Sensorless Direct Torque Control of PMSM

using Unsected Kalman Filter ⋆

Zhen Chen ∗ Liang Wang ∗ Xiangdong Liu ∗

∗ School of Automation, Beijing Institute of Technology, Beijing100081, P.R.China

(e-mail: chenzhen76, nicholas1233, xdliu@ bit.edu.cn).

Abstract: This paper considers the sensorless direct torque control of a Permanent MagnetSynchronous Motor (PMSM) by using Unscented Kalman Filtering (UKF) to estimate theflux-linkage, the rotor speed and the position. UKF adapts the unscented transformation(UT) algorithm and symmetrical sampling strategy. The UKF algorithm represents a seriousstrongpoint to the Extended Kalman Filter (EKF) for the highly nonlinear systems. UKFensures high estimation precision and simple calculation. The sensorless Direct Torque Control(DTC) for PMSM is designed in this paper. The estimating errors when using UKF and EKFare compared. The effectiveness of this method is verified through several simulations. The DTCbased on UKF for PMSM has good speed and torque response performances. This method hasrapid astringency of estimated error.

Keywords: Direct Torque Control, PMSM, Unscented Kalman Filtering, Flux LinkageEstimation, Sensorless.

1. INTRODUCTION

PMSM is becoming a hot research topic in the field ofelectricity transmission, due to its various advantages suchas high power density, high efficiency, high reliability,and wide implementation in many occasions requiringhigh performance. As a high performance control scheme,DTC has been widely used in induction motor drives,also has witnessed many research results in PMSM. Inmany circumstances, both the rotor speed and positionuse encodes or sensors, whose performance is proportionalto the cost. Therefore, using low-performance sensors andencodes inevitably cause system noise and poor precision,while improving the cost of devices may make the relia-bility and robustness become worse. There are also harshenvironments where the position and velocity encoders cannot be simply installed. One typical example is the singlegimbal control moment gyro servo drive system, whichrequires high reliability and low speed stability. Utilizing ahigh-precision encoder will certainly increase the cost, anddecrease reliability for the harsh environments in space.Therefore, sensorless motor control has drawn attentionfrom many research scholars.

So far, the research of sensorless technology mainly concen-trates on estimating the position and velocity, specificallyfor an AC servo motor, on the estimation of the positionand speed estimation of the vector control system. Forthe DTC system, several scholars have proposed varioussensorless control strategies by using different estimationmethods (Lascu et al. (2004); Feng and Xu (2006); Zhanget al. (2007); Huang et al. (2006)). EKF was used widelyin these methods. Borsje et al. (2005) and Nanga-Ndjana

⋆ This paper is supported by National Natural Science Foundationof China (Grant No. 10872030) .

and Lautier (2006) developed EKF and UKF algorithmsfor the drives modes of PMSM-SVPWM. However, tothe knowledge of the authors, utilizing UKF to estimatePMSM-DTC flux hasnt been yet studied. Despite the wideapplications of EKF, there are some shortcomings of EKF,e.g, the high-order terms neglected while using Taylorexpansion to linearize a nonlinear function may lead toa fairly large estimation error, and sometimes cause thefilter unstable. Besides, the Jacobian matrix needs to becalculated at each control period, which becomes moredifficult as the dimension of the system is higher.

In order to tackle the disadvantages of using EKF to dealwith the nonlinear filtering problems, Julier et al. (1995)proposed UKF method in 1995 for the first time. Wanand VanDer-Merwe (2000) and VanDer-Merwe and Wan(2001) further studied UKF. Pan et al. (2005) pointed outthe following advantages of UKF over EKF:

(1) approximate the probability density function of non-linear distribution, instead of the nonlinear functionitself.

(2) the calculation accuracy of nonlinear distributionstatistics is up to at least 2 order. Even higher-orderaccuracy can be achieved if special sampling strate-gies are adapted, e.g., 4-order sampling of gaussiandistribution and sample skewness.

(3) it does not need to compute the Jacobian matrix,and moreover, it can handle non-additive noises anddiscrete systems, thereby extending the applicationarea.

(4) the calculation amount is of the same order withEKF.

(5) it avoids the particle recession problem, for it usesthe deterministic sampling approach, rather than therandom sampling of ions filters.

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This paper uses UKF to estimate the flux-linkage, rotorspeed and position, and further proposes a sensorless directtorque control system of PMSM. Therefore, the study ofthis paper has significant potential practice value.

2. THE PRINCIPLE OF UKF ESTIMATOR

As UKF approximates the probability distribution of anonlinear function rather than approximates the linearfunction itself, it uses a deterministic sampling approach tocapture the mean and covariance estimates with a minimalset of sample points. In order to avoid the linearizationerror of EKF, UKF uses a more accurate approximationof UT, which, based on the prior state of input, usesdeterministic sampling approach to capture the meanand covariance, and to calculate the point set of Sigma,then makes of nonlinear transformation of each Sigmapoint. Julier et al. (2000) discussed the involved samplingstrategies. This paper adapts the symmetrical samplingstrategy, taking into consideration the calculation amountand the realization problem.

Consider the following non-linear system:

Xk+1 = Fk(Xk, Uk) + wk

yk+1 = Hk(Xk+1) + vk(1)

where Xk is the L-dimensional state vector of the system,Fk(Xk, Uk) is the state propagation equation, Uk is thecontrol input vector, wk is the process noise, which isassumed to be the zero-mean Gaussian white noise withQk as its variance (denoted by wk ∼ N(0, Qk)), yk is themeasurement value, Hk(Xk) is the measurement equation,and vk is the measurement noise, which also satisfiesvk ∼ N(0, Rk). Suppose that X0 and P0 are, respectively,the initial value and variance of the system’s state vector.

The UKF algorithm mainly consists of three steps:

(1) the sampling and calculation of Sigma points.(2) time update (the prediction step).(3) measurement update (estimation correction ).

Specifically,

Step 1 Initialization, computing Sigma points andchoosing weights

The initial states and the covariance matrices are asfollows:

X0 = E [X0]

P0 = chol

E(

X0 − X0

) (

X0 − X0

)T

(2)

According to the optimal state estimate Xk−1 of time k−1,we can obtain the 2L+1 sets of Sigma points as xi,k|k−1,by using the symmetrical sampling approach. the set ofSigma points are computed as follows:

χ0,k|k−1 = Xk−1

χi,k|k−1 = Xk−1 + (√

(L+ λ)Pk−1)i

χi+L,k|k−1 = Xk−1 − (√

(L+ λ)Pk−1)i

(3)

with i = 1, 2, · · · , L.

The coefficients of the mean and covariance are weightedas:

W(m)0 =

λ

L+ λ

W(c)0 =

λ

L+ λ+ 1 − α2 + β

W(m)i = W

(c)i =

1

2(L+ λ)

(4)

where the superscript (m) and (c) denote, respectively, theweight for the mean and covariance, λ is the proportionalparameter, used to adjust the distance between Sigmapoints and X, which is taken as λ = α2(L + κ) − L,the constant α determines the spread of Sigma points,usually is chosen to be a small positive value (e.g., 10−4 ≤α ≤ 1), κ is a constant, usually being either 0 or 3 − L,β is a parameter related to the prior knowledge of thedistribution of state (β = 2 is the optimal, when the state

is the Gaussian distribution), (√

(L+ λ)Pk)i is the i−throw or column of the square root of (L+ λ)Pk, and Wi isthe weight associated with the i−th Sigma point.

Based on the above discussions, (3) and (4) can be rewrit-ten as:

χ0,k|k−1 = Xk−1

χi,k|k−1 = Xk−1 + (α√

(L+ κ)Pk−1)i

χi+L,k|k−1 = Xk−1 − (α√

(L+ κ)Pk−1)i

(5)

and

W(m)0 = 1 −

L

α2 (L+ κ)

W(c)0 = 1 −

L

α2 (L+ κ)+ 1 − α2 + β

W(m)i = W

(c)i =

1

2α2(L+ κ)

(6)

Step 2 Time update (the prediction step)

The L×(2L+1) matrix χk−1 of Sigma points is constitutedby using the 2L+1 sets of Sigma points χi,k|k−1. In lightof the first (1), we can get prediction matrix χk|k−1 bydoing a non-linear transformation for each sample pointas:

χi,k|k−1 = Fd (χi,k−1, Uk−1) , i = 0, · · · , (2L+ 1) (7)

Based on the weights, the mean and covariance matrix ofthe state are predicted as follows:

Xk|k−1 =

2L∑

i=0

W(m)i χi,k|k−1 (8)

Pk|k−1 =2L∑

i=0

W(c)i

[

χi,k|k−1 − Xk|k−1

] [

χi,k|k−1 − Xk|k−1

]T

+Q

(9)

According to the second equation of (1), the value ofoutput measurement can be obtained by calculating eachof prediction points χi,k|k−1:

yi,k|k−1 = Hk

(

χi,k|k−1

)

, i = 0, · · · , (2L+ 1) (10)

According to the weight of the mean coefficient, the outputmeasurement update is:

yk|k−1 =

2L∑

i=0

W(m)i yi,k|k−1 (11)

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Step 3 Measurement update

Compute the error variance matrix of the update measure-ment as:Pykyk

=2L∑

i=0

W(c)i

[

yi,k|k−1 − yk|k−1

] [

yi,k|k−1 − yk|k−1

]T+R

(12)

Then, calculate the error variance matrix between pre-dicted states and update measurement:

PXkyk=

2L∑

i=0

W(c)i

[

χi,k|k−1 − Xk|k−1

]

[

yi,k|k−1 − yk|k−1

]T (13)

Further, compute the Kalman gain:

Kk = PXkykP−1

ykyk(14)

Finally, compute the optimal estimation of the state vectorand the covariance matrix of time k:

Xk = Xk|k−1 +Kk

(

yk − yk|k−1

)

(15)

Pk = Pk|k−1 −KkPykykK−1

k (16)

The detailed realization of UKF will be discussed, by com-bining with the state estimation of PMSM-DTC system.

3. THE MODEL OF PMSM-DTC

In PMSM-DTC system, the amplitude of flux linkage andelectromagnetic torque are primarily to be controlled. ThePMSM voltage and flux linkage equations in the two-axisstationary αβ reference frame are:

usα =dψsα

dt+Rsisα

usβ =dψsβ

dt+Rsisβ

(17)

ψsα = Lsisα + ψf cos θr

ψsβ = Lsisβ + ψf sin θr(18)

where usα, usβ , isα, isβ are the voltages and currents onthe α and β axis, respectively, ψsα, ψsβ are the statorflux linkage on the α and β, ψf is the magnet fluxlinkage, Ls, Rs are the stator inductance and resistance,respectively, and θr is the relative position between statorflux and rotor flux, which is called the rotor electricalangle.

Substituting (18) into (17) gives:

dψsα

dt= −

Rs

Ls

ψsα +Rs

Ls

ψf cos θr + usα

dψsβ

dt= −

Rs

Ls

ψsβ +Rs

Ls

ψf sin θr + usβ

(19)

where Ls is the stator inductance of PMSM, Rs is thestator resistance.

In order to design the UKF observer, the angular velocityωe is regarded as constant within the sampling period Ts,and ωe is the changing rate of angle θr, i.e.,

dωe

dt= 0,

dθr

dt= ωe

The UKF estimator is independent of the motor mechan-ical parameters, when using dωe/dt = 0 as the dynamic

equation of rotor speed, thereby overcoming the impact ofthe uncertainties of the mechanical parameters on estima-tor. Here, the mechanical parameters include load torquedisturbance, viscous friction coefficient, rotor inertia, andso forth.

Then, we get the nonlinear mathematical model of PMSMas:

X = f(X) +BUy = h (X)

(20)

where X = [ ψsα ψsβ ωe θe ]T

is the state variable, U =

[ usα usβ ]T

is the control input, and y = [ isα isβ ]T

isthe output vector. In the PMSM-DTC system, currentand voltage sensors measure the three-phase current andvoltage, so the stator current isα, isβ and voltage usα, usβ

can be obtained through transformation between three-axis stationary ABC coordinate and two-axis stationaryαβ coordinate. The system state matrices are defined as:

f (X) =

−Rs

Ls

ψsα +Rs

Ls

ψf cos θe

−Rs

Ls

ψsβ +Rs

Ls

ψf sin θe

0ωe

, B =

1 00 10 00 0

h (X) =

1

Ls

(ψsα − ψf cos θe)

1

Ls

(ψsβ − ψf sin θe)

Equation (20) can be is expressed in the discrete dynamicequation form of UKF estimator.

Xk+1 = Fk(Xk) +BkUk + wk

yk = Hk(Xk) + vk(21)

where

Fk(Xk) = Xk + f(Xk)Ts

Bk = BTs

Hk(Xk) = h(Xk)

wk and vk are the noise and measurement noise, respec-tively, which are independent zero-mean Gaussian process,Ts is the sampling time, and the expressions of Fk(Xk) andHk(Xk) are as follows:

Fk(Xk) =

(

1 −Rs

Ls

Ts

)

ψsα +Rs

Ls

ψfTs cos θr

(

1 −Rs

Ls

Ts

)

ψsβ −Rs

Ls

ψfTs sin θr

ωe

θr + ωeTs

X=Xk

(22)

Hk (Xk) = h (Xk) =

1

Ls

(ψsα − ψf cos θe)

1

Ls

(ψsβ − ψf sin θe)

X=Xk

(23)

4. PMSM-DTC SYSTEM BASED ON UKF

The ectromagnetic torque Te of PMSM is:

Te =3Pm

2Ls

ψf (ψsβ cos θr − ψsα sin θr) (24)

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The calculation formula for the square of stator fluxamplitude is:

|ψs|2

= (ψsα)2+(ψsβ)

2(25)

According to the UKF algorithm and the dynamic modelof PMSM-DTC, the flux-linkage, rotor speed and positionof PMSM-DTC system are estimated. Then according to(24) and (25), PMSM-DTC of torque and flux-linkageamplitude closed-loop system are established through cal-culating the feedbacks of DTC. The block diagram of thePMSM-DTC based on UKF is shown as Fig.1.

Fig. 1. Diagram of the PMSM DTC based on UKF

The direct torque control strategy is that the deviationbetween reference speed and the estimated speed of UKFthrough speed controller, then the electromagnetic torqueT ∗

e is obtained, and then ∆Te is known by subtractingthe estimated electromagnetic torque Te of UKF, ∆Ψs isobtained through reference flux-linkage and the estimatedflux-linkage. The torque deviation and the amplitude de-viation of flux-linkage through the controller, and twooutputs t and f are derived, through combining with theinteraction of estimated rotor position of UKF to select theappropriate voltage vector, and finally PMSM is derivedthrough the voltage source inverter.

5. SIMULATION AND EXPERIMENT RESULTS

According to the PMSM-DTC system and UKF algo-rithm, one will verify the state estimation performance ofUKF. Here, the parameters of PMSM are given in table.1.System sampling period is 50µs and the initial speed is1500r/min. Then the speed is changed to 700r/min at 0.1s,and further changed to 1000r/min at 0.2s. The experimen-tal platform is shown in Fig.2. and the simulation resultsare displayed in Fig.3–Fig.9.

Table 1. The parameters of PMSM

Parameter name Parameter value

Stator resistance (Rs) 0.93ΩRotor inertia (J) 5× 10−4Kg.m2

Stator inductance (Ls) 0.27 mHPermanent magnet flux (Ψf ) 0.01 Wb

Pole pairs (Pm) 4Bus voltage (Udc) 15V

The estimated speed of UKF and the actual speed arecompared in Fig.3. Fig.4 is the error curve between theestimated speed of UKF and the actual speed, which showshigh tracking accuracy. The relative average error is about1.93% when the speed is changing, then it turns to beabout 0.13% in the steady state. Fig.5 shows the rotor

position estimation of UKF. Fig.6 shows the error curveof estimated position, from which can easily be observedthat the position estimation error is slightly larger whenthe speed is changing state than it is steady. Fig.7 showsthe estimated stator flux-linkage of UKF. Fig.8 showsthe error curve of the estimated stator flux-linkage, inwhich ∆Ψsα,∆Ψsβ are, respectively, the error curve of theestimated stator flux-linkage on the α and β axis. Whenadding 0.05N.m load disturbance at 0.25s, the estimatedelectromagnetic torque curve is shown as Fig.9, whichshows that PMSM-DTC has fast torque response, andthe disturbance is reflected only when the electromagnetictorque and the stator current change. It can be seenfrom Fig.3 that the speed adjusts to the given valuerapidly at 0.25s, which improves the smooth of speed andthe disturbance rejection capability for the direct torquecontrol of servo system.

Fig. 2. Experimental platform of PMSM-DTC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−200

0

200

400

600

800

1000

1200

1400

1600

Sp

ee

d(r

/min

)

Time (sec)

Estimated speed Encoder speed

Fig. 3. Estimated speed by UKF

As analyzed in Julier et al. (2000), when EKF is used todeal with the filtering problem of nonlinear systems, thenonlinear function is linearized by Taylor expansion whichneglects the higher-order terms, thus it has only first-orderaccuracy. By contrast, UKF approximates the probabilitydensity function of the nonlinear function, rather than thenonlinear function itself. As pointed out in Julier et al.(2000), the calculation accuracy of nonlinear distribution

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−30

−20

−10

0

10

20

30

Sp

ee

d e

rro

r(r/

min

)

Time (sec)

Fig. 4. Estimated speed error by UKF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

150

200

250

300

350

400

An

gle

po

sitio

n(d

eg

)

Time (sec)

Estmated angle by UKF Encoder angle

Fig. 5. Estimated position by UKF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

An

gle

err

or(

de

g)

Time (sec)

Fig. 6. Estimated position error by UKF

statistics is at least 2-order. This paper adopts symmetricsampling for the Sigma points to ensure the approximationaccuracy of any distribution reaches 2-order of Taylorexpansion, therefore, UKF gives more accurate estimationthan EKF. The experimental results are shown in Fig.10–Fig.12.

Fig.10 compares the filter prediction error when usingUKF and EKF to estimate the speed. The estimation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Estim

ate

d s

tato

r F

lux(W

b)

Time (sec)

ψsα

ψsβ

Fig. 7. Estimated flux linkage of stator by UKF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

Estim

ate

d F

lux e

rro

r(W

b)

Time (sec)

∆ψsα

∆ψsβ

Fig. 8. Estimated flux linkage error of stator by UKF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

To

rqu

e (

N.m

)

Time (sec)

Fig. 9. Torque curve adding load disturbance at 0.2 s

accuracy of UKF is higher than that of EKF. The relativemean of the EKF estimated speed error is about 3.27%,while that of UKF is 1.93%. In the steady state, theestimation of UKF is more accurate with small fluctuationsin speed, especially when the sudden change of speedcaused by the load disturbance at 0.25s, the estimationaccuracy of UKF is 1.9 times than that of EKF, so UKFhas both good dynamic and static performances. Fig.11shows the error curves when using respectively UKF and

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−80

−60

−40

−20

0

20

40

60

80

Estim

ate

d s

pe

ed

err

or(

r/m

in)

Time (sec)

Estimated by UKF Estimated by EKF

Fig. 10. Error curves of estimated speed by UKF and EKF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Estim

ate

d a

ng

le e

rro

r(d

eg

)

Time (sec)

Estimated by UKF Estimated by EKF

Fig. 11. Prediction error of estimated position by UKF andEKF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−3

−2

−1

0

1

2

3x 10

−4

Estim

ate

d F

lux e

rro

r(W

b)

Time (sec)

Estimated by UKF Estimated by EKF

Fig. 12. Error curves of estimated flux-linkage by UKF andEKF

EKF to estimate the position and Fig.12 depicts the errorcurves when using respectively UKF and EKF to estimatethe flux linkage.

6. CONCLUSIONS

This paper presents a sensorless direct torque control ofa PMSM using UKF for flux-linkage, rotor speed andposition estimation. Comparison with EKF is done. Asseen from the experiment and simulation results, theeffectiveness of UKF for the state estimation of PMSM-DTC is verified. Meanwhile, it shows that the PMSM-DTC system based on UKF has a quick dynamic responseand a satisfactory steady-state performance. Therefore,the outstanding performance of UKF in sensorless controlis illustrated.

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