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The comparison of Sensorless Estimation

Techniques for PMSM between Extended

Kalman Filter and Flux-linkage Observer

M.C. Huang1 and A.J.Moses1 and F. Anayi1 1 Cardiff University/School of Engineering, Cardiff, UK

Abstract— This paper compares two main sensorless control

techniques for permanent magnet synchronous motors

(PMSM): Flux-linkage observer (FLO) and Extended

Kalman Filter(EKF). This paper investigates experimentally

five aspects via FLO and EKF: the error of estimated

position, speed response, variable speed tuning , the ability

of speed zero crossing and convergence in the estimation for

position transition in four-quadrants.

I.

II.

I NTRODUCTION High performance industrial servo drives mostly employ permanent magnet synchronous motors with position

sensors such as optical encoders or revolvers for feedback.

Although sensorless technique has been developing for 15

years, it is still difficult for them to be applied in drives because of some inherent drawbacks: initial position

judgment at start-up, operation in four-quadrant region,

etc. Flux linkage observer (FLO) [1, 2] is a sensorless

estimation for permanent magnet synchronous motors(PMSM) based on the integration of back EMF, with a

simple start-up method, an explicit and fast computing

scheme, and reliable precision of position. However,

initially the voltage vector must be applied to terminals of

the motor winding [3, 4], and FLO is only used onsurface-mounted permanent magnet synchronous motors

(SMPMSM), not on salient-pole permanent magnet

motors such as interior PM motors. However, surface-mounted PMSMs are mostly used in AC servo drives;

therefore FLO is still the low cost and reliable scheme for

motor drive manufactures. The Extended Kalman Filter

(EKF) [5, 6] is another stochastic state observation basedon least–square variance estimation. Due to automatic on-

line adjustment for least–square variance from system

state variables, EKF has strong auto-search capability of

the initial position [7-8], which can correct the wrongconvergence in the start-up. The initial voltage vector

needs not be applied to the motor. As EKF can directly

estimate the angular velocity, sensorless operation based

on EKF for PMSM certainly gives a more rapid speedresponse than any other sensorless methods based on

incremental calculations. However, EKF requires heavy

on-line matrix computing, it is huge challenge for most

motor drive manufacturers which are using fix-point processors. Besides, although EKF can actively search for

initial position, it inevitably has slower response than

FLO. In addition, incorrect convergences occasionally

happens when sensorless control based on EKF for PMmotor start up in the anticlockwise direction or operation

in four-quadrants. This paper compares the two classicsensorless methods in start-up, speed response and zero

crossing in four-quadrants.

SENSORLESS CONTROL BASED O N FLUX LINKAGE

OBSERVER

The flux linkage observer is based on the phasor diagram

of a PM synchronous motor shown in figure 1, where Uand I are voltage and current vector, ψs is stator flux-

linkage, ψf is the excitation flux linkage by PM fixed onthe d-axis, L and R are phase inductance and resistance

respectively.

Figure 1 The phasor diagram for PMSM [9]

The angle between d-axis and α-axis is the estimated

rotor oriented angle. The first step is to obtain ψ froms

)0(0)( s

t

s dt I RU ψ ψ &&&& +⋅−= ∫ (1)

The investigation of stator flux linkage needs to be

executed in a stationary coordinate frame α-β while the

rotor flux linkage is certainly studied in d-q coordinates.

The angle between d-axis and α-axis is the estimated rotor

position angle. Obviously, position estimation relied on

ψf α, ψf β. Those components of ψf are calculated by the

integration of the respective EMFs, which are

correspondingly obtained by subtracting the voltage dropson the winding resistance from the voltages Uα, Uβ

computed and transformed from the DC link terminal

voltage based on space vector PWM theory.

ψs(0) represents the initial value for the synthesized flux

linkage, because at the initial instant the rotor is aligned

by applying one voltage vector on the stator. ψs(0) is

considered equal to the initial stator flux linkage vector

since at that moment the current in the stator winding is

zero. The pure integration from (1) will accumulate any dc

6540-7803-9547-6/06/$20.00 ©2006 IEEE.



offset and error from the measured current so as ψs enters

the saturation region. Thus, the result cannot keep the

deduced ψ stable. A high-pass filter [10] whose transfer function is s/(s+ω0), can be inserted at the front of the

integrated quantities. Since the integration can be

represented in the form of 1/s as a “Laplace”

transformation, the eventual transfer function is 1/(s+ω0)

which is actually equivalent to a low pass filter. Thereby,the corresponding integration equation is amended to (2):

f

)0(00

)]([ s s

t

s dt I RU ψ ψ ω ψ &&&&& +⋅−+−= ∫ (2)

where ω0 is cutoff frequency.

The second step is to calculate ψf . From figure 1, ψf is

obtained by subtracting the voltage drop in phase

inductance from ψs. If the difference between inductance

Lα and Lβ can be ignored and Lα= Lβ= Ls, the equation for

ψf can be simplified to:

I L s s f &&& −=ψ ψ (3)

Therefore, in the α-β coordinate, (3) can be used to

calculate ψf α, ψf β components of ψf , as shown in figure 1.

Finally, the rotor positionr

θ can be computed by equation

(4) as:

)arctan(α

β

ψ

ψ θ

f

f

r = (4)

The final deduced experimental components of fluxlinkage are displayed in figure 2. The average flux linkage

is 0.081Wb. It illustrates in figure 1that the locus of flux

linkage of PM body is almost circular as it should be

theoretically.

Flux linkage in the stat ionary co-ordination

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0 0.02 0.04 0.06 0.08 0.1

Time(s)

F l u x l i n k a g

e ( W b )

Flux linkage of alpha axis flux linkage of beta linkage

(a)

Flux linkage cycle by PM body

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

-0.10 -0.05 0.00 0.05 0.10

flux linkage(Wb)

f l u x l i n k a g e ( w b )

Flux linkage cycle by PM body

(b)

Figure 2 Components ψf α, ψf β (a) and the locus of vector ψf (b)

III. SENSORLESS CONTROL BASED O N EXTENDED K ALMAN

FILTER

The EKF is an optimal estimator in the least squarevariance for estimating the states of dynamic nonlinear

systems. In order to straightforwardly apply the EKF to

sensorless operation for PMSM, the PM motor nonlinear

state equations are written as (5):

⎩⎨⎧

+=

++=

)())(()(

)()()]([)(

t t xht y

t t Bvt x f t x

k k μ

σ &

(5)

where σ(t) is the system noise, μ(t) is the measured noise ,

both σ(t) and μ(t) have a zero-mean white Gaussiandistribution; Q(t) is the covariance of σ(t); R(tk ) is the

covariance of μ(t). PMSM has been modeled in a two-axis stationary reference frame (α, β). The system state

variable ][ ′θ ω β α ii x , Input vector v isis

][ ′ β α vv and the system output variable is [ y ]′ β α ii .

The system matrices can be written as:

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

+−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

β

α

β

α

ω

θ ωψ

θ ωψ

i

i xh

L

L

B L L

i R

L L

i R

f

f

f

f

x f s

s

s s

s

s s

s

)(

,

00

00

10

01

,

0

cos

sin

)(

4

3

2

1

where R S is the motor phase resistance, Ls is motor

synchronous inductance, ψ is the PM flux linkage.Obviously, this system is nonlinear and the variables in

the system state are closely coupled. If EKF is used to

rewrite this nonlinear model, the “Jacobian” matrices are :

)6(0010

0001))((

0100

0000

sincos0

cossin0

))((

)(

)(

⎥⎦

⎤⎢⎣

⎡=

∂

∂=

⎥

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−

=∂

∂=

=

=

t x x

s s s

s

s s s

s

t x x

x

ht x H

L L L

R

L L L

R

x

f t x F θ

ωψ θ

ψ

θ ωψ

θ ψ

After relative linearization of the system equation (5),there is a partial linear relationship between states and

outputs as given in equation (7) and (8), which can be

discretized into equation (9) with constant period TC.

(7))()(ˆ))(()(ˆ t t xt x F t x σ +=&

)()(ˆ)( t t x H t y μ += (8)

))()(ˆ))(,,()(ˆ 1111 −−−− +Φ= k k k k k k t t xt xt t t x υ (9)

Ф is the state transfer matrix for the linear system (7) and

it is an exponential matrix. It approximately can be

simplified in [5] to be given by:ck k k FT I t xt t +≅Φ −− ))(,,( 11 (10)

For a given sampling time tk , both the optimal state

estimated sequencek k

x~ and its covariance matrixk k

P ~

are

generated by EKF through two steps: The first one is a prediction step, performing a prediction of both quantities

based on the previous estimates11

~−− k k

x and mean voltage

vector vk-1 actually applied to the system in the periodfrom tk-1 to tk .

655



A simple rectangular integration technique is used to

produce the following recursive difference equations:

)11()

~~(

~ˆ

)]()~([~ˆ

111111111

11111

⎪⎩

⎪⎨⎧

+′++=

++=

−−−−−−−−−

−−−−−

d ck k k k k k k k k k

ck k k k k k k

QT F P P F P P

T B x f x x υ

where Fk-1 is computed for x=11

~−− k k

x .

The second step is an innovation step, correcting the

predicted state estimate and its covariance matrix througha feedback correction scheme that makes use of the actual

measured quantities; it is realized by the following

equations .

1

)ˆ(ˆ

)12(ˆˆ~)ˆ(ˆ~

11

11

11

−

+′′=

⎪⎩

⎪⎨⎧

−=

−+=

−−

−−

−−

R H P H H P K

P H K P P

x H y K x x

k k k k k

k k k k k k k

k k k k k k k k

IV. EXPERIMENTAL SETUP

Figure 3 DSP-control PMSM experimental rig

Sensorless algorithms of EKF and RLKF are implemented

in the experimental set-up including : PMSM ACM2n320-

4/2-3, TMS320C31-50 DSP DSK , MOSFET inverter andDC shunt motor as a load generator. PMSM ACM2n320-

4/2-3 has the specification given in table 1:

TABLE I. PMSM SPECIFICATION

Rated power 1.4kW

Rated DC-link voltage 350V

Pole pair number 3

Phase resistance 1Ω

Phase inductance 5.5mH

Rated speed 4000rpm

The controller chosen is TMS320C31DSK which embeds

DSP TMS320C31-50 on board; The peripheral out-data/acquisition cards are respectively based on 12- bit

Analogue-to-Digital converter AD678 and Digital-to-

Analogue converter DA767, the system sub-rack frameobeys the Euro-card standard. The operation frequency of DSP is 50MHz. The position sensor is an optical encoder

which has a precision of 2048 pulse per turn.

V. EXPERIMENTAL R ESULT

A. Estimation Error Comparision:

The average error for rotor position by FLO is 8.09º/cycle

shown in figure 4. The error analysis shows that the angleerror via FLO is less than 10° in absolute position angle [-

180°, 10°] while the error via FLO falls into range [10°,

20°] in the absolute position angle [10°, 180°]. Inevitably,the maximum error in the sensorless controlled PMSM

always occurs at the start and end of the cycle when the

error can be up to ±360°. This type of edge error

occasionally occupies about 10° of a cycle for FLO.

position estimation via flux linkage observer (FLO)

-360.0

-310.0

-260.0

-210.0

-160.0

-110.0

-60.0

-10.0

40.0

90.0

140.0

190.0

0 0.005 0.01 0.015 0.02

time(s)

p o s i t i o n ( d e g r e e

real position es timated pos ition error

position estimated error in flux linkage observer (FLO)

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

0.00 0.01 0.01 0.02 0.02

time(s)

p o s i t i o n ( d e g r e e )

Real position Estimated position error

Figure 4 Estimated position (top) and error (bottom) via FLO

Figure 5 illustrates the position estimation via EKF anderrors. P40 and Q40 represent the initial value of state

covariance P0(4) and noise covariance Q0(4), which

determines the convergence precision for EKF. Explicitly

the estimation error via EKF is distributed quite even in

amplitude other than that of FLO. The average error is

about 12.4°/ cycle. The error analysis denotes that mosterrors fall in the range [10°, 20°], a minority of errors

occur beyond the average value. The edge error shift

where maximum error occurs is about 13°.

Position estimation via EKF P40=2E-12,Q40=-12

-400.0

-300.0

-200.0

-100.0

0.0

100.0

200.0

0.000 0.005 0.010 0.015 0.020 0.025 0.030

time(s)


rea l pos it ion est ima ted pos it ion error

(a)

656



estimated pos ition error via EKF P40=2E-12,Q40=5E-12

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

0.002 0.007 0.012 0.017 0.022

time(s)


real pos ition es timated position error

(b)

Figure 5 Estimated position (a) and error (b) via EKF

B. Speed Response to unit step instruction:

Because FLO estimates angular velocity through

incremental computation, obviously EKF implies a faster

speed response than FLO as shown in figure 6. The reason

is apparent that EKF can directly estimate speed throughthird state variable in equation (5) and (7). Figure 6

especially verified that the speed estimated by EKF cantake longer than that of FLO to reach steady-state. Figure

6 illustrates the overshoot for FLO would last longer thanthat of EKF.

Speed Response to unit step via Flux-linkage

observer through estimated position incrementation

-100.0

100.0

300.0

500.0

700.0

900.0

1100.0

1300.0

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

time(s)

s p e e d ( r p m )

Speed Command

estimated speed via FLO

Real Speed via encoder

(a)

Speed Res ponse to Unit Step Command via EKF

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

3.15 4.15 5.15 6.15 7.15 8.15

time(s)

s p e e d ( r p m )

S p ee d Co mma n d S pe e d Re s po ns e fro m EKF

(b)

Figure 6 Speed response via FLO(top) and EKF(bottom)

C. Unilateral Speed adjustment ability comparison :

speed response via Flux-linkage observer through

estimated position incrementation

0.0

300.0

600.0

900.0

1200.0

1500.0

0 5 10 15 20

time(s)

s p e e d ( r p m )

Sp eed command es timated s peed via F LO real s peed v ia encoder

(a)

Speed Res ponse to Unilater al command via EKF

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

0.00 5.00 10.00 15.00 20.00

Time(s)

S p e e d ( r p m

estimated spee d

unilateral speed command

real spee d

(b)Figure 7 Dynamic response to unilateral command

As shown in figure 7 , the unilateral speed command is asquare wave whose top instruction is 1100rpm and bottom

instruction is 550rpm, and load torque is 0.9 N.m.

Experimental results indicate that FLO and EKF both can

be applied in unilateral velocity adjustment. Figure 7 (a)

denotes that there are serious ripples and offsets levels inthe estimated speed through increment via FLO while

speed response via EKF explicitly tracks the real speed

locus without much high frequency ripples or offset as

shown in figure 7 (b).

D.Speed cross-zero ability comparision:

Speed response via flux-linkage observer in 4

quadrant operation

-1500.0

-1000.0

-500.0

0.0

500.0

1000.0

1500.0

0 5 10 15 20

time(s)

s p e e d ( r p

m )

Speed Command Speed Response

(a)

657



Figure 9 indicates that FLO and EKF both have the

capability of tracking instantaneous position transition

caused by operation in four-quadrant. However, FLOwould produce fugacious oscillation when speed

switching in 4-quadrant occurs in the angular positionwhich is not only the peak value (+π) in the clockwise

direction but also the same value (-π) in the anticlockwise

direction. When speed via FLO passes zero-speed, theinitial error caused via FLO is relatively larger than thatvia EKF. This also proves that EKF has a very strong self

start-up capability. The phase shift caused via FLO is still

obviously smaller than that via EKF regardless of the error

analysis mentioned above.

Speed Response in 4 QUADRA EKF Operation

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

1500.00

16 21 26 31

time(s)

S p e e d

( r p m )

spe ed c omma nd S pee d Respo nse via EK F

(b) Figure 10 shows that FLO and EKF both track real

position transition at the negative peak value – π where

PMSM passes zero-speed from clockwise to anticlockwisedirection. Fugacious oscillation happened in figure 9 (a)

also can be seen in figure 10 (a). EKF causes phase lag for

positive jump to π while FLO causes no phase lag.However, FLO takes a little longer time to get close to

real track while EKF has no lag to converge into real track

promptly.

Figure 8 Speed response in four-quadrant region by FLO(a) and EKF(b)

The modern servo industry requires actuators such as

PMSM to acquire the capability of speed zero crossing,which allow actuators to switch among 4 quadrants.

Actuators should go through zero velocity promptly

without any delay from the switch instructs from

microprocessor. Figure 8 compares the characteristics of

the velocity zero-crossing ability of FLO and EKF.Bidirectional speed command would lead PMSM to pass

zero from +1000rpm to -1000rpm or from -1000rpm to

+1000rpm.Figure 8 (b) shows that estimated speed via EKF smoothly

crosses zero-speed and quickly reaches state-steady

command without any overshooting, or severe ripples,

even minor offset while estimated speed via FLO passeszero with high ripple and finally enter the state-steady

situation with a large offset.

E. Estimated Position transition for cross zero-speed:

Estimated position transition for rising speed

cross-zero via FLO

-250.0

-200.0

-150.0

-100.0

-50.0

0.0

50.0

100.0

150.0

200.0

250.0

0.125 0.175 0.225 0.275 0.325 0.375

time(s)

P o s i t i o n ( d e g r e e )

Real Estimated

(a)

Estimated position transition for rising speed cross-zero

via EKF

-250.0

-200.0

-150.0

-100.0

-50.0

0.0

50.0

100.0

150.0

200.0

250.0

0.7 0.9 1.1 1.3 1.5

Time(s)


Real Estimated

(b)

Figure 9 Estimated position transition for rising speed cross-zero

Estimated position transition for falling

spee d cross-zero via FLO

-250.0

-200.0

-150.0

-100.0

-50.0

0.0

50.0

100.0

150.0

200.0

250.0

0 .03 0 .08 0.13 0 .18 0.23

Time(s)


R ea l P o sit io n E st ima te d P o sit io n via FLO

(a)

Estimated position transition for falling speed cross-

zero via EKF

-250.0

-200.0

-150.0

-100.0

-50.0

0.0

50.0

100.0

150.0

200.0

250.0

0.3 0.35 0.4 0.45 0.5 0.55 0.6

Time(s)


Real Estimated

(b)

Figure 10 Estimated Position Transition for the falling speed cross-zero

VI. CONCLUSION

Sensorless control based on FLO and EKF both can be

applied in AC servo drives, EKF needs to solve parameter

tuning so as to match the requirement in the four-quadrant. EKF has a faster speed response than FLO. FLO

implemented by cheap floating point processors is also an

658



alternative method for time –tight development tasks. In

four quadrant regions, position error from FLO obviously

is smaller than EKF. Generally, the estimated position precision via FLO is a little higher than that of EKF.

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