result obtained during steady state at 15 Hz

a Dl ¼ 0.001b Dl ¼ 0.002

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

DC current

A

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6850

900

950

1000

1050

1100measure and estimate speed

rpm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−200

−150

−100

−50

0

speed estimation error

time (s)

rpm

DC current

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

A

measure and estimate speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6850

900

950

1000

1050

1100

rpm

speed estimation error

time (s)

a b

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−200

−150

−100

−50

0

rpm

Fig. 9 Experimental result obtained during steady state at 15 Hz with þ20% of initial error

a Dl ¼ 0.001b Dl ¼ 0.002

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1−0.02

−0.01

0

0.01

0.02output signal of the band−pass filter

A

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1−1

−0.5

0

0.5

1x 10

−3 output signal of the double derivative filter

A (

rad/

s)2

time (s)

Fig. 10 Output signal of the two filters for a mechanical speed of600 rpm

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is valid. Therefore the steady state is reached and the vari-able N is incremented by 1 at each sample period. When thevariable N reaches the value of 2500 (after 0.5 s), the switchINT1 changes from position 0 to position 1 (Fig. 12). Thismeans that the frequency estimator is autonomous. In thisconfiguration, the used speed in the speed control loopalways comes from the SKF. If the machine is still insteady state after 1 s (N ¼ 5000), then the second logicalvariable RP2 is equal to 1 and the switch INT2 passesfrom position 0 to position 1. In this configuration, theused speed in the speed control loop comes from the fre-quency estimator. The static error between the referencespeed and the real mechanical speed is zero.

Fig. 14a represents the measured mechanical speed, 14bthe speed estimated by the SKF and 14c the speed estimatedby the frequency estimator with a load torque disturbance.The second section of Fig. 14a with the variables RP1and RP2 equal to 1 and 0, respectively, means that the

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−0.2

0

0.2

0.4

DC current

A

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

500

1000

1500

2000measure and estimate speed

rpm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−200

−150

−100

−50

0

50speed estimation error

time (s)

rpm

DC current

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−0.2

0

0.2

0.4

A

measure and estimate speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

500

1000

1500

2000

rpm

speed estimation error

time (s)

a b

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−40

−20

0

20

40

rpm

Fig. 11 Experimental result obtained during speed variation

a Dl ¼ 0.001b Dl ¼ 0.002

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selection algorithm knows that the machine works in steadystate. After 1 s of steady state (between the instants 2 and3 s), the variables RP1 and RP2 are equal to 1. Thismeans that the machine is still in steady state and thespeed estimated by the frequency estimator is used in thecontrol loop.

We can also add the following:

1. The estimated speed by the frequency estimator is onlyused during the steady state and the low-transient operatingmode. The filter bandwiths can be reduced to 2 or 3 Hz inorder to decrease the variance upon the speed estimationwithout risk of divergence.2. During the transient state, the placement of the band-passfilter and the calculation of the coefficients of the FIR filterare made, thanks to the speed estimation by the SKF. Thishas some advantages.

(a) The tracking capability of the frequency estimator isincreased because the estimator is in open-loop.(b) The frequency estimator diverges when the speed ofrotation is low because it is not capable of extracting onlyone harmonic in the signal although the band-pass filterand the double derivative filter parameters are madewith the speed estimation by the SKF. Therefore theinjection of the speed estimation by the SKF allows

Fig. 12 Diagram of the selection algorithm

Fig. 13 Selection algorithm

Condition 2 is jvref2 vfj , 200 and jvfj . 50, condition 3 is jvref2vskfj , 60 rpm and jvfj . 50 rpm

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the frequency estimator to converge very quickly to the‘exact’ value of the mechanical speed after a divergence.This is illustrated in Fig. 15.

Fig. 16a represents the measured mechanical speed, thespeed estimated by the SKF and the speed estimated bythe frequency estimator for a periodic speed referencechange from 600 to 2600 rpm. The selection algorithm

Table 3: Synthesis of the speed estimation selectedfollowing the operating mode of the DC-motor

Condition Stationary

Kalman filter

Frequency

estimator

steady state jvj � 50 rpm �

jvj � 50 rpm �

transient state jvref2 vj � 200 rpm �

jvref2 vj � 200 rpm �

Fig. 14 Experimental result obtained during a sensorless controlwith load torque disturbance

a Measure speedb Suboptimal kalman filter speed estimationc Frequency speed estimation

13 13.5 14 14.5 15 15.5 16−400

−300

−200

−100

0

100

200

300

400measure and estimate speed (frequency estimator)

rpm

time (s)

ω

ωf

convergence

divergence

critical zone

Fig. 15 Open-loop tracking capability of the frequencyestimator

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0 2 4 6 8 10 12 14 16 18 20

−500

0

500

measure speed

rpm

0 2 4 6 8 10 12 14 16 18 20

−500

0

500

suboptimal Kalman filter speed estimation

rpm

0 2 4 6 8 10 12 14 16 18 20

−500

0

500

frequency speed estimation

rpm

time (s)

0 2 4 6 8 10 12 14 16 18 20−400

−200

0

200

400speed estimation error

rpm

8 8.5 9 9.5 10 10.5 11 11.5 12−20

−10

0

10

20zoom between 8 and 12 s

rpm

time (s)

a b

Fig. 16 Experimental result obtained during a sensorless control for a periodic speed reference change from 600 to 2600 rpm

5 10 15 20 25 30 35 40

0

200

400

600

measure and reference speed

rpm

5 10 15 20 25 30 35 40

0

200

400

600

speed estimation by suboptimal Kalman filter

rpm

5 10 15 20 25 30 35 40

0

200

400

600

speed estimation by frequency estimator

rpm

time (s)

a

0 5 10 15 20 25 30 35 40

−50

0

50

100

150

200

sledging error = reference speed − measure speed

rpm

time (s)

b

zero speed error

Fig. 17 Experimental result obtained during a sensorless control for a periodic speed reference change to 0–600 rpm

allows to make important changes of speed and allows zerotracking error at steady state (Fig. 16b).

Fig. 17a a shows a speed transient between 0 and600 rpm. Three important points can be seen in this figure.

1. The use of dependent parameter speed observer (here theSKF) leads to a bias estimation error (nearly 32 rpm atsteady state in this example) because parameters are notwell known and could fluctuate (see also Fig. 16b formore details).2. The use of frequency estimator leads to zero speed esti-mation error at steady state (see Fig. 17b between the instant5 and 10 s).3. On the other hand, high-transient behaviour can berealised with dependent parameter speed observer com-pared to frequency estimator.

Fig. 18 represents an experimental result obtained duringa sensorless control with an abrupt load torque disturbance.As we can see at the instant 3 s, the frequency estimatorcannot estimate the frequency of the harmonic at 16 times

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2 2.5 3 3.5 4 4.5 5 5.5

0

200

400

600

measure speed

rpm

2 2.5 3 3.5 4 4.5 5 5.5

0

200

400

600

suboptimal Kalman filter speed estimation

rpm

2 2.5 3 3.5 4 4.5 5 5.5

0

200

400

600

frequency speed estimation

rpm

time (s)

30 rpm statique error 0 rpm statique error

Fig. 18 Experimental result obtained during a sensorless controlwith an abrupt load torque disturbance

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the rotational frequency of the machine because of itsmedium tracking capability. In fact, this estimator is tunedso that the speed estimation is accurate in steady state(small variance), and so that the algorithm tracks theharmonic during low-transient operating modes. Thus, toreach high-transient behaviour and also zero speed estimationerror at steady state, we hold the plurality of the twoestimators.

5 Conclusion

The use of rotor slot harmonics for sensorless speed esti-mation is a difficult problem, notably because of thelocations of these harmonics towards power harmonics ofthe converter and the mechanical frequency oscillation.This approach is attractive because the speed estimator isnot parameter-dependent compared to standard techniquesbased on state space model. Experimental results showthat good tracking capabilities during transient and zerospeed error at steady state is ensured. This method is veryaccurate for some operating mode, so it is suitable forprecise feedback speed control. This allows us to concludethat our approach should be able to replace a mechanicalspeed sensor in a large variety of applications. Electricalpropulsion ship and traction railways could be an appli-cation field because of the slow-speed dynamique.Moreover, the frequency estimator could also be used incomplement of a low-cost mechanical sensor for fault-tolerant motor control to increase the reliability of thesystem. The impossibility to know the spin direction withthese methods can be pointed out. But this informationcan easily be found by a simple parameter-dependentspeed estimator. The computional cost of this speed estima-tor is reduced and allows to consider this algorithm as acomplementary speed estimator for a two spin directionhigh performance sensorless drive.

To reach high-transient behaviour and also zero speedestimation error at steady state, a commutator algorithm isproposed. This algorithm selects the best speed estimationaccording to the performance of the two speed estimatorsand the operating mode.

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