Evaluation of Different Speed and Position Estimation Methods Suitable for Control of a PMSM

IN

DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2016 ,

Sensorless Control of a PMSM

Evaluation of Different Speed and Position Estimation Methods Suitable for Control of a PMSM

ISAK WESTIN

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

.

Master of Science Thesis MMK 2016:168 MDA569 Sensorless Control of a PMSM

Isak Westin

Approved: Examiner: Supervisor:

2016-09-29 Hans Johansson Bengt Eriksson

Commissioner: Contact person:

Husqvarna AB Martin Lars´en

.

Abstract

.

This thesis is done together with KTH and Husqvarna AB. Husqvarna is one of the leading producers of outdoor power equipment in the world. A chainsaw developed by Husqvarna, that is driven by a permanent magnet synchronous motor (PMSM), is currently controller by a sensorless brushless DC-based controlling strategy. But to increase efficiency of the motor and to get a more flexible current control, Husqvarna wants to explore the feasibility to use a sensorless PMSM-based controlling strategy instead, called field-oriented control (FOC).

Sensorless control is achieved when the speed and position information used in the control is given by an estimator instead of a sensor. The aim of this thesis is thus to evaluate different rotor speed and position estimators that are applicable to the FOC scheme.

The thesis will include a case study which is a PMSM that is suitable to actuate a battery operated chainsaw. The thesis is then conducted in two steps. First, a literature study was performed to get an overview of different speed and position estimation methods and to get enough knowledge to determine which methods that are applicable to the case study. The different methods are compared based on a few predefined performance aspects that represents desired characteristics in an estimator. The second step is to model and simulate the one or two methods that according to the literature study seems best suited. In the simulations, each method will be controlling the motor while running through a set of test cases. The test cases are designed to imitate real potential scenarios for the motor that could occur when using the end-product. The different methods are also modelled both in continuous mode, to see if it works, and discrete mode to get closer to reality. The combined results from the simulations and literature study should indicate which method that is most appropriate to use in a sensorless control strategy for the motor of this case study.

From the literature study, the model reference adaptive system (MRAS) and the sliding mode observer (SMO) were chosen to be modelled and simulated. Both methods show good results in continuous mode simulations. When designed for discrete mode however, the SMO struggles and causes the whole control system to go unstable. The MRAS on the other hand, shows almost as good results as in continuous mode. The MRAS also shows an overall better estimation performance, both for different load cases and for motor parameter variations.

Considering the MRAS is also better suited to include a resistance estimation, which is a

useful feature, it will be proposed as the better option in the sensorless control of this case

study.

.

Examensarbete MMK 2016:168 MDA569 Sensorl¨ os reglering av en PMSM

Isak Westin

Godk¨ant: Examinator: Handledare:

2016-09-29 Hans Johansson Bengt Eriksson

Uppdragsgivare: Kontaktperson:

Husqvarna AB Martin Lars´en

.

Sammanfattning

.

Det h¨ ar examensarbetet ¨ ar gjort tillsammans med KTH och Husqvarna AB. Husqvarna ¨ ar en av v¨ arldens fr¨ amsta tillverkare av utomhusprodukter. En motors˚ ag som ¨ ar utvecklad av Husqvarna, som drivs av en permanent magnet synchronous motor (PMSM), regleras f¨ or tillf¨ allet med en sensorl¨ os brushless DC-baserad regleringsstrategi. F¨ or att ¨ oka motorns effektivitet, och f¨ or att f˚ a en mer flexibel str¨ omreglering, s˚ a vill Husqvarna utforska m¨ ojligheten att anv¨ anda en sensorl¨ os PMSM-baserad regleringsstrategi ist¨ allet som kallas field-oriented control (FOC). En reglering klassas som sensorl¨ os n¨ ar informationen om motoraxelns position och hastighet som anv¨ ands i regleringen ges av en estimator ist¨ allet f¨ or en sensor. M˚ alet med det h¨ ar examensarbetet ¨ ar allts˚ a att utv¨ ardera olika hastighets- och positionsestimatorer som g˚ ar att till¨ ampa i en FOC.

Arbetet kommer att innefatta en fallstudie som best˚ ar av en PMSM som ¨ ar l¨ amplig f¨ or att driva en batteridriven motors˚ ag. Arbetet ¨ ar sedan uppdelat i tv˚ a steg. F¨ orsta steget ¨ ar att g¨ ora en litteraturstudie f¨ or att f˚ a en ¨ overblick ¨ over olika hastighets- och positionsestimatorer och f¨ or att f˚ a tillr¨ ackligt med kunskap f¨ or att kunna avg¨ ora vilka metoder som kommer fungera i den h¨ ar fallstudien. De olika metoderna j¨ amf¨ ors i f¨ orh˚ allande till n˚ agra f¨ orbest¨ amda aspekter som ska representera ¨ onskade egenskaper hos en estimator. N¨ asta steg ¨ ar att modellera och simulera de en eller tv˚ a metoderna som verkade mest l¨ ampliga enligt litteraturstudien. I simuleringarna s˚ a reglerar varje metod motorn n¨ ar den k¨ or igenom ett antal testfall. Testfallen

¨

ar konstruerade att avbilda riktiga anv¨ andningsscenarion f¨ or motorn som skulle kunna intr¨ affa vid anv¨ andande av slutprodukten. De olika metoderna modelleras i b˚ ade kontinuerlig form, f¨ or att se att de fungerar, och diskret form f¨ or att komma n¨ armare verkligheten. De kombinerade resultaten fr˚ an simulationerna och litteraturstudien ¨ ar t¨ ankta att indikera vilken metod som skulle passa b¨ ast i en sensorl¨ os regleringsstrategi f¨ or motorn i den h¨ ar fallstudien.

Baserat p˚ a litteraturstudien s˚ a valdes model reference adaptive system (MRAS) och slid-

ing mode observer (SMO) till att bli modellerade och simulerade. Simuleringarna f¨or b˚ ada

metoderna visar bra resultat i kontinuerlig form. Men n¨ ar SMO simuleras i diskret form s˚ a

blir den v¨ aldigt brusig och g¨ or s˚ a att hela reglersystem blir instabilt. MRAS visar ˚ a andra

sidan lika bra resultat i diskret form som i kontinuerlig. MRAS visar ocks˚ a en ¨ overlag b¨ attre

estimeringsprestanda, b˚ ade f¨ or olika lastfall och f¨ or parametervariationer i motorn. Eftersom

MRAS ¨ aven l¨ ampar sig f¨ or resistansuppskattning, vilket ¨ ar en anv¨ andbar funktion, s˚ a kommer

den att bli f¨ oreslagen som det b¨ attre valet i det sensorl¨ osa reglersystemet f¨ or fallstudien.

Acknowledgements

This master thesis was done by a single student from the Machine Design department at KTH Royal Institute of Technology, Stockholm. It was made in cooperation with Husqvarna AB in Huskvarna. I would like to sincerely thank my supervisor from Husqvarna, Martin Lars´en.

His experience and commitment has been a valuable support throughout the project. I would

also like to thank my supervisor and examiner at KTH, Bengt Eriksson and Hans Johansson,

for the help and feedback they have given me during this project. Finally I would like to

thank the opponent at my final presentation, Yuchao Li, for pointing out different ways to

improve the research and the final report.

Contents

Abstract iii

Sammanfattning v

Acknowledgements vii

List of Figures xi

List of Tables xiii

Acronyms xv

Nomenclature xvii

1 Introduction 1

1.1 Background . . . . 2

1.2 Problem Definition . . . . 2

1.3 Scope . . . . 3

1.4 Methodology . . . . 3

1.5 Content . . . . 4

2 Theory 5 2.1 PMSM model . . . . 5

2.1.1 Clark and Park Transformations . . . . 6

2.1.2 PMSM model in abc-frame . . . . 7

2.1.3 PMSM model in αβ-frame . . . . 8

2.1.4 PMSM model in dq-frame . . . . 8

2.2 Field-Oriented Control . . . . 9

2.3 Summary . . . 10

3 Literature Study 13 3.1 Back-EMF Estimation . . . 14

3.2 Model Reference Adaptive System . . . 16

3.3 Extended Kalman Filter . . . 18

3.4 Sliding Mode Observer . . . 20

3.5 Discussion . . . 23

3.6 Summary . . . 25

4 Simulation 27

4.1 Simulink Model . . . 27

4.2 Field-Oriented Control Design . . . 28

4.3 Test Cases . . . 29

4.4 MRAS Estimation Model . . . 30

4.4.1 Model Design . . . 31

4.4.2 Simulink Model with Variable Time Step Simulation . . . 32

4.4.3 Simulink Model with Fixed Step Size . . . 36

4.5 SMO Estimation Model . . . 41

4.5.1 Model Design . . . 41

4.5.2 Simulink Model with Variable Step Size . . . 43

4.5.3 Simulink Model with Fixed Step Size . . . 46

4.6 Discussion . . . 47

4.7 Summary . . . 54

5 Conclusion 55 5.1 Future Work . . . 56

Bibliography 57

A Matlab script of the PMSM 61

B Popov hyper-stability theorem 63

List of Figures

2.1 Different PMSM rotor designs. SPMSM to the left and IPMSM to the right. . 5

2.2 Stator-fixed abc- and αβ-reference frames. . . . 6

2.3 Rotor-fixed dq-reference frame in relation to the stator-fixed αβ-reference frame. ω

_el

is the rotor electrical speed and θ

_el

is the rotor electrical position. . . . . . 7

2.4 field-oriented control (FOC) control scheme. . . . 9

2.5 Sensorless version of FOC. . . 10

3.1 Illustration of the model reference adaptive system. . . 16

3.2 Illustration of the model reference adaptive system using the permanent magnet synchronous motor (PMSM) as reference model. . . 17

3.3 Illustration of the sliding mode observer (SMO) principle. ˆ x is the estimated state variable and x is the real (or measured) state variable. . . 20

4.1 Simulink model of PMSM and FOC. . . 27

4.2 Error between mathematical model and Simulink model of PMSM and FOC. . 28

4.3 Speed response of the tuned FOC. . . 29

4.4 The Simulink model of model reference adaptive system (MRAS) estimator. . . 32

4.5 Simulink model of PMSM and FOC extended with a MRAS estimation. . . 32

4.6 Speed response for the MRAS control running test cases 1, 2, and 6. . . 33

4.7 Estimation errors for the MRAS control running test cases 1, 2, and 6. . . 33

4.8 Speed response for the MRAS control running test cases 3, 4, and 5. . . 34

4.9 Estimation errors for the MRAS control running test cases 3, 4, and 5. . . 35

4.10 Speed response for the MRAS control running test case 7. . . 35

4.11 Estimation error for the MRAS control running test case 7. . . 36

4.12 Simulink model of the discrete MRAS-based FOC. . . 37

4.13 Simulink model of the PMSM extended with a discrete MRAS-based FOC. . . 37

4.14 Speed response for the discrete MRAS control running test cases 1, 2, and 6. . 38

4.15 Estimation errors for the discrete MRAS control running test cases 1, 2, and 6. 38 4.16 Speed response for the discrete MRAS control running test cases 3, 4, and 5. . 39

4.17 Estimation errors for the discrete MRAS control running test cases 3, 4, and 5. 39 4.18 Speed response for the discrete MRAS control running test case 7. . . 40

4.19 Estimation error for the discrete MRAS control running test case 7. . . 40

4.20 Simulink of the SMO. . . 42

4.21 Simulink model of PMSM and FOC extended with a SMO estimator. . . . 43

4.22 Speed response for the SMO control running test cases 1, 2 and 6. . . 43

4.23 Estimation errors for the SMO control running test cases 1, 2 and 6. . . 44

4.24 Speed response for the SMO control running test cases 3, 4 and 5. . . 44

4.25 Estimation errors for the SMO control running test cases 3, 4 and 5. . . 45

4.26 Speed response for the SMO control running test case 7. . . 45

4.27 Estimation errors for the SMO control running test case 7. . . 46

4.28 Simulink model of the discrete SMO-based FOC. . . 46

4.29 Speed response for the discrete SMO. . . 47

4.30 Estimation errors of the SMO using different low-pass filter (LPF)s. . . 48

4.31 Estimation errors of the discrete MRAS using different sample frequencies. . . 49

4.32 Speed response of discrete MRAS controller running test case 4 with different values for the start load. . . 50

4.33 Speed response of SMO controller running test case 4 with different values for the start load. . . 50

4.34 Speed response of discrete MRAS controller running test case 4 with different values for the start load and a anti-windup functionality for the FOC speed controller. . . 51

4.35 Speed response of SMO controller running test case 4 with different values for the start load and a anti-windup functionality for the FOC speed controller. . . 51

4.36 Speed response of discrete MRAS controller running test cases 4 and 5 combined with different values for the start load. . . 52

4.37 Estimation errors of discrete MRAS controller running test cases 4 and 5 com-

bined with different values for the start load. . . 53

List of Tables

1.1 Motor characteristics. . . . 4 3.1 Summary of literature study results. Green means the estimation method per-

forms good in relation to respective performance aspect, yellow means the

method performs satisfactory and red means it performs bad. . . 23

4.1 Computational power evaluation. . . 49

Acronyms

AC alternating current.

ANFIS adaptive neuro-fuzzy inference system.

ANN artificial neural network.

back-EMF back electromotive force.

BLDC brushless DC.

DC direct current.

EKF extended kalman filter.

FIR finite impulse response.

FIS fuzzy inference system.

FOC field-oriented control.

IPMSM interior permanent magnet synchronous motor.

LPF low-pass filter.

MRAS model reference adaptive system.

PMSM permanent magnet synchronous motor.

PSO particle swarm optimization.

Q-PLL quadrature-phase locked loop.

SGA simple genetic algorithm.

SMO sliding mode observer.

SPMSM surface permanent magnet synchronous motor.

SVM space vector modulation.

Nomenclature

v

sabc

Phase voltage for respective phase a, b and c.

v

_α

, v

_β

α and β components of the phase voltages when expressed in αβ-coordinates.

v

d

, v

q

d and q components of the phase voltages when expressed in dq-coordinates.

i

_sabc

Stator winding current for respective phase a, b and c.

i

_α

, i

_β

α and β components of the stator currents when expressed in αβ-coordinates.

i

d

, i

q

d and q components of the stator currents when expressed in dq-coordinates.

e

_abc

Induced electromotive force in the stator by the rotor for each phase a, b and c.

e

α

, e

β

α and β components of the Back-EMF when expressed in αβ-coordinates.

e

_d

, e

q

d and q components of the Back-EMF when expressed in dq-coordinates.

φ

_sabc

Total flux induced in the stator windings for respective phase a, b and c.

φ

rabc

Flux induced in the stator by the rotor magnets for each phase a, b and c.

φ

_r

Amplitude of the flux induced in the stator by the rotor permanent magnets.

ω

el

Electrical rotation speed of the rotor.

ω

r

Mechanical rotation speed of the rotor.

θ

_el

Electrical position of the rotor.

θ

r

Mechanical position of the rotor.

L

_ss

Stator inductance matrix.

L

s

Stator inductance.

R

s

Resistance in the stator windings in the PMSM.

p Number of pole-pairs of the permanent magnet in the rotor.

J Rotor inertia.

f Viscous damping constant.

T

em

Electromagnetic torque produced by the motor.

T

_L

Load torque applied to the motor.

currK

_p^{F OC}

Proportional gain constant of the current controller in the FOC.

currK

_i^{F OC}

Integral gain constant of the current controller in the FOC.

spdK

_p^{F OC}

Proportional gain constant of the speed controller in the FOC.

spdK

_i^{F OC}

Integral gain constant of the speed controller in the FOC.

K

_p^{M RAS}

Proportional gain constant of the MRAS estimator.

K

_i^{M RAS}

Integral gain constant of the MRAS estimator.

k

^{SM O}

Switching gain constant of the SMO estimator.

l

^{SM O}

Observer gain constant of the SMO estimator.

T

s

Time of sampling period.

Chapter 1

Introduction

AC electrical motors are widely used in industry. An AC motor is a machine that converts alternating current (AC) into torque, contrary to the DC motor which produces torque from direct current (DC). Transportation (like trains or cars), washing machines and industrial cranes are some applications where AC motors are used [1].

There are two main classifications of AC motors: induction and synchronous [1]. The difference is that for an induction motor, current are induced in the rotor windings whenever the speed of the rotor differs from the speed of the rotating magnetic field generated from the currents in the stator windings. While for synchronous motors, the rotor always rotates at the same speed as the rotating magnetic field. The synchronous speed can be achieved by either injecting current into the rotor or using permanent magnets in the rotor.

Of the two, induction motors are the most popular in industry, mainly because they are small, robust and cheap [2]. The drawback with the induction motor is that it is difficult to control because of its complexity and nonlinear behavior. One particular synchronous motor, the PMSM, are however gaining in popularity due its better power/mass ratio and higher efficiency. Due to the permanent magnets, there are no energy losses in the rotor. However, the PMSM is still more expensive and thus mostly used for high performance applications [3].

A motor that is very similar to the PMSM, and perhaps more used, is the brushless DC (BLDC) motor. Both have permanent magnets in the rotor and require alternating stator currents to produce constant torque [4]. The difference is that the back electromotive force (back-EMF) in the BLDC motor has a trapezoidal waveform while the counterpart in the PMSM is sinusoidal. This gives a difference in the operating requirements of the two motors.

The BLDC motor is controlled with a six-step current and the implementation of such a controller is relatively straight forward The PMSM is harder since it should be controlled with a three-phase sinusoidal current. This requires a much faster control loop which puts a lot of demands on the hardware. Additionally, all three phases are used simultaneously which makes it harder to do correct measurements. The rotor position must also be accurately known.

A mature and well-used control strategy for PMSMs is the FOC. The idea with FOC is to separately control the motor flux and torque [5]. This is done by transforming the three stator currents, represented in a three-coordinate reference frame, into a rotating two- coordinate reference frame. To do this, the exact position of the rotor needs to be known. The two current components are then controlled independent on each other. The control output, that is the new motor voltage command, is then transformed back and instructs the voltage inverter to produce the sinusoidal voltages that will be fed to the motor. This way, FOC makes the AC control behave like a DC control [6].

Traditionally for speed dependent applications, some kind of sensor is used to read the

motor speed and feed the value back to the controller. The rotor position is also needed when using FOC. However, extra sensors require extra physical space in the application and it also introduces another source of failure in the system. So, with the additional purposes of reducing cost and maintenance needs, the sensor can be replaced by an estimator that mathematically estimates the speed or position of the rotor. This is called sensorless control [2], [3].

1.1 Background

Husqvarna AB is among the worlds biggest producers of outdoor power equipment. They develop and sell products like chainsaws, lawn mowers and blowers. One of the electrical chainsaws developed by Husqvarna is driven by a PMSM which is currently controlled with a sensorless BLDC six-step controlling approach. In an attempt to improve the performance of the PMSM in this application, Husqvarna wants to investigate the feasibility to replace the BLDC controller with a sensorless PMSM based controlling method.

By using a PMSM based controlling method, the current control, using FOC, and motor dynamics get more flexible. So if applied correctly, the application should improve in efficiency and different driving behaviors will be enabled. The implementation is however more complex and has higher demands on the time frame of the controller and the computational power.

By switching controlling method, there will also be a need to design a new speed and position estimator. This thesis will thus evaluate different speed and position estimation methods in order to propose the method that would be most suitable for the new controlling method.

1.2 Problem Definition

The field of motor speed and position estimation is not new. There exists a lot of literature of this technique with several different methods used in applications. However, there are few comparisons made and there are no general guideline to follow for different applications. This thesis will therefore contribute with such a guideline for a specific application and aim to answer following questions:

1. What is the difference between the speed and position estimator solutions for sensorless control of a PMSM used today?

2. Which solution is best suited for the platforms used by Husqvarna in regard to accuracy and computational efficiency?

The first objective in this thesis is a comprehensive literature study. The result of the literature study will be an analysis and evaluation of different position and speed estimators that can be used to provide the position and speed information of the rotor to the controller in all situations. The literature study aims to answer research question 1 and parts of research question 2.

The performance of the estimation methods that, according to the analysis, seems suitable

for the application in this thesis will then be further studied. The methods will be modeled

and simulated in Simulink and compared by a few predefined test cases that should represent

the application in use. The simulations aims to provide further answers to research question 2.

1.3 Scope

The research area of sensorless control is huge and in order to achieve a valuable result in time, there need to be some delimitations. Since the literature study will be the base of the thesis, that is where most delimitations will be applied.

The primary estimation methods that will be studied are the ones that are designed for the high speed range. It is a common problem for high speed estimators to perform unsatisfactory at low speed and vice versa. Since low speed estimators and high speed estimators are two different (but related) research areas, this thesis will focus on the latter one. Also, both rotor speed and rotor position estimators will be investigated since one can be derived from the other.

Considering the application in this thesis involves a surface permanent magnet synchronous motor (SPMSM), only estimation methods that can be applied to such a motor will be stud- ied. Many methods is applicable to both interior permanent magnet synchronous motors (IPMSMs) and SPMSMs but methods that are applicable to IPMSMs only will be disre- garded.

When talking about sensorless control, it is preferable to separate the motor control and the speed estimation. Since the focus in this thesis is the estimation part, the motor control is decided to be FOC, see Section 2.2. Thus, only methods that are applicable to FOC will be considered.

If the result from the literature study shows that several estimation methods could be suitable for the application, only the two methods with highest potential will be further investigated. This limitation is needed to ensure efficiency in the simulation analysis.

The literature study focus on finding both theory and applications of the different esti- mation methods. Theory is necessary to understand the equations and how they are used to estimate the rotor speed or position. Applications are useful to discover practical limitations and to get a hint of the overall performance of each method.

1.4 Methodology

Since this area of research is somewhat established, this thesis will mostly be based on existing theory. Thus, a deductive researching approach is suitable. First, a set of research questions will be defined, see Section 1.2. The objective of this thesis is to find answers to these questions. Based on the research questions, the literature study and simulations, one optimal solution for the given application will be suggested as the best solution.

This thesis outline suggests a mixed methods approach to be adopted [7]. The literature study will be purely qualitative and will be the greater part of this thesis. The methods from the literature study will be compared using certain performance aspects that are qualitatively chosen. The comparison result should indicate which methods are most suitable to be subject for the following case study. The testing in the case study will then be conducted in a quantitative manner. Test cases will be qualitatively predefined and the results will summarize the performance of each estimation method.

The case study will be based on a motor specification that is suited for a battery oper- ated chainsaw driven by a SPMSM and controlled by FOC. The motor characteristics are summarized in Table 1.1.

The result from this thesis will then serve as a guideline for sensorless control in similar

applications. Similar applications include sensorless speed control of a SPMSM using a low-

cost microprocessor.

Table 1.1: Motor characteristics.

Number of poles in the rotor 14 Number of slots in the stator 12

Rated power output 1.2 kW

Rated torque output 1.5 Nm

Rated speed 8500 rpm

No load speed 11000 rpm

Rated DC link voltage 30 V

Maximum current 60 A

Before conducting this study, some potential ethical issues are considered. Firstly, plagia- rism is not acceptable and will be considered throughout the whole project. Also, since the end user of Husqvarna’s products mostly are humans, there should always be some concern about their safety and integrity when using the final product.

1.5 Content

The rest of the report is organized as follows. Chapter 2 presents some previous knowledge

and theory that this thesis will build upon. The theory includes PMSM dynamical equations,

Clarke and Park vector transformations and the FOC. Chapter 3 presents different sensorless

control methods found in the literature. Different applications for these methods will be

discussed and the feasibility for this case study is evaluated. The simulation procedure,

simulation model, test cases and simulation results will be presented in Chapter 4. The work

will then be concluded and future work will be presented in Chapter 5.

Chapter 2

Theory

In this chapter, theories and equations are presented that will be used throughout this thesis.

This will be a mixture of former knowledge and knowledge acquired for the purpose of this thesis. These equations will form the ground on which following control system and estimation methods will be based upon.

2.1 PMSM model

There are many different kinds of PMSMs that differ in the physical structure of the machine.

Two of the most used types can be seen in Figure 2.1 and are the SPMSM and the IPMSM [2].

(a) SPMSM (b) IPMSM

Figure 2.1: Different PMSM rotor designs. SPMSM to the left and IPMSM to the right.

For the SPMSM, the magnets are placed on the surface of the rotor. The magnets are evenly distributed on the surface so the stator inductances does not depend on the rotor position. This type of motor is the easier one to produce and control but the magnets are more exposed for damage and the machine lack the saliency information that could be used to find the rotor position. In the IPMSM, the magnets are integrated in the rotor which contributes to more mechanical durability and robustness. This motor is a saliency pole type and the inductance do depend on the rotor position. However, it is more expensive to manufacture and more complex to control.

For the case study in this thesis, a SPMSM motor type is used and it will be the only type

studied. Therefore, for the rest of this report, when referring to a PMSM it is the SPMSM

type that is meant if nothing else is stated.

2.1.1 Clark and Park Transformations

The three-phase motor dynamics can be described from different point of views by using different reference frames. The most standard frame is the three-phase stationary reference frame, also called abc-frame. This is the physical reference frame of the motor where the three axes a, b and c represents the three electrical phases in the stator. Current and voltage measurements are also done in this reference frame. The abc-frame is thus fixed to the stator.

When using FOC as the control structure, the measured three-phase stator currents needs to be transformed into the rotating two-coordinate reference frame, called dq-frame. This transformation is called the Park transformation [1]. Furthermore, the voltage control signal needs to be transformed back into abc-coordinates. Estimators usually also use transformed values.

The Park transformation is done in two steps. The first step is to project the the three- phase variable onto the complex two-coordinate αβ-reference frame, called the Clark trans- formation. Like the abc-frame, the αβ-frame is also fixed to the stator but the number of axes are reduced to two as seen in Figure 2.2.

α a b

c

β

Figure 2.2: Stator-fixed abc- and αβ-reference frames.

The Clark transformation is defined as follows [1]

x

α

x

_β



= r 2 3

"

1 −

¹₂

−

¹₂

0

√3

2

−

√3 2

# 

 x

a

x

_b

x

_c



 . (2.1)

The second step in the Park transformation is to transform the αβ- representation of the

variable into the rotating dq-reference frame which is fixed to the rotor. This transformation

is dependent on the rotor electrical position which is defined as the angle between the stator

α-axis and the rotor d-axis, see Figure 2.3. The ”direct” d-axis is aligned with the rotor flux

and the ”quadrature” q-axis is positioned at 90

^◦

in the positive rolling direction[8].

α β

d q

θ

el

ω

el

Figure 2.3: Rotor-fixed dq-reference frame in relation to the stator-fixed αβ-reference frame.

ω

_el

is the rotor electrical speed and θ

_el

is the rotor electrical position.

The rotation transformation is defined as [1]

i

d

i

_q



=  sin θ

el

cos θ

_el

cos θ

_el

− sin θ

_el

 i

α

i

_β



. (2.2)

Depending on the method, the speed and position estimator may use both Clarke transformed or Park transformed variables.

2.1.2 PMSM model in abc-frame

The three-phase stationary frame representation of a PMSM is given by [2]

v

_sabc

= R

_s

i

_sabc

+ d

dt φ

_sabc

(2.3)

where v

_sabc

is the stator voltage in respective phase, R

s

is the stator winding resistance, i

_sabc

is the current in the stator windings in respective phase and φ

_sabc

is the flux within the stator windings in respective phase. The stator flux is given by

φ

sabc

= L

ss

i

sabc

+ φ

rabc

. (2.4)

The first term is the flux produced by the currents in the stator windings; L

ss

is the stator inductance matrix which depend on the type of machine. Since this thesis only deals with the SPMSM, the inductance of the d-axis is equal to the inductance in q-axis. This means that a constant value can be used for the stator inductances in the transformed representations of the machine [2]. The second component, φ

_rabc

, is the flux induced in each stator phase by the permanent magnets in the rotor. It has a sinusoidal waveform and depends on the rotor position.

Equation (2.3) can then be rewritten as v

_sabc

= R

_s

i

_sabc

+ d

dt (L

_ss

i

_sabc

) + ω

_el

d

dθ

_el

φ

_rabc

. (2.5)

The last component represents the back-EMF induced by the rotor according to e

_abc

= ω

_el

d

dθ

el

φ

_rabc

. (2.6)

Both electrical speed and position has the same relation to their mechanical counterparts, see Equations (2.7) and (2.8).

ω

el

= pω

r

(2.7)

θ

el

= pθ

r

(2.8)

where ω

_r

is the mechanical speed of the rotor, θ

_r

is the mechanical position of the rotor and p is the number of pole pairs of the magnets in the rotor.

2.1.3 PMSM model in αβ-frame

By using Equations (2.5) and (2.1) the current dynamic equations of the PMSM expressed in αβ-coordinates can be obtained as

d

dt i

αβ

= − R

s

L

s

i

αβ

+ 1 L

s

(v

αβ

− e

_αβ

) (2.9)

where i

_αβ

= [i

_α

i

_β

]

^T

are the currents in the stator windings, v

_αβ

= [v

_α

v

_β

]

^T

are the applied stator voltages, L

s

is the constant value of the stator inductance in d- and q-axes, and e

αβ

= [e

α

e

_β

]

^T

is the induced back-EMF for each axis and is given by

e

α

= − r 3

2 φ

r

ω

el

sin(θ

el

) = −k

e

ω

el

sin(θ

el

) (2.10) e

_β

= r 3

2 φ

r

ω

_el

cos(θ

_el

) = k

e

ω

_el

cos(θ

_el

) (2.11) where φ

r

is the amplitude of the flux produced by the rotor magnets.

2.1.4 PMSM model in dq-frame

By using Equations (2.5), (2.1) and (2.2) the current dynamics of the PMSM in dq-coordinates can be expressed as

d

dt i

_dq

= − R

s

L

_s

i

_dq

+ 1

L

_s

(v

_dq

− e

_dq

) (2.12)

where i

dq

= [i

d

i

q

]

^T

are the currents in the stator windings, v

dq

= [v

d

v

q

]

^T

are the applied stator voltages and e

_dq

= [e

_d

e

q

]

^T

are the induced back-EMF for each axis and are given by

e

d

= −L

s

i

q

ω

el

(2.13)

e

q

= L

s

i

d

ω

el

+ φ

r

ω

el

. (2.14) Equations (2.12)-(2.14) can also be represented in state space form with state vector x = [i

d

i

q

]

^T

according to

˙x = Ax + Bu

y = Cx (2.15)

with

A =

"

−

^R_L^s

s

ω

el

−ω

_el

−

^R_L^s

s

#

; Bu =



₁

Ls

v

d 1

Ls

(v

q

− φ

r

ω

_el

)



; C = 1 0 0 1



.

The electro magnetic torque produced by the PMSM in dq-coordinates, T

em

, is given by T

_em

= 3

2 pφ

_r

i

_q

. (2.16)

The mechanical dynamic equation of the PMSM is given as J d

dt ω

r

= T

em

− f ω

r

− T

L

(2.17)

where J is the rotor inertia, f is the viscous damping constant and T

_L

is the load torque.

2.2 Field-Oriented Control

The principle with field-oriented control is that it controls the motor current in dq-frame, instead of directly in abc-frame. The benefit with this is that since the dq-frame rotates together with the rotor, the electrical state of the motor does not depend on the rotor position.

The currents is thus perceived as DC currents which are easier to control. Another advantage with FOC is that in the PMSM, the d-component of the current is proportional to the flux and the q-component is proportional to the torque [5], [9]. The FOC can thus manage to control both flux and torque by controlling the currents in d- and q-coordinates separately. A FOC control scheme applied on the PMSM is illustrated in Figure 2.4.

SVM

PMSM PI

PI PI

dtd_

Voltage inverter

abc

dq ia

position speed

speed ref

id

vq

vd

iq

i refq

i refd

ib

i_c

+_

+_ +_

Figure 2.4: FOC control scheme.

The measured three-phase currents are transformed into dq-coordinates using the rotor

position as described in Section 2.1.1. The d and q components of the current are then fed

to respective d and q current controller. Both current controllers are usually traditional PI

controllers. The PI controllers outputs the dq-components of the voltages which are fed to a

space vector modulation (SVM) function. The SVM instructs the voltage inverter to produce

the sinusoidal voltages that should be applied on the motor.

As can be seen in Equation (2.16), the q component of the current alone controls the motor torque. Thus, it is clear that to get maximum torque, the d component of the current should be zero and this is often the case in applications. However, if the d-axis is controlled with a non-zero current, the speed range of the motor changes at the cost of torque. This technique is called flux-weakening control [8].

When using FOC as a speed controller, an extra speed PI controller is added which decides the reference signal for the q current controller, as seen in Figure 2.4. The feedback signal for the speed controller is often obtained by differentiating the position signal from an encoder mounted on the motor.

For a sensorless control scheme, the position encoder on the motor is replaced by an esti- mator that only uses measurable electrical quantities like current and voltage. The sensorless FOC scheme is illustrated in Figure 2.5. The estimator then provides rotor position for the transformations as well as rotor speed for the speed control.

SVM

PMSM PI

PI PI

Voltage inverter

abc

dq i_a

i_abc v_abc speed ref

position speed

i_d v_q v_d

i_q i ref_q

i ref_d

i_b i_c

+_

+_ +_

Speed and position estimator

Figure 2.5: Sensorless version of FOC.

To make it easier for the controller a feed-forward part can be added to the FOC. The feed- forward compensates each component of the voltage control signal for known disturbances.

The compensation is calculated with Equations (2.13) and (2.14).

2.3 Summary

In this chapter the theoretical background for this thesis is presented. Usually, the equations of a three-phase AC motor is described in a two dimensional three-axis coordinate system, called abc-frame, where each axis represents one phase in the stator. To make the control of such a motor easier, the control system called FOC transforms the motor quantities (e.g.

stator current and stator voltage) into a two dimensional two-coordinate rotating frame called

dq-frame.

In dq-frame, the current control resembles the easy control of a DC motor. Also, d- and q- currents relates to different motor characteristics independent on each other. This also means that they can be controlled separately.

The FOC control needs the value of the rotor position for a successful coordinate trans- formation. Typically this value is given by a position encoder mounted on the motor shaft.

To remove the weight, space and source of failure given by an encoder, the FOC can instead

receive values of the rotor position from an estimator that mathematically approximates the

rotor position using measured motor states (i.e. stator current and stator voltage). By using

an estimator to produce rotor position values, the FOC operates completely sensorless.

Chapter 3

Literature Study

In this chapter several different existing sensorless control methods for PMSMs will be an- alyzed and discussed. The methods are found in existing literature and only methods that are applicable for this case study will be included, as described in Section 1.3. Also, methods with small coverage, i.e. methods that are immature or unusual, will be disregarded. This is because it is hard to do a proper analysis based on literature without sufficient amount of data.

In the analysis, the different methods will be compared through some specific aspects.

The aspects are chosen together with supervisors to best expose the benefits and drawbacks for each method in relation to the case study. The considered aspects are:

• The requirements put on the processor. Determine how much computational power is needed. To decrease the cost of the final product, the hardware cannot be too expensive.

Thus, the lower computational power needed, the better.

• The ability to estimate position and speed of the rotor at all speed ranges. This does somewhat correspond to the estimation performance of the method. The estimation error should be as low as possible.

• Sensitivity to motor parameter variations. Since the motor parameters varies when the motor operates, mainly because of the temperature increase, it could affect the correctness of the estimation. If the methods are too reliant on the motor parameters, it will introduce a problem when implemented on a real motor.

– A favorable feature of a method would be the ability to estimate the stator re- sistance. The knowledge of the stator resistance could be used to improve the estimation accuracy and to estimate the temperature of the motor. This will how- ever not exclude any method, it will only be considered a benefit.

• The capability to re-tune for different motors. If the implementation of a method is found beneficial there might be an interest to incorporate it into other products as well.

In that case, the method cannot depend too much on tuned parameters. That will make the integrating process too time consuming.

Sensorless control can be divided into three main strategies according to [10]. The first

one is open loop methods. The methods in this category are straight forward and relatively

easy to implement. They are however not really reliable since they have no real protection

against parameter variations and measurement noise. These methods are thus seldom used

in applications. The second category is closed loop methods that are superior to open loop

methods. Since these methods use the error signal between measured and estimated quantities, the convergence can be guaranteed. Here some of the most used high speed estimation methods are found. The main drawback is that most of the methods depends on the motor parameters or the back-EMF which is small at low speed. Hence, they are mostly used for high speed applications only. The last category targets control of the standstill and low speed region of the motor. This is the methods not based on the motor fundamental equations. These methods are however not suitable for high speed control.

This thesis will focus on the closed loop methods since they usually performs best and are by far the most used methods in high speed applications.

3.1 Back-EMF Estimation

The idea with the back-EMF estimation method is to use the electrical equations of the machine to estimate the induced back-EMF. As seen in Equations (2.10) and (2.11), the back- EMF contains information about the rotor position and speed. Combining these equations gives the rotor position according to

θ

_el

= − tan

⁻¹

 e

α

e

_β



. (3.1)

The back-EMF estimation method is probably the most common method for sensorless PMSM control [1], [11]. It is simple and easy to calculate while still showing great performance at high speed control applications [12], [13], [14].

However, a well known problem with this type of method is that the back-EMF is depen- dent on the rotor speed. This means that at low speeds, the back-EMF is very small and hence difficult to estimate correctly [12], [13], [14], [15], [16]. The most common way to solve this problem is to use some kind of start-up strategy or another type of estimation method that provide the control with rotor speed and position information in the low-speed range. When the back-EMF value is large enough, at a specific speed, the back-EMF method takes over the estimations. In [16] for example, a simple start-up strategy is designed using a predefined reference speed input. The rotor position is estimated by integrating the reference speed.

This method can only be used in the start of an operation. The estimations will deviate with time and a better estimator will be needed. In [15], an active flux observer is used for position estimation at low speed. Here, the flux is estimated and used to calculate the rotor position.

Other mentioned disadvantages with the back-EMF estimation method are the sensitivity against parameter uncertainties, measurement noise and inverter irregularities [11], [13], [14].

The most straightforward approach to use the back-EMF for speed and position estima- tions is by using it as an open loop method, i.e. using the motor equations to directly calculate the back-EMF from measured values as in [16]. But as mentioned above, this is not a reliable choice since it heavily depends on correct motor parameters and noise-free measurements.

Neither of these problems are discussed and the method is only verified through simulations.

Another, more robust, way is to use an observer, making it a closed loop method. When using an observer to estimate the back-EMF there are two different approaches: indirect and direct estimation [17].

In indirect estimation, the back-EMF is considered a disturbance and is obtained by observing the stator currents and matching them with the measured currents. In [18], a Luenberger observer is observing the stator currents in αβ-frame, using Equation (2.9). The observer is defined as

d

dt ˆi

_αβ

= − R

_s

L

s

ˆi

_αβ

+ 1 L

s

(v

αβ

− e ˆ

_αβ

) + K

1

(i

αβ

− ˆi

_αβ

) (3.2)

where ˆi

αβ

= [ˆi

α

ˆi

β

]

^T

are the observed currents, i

αβ

= [i

α

i

β

]

^T

are the measured currents, v

_αβ

= [v

α

v

_β

]

^T

are the measured voltages and ˆ e

_αβ

= [ˆ e

α

ˆ e

_β

]

^T

are the estimated back-EMF.

K

₁

is the observer gain.

The back-EMF is here treated like a disturbance and are estimated using the current observer error according to

d

dt e ˆ

_αβ

= K

2

(i

_αβ

− ˆi

_αβ

) (3.3) where K

2

is the estimation gain.

The rotor position and speed are then extracted from the back-EMF estimations by the means of quadrature-phase locked loop (Q-PLL) [18]. This is to reduce the risk of noise issues that could occur when calculating the rotor position directly from the back-EMF, as in Equation (3.1).

Similarly, in [11] the back-EMF is seen as a disturbance and the currents are estimated by a simple observer. The estimation error is then used to derive the back-EMF. The observer is defined using the electrical equations of the PMSM in a specific Park δγ-reference frame.

A thorough robustness study of the method shows that it is sensitive to motor parameter uncertainties, specifically the stator resistance variations. It is also sensitive to measurement errors and inverter irregularities, especially at low speeds. An estimator of stator resistance is added to address these uncertainties which makes the method more robust.

In direct estimation, the back-EMF is considered a state variable and is directly estimated by the observer. In [15], the PMSM is presented in state space form in αβ-frame using the state vector x = [i

α

i

β

e

α

e

β

]

^T

. The state vector, and thus the back-EMF components, are estimated using a Luenberger observer. The approach seems stable and accurate for low speeds. It is however only tested up to 150 rad/s which is less than 32% of the rated speed of the used motor. The switch between low speed control and high speed control also causes an overshoot.

In [19], the PMSM is modeled in αβ-frame with the same state vector as above. Two different observers are then used to estimate the state vector: Luenberger observer and SMO.

The performance of both observers are then compared and the SMO comes out as the better choice (Section 3.4 will give a closer look at the SMO). A Kalman filter is also used to filter out noise from the estimated back-EMF components and and estimate the rotor speed.

Looking at the performance aspects, the back-EMF estimation method is seen as one of the kinder methods in terms of computational requirements. Since it only consists of one observer and some straight forward calculations, it is not so heavy for the processor. The method also seems to perform satisfactory at high speed but requires a complementary estimation method for standstill and low speed.

According to many [11], [13], [14], the back-EMF method is highly sensitive to variations of the motor parameters. Since the motor parameters varies during operation, the performance depends quite a lot on this problem. There are however some proposed solutions. For example, [18] calculates the speed and position using the Q-PLL which reduces the dependability on parameter correctness. In [16], an artificial neural network (ANN) is added to the FOC to improve accuracy and robustness.

In [11], an additional reference model is used to estimate the stator resistance. The estimated resistance is adjusted on-line when estimating the back-EMF. Results shows that while the resistance estimation is erroneous, it still improves the performance of the control.

For the aspect of using the control on several different motors, the back-EMF estimation

method seems neither bad nor perfect. Of course, the motor parameter used in the motor

equations has to be updated. The gain-values for the observer will probably also have to be

re-tuned. If some other method is used together with the back-EMF method, the parameters of that method might also need to be changed.

Overall, the back-EMF estimation method seems good much because of its simplicity.

However, it struggles a lot at standstill and low speed region and will in most cases require an additional method for that purpose. Since it is based upon the motor equations, the dependability on correct motor parameters are somewhat inevitable. To improve in these areas, some extra methods are preferable. If that is the case, the simplicity of this method is somewhat lost and it seems rather difficult to get it working correctly.

3.2 Model Reference Adaptive System

Another closed loop method to estimate the speed and position of the rotor is the MRAS. The basics of MRAS is to use two independent models: One reference model which is independent on the variable to be estimated and one adjustable model which is dependent on the variable to be estimated. The two models uses different sets of inputs to calculate the same state variables which are in turn fed to a certain adaption mechanism. The adaption mechanism uses the difference between the two signals to tune the estimated variable and feed it back to the adjustable model. The estimated value will this way be driven to its true value [10], [13].

The MRAS scheme is illustrated in Figure 3.1.

Reference model

Adjustable model

Adaptation mechanism

Input

Output Estimated variable

Error

Figure 3.1: Illustration of the model reference adaptive system.

The MRAS can be implemented in several different ways depending on the state variable to be estimated, choice of reference model, choice of adjustable model and choice of adaptive mechanism. When using MRAS to estimate the rotor speed and position of a PMSM, a motor variable, e.g. stator current or power, should be chosen as the state variable [9]. Two different models are then used: The reference model calculates the state variable without using the rotor speed or position and the adjustable model calculates the same state variable with the help of the rotor speed or position value given by the adaptive mechanism.

Similar to the back-EMF method, the MRAS thrives on its simplicity. The method is

popular because of its simple structure and good dynamic performance while its still easy

to implement [3], [20], [21], [22]. It is regarded as a good estimator for both speed and position [12] and it requires low computational effort [23].

The most frequent criticism against MRAS is its sensitivity to motor parameter varia- tions [24], [25]. Since the models used mostly includes mathematical equations of the motor the parameter accuracy is important. However, according to [10], this problem can be some- what reduced by using the motor itself as a reference model, see Figure 3.2. This makes the implementation easier and the stability can be guaranteed. This strategy is also well adopted in studies and applications.

Plant (PMSM)

Adjustable model

Adaptation mechanism

Input

Output Estimated variable

Error

Figure 3.2: Illustration of the model reference adaptive system using the PMSM as reference model.

The by far most common approach is to use the electrical model of the PMSM in dq-frame, see Equation (2.12), as the adjustable model. As can be seen, the equations depend on the rotor electrical speed. In the adjustable model, the rotor speed is replaced by the estimated value of the rotor speed, ˆ ω

el

. The electrical motor equations estimate the stator currents which are then compared to the measured ones [20].

There are other solutions as well. In [26], the adjustable model is given by the PMSM electrical equations in dq-frame together with a Luenberger observer. A somewhat different approach can be found in [9]. In the reference model, the measured currents and voltages of the motor are used to calculate the active power, P , according to

P = v

d

i

d

+ v

q

i

q

. (3.4)

The same equation is then also used in the adjustable model. The voltages are given by the PMSM electrical equations supplied with the measured currents. The two values of actual power is then used to estimate the rotor speed.

The choice of adaptation mechanism shows a little more diversity. The simplest and most

common method in applications is to tune the estimated speed value with a PI adaptive mech-

anism [13], [24], [26]. This is basically a PI controller executing on a specific expression given

by the actual and estimated state variables. By using Popov’s hyper-stability theorem [27],

this rule can be chosen such that the stability of the estimator is guaranteed. This is one reason why the PI adaptation mechanism is so popular [25].

As an attempt to improve the estimation performance, some other approaches for the adaptation mechanism can be found. In [9], an ANN is used as adaptation mechanism to tune the speed estimation value. In the ANN, measured stator currents and voltages together with the previous speed value and the active power estimation error goes through a network of neurons that is capable of machine learning and pattern recognition. The network then outputs one single value, the estimated rotor speed, and feed it back to the adjustable model.

The ANN based speed estimation is also compared to the PI and the particle swarm optimiza- tion (PSO) based PI estimation method. Results show that the ANN estimation performs better than its competitors.

Another type of adaptive mechanism is employed in [20] and is based on sugeno fuzzy in- ference system (FIS). The sugeno FIS uses the error and derivative error between the reference model and adjustable model to estimate the rotor speed.

In [12], a combination of sugeno FIS and a hybrid learning algorithm, named adaptive neuro-fuzzy inference system (ANFIS), is used as the adaption mechanism. The mechanism uses the estimated currents from the adjustable model to estimate the rotor speed. The error between measured and estimated current is then used to tune the estimation parameters.

As already mentioned, the MRAS estimation method uses relatively straight forward cal- culations which makes it simple and not so demanding for the processor. This will of course depend on the choice of adaptation mechanism. The MRAS should also give a good perfor- mance at all speed range.

A well known problem with this method is the sensitivity for parameter variations which could also affect the estimation performance, especially at low speed. One way to handle this problem is presented in [26] where the MRAS is used to estimate the rotor speed and the stator resistance simultaneously and adjust the resistance parameter on-line. The approach is simply to let the adaptation mechanism tune both speed and resistance in the adjustable model.

The same approach is taken in [23]. The stator resistance parameter in the adjustable model is seen as an estimated value which is tuned by the adaptation mechanism. The stability of this estimator is here guaranteed by the Popov hyper-stability theorem.

As for the ability to tune for different motors, this also depends on the choice of adaptation mechanism. When using PI adaptation, the proportional and integral gain constants affect the tracking performance of the estimation and will probably have to be re-tuned. When using artificial intelligence adaptation, the estimator parameters are often decided by some off-line training. If the motor is changed, this training will probably have to be done again.

It is however often done automatically.

In summary, the MRAS estimation method seems like a good choice for a simple and good performing rotor speed and position estimator. It also seems viable for the entire speed range. Since the stability can be guaranteed when using the motor as reference model, which is applicable in this case study, this method looks promising. A big advantage is also of course if it is relatively easy to estimate the stator resistance and adjust it on-line, without sacrificing too much processing time, since this will most certainly improve the performance.

3.3 Extended Kalman Filter

The kalman filter is a state observer which estimates the states of a dynamic linear system

based on least-square optimization. The extended kalman filter (EKF) is an extension of the

kalman filter that handles nonlinear systems [10]. The nonlinearity is handled by linearizing

the system at each sample point. The keystone of the EKF is that it takes care of model inaccuracies and measurement noise in a system by assuming them to be zero-mean white Gaussian noise [25].

The EKF uses a nonlinear system of the form

˙x(t) = f (x(t), u(t)) + w(t)

y(t) = h(x(t)) + v(t) (3.5)

where x(t) is the state vector, u(t) is the input vector and y(t) is the output vector. Both w(t) and v(t) are zero-mean white Gaussian noise. w(t) is related to model uncertainties and v(t) is related to measurement noise.

The estimation is then conducted in two steps: [28]

1. Prediction step. The state estimate is predicted based on previous state and new input values. The state covariance matrix is also predicted using the linearized system.

2. Correction step. The EKF gain matrix is calculated using the linearized output of the system. The state estimate and state covariance matrix is then corrected based on the predicted values and the EKF gain matrix.

Since EKF is a state observer, the rotor speed and position has to be added as state variables in order to get estimated. Furthermore, the EKF has the option to be implemented in both αβ-frame (state vector x = [i

α

i

β

ω

el

θ

el

]

^T

) or dq-frame (state vector x = [i

d

i

q

ω

el

θ

el

]

^T

).

It is not clear, however, which frame is best. By implementing in αβ-frame, the EKF might converge to the wrong solution [29]. Some hardware implementation solutions for sine and cosine functions will also be required [25]. On the other hand, when implementing in dq-frame the measured currents need to be transformed using the estimated position from previous sample time. This will result in a constant position estimation error [29].

Since the EKF is based on least-square optimization, it finds the optimal values for the state variables. This generally gives a very good estimation [25], [30]. Additionally, the EKF behaves like a LPF, it efficiently reduces input noise. This includes both system noise and measurement noise [10], [25], [31]. The filtering is so efficient that in some cases, the estimated currents are used as feedback in the FOC control, instead of the measured ones [28].

Other mentioned advantages with the EKF is robustness against parameter variations [10], [32], good performance against load interference [25] and that it does not need knowledge about mechanical parameter or initial position [33].

One acknowledged disadvantage with the EKF regarding applications on a PMSM is the high requirements on the processor. Since there are several matrix multiplications, the EKF is very computationally costly and time inefficient compared to other methods [10], [29], [30].

This will make it hard to implement on an electrical drive if the processor used is not very high performing.

The tuning of the EKF is done by choosing initial values for the state and noise covariance matrices. This is perhaps the most important part of the EKF design, especially the tuning of the noise covariance matrices [10], [32]. The tuning is also one of the most common criticism directed towards the EKF. There is no obvious procedure to do the tuning. The most common approach is the tune it in a trial and error process [28], [33], [34]. Since there are a lot of initial values to set, this process is really hard and time consuming [25], [30].

There is however one attempt to establish a defined tuning process to cope with this

problem [35]. By normalizing the system and the EKF algorithm, an effective initial guess

of the covariance matrices can be found. It is claimed that if a coherent normalization of

both the system and the EKF is accomplished, the found covariance matrices will fit almost

all standard PMSM drives. The normalization is done by first defining a set of base values, then dividing each variable of interest by the respective base value. This approach has been recognized in [34] where it is combined with the simple genetic algorithm (SGA) to optimize the covariance matrices on-line.

With good accuracy on estimating the rotor speed and position and the good robustness against motor parameter variations, the EKF seems a profitable estimation method. As noticed in [25], the EKF can struggle at very low speeds since, at that point, the noise is almost as large as the control signals.

No specific stator resistance estimation using the EKF has been found. It should however be possible to add the stator resistance as a state variable and assume it constant during the sample period. This is similar as [28] has done to estimate the torque load. At the same time, by increasing the amount of state variables, the tuning of the EKF will get heavier. The tuning problem will also make it harder to apply the filter on other motors as well since the tuning has to be done for each motor.

With regard to the requirements put on the processor, the EKF seems unfavorable for this case study. The heavy calculation made by the EKF does not fit into the picture of using an efficient low-cost micro-controller. An attempt to improve in this aspect is a solution where the EKF gain matrix is calculated off-line [28]. This will however add the need of a second parallel computing device which is not really applicable in this case.

3.4 Sliding Mode Observer

The idea of the SMO is to define a sliding mode equation, use the sliding mode equation to define a so called sliding surface, which represents the real state variables, and let a high- frequency, discontinuous switching function force the state variables of the observer towards the sliding surface until sliding mode occurs [36]. SMO principle can be seen in Figure 3.3.

Sliding surface

Sliding mode reached

Sliding mode equation

x x ˆ

Figure 3.3: Illustration of the SMO principle. ˆ x is the estimated state variable and x is the real (or measured) state variable.

As can be seen, with an infinite switching frequency in the sliding mode, the sliding surface

is reached and the observed state variables equals the real ones. However, infinite frequency