Comparison of a Linear and a Non-linear StateObserver for Sensorless Control of PM Machines

F. Demmelmayr, M. Troyer, M. SchroedlInstitute of Energy Systems and Electrical Drives

Vienna University of Technology, Vienna, Austriaflorian.demmelmayr@tuwien.ac.at, http://www.ieam.tuwien.ac.at/

Abstract—This paper presents two different state observersfor position-sensorless control of permanent magnet synchronousmachines (PMSMs). Both methods improve the angular rotorposition and determine the machine speed of a back electromotiveforce (back-EMF) model [1].The Arctan observer calculates the electrical rotor position fromreal and imaginary flux-linkage components with a commonarctangent function. A novel observer with non-linear feedbackcoefficients, called Sin/Cos observer, omits this calculation andimproves the quality of the position information. Dynamic mea-surements on an outer rotor traction machine [2] verify thecharacteristics of the presented observer techniques.

I. INTRODUCTION

The actual angular rotor position is required for field-oriented control of permanent magnet synchronous machines(PMSMs). Special sensors, such as encoders or resolvers candetect the rotor angle. But these sensors have many drawbacks.Especially for traction machines, space limitations or harshenvironmental conditions could make their usage impossible.Furthermore, high distances between a machine and powerelectronics complicate the position detection directed at themachine. Additionally, sensitive electronic devices on themachine decrease the reliability of the whole drive [3].

Sensorless techniques can avoid these weaknesses. Theycalculate the position information directed at the power elec-tronics. The PMSM itself is used as the position sensor.

The machine speed influences the properties of sensorlesscontrol. At medium and high speed, back electromotive-force(EMF) methods are state-of-the-art [1]. They identify the rotorposition with measurements of the induced voltage of therotating machine.

Signal noise and measurement uncertainties impair thequality of angular-position calculation. Machine speed canbe received by differentiation of these faulty position values.More reliable approaches use mechanical observers to improvethe position quality and estimate the angular velocity [4], [5].

Common back-EMF models are based on two steps. First,the real and imaginary flux-linkage components due to thepermanent magnets are calculated. Second, the rotor positionresults from a division and the arctangent function. The firstpart of this paper introduces a state-space observer based onthese calculations called Arctan observer (Section II).

A disadvantage of this angular calculation is the divisionof the two parts of the flux-linkage space-phasor. Especially

for small real parts, measurement uncertainties and noisecause significant errors. Additionally, divisions and arctangentfunctions require long computing time. The Sin/Cos observer(Section III) represents a useful alternative without thesedrawbacks. It uses non-linear feedback factors to estimate rotorspeed and actual angular position.

Measured load steps and speed behavior prove the compa-rable performance of the Arctan and the Sin/Cos observer. Thestandard deviation of the difference between encoder angle andsensorless observer displays the high quality from low speedup to high speed values.

All measurements are implemented on an outer rotor trac-tion drive presented in [3]. This machine is a prototype for anelectric tram (Figure 1, data in Table I). A standard tractioninverter operates the whole system.

Figure 1. Picture of the used PMSM [3]

reference current (continuous current) 162Armscontinuous output power 43.8kWcontinuous torque 1945Nmreference speed 545rpm

Table IDATA OF THE USED PMSM

II. ARCTAN OBSERVER

The input of this state observer is the calculated rotorposition of a back-EMF model, presented in [6]. An arctangentfunction yields this information from real and imaginary

ΨM,β

ΨM,α ∆γKγ

Kω

tct

ω̂

−

z−1

z−1

γ̂γ

ω∗

γ∗

arctan÷

1

Figure 2. Arctan observer structure

space-phasor components of the permanent magnet flux link-age in the stator oriented αβ - reference frame.

γ = arctan(

ℑ{ΨM}ℜ{ΨM}

). (1)

A. Observer ModelThe state-space model in discrete representation requires

only a single input value (electrical position γ(k)).

∆γ(k) = γ(k)− γ∗(k) (2)

The observed system states of this model contain rotor position(γ̂(k)) and machine speed (ω̂(k)). The observer deals with twofeedback factors, one for the position (K1γ) and the second thefor speed (K1ω) estimation.(

γ̂(k)ω̂(k)

)=

(γ∗(k)ω∗(k)

)+

(K1γ

K1ω

)∆γ(k) (3)

The prediction (denoted with ∗) of the next system taskincludes the normalized dead time (tct) which considers thetime between two tasks of the used controller. The speedbetween two software tasks is assumed constant.(

γ∗(k+1)ω∗(k+1)

)=

(1 tct0 1

) (γ̂(k)ω̂(k)

)(4)

Finally, the complete observer model requires a prediction ofthe rotor position for the next task to build the differencebetween measurement and prediction.

γ∗(k+1) =

(1 0

) ( γ∗(k+1)ω∗(k+1)

)(5)

B. Feedback Factor DeterminationAdequate feedback factors are determined by error analysis.

The observer error vector is the difference between real andpredicted system state.

The two gain elements are chosen to

K1γ = Kγ− tct K1ω and K1ω = Kω. (6)

Therefore, position (eγ) and speed (eω) error of the Arctanobserver between two tasks decrease with

∆eγ(k) =−Kγ eγ(k)+ tct eω(k) (7)

and∆eω(k) =−Kω eγ(k). (8)

The implemented structure is presented in Figure 2.

C. Error analysis

The following error analysis calculates the expected value ofthe observer error at a rotor position of 45◦. The result servesfor quality characteristics of the different observer methods.The analysis compares the difference between measured andestimated rotor position rated with the gain factor. Below, theexpression (k) is omitted.The measured values (ΨM,α, ΨM,β) are described by the sum

eα

γ∗

ΨM

eγ

γ0

eβ

ΨM,0

ΨM,β

ΨM,β,0

ΨM,αΨM,α,0

β

α

Ψ∗M

γ

1

Figure 3. Representation of faulty measurement and prediction

of their true value (ΨM,α,0, ΨM,β,0) and their measurementerrors (eα, eβ) caused by uncertainties and noise. Also thepredicted rotor position (γ∗) is decomposed in its real part (γ0)and the uncertainty of the prediction (eγ). Figure 3 depicts thisrepresentation.This calculation substitutes the arctangent function by its firsttwo elements of the Fourier series. Furthermore, higher ordererrors are neglected. Therefore, the error of the Arctan ob-server at γ0 = 45◦ approximately depends on three independenterrors of measurement and prediction.

Kγ

(arctan

(ΨM,β

ΨM,α

)− γ∗)∣∣∣∣

γ0=45◦≈

≈ Kγ√2 |ΨM|

(−eα + eβ−

√2 |ΨM|eγ

) (9)

The different errors are statistically independent. Therefore,the standard deviation of the position error of the Arctan

|ΨM|

inverse

Kγ

γ

tctz−1

γ̂

γ∗

Kω ω̂

z−1ω∗

arctan

|Ψ∗M

|

|Ψ̂M|

z−1

−abs

−

sin

ΨM,α

ΨM,β

cos

−

−

−

∆|ΨM

|KΨ

1

Figure 4. Sin/Cos observer structure and additional estimation of the flux-linkage magnitude (broken lines)

observer results to

σArctan(∆γ)|γ0=45◦ ≈

Kγ√2 |ΨM|

√(e2

α + e2β+2 |ΨM|2 e2

γ

)(10)

III. SIN/COS OBSERVER

The Sin/Cos observer calculates the rotor position andthe machine speed with non-linear factors [7]. The feedbackcoefficients contain sine and cosine functions of the estimatedrotor information. The presented state observer uses the realand imaginary components of the flux linkage in the statororiented reference frame [8], [9]. In comparison to otherapproaches [10], [11], it requires no arctangent calculation.

A. Observer Model

Measurement inputs of the state-space structure are thestator oriented αβ-components of the permanent magnet flux-linkage (ΨM,α(k), ΨM,β(k)). Its absolute value (|ΨM(k)|) en-ables the consideration of long-term temperature drifts.∆ΨM,α(k)

∆ΨM,β(k)∆|ΨM(k)|

=

ΨM,α(k)ΨM,β(k)|ΨM(k)|

−|Ψ∗M(k)|cos(γ∗(k))|Ψ∗M(k)|sin(γ∗(k))

|Ψ∗M(k)|

(11)

γ̂(k)ω̂(k)|Ψ̂M(k)|

=

γ∗(k)ω∗(k)|Ψ∗M(k)|

+

+

K1γ K2γ 0K1ω K2ω 0

0 0 KΨ

∆ΨM,α(k)∆ΨM,β(k)∆|ΨM(k)|

(12)

The state prediction of the next task is similar to theArctan observer. Additionally, constant flux-linkage magnitudebetween two tasks is assumed. γ∗(k+1)

ω∗(k+1)|Ψ∗M(k+1)|

=

1 tct 00 1 00 0 1

γ̂(k)ω̂(k)|Ψ̂M(k)|

(13)

System state and output display non-linear behavior.Ψ∗M,α(k+1)Ψ∗M,β(k+1)|Ψ∗M(k+1)|

=

|Ψ∗M(k+1)| cos(γ∗(k+1))|Ψ∗M(k+1)| sin(γ∗(k+1))

|Ψ∗M(k+1)|

(14)

B. Feedback Factor Determination

The substitution of the elements of the gain matrix withposition-dependent coefficients characterizes the Sin/Cos ob-server. The feedback factor (KΨ) of the flux-linkage magnitudeis independent from the other values. It is experimentallydetermined by machine temperature characteristics. The im-plemented structure of the Sin/Cos observer is presented inFigure 4.

Figure 4 distinguishes between the angle and positionestimation (full lines) and a separate estimation of the flux-linkage magnitude (broken lines). The latter is an optional partof the Sin/Cos Observer to deal with long term changes of theflux-linkage magnitude. Its feedback loop can be separatelydesigned according to Equation 12.

K1γ =−Kγ

sin(γ∗)|Ψ∗M(k)| − tct K1ω

K2γ = Kγ

cos(γ∗)|Ψ∗M(k)| − tct K2ω

K1ω =−Kω

sin(γ∗)|Ψ∗M(k)|

K2ω = Kω

cos(γ∗)|Ψ∗M(k)|

(15)

This choice leads to equal differences of both observers withrespect to position and velocity error.

∆eγ(k) =−Kγ eγ(k)+ tct eω(k) (16)

∆eω(k) =−Kω eγ(k) (17)

C. Error Analysis

The statistical error analysis determines the expected valueof the observer error at 45◦ similar to the Arctan-observer con-sideration. It compares the rated difference between measuredand estimated rotor position.

175 200 225 250 275 300 325−30

−20

−10

0

10

20

30

time / ms

∆ γ el

/ °

∆ γEMF

∆ γSin/Cos

∆ γArctan

−1

−0.5

0

0.5

1

i q

(a) encoder control

0 100 200 300 400 500 600 700−30

−20

−10

0

10

20

30

time / ms

∆ γ el

/ °

−1

−0.5

0

0.5

1

i q

(b) back-EMF model control

0 100 200 300 400 500 600 700−30

−20

−10

0

10

20

30

time / ms

∆ γ el

/ °

−1

−0.5

0

0.5

1

i q

(c) Arctan observer control

0 100 200 300 400 500 600 700−30

−20

−10

0

10

20

30

time / ms

∆ γ el

/ °

−1

−0.5

0

0.5

1

i q

(d) Sin/Cos observer control

Figure 5. Dynamic characteristics of the different rotor position sources for machine control at a torque step (iq = 0 to iq = 1); Difference between encoderangle and sensorless method at ω = 0.2 (per unit values)

The abstract elements are replaced by their non-linear ex-pressions (Equation (15)). The small error vector permits theapproximation of the angular functions by their first elementsof the Tailor series only. The error of the Sin/Cos observerat γ0 = 45◦ also depends on the three statistical independenterrors of measurement and prediction.

K1γ

(ΨM,α−Ψ

∗M,α

)+K2γ

(ΨM,β−Ψ

∗M,β

)∣∣∣γ0=45◦

≈

≈ Kγ√2

(− eα + eβ−

√2 |ΨM|eγ

) (18)

The standard deviation of the position error of the Sin/Cosobserver results to

σSin/Cos(∆γ)∣∣γ0=45◦ ≈

Kγ√2

√(e2

α + e2β+2 |ΨM|2 e2

γ

)(19)

Standard deviations of position errors of the Arctan and theSin/Cos observer are equal at |ΨM|= 1.

IV. EXPERIMENTAL RESULTS

A. Dynamic characteristics

Figure 5 presents the characteristics of different position-information sources. The differences between sensorless esti-mation and rotor angle from encoder are recorded at a nominal

machine speed of ω = 0.2. The figures show the dynamicbehavior of a torque step (iq = 0 to iq = 1). The performancesat different position sources for machine control are displayed(input source 5(a): encoder, 5(b): back-EMF model only, 5(c):Arctan observer, 5(d): Sin/Cos observer).Table II contrasts the standard deviation of the differencebetween encoder and sensorless rotor position. The Arctanobserver improves the quality of the position information ofthe back-EMF method by 1.2◦. Additionally, the Sin/Cos ob-server presents the best dynamic characteristic of the presentedmethods.

error ∆γel encoder control sensorless control(fig. 5(a)) (fig. 5(b) - 5(d))

back-EMF model 2.5◦ 5.0◦

Arctan observer 2.4◦ 3.8◦

Sin/Cos observer 2.4◦ 3.4◦

Table IISTANDARD DEVIATION OF DIFFERENCE BETWEEN ENCODER AND

SENSORLESS ROTOR POSITION

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72

2.5

3

3.5

4

4.5

5

normalized rotor speed

stan

dard

dev

iatio

n / °

el

Sin/Cos Arctan

(a) no load (iq = 0)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72

4

6

8

10

12

normalized rotor speed

stan

dard

dev

iatio

n / °

el

Sin/Cos Arctan

(b) nominal load (iq = 1)

Figure 6. Standard deviation of the difference between observer angle and rotor position from encoder

B. Speed characteristics

Both observers present comparable quality characteristicsover machine speed. The sensorless position estimation showsgood results over about 4% of rated speed at no-load andabout 8% of rated speed at nominal load. Beyond these speedvalues, their performances are almost independent from themachine speed (Figure 6).

V. CONCLUSION

The Sin/Cos observer with non-linear feedback coefficientspresents a useful alternative to observers which calculate therotor position from the back-EMF model with the arctangentfunction. Both, the standard deviation at load steps and theperformance over machine speed prove a comparable qualitycharacteristic of the Arctan and the Sin/Cos state observer.

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