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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Parameter Estimation in a Permanent Magnet

Synchronous Motor

Examensarbete utfört i Reglerteknik vid Tekniska högskolan vid Linköpings universitet

av Mikael Tenerz LiTH-ISY-EX- -11/4495- -SE

Linköping 2011

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

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Parameter Estimation in a Permanent Magnet

Synchronous Motor

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Mikael Tenerz LiTH-ISY-EX- -11/4495- -SE

Handledare: Andre Carvalho Bittencourt

isy, Linköpings universitet

Stig Moberg

ABB Robotics, Västerås

Examinator: Mikael Norrlöf

isy, Linköpings universitet

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Avdelning, Institution

Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2011-08-30 Språk Language  Svenska/Swedish  Engelska/English   Rapporttyp Report category  Licentiatavhandling  Examensarbete  C-uppsats  D-uppsats  Övrig rapport  

URL för elektronisk version

http://www.control.isy.liu.se http://www.ep.liu.se ISBNISRN LiTH-ISY-EX- -11/4495- -SE

Serietitel och serienummer

Title of series, numbering

ISSN

Titel

Title

Estimering av parametrar i en permanentmagnetisk synkronmotor Parameter Estimation in a Permanent Magnet Synchronous Motor

Författare

Author

Mikael Tenerz

Sammanfattning

Abstract

This thesis adresses the problem of estimating the parameters in a permanent magnet synchronous motor (PMSM). There is an uncertainty about the param-eters, due to age and tolerances in the manufacturing process. Parameters such as the resistance and the current to torque factor Kt, changes with respect to

temperature as well. The temperature in the motor varies in normal motor opera-tion, due to variations in angular velocity and torques. Online estimation methods with the model reference adaptive systems technique (MRAS) and offline meth-ods are presented. The estimation algorithms are validated in simulations with Matlab/Simulink and also evaluated with experimental data. Experiments were performed on a range of different motors, in realistic scenarios. Relevant factors such as the angular velocity of the rotor and the impact of the gravity force are investigated. The results show that it is possible to estimate the motor factor Kt,

with an accuracy of two percentage from its reference value in normal industry conditions. The estimated value of the motor inductance is within 25 percentage of the calculated reference value. The resistance however is affected by the resis-tance in the cables from the motor to the measurement device. With the cable resistance included in the calculations, the estimate still often exceeds double the value of the reference value.

Nyckelord

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Abstract

This thesis adresses the problem of estimating the parameters in a permanent magnet synchronous motor (PMSM). There is an uncertainty about the param-eters, due to age and tolerances in the manufacturing process. Parameters such as the resistance and the current to torque factor Kt, changes with respect to

temperature as well. The temperature in the motor varies in normal motor opera-tion, due to variations in angular velocity and torques. Online estimation methods with the model reference adaptive systems technique (MRAS) and offline meth-ods are presented. The estimation algorithms are validated in simulations with Matlab/Simulink and also evaluated with experimental data. Experiments were performed on a range of different motors, in realistic scenarios. Relevant factors such as the angular velocity of the rotor and the impact of the gravity force are investigated. The results show that it is possible to estimate the motor factor Kt,

with an accuracy of two percentage from its reference value in normal industry conditions. The estimated value of the motor inductance is within 25 percentage of the calculated reference value. The resistance however is affected by the resis-tance in the cables from the motor to the measurement device. With the cable resistance included in the calculations, the estimate still often exceeds double the value of the reference value.

Sammanfattning

Det här examensarbetet behandlar problemet med parametervariationer i PMSM. Parametervariationerna är en konsekvens av temperaturskiftningar i motorn. Un-der en vanlig arbetscykel varierar motorns varvtal och moment. Därmed sker även variationer i värmeförluster således temperaturen i motorns komponenter. Det finns även en osäkerhet på parametervärderna i rumstemperatur på grund av ålder och toleranser i tillverkningen. I examensarbetet presenteras skattningsmetoder i form av observatörer som skattar parametrarna online, såväl som metoder där skat-tningen sker offline. Valideringen av de utvecklade algoritmerna sker sedan genom simulering i Matlab/Simulink, och slutligen genom experiment på roboten. Exper-imenten är utförda på motorer av olika storlek i realistiska arbetsscenarion. Fak-torer som är viktiga att ta hänsyn till är rotorns hastighet och gravitationskraften. Resultaten från experimenten visar att det är möjligt att skatta motorkonstanten

Kt med två procents noggranhet under normala industriella arbetsförhållanden.

Skattningen av resistans och induktans är mer osäker. Induktansens skattade värde ligger vanligtvis inom 25 procents avvikelse från det beräknade referensvärdet. Re-sistansskattningen påverkas av resistansen i kablarna från motorn till mätsensorn.

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vi

Men även om kabelresistansen tas med i beräkningarna, så överstiger ändå det skattade värdet oftast dubbla referensvärdet.

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Acknowledgments

The author would like to thank the supervisors Dr. Stig Moberg at ABB and graduate student Andre Carvalho Bittencourt at Linköping University for their valuable guidance and help during the thesis. The author would also like to thank the examiner Prof. Mikael Norrlöf for his insights during the thesis and help with the writing.

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Contents

1 Introduction 1 1.1 Background . . . 1 1.2 Problem Description . . . 1 1.3 Purpose . . . 2 1.4 Related work . . . 2 1.5 Methodology . . . 2 1.6 Outline . . . 3 2 Physical analysis 5 2.1 The principle of the PMSM . . . 5

2.2 Torque generation . . . 5

2.3 Temperature effects of the motor constant Kt . . . 6

3 Mathematical model 9 3.1 DQ reference frame . . . 9 3.2 Assumptions . . . 10 3.3 PMSM . . . 10 3.4 Temperature effects . . . 13 3.5 Sensitivity Analysis . . . 13 4 Control system 15 4.1 Overview . . . 15 4.2 Current controller . . . 17 4.3 Torque maximization . . . 18 5 Estimation methods 19 5.1 Model reference adaptive system (MRAS) . . . 19

5.1.1 Design using Lyapunov theory . . . 19

5.1.2 Constant adaption gain . . . 25

5.2 Offline estimation methods . . . 26

5.2.1 Least squares method . . . 27

5.2.2 Weighted least squares . . . 27

5.2.3 Non linear greybox . . . 27

5.3 Parametric error index . . . 28 ix

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x Contents

6 Experiment design and data evaluation 29

6.1 Experimental design . . . 29 6.2 Disturbances . . . 32 6.2.1 Measurement disturbances . . . 32 6.2.2 Model disturbances . . . 32 6.3 Filtering . . . 33 6.4 Scaling of parameters . . . 34

7 Results and analysis 35 7.1 Model validation . . . 35

7.2 Online estimation methods . . . 36

7.3 Offline estimation methods . . . 41

7.3.1 Results of Exp 3-5 . . . 41

7.3.2 Weighting criterion . . . 45

7.3.3 Results of Exp 6 . . . 46

7.4 Data excitation . . . 49

7.4.1 Parameter error index . . . 49

7.4.2 Gravity force . . . 49 8 Conclusion 51 8.1 Discussion . . . 51 8.2 Future work . . . 53 Bibliography 55 A Lyapunov theory 57

B Parametric error index 59

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Chapter 1

Introduction

1.1

Background

This thesis was performed at ABB Robotics in Västerås. ABB Robotics is one of the world leaders in development and manufacturing of industrial robots. In-dustrial robots operate under a wide range of applications, often with demands of high accuracy. Each axis of the robot is operated by a permanent magnet syn-chronous motor. The PMSM (permanent magnet synsyn-chronous motor) is a popular choice of electric motor in many different areas and applications. In comparison to other electric motors the PMSM offers many advantages. It has the highest power density of all the motor types, due to the permanent magnet. The PMSM requires no excitation circuit and the elimination of copper losses in the rotor can improve the efficiency with up to 10 % compared to an induction machine [6]. In addition the PMSM has a high reliability and can be operated over a wide speed range. As the motor operates, internal losses causes the motor temperature to rise and certain parameters in the motor tend to drift from their nominal values specified at room temperature. The focus of this thesis is to estimate and update these parameters.

1.2

Problem Description

The nominal motor parameters are calibrated at room temperature (20◦C). The temperature in the motor rises due to heat losses during the motor operation, causing the parameters to change. As a consequence, the mathematical model used in the control system drifts from the real system. This leads to larger control errors, and a more sensitive control system. The temperature effects in the motor are well-known, but there is typically no thermal sensor available for measurement in a standard PMSM. There is also an uncertainty of the nominal parameter, due to tolerances in the manufacturing process and age of the motor. The parameters of interest are the resistance, inductance and the current to torque factor Kt.

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2 Introduction

1.3

Purpose

It is desirable to gain as much information about the system as possible, but the primary objective is to estimate the current to torque factor Kt. An online Kt

estimation would generate advantages in several areas. The control system would be more accurate, with smaller control errors in the outer loops. Smaller control errors could lead to slower transients and a more energy efficient system, which in turn would reduce the costs. Other areas in which this information could be useful is diagnostics and supervision, in applications such as collision detection.

1.4

Related work

Control of electric motors has been studied widely, but the issue of parameter drift is often neglected. The mathematical model describing the permanent mag-net synchronous motor is based on Chiasson’s model and is widely used in different studies, see [2]. Instinctively, a way to estimate the parameters is to design a ther-mal model of the permanent magnet motor and integrate this in the control system. The thermal model usually gets very complex, if the objective is to estimate the temperature of each component of the motor. Thermal models are considered to be the most accurate way of estimating the temperature and in turn the parameters. The model is usually derived using numerical finite element analysis, which makes it computational heavy and time consuming [6]. Knowledge about the parameter of the individual motor at 20◦C values is also needed, when using the thermal model. Another method for parameter estimation is the model reference adaptive sys-tem (MRAS), which is an observer design that uses an adaptive mechanism to adjust the parameters in the model. The MRAS observer updates the parameters at each time sample. Chung [2] uses the MRAS technique for estimation of in-stantaneous torque. Piippo and Hinkkanen amongst others write about adaption of motor parameters in a sensorless PMSM [12]. The MRAS method has been proven successful in computer simulations [2, 3].

Another way of identifying the parameters is to use an offline algorithm on a batch of data. In System Identification [11] , several offline identification methods are presented. An advantage with the offline identification methods is that the same stability analysis as the online algorithm is not necessary.

1.5

Methodology

Initially, a literature study of the previous work in the related field is presented. Methods of interest are chosen to investigate further and an appropriate matical model of the PMSM is derived, based on the literature study. The mathe-matical model of the PMSM and the online estimation methods using the MRAS technique are then implemented and simulated in Matlab/Simulink in continuous time. Offline identification methods are also derived as an alternative to the MRAS

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1.6 Outline 3

technique. The control system including the PMSM and the estimator are then discretized and simulated with the sample time used by the drive unit of ABB. When the simulations are proven successful in the simulations, an experiment plan is developed for experiments on the robot. The identification methods are then evaluated, based on the experimental data. Improvements in terms of tuning of design parameters and filters are made, before the methods are compared with each other.

1.6

Outline

Chapter 1 contains an introduction with the background, purpose, methodology of the thesis and the previous work in the related field. An analysis of the physics in the PMSM, including the torque generation and temperature effects are pre-sented in Chapter 2. The mathematical model is derived in Chapter 3, including a sensitivity analysis with respect to the parameter drift. In Chapter 4 the role of the estimator in the control system is explained. Chapter 5 presents the esti-mation methods and the results of the computer simulations. The design of the experiments and data evaluation is discussed in Chapter 6. Chapter 8 focuses on the results and analysis of the experiments. Chapter 8 includes the conclusions of the thesis and the possibilities for future work.

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Chapter 2

Physical analysis

This Chapter goes deeper into the physics of the permanent magnet synchronous motor, which in Chapter 3 leads to the mathematical model. The primary source for the physical background is [1].

2.1

The principle of the PMSM

A permanent magnet synchronous motor (PMSM) consists of an iron core stator with copper windings and permanent magnets mounted on the rotor, shown in Figure 2.1. The permanent magnets establishes a magnetic field, due to the pattern of the atoms. This is easiest explained in an atomic scale. The motion of the electrons of the atoms in a magnetic material align themselves in the same plane. The electron paths can be interpreted as current loops. Using the right-hand rule one comes to the conclusion that the current loops generate a magnetic dipole moment perpendicular to the loop, which results in an external magnetic field from the south pole to the north pole. The stator generates a rotating magnetic field by inductance, when current is driven through the stator windings. The principle of the synchronous machine is that the rotor is pulled along by a rotating magnetic field produced by the stator. Both the rotor and stator then rotate synchronously at the same angular speed. The stator windings in Figure 2.1 are divided into three phases which overlap. Each group of stator windings in Figure 2.1, represent a stator slot.

2.2

Torque generation

The torque generated in a PMSM is usually divided into three parts.

• The electromagnetic torque arises due to the magnetic force generated by the magnetic field. The magnetic force is perpendicular to the magnetic field, which is usually seen as the dominant component of the three parts.

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6 Physical analysis

Figure 2.1. Cross-Section of a surface mounted permanent magnet synchronous motor.

• The cogging torque is defined as the oscillating torque created by the geom-etry of the stator slots. This component generates an angular variation in the torque.

• The surface mounted PMSM is considered to have a non-salient rotor. A non-salient rotor means a cylindrical shape with uniform thickness of the airgap. The reluctance torque is created by the irregularity of the magnetic field caused by the rotor geometry and is therefore negligible.

The angular variation in the torque, is generally referred to as torque ripple. Both the cogging torque and the reluctance torque contributes to the torque ripple, since it is impossible to make a slot-less non-salient PM motor. The torque ripple can be reduced by using more poles or phases [8]. The PMSM should be designed so that the number of poles is inversely proportional the maximum speed of the motor. High-speed applications such as robotics, usually has about 4 pole-pairs.

2.3

Temperature effects of the motor constant K

t

The primary source for this Section is [14]. The temperature variation of the com-ponents appears when the motor operates at various speeds and different torques. The temperature is affected by the power losses, such as friction and heat transfer from the copper windings. These factors affect the torque capability and efficiency of the PMSM. There are certain characteristics of the PMSM that are important for the sensitivity with respect to the temperature. Two important factors of in-terest are the magnetic material in the rotor and the amount of stator windings

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2.3 Temperature effects of the motor constant Kt 7

per motor unit. Different permanent magnet materials have different temperature characteristics.

When the temperature rises in the permanent magnet, a demagnetization pro-cess of the permanent magnet begins. The atoms start to move more randomly, due to the increased energy in the permanent magnet [1]. Naturally, the magnetic field gets weaker. The residual flux density and the intrinsic coercivity Hciof the

permanent magnet decreases. The residual flux density is defined as the magnetic field per area unit and is proportional to the torque capability of the motor. The intrinsic coercivity Hci can be seen as a measurement of the permanent magnets

ability to resist demagnetization [13, 8]. This process is reversible and linear up to a critical temperature (150◦C for Ne-Fe-B), sometimes referred to as the Curie temperature [8]. If the temperature of the permanent magnet exceeds Tcrit, the

de-magnetization process increases exponentially. The dede-magnetization process over

Tcrit is irreversible and results in a permanently damaged rotor. Studies of the

demagnetization beyond the critical temperature are not easily found. Sebastian [14] has documented the residual flux density coefficient s for temperatures above

Tcritfor Ne-Fe-B magnets. Since residual flux density is proportional to the motor

constant Kt, the temperature coefficient is the same. The coefficient is constant

at s = 0.11 [%/C] up to Tcrit. A measurement at 175◦C results in a coefficient

of s = 0.15 [%/C]. Figure 2.2 shows an estimate of the motor constant Kt

vari-ation with temperature, if the coefficient changes at the same rate (dsdt = 0.0016) above Tcrit.

The age of the permanent magnet, also affects the strength of the magnetic

Figure 2.2. Temperature dependence of Kt, with dTds = 0.0016 above Tcrit= 150◦C.

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8 Physical analysis

the temperature [4]. The demagnetization process of the permanent magnet due to age is exponential. The loss in magnetic flux is found to be about twice as much after 30 years, than one hour after the magnetization. Figure 2.2 shows how Kt

changes with respect to the temperature. The age simple lowers the curve about one or two percent.

The resistance of the stator phase windings increases linearly with temperature [17]. The stator resistance may change up to 70 % under heavy duty [5]. Usually copper is used for the stator windings.

A PMSM used in a robot operates at various speeds and torques, depending on the working cycle. The temperature is therefore difficult to predict. In other ap-plications with a fixed operating speed such as a fan or a pump, the prediction of the operating temperature is easier.

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Chapter 3

Mathematical model

This chapter presents the mathematical model of the PMSM used in the thesis. The mathematical model is derived based primarily on [1, 2]. In order to achieve accurate estimations, a good model is important.

3.1

DQ reference frame

When controlling the PMSM it is preferred to navigate in a rotating coordinate sys-tem fixed to the rotor. This coordinate syssys-tem is referred to as direct-quadrature

Figure 3.1. Geometric interpretation of the Park transformation.

reference frame. The stator phase current is described in the stationary abc-reference plane. The Park transform is used for transformations between the two coordinate systems, where the direct axis is the radial component and the quadra-ture axis is aligned orthogonally to the direct axis. An illustration of the two coordinate systems is found in Figure 3.1. One reason for this transformation is that the magnetic field is in the same direction as the direct axis at all times, which simplifies the torque calculations. The Park transform is popular because the magnitude of the vector variables (current, voltage and magnetic flux) remains

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10 Mathematical model

through the transformation. On the other hand a factor of (3/2) will appear in the torque equation, to keep the 2-phase circuit equivalent with the 3-phase circuit. The reason for the factor is to keep the power input equation

Pin= uaia+ ubib+ ucic = (3/2)(uqiq+ udid) (3.1)

equivalent. Another commonly used transformation is the extended Park or the DQ0-transformation. This transformation is power invariant, but the current and voltages signals are scaled with the factorp3/2.

3.2

Assumptions

When designing a mathematical model, there is always a balance between com-plexity and usefulness. The difficulty is knowing what assumptions to make and which parts of the model that are essential for the purpose of the project. The following assumptions are made:

• the machine is symmetric and balanced, which means sinusoidal distributed windings and the sum of the phase currents is zero,

• there is no leakage flux,

• the eddy currents are negligible,

• there are no magnetic saturations in the stator core,

• the temperature of the permanent magnet is kept below the critical temper-ature i.e, reversible demagnetization,

• the stator windings are sinusoidally distributed i.e, no torque ripple, • the magnetic field has the same intensity throughout the airgap.

3.3

PMSM

The motor considered in the thesis is the surface-mounted three-phased PMSM. The magnetic field established by the permanent magnets on the rotor if the magnetic field is calculated at the stator side of the airgap (r = rS) is expressed

as, ~ BR(r = rS, θ − θR) = κBm rR rS cos (θ − θRr (3.2)

where rR and rS are the rotor and stator radius, θ is the angle position, θR is

the angle position of the magnetic field and Bm is the magnetic field inside the

permanent magnet. r is the position in the radial vector ˆr, where the magnetic

field is calculated. Bm is dependent on the magnet material and varies with

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3.3 PMSM 11

is assumed to be one. The stator magnetic field Bsis the vector sum of the fields

established by each phase and is expressed as,

~ BS(ia, ib, ic, θ) = µ0Ns 2g rR r [iacos θ + ibcos (θ − 3 ) + iccos (θ − 3 )] (3.3) where µ0is slope of the magnetic field in air with respect to the current, NS is the

turns of the stator windings in total and g is the length of the airgap. iabc is the

stator phase currents. The total magnetic field on the stator side of the airgap is,

~

B(ia, ib, ic, θ, θR) = ~BS(ia, ib, ic, θ) + ~BR(r, θ − θR). (3.4)

The flux linkage is here defined as the total flux through the surface of the rotor. The stator flux linkage is expressed as

ψa(t) = π Z 0 Ns 2 sin θ θ Z θ−π l1rSB(i~ a, ib, ic, θ0, θR)dθ0 (3.5)

where ψa(t) is the total flux linkage of phase a at the time t, and is calculated the

same way for each phase. Equation (3.5) can be rewritten as

ψa(t) = Ls(ia+ ibcos

3 + iccos

3 + Kecos θR) (3.6) and the stator flux linkages in each phase can be expressed as

  ψa(t) ψb(t) ψc(t)  = Ls   1 cos3 cos3 cos3 1 cos3 cos3 cos−2π3 1     ia(t) ib(t) ic(t)  + Ke   cos θR cos (θR3) cos (θR3)  

in matrix form. Ke is referred to as the back emf constant and is expressed as

Ke=

πl1l2BmNs

4 . (3.7)

Equation (3.7) gives a good understanding of the parameters affecting Ke. The

properties of the PMSM have an important role for Ke, such as length l1, diameter

l2 of the rotor and the number of turns in the copper windings of the stator Ns.

Bmis the only temperature dependent parameter, assuming there is no elongation

in the rotor due to temperature. The inductance can be described as

Ls=

πµ0l1l2Ns2

8g (3.8)

where the inductance should not vary much with temperature. As the rotor has no windings, the torque on the rotor is computed from the stator phase windings by letting ~τR= −~τS, [1]. The mathematical model of the PMSM in the stationary

reference frame abc can be expressed as

uabc(t) = Rsiabc+

dψabc(t)

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12 Mathematical model

using Faraday’s law, where uabc is the stator phase voltage. The Park transform

is used for transformating to the dq-reference frame and is expressed as   sq sd s0   = (2/3)  

cos θ cos (θ − 2π/3) cos (θ − 4π/3) sin θ sin (θ − 2π/3) sin (θ − 4π/3)

1/2 1/2 1/2     sa sb sc  

where sabc is the vector variables. The magnetic flux does not vary that much

with position around the rotor. This causes the difference in induction between the d and q-axis to be insignificant. The following assumption can be made,

Ld = Lq = Ls, [3]. The final expression used for the electrical equations in the

dq-plane are expressed as,

ud = Rsid+ Ls did dt − Lsiqnpωr (3.10) uq = Rsiq+ Ls diq dt + Lsidnpωr+ ωr r 2 3Ke (3.11)

where ωr is the mechanical angular velocity, np is the number of pole-pairs, ud is

the direct voltage and uq is the quadrature voltage. The electromagnetic torque

after the Park transformation can be described as

Tel= 3 2 r 2 3Keiqs = r 3 2Keiq= Ktiq. (3.12) As seen in (3.12), the back emf constant Keis proportional to the current to torque

constant Kt. The mechanical equation can be described as,

dωr

dt =

1

J(Te− Tload− Tf riction− Tcogging). (3.13)

Neglecting the cogging torque Tcoggingsince the stator windings are assumed to be

sinusoidally distributed and including Tf rictionand Tloadin τL, the torque equation

is expressed as

Jdωr

dt = Ktiq− τL. (3.14)

The equations expressed in state-space form,

did dt = 1 Ls [−Rsid+ npωrLsiq+ ud] (3.15) diq dt = 1 Ls [−Rsiq− npωrLsid+ uq− r 2 3Keωr] (3.16) dωr dt = 1 J[Ktiq− τL] (3.17) dt = ωr. (3.18)

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3.4 Temperature effects 13

3.4

Temperature effects

When validating the estimation methods in simulations there is a need for a realis-tic model of the temperature effects. As mentioned in Chapter 2, the weakening of the magnetic field inside the permanent magnet is linear [14] up to the critical tem-perature. This temperature is not exceeded under normal operating conditions. The magnetic field is described by

Bm= Bm,ambient+ s(T − Tambient) (3.19)

where s is the slope of Bmwith respect to the temperature. The slope for

NdFeB-magnets is about s = 0.001. Bm,ambient and Tambient is the magnetic field and

temperature in ambient temperature. The relation between the stator winding resistance and the temperature is considered to be linear and can be described as

rs= rs,ambient+ α(T − Tambient) (3.20)

where α = 0.00393 is the temperature coefficient of copper [17] and rs,ambient is

the stator windings resistance.

3.5

Sensitivity Analysis

This section analyzes the sensitivity of the mathematical models with respect to changes in temperature. Bmis proportional to Keas seen in (3.7), and also varies

linearly with temperature. The instantaneous resistance and Ke with respect to

the temperature is expressed as

Rs= Rs,nom+ ∆Rs (3.21)

Ke= Ke,nom+ ∆Ke (3.22)

where ∆Rsand ∆Keare the errors in resistance and Kedue to the parameter drift.

Ke,nom and Rs,nom are the nominal values of the parameters. Adding equations

(3.21) and (3.22) into the state equations result in,

did dt = 1 Ls [−(Rs,nom+ ∆Rs)id+ npωrLsiq+ ud] (3.23) diq dt = 1 Ls [−(Rs,nom+ ∆Rs)iq− npωrLsid+ uq − r 2 3(Ke,nom+ ∆Ke)ωr] (3.24) where the error in resistance influences both equations as seen in (3.23) and (3.24).

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Chapter 4

Control system

4.1

Overview

An overview of a typical control system with the Ktestimator included, is shown

in Figure 4.1. The controller consists of current, a speed and a position regulatora, a pulse width modulator (PWM) and the Kt estimator. The control system is of

cascade design and has three control loops. For this design to work, the inner loop has to be faster than the outer loop [16]. The current controller has the fastest dynamics followed by the speed and position controller. The Ktestimator

is integrated in the current loop, with the purpose of reducing control errors in the position loop. Sensors are available for measuring the stator phase current and the motor angle. The voltage signals used as input signals is also available. PWM is commonly used for controlling power to inertial electrical devices. The power is controlled by sending short current impulses at high frequencies. The longer the current pulse is the higher the mean value gets, and in turn the power input. The controllers navigate in a coordinate system fixed to the rotor. Park transforms are used to switch from the rotor fixed coordinates to the stator fixed, as seen in Figure 4.1. This is explained in detail in Chapter 3.

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16 Control system

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4.2 Current controller 17

4.2

Current controller

A schematic picture of the current controller is shown in Figure 4.2. The current

Figure 4.2. Schematic Figure of the current controller.

controller uses a PI-regulator, with a feed forward link. The direct current and quadrature current are controlled separately and the feed forward link is needed to cancel out the cross-coupling effects. As seen in Equation (3.15), an increase in the direct current also affects the quadrature current equation. Lambda tuning is a technique that specifies design parameters in terms of the time constants of the system, in order to get a smooth response with no overshoot [16]. Lambda tuning is implemented in the current controller. This is seen in Figure 4.3, where a step

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18 Control system

response is simulated. When the reversible demagnetization of the permanent magnet starts, Kt decreases and the current controller is unable to track the

reference torque, see Figure 4.4. A simple proportional regulator is implemented

Figure 4.4. The effects on the electro magnetic torque, due to the decreasing of the magnetic flux.

in the position loop and a PI regulator in the speed loop.

4.3

Torque maximization

To fully utilize the motor’s capacity, the torque has to be maximized. The power input (3.1) is dependent on the electromagnetic torque and the angular velocity of the rotor,

Pin= Telωr= (3/2)(uqiq+ udid) (4.1)

where the electromagnetic torque is proportional to iq, see equation (3.12). The

maximum torque from the PMSM is achieved, when the direct current id is

con-trolled to zero and the maximum voltage and current are constrained by

Vmax≥ q u2 q+ u2d (4.2) Imax≥ q i2 q+ i2d. (4.3)

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Chapter 5

Estimation methods

This chapter presents two online and some offline estimation methods. The online methods are variations of the MRAS-technique and the offline methods are identi-fication tools such as linear least squares and nonlinear greybox methods. Figure 4.1 in Chapter 4, shows how the Ktestimator is integrated in the control system.

As mentioned in Chapter 2, the parameter drift appears as non linearities in the system. It is often difficult to guarantee stability with non linear observers [10]. In some cases, stability can be proven with Lyapunov functions. Another difficulty is the low or non-existing current in the direct axis, caused by the maximiza-tion of the torque. One consequence of the torque maximizamaximiza-tion is that only the quadrature equation (3.16) contains information about the unknown parameters.

5.1

Model reference adaptive system (MRAS)

The basic idea with adaptive control is to estimate and adjust uncertain plant parameters online. The adaptive mechanism is the algorithm applied for the pa-rameter adjustment. The adaptive mechanism is sometimes referred to as the adaptive law. Adaptive control systems are non linear and have to be analyzed as such, even if the plant itself should be linear [9]. The initial MRAS design is based on [2] and [9]. A variation of the initial design that uses a constant adaption of the parameters is also presented.

5.1.1

Design using Lyapunov theory

The idea is to design the adaption mechanism based on the Lyapunov function, and find design parameters that fit the criterion for stability. According to the sensitivity analysis in Chapter 3, the resistance affects both electrical Equations (3.15) (3.16), whereas Keonly affects the equation in quadrature direction. The

MRAS observer is divided into one resistance estimator and one Kt-estimator.

The estimated resistance from the resistance estimator is used as an input signal to the Kt-estimator, as seen in Figure 5.1.

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20 Estimation methods

Figure 5.1. Schematic figure of the MRAS observer.

The Kt estimator

The mathematical model of the PM synchronous motor derived in Chapter 3, can be represented by ˙ x = Ax + Bu + DKe (5.1) x = id iq  u = ud uq 

in state space form. The state-space matrices are expressed as,

A = "−R s Ls −npωr npωr −RLs s # B = 1 Ls 0 0 L1 s  D = " 0 −q2 3 ωr Ls. #

The Kt-estimator has one feedback loop from the controller error and one outer

loop for adaption of the parameters [9]. The feedback gain can be interpreted as the speed of the observer and the adaption gain the speed of the parameter adaption. The estimator is described as,

˙ˆx = Aˆx + Bu + D ˆKe+ F e (5.2)

where F is the feedback gain, and e is the error e = x − ˆx. The back-emf constant Keis seen as a disturbance. The reason for this is that all the states in the observer

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5.1 Model reference adaptive system (MRAS) 21

in Appendix A. The following adjustment law is chosen according to the Lyapunov theory [9] and [2]

˙ˆ

Ke= γKeD

TP e (5.3)

where γKe is the adaption gain and

P = 1 0 0 1 

is the solution of the Lyapunov equation in Appendix A, with the stability anal-ysis. As seen in Equation (5.3), the adaption gain is proportional to the angular velocity of the rotor. The criterion for the feedback gain F is also found in the stability analysis. With the Lyapunov solution in the adaption rule, the adaption mechanism is proven stable [9]. Ktis obtained through equation (3.12), given the

estimated Ke. The inclusion of matrix D in the adaptive mechanism ensures that

only the error in quadrature current is used to adjust Ke. A schematic Figure of

the Kt-estimator is seen in Figure 5.3.

Figure 5.2. Schematic figure of the Kt-estimator.

The resistance estimator

A similar strategy is used for the design of the resistance estimator. The error in direct current is dependent only on the resistance assuming the mathematical model is perfect. When the direct current error is driven to zero in the resistance estimator, the value for the resistance should be correct. The adaption mechanism applied in the resistance estimator is

˙ˆ

Rs= γRsed (5.4)

where γRs is a constant adaption gain. A scheme of the resistance estimator is

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22 Estimation methods

Figure 5.3. Scheme of the resistance estimator.

The solver used for the simulations is ode45, and the discretization method is zero order hold. The PMSM model is simulated as a continuous system. Ode 45 manages the issue of handling both continuous and discrete subsystems, which is necessary in the experiments. Zero order hold makes the corresponding value con-stant during the sample time. The observer has the same sample time (ts= 0.126

[ms]), as the current loop. Two experiment scenarios are simulated, where the purpose of Experiment one (Exp 1) is to validate the adaption mechanism and the purpose of Experiment two (Exp 2) is to prove that the control errors in the outer loops are reduced. Tuning of the design parameters is more relevant on the experiments with real data. The conditions of Exp 1 are,

• The torque reference is set to 100 Nm after one second.

• The nominal Kt value in the estimator is set to 80% of Kt in the model.

The Ktvalue decreases linearly five seconds into the simulation and remains

constant 2 seconds later.

• The nominal resistance value is set to 80% of the resistance in the model. The resistances increases linearly nine seconds into the simulation and remains constant two seconds later

• The direct current reference value is set to 20% of the quadrature current reference value.

• The design parameters are set as γKe= 0.001, γRs = −10 and the feedback

gain

F = 100 0

0 100



The MRAS technique is successful in simulation when the direct current is not controlled to zero, see Figures 5.4 to 5.7. The resistance- and Ktestimates converge

to the correct value. If the direct current reference value would have been set to zero (torque maximization), there would not have been enough information for the resistance estimator to update the resistance. The error in resistance also affects

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5.1 Model reference adaptive system (MRAS) 23

Figure 5.4. Exp 1: Electromagnetic torque.

Figure 5.5. Exp 1: Kt estimation.

the estimation of the parameter Kt, since the calculation of Kt is based on the

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24 Estimation methods

Figure 5.6. Exp 1: Resistance estimation.

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5.1 Model reference adaptive system (MRAS) 25

As mentioned in the introduction, a more accurate Ktwould reduce the control

errors in the outer loops. A simulation with the speed and position loop included in the control system, was performed to validate this theory. Exp 2 is a simple step in position, with a speed and acceleration limit for the motor. Exp 2 was performed with continuous signals, due to issues in the discrete implementation. Figure 5.8 shows a reduction of the speed control error. The motor reaches its maximum speed after five seconds, hence the control error goes to zero. In Exp 2, the nominal motor constant Kt is set 20% higher than its correct value. Kt is

adjusted, with the adaptive mechanism implemented and the current controller is able to compensate the error in speed faster than the speed and position loop.

Figure 5.8. Exp 2: Control error in angular velocity of the rotor ωr.

When the direct current is controlled to zero, there is no sufficient information to adjust the resistance according to the simulations. One idea is to implement an injection of direct current every ten minutes or so to update the resistance. The factors to investigate further are how much current that is needed and for how long time the current is needed, in order to adjust the resistance. This injection does not have to be longer than one second, depending on the adaptive gain and the magnitude of the direct current. The interval between the injection is dependent on the thermal time constant of the system which is about 10 to 30 minutes.

5.1.2

Constant adaption gain

The Ktestimator in the Lyapunov based observer, adjusts the adaption gain with

respect to the angular velocity of the rotor. The reason for this is the inclusion of the matrix D, in the adaption mechanism. This might not be the optimal

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26 Estimation methods

approach, considering the slow dynamics of the temperature. A variation of the Lyapunov based adaption mechanism, is introduced with a constant adaption gain. To achieve the constant adaption gain, a relay is added to the adaption mechanism in the Ktestimator. The relay is implemented with respect to the angular velocity

in the D matrix, as

ωr=



1 if w ≥ 0

−1 if w < 0

Apart from the relay, the adaption mechanism is the same as before. Simulations with a step response give similar results as the Lyapunov solution. The adaption gain however has to be increased, to obtain the same adaption speed. The effects of the relay, is clearly seen in the experiments with real data when the angular velocity is varying.

5.2

Offline estimation methods

Another approach for parameter estimation is to consider the offline estimation methods. The offline estimation methods presented here are primarily based on [11]. There is a wide range of methods and approaches in this area. The first step is to collect a data set and then use an algorithm to minimize the prediction error. The same mathematical model for the electrical system as before is used

ud = Ls did dt + Rsid− npLsωriq (5.5) uq = Ls diq dt + Rsiq+ Lsnpωrid− r 2 3Keωr. (5.6) The unknown parameters are defined as,

θT = Rs Ls Ke



and the prediction error is defined as,

(t, θ) = y(t) − ϕT(t)θ (5.7) where y(t) is the output signal, ϕT(t) is the regressor. These are measured signals,

received from the sensors. The prediction error can be filtered through a pre-filter

L(q), which has the same effect as filtering the input-output. This filter can be

used to reduce the effect of high frequency disturbances or slow drift terms.

F(t, θ) = L(q)(t, θ) (5.8)

Then the following expression is used,

VN(θ) = 1 N N X t=n l(F(t, θ)) (5.9)

where l(·) is scalar function, which is defined for different identification methods. The estimate is then found by minimizing the argument of (5.9),

ˆ

θ = arg min

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5.2 Offline estimation methods 27

5.2.1

Least squares method

The linear least squares method is commonly used in the industry because of it’s simplicity. The pre-filter L(q) is chosen as a low pass filter of fourth order with the cutoff frequency 0.1 Hz. When choosing the cut-off frequency for L(q), it is important to find the balance between information losses and noise reduction. Least squares methods uses the quadratic norm, l(F) = 122F. The output signal

is defined as yT(t) = [uq(t)ud(t)]. uq(t) is a vector of the quadrature voltages

collected from the time t0to tN, where N is the number of samples. The regressor

ϕ(t) is defined as, ϕ(t) = iq(t) diq dt(t) + npωrid(t) p2/3ωr(t) id(t) didtd(t) − npωriq(t) 0 

where the time derivative of the current signals is calculated using diff in Matlab. The output signal vector and regressor were initially only based on the quadrature equation. Since all the parameters are included in the quadrature equation, it is of interest to see if the direct equation is needed for the estimation.

5.2.2

Weighted least squares

Measurements at different times sometimes have different quality for identification. As mentioned in Chapter 2, the effects of Ktare larger at high speeds. One could

then apply a weighting function, which weights the information in the incoming data and thereby the calculation. The weighting function β(w), is dependent on the angular velocity ωr. The minimizing function with the weighting function

included is expressed as VN(θ, ZN) = 1 N N X t=n β(w)l(F(t, θ)). (5.11)

5.2.3

Non linear greybox

Greybox identification is a mixture between whitebox and blackbox identification. The whitebox model is purely physical whereas the blackbox model is non-physical. The same mathematical model as (3.15) and (3.16) are specified, with ud, uq and

ωr as input signals, id and iq as states and also the unknown parameters Rs, Ls

and Ke. An initial guess of the parameters and the initial value for the states is

also specified. The unknown parameters are then found, with a search algorithm that uses iterative search to find the minimum gradient of the prediction error. The Matlab commands idnlgrey and pem are used for stating the model and calculating the parameters. It is also possible to add a weighting function to the non linear greybox algorithm.

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28 Estimation methods

5.3

Parametric error index

The parametric error index is a measurement of how sensitive the prediction error is with respect to a change of a certain parameter, see [1]. The index is based on a calculation of how much in percentage the parameter needs to be changed in order to double the minimum prediction error. A low percentage corresponds to a good excitation in the data. The parameter index is primarily used for comparison between the different parameters and can give an explanation to the accuracy of the parameter estimation. It is obtained by the following calculations, assuming the estimation interval is [n, N ]. The residual error on the estimation interval is defined as E2(ˆθ) = N X t=n (t, ˆθ)T(t, ˆθ). (5.12) The error (t, θ) is replaced with the filtered error l(F(t, θ)), if the pre-filters are

applied. δθi is the deviation of parameter θi that doubles the residual error and

is defined as δθi= v u u tE2(ˆθ)( N X t=n ϕ(t)Tϕ(t))−1 ii (5.13)

where E2(ˆθ) is the residual error with the estimated parameters and ϕ(nT ) is the

regressor. Equation (5.13) is derived in Appendix B. The parametric error index is defined as Pindex= 100δθi ˆ θi (5.14) where ˆθi is the estimated parameter and δθi is the change in parameter.

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Chapter 6

Experiment design and data

evaluation

When evaluating the experimental data, there are several aspects to keep in mind. Naturally, the first step is to take a closer look at the data to see if there is any unexpected behavior. An appropriate filter is then applied before it is time to validate the mathematical model and evaluate the results of the different methods.

6.1

Experimental design

There are a few considerations that have to be made when designing the experi-ments. The purpose is to evaluate the quality of the parameter estimation with the different methods. The experiment scenarios are designed for comparisons of certain factors. Factors of interest are the importance of angular velocity, accel-eration of the rotor and the temperature. In addition, the parametric error index is calculated as an indication of the difference in excitation among the parame-ters. Finally the online and offline estimations are tested in two realistic operating scenarios. Experiment one and two are computer simulations. The experiments performed on robots are described as,

Experiment three (Exp 3): is designed to recognize the influence of the angu-lar velocity of the rotor. To evaluate the speed as a factor, single operating cycles were repeated with increasing speed. The cycle consists of a simple back and forth motion and approximately 20 cycles were evaluated on each of the 6 axes. Conditions like gravity on axes 2 and 3 and the proportions of the motor could possibly influence the results. Exp 3 is performed on cold robots.

Experiment four (Exp 4): is performed to evaluate the influence of temper-ature. The purpose of Exp 4 is to validate that the parameter estimates are changing, when the temperature is changing. The experiment is designed

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30 Experiment design and data evaluation

similar to Exp 3, with single cycles. Before the experiment begins, the mo-tors have been operating for one hour so that the momo-tors have an increased temperature. It is preferable to measure the temperature as soon as possible after the experiment, so the measured temperature is the same as when the experiment is performed. Therefore Exp 4 consists of fewer operating cycles than Exp 3.

The temperature of the permanent magnet and stator windings is rather difficult to measure since they are covered by the housing of the motor. For practical reasons, the temperature of the motor is assumed to be the same in all components of the motor and the temperature is measured with a thermometer on the housing. As the motors on axes 3-6 are shielded, the temperature could not be measured there. For that reason, data were only collected for axes 1-3. The temperature on axis 3 is assumed to be similar to axis 1 and 2.

Experiment five (Exp 5): is designed to find out if the acceleration of the rotor is an important factor for the data excitation. Cycles with the same operating speeds as Exp 3 are used, so the estimation results are comparable. The acceleration of the motor is lower than in Exp 3.

Experiment six (Exp 6): is designed to get a sense of how well the parameters could be estimated in normal industrial conditions. These experiments are performed with cold robots. Examples of common tasks for the robots in industrial applications are material handling and water cutting. The charac-teristics of the movement differ a lot between the two scenarios. The function

Figure 6.1. The tracks of the realistic scenario programs of Exp 6.

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6.1 Experimental design 31

place it at point. Picking up and place the material is usually executed at slow speed, because of the precision that is needed. The transfer from the nearby area of point A to point B however, does not have to be as precise and is therefore executed at a higher speed. Figure 6.1 shows an overview of the material movement program.

Water cutting is an application, with high precision demands. In this case, a fixture shown in Figure 6.1 is created with the program. The PMSM is operating at various speeds in the water cutting scenario. The movement during the cutting section is relatively slow in comparison to the material movement scenario. When the fixture is ready, the tool continues the track a bit beyond point D and then returns to the starting point C. Point C is the waiting position between the cuttings. The transfer to point C can be executed at higher speeds, but the distance is short and the PMSM will not have the time to reach very high speeds. Figure 6.1 shows an overview of the water cutting program. The motors operate simultaneously in Exp 6, which could possibly affect the results.

The results of the experiments are shown in Chapter 7, for axis 1. The results of the other axes are found in Appendix C.

Figure 6.2. Data selection.

Each operating cycle was evaluated individually with the offline estimation meth-ods and the whole experiment evaluated with the MRAS technique. Figure 6.2 shows the data set and the single cycle. This data selection approach ensures a good comparability between the axes and speeds. Experiments with lower accel-eration were also performed. Different operating cycles have different excitation and the results will therefore not be as comparable.

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32 Experiment design and data evaluation

temperature are available for validation. In order to compare the estimated param-eters with these values, (3.19) and (3.20) were used to get the validation paramparam-eters at the correct temperature.

6.2

Disturbances

It is important to be aware of the disturbances, when analyzing a system. The dis-turbances can be divided into measurement disdis-turbances and model disdis-turbances, based on the physical origin [10]. There is always an error between the measured signal of the sensors and the actual value. This is defined as the measurement disturbance. The model disturbances arises due to the difference between the mathematical model and the actual behavior of the system. Assumptions made in the mathematical model and unknown characteristics causes model disturbances. System disturbances can also be included in the model disturbances. System dis-turbances are disdis-turbances that is difficult to predict, for example if an object hits the robot or an earthquake.

6.2.1

Measurement disturbances

The measurement disturbances usually have high frequency with low amplitude. These errors should if possible be filtered out. Therefore it is useful to make a distinction between the measurement noise and the system error. A resolver is used to extract the rotor position. Factors that affect the precision of the resolver are the position of the resolver rotor, the material, the number of counts per rev-olution and the sample frequency [1]. Unevenness of the resolver poles causes a higher frequency error. Since the rotor velocity is needed in the control system, the time derivative of the position signal is used. The time derivative amplifies the measurement disturbances.

The resistance of the cable from the motor to the drive unit is a measurement disturbance for the motor resistance. The cables are of different length, depend-ing on the location of the robot and the placement of the motor. The length of the cable is usually about 20 meters, but stretches from about 7 to 50 meters in extreme cases. Assuming that the cables are 20 meters, calculations on the test robot show that the magnitude of the cable resistance per phase is from 50% to 70% of the motor resistance. If the cable length is 50 meters, the resistance of the cable is up to 180% of the motor resistance on axis 2. A short cable of seven meters has up to 25% resistance of the motor resistance. In other words, the cable resistance is definitely worth taking into account when estimating the resistance. As mentioned in Exp 3, the temperature of the permanent magnet and the stator windings is assumed to be the same as the housing of the motor.

6.2.2

Model disturbances

There is an advantage to have as much knowledge of the disturbances as possi-ble, even if they are not included in the model. This information is useful when

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6.3 Filtering 33

choosing the appropriate filter for the signals. Known model disturbances can be related to the assumptions in the mathematical model. The torque ripple for ex-ample, which is caused by the cogging and reluctance torque and has a frequency which is dependent of the number of stator slots per pole-pair and the velocity. The reluctance torque appears at frequencies in multiples of ωr.

6.3

Filtering

Relevant disturbances to the system are reviewed in Section 6.2. The measurement

Figure 6.3. Signals filtered through a fourth order butterworth filter, with a cut-off frequency of 100 [Hz].

disturbances have high frequencies and are usually filtered out with a low-pass filter. When deciding the cut-off frequency of the low-pass filter, one has to con-sider the system. Important information relevant for the purpose of identification, should not be filtered out. The higher frequencies are not that important when identifying the motor parameters. A butterworth filter of fourth order is applied with a cut-off frequency of 100 [Hz]. The time delay increases with the order of filter and decreases with the cut-off frequency. Figure 6.3 shows the unfiltered and filtered signals. As seen in the Figure 6.3, there is not too much time delay. The highest frequencies are also filtered out. This is of extra importance since it is required to compute derivatives of the current signals in the identification algo-rithm. The high-frequency noise gets amplified when using the time derivative in the calculation [1].

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34 Experiment design and data evaluation

6.4

Scaling of parameters

Because of the design of the PMSM, it is natural to measure the line to line voltages instead of the phase voltages. These are also the parameters used in the software to get a better overview. The phase parameters have to be multiplied by a factor √

3 to get their equivalent line to line values [15]. One also have to consider if the currents are measured in top values or in effective values. Divide the top values with the factor√2, in order to get the effective values.

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Chapter 7

Results and analysis

This chapter contains the results and analysis of the experiments presented in Chapter 6. The mathematical model is validated in Section 7.1. The methods are divided into online and offline estimation and evaluated starting with the online estimation methods, that uses the MRAS technique. There is also a section about the excitation of the data and finally a discussion about the results of the thesis.

7.1

Model validation

Figure 7.1 shows a validation of the mathematical model. The voltage signal is used as output in this simulation, similar to the least squares problem. Since the motor is cold, the parameters should not be affected by the temperature. The current derivatives are obtained using diff in Matlab. As seen in the figure, the model does not describe the data well at low voltage levels. The data is taken from the water cutting scenario.

Figure 7.1. Model validation in the water cutting scenario, with the nominal parameter values.

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36 Results and analysis

7.2

Online estimation methods

There were two estimation methods with the MRAS design presented in Chapter 5. The main difference between the Lyapunov based observer and the constant adaption gain method, is the adaption gain in the Ktestimator. The Lyapunov

based observer has an adaption gain proportional to the angular velocity, unlike the other version which has a constant adaption gain as the name of the method implies. Since the torque is maximized in these experiments, there is only a low noisy signal in the direct current. It became obvious that these were not sufficient information to estimate the resistance, when Exp 3 was performed. The resistance estimate oscillates and does not converge. For this reason the adaption gain for the resistance is tuned close to zero γRs ≈ 0, so the change in estimated resistance

is negligible for the outcome of the Ktestimation. The increase in resistance with

respect to the temperature during the testing period is negligible. The feedback gain is fixed at F = 1000, so the estimation methods can be compared with each other.

Exp 3 which includes the speed test is good for tuning the observers, since the adaption gain is dependent on the speed of the rotor. The data is taken from Exp 3 on axis 1. Ideally, Kt should have low deviation from the calculated Kt and

slow transients. The slow transients should depend on the variation in tempera-ture during the testing period, which should not be more than one degree which answers to 0.1% change in Kt. Figures 7.2 to 7.4 shows the estimation of Kt

Figure 7.2. Exp 3: Lyapunov based observer with adaption gain γKe= 10

−4

.

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valida-7.2 Online estimation methods 37

Figure 7.3. Exp 3: Lyapunov based observer with adaption gain γKe= 0.5 ∗ 10

−5

.

Figure 7.4. Exp 3: Lyapunov based observer with adaption gain γKe= 10

−6

.

tion implies that the mathematical model is more reliable at high speeds, which also is in line with the results. The speed-accuracy relation is analyzed further in

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38 Results and analysis

Section 7.3.1. A large adaption gain results in fast transients at high speeds, as shown in Figure 7.2. However, the mean value of the estimated Kt is closer to

the calculated value with larger adaption gain. This causes a conflict of interest, between slow transient and low deviation. The speed has to be relatively high over a longer period of time for the parameter Ktto converge to the correct value with

a smaller adaption gain, as seen in Figure 7.4. Figure 7.4 shows that the estimate actually have not converged to the reference value, although if the speed is kept high during a longer period of time the estimate should get closer and closer to reference value. The results of the material movement scenario gives a clearer view on converging process in a realistic scenario. The difference between the constant

Figure 7.5. Exp 3: Constant adaption method with adaption gain γKe= 10

−3

.

adaption and the speed dependent adaption is clearly seen in the figures. If the statement is true that Ktis not dependent on the velocity and the overestimated

Ktvalues only depend on errors in the mathematical model and data excitation,

then neither of the MRAS estimation methods are useful. The MRAS observers updates Kteach sample and the estimation error is large at low speeds regardless

of the tuning. The Lyapunov solution however is preferable, since the adaption speed of Kt is faster when the estimation is more correct. For that reason the

Lyapunov based observer with adaption gain γKe= 10

−5 will be evaluated in the

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7.2 Online estimation methods 39

Figure 7.6. Exp 3: Constant adaption method with adaption gain γKe = 10

−4

.

Figure 7.7. Exp 3: Constant adaption method with adaption gain γKe = 10

−5

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40 Results and analysis

As seen in Figures 7.8 and 7.9, the Lyapunov based observer’s performance is dependent on speed. Since the adaption gain is proportional to the speed, the

Figure 7.8. Kt estimate on the material movement scenario. The Lyapunov based

observer is used with the adaption gain γKe = 10

−5

.

Figure 7.9. Kt estimate on the water cutting scenario. The Lyapunov based observer

is used with the adaption gain γKe = 10

−5

.

estimated Ktdoes not return to its initial value when the speed is lower. With the

proper tuning and one high speed section in the operating cycle one can estimate the nominal Kt, as seen in Figure 7.8.

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7.3 Offline estimation methods 41

7.3

Offline estimation methods

Two methods of linear least squares were presented in Section 5.2.1. The difference between them is that one of them only includes the equation in the quadrature direction and the other includes both equations. Exp 3 shows that both equations are needed, to estimate the parameters. The linear least squares refers to the version with both equations.

7.3.1

Results of Exp 3-5

Figures 7.10-7.16 show the experiment results for the offline estimation methods, on axis 1. The results from the other axes can be found in Appendix C. Values used as reference are calculated from the motor housing temperature and the nom-inal value. Figures 7.11, 7.14 and 7.16 are showing the results of Exp 4, which is performed on PM-motors with increased temperature. As mentioned in Chapter 6, the warm test is primarily used to validate that the Ke-estimate actually

de-creases with temperature. Exp 3 evaluates the speed dependence and is also used when determing the preferred weighting criterion. More details on the experiment procedure can be found in Chapter 6.

Figure 7.10. The resistance estimation from Exp 3 and Exp 5, with the offline estima-tion methods.

Figure 7.10 and Figure 7.11 shows the resistance estimation on experiments three to five. The estimation converges to about double the expected reference values, at speeds above 100 [rad/s]. The same tendencies are seen on the other axes, see Appendix C.9 and Appendix C.10. The deviation from the reference value differs

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42 Results and analysis

Figure 7.11. The resistance estimation from Exp 4, with the offline estimation methods.

significantly from motor to motor. One can also see a larger standard deviation among the resistance estimates, compared to inductance and Ke. Since the

devi-ation from the reference value is large, it is difficult to come to any conclusions concerning Exp 5. Exp 4 shows that a lower acceleration does not improve the es-timation results on the resistance. Figure 7.12 shows that the deviation is smaller on axis 1, when the cable resistance is included in the calculation of the reference value. The difference between the estimated resistance and the reference value is still too large for the estimation to be considered reliable.

The results of Exp 4 shows that the inductance is not affected by the temper-ature, which is seen the diagrams in Figure 7.13 and 7.14. Exp 3 indicates a similar speed-accuracy relation as the resistance with convergence at speeds above 100 [rad/s], see Figure 7.13. The deviation is spread from 2-25 % at maximum speed, which is seen in Appendix C.12 and C.13. Lower acceleration does not improve the accuracy for the inductance.

Keis the most important parameter to estimate, since it is used in the feed forward

link Kt=

q

3

2Keto obtain torque control from current control. Exp 3 shows that

Kealso converges with increasing speed, as seen in Figure 7.15 and 7.16. However,

the accuracy with respect to the reference value is better. Figure 7.15 and 7.16 shows a deviation around 1% with maximum speed, for axis 1. Exp 4 show that the deviation is similar regardless of temperature, which indicates that the method works. Figures C.14, C.15 and C.18 show deviation of 0.25 % to 2,1 % at maximum speed. It is also noticeable that the estimates of Keare lower than their

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respec-7.3 Offline estimation methods 43

Figure 7.12. The resistance estimation from Exp 3 and Exp 5, with the cable resistance included. The cable is assumed to be 20 meters.

Figure 7.13. The inductance estimation on from Exp 3 and Exp 5, with the offline estimation methods.

tive reference values for axes 1 and 2. As mentioned in Chapter 6, it is likely that the temperature of the permanent magnet is higher than the housing of the motor.

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44 Results and analysis

Figure 7.14. The inductance estimation from Exp 4, with the offline estimation meth-ods.

Figure 7.15. The Ke estimation from Exp 3 and Exp 5, with the offline estimation

methods.

References

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Parallellmarknader innebär dock inte en drivkraft för en grön omställning Ökad andel direktförsäljning räddar många lokala producenter och kan tyckas utgöra en drivkraft

Närmare 90 procent av de statliga medlen (intäkter och utgifter) för näringslivets klimatomställning går till generella styrmedel, det vill säga styrmedel som påverkar

I dag uppgår denna del av befolkningen till knappt 4 200 personer och år 2030 beräknas det finnas drygt 4 800 personer i Gällivare kommun som är 65 år eller äldre i

På många små orter i gles- och landsbygder, där varken några nya apotek eller försälj- ningsställen för receptfria läkemedel har tillkommit, är nätet av

Detta projekt utvecklar policymixen för strategin Smart industri (Näringsdepartementet, 2016a). En av anledningarna till en stark avgränsning är att analysen bygger på djupa

DIN representerar Tyskland i ISO och CEN, och har en permanent plats i ISO:s råd. Det ger dem en bra position för att påverka strategiska frågor inom den internationella