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Preface

The research work was carried out between April 2015 and April 2019 in ElectricDrives research group, Department of Electrical Engineering and Automation,Aalto University. This work has been financed in part by the Doctoral Pro-gramme of Electrical Engineering, Aalto University and in part by ABB OyDrives, Helsinki, Finland. The research grants awarded by the Walter AhlstömFoundation and Kansallis-Osake-Pankki fund are most gratefully acknowledged.

First of all I would like to express my deepest gratitude to my supervisor Prof.Marko Hinkkanen for providing me with an amazing opportunity to be a part ofan ongoing research project on the control of electric drives. I am thankful for hisguidance and support during this research work. I wish to thank Prof. FernandoBriz, Prof. Radu Bojoi, and Prof. Gianmario Pellegrino for their cooperationduring completion of Publications I and III. The collaboration with Dr. SeppoSaarakkala was really helpful as he was always there to help me in the lab. Iwish to express my thanks to all my colleagues at the Department of ElectricalEngineering and Automation for providing an amazing working environment. Iam thankful for the people at ABB for their constructive remarks and helpfulsuggestions on the research work.

Finally, I would like to thank my parents for their endless support throughoutmy life, and I would like to thank my wife Reeja for her love and support duringthis journey. Loads of love for our daughter Zainab, you are the biggest reasonfor joy and happiness in our lives.

Espoo, September 30, 2019,

Hafiz Asad Ali Awan

1

Preface

2

Contents

Preface 1

Contents 3

List of Publications 5

Author’s Contribution 7

Abbreviations 9

Symbols 11

1. Introduction 131.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Objective and Outline of the Thesis . . . . . . . . . . . . . . . . . 14

2. System Model 172.1 Space Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Inverter Model and Pulse-Width Modulation . . . . . . . . . . . 172.3 Continuous-Time Motor Model . . . . . . . . . . . . . . . . . . . 19

2.3.1 Saturation Characteristics . . . . . . . . . . . . . . . . 192.3.2 Voltage and Electromagnetic Torque . . . . . . . . . . 21

2.4 Discrete-Time Motor Model . . . . . . . . . . . . . . . . . . . . . 222.5 Continuous-Time Motor Model in Stator Flux Coordinates . . . 22

3. Control Schemes 253.1 Continuous-Time Current Control . . . . . . . . . . . . . . . . . 25

3.1.1 Current as a State Variable . . . . . . . . . . . . . . . 273.1.2 Flux as a State Variable . . . . . . . . . . . . . . . . . . 283.1.3 Digital Implementation . . . . . . . . . . . . . . . . . . 30

3.2 Discrete-Time Current Control . . . . . . . . . . . . . . . . . . . 303.2.1 Inclusion of the Control Delay . . . . . . . . . . . . . . 313.2.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . 31

3

Contents

3.3 Anti-Windup Mechanism . . . . . . . . . . . . . . . . . . . . . . . 333.4 Stator-Flux-Oriented Control . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . 343.4.2 Anti-Windup Mechanism . . . . . . . . . . . . . . . . . 36

3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 36

4. Flux Observers and Sensorless Control 414.1 Continuous-Time Flux Observer . . . . . . . . . . . . . . . . . . 41

4.1.1 Simple Observer with Speed Feedback . . . . . . . . . 414.1.2 Flux Observer with Speed Estimation . . . . . . . . . 424.1.3 Discrete-Time Implementation . . . . . . . . . . . . . 43

4.2 Discrete-Time Flux Observer . . . . . . . . . . . . . . . . . . . . 434.2.1 Observer Structure . . . . . . . . . . . . . . . . . . . . 454.2.2 Estimation-Error Dynamics . . . . . . . . . . . . . . . 464.2.3 Gain Selection . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5. Optimal Reference Generation 495.1 Conventional Methods . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Feedback Field-Weakening Methods . . . . . . . . . . 505.1.2 Feedforward Field-Weakening Methods . . . . . . . . 505.1.3 Plug-and-Play Startup . . . . . . . . . . . . . . . . . . 51

5.2 Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2.1 Feedforward Field-Weakening Scheme . . . . . . . . . 525.2.2 Current References . . . . . . . . . . . . . . . . . . . . 53

5.3 Look-Up Table Computation . . . . . . . . . . . . . . . . . . . . . 555.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6. Experimental Setup 57

7. Summaries of Publications 617.1 Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . . 63

8. Conclusions 65

References 67

Publications 73

4

List of Publications

This thesis consists of an overview and the following publications which arereferred to in the text by their Roman numerals.

I M. Hinkkanen, H. A. A. Awan, Z. Qu, T. Tuovinen, and F. Briz. Currentcontrol for synchronous motor drives: Direct discrete-time pole-placementdesign. IEEE Transactions on Industry Applications, vol. 52, no. 2, pp.1530–1541, March/April 2016.

II H. A. A. Awan, S. E. Saarakkala, and M. Hinkkanen. Flux-linkage-basedcurrent control of saturated synchronous motors. IEEE Transactions onIndustry Applications, vol. 55, no. 5, pp. 4762–4769, September/October2019.

III H. A. A. Awan, M. Hinkkanen, R. Bojoi, and G. Pellegrino. Stator-flux-oriented control of synchronous motors: A systematic design procedure.IEEE Transactions on Industry Applications, vol. 55, no. 5, pp. 4811–4820,September/October 2019.

IV M. Hinkkanen, S. E. Saarakkala, H. A. A. Awan, E. Mölsä, and T. Tuovinen.Observers for sensorless synchronous motor drives: Framework for designand analysis. IEEE Transactions on Industry Applications, vol. 54, no. 6,pp. 6090–6100, November/December 2018.

V H. A. A. Awan, T. Tuovinen, S. E. Saarakkala, and M. Hinkkanen. Discrete-time observer design for sensorless synchronous motor drives. IEEETransactions on Industry Applications, vol. 52, no. 5, pp. 3968–3979,September/October 2016.

VI H. A. A. Awan, Z. Song, S. E. Saarakkala and M. Hinkkanen. Optimaltorque control of saturated synchronous motors: Plug-and-play method.IEEE Transactions on Industry Applications, vol. 54, no. 6, pp. 6110–6120,November/December 2018.

5

List of Publications

6

Author’s Contribution

Publication I: “Current control for synchronous motor drives: Directdiscrete-time pole-placement design”

The author performed the simulations, experiments, robustness analysis, andparticipated in the writing of the paper.

Publication II: “Flux-linkage-based current control of saturatedsynchronous motors”

The author wrote the paper under the guidance of Prof. Hinkkanen. Dr. Saarakkalacontributed by helping with the measurements.

Publication III: “Stator-flux-oriented control of synchronous motors:A systematic design procedure”

The author wrote the paper under the guidance of Prof. Hinkkanen. Prof. Bojoiand Prof. Pellegrino contributed by commenting on the manuscript.

Publication IV: “Observers for sensorless synchronous motordrives: Framework for design and analysis”

The author performed the experiments with the help of Mr. Mölsä and partici-pated in the writing of the paper.

7

Author’s Contribution

Publication V: “Discrete-time observer design for sensorlesssynchronous motor drives”

The author wrote the paper under the guidance of Prof. Hinkkanen. Dr. Tuovi-nen did the initial design of the discrete-time flux observer and wrote the initialdraft of the paper. Dr. Saarakkala contributed by helping with the measure-ments.

Publication VI: “Optimal torque control of saturated synchronousmotors: Plug-and-play method”

The author wrote the paper under the guidance of Prof. Hinkkanen. Dr. Songand Dr. Saarakkala contributed by helping with the measurements and com-menting on the manuscript.

8

Abbreviations

2DOF Two-degrees-of-freedom

AC Alternating current

DC Direct current

DFVC Direct-flux vector control

DTC Direct torque control

EMF Electromotive force

IMC Internal model control

MTPA Maximum torque-per-ampere

MTPV Maximum torque-per-volt

PI Proportional-integral

PM Permanent magnet

PM-SyRM Permanent-magnet synchronous reluctance motor

PWM Pulse-width modulation

SyRM Synchronous reluctance motor

ZOH Zero-order hold

9

Abbreviations

10

Symbols

Matrices are denoted by boldface upper-case letters and vectors by boldfacelower-case letters. The vectors in stationary coordinates are marked with thesuperscript s and in stator flux coordinates with the superscript f. Referencevalues are marked with the subscript ref, and limited reference values aremarked with overline.

O Zero matrix[

0 00 0

]G, G′, K Observer gain matrices

Gd, Kd Discrete-time observer gain matrices

Gi, K i Integral gain matrices

Gt, Kt Reference-feedforward gain matrices

G1, K1, K2 State-feedback gain matrices

I Identity matrix[

1 00 1

]i Stator current vector in rotor coordinates

id, iq Stator current components in rotor coordinates

J Orthogonal rotation matrix[

0 −11 0

]k Discrete-time index

L Inductance matrix

Ld, Lq Direct- and quadrature-axis inductances

Li Incremental inductance matrix

R Stator resistance

T Electromagnetic torque

T Nonlinear transformation matrix

11

Symbols

Ts Sampling time

u Stator voltage vector in rotor coordinates

ud, uq Stator voltage components in rotor coordinates

udc DC-bus voltage

α Closed-loop bandwidth

β Discrete-time pole exp(−αTs)

δ Stator flux vector angle in rotor coordinates

ϑm Electrical angular position of the rotor

Φ, Γ, γ Discrete-time system matrices

ψ Stator flux vector in rotor coordinates

ψd, ψq Stator flux components in rotor coordinates

ψf Permanent-magnet flux linkage

ωm Electrical angular speed of the rotor

12

1. Introduction

1.1 Background

Synchronous motors with a magnetically anisotropic rotor, such as interiorpermanent-magnet (PM) synchronous motors, synchronous reluctance motors(SyRMs), and permanent-magnet synchronous reluctance motors (PM-SyRMs),are more and more applied in hybrid and electric vehicles, heavy-duty workingmachines, and industrial applications. Compared to induction motors, theabsence of the rotor winding in synchronous motors makes the resistive lossessmall in the stator and negligible in the rotor. Apart from the increase inefficiency, synchronous motors provide high torque-per-ampere ratio, smallinertia, and high power density compared to induction motors of the same size(Boglietti and Pastorelli, 2008).

In applications with low dynamic performance, constant volts-per-hertz con-trol or scalar control can be used. This control may have unstable operatingregions when applied to synchronous motors (Perera et al., 2003). The vectorcontrol methods are preferred for synchronous motors due to their ability toeffectively control the electromagnetic torque, improved efficiency, and betterdynamic performance. Furthermore, it is easier to implement the current lim-itation in vector control methods. These methods include the field-orientedcontrol (Blaschke, 1972; Bayer et al., 1972) and direct torque control (DTC)(Takahashi and Noguchi, 1986; French and Acarnley, 1996) or direct self control(Depenbrock, 1988). The stator current vector is controlled in the field-orientedcontrol methods. The estimate of the electromagnetic torque and stator fluxmagnitude are controlled using the hysteresis controllers in DTC. Other controlmethods include predictive control (Bolognani et al., 2009; Geyer et al., 2010;Mariethoz et al., 2012; Carlet et al., 2019). The focus of this thesis is on themodel-based vector control methods equipped with a pulse-width-modulator(PWM). Hysteresis control, DTC, and predictive control are left out of the scopeof this thesis.

The rotor position information is essential for vector control methods. The

13

Introduction

rotor position can be measured using a position sensor or it can be estimated.In many applications, such as pumps, fans, and conveyors, the drive shouldobviously be sensorless for cost savings. Furthermore, interior PM motors andPM-SyRMs are increasingly used in electric vehicles (Pellegrino et al., 2012b;Bianchi et al., 2016). Even though the drives in electric vehicles are typicallyequipped with a motion sensor, a sensorless mode is beneficial for providingtolerance to sensor failures. In this thesis, the focus is on back electromotiveforce-(back EMF) based observers that estimate the rotor speed and positionbased on the mathematical motor model.

Energy-efficient and sensorless control methods for saturated synchronousmotors, such as SyRMs and PM-SyRMs, rely heavily on the knowledge of themagnetic saturation characteristics. These characteristics can be highly nonlin-ear, and cross-coupling between the d- and q-axis exists. These characteristicsshould be properly taken into account in the control system in order to reachhigh performance. For optimal control, the motor is driven at the maximum-torque-per-ampere (MTPA) locus, in the field-weakening region, and at themaximum-torque-per-volt (MTPV) limit. Typically, look-up tables are used forthese control characteristics (Meyer and Böcker, 2006; Cheng and Tesch, 2010).In this thesis, a look-up table computation and optimal reference calculationmethod is developed. When combined with an identification method for themagnetic model, the proposed method enables the plug-and-play startup ofan unknown motor. Various identification methods for the magnetic model ofsynchronous motors are already available (Bedetti et al., 2016; Hinkkanen et al.,2017; Pescetto and Pellegrino, 2017; Odhano et al., 2019; Antonello et al., 2019).Therefore, the identification methods are left out of the scope of this thesis.

1.2 Objective and Outline of the Thesis

The goal of this thesis is to develop control methods for synchronous motors withthe following properties:

• The proposed control methods can be automatically tuned with minoreffort from the user if the magnetic model of the motor is known.

• The control methods should provide high dynamic performance and stableoperation in both the sensored and speed sensorless operation. Apart fromstable operation at low speed, the control methods should be stable at lowsampling to fundamental frequency ratios (below ten).

• The optimal control characteristics, such as the MTPA locus, the MTPVlimit, and the field-weakening region should be included in the controldesign.

• The proposed methods can be implemented in the embedded processor ofa converter.

14

Introduction

This thesis focuses on model-based control methods. The rotor-oriented andstator-flux-oriented control methods are used.

This thesis consists of an overview and six publications. The overview isorganized as follows. Chapter 2 presents the system model, while Chapter3 reviews the controllers proposed in the literature. Chapter 4 reviews thesensorless control methods. Chapter 5 reviews the optimal state referencecalculation methods. Chapter 6 describes the experimental setup used in thepublications. The summaries of the publications and the scientific contributionsof this thesis are presented in Chapter 7. Chapter 8 concludes the thesis.

15

Introduction

16

2. System Model

2.1 Space Vectors

In order to model synchronous motors, real space vectors are used throughoutthe thesis. Vectors are denoted using boldface lowercase letters and matricesusing boldface uppercase letters. A two-axis model is used to represent thebehavior of the system. Space vectors in stator coordinates are marked with thesuperscript s, no superscript is used for space vectors in rotor coordinates. Forexample, the stator-current vector in stationary coordinates is

is =[

iαiβ

]=[

23 −1

3 −13

0 1�3

− 1�3

]⎡⎢⎢⎣

ia

ib

ic

⎤⎥⎥⎦ (2.1)

where iα and iβ are the orthogonal components of the real-valued stator currentvector in stator coordinates, and ia, ib, and ic are the three phase currents. Thesum of phase currents is zero for synchronous motors considered in this thesis.Other space vectors are defined similarly.

The stator current vector is can be transformed from stationary coordinates torotor coordinates, rotating at the speed ωm, as

i =[

id

iq

]= e−ϑmJ

[iαiβ

]=[cos(−ϑm)I+sin(−ϑm)J

][iαiβ

](2.2)

where id and iq are the components of the stator current, ϑm = ∫ωmdt is the

rotor position, I = [1 00 1] is the identity matrix, and J = [0 −1

1 0 ] is the orthogonalrotation matrix. The matrix transpose is marked with the superscript T.

2.2 Inverter Model and Pulse-Width Modulation

Fig. 2.1 shows the model of a three-phase two-level inverter. The inverterAC voltage is pulse-width modulated. Fig. 2.2 illustrates a PWM update and

17

System Model

udc

a b c

Figure 2.1. Model of a three-phase two-level inverter.

Tsw

t/Ts

da(k)

carrier

k k+1 k+2

sample

updatereferences

updatereferences

currentssample

currentssample

currents

qa

Figure 2.2. Single-update PWM with sampling at the start of the switching period Tsw. Thetriangular carrier signal, a-phase duty ratio da(k), and resulting a-phase gate signalqa are illustrated. The duty ratios db(k) and dc(k) (not shown) of the other twophases are updated simultaneously with da(k) and compared to the same carriersignal.

sampling scheme used in this thesis. The sampling period is denoted by Ts

and the discrete-time index by k. At every time step, the duty ratios dabc =[da,db,dc]T for each phase are calculated from the stator voltage referenceus

ref by means of the space-vector PWM algorithm. The gate signals qabc =[qa, qb, qc]T are obtained by comparing the duty ratios dabc with the carriersignal. It is also worth mentioning that the methods developed in thesis aredirectly compatible with the double-update PWM with sampling twice per theswitching period.

For the control design, switching-cycle-averaged voltage is assumed. Theaverage of the stator voltage over the sampling period Ts is

us(k)= 2udc

3

[1 −1

2 −12

0�

32 −

�3

2

]dabc(k) (2.3)

where udc is the dc-link voltage. The stator voltage in stator coordinates isassumed to be piecewise constant between two consecutive sampling instants,which corresponds to the zero-order hold (ZOH) in stator coordinates. In otherwords, the stator voltage us(t) is constant during kTs < t < (k+1)Ts. Figure2.3 shows the switching-cycle-averaged model of a PWM inverter in statorcoordinates. The model consists of the voltage saturation (corresponding to thevoltage hexagon), the computational delay of one sampling period, and the ZOH.

Fig. 2.4 illustrates the maximum available voltage, which corresponds tothe border of the voltage hexagon. In the first sector, the maximum voltage

18

System Model

udc

z−1us

refZOH

ususref

Figure 2.3. Model of a PWM inverter in stator coordinates. The model consists of the voltagesaturation, the computational delay z−1, and the ZOH.

usref

usref

ϑuuα

uβ

udc�3

Figure 2.4. Voltage hexagon of a two-level PWM inverter in stator coordinates.

magnitude is (Khambadkone and Holtz, 2002)

umax = udc�3sin(2π/3−ϑu)

(2.4)

where ϑu = [0,π/3] is the angle of the reference voltage usref. The above equation

can be easily applied in other sectors as well. The realizable voltage reference inrotor coordinates can be calculated as

uref =⎧⎨⎩uref, if ‖uref‖ ≤ umax

uref

‖uref‖umax, if ‖uref‖ > umax

(2.5)

where uref is the reference voltage in rotor coordinates.

2.3 Continuous-Time Motor Model

The continuous-time model of synchronous motors is presented in the followingsubsections. For the interior PM motor and the PM-SyRM, the d-axis of thecoordinate system is fixed to the direction of the PMs. Unless stated otherwise,the d-axis of the SyRM is fixed along the maximum inductance axis.

2.3.1 Saturation Characteristics

Generally, the stator flux linkage ψ of a synchronous motor is a nonlinearfunction of the stator current i, i.e.,

ψ=ψ(i)=[ψd(id, iq)

ψq(id, iq)

](2.6)

19

System Model

(a) (b) (c)

Figure 2.5. Saturation characteristics of a 6.7-kW SyRM: (a) ψd =ψd(id, iq); (b) ψq =ψq(id, iq);(c) ψd =ψd(id,0). The saturation characteristics are obtained by fitting the measureddata to an algebraic magnetic model (2.10).

The reciprocity condition ∂ψd/∂iq = ∂ψq/∂id should hold, since the nonlinearinductor does not generate or dissipate energy (Vagati et al., 2000). As anexample, Fig. 2.5 illustrates the nonlinear saturation characteristics of a 6.7-kWSyRM. In the special case of an unsaturated motor, the stator flux linkage is

ψ= Li+ψf (2.7)

where the inductance matrix and the PM-flux vector, respectively, are

L =[

Ld 0

0 Lq

]ψf =

[ψf

0

](2.8)

with the constant inductances Ld and Lq and constant PM-flux linkage ψf. It isworth noticing that the effect of PMs is inherently included in (2.6). Alternatively,the saturation characteristics can be defined using the inverse function of (2.6),i.e.,

i = i(ψ)=[

id(ψd,ψq)

iq(ψd,ψq)

]. (2.9)

The magnetic saturation can be modeled using generic saturation characteris-tics in the form of (2.6) or (2.9). The advantage of these forms is that they aredirectly compatible with both look-up tables and explicit functions.

Algebraic Magnetic Model: An ExampleThe magnetic saturation of synchronous motors can be modeled using an alge-braic magnetic model. An algebraic magnetic model (Hinkkanen et al., 2017),originally developed for SyRMs, is extended to the PM synchronous machines as

id =(

ad0 +add|ψd|α+adq

δ+2|ψd|γ|ψq|δ+2

)ψd − if (2.10a)

iq =(

aq0 +aqq|ψq|β+adq

γ+2|ψd|γ+2|ψq|δ

)ψq (2.10b)

where ad0, add, aq0, aqq, and adq are non-negative coefficients and α, β, γ, and δ

are non-negative exponents. The constant current source if is used to represent

20

System Model

1s

i(ψ)

R

ωmJ

u iψ

Figure 2.6. Model of a saturated synchronous motor in rotor coordinates.

the magnetomotive force (MMF) of the PMs (Jahns et al., 1986). The structureis based on the assumption that the MMFs of the d-axis current and of the PMsare in series.

The coefficient ad0 is the inverse of the unsaturated d-axis inductance andthe coefficient aq0 is the inverse of the unsaturated q-axis inductance. Thecoefficients add and aqq take the self-axis saturation characteristics into account,while adq takes the cross-saturation into account. The functions (2.10) fulfill thereciprocity condition. The model is invertible: for any given values of id and iq,the corresponding values of ψd and ψq can be obtained by numerically solving(2.10).

If the saturation effects are omitted, i.e. add = aqq = adq = 0, the magneticmodel equals the constant inductance model (2.7), with Ld = 1/ad0, Lq = 1/aq0,and ψf = if/ad0.

2.3.2 Voltage and Electromagnetic Torque

The voltage equation of a synchronous motor in rotor coordinates is

dψdt

= u−Ri−ωmJψ (2.11)

where u is the stator voltage, R is the stator resistance, and ωm is the electricalangular speed of the rotor. The voltage equation (2.11) together with the sat-uration characteristics (2.9) forms a nonlinear state-space model of the motor,shown in Figure 2.6.

The voltage equation with the stator current as a state variable is obtained bysubstituting (2.6) into (2.11)

Lididt

= u−Ri−ωmJψ (2.12)

where the incremental inductance matrix is

Li = Li(i)=[

∂ψd(id,iq)∂id

∂ψd(id,iq)∂iq

∂ψq(id,iq)∂id

∂ψq(id,iq)∂iq

]=[

Ldi(id, iq) Ldqi(id, iq)

Ldqi(id, iq) Lqi(id, iq)

](2.13)

The matrix is symmetric due to the reciprocity condition.The electromagnetic torque is

T = 3p2

iTJψ= 3p2

(ψd iq −ψq id

)(2.14)

where p is the number of pole pairs.

21

System Model

i(ψ)Φi

z−1Tsu ψ

Figure 2.7. Hold-equivalent motor model in rotor coordinates. The state-transition matrix isΦ= exp(−TsωmJ).

2.4 Discrete-Time Motor Model

The switching cycle averaged quantities are considered. Hence, the actual statorvoltage us(t) in stator coordinates is piecewise constant between two consecutivesampling instants, which corresponds to the ZOH in stator coordinates, cf. Figure2.3. The stator resistance R = 0 is assumed. Under these assumptions, the exacthold-equivalent discrete-time model in rotor coordinates can be derived from(2.11), leading to

ψ(k+1)=Φψ(k)+TsΦu(k) (2.15)

where Ts is the sampling period. The state-transition matrix is

Φ= exp(−TsωmJ) (2.16)

Figure 2.7 shows the hold-equivalent motor model in rotor coordinates. As canbe seen in Figure 2.7, the saturation characteristics appear only in the outputequation due to the assumption R = 0. Therefore, generic nonlinear saturationcharacteristics i = i(ψ) can be used. The discrete-time model (2.15) is presentedin Publication II. The discrete-time model with the assumption of constantmotor parameters is reported in Publication I. The model is derived with theassumption of linear magnetics given in (2.7).

The discrete-time motor model in Publication I is more suitable to be usedwith flux observers. It is due to the fact that the stator resistance is included inthe model and the low-speed behavior of observers are greatly affected by thestator resistance. The model in Publication II is derived with the assumptionof R = 0. Unlike the model in Publication I, the magnetic saturation can beproperly included in the motor model presented in Publication II. It is moresuitable to be used with current controllers, where the effect of stator resistancecan be easily compensated for by the integral action of the controller.

2.5 Continuous-Time Motor Model in Stator Flux Coordinates

Bilewski et al. (1993) suggested that a stator-flux-oriented reference frame canbe adopted for control robustness of interior PM motors. The motor model instator flux coordinates is a starting point for designing and analyzing controllersin this coordinate system. Fig. 2.8 shows stator flux coordinates (ψτ), whoseψ-axis is parallel to the stator flux vector. The vectors in these coordinates are

22

System Model

iq

d-axis

q-axis

ψ-axis

τ-axis

δ

iτ

id ψd

ψψq

iψ

Figure 2.8. Rotor coordinates (dq) and stator flux coordinates (ψτ). Flux and current componentsare depicted in both coordinates.

marked with the superscript f, e.g.,

ψf =[ψ

0

]= e−δJψ if =

[iψiτ

]= e−δJi (2.17)

where δ is the angle of the stator flux vector in rotor coordinates. Other vectorsare transformed to stator flux coordinates similarly. The coordinate transforma-tion can be written as

exp(δJ)=[

cosδ −sinδ

sinδ cosδ

](2.18)

The matrix elements are cosδ=ψd/ψ and sinδ=ψq/ψ, where the flux magnitudeis ψ= (ψ2

d+ψ2q)1/2. In stator flux coordinates, the torque expression (2.14) reduces

toT = 3p

2ψiτ (2.19)

The stator-flux magnitude and the torque-producing current component arepacked into a state vector

xf =[ψ

iτ

](2.20)

Using (2.7), (2.11), and (2.17), a nonlinear model with these state variables isobtained (Pellegrino et al., 2009)

dxf

dt=[

1 0

a/Ld b/Ld

](uf −Rif −ωmJψf) (2.21)

where the factors are

a = 12

(Ld

Lq−1

)sin2δ b = ψf

ψcosδ+

(Ld

Lq−1

)cos2δ (2.22)

It is to be noted that the condition b = 0 corresponds to the maximum-torque-per-volt (MTPV) limit (Bilewski et al., 1993).

23

System Model

24

3. Control Schemes

The focus of this chapter is on the design of controllers. The calculation ofreferences is discussed in detail in Chapter 5. Three different controllers arepresented in this chapter. The first controller is designed in the continuous-timedomain and operates in rotor coordinates. The second controller is designeddirectly in the discrete-time domain and operates in rotor coordinates. The thirdcontroller is designed in the continuous-time domain and operates in stator fluxcoordinates.

Figure 3.1 shows an overall control system for a typical current-controlleddrive. The current controller operates in rotor coordinates. It is also worthmentioning that instead of controlling the measured current, estimated currentsor flux linkages could be controlled. The only change is the addition of a currentor flux observer. Furthermore, for speed sensorless operation, the control systemcan be augmented with a speed adaptation law. A speed-adaptive full-orderobserver for sensorless interior PM motor and PM-SyRM drives designed directlyin the discrete-time domain is presented in Chapter 4.

3.1 Continuous-Time Current Control

In the early days of AC motor drives, the hysteresis controllers were usedfor controlling three-phase currents. The hysteresis controller has a simplestructure and provides a fast response, but the switching frequency depends onthe operating point. The stator-frame proportional-integral (PI) controller hasalso been used, but it suffers from non-zero steady-state error. More informationon these controllers can be found in review papers (Brod and Novotny, 1985;Kazmierkowski and Malesani, 1998).

The synchronous frame current regulators have been considered an industrystandard for controlling inverter-fed drives for more than three decades (Rowanand Kerkman, 1986). The synchronous or rotor coordinate system is a naturalselection since the controllable quantities are DC in steady-state, the inductancematrix and the PM flux vector are (ideally) constant, and other parts of thecontrol system typically operate in rotor coordinates. The most widely used

25

Control Schemes

isi

udc

usref

Motorus

PWMinverter

e−ϑmJ

ddt

ωm ϑm

urefTref

udc

eϑmJ

irefReferencecalculation

controllerCurrent

Figure 3.1. Overall control system of a typical current-controlled drive. The control system mayinclude a speed controller, which provides the torque reference.

approach is a synchronous-frame PI controller. Unlike the hysteresis controller,the synchronous-frame current controller allows a constant switching frequency,and unlike the stator-frame current controller, the steady-state control error iszero (Harnefors and Nee, 1998). Even with these control benefits, the controllersin the synchronous reference frame still suffer from cross-coupling between thed- and q-axis quantities. Harnefors and Nee (1998) introduced a decouplingscheme to remove the cross-coupling by a state-feedback. Similar schemes havealso been used by others for cross-coupling decoupling (Springob and Holtz, 1998;Briz del Blanco et al., 1999). Soricellis et al. (2018) replaced the model-basedcross-coupling with a PI-based observer. They claimed that the observer-basedcurrent controller reduces the overall noise content and the overshoot in thecurrent components during transients. The observer-based current controllerhas a higher order and more tuning effort is needed for the control design.

The general reference tracking objectives for current controllers are as follows:1) no cross coupling between the d- and q-axes and 2) the same closed-loopdynamics for both axes. Harnefors and Nee (1998) proposed a synchronous-frame PI controller with the first-order response using an internal model control(IMC) method. Unlike the classical PI controllers that involve the adjustmentof two parameters, the IMC design method requires the selection of only oneparameter, i.e., the desired control bandwidth of the controller. Similar to theIMC design, a complex vector design was introduced by Briz del Blanco et al.(1999). The complex vector modeling approach requires the AC motor to besymmetric, i.e., same parameters in the d- and q-axis. Therefore, the originalmethod is only suitable for nonsalient AC machines, e.g., induction motors andsurface PM synchronous motors.

The disturbance rejection of synchronous-frame PI controllers was furtherimproved with additional feedback from the stator current, referred to as anactive resistance or active damping in the literature (Briz del Blanco et al., 1999;Harnefors, 2001). It is a fictitious resistance included in the decoupling loopand implemented using signals only. Therefore, the overall losses of the systemdo not increase. The complex vector design was generalized for salient motorsby Kim and Lorenz (2002) and Kim and Lorenz (2004), but the detailed design

26

Control Schemes

uref

G1

Gt

Giiref

i

1s

ξ

ψ(i)

ωmJ

Li(i)

Liξ

Figure 3.2. Current controller with current as a state variable.

guidelines and generalization methods were not presented. An attempt hasalso been made to generalize the IMC and complex vector designs for salientsynchronous motors using a transfer-matrix approach (Jeong and Sul, 2005).

Most of these two-degrees-of-freedom (2DOF) PI current controllers can also berepresented as full-state feedback controllers with integral action and referencefeedforward. This framework simplifies the systematic design and analysisof controllers. Continuous-time 2DOF PI current controller designs for SPMdrives (Briz del Blanco et al., 1999; Briz et al., 2000) were extended to interiorPM motor drives in Publication I. A basis for the generalization is that 2×2coefficient matrices of the transfer-function matrix are analogous to complexcoefficients of the complex transfer function. This approach is kin to block-poleplacement of multi-input-multi-output systems (Shieh and Tsay, 1982; Shiehet al., 1983).

The control schemes discussed so far have been developed by assuming a linearmagnetic circuit, i.e., constant d- and q-axis inductances. SyRMs and PM-SyRMssaturate heavily during operation. If the gains of the current controller arenot adapted to varying motor parameters, the control response will becomeoscillatory or even unstable. Kim and Lorenz (2002) used an online estimationalgorithm to update the values of Ld and Lq during operation. This increasedthe complexity of the overall control system. Furthermore, it is not clear howthe dynamics of the inductance estimation algorithm affect the control dynamics.The controllers presented in Publication II properly take the magnetic satura-tion into account. The control schemes developed in Publication II are brieflypresented in the following subsections.

3.1.1 Current as a State Variable

Figure 3.2 shows the current control structure, where the stator current is usedas a state variable. The magnetic saturation is taken into account based on(2.12). The voltage reference is given by

uref =ωmJψ+Liξ (3.1)

27

Control Schemes

K1

Kt

ψref

Kiiref

i ψψ(i)

ψ(i) 1s

uref

Figure 3.3. Current controller with flux linkage as a state variable.

where ξ is an auxiliary control variable, obtained from a linear controller to bedesigned subsequently. Assuming u = uref and R = 0 and substituting (3.1) into(2.12) leads to

didt

= ξ (3.2)

The closed-loop poles can now be easily placed via the linear part of the currentcontroller. Here, a state-feedback controller with integral action and referencefeedforward is used,

ξ=Gtiref +Gi

s(iref − i)−G1i (3.3)

where s = d/dt is the differential operator, Gt is the reference-feedforward gain,Gi is the integral gain, and G1 is the state-feedback gain. These gains can bechosen in various ways. As an example, the gains corresponding to the IMCdesign are

Gt =αI Gi =α2I G1 = 2αI (3.4)

In this case, all the closed-loop poles are placed at s = −α. The closed-loopreference-tracking dynamics are

i = α

s+αiref (3.5)

where α is the bandwidth.It was pointed out in Publication II that for implementing the control law

(3.1), five nonlinear functions ψd(id, iq), ψq(id, iq), Ldi(id, iq), Lqi(id, iq), andLdqi(id, iq) should be implemented, typically with look-up tables, which compli-cates the control system.

3.1.2 Flux as a State Variable

Figure 3.3 shows the current control structure given in Publication II, wherethe flux linkage is chosen as a state variable. The measured current and thereference current can be easily mapped to the corresponding flux linkage vari-ables using the known saturation characteristics. These flux linkages can thenbe used in the current controller, as originally proposed for a simple proportionalcontroller in an early paper (Haylock et al., 1999). The flux-linkage-based cur-rent controller takes the saturation effects inherently into account, yielding asimple implementation and a robust controller.

28

Control Schemes

tTs

ψd(k)= 1−β

z(z−β)ψd,ref(k)

ψd

ψd,ref

ψd = α

s+αψd,ref

Figure 3.4. Step responses of the continuous-time system (3.9) and the discrete-time system(3.13), where β = exp(−αTs). Setting β = 0 would give the dead-beat response.The green dashed line shows the response of the discrete-time system without thecomputational delay.

Similarly to (3.3), a state-feedback controller with integral action and referencefeedforward is used,

uref = Ktψref +K i

s(ψref −ψ

)−K1ψ (3.6)

where Kt is the reference-feedforward gain, K i is the integral gain, and K1 isthe state-feedback gain. It is also worth noticing that the controller (3.6) doesnot see the magnetic saliency of the motor due to the mapping of the current tothe flux linkage. Therefore, the controller (3.6) and the resulting flux-linkagedynamics could be described using complex space vectors and complex gains,instead of real space vectors and gain matrices.

Two typical selections for the these gains are the IMC design (Harnefors andNee, 1998)

Kt =αI K i =α2I K1 = 2αI−ωmJ (3.7)

and a design similar to the complex vector design (Briz del Blanco et al., 1999)

Kt =αI K i =αI(αI+ωmJ) K1 = 2αI (3.8)

Both these designs lead to the first-order closed-loop system

ψ= α

s+αψref (3.9)

However, the disturbance rejection characteristics of the two designs are differ-ent, leading also to different sensitivities to parameter errors, as explained inPublication I. The step response corresponding to (3.9) is shown in Figure 3.4.

As pointed out in Publication II, the control structure in Figure 3.3 needs onlytwo look-up tables, ψd(id, iq) and ψq(id, iq), and it inherently takes the effects ofthe incremental inductances into account. It is also worth noticing that the twocontrollers in Figs. 3.2 and 3.3 are mathematically equivalent if the magneticsaturation is omitted and if the same pole locations are chosen.

29

Control Schemes

3.1.3 Digital Implementation

For digital implementation, continuous-time control algorithms have to be dis-cretized using methods, such as the Euler or Tustin. Unfortunately, unless thesampling frequency is much higher than the closed-loop bandwidth and the max-imum operating frequency, the actual closed-loop system deviates significantlyfrom (3.5) and (3.9) due to discretization errors, leading to the cross-couplingbetween the d- and q-axes, oscillations, or even instability (Yim et al., 2009; Kimet al., 2010). The performance of continuous-time designs is acceptable if thesampling frequency is about 20 times higher than the closed-loop bandwidthand the operating frequency (Briz and Hinkkanen, 2018).

Different schemes have been presented in the literature to compensate forthe discrete-time effects for the current controllers designed in the continuous-time domain. Apart from the one sampling period computational delay, halfa sampling period delay is caused by the digital implementation of the PWM.These time delays cause an angle error in the coordinate transformation (fromrotor to stator coordinates), which ultimately cause a phase and magnitude errorin the output voltage. Bae and Sul (2003) developed a method to compensate forthe effect of computational delay and delay caused by the PWM.

The delay compensation by Bae and Sul (2003) is effective for continuous-timedesigns if the ratio between the sampling and fundamental frequencies is about20 (Briz and Hinkkanen, 2018). Yim et al. (2009) presented a compensationmethod to further increase the operation range of continuous-time controllersto higher fundamental frequencies. It was suggested to use one-step predictionof the stator current in the active damping terms of current controllers. Thecompensation methods by Bae and Sul (2003) and Yim et al. (2009) are onlyeffective if the sampling frequency is high enough compared to the fundamentalfrequency.

3.2 Discrete-Time Current Control

Higher maximum speeds, higher dynamic performance, and better robustness ata given sampling frequency can be achieved by designing the controller directlyin the discrete-time domain (Huh and Lorenz, 2007; Kim et al., 2010; Peterset al., 2011; Peters and Böcker, 2013). For the control design, a hold-equivalentdiscrete-time motor model is needed. Furthermore, the computational delayshould be included in the control design.

A few direct discrete-time current controller designs for interior PM motorsare available. Huh and Lorenz (2007) proposed a discrete-time current controllerbased on the exact (but numerically evaluated) hold-equivalent discrete-timemotor model. However, the controller order is unnecessarily high. The discrete-time current controllers in (Peters et al., 2011) and (Peters and Böcker, 2013)are based on discretized motor models. The motor model is discretized using the

30

Control Schemes

Euler method in stator coordinates and then transformed to rotor coordinatesfor the control design. Altomare et al. (2016) used a similar approach for SyRMs,i.e., using a discretized motor model for designing the discrete-time currentcontroller.

In Publication I, an analytical direct discrete-time design method for a 2DOFPI current controller is proposed for magnetically salient synchronous motors. Astate-feedback controller with integral action and reference feedforward is usedas a framework. The method is based on an exact analytical hold-equivalentdiscrete-time motor model and the computational delay is taken into account inthe state-feedback law. The proposed method is easy to apply: only the desiredclosed-loop bandwidth and the three motor parameters (Ld, Lq, R) are needed.

The discrete-time controllers discussed so far have been designed by eitherconsidering linear magnetics or approximations have been used for the inclusionof magnetic saturation. Peters and Böcker (2013) used a diagonal incrementalinductance matrix to include the magnetic saturation, whereas the effect ofcross-saturation is neglected. Altomare et al. (2016) used look-up tables toupdate the values of Ld and Lq, but the effect of cross-saturation is neglected.Furthermore, it is not clear whether the incremental inductances or the chord-slope inductances were used. In Publication I, the values of inductances areupdated during each sampling period using known saturation characteristics.In Publication II, a control framework is presented that properly takes themagnetic saturation into account. This approach is analogous to its continuous-time counterpart in Section 2.4. In the following subsections, the discrete-timecontroller presented in Publication II is briefly reviewed.

3.2.1 Inclusion of the Control Delay

Due to the finite computational time, the digital control system typically has atime delay of one sampling period, i.e., us(k)= us

ref(k−1). For control design, thecomputational delay can be easily included in the plant model (2.15) as follows[

ψ(k+1)

u(k+1)

]=[Φ TsΦ

O O

][ψ(k)

u(k)

]+[

O

Φ

]uref(k) (3.10)

where O= [0 00 0] is the zero matrix. Due to this computational delay, the order of

the discrete-time plant model is higher than the order of the continuous-timemodel. An advantage of the discrete-time control design is that the delay canbe easily taken into account in the controller. It is to be noted that the statorresistance R = 0 is assumed in the plant model (3.10).

3.2.2 Control Design

Similarly to the continuous-time design in Section 3.1.2, the stator currentsare mapped to the flux linkages. Furthermore, a state-feedback controller with

31

Control Schemes

uref

K−1t

K1

Kt

ψref

KiTs

z−1

z−1K2

iref

i ψψ(i)

ψ(i)

uref −uref

Figure 3.5. Discrete-time current controller. The voltage saturation and the anti-windup mecha-nism are also shown. The limited voltage reference uref can be calculated as a partof the current control algorithm or it can be provided by the PWM algorithm. In thelinear modulation range, uref = uref.

integral action and reference feedforward is applied. The control algorithm is

ui(k+1)= ui(k)+TsK i[ψref(k)−ψ(k)

](3.11a)

uref(k)= Ktψref(k)−K1ψ(k)−K2uref(k−1)+ui(k) (3.11b)

where ui is the integral state, K i is the integral gain, Kt is the reference-feedforward gain, and K1 and K2 are the state-feedback gains. Figure 3.5 showsthe corresponding block diagram, where also the anti-windup mechanism is in-cluded. It can be seen that the discrete-time control design is very similar to thecontinuous-time design. The computational delay is omitted in the continuous-time design, while it is taken into account in the discrete-time design by meansof K2. The controller output is stored in the memory and the delayed outputis then used as an additional state in the state-feedback law (Franklin et al.,1997).

The control law similar to (3.11) is presented in Publication I with the statorcurrent as a state variable. The design of the current controller is based onthe motor model with the assumption of constant parameters. Unlike thecontroller in Publication I, the controller in Publication II takes the saturationcharacteristics properly into account. Furthermore, the effect of stator resistanceis compensated for by the integral action of the controller (3.11).

Based on (3.10) and (3.11), the closed-loop reference-tracking dynamics can beexpressed as

ψ(k)= (z3I+ z2A2 + zA1 + A0)−1(zB1 +B0)ψref(k) (3.12)

where z is the forward-shift operator and A0, A1, A2, B0, and B1 are thecoefficient matrices. The coefficient matrices can be selected according to theIMC or the complex vector design. The discrete-time counterparts of the IMCand the complex vector designs are presented in Publication II. In both cases,the closed-loop reference-tracking dynamics (3.12) reduces to

ψ(k)= 1−β

z(z−β)ψref(k) (3.13)

32

Control Schemes

isi

udc

usref

Motorus

PWMinverter

e−ϑmJ

ddt

ωm ϑm

urefTref

udc

eϑmJ

xfref

Referencecalculation

controller

Stator-flux-oriented

Figure 3.6. Overall control system of a typical stator-flux-oriented-controlled drive. The outputof the reference calculation block is xf

ref = [ψref, iτ,ref]T.

where β= exp(−αTs) is the exact mapping in the discrete domain of the intendedreal pole of the system. The step response corresponding to (3.13) is shown inFigure 3.4.

3.3 Anti-Windup Mechanism

In order to prevent integrator windup of the current controllers presented inSections 3.1 and 3.2, an anti-windup mechanism should be implemented. As anexample, an integrator anti-windup mechanism is shown for the discrete-timecurrent controller presented in Section 3.2. Figure 3.5 shows an integratoranti-windup mechanism, which is based on the realizable voltage referenceuref (Peng et al., 1996). This mechanism is not active in the linear modulationrange where uref = uref holds. The realizable voltage reference can be eithercalculated in the current controller using (2.4) and (2.5) or it can be obtainedfrom the PWM. Harnefors and Nee (1998) used a similar approach to preventan integrator windup in synchronous frame PI controllers using the so-calledback-calculation method.

3.4 Stator-Flux-Oriented Control

Bilewski et al. (1993) introduced a control method for the control of AC motors instator flux coordinates. It was suggested that the stator-flux magnitude and thetorque-producing current component can be selected as the controlled variables.This choice simplifies the calculation of the state references. Only the maximumtorque-per-ampere (MTPA) and MTPV features have to be implemented, but notwo-dimensional look-up tables are needed for the calculation of the current orflux references. Pellegrino et al. (2009) used the term direct-flux vector control(DFVC) for a similar controller. Hofmann et al. (2004) used the stator-flux-oriented vector control method for the maximum efficiency control of SyRMs.

33

Control Schemes

It is worth mentioning that the stator flux magnitude and the electromagnetictorque can be controlled using the hysteresis or PI controllers (Zhong et al.,1997; Rahman et al., 2003; Buja and Kazmierkowski, 2004; Ekanayake et al.,2018; Moradian et al., 2019).

Figure 3.6 shows an overall control system for a stator-flux-oriented-controlleddrive. Typically, two separately tuned proportional-integral (PI) controllers areused for controlling the stator-flux magnitude and the torque-producing currentcomponent (Bilewski et al., 1993). Pellegrino et al. (2009) studied the effect ofDC-link voltage variation on the control. Bojoi et al. (2010) used DFVC for thecontrol of traction drives. Pellegrino et al. (2011) unified DFVC for inductionand synchronous motors. Pellegrino et al. (2012a) discussed the performanceof DFVC along the MTPV limit. The load angle was limited to a predefinedmaximum value corresponding to the MTPV limit. Furthermore, DFVC has alsobeen used in connection with the predictive control (Boazzo and Pellegrino, 2015;Pellegrino et al., 2015).

A drawback of these stator-flux-oriented schemes is that the torque-producingcurrent control loop is nonlinear (even in the case of linear magnetics), whichcomplicates the tuning procedure. The control performance for constant gainsdepends on the operating point due to the nonlinear dynamics. To avoid anoscillatory response, the control design can be performed for the best case in asuboptimal manner (Pellegrino et al., 2009).

The stator-flux-oriented control schemes discussed so far do not present asystematic design procedure for the controller and gain selection. PublicationIII presents a feedback-linearization stator-flux-oriented control method andits systematic design procedure. An exact input-output feedback linearizationcontroller structure is derived, yielding a completely decoupled and easy-to-tune system. A state-feedback controller with integral action and referencefeedforward is designed and the design guidelines and tuning principles arepresented. Furthermore, an anti-windup mechanism is developed, taking intoaccount the nonlinear structure of the controller. The following subsection brieflyreviews the controller presented in Publication III.

3.4.1 Control Design

Figure 3.7 shows the overall structure of the stator-flux-oriented controller. Ithas two main parts, i.e., a state-feedback controller and a flux observer. Thedesign of a state-feedback controller is presented, whereas the design of a fluxobserver is discussed in Chapter 4. Alternatively, the stator flux linkage canbe estimated directly using the flux model (2.7) without any observer. Then,the state vector xf is obtained using (2.17) and (2.20). An advantage of thisapproach is that the order of the whole control system is not increased due tothe flux estimation and no additional gains are needed. In practice, applyinga flux observer is preferred, since it reduces the sensitivity to the errors in themagnetic model (2.7) and to the measurement noise.

34

Control Schemes

uref

K−1t

uref −uref

K1

Kt

xfref Ki

xf

vf

ωmJ

T

iψ

uref1s

Fluxobserver

u

T−1

Figure 3.7. Stator-flux-oriented controller, including an anti-windup scheme (shaded region).The nonlinear transformation matrix T = T(ψ) is given in (3.17). Here, the fluxobserver operates in rotor coordinates, but stator coordinates could be used as well.The compensation for the resistive voltage drop has been omitted.

Nonlinear State FeedbackAn exact input-output feedback linearization (Sastry and Isidori, 1989) is appliedto tackle the nonlinearity in the model (2.21). Inserting the control law

ufref = Rif +ωmJψf +

[1 0

−a/b Ld/b

]vf (3.14)

into (2.21) leads to a simple linear system

dxf

dt= vf (3.15)

where vf is the transformed input vector, obtained from an external linearcontroller to be designed subsequently. The control law (3.14) can be transformedto rotor coordinates, leading to

uref = Ri+ωmJψ+Tvf (3.16)

where

T= eδJ

[1 0

−a/b Ld/b

](3.17)

This nonlinear transformation matrix includes both the coordinate transfor-mation (from stator flux coordinates to rotor coordinates) and the feedbacklinearization.

Linear ControllerThe relation (3.15) between the transformed input and the output can be rewrit-ten as

xf = vf

s(3.18)

Any linear controller can be easily designed for the system (3.18). As an exam-ple, a simple proportional controller would suffice if steady-state errors were

35

Control Schemes

acceptable. Here, a state-feedback controller with reference feedforward andintegral action is used,

vf = Ktxfref +

K i

s(xf

ref − xf)−K1xf (3.19)

where Kt is the reference feedforward gain, K i is the integral gain, and K1 isthe state-feedback gain. The gains can be selected as Kt = αI, K i = α2I, andK1 = 2αI, leading to the first-order closed-loop response

xf = α

s+αxf

ref (3.20)

where α is the bandwidth. If desired, the controller could be easily modifiedin such a way that the flux and torque channels have different bandwidths.It is worth noticing that the effects of the parameter errors in the nonlineartransformation (3.17) on the steady-state accuracy are compensated for by theintegral action of the linear controller (3.19).

3.4.2 Anti-Windup Mechanism

Figure 3.7 shows the stator-flux-oriented controller equipped with an anti-windup technique, which is based on the realizable reference (Peng et al., 1996).The realizable voltage reference can be either calculated in the controller using(2.4) and (2.5) or obtained from the PWM. It is important to notice that theeffect of the nonlinear transformation in (3.17) has to be properly included inthe anti-windup scheme, as shown in the figure.

3.5 Results and Discussion

The overall control systems shown in Figures 3.1 and 3.6 are similar, even thoughthe controlled variables are different and the fastest control loop operates indifferent coordinates. Comparing the shaded region in Figures 3.1 and 3.6, itcan be seen that the inputs (Tref, ωm, udc, and i) and the output uref of bothcontrollers are the same.

If the magnetic saturation and the speed changes are omitted, the dynamicsseen by the current controller are linear and the closed-loop system can bemade comparatively robust. As pointed out earlier, the torque-producing currentcontrol loop in the stator-flux-oriented controller is nonlinear (even in the case oflinear magnetics). This nonlinearity was addressed and removed in PublicationIII. It is worth mentioning that the MTPV limit as well as the zero-flux conditionare singularities in stator-flux-oriented control. These singularities can be easilyavoided with negligible losses in the energy efficiency and in the maximumtorque capability. In the case of current-controlled drives, these singularitiesdo not exist. Furthermore, the current limitation is easier to implement in thecurrent-controlled drives.

36

Control Schemes

A detailed set of results is presented in Publications I, II, and III. Selectedrepresentative results are shown as follows. The robustness of the currentcontroller and the stator-flux-oriented controller against parameter errors isstudied by means of simulating a 2.2-kW interior PM motor. The discrete-timecurrent controller shown in Figure 3.5 and the stator-flux-oriented controllershown in Figure 3.7 with and without the flux observer are selected. The fluxobserver (4.1) appying the measured speed, to be explained in detail in Chapter 4,is used for stator-flux-oriented control. For the case without the flux observer, thestator flux linkage is estimated using the magnetic model (2.7). The closed-looppoles are placed using the IMC-based design for all the controllers.

The parameter estimates given in Chapter 6 are used in the control systemand the errors are introduced in the plant model. The desired control bandwidthfor all the controllers is set to α= 2π ·100 rad/s and the sampling (switching)frequency is 5 kHz. The motor operates at a constant speed of ωm = 0.75 p.u. andthe torque reference is stepped from 0 to the rated torque with the incrementsof 25% of the rated torque.

Figures 3.8 and 3.9 show the simulation results for the 2.2-kW interior PMmotor with parameter errors. For the results in Figure 3.8, the actual motorparameters are: Ld = Ld, Lq = 2Lq, and ψf = 0.5ψf. Figure 3.8(a) shows theresults for the current controller and Figure 3.8(b) for the stator-flux-orientedcontroller without the flux observer. As expected, there is a steady-state error inthe actual torque due to the parameter errors. The control response for both thecontrollers is similar. Figure 3.8(c) shows the results for the stator-flux-orientedcontroller with the flux observer. The percentage error in the actual torque ismore compared to the cases in Figures 3.8(a) and 3.8(b). It is mainly due tothe errors in the flux estimation originating from the large parameter errors.Similar conclusions can be drawn for the results in Figure 3.9. The actual motorparameters for the results in Figure 3.9 are: Ld = 0.5Ld, Lq = Lq, and ψf = 0.5ψf.

To summarize, the robustness of a current-controlled drive and of a stator-flux-orientation controlled drive (without the flux observer) against parameter errorsis similar if their closed-loop poles are placed similarly, and if the samplingfrequencies are high enough. If a very low ratio of the sampling frequency tothe maximum speed is required, the current control is more robust compared tothe stator-flux-oriented controller due to the direct discrete-time control design.This design option is not yet available for stator-flux-oriented control.

For controlling highly saturated machines, the magnetic saturation modelcan be incorporated into the stator-flux-oriented controller, as was done in theexperimental systems of Publication III. Compared to the standard currentcontroller with the same magnetic model, the proposed stator-flux-oriented con-troller works better in transients, since the flux magnitude is used as anothercontrolled state variable. Even better robustness against the magnetic satu-ration can be achieved by applying the flux-linkage-based current controller,where the d- and q-axis flux components are controlled (Publication II).

37

Control Schemes

(a)

(b)

(c)

Figure 3.8. Simulation results for the 2.2-kW PM motor with parameter errors: (a) currentcontroller; (b) stator-flux-oriented controller without the flux observer; (c) stator-flux-oriented controller with the flux observer (4.1). The actual motor parameters are:Ld = Ld, Lq = 2Lq, and ψf = 0.5ψf.

38

Control Schemes

(a)

(b)

(c)

Figure 3.9. Simulation results for the 2.2-kW PM motor with parameter errors: (a) currentcontroller; (b) stator-flux-oriented controller without the flux observer; (c) stator-flux-oriented controller with the flux observer (4.1). The actual motor parameters are:Ld = 0.5Ld, Lq = Lq, and ψf = 0.5ψf.

39

Control Schemes

40

4. Flux Observers and SensorlessControl

In this thesis, the focus is on back-EMF-based observers that estimate the rotorspeed and position based on the mathematical motor model. The observer shouldprovide (locally) stable and sufficiently fast estimation-error dynamics at allspeeds and loads. It should also be robust against the measurement noise andparameter errors. Crossing the zero speed even with large load torque as well assmooth starting and stopping in a no-load condition should be possible withoutadditional algorithms. These requirements are not trivial to fulfill, since theestimation-error dynamics unavoidably become nonlinear.

4.1 Continuous-Time Flux Observer

The continuous-time flux observers are reviewed in Publication IV. A unifieddesign and analysis framework for a class of back-EMF-based observers isdeveloped and the links between apparently different estimation methods arebrought out. State observers equipped with a speed-adaptation law were shownto be mathematically equivalent to voltage-model-based flux observers equippedwith a position-tracking loop. The error signal driving the adaptation law or thetracking loop is presented in a generalized form.

4.1.1 Simple Observer with Speed Feedback

If the drive is equipped with a position sensor, the flux linkage can be estimatedusing a simple state observer in rotor coordinates (Vagati et al., 1997),

dψdt

= u−Ri−ωmJψ+G′(Li+ψf −ψ) (4.1)

where G′ is the observer gain matrix. Based on (2.7), (2.11), the dynamics of theestimation error ψ=ψ−ψ are governed by

dψdt

=−(ωmJ+G′)ψ (4.2)

Therefore, any desired closed-loop system matrix can be easily set via theobserver gain G′. If a constant gain matrix G′ = gI is used, the observer behaves

41

Flux Observers and Sensorless Control

kp + kis

ωm 1s

ϑmu

iεψ

Eq. (4.5)Eq. (4.3)

Figure 4.1. Structure of a sensorless flux observer in estimated rotor coordinates.

as the voltage model at higher speeds and as the flux model at low speeds(Guglielmi et al., 2004). The parameter g defines the corner frequency (typicallyg = 2π ·15. . .30 rad/s).

4.1.2 Flux Observer with Speed Estimation

A voltage-model-based flux observer is adopted for the framework in PublicationIV. This structure leads to the simplest form of the equations and simplifies theinclusion of the magnetic saturation model in the observer. The flux observerin Publication IV is briefly presented in this subsection. The flux observer inestimated rotor coordinates is defined by

dψdt

= u−Ri− ωmJψ+K(Li+ψf −ψ

)(4.3)

where K is a 2×2 observer gain matrix and estimates are marked with a hat. Insensorless drives, the actual rotor position ϑm is naturally unknown. Therefore,the correction vector generally differs from the real flux estimation error ψ−ψ

during transients, even if the accurate model parameter estimates are assumed.

Speed EstimationAs seen in Figures 4.1, the PI mechanism is used to drive the error signal ε tozero by adjusting the speed estimate, which is further fed to the integrator forgetting the position estimate,

ωm = kpε+∫

kiεdt ϑm =∫

ωmdt (4.4)

where kp and ki are the gains. The generalized error signal is defined by meansof the scalar product

ε=λTJ(Li+ψf −ψ

)(4.5)

where the projection vector λ can be a constant vector or it may depend on ψ

and i.

Equivalent State Observer with a Speed-Adaptation LawAn open-loop flux observer augmented with an output-error-based correctionterm is presented by Piippo et al. (2008) and Tuovinen et al. (2012)

dψdt

= u−R i− ωmJψ+G(i− i

)(4.6a)

i = L−1(ψ−ψf)

(4.6b)

42


It is easy to show that (4.6) is mathematically equivalent to (4.3) if

K =GL−1 +RL−1 (4.7)

The speed-adaptation law used in (Piippo et al., 2008; Tuovinen et al., 2012)equals (4.4) and (4.5). The error signal ε = iq − iq is obtained by substitutingλ= [1

0]/Lq in (4.5).

4.1.3 Discrete-Time Implementation

Usually, a speed and position observer is first designed in the continuous-timedomain and then discretized for a digital processor. The simplest option would beto apply the Euler approximation, but it leads to severely distorted pole locationsand instability issues at higher speeds, unless a comparatively high samplingfrequency is used.

An alternative approach was introduced in Publication IV. An exact hold-equivalent motor model is used in a discretized flux observer. The state-observerform (4.6) is chosen as a starting point for discretization, making it possible toapply an exact hold-equivalent motor model (or its series approximation) for theopen-loop observer part. Hence, the structure of the discrete-time observer isanalogous to (4.6),

ψ(k+1)=Φψ(k)+Γfψf +Γu(k)+Gd[i(k)− i(k)

](4.8a)

i(k)= L−1[ψ(k)−ψf]

(4.8b)

where Gd is the observer gain matrix and Φ, Γf, and Γ are the system matrices.The system matrices depend on the speed estimate, e.g., Φ = Φ[ωm(k)]. Theexact expressions for system matrices can be found in Publication IV. The samesystem matrices are used in the discrete-time controller in Publication I. If thecontrol system has a typical computational delay of one sampling period Ts, thevoltage in (4.8) is

u(k)= e−TsωmJuref(k−1) (4.9)

where uref is the reference voltage to the PWM in rotor coordinates. The matrixΓ inherently takes the PWM delay into account and no additional compensationshould be used. The observer gain K is mapped to the discrete-time domain as,cf. (4.7),

Gd = Ts (KL−RI) (4.10)

If the exact system matrices (or their second-order series approximations) areused, the estimation-error dynamics remain stable at very low ratios betweenthe sampling frequency and fundamental frequency.

4.2 Discrete-Time Flux Observer

In this section, the discrete-time flux observers are discussed. Higher funda-mental frequencies and improved robustness at a given sampling frequency

43


can be achieved by designing the control system directly in the discrete-timedomain (Kim et al., 2010; Lee et al., 2011; Xu and Lorenz, 2014). For the directdiscrete-time control design, a hold-equivalent discrete-time model of the motoris needed, including the effects of the ZOH and sampler.

Lee et al. (2011) used an approximate discrete-time motor model to designa discrete-time current and flux observer to be used with a deadbeat-directtorque and flux controller. Xu and Lorenz (2014) used a similar approach todesign a discrete-time flux observer, but the approximate discrete-time motormodel is more complex and has unnecessarily high number of model parameters.Furthermore, the selection of the observer gain is not discussed. Guoqianget al. (2014) presents an active flux based full-order discrete-time sliding modeobserver for the sensorless control of interior PM motor drives, but the motormodel is discretized using the forward Euler method. Yang and Chen (2017)developed a discrete-time current observer for the high-speed surface-mountedPM motor. The observer is based on the approximate discrete-time motor modeland it is not directly applicable to the salient synchronous motors.

In Publication V, a speed-adaptive full-order observer is designed and analyzeddirectly in the discrete-time domain. The observer design is based on the exactdiscrete-time motor model derived in Publication I, which inherently takes thedelays in the control system into account. A linearized model for the discrete-time estimation-error dynamics is derived and a stabilizing observer gain isproposed. The proposed design decouples the speed-estimation dynamics fromthe flux-estimation dynamics, which simplifies the tuning procedure. This de-coupling approach was originally proposed for the continuous-time flux observerby Tuovinen et al. (2012). Based on the results, performance improvementsobtained via the direct discrete-time design, compared to the correspondingcontinuous-time designs, are significant if the ratio between the sampling fre-quency and the fundamental frequency is low. The ratio below 10 between thesampling and fundamental frequencies is achieved in experiments with theproposed discrete-time design.

Since the availability of Publication I and V, some authors have used the exacthold-equivalent motor model in the design of discrete-time flux observers. Onthe basis of Publication I, Zhang et al. (2017) developed a complex-vector basedhold-equivalent model of interior PM motors and used it to design a full-orderobserver in rotor coordinates. The observer design totally omits the effect ofthe speed estimation. In reality, the speed and flux estimation dynamics arenonlinear and coupled. A recent review paper lists most of the available discrete-time flux observers for interior PM motors (Xu et al., 2018). According to Xu et al.(2018), the research on discrete-time flux and speed estimation for interior PMmotors at low sampling to fundamental frequency ratios lags behind inductionmotors.

To the best knowledge of the present author, before the availability of Publica-tion V, no discrete-time designs of flux observers based on exact hold-equivalentmotor model were available for interior PM motors in the literature. The discrete-

44


iref

ϑm

is

M

uref

ωmi

i

u

PWM

usref qabc

ψe−ϑmJ

e(ϑm+ωmTs)JCurrent

controller

z−1

Observer

z−1

Figure 4.2. Sensorless control system. The discrete-time plant model includes the grey blocks:motor, PWM, and computational time delay z−1. The white blocks represent thediscrete-time control algorithm. For discrete-time modeling, the PWM is replacedwith the ZOH and the gate signals qabc are replaced with the duty ratios dabc.

time speed-adaptive full-order observer developed in Publication V is brieflyreviewed below.

4.2.1 Observer Structure

Figure 4.2 shows a sensorless control system, which is the framework for thediscrete-time observer. The discrete-time observer in estimated rotor coordinatesis defined by

ψ(k+1)= Φψ(k)+ Γu(k)+ γψf +Kd i(k) (4.11a)

i(k)=Cψ(k)+dψf (4.11b)

where Kd is the gain matrix, C = L−1, d = [−1/Ld,0]T, and i = i− i. It is worthnoticing that the matrices Φ, Γ, and γ in the discrete-time observer are functionsof the estimated speed ωm. The closed-form expressions for system matrices canbe found in Publications I, IV, and V.

A discrete-time rotor-position estimation is

ϑm(k+1)= ϑm(k)+Tsωm(k) (4.12)

and the speed-adaptation law is

ωmi(k+1)= ωmi(k)+Tski i(k) (4.13a)

ωm(k)= ωmi(k)+kp i(k) (4.13b)

where kp = [0,kp] and ki = [0,ki] are the gain vectors. The integral state ωmi canbe used as an input signal in outer control loops (e.g., in the speed controller),while the observer state matrices in (4.11) and the rotor position estimation in(4.12) depend on ωm.

45


ωm(z)

ωm(z)Gϑ(z)

i(z)

ωm(z)H(z)

Tsz−1

ϑm(z)

Gω(z)

Figure 4.3. Linearized estimation-error dynamics for the discrete-time observer design. Thedashed line disappears if bω = 0 is assumed.

4.2.2 Estimation-Error Dynamics

Nonlinear DynamicsFor analyzing the estimation-error dynamics, the plant model in estimated rotorcoordinates is

ψ(k+1)=Φ′ψ(k)+Γ′u(k)+γ′ψf (4.14a)

i(k)=C′ψ(k)+d′ψf (4.14b)

where

Φ′ = e−ϑm(k+1)JΦeϑm(k)J Γ′ = e−ϑm(k+1)JΓeϑm(k)J

γ′ = e−ϑm(k+1)Jγ C′ = e−ϑm(k)JCeϑm(k)J d′ = e−ϑm(k)Jd (4.15)

The nonlinear estimation-error dynamics become

ψ(k+1)= (Φ+KdC)ψ(k)+ (Φ+KdC)ψ(k)+ (γ+Kdd)ψf + Γu(k) (4.16a)

i(k)=Cψ(k)+ dψf + Cψ(k) (4.16b)

where Φ= Φ−Φ′ and other matrices are defined similarly.

Linearized DynamicsLinearization of (4.16) leads to

ψ(k+1)= Aψψ(k)+bϑϑm(k)+bωωm(k) (4.17a)

i(k)=Cψ(k)+dϑϑm(k) (4.17b)

where the system matrices are

Aψ =Φ0 +KdC, dϑ = (JC−CJ)ψs0 +Jdψf

bω =(

∂Φ

∂ωm

∣∣∣0+TsJΦ0

)ψs0 +

(∂γ

∂ωm

∣∣∣0+TsJγ0

)ψf +

(∂Γ

∂ωm

∣∣∣0+TsJΓ0

)us0

bϑ = (JΦ0 −Φ0J)ψs0 +Jγ0ψf +Kddϑ+ (JΓ0 −Γ0J)us0. (4.18)

46


The elements of bω approach zero as the sampling period Ts approaches zero.The transfer-function matrix from ϑm(z) to i(z) is

Gϑ(z)=C(zI− Aψ)−1bϑ+dϑ (4.19)

and the transfer-function matrix from ωm(z) to i(z) is

Gω(z)=C(zI− Aψ)−1bω. (4.20)

The transfer-function matrix from i(z) to ωm(z), corresponding to the adaptationlaw (4.13), is

H(z)= kp +Tski

z−1(4.21)

Figure 4.3 presents the block diagram of the linearized estimation-error dynam-ics.

4.2.3 Gain Selection

The linearized system in Figure 4.3 is of the fourth order, and in general, explicitexpressions for the gain selection may not exist. In order to obtain an approx-imate solution, bω = 0 is assumed. It is also worth noticing that bω does notdepend on the observer gain, i.e., it cannot be affected by the gain selection. Onthe other hand, bϑ = 0 can be forced if the observer gain is selected as

Kd =[

Ldk1 Lq(v−βk1)

Ldk2 Lq(w−βk2)

](4.22)

with

v = [uq(γ11 −γ22)−ud(γ12 +γ21)+ (φ11 −φ22)ψq −γ2ψf

]/ψ′

f (4.23a)

w = [ud(γ11 −γ22)+uq(γ12 +γ21)+ (φ11 −φ22)ψd +γ1ψf

]/ψ′

f (4.23b)

where β= (Ld−Lq)iq/ψ′f and ψ′

f =ψf+(Ld−Lq)id. The term ψ′f can be interpreted

as a fictitious flux (Koonlaboon and Sangwongwanich, 2005; Agarlita et al.,2012).

The fourth-order characteristic polynomial is expressed as a product of twosecond-order polynomials (z2+bz+c)(z2+dz+e), where the first part correspondsto the flux estimation and the second part corresponds to the speed adaptation.The resulting stabilizing gain selection is

k1 =− (φ211 +bφ11 −φ2

21 +φ21v+ c)β+ (φ11 +φ22 +b+w)(v−φ21)v−φ21(1+β2)+ (φ11 −φ22 −w)β

(4.24a)

k2 = φ221 −φ21v− c− (φ22 +w)(φ22 +b+w)− (φ11 +φ22 +b+w)φ21β

v−φ21(1+β2)+ (φ11 −φ22 −w)β(4.24b)

The speed-adaptation gains are

kp =Lq(d+2)

Tsψ′f

, ki =Lq(d+ e+1)

T2sψ

′f

. (4.25)

47


In order to guarantee stable operation, the discrete-time design parameters b, c,d, and e have to be selected so that the roots of the corresponding polynomials,z2 +bz+ c and z2 +dz+ e, remain inside the unit circle in every operating point.

4.3 Discussion

Tuovinen et al. (2012) presented a continuous-time gain design for a speed-adaptive full order observer. The design guarantees the local stability of theestimation error dynamics at every operating point (except at zero speed) inideal conditions. However, the effects of the digital implementation were notconsidered. If the ratio between the sampling frequency and the fundamentalfrequency is low, the stability conditions derived in (Tuovinen et al., 2012) arenot valid and the system can become unstable.

The low-speed performance of the continuous-time design and the discrete-timedesign is quite similar. The low-speed results corresponding to the continuous-time design without the signal injection were presented in (Tuovinen et al., 2012)and with signal-injection in (Tuovinen and Hinkkanen, 2014). The correspondingdiscrete-time results are presented in Publication V.

Simulation and experimental results at higher fundamental frequency for fluxobservers designed in the continuous-time domain and the discrete-time domainare presented in Publication V. The flux observer designed in the continuous-time domain is unstable at higher fundamental frequencies. The correspondingdiscrete-time observer gives a stable response. Based on the results, performanceimprovements obtained via the direct discrete-time design, compared to thecorresponding continuous-time designs, are significant if the ratio between thesampling frequency and the fundamental frequency is low.

48

5. Optimal Reference Generation

This chapter focuses on the calculation of the state references for the fastestcontrol loop, i.e., the current references id,ref and iq,ref.

5.1 Conventional Methods

The motor is driven at the MTPA locus, at the current limit, at the MTPVlimit, or in the field-weakening region (which is the region bounded by theabove-mentioned locus and limits), depending on the torque reference Tref, theoperating speed ωm, and the DC-bus voltage udc. An example of these controlcharacteristics for a 6.7-kW SyRM are shown in Figure 5.1. In existing controlmethods, the MTPA locus and the MTPV limit are either computed off-line basedon the known magnetic saturation characteristics or measured with a suitabletest bench. The resulting look-up tables are then implemented in the real-timecontrol system. The off-line computation as well as test-bench measurementsare typically time-consuming processes. Instead of using an MTPA look-uptable, the MTPA locus could be tracked using signal injection (Kim et al., 2013),which, however, causes additional noise and losses. In some applications, an

(a) (b)

Figure 5.1. MTPA locus (green), MTPV limit (red), and current limit (blue) for the 6.7-kW SyRM:(a) id–iq plane; (b) ψd–ψq plane. The dashed lines show the loci, when the magneticsaturation is not taken into account (inductances correspond to the rated operatingpoint). The black dashed line corresponds to the constant current circle.

49

Optimal Reference Generation

approximate MTPV limit could be searched for in an iterative manner by meansof repeated acceleration tests (Pellegrino et al., 2012a).

5.1.1 Feedback Field-Weakening Methods

Field-weakening methods can be broadly divided into feedback methods andfeedforward methods. The feedback field-weakening methods apply the differ-ence between the reference voltage and the maximum available voltage. Thesemethods use the maximum voltage in the field-weakening operation, but theydo not necessary guarantee minimum losses. Kim and Sul (1997) presentedthe feedback field-weakening method for interior PM motors, but the MTPVlimit was not included. Kwon et al. (2012) improved the transient performanceof the feedback field-weakening methods by dynamically adjusting the currentreferences during transients. Lin et al. (2014) extended the operation of thefeedback field-weakening methods along the MTPV limit. The current referenceswere limited by analytically solving the MTPV limit with constant motor pa-rameters. Hoang and Aorith (2015) also combined the feedback field-weakeningmethod with analytically solved MTPV and current limits with constant motorparameters. Manzolini et al. (2018) extended the operation range of the feed-back field-weakening methods to the MTPV limit. For including the magneticsaturation, it was suggested to use two model dependent nonlinear functionsor look-up tables for the MTPA locus and the MTPV limit. Furthermore, thevoltage control loop of the feedback field-weakening methods has to be properlytuned and should have a much lower bandwidth than the innermost currentcontroller (Bedetti et al., 2015).

5.1.2 Feedforward Field-Weakening Methods

The feedforward field-weakening methods provide the optimal references with-out delays. Due to the feedforward nature of these methods, the dynamics of theinner control loop remain intact and the noise content in the state referencesis minor. However, modeling inaccuracies may reduce the available torque orincrease the losses. Some feedforward field-weakening methods are based onanalytical solutions of the intersection of a voltage ellipse and a torque hyperbola(Morimoto et al., 1990; Jung et al., 2013; Jeong et al., 2013). A disadvantage ofthese methods is that they do not take the magnetic saturation into account (andthe saturation effects cannot be properly taken into account afterwards, sincethe saturation deforms the shape of the voltage ellipses and torque hyperbolas).Other feedforward methods are based on off-line computed look-up tables (Meyerand Böcker, 2006; Cheng and Tesch, 2010; Peters et al., 2015; Huber et al., 2015).The magnetic saturation can be included properly in these methods, but theoff-line data processing is difficult and time consuming. Feedforward methodsin (de Kock et al., 2010; Mohammadi and Lowther, 2017) are based on thefinite-element method (FEM) and rely on the knowledge of the motor geometry.

50


id,ref

ϑm

iabc

M

Currentcontrol

udc

Tref Referencecalculation

ddt

ωm

Magneticmodel

identification

Look-uptable

computation

Plug-and-play start-up

iq,ref

Figure 5.2. Control system of a typical current-controlled drive (the lower part), augmented witha plug-and-play start-up method (the upper part). The start-up method consistsof two stages, magnetic model identification and look-up table computation, whichcan be run in the embedded processor of the drive. Look-up tables are then used inthe reference calculation. The control system may include a speed controller, whichprovides the torque reference.

Furthermore, the feedforward methods can be augmented with an additionalvoltage controller in order to be able to dynamically adjust the voltage margin(Peters et al., 2015; Huber et al., 2015).

The focus of the rest of the chapter is on the feedforward field-weakening meth-ods. The following subsection briefly reviews a plug-and-play start-up methodfor optimal reference calculation presented in Publication VI. Furthermore, acomparison is presented with the conventional methods.

5.1.3 Plug-and-Play Startup

Figure 5.2 exemplifies a control system of a typical current-controlled drive, aug-mented with a plug-and-play start-up method (the grey part). A plug-and-playstart-up method for optimal reference calculation was presented in PublicationVI. A look-up table computation method is proposed. When combined with anidentification method for the magnetic model, the proposed method enables theplug-and-play start-up of an unknown motor. Various identification methods forthe magnetic model of SyRMs, PM-SyRMs, and interior PM motors are alreadyavailable (Bedetti et al., 2016; Hinkkanen et al., 2017; Pescetto and Pellegrino,2017; Odhano et al., 2019). The identification methods are left out of the scopeof this thesis.

The magnetic model identification and look-up table computation are executedoff-line during the start-up, preferably in the embedded processor of the drive.After the start-up has been completed, the control look-up tables are ready to beused in the real-time control.

51


Tref

ωm

min(·)

ψmtpa

ψref

Tmax

udc

min(·)Tref

sign(·)

| · |

| · |

umaxku/

�3

ψmax

Figure 5.3. Feedforward field-weakening scheme including the MTPA locus and the current andMTPV limits. The optimal flux magnitude is ψref and the limited torque referenceis Tref. The torque limit is Tmax =min(Tmtpv,Tlim), where Tlim corresponds to themaximum current and Tmtpv corresponds to the MTPV limit. The factor ku definesthe voltage margin.

5.2 Control System

Figure 5.2 depicts the overall structure of the current-controlled drive system,which is used as an example in this chapter. The reference calculation is ex-plained in the following subsections.

5.2.1 Feedforward Field-Weakening Scheme

Figure 5.3 shows a conventional feedforward field-weakening scheme includingthe MTPA locus and the current and MTPV limits (Meyer and Böcker, 2006;Peters et al., 2015; Huber et al., 2015). From (2.11), the stator voltage magnitudecan be expressed as

u =√

u2d +u2

q

=√(

ωmψq −Rid −dψd

dt

)2

+(ωmψd +Riq +

dψq

dt

)2

(5.1)

If R = 0 and the steady state are assumed, this expression for the voltagemagnitude reduces to u = |ωm|ψ, which is used to convert the maximum availablevoltage umax to the maximum available flux magnitude ψmax in the schemeshown in Figure 5.3.

The maximum voltage umax = kuudc/�

3 is calculated from the measured DC-link voltage udc. Hence, any sudden variations in udc are directly translatedinto the references. The factor ku defines the voltage margin. As can be realizedfrom (5.1), some voltage reserve is necessary for the resistive voltage dropsand for changing the flux linkages in transient conditions. If ku = 1 is chosen,the overmodulation region may be entered even in the steady state (due tothe resistive voltage drops), which causes the sixth harmonics in the statorvoltage. To avoid entering the overmodulation region in the steady state, the

52


Tref

id,ref

ψref

iq,ref

sign(·)

| · |

Tref

ψd,ref

ψq,refsign(·)

√ψ2

ref −ψ2d,ref

| · |

ψref

id,ref

iq,refEq.

(2.10)

(b)

(a)

Figure 5.4. Current references: (a) conventional method (Meyer and Böcker, 2006; Cheng andTesch, 2010; Peters et al., 2015; Huber et al., 2015); (b) proposed method with onlyone two-dimensional look-up table.

factor ku < 1 should be chosen. Alternatively, the resistive voltage drops could becompensated for by means of the measured current and the resistance estimate,cf. e.g. (Pellegrino et al., 2012a). However, applying this feedback action inthe reference calculation may introduce increased noise content and ringingphenomena. It is also worth noticing that the factor ku could be dynamicallyadjusted by means of an additional voltage controller (Peters et al., 2015; Huberet al., 2015). For simplicity, a constant value for ku is used in this thesis.

As seen in Figure 5.3, the optimal MTPA flux magnitude ψmtpa is read froma look-up table, whose input is the torque reference. The MTPA flux is limitedbased on the maximum flux ψmax, yielding the optimal flux magnitude ψref

under the voltage constraint. The torque reference Tref is limited by the torqueTmax corresponding to the combined MTPV and current limits, yielding thelimited torque reference Tref. An advantage of the scheme shown in Figure 5.3 isthat the optimal reference values are obtained without any delays. The schemecan be used directly with stator-flux-oriented control.

5.2.2 Current References

If the current-controlled drive is used, the optimal flux reference and the limitedtorque reference have to be mapped to the corresponding current references. Fig-ure 5.4(a) shows the conventional current reference calculation method (Meyer

53


MTPV

Output list: {Tmtpv(m)}

M

compute MTPV look-up tablefor m = 1 : M

Input list: {ψ(m)}

MTPA

Output lists: {ψmtpa(l)}, {Tmtpa(l)}

L, imax

compute MTPA look-up tablefor l = 1 : L

Input list: {i(l)}

ψm

tpa (L)

Current limit

Output list: {Tlim(m)}

for m = 1 : M

Input list: {ψ(m)}

compute current limit look-up table

{ψd,m

tpv (m)}

Reference look-up table

Output table: {ψd,ref(m,n)}

for m = 1 : M

Input lists: {ψ(m)}, {Tref(n)}

for n = 1 : M

compute ψd,ref look-up table

{Tm

tpv(

m)}

Figure 5.5. Look-up table computation procedure. The parameters of the magnetic model (2.10)are also needed in each stage.

and Böcker, 2006; Cheng and Tesch, 2010; Peters et al., 2015; Huber et al., 2015),based on the two two-dimensional look-up tables. Interpolation is used to getthe values of id,ref and iq,ref from the look-up tables.

Figure 5.4(b) shows the proposed current reference calculation method. Asingle two-dimensional look-up table is used to determine ψd,ref. Then, the valueof the q-axis flux ψq,ref is obtained by means of the Pythagorean theorem. Theflux-linkage references are mapped to the current references using (2.10). Sinceonly one two-dimensional look-up table is needed, the memory requirementsof the control system are less than in the conventional method. An interpola-tion procedure, applicable to the conventional method as well, is given in theAppendix of Publication VI.

54


5.3 Look-Up Table Computation

Figure 5.5 shows an overall diagram of the look-up table computation methodproposed in Publication VI. The look-up table computation method is dividedinto four stages, i.e., the MTPA, MTPV, current limit, and reference look-up table.A step-by-step method to compute each look-up is presented in Publication VI.The saturation model (2.10) is used to represent the magnetic model of SyRMs,PM-SyRMs, and interior PM motors.

The look-up table computation method is not limited to the saturation model(2.10). Different magnetic models or even look-up tables could be used insteadif they are physically feasible and invertible in the relevant operation range.Furthermore, the reference calculation method corresponding to Figures 5.3and 5.4(b) is used as an example, but the proposed look-up table computationmethod can be easily modified for other reference calculation structures as well.

5.4 Discussion

The reference state vector for stator-flux-oriented control can be calculated usingψref and Tref from Figure 5.3 and (2.19), i.e.,

xfref =

[ψref

iτ,ref

]=[

ψref2

3p Tref

](5.2)

It can be seen from Figure 5.3 that only the MTPA locus, the current limit, andthe MTPV limit are needed for the reference calculation.

Much simpler reference calculation methods can be used for stator-flux-orientedcontrol than for rotor-oriented current control. Typically, a current-controlleddrive system requires at least one two-dimensional look-up table (in additionto the MTPA and MTPV look-up tables), which is computed off-line using acomplicated special algorithm (Publication VI). Furthermore, two-dimensionallook-up tables require more memory in the embedded processor and algorithmsincrease the computational burden. To summarize, the state reference calcula-tion methods for stator-flux-oriented control are simpler than those for currentcontrol.

55


56

6. Experimental Setup

The block diagram of the experimental setup is shown in Figure 6.1. The DC-buses of the load drive and the drive under test are connected in parallel. Thestudied motors are a 6.7-kW four-pole SyRM and a 2.2-kW six-pole interior PMmotor. The data of the SyRM are given in Table 6.1 and for the interior PMmotor in Table 6.2. The load motor is a 6.7-kW servo induction motor (IM). Thecontrolled motor is fed by an ABB ACSM1 frequency converter. The originalcontrol board of the ACSM1 has been replaced with an interface board for usewith the dSPACE system. The interface board passes the switching signals tothe frequency converter and provides protection against the over-current andover-voltage. Furthermore, the dead-time is also generated in the interfaceboard before passing switching signals to the converter. The load IM is fedby an ABB ACSM1 converter in Publications I and V and by an ABB ACS880converter in all the other Publications. The technical data of the hardware usedin the experimental setup is listed in Table 6.3. A photograph of the laboratoryexperimental setup is shown in Fig. 6.2.

The control algorithms for the drive under test were implemented in a rapidprototyping dSPACE system consisting of DS1104, DS1006, or DS1202 control

M

M

Load drive

ωm

400 V50 Hz

dSPACE system

udc iabcPWM

Drive under test

Figure 6.1. The block diagram of the experimental setup used in the measurements.

57

Experimental Setup

Table 6.1. Data of the 6.7-kW SyRM

Rated values

Phase voltage (peak value)�

2/3·370 V 1.00 p.u.

Current (peak value)�

2·15.5 A 1.00 p.u.

Frequency 105.8 Hz 1.00 p.u.

Speed 3175 r/min 1.00 p.u.

Torque 20.1 Nm 0.67 p.u.

Parameters at the rated operating point

d-axis inductance Ld 45.6 mH 2.20 p.u.

q-axis inductance Lq 6.84 mH 0.33 p.u.

Stator resistance R 0.55 Ω 0.04 p.u.

Table 6.2. Data of the 2.2-kW interior PM motor

Rated values

Phase voltage (peak value)�

2/3·370 V 1.00 p.u.

Current (peak value)�

2·4.3 A 1.00 p.u.

Frequency 75 Hz 1.00 p.u.

Speed 1500 r/min 1.00 p.u.

Torque 14 Nm 0.80 p.u.

Parameters at the rated operating point

d-axis inductance Ld 36 mH 0.34 p.u.

q-axis inductance Lq 51 mH 0.48 p.u.

Stator resistance R 3.6 Ω 0.073 p.u.

PM flux ψf 0.545 Vs 0.85 p.u.

58

Experimental Setup

ACSM1 ACS880

Voltage sensor

Currentsensors

(a) (b)

SyRM IM

(c)

Figure 6.2. Photograph of the laboratory experimental setup: (a) frequency converters; (b) dSpaceMicroLabBox; (c) SyRM and IM mechanical assembly.

boards. First, the control algorithms were developed in Matlab/Simulink envi-ronment and then compiled, and the resulting files were uploaded to the dSPACEsystem. The PWM interface of the dSPACE system was used for generatingthe switching signals for the frequency converter under test. The input to thePWM interface is the sampling period and the duty ratios dabc. The duty ratiosfor each phase were calculated from the stator voltage reference us

ref by meansof the space-vector PWM algorithm. The phase currents and DC-link voltagewere measured using external Hall-effect transducers. The data of current andvoltage transducers used in the experimental setup are given in Table 6.3. Themeasured quantities are sampled in synchronism with the PWM.

The motor speed is measured using an incremental encoder for the feedbackor monitoring purposes in Publications I, IV, V, and VI and using a resolver inPublications II and III. The measured speed is fed to the dSPACE system to beused in the control algorithms. The initial rotor position (when an incrementalencoder was used) was set simply by supplying the DC-current vector to thedirection of the a-phase magnetic axis, causing the rotor to rotate into thisdirection.

59

Experimental Setup

Table 6.3. Technical data of the experimental setup. The current and voltage values are rmsvalues. The three-phase voltage is phase-to-phase voltage and the current is phasecurrent.

SyRM ABB

Rating 370 V, 15.5 A, 105.8 Hz

3175 r/min, 6.7 kW, cosϕ= 0.74

Interior PM motor ABB M2BJ 100L 6 B3

Rating 370 V, 4.3 A, 75 Hz

1500 r/min, 2.2 kW, cosϕ= 0.9

IM ABB HDP Servo motor

Rating 400 V, 17.1 A, 152.1 Hz

3000 r/min, 6.7 kW, cosϕ= 0.62

Freq. converter for SyRM/interior PM motor ABB ACSM1-04AS-031A-4

Supply voltage 380. . . 480 V (50/60 Hz)

Output voltage 0. . . 100 % of supply voltage

Output current, normal use 31 A

Output frequency 0. . . 500 Hz

Freq. converter for IM ABB ACS880-01-038A-3

Supply voltage 380. . . 415 V (50/60 Hz)

Output voltage 0. . . 100 % of supply voltage

Output current, normal use 38 A

Output frequency 0. . . 500 Hz

Current transducers LEM LA 55-P

Bandwidth 0. . . 200 kHz (−1 dB)

Accuracy (at 25oC, rated current) ±0.9 %

Voltage transducer LEM LV 25-P

Accuracy (at 25oC, rated current) ±0.9 %

Incremental encoder Leine & Linde 392911-50

Line counts 2048 ppr

Resolver Smartsyn TS2641N11E64

Control boards

dSPACE DS1104 Used in Publications I and V

dSPACE DS1006 Used in Publications IV and VI

dSPACE MicroLabBox Used in Publications II and III

60

7. Summaries of Publications

7.1 Abstracts

The abstracts of the publications are given in this section.

Publication I

This paper deals with discrete-time models and current control methods forsynchronous motors with a magnetically salient rotor structure, such as inte-rior permanent-magnet synchronous motors and synchronous reluctance mo-tors (SyRMs). The dynamic performance of current controllers based on thecontinuous-time motor model is limited, particularly if the ratio of the sam-pling frequency to the fundamental frequency is low. An exact closed-formhold-equivalent discrete motor model is derived. The zero-order hold of thestator-voltage input is modeled in stationary coordinates, where it physicallyis. An analytical discrete-time pole-placement design method for two-degrees-of-freedom proportional–integral current control is proposed. The proposedmethod is easy to apply: only the desired closed-loop bandwidth and the threemotor parameters (R, Ld, Lq) are required. The robustness of the proposedcurrent control design against parameter errors is analyzed. The controller isexperimentally verified using a 6.7-kW SyRM drive.

Publication II

Magnetic saturation characteristics of synchronous reluctance motors (SyRMs),with or without permanent magnets (PMs), are highly nonlinear. These nonlin-ear effects can be included in the current controller by changing its state variablefrom the stator current to the stator flux linkage using the known saturationcharacteristics. A direct discrete-time variant of the flux-linkage-based currentcontroller is developed in a state-space framework. If the magnetics are modeledto be linear, the proposed control structure reduces to the standard current

61

Summaries of Publications

controller in this special case. Experimental results on a 6.7-kW SyRM drivedemonstrate that the proposed flux-linkage-based controller enables a higherclosed-loop bandwidth and is more robust against parameter errors, as comparedto the standard current controller.

Publication III

This paper deals with stator-flux-oriented control of permanent-magnet (PM)synchronous motors and synchronous reluctance motors (SyRMs). The variablesto be controlled are the stator-flux magnitude and the torque-producing currentcomponent, whose references are easy to calculate. However, the dynamics ofthese variables are nonlinear and coupled, potentially compromising the controlperformance. We propose an exact input-output feedback linearization structureand a systematic design procedure for the stator-flux-oriented control method inorder to improve the control performance. The proposed controller is evaluatedby means of experiments using a 6.7-kW SyRM drive and a 2.2-kW interior PMsynchronous motor drive.

Publication IV

This paper deals with the speed and position estimation for synchronous re-luctance motors (SyRMs) and interior permanent-magnet synchronous motors(IPMs). A unified design and analysis framework for a class of back electro-motive force based observers is developed and the links between apparentlydifferent estimation methods are brought out. State observers equipped witha speed-adaptation law are shown to be mathematically equivalent to voltage-model-based flux observers equipped with a position-tracking loop. The errorsignal driving the adaptation law or the tracking loop is presented in a gener-alized form. Using the framework, a stabilizing gain design is reviewed anddetailed design guidelines are given. Selected observer designs are experimen-tally evaluated using a 6.7-kW SyRM drive and a 2.2-kW IPM drive.

Publication V

This paper deals with the speed and position estimation of interior permanent-magnet synchronous motor (IPM) and synchronous reluctance motor (SyRM)drives. A speed-adaptive full-order observer is designed and analyzed in thediscrete-time domain. The observer design is based on the exact discrete-timemotor model, which inherently takes the delays in the control system into ac-count. The proposed observer is experimentally evaluated using a 6.7-kW SyRMdrive. The analysis and experimental results indicate that major performanceimprovements can be obtained with the direct discrete-time design, especiallyif the sampling frequency is relatively low compared with the fundamental fre-quency. The ratio below 10 between the sampling and fundamental frequencies

62


is achieved in experiments with the proposed discrete-time design.

Publication VI

This paper deals with the optimal state reference calculation for synchronousmotors having a magnetically salient rotor. A lookup table computation methodfor the maximum torque-per-ampere locus, maximum torque-per-volt limit, andfield-weakening operation is presented. The proposed method can be used duringthe start up of a drive, after the magnetic model identification. It is computa-tionally efficient enough to be implemented directly in the embedded processorof the drive. When combined with an identification method for the magneticmodel, the proposed method enables the plug-and-play startup of an unknownmotor. Furthermore, a conventional reference calculation scheme is improved byremoving the need for one two-dimensional lookup table. A 6.7-kW synchronousreluctance motor drive is used for experimental validation.

7.2 Scientific Contributions

The main scientific contributions of this thesis are summarized as follows:

• An exact closed-form hold-equivalent discrete-time motor model with theassumption of constant parameters is derived for the magnetically salientsynchronous motors in Publication I. A modified hold-equivalent discrete-time motor model with the assumption of R = 0 is derived in PublicationII. This model takes the nonlinear saturation characteristics into accountproperly.

• An analytical direct discrete-time design method for a state-feedback cur-rent controller with integral action and reference feedforward is proposedin Publication I. The controller is based on the discrete-time motor modeland the computational delay is taken into account in the state feedbacklaw. The proposed method is easy to apply: only the desired closed-loopbandwidth and the three motor parameters (Ld, Lq, R) are needed.

• A current control for highly saturated synchronous motors is presentedin Publication II. A simple control structure is obtained by changing thecontrolled variable from the stator current vector to stator flux-linkagevector. In this way, the effects of the magnetic saturation are inherentlyincluded in the controller. The two controllers are designed in both thecontinuous and discrete-time domains.

• An exact input-output feedback linearization structure and a systematicdesign procedure for the stator-flux-oriented control method is proposedin Publication III. Design guidelines and tuning principles are presented

63


for a state-feedback controller. Apart from the motor parameters, theproposed method requires only the desired closed-loop bandwidth for thecontrol.

• A speed-adaptive full-order observer is designed and analyzed in thediscrete-time domain in Publication V. The observer design is based on theexact discrete-time motor model derived in Publication I, which inherentlytakes the delays in the control system into account. A linearized model forthe discrete-time estimation-error dynamics is derived and a stabilizingobserver gain is proposed. The proposed design decouples the speed-estimation dynamics from the flux-estimation dynamics, which simplifiesthe tuning procedure.

• An optimal state reference calculation method for synchronous motorsis presented in Publication VI. A lookup table computation method forthe MTPA locus, MTPV limit, and field-weakening operation is presented.The proposed method can be used during the start up of a drive, afterthe magnetic model identification. When combined with an identificationmethod for the magnetic model, the proposed method enables the plug-and-play startup of an unknown motor.

64

8. Conclusions

Control methods suitable for magnetically anisotropic synchronous motors weredeveloped in this thesis. Higher fundamental frequencies and improved robust-ness against parameter errors at a given sampling frequency can be achievedwith the proposed methods. Two analytical discrete-time motor models forsalient synchronous motors were developed. These models were then used in thedesign and analysis of current controllers and flux observers. The discrete-timemotor model in Publication I is derived with the assumption of constant motorparameters. The model in Publication II is derived with the assumption of zerostator resistance. The magnetic saturation can be properly included in thismodel. The motor model in Publication I is suitable for the design and analysisof flux observers, whereas the motor model in Publication II is more suitablefor the design of current controllers. Before the availability of Publication I, noexact closed-form expressions valid for magnetically salient synchronous motorswere available in the literature.

Two current controllers based on the discrete-time motor models were designedin the discrete-time domain. A state-feedback controller with integral action andreference feedforward was used as a framework. The discrete-time delays wereproperly included in the design. The proposed methods are easy to apply: onlythe desired closed-loop bandwidth and the motor model are needed. The SyRMsand PM-SyRMs have highly nonlinear saturation characteristics, which shouldbe properly taken into account in the control system in order to reach highperformance. Unlike in the controller in Publication I, the magnetic saturationcan be properly included in the controller in Publication II.

A plug-and-play startup method for optimal reference calculation was devel-oped in this thesis. A look-up table computation method is proposed for theMTPA locus, the MTPV limit, the current limit, and field-weakening region. Themethod is capable of providing the optimal current or flux references without anydelays, depending on the torque reference, the operating speed, and the DC-busvoltage. Any sudden variations in the DC-bus voltage are directly translatedinto references. The proposed method is computationally efficient enough to beimplemented directly in the embedded processor of the drive. Since only onetwo-dimensional lookup table is needed, the memory requirements of the control

65

Conclusions

system are less than in the conventional methods.Apart from the current controllers, a feedback-linearization stator-flux-oriented

control method and its systematic design procedure was also proposed in thisthesis. An exact input-output feedback linearization scheme is developed andcombined with a simple linear control law. The computational burden of the pro-posed controller is comparable to the conventional stator-flux-oriented controller.Furthermore, the proposed controller provides better dynamic performance, ismore robust against parameter errors, and is easier to tune.

The robustness of a conventional current-controlled drive and of a proposedstator-flux-orientation controlled drive against parameter errors is similar iftheir closed-loop poles are placed similarly and if the sampling frequencies arehigh enough. If a very low ratio of the sampling frequency to the maximum speedis required, more robust discrete-time current controllers developed in this thesisshould be used. The discrete-time design option is not yet available for stator-flux-oriented control. The main benefit of using the stator-flux-oriented controlis a simple reference calculation method, compared to the rotor-oriented currentcontrol. To summarize, the simplicity and robustness of stator-flux-orientedcontrol is tempting for many applications, while better control performance canbe achieved with the proposed discrete-time control designs.

For the speed sensorless operation, the control schemes developed in this thesiscan be augmented with a speed-adaptation law. In this thesis, a speed-adaptivefull-order observer was designed and analyzed directly in the discrete-timedomain. A linearized model for the discrete-time estimation-error dynamics werederived and a stabilizing observer gain was proposed based on the linearizedmodel. The speed-estimation dynamics were decoupled from the flux-estimationdynamics, which simplifies the tuning procedure. There are only four designparameters to be selected for the stable operation of the sensorless controlsystem.

A suitable topic for future research is to design a direct discrete-time versionof the stator-flux-oriented controller. The discrete-time flux observer can beaugmented with signal injection methods for sustained operation in loadedconditions at zero speed. Furthermore, online estimation of the stator resistanceand PM-flux can be included in the discrete-time flux observer.

66

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