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SPEED CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR

USING EXTENDED HIGH-GAIN OBSERVER

By

Abdullah Ahmad Alfehaid

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Electrical Engineering—Master of Science

2015

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ABSTRACT

SPEED CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR

USING EXTENDED HIGH-GAIN OBSERVER

By

Abdullah Ahmad Alfehaid

Feedback linearization is used to regulate and shape the speed of a surface mount

Permanent Magnet Synchronous Motor (PMSM). An extended high-gain observer, which is

driven by the measured position of the PMSM’s rotor, is also used to estimate both the speed of

the motor and the disturbance present in the system to recover the performance of feedback

linearization. Two methods are presented to design the extended high-gain observer. The first

method is based on the full model of the PMSM and the second method is based on a reduced

model of the PMSM. The reduction of the model is made possible by creating fast current loops

that allowed the use of singular perturbation theory to replace the current variables by their

quasi-steady-state equivalent. The design of the speed controller and the extended high-gain

observer is based on the nominal parameters of the PMSM. The disturbance is assumed to be

unknown, and time-varying but bounded. Stability of the output feedback system is shown.

Finally, simulation and experimental results confirm stability, robustness, and performance of the

system.

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Copyright by

ABDULLAH AHMAD ALFEHAID

2015

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To my family

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ACKNOWLEDGMENTS

I would like to express my sincere appreciation to the following people:

To my advisor Dr. Hassan K. Khalil who patiently and excellently guided me through my

research. He is truly an honest, patient and a humble person. His knowledge and passion for the

subject is inspiring. Indeed, his advice and suggestions have been and will be of great help to me.

I am honored to have been his student.

To Dr. Elias G. Strangas who offered his laboratory for conducting the experiment and

for his excellent insights and assistance throughout the experiment.

To my parents (Ahmad Alfehaid and Moody Alhomaidy) who supported me, encouraged

me, and believed in me. I sincerely thank them for teaching me the love of seeking knowledge.

And to my wife (Yara Almani) who stood by me during my studies and created a

comfortable atmosphere. I could not have done it without you.

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TABLE OF CONTENTS

LIST OF TABLES ....................................................................................................................... viii

LIST OF FIGURES ....................................................................................................................... ix

CHAPTER 1 ................................................................................................................................... 1

Introduction ..................................................................................................................................... 1

1.1 Mathematical Model of PMSM........................................................................................ 4

1.2 Preliminaries..................................................................................................................... 7

CHAPTER 2 ................................................................................................................................. 11

Control Algorithm ......................................................................................................................... 11

2.1 Full Model Approach ..................................................................................................... 11

2.1.1 Extended High-Gain Observer ................................................................................... 13

2.2 Reduced Model Approach .............................................................................................. 15

2.2.1 Current Loops ............................................................................................................. 16

2.2.2 Extended High-gain Observer .................................................................................... 18

2.2.3 Feedback Linearization ............................................................................................... 20

2.2.4 Closed Loop analysis .................................................................................................. 21

CHAPTER 3 ................................................................................................................................. 27

Simulation ..................................................................................................................................... 27

3.1 Simulation Setup ............................................................................................................ 27

3.2 Simulation Results.......................................................................................................... 28

3.2.1 Simulation I ................................................................................................................ 28

3.2.2 Simulation II ............................................................................................................... 30

3.2.3 Simulation III .............................................................................................................. 31

3.2.4 Simulation IV ............................................................................................................. 32

CHAPTER 4 ................................................................................................................................. 33

Experiment .................................................................................................................................... 33

4.1 Experiment Setup ........................................................................................................... 33

4.1.1 Current Measurement ................................................................................................. 35

4.1.2 Incremental Encoder Interface .................................................................................... 37

4.1.3 Incremental Encoder’s Digital Filter .......................................................................... 44

4.1.4 PWM Controller Circuit ............................................................................................. 44

4.1.5 Control Algorithm Loop ............................................................................................. 49

4.1.5.1 Three Phase to α-β Transformation ..................................................................... 50

4.1.5.2 α-β to d-q Transformation ................................................................................... 51

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4.1.5.3 Extended High-Gain Observer ............................................................................ 52

4.1.5.4 Feedback Linearization ....................................................................................... 54

4.1.5.5 PI Controller ........................................................................................................ 55

4.1.5.6 d-q to α-β Transformation ................................................................................... 56

4.1.5.7 α-β to Three Phase Transformation ..................................................................... 57

4.2 Experimental Results...................................................................................................... 58

4.2.1 Experiment I ............................................................................................................... 60

4.2.2 Experiment II .............................................................................................................. 60

4.2.3 Experiment III ............................................................................................................. 61

CHAPTER 5 ................................................................................................................................. 63

Conclusion and Future Work ........................................................................................................ 63

5.1 Conclusion ...................................................................................................................... 63

5.2 Future Work ................................................................................................................... 64

5.2.1 Field Weakening ......................................................................................................... 64

5.2.2 Sensorless Control ...................................................................................................... 65

APPENDIX ................................................................................................................................... 66

BIBLIOGRAPHY ......................................................................................................................... 68

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LIST OF TABLES

Table 2.1. Routh’s array for the characteristic equation ( 2.46 ). ................................................. 25

Table 3.1. Nominal parameters of the used PMSM. ..................................................................... 28

Table 4.1. Truth table for the driving clock clk of the edges counter. .......................................... 40

Table 4.2. State transition table for the state diagram of channel A and B and the direction of

rotation. ........................................................................................................................ 42

Table 4.3. State transition table for the high-side switching signal. ............................................. 48

Table 4.4. State transition table for the Low-side switching signal. ............................................. 49

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LIST OF FIGURES

Figure 1.1. Cross-sectional view of a surface mounted PMSM [2]. ............................................... 2

Figure 1.2. Relationship between the stator and the rotor frame of references. ............................. 6

Figure 2.1. Block diagram of the proposed control algorithm. ..................................................... 16

Figure 3.1. (a) Speed of PMSM using nominal parameters, (b) Speed deviation of PMSM from

target speed. ................................................................................................................ 29

Figure 3.2. (a) Speed of PMSM using a 20% increase in the nominal parameters, (b) Speed

deviation of PMSM from target speed. ....................................................................... 30

Figure 3.3. (a) Speed of PMSM before and after the external load was applied, (b) Applied

external load and its estimate. ..................................................................................... 31

Figure 3.4. Error between target speed and motor speed as �� → 0 & � → 0. ............................. 32

Figure 4.1. Block diagram of the experimental setup. .................................................................. 34

Figure 4.2. First order RC low-pass filter. .................................................................................... 37

Figure 4.3. Incremental encoder output signals. ........................................................................... 38

Figure 4.4. State diagram for channel A and B and the direction of rotation. .............................. 41

Figure 4.5. Incremental encoder interface circuit. ........................................................................ 43

Figure 4.6. High frequency corruption of Channel A and B. ....................................................... 43

Figure 4.7. Implementation of the digital filter circuit in LabView. ............................................ 45

Figure 4.8. Signals generation by the PWM controller circuit. .................................................... 47

Figure 4.9. Implementation of the PWM controller circuit for one phase pole. ........................... 50

Figure 4.10. Implementation of the three phase to α-β transformation in LabView. ................... 51

Figure 4.11. Implementation of α-β to d-q transformation in LabView. ...................................... 52

Figure 4.12. implementation of the discrete extended high-gain observer in LabView. .............. 53

Figure 4.13. Implementation of the speed controller ( 2.37 ) with saturation in LabView. ......... 54

Figure 4.14. Block diagram of the PI controller. .......................................................................... 55

x

Figure 4.15. Implementation of the discrete PI controller with anti-winding in LabView. ......... 56

Figure 4.16. Implementation of d-q to α-β transformation in LabView. ...................................... 57

Figure 4.17. Implementation of α-β to three phase transformation in LabView. ......................... 58

Figure 4.18. (a) Simulation and experimental speed of PMSM using nominal parameters, (b)

simulation and experimental speed deviation from target speed. ............................. 59

Figure 4.19. (a) Simulation and experimental speed of PMSM when the nominal parameters are

increased by 20%, (b) simulation and experimental speed deviation from target

speed when the nominal parameters are increased by 20%. ..................................... 61

Figure 4.20. Speed of PMSM before and after the external load was applied. ............................ 62

1

CHAPTER 1

Introduction

Permanent Magnet Synchronous Motors (PMSM) are increasingly used in the industries

and rapidly replacing induction and DC motors particularly in servo application such as CNC

machines and robotic systems. PMSM are popular due to their efficiency, high power density,

light weight, maintenance-free, and small size comparing to DC and induction machines [1].

There are two types of three phase AC PMSMs: the surface mounted PMSMs and the

interior magnet PMSM. The surface mounted PMSMs are built with magnets mounted on the

surface of the rotor while the interior magnet PMSMs are built with magnets embedded in the

rotor. This structural difference leads to different mathematical models and hence leads to

different control approaches. Throughout this document, only the surface mounted PMSM will

be considered. Figure 1.1 shows a cross-sectional view of a four pole surface mounted

PMSM [2].

PMSMs are not easy to control because they exhibit time-varying nonlinear dynamic

behavior. The parameters of PMSMs are prone to temperature changes and variation in operating

points, e.g the stator winding resistance can vary by as much as 200% of its nominal value and

the rotor flux linkage can vary by as much as 20% of its nominal value [3]. PMSM popularity in

the recent years has triggered the interest in the control community which led to many control

approaches.

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In the industry, linear Proportional and Integral (PI) controllers have been largely used in

PMSM drives. It is considered to be one of the simplest control techniques that offer an adequate

performance. However, PI controllers are not a great choice in applications where high

performance and high precision are required.

Figure 1.1. Cross-sectional view of a surface mounted PMSM [2].

Sliding mode Control (SMC) is becoming popular in PMSM drives due to its robustness

to parameter variations. However, in the presence of disturbance and system parameter variation,

the gains of the SMC are increased to guarantee robustness. This causes the system to exhibit a

phenomenon called chattering. Improvements to SMC have taken place to reduce chattering such

as using reaching laws and disturbance estimators. In [4] and [5], reaching laws are used to

decrease chattering but this causes reduction in SMC robustness near the sliding surface and also

increases the reaching time. In [5], an extended SM observer is used to estimate the disturbance

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and then cancel it in the control law. A key difference between this thesis and [5] is that this

work includes the position measurement in the closed loop analysis as opposed to assuming the

speed is measured. Furthermore, the proposed control method in this work is capable of shaping

the transient response, whereas the work presented in [5] provides no means for shaping the

transient response.

Adaptive control has been used to control the speed of PMSM. In [6], Model Reference

Adaptive Control (MRAC) is used with disturbance estimator to avoid estimating each parameter

of the motor separately. This work is similar to our work in two ways, 1) the disturbance is

estimated and then canceled in the control law, 2) The transient response is shaped by the

MRAC. However, our work is different in that we assume a time-varying non-vanishing

disturbance that could be dependent on both states and time, and we do not assume that the speed

is directly measured. In [6], the disturbance depends only on time and its derivative converges to

zero as time tends to infinity.

Feedback linearization has been also used to control the speed of PMSM. It is one of the

best achievements of nonlinear control theory because it allows the use of linear control

techniques to design controllers. In [7], feedback linearization is used with a PI controller to

regulate the speed of PMSM. However, in real applications, feedback linearization, with or

without PI controller; fails in shaping the transient response in the presence of model uncertainty

and unknown disturbance. Therefore, other tools must be used with feedback linearization to

guarantee both stability and performance. In [8], feedback linearization is used with an extended

observer to estimate speed and disturbance. This thesis differs from [8] in that this work assumes

time-varying and state-dependent disturbance, the model of the PMSM is reduced hence

reducing the order of the observer, the controller design is based on the nominal parameters of

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the PMSM, and in this thesis PI controller are not used for the speed loop but it relies on the

integral action provided by the controller.

1.1 Mathematical Model of PMSM

The mathematical model of a surface mount PMSM in the two-phase equivalent stator

frame of reference, the α-β coordinates, is as follows [10]:

� ���� = − �� + �� sin������ + �� ( 1.1 )

� ����� = − ��� − �� cos������ + ��� ( 1.2 )

� ��� = ���−�� sin����� + ��� cos������ − � ( 1.3 )

��� = � ( 1.4 )

where �� and ��� are the two-phase equivalent stator currents, �� and ��� are the two-phase

equivalent stator voltages, � is the mechanical rotor speed, � is the rotor position, � is the

external load, is the stator winding resistance, � is the stator inductance and it is defined to be

the sum of the magnetizing inductance and the leakage inductance of the stator, �� is the number

of pole pairs, �� is the rotor magnetic flux linkage, and � is the moment of inertia of the rotor.

The relationship between the three phase voltages and their two-phase equivalent voltages is

given by,

"������# $ = %23()))))* 1 −12 −120 √32 −√321√2 1√2 1√2 -..

.../"�0�1�2$

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where �0, �1, and �2 are the three phase voltages, and �# is the zero-sequence voltage which is

identically zero for a balanced three phase system. Furthermore, the relationship between the

three phase currents and their two-phase equivalent currents is governed by,

3������# 4 = %23()))))* 1 − 12 −120 √32 −√321√2 1√2 1√2 -..

.../"�0�1�2$

where �0, �1, and �2 are the three phase currents, and �# is the zero-sequence current which is

identically zero for a balanced three phase system. The model of the PMSM shown above is

nonlinear and is hard to control. However, it is much easier to control the motor in the rotor’s

frame of reference, the d-q coordinates, which is a rotating frame of reference. Figure 1.2 shows

the relationship between the stator and the rotor frame of references. From Figure 1.2, the

transformation from the α-β coordinates to the d-q coordinates is achieved by the following

relationship,

5�6�78 = 9 cos����� sin�����− sin����� cos�����: 5�����8 and

;�6�7< = 9 cos����� sin�����− sin����� cos�����: ;�����< where �6 is the direct-axis input voltage, �7 is the quadrature-axis input voltage, �6 is the direct-

axis current, and �7 is the quadrature-axis current. Now, the system ( 1.1 )-( 1.4 ) can be rewritten

in the rotor’s frame of reference (d-q coordinates) as shown in ( 1.5 )-( 1.8 ).

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Figure 1.2. Relationship between the stator and the rotor frame of references.

� ��6� = − �6 + �����7 + �6 ( 1.5 )

� ��7� = − �7 − �����6 − ��� + �7 ( 1.6 )

� ��� = ���7 − =� − � ( 1.7 )

��� = � ( 1.8 )

The mathematical model of the PMSM in the rotor’s frame of reference is still nonlinear;

however, it is easier to control. Controlling the motor in this frame of reference is called Field

Oriented Control (FOC) because stator currents are projected onto the rotor’s magnetic field.

This transformation reveals a very important piece of information that �7 is the only torque

producing current as seen in ( 1.7 ). Hence, the current �6 should be regulated to zero to increase

the efficiency of the system.

The mathematical model of the PMSM is subject to practical constraints. The stator

voltages and currents cannot exceed a certain limit; that is,

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�6> + �7> ≤ @�0A> and

�6> + �7> ≤ B�0A>

where @�0A and B�0A are the maximum stator voltage and current, respectively. The voltage

constraint is often imposed on the model by the used DC-link that is utilized by the inverter. The

current constraint, on the other hand, is imposed on the model by the current rating of both the

used inverter and the PMSM. It is very important not to violate these limitations otherwise it

would cause serious damage to the motor as well as to the inverter. Therefore, the controller

design must account for this limitation.

1.2 Preliminaries

Consider the following single-input-single-output nonlinear system in the normal

form [9]:

CD = EC + =[GHC,JK + LHC,JK�], N = OC ( 1.9 )

where C ∊ ℝR is the state trajectory, � ∊ ℝ is the control input, N ∊ ℝ is the measured output, J

is the disturbance input and it belongs to a known compact set S ⊂ ℝℓ, LHC,JK ≥ L# > 0, and

E, =, and O are defined as follows:

E =()))*0 1 0 ⋯ 00 0 1 ⋯ 0⋮ ⋮ ⋱ ⋱ ⋮0 0 ⋯ 0 10 0 ⋯ ⋯ 0-..

./ ∊ ℝR×R, = =()))*00⋮01-.../ ∊ ℝR

O = [1 0 ⋯ ⋯ 0] ∊ ℝ\×R

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The objective here is to design an output feedback controller that not only stabilizes the origin

C = 0 but also drives the system trajectories to match that of a target system. A natural choice

for the target system would be:

CD ⋆ = HE − =^KC⋆, N = OC⋆

where ^ is chosen such that HE − =^K is Hurwitz and C⋆ is the state of the target system whose

trajectories meet the desired transient response. If the state C were available for measurement and

the functions LHC,JK and GHC,JK were exactly known, then a control input � that achieves the

objective via feedback linearization would be given by:

� = −GHC,JK − ^CLHC,JK

However, in real applications, two problems arise. First, only the nominal values of LH·K and GH·K are known. Second, some states of the system may not be accessible for measurement or simply

we choose not to measure them due to technical or economic reasons. Therefore, state observers

are usually utilized to solve these problems. Here, the following extended high-gain observer is

used:

C̀D = EC̀ + =ab̀ + GcHC̀K + L̀HC̀K�d + 5e\� ⋯ eR�R8f HN − OC̀K �b̀� = eRg\�Rg\ HN − OC̀K ( 1.10 )

where C̀ is the estimate of C, L̀HC̀K and GcHC̀K are nominal values of LHC,JK and GHC,JK, respectively, b̀ is the estimate of the disturbance, � > 0 is a small parameter, and e\,.…, eRg\

are chosen such that the polynomial

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hRg\ + e\hR +⋯+ eRg\

is Hurwitz. The control input � can now be taken as:

� = −b̀ − GcHC̀K − ^C̀L̀HC̀K

It is assumed that L̀HC̀K ≥ L# > 0. To protect the system from the peaking phenomenon of high-

gain observers [11], the control law � is saturated outside a compact set, that is,

� = ihL j−b̀ − GcHC̀K − ^C̀iL̀HC̀K k

where hLH∙K is the saturation function and it is defined as hLHmK = n��o1, |m|qh�r�HmK, and i

is a scaling constant given by,

i > maxA∈wx,J∈S y−GHC,JK + ^CLHC,JK y where Ω2 is a compact set given by,

Ω2 = o@HCK ≤ {q where @HCK is a Lyapunov function defined by,

@HCK = Cf|C

where | = |f > 0 is the solution of the Lyapunov equation |HE − =^K + |HE − =^Kf| = −}

for some } = }f > 0. The constant { is chosen large enough such that any given compact subset

of ℝR can be included in the interior of Ω2. Under this control law, it is shown in [9] that not

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only does the control law stabilize the origin C = 0 but it also recovers the performance of

feedback linearization in the presence of both model uncertainty and unknown disturbance.

11

CHAPTER 2

Control Algorithm

The goal is to design a feedback controller that can achieve the following objectives:

1) Regulating the speed of the PMSM to a reference signal �~�� in the presence

of both bounded external load � and parameters uncertainty.

2) The ability to shape the speed transient response.

The aforementioned objectives can be realized using the method described in [9] with two

different approaches. The first approach is a direct application of the method described in chapter

1 and it is based on the complete model of the PMSM. The second approach is based on a

reduced mathematical model of the PMSM that is obtained by utilizing the singular perturbation

method; consequently, requiring a lower order extended high-gain observer. In both cases the

rotor position � and the three phase currents �0. �1, and �2 are measured, thus �6 and �7 are

known.

2.1 Full Model Approach

To apply the control method described in [9], the mathematical model of the

PMSM ( 1.5 )-( 1.8 ) must first be put in the normal form as in ( 1.9 ). Since there are two

control inputs to the system �6 and �7, then it can be treated as two separate systems. The first

system takes ( 1.5 ) and the second system takes ( 1.6 )-( 1.8 ); that is,

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System I ����� = − � �� + ����� + 1��� ( 2.1 )

���������� = ���� = �n� �� −=� �− 1� ������ = − � �� − ����� − �n� �+ 1���

( 2.2 )

System II ( 2.3 )

( 2.4 )

It can be seen that system I is already in the normal form and needs no further manipulation. On

the other hand, System II requires additional steps to be put in the normal form. As a first step,

define the acceleration as,

� = ��� ( 2.5 )

then substituting ( 2.3 ) into ( 2.5 ) yields,

� = ��� �7 − =� � − 1� � ( 2.6 )

The second step is to take the derivative of ( 2.6 ) with respect to time which brings 6��6� , thus

bringing the control input �7 with it, that is,

��� = ��� ��7� − =� ��� − 1� �� � ( 2.7 )

Substituting ( 2.4 ) and ( 2.5 ) into ( 2.7 ) yields,

��� = −�� �� �7 − ����� ��6 − ��>�� � + ���� �7 − =� � − 1� �� �

Now system II becomes,

��� = � ( 2.8 )

13

��� = � ( 2.9 )

��� = −�� �� �7 − ����� ��6 − ��>�� � − =� � − 1� �� � + ���� �7 ( 2.10 )

which is in the normal form.

2.1.1 Extended High-Gain Observer

Since the system was divided into two parts, then each part requires its own extended

high-gain observer. Also, since the external load � is not known and only the nominal

parameters are known, then system I and II are rewritten as,

System I ����� = − ��� �� + ����� + b1 + 1�� �� ( 2.11 )

���������� = ���� = ���� = −�n� ��c�� �� − �n����c ��� − �n� 2�c�� �− =��c � + b2 + �n��c�� ��

( 2.12 )

System II ( 2.13 )

( 2.14 )

Where c, �c, ��� , =c , and �� are the nominal values of , �, ��, =, and �, respectively. b\ and b>

are the disturbances and they are defined as,

b\ = −j � − c�ck �6 b> = −j�� �� − ��� c���c k �7 − j��� − ����� k ����6 − ���>�� − ��� >

���c �� − j=� − =c��k �+ j���� − ������c k �7 − 1� �� �

14

The current �6 is a measured quantity and it will be used to drive the extended high-gain observer

of system I. Furthermore, the extended high-gain observer of system II will be driven by the

measured position of the rotor �. Both extended high-gain observer are designed based on the

nominal model ( 2.11 ) and ( 2.12 )-( 2.14 ) and both take the form ( 1.10 ), that is,

System I

Extended

High-Gain

Observer

����� ����� = − ��� ��� + ���� �� + b�1 + 1���� + e1�1 H�� − ���K�b�1� = e2�12 ��� − ���� ( 2.15 )

System II

Extended

High-Gain

Observer ��������� ���� = �� + e3�2 �� − ������� = �c + e4�22 �� − ���

��c� = −�n� ��c�� �� − �n����c �� �� − �n� 2�c�� �� −=��c �c + b�2 + �n��c�� �� + e5�23 �� − ����b�2� = e6�24 �� − ��� ( 2.16 )

where �6̂, b̀\, �c, ��, �̀, and b̀> are the estimates of �6, b\, �, �, �, and b>, respectively, e\and e>

are chosen such that

h> + e\h + e> = 0

is Hurwitz, e�, e�, e�, and e� are chosen such that

h� + e�h� + e�h> + e�h + e� = 0

is Hurwitz, and �\, �> > 0 are small parameters. The control law that achieves the objectives can

now be derived using feedback linearization and based on the designed extended high-gain

observers ( 2.15 ) and ( 2.16 ). As drawbacks of using the full model to design the extended high-

gain observers is the amplified measurement noised due to the large gain associated with the

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term \��� [11]. In addition, Implementation of such an observer would require sophisticated and

fast hardware which is expensive. These problems can be solved by reducing the model of the

PMSM for the design of the extended high-gain observer.

2.2 Reduced Model Approach

The mathematical model of the PMSM contains two groups of physical quantities,

electrical and mechanical. The electrical quantities are represented by the currents �6 and �7, and

the mechanical quantities are represented by the speed of the rotor � and the rotor position �.

The electrical time constant of ( 1.5 ) and ( 1.6 ) is found to be �� = �, while the mechanical time

constant of ( 1.7 ) is given by �� =  ¡. Typically, the electrical time constant is much smaller

than the mechanical time constant. Subsequently, the electrical states are much faster than the

mechanical states. A conclusion can be made that the mathematical model of the PMSM is a two

time scale system.

To exploit the time scale property, fast currents loops are created by using PI controllers.

Consequently, the electrical variables will reach quasi-steady-state much faster than the

mechanical variables. Then, the method of singular perturbation is utilized to replace the

electrical variables by their equivalent quasi-steady-state. Hence, the model of the PMSM is

reduced for the design of the extended high-gain observer. As a result, a third order extended

high gain observer is obtained.

The proposed control algorithm that is based on the reduced model consists of three main

parts, fast inner current loops, speed and disturbance estimation using the measured position via

16

an extended high-gain observer, and speed regulation via feedback linearization. The block

diagram of the control algorithm that is based on the reduced model is shown in Figure 2.1.

Figure 2.1. Block diagram of the proposed control algorithm.

2.2.1 Current Loops

The fast current loops are made possible by the smallness of the electrical time constant

�� = � and the use of PI controllers for �6 and �7. For instance, define the current tracking errors

as:

¢6 = �6£¤¥ − �6 ( 2.17 )

17

¢7 = �7£¤¥ − �7 ( 2.18 )

where ¢6 and ¢7 are the direct and quadrature current-tracking errors, respectively; �6£¤¥ and

�7£¤¥ are the direct and quadrature current reference signals, respectively. The control inputs �6

and �7 are chosen as follows:

�6 = ��¢6 + C6

�7 = ��¢7 + C7

with

C6 = ��¦¢6 � ( 2.19 )

C7 = ��¦¢7 � ( 2.20 )

where �� is the proportional gain, �� is the integral gain, and C6 and C7 are the integrals of ¢6

and ¢7, respectively. Substituting �6 and �7 into ( 1.5 ) and ( 1.6 ) yields the following current

tracking error equations:

� �¢6� = � ��6£¤¥� + � + ��� �6£¤¥ − ¢6 − ���� §�7£¤¥ − ¢7¨ − 1� + ��� C6 ( 2.21 )

� �¢7� = � ��7£¤¥� + � + ��� �7£¤¥ − ¢7 + ��� + ���� + ���� §�6£¤¥ − ¢6¨− 1� + ��� C7

( 2.22 )

where � = �g©ª.

18

2.2.2 Extended High-gain Observer

By the proper choice of the current controller gains �� and ��, ¢�« and ¢�� are made fast

and they will reach quasi-steady-state much faster than other state variables in the system. This

induces a two time scale system, with fast and slow dynamics, which gives us an advantage and

invites the use of the singular perturbation method [11] to reduce the model and then design the

extended high-gain observer.

The quasi-steady-state of the fast variables ¢�« and ¢��, obtained by setting � = 0 in ( 2.21 ) and ( 2.22 ) is,

¢6 = 1 + �� § �6£¤¥ − C6¨ ( 2.23 )

¢7 = 1 + �� § �7£¤¥ + ��� − C7¨ ( 2.24 )

Substitute ( 2.18 ) into ( 1.7 ) to obtain the equation:

��� = ��� §�7£¤¥ − ¢7¨ − =� � − 1� � ( 2.25 )

where �7£¤¥ is viewed as the control input. Now, ( 2.19 ), ( 2.20 ), ( 2.23 ), ( 2.24 ), and ( 2.25 )

are used to arrive at the following slow dynamics of the system:

�C6� = �� + �� § �6£¤¥ − C6¨ ( 2.26 )

�C7� = �� + �� § �7£¤¥ + ��� − C7¨ ( 2.27 )

��� = ��7£¤¥ − ¬� + ­C7 − 1� � ( 2.28 )

19

where � = ©®©ª ��g©ª� , ¬ = ©®� ��g©ª�+ ¡  , ­ = ©® ��g©ª�. Since the external load � is unknown and

only the nominal parameters of the PMSM are known, equation ( 2.28 ) is rewritten as:

��� = �̀�7£¤¥ − ¬̀� + ­̂C7 + b ( 2.29 )

where �̀, ¬̀ and ­̂ are the nominal values of �, ¬ and ­, and b is the disturbance, which is defined

as follows:

b = H� − �̀K�7£¤¥ − H¬ − ¬̀K� + H­ − ­̂KC7 − 1� �

Now, the measured rotor position is used together with ( 1.8 ) and ( 2.29 ) to design the following

extended high-gain observer:

��c� = �� + e\� �� − �c� ( 2.30 )

���� = �̀�7£¤¥ − ¬̀�� + ­̂C7 + b̀ + e>�> �� − �c� ( 2.31 )

�b̀� = e��� �� − �c� ( 2.32 )

where �c, ��, and b̀ are the estimates of �, �, and b, respectively, e\, e>, and e� are chosen such

that

h� + e\h> + e>h + e� = 0 ( 2.33 )

is Hurwitz, and � > 0 is a small parameter. If the singular perturbation method were not used to

reduce the model, the extended high-gain observer would have been a 4th

order as shown in

section 2.1.1, which is harder to implement.

20

2.2.3 Feedback Linearization

The method of feedback linearization is used to regulate the speed of the PMSM to a

reference signal �~��. It also provides means to shape the transient response of the speed. The

speed tracking error is define as:

¢¯ = �~�� −� ( 2.34 )

Using ( 2.29 ) and ( 2.34 ) result in the following:

�¢¯� = ��~��� + ¬̀�~�� − �̀�7£¤¥ − ¬̀¢¯ − ­̂C7 − b ( 2.35 )

In this case, it is desired that the transient response of the speed tracking error matches that of the

following target system:

�¢¯⋆� = −�¯¢¯⋆ ( 2.36 )

where ¢¯⋆ is the trajectory of the target system, and �¯ > 0. The control law can be derived

using ( 2.35 ) and the estimate b̀ with �7£¤¥ viewed as the control input. The control law is given

by:

�7£¤¥ = 1�̀ ;��~��� + ¬̀�~�� + H�¯ − ¬̀K¢°̄ − ­̂C7 − b̀< where ¢°̄ is the speed tracking error estimate, defined as:

¢°̄ = �~�� − ��.

To protect the system from the peaking phenomenon of high-gain observers [11], the control law

�7£¤¥ is saturated outside a compact set as shown in section 1.2. The saturation will also serve to

21

protect the motor from overcurrent. In this case, �7 must not exceed the maximum current that

the PMSM can withstand. Therefore, the commanded current �7£¤¥ must be saturated at the

maximum current ��0A, that is,

�7£¤¥ = ��0AhL j 1�̀��0A ;��~��� + ¬̀�~�� + H�¯ − ¬̀K¢°̄ − ­̂C7 − b̀<k ( 2.37 )

2.2.4 Closed Loop analysis

To analyze the stability of the system, the closed loop system is first put in the singularly

perturbed form:

�C6� = ��¢6 ( 2.38 )

�C7� = ��¢7 ( 2.39 )

�¢¯� = §� − L� ¨��~��� + §G − L¬� ¨�~�� − 5L� H�¯ − ¬K + G8 ¢¯

+ L� H�¯ − ¬̀K�±> + L­� C7 − L� ±� + L¢7 + ²� − L�� ³�

( 2.40 )

� �±\� = ±> − e\±\ ( 2.41 )

� �±>� = −§� − L� ¨ ��~��� − §G − L¬� ¨�~�� − L­� C7 + L� ±� − L¢7 − e>±\

− 5�¯ − L� H�¯ − ¬K − G8 ¢¯ − 5L� H�¯ − ¬̀K − �¯8 �±> − ²� − L�� ³ � ( 2.42 )

22

� �±�� = −²� − �̀�̀ + 1³ e�±\

+� \́ j¢7 , ±\, ±>, ±�, C7 , ¢¯, �>�~���> , ��~��� , �~�� , � , �� � k

( 2.43 )

� �¢6� = −¢6 − 1 + �� C6 + + �� �6£¤¥

+� >́ j¢7 , ±>, ±�, C7 , ¢¯, ��6£¤¥� , ��~��� , �~��, � k

( 2.44 )

� �¢7� = −¢7 + + �� ;1� ��~��� + ²¬� + �� ³�~�� − 1� ±� − ²­� + 1 ³ C7+ ²1� H�¯ − ¬K − �� ³ ¢¯ − �� H�¯ − ¬̀K±> + 1�� � < + �� e��̀ ±\

+� �́ j¢6, ¢7 , ±\, ±>, ±�, ¢¯ , �6£¤¥ , �>�~���> , ��~��� , �~��k

( 2.45 )

where ±\ = \�� �� − �c�, ±> = \� H� − ��K, ±� = b − b̀ are the scaled estimation error, \́,>,�H∙K are

continuously differentiable functions and they are explicitly shown in Appendix. A, L = ©®  , and

G = ¡  . We assume that � ≪ � ≪ 1; hence inducing a multi-time scale system which allows the

use of singular perturbation theory to analyze the stability of the system. As a result of the later

assumption, ¢6 and ¢7 are considered superfast variables, ±\, ±>, and ±� are fast variables, and

C6, C7, and ¢¯ are the slow variables of the system. The boundary layer of ( 2.44 ) and ( 2.45 ),

given by:

� �¢6� = −¢6

23

� �¢7� = −¢7

is exponentially stable and ¢6 and ¢7 reach quasi-steady-state much faster than the other

variables in the system. Therefore, by singular perturbation theory [11], the closed loop system

can be reduced by substituting the quasi-steady-state of ¢6 and ¢7 into ( 2.38 )-( 2.43 ). The

quasi-steady-state of ¢6 and ¢7, obtained by setting � = 0 and ¶� = 0 in ( 2.44 ) and ( 2.45 ), is

given by:

¢6 = 1 + �� § �6£¤¥ − C6¨

¢7 = + �� ;1� ��~��� + ²¬� + �� ³�~�� + ²1� H�¯ − ¬K − �� ³ ¢¯ − �� H�¯ − ¬̀K±>− ²­� + 1 ³ C7 + 1� ±� + 1�� � <

After substituting quasi-steady-state of ¢6 and ¢7 into ( 2.38 )-( 2.43 ), the closed loop system is

reduced to the following:

�C6� = �� + �� § �6£¤¥ − C6¨

�C7� = �� + �� ;1� ��~��� + ²¬� + �� ³�~�� + ²1� H�¯ − ¬K − �� ³ ¢¯ − �� H�¯ − ¬̀K±>− ²­� + 1 ³ C7 + 1�� � <

�¢¯� = −�¯¢¯ − ±� + �H�¯ − ¬̀K±>

24

� �±\� = ±> − e\±\

� �±>� = ±� + �¬̀±> − e>±\

� �±�� = −²� − �̀�̀ + 1³ e�±\ + � �́ j¢7 , ±\, ±>, ±�, C7 , ¢¯, �>�~���> , ��~��� , �~�� , � , �� � k

where �́H∙K is a continuously differentiable function and it is explicitly shown in Appendix. A.

The reduced system is a two time scale system. We exploit the fact that ±\,±>, and ±� are faster

than C6, C7, and ¢¯, by using singular perturbation theory. The boundary layer of ±\,±>, and ±�

is given by:

� �±\� = ±> − e\±\

� �±>� = ±� − e>±\

� �±�� = −²� − �̀�̀ + 1³ e�±\

The boundary layer of ±\,±>, and ±� is linear and its stability can be analyzed using Routh-

Hurwitz criterion. The characteristic equation of the boundary layer of ±\,±>, and ±� is,

··hB − ())))* −

e\� 1� 0−e>� 0 1�−e����̀ 0 0-..

../·· = ��h� + �>e\h> + �e>h + e���̀ ( 2.46 )

where B is a 3 × 3 identety matrix. Table 2.1 shows Routh’s array for the characteristic equation

( 2.46 ). To guarantee stability, the terms in the middle column of Table 2.1 must not change

25

signs. Since � > 0 by assumption, then the terms in the middle column of Table 2.1 must be all

positive; that is,

�>e\ > 0

��e\e> − ��e���̀�>e\ > 0

e���̀ > 0

given � > 0, �̀ > 0, and � < {, it can be concluded that the boundary layer of ±\,±>, and ±� is

exponentially stable provided that the following conditions are satisfied,

e\ > 0 ( 2.47 )

e� > 0 ( 2.48 )

e> ≥ e�e\ { > e��e\�̀ ( 2.49 )

h� �� �e>

h> �>e\ e���̀

h\ ��e\e> − ��e���̀�>e\

0

h# e���̀ 0

Table 2.1. Routh’s array for the characteristic equation ( 2.46 ).

26

where { is a positive constant defined as � < { and it is used to impose a bound to how much

the nominal parameter �̀ can vary from the true parameter�. The closed loop system can further

be reduced by using the quasi-steady-state of ±\,±>, and ±�, obtained by setting � = 0, which

yields:

±\ = ±> = ±� = 0

The closed loop system reduces to:

�C6� = �� + �� § �6£¤¥ − C6¨ ( 2.50 )

�C7� = �� + �� ;1� ��~��� + ²¬� + �� ³�~�� + ²1� H�¯ − ¬K − �� ³ ¢¯ − ²­� + 1 ³ C7+ 1�� � <

( 2.51 )

�¢¯� = −�¯¢¯ ( 2.52 )

which is exponentially stable because it is a linear system and equations ( 2.50 ) and ( 2.52 ) are

decoupled. By singular perturbation theory, we can conclude that for sufficiently small ¶� and �,

the closed loop system is exponentially stable. Furthermore, ¢¯ of the full system approaches the

solution of ( 2.52 ), which is the target system ( 2.36 ), as ¶� → 0 & � → 0; that is,

|¢¯⋆HK − ¢¯HK| → 0 as ¶� → 0 & � → 0∀ ≥ 0.

27

CHAPTER 3

Simulation

Computer simulation is a very important and powerful tool in confirming theoretical

results. Furthermore, Computer simulation is often used to determine the feasibility of an

experiment to be conducted in real life. It is considered one of the most important steps to

perform before moving on to experimentation. It can also be used to create and test scenarios that

are either extremely difficult to be carried out as an experiment or very expensive to replicate in

real life. This chapter is concerned with the simulation of the proposed control method and it is

divided into two sections. Section 3.1 describes the simulation setup and section 3.2 shows the

simulation results.

3.1 Simulation Setup

To evaluate the performance of the proposed control method that is based on the reduced

model, the system is simulated using MATLAB Simulink. The two-phase equivalent model of

the PMSM ( 1.1 )-( 1.4 ) is used. The nominal parameters of the used surface mount PMSM are

shown in Table 3.1 and they are obtained experimentally by using the method described in [12].

The proportional and integral gains of the PI current controllers are: �� = 20, and �� = 1200,

respectively; and they are chosen such that the current response is fast and has a minimal

overshoot to protect the motor from overcurrent. The constant �¯ = 25. The parameters of the

28

extended high-gain observer are: e\ = 3, e> = 3, and e� = 1, and they are chosen such that the

conditions ( 2.47 )-( 2.49 ) are satisfied with � < 9. With � = 2.145 ∗ 10½�, the parameter � of

the extended high-gain observer is chosen to be 0.01 so that the assumption � ≪ � ≪ 1 holds.

Parameter Value

Rated Voltage 200 VACL-L

Rated Current 5.1 A

Rated Torque 3.18 N.m

Rated Speed 3000RPM

Inductance ¾ 4.47 mH

Per Phase Winding Resistance ¿ 0.835 Ω

Torque Constant ÀÁ 0.859 @ · h

Number of Pole Pairs ÂÃ 4

Viscosity Coefficient Ä 0.0011 Å·�·�~06

Moment of Inertia Æ 0.0036 ^r · n>

Table 3.1. Nominal parameters of the used PMSM.

3.2 Simulation Results

There are six computer simulations performed. Some of the simulations will be

conducted later as an experiment and some have scenarios that are difficult to replicate as an

experiment. Simulation I is performed using the nominal parameters of the PMSM, simulation II

is carried out with 20% change in the nominal parameters, in simulation III an external load is

applied while the PMSM is running at constant speed, and in simulation IV the error between the

target speed and the motor speed as ¶� → 0 is shown.

3.2.1 Simulation I

In this case, the motor is at standstill when a step command of 100rad/s is applied at

t = 0.1s. The nominal parameters in Table 3.1 are used in the controller and the motor in this

29

case is not externally loaded. Figure 3.1.a shows the commanded speed, the speed of the motor,

and the trajectory of the desired target system. It can be seen that the control law was able to

regulate the speed and shape the transient response of the speed to the desired trajectory.

Figure 3.1.b shows the error between the target speed and the speed of the motor. A maximum

deviation from the target trajectory 0f about 0.7rad/s can be seen which clearly demonstrates

the performance of the control method.

Figure 3.1. (a) Speed of PMSM using nominal parameters, (b) Speed deviation of PMSM from

target speed.

Time (s)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

50

100

Commanded Speed

Motor Speed

Target Speed

Time (s)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.5

0

0.5

1

30

3.2.2 Simulation II

In this case, the nominal parameters of the PMSM are increased by 20% and then used in

the controller and the observer. Then, a step command of 100rad/s is applied at t = 0.1s while

the PMSM is at standstill. In addition, the motor in this case is not externally loaded. Figure 3.2.a

shows the commanded speed, the speed of the motor, and the trajectory of the desired target

system. Even though there is a 20% increase in the nominal parameters, the control law was able

to regulate the speed, and to a great extent, shape the transient response of the speed to the

desired trajectory. Figure 3.2.b shows the error between the target speed and the speed of the

motor. A maximum deviation from the target trajectory of about 5rad/s can be seen which is

expected to be higher than the case where the exact nominal values were used in the controller.

Figure 3.2. (a) Speed of PMSM using a 20% increase in the nominal parameters, (b) Speed

deviation of PMSM from target speed.

Time (s)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

50

100

Commanded Speed

Motor Speed

Target Speed

Time (s)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-5

0

5

31

3.2.3 Simulation III

In this case, the motor is rotating at a constant speed of 100rad/s. Then, a step of

external load of 2N.m is applied at about t = 0.5s and then removed at t = 1.1s. Figure 3.3.a

shows the motor’s speed before and after the external load was applied and removed. At the

moment the external load was applied the speed of the motor dropped to about 85rad/s but

recovered quickly within 0.2s. Furthermore, the speed of the motor increased to about 115rad/s when the external load was removed. Figure 3.3.b shows the applied external load and its

estimate. It can be seen that it took the extended high-gain observer about 100ms to estimate the

external load.

Figure 3.3. (a) Speed of PMSM before and after the external load was applied, (b) Applied

external load and its estimate.

Time (s)

(a)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.380

100

120

Commanded Speed

Motor Speed

Time (s)

(b)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0

1

2

3

32

3.2.4 Simulation IV

In this case, three simulations were performed to demonstrate the performance recovery

property of the proposed control method. This property can be shown by calculating the

difference between the target speed and the motor speed as ¶� → 0 & � → 0. In all three

simulations, the motor is at standstill when a step command of 100rad/s is applied at t = 0.1s. The nominal parameters in Table 3.1 are used in the controller and the motor in this case is not

externally loaded. The first simulation uses ¶� = 0.003 and � = 0.1. The second simulation uses

¶� = 7.5 ∗ 10½� and � = 0.01. The third simulation uses ¶� = 1.875 ∗ 10½� and � = 0.001.

Figure 3.4 shows the error between the target speed and the motor speed for the three cases. It

can be seen that the error decreases as ¶� → 0 & � → 0 which confirms the performance recovery

property of this control method.

Figure 3.4. Error between target speed and motor speed as ¶� → 0 & � → 0.

Time (s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

e*- e (rad/s)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

/ =3e-3 and =0.1

/ =7.5e-4 and =0.01

/ =1.875e-4 and =0.001

33

CHAPTER 4

Experiment

Theoretical and simulation results are very important in determining the feasibility of an

idea to be implemented in real life application. However, most of the time they are not enough to

fully test and validate an idea that is meant to be applied to a real life application such as motor

control where elements like noise, temperature fluctuation, and parameters uncertainty are

present. It is nearly impossible to replicate an experiment in a simulated environment since the

world we live in is to some degree unpredictable. Therefore, an experiment is conducted to

assess the proposed control method in real life. This chapter is concerned with the application of

the proposed method and it is divided into two sections. Section 4.1 walks the reader through the

setup of the experiment and section 4.2 shows the results of the experiment.

4.1 Experiment Setup

Figure 4.1 shows the block diagram of the experiment. The host computer is used to

perform multiple functions such as, providing user interface, plotting measured quantities of

interest in real-time, and building graphical programs on LabVIEW and deploying them on the

Target PC. The target PC uses National Instruments’ real-time operating system (RTOS) to

execute the graphical programs in real-time. The target PC communicates with the inverter and

the incremental encoder through the NI PCIe-7852R card which is a real-time multifunction Data

Acquisition (DAQ) card.

34

Figure 4.1. Block diagram of the experimental setup.

35

Two modules of the NI PCIe-7852R card are utilized in the experiment: 1)-the 16-bit

Analog to Digital Converter (ADC) module, and the Field Programmable Gate Array (FPGA)

module. The ADC is used to measure the phase currents, temperature of the IGPT’s used in the

inverter, and the DC-link voltage. The FPGA is used to interface the incremental encoder and

provide switching signals to the inverter via a Pulse Width Modulation (PWM) controller circuit.

The FPGA module on the NI PCIe-7852R runs on an on-board 40MHz oscillator which will be

referred to as the system’s clock.

The incremental encoder is connected to the shaft of the PMSM which is also connected

to the Induction Motor. The connection between the PMSM and the induction motor is made

with a jaw coupler that is cushioned with a rubber spider. Here, the induction motor is used to

apply load on the PMSM to assess the proposed controller’s ability to cope with external

disturbance. The induction motor is driven with a Texas Instruments’ RDK-ACIM board.

4.1.1 Current Measurement

The phase currents of the PMSM are measured using hall-effect sensors. The output

signal of each hall-effect sensor is passed through a first order RC low-pass filter to reduce

noises present at the measurement pin of the NI PCIe-7852R card. The circuit of the low-pass

filter used in the experiment is shown in Figure 4.2. The cutoff frequency of the low-pass filter is

selected based on the rated speed �~0��6 and number of pole pairs �� of the used PMSM, and

the switching frequency �́. Since the used motor is of the synchronous kind, then at steady-state

operation of the PMSM the fundamental frequency �́ of the input voltage of the PMSM is

related to the speed of the motor by,

36

�́ = ���2Ï

where �́ is the fundamental angular frequency of the input voltage to the PMSM. Also, since the

proposed control method is designed to regulate the speed of the PMSM below or at the rated

speed of the motor, then it is expected to measure currents with maximum fundamental electrical

frequency �́,�0A which occurs at the rated speed. Hence, the cutoff frequency of the low-pass

filter can be chosen to be larger than the expected maximum fundamental electrical frequency

and much less than the switching frequency. The used PMSM has a rated speed �~0��6 =3000 |i and �� = 4 making the expected maximum fundamental electrical frequency to be:

�́,�0A = ���~0��62Ï

�́,�0A = 4 ∗ 3000 |i60h

�́,�0A = 200Ðm

With a switching frequency of 10�Ðm, the cutoff frequency 2́ should be chosen to satisfy the

following inequality,

200Ðm < 2́ ≪ 10�Ðm

The values of the resistor and the capacitor are chosen according to the following

equation [13]:

2́ = 12Ï O ( 4.1 )

37

where is the resistance and O is the capacitance. Here, the capacitance is selected to be 10�´

and the cutoff frequency 2́ = 2 �́,�0A = 400Ðm. The resistance is then calculated using

equation ( 4.1 ),

= 12ÏO 2́

= 12Ï ∗ 10�´ ∗ 400Ðm

= 39.8�Ω

the calculated resistance of 39.8�Ω is not a standard value, so the closest standard value of 39�Ω

is used instead. This causes a very slight change in the cutoff frequency.

Figure 4.2. First order RC low-pass filter.

4.1.2 Incremental Encoder Interface

The incremental encoder is used to measure the position of the rotor which is needed for

both performing Park’s transformation and driving the extended high-gain observer. The rotary

displacement of the incremental encoder is converted into pulse signals with each pulse

signifying a resolution increment. The encoder has three digital outputs, channel A, channel B,

38

and channel Z. The outputs of channels A and B are shown in Figure 4.3. It can be seen that

channel A and channel B are always 90 degrees apart, thus, providing information about the

direction of rotation. Channel Z is used to locate the origin of rotation and it is only asserted once

every 360 degrees. The accuracy of incremental encoders is measured by Pulses Per Revolution

(PPR) per channel. Therefore, the resolution of incremental encoders per channel is governed by

the following equation:

Δ� = 2Ïn

Where Δ� is the resolution of the incremental encoder, n is the number of pulses per revolution

per channel.

Figure 4.3. Incremental encoder output signals.

There are two different digital circuits that interface the incremental encoder, the x2 and

the x4 circuits. The x2 circuit uses either rising or falling edges of channels A and B to detect

position change and provide information about the direction of rotation. The x4 circuit, on the

other hand, uses both the rising and the falling edges of channels A and B to detect position

39

change and provide information about the direction of rotation. Hence, the resolution delivered

by the x4 circuit is twice more accurate than that of the x2 circuit and it is governed by:

Δ� = Ï2n

Therefore, the x4 circuit will be designed and used in the experiment.

The objective of the x4 circuit is to count the rising and falling edges on both channels A

and B and also to determine the direction of rotation. The x4 circuit creates a clock signal

whenever an edge is detected. The clock signal will drive a counter that keeps track of detected

edges. To detect the edges, two D-flip flops are used to continually store the latest sample of

channel A and B. LetEH�K = E and =H�K = = be the current samples of channel A and B,

respectively. Let the output of the D-flip flops be EH� − 1K = L and =H� − 1K = G. Furthermore,

let {Ò� be the clock to the edges’ counter. Table 4.1 shows the truth table for the driving clock

{Ò� of the edges counter. In Table 4.1, {Ò� = 1 whenever E ≠ L, and = ≠ G indicating an edge

detection. Exception is when E ≠ L and at the same time = ≠ G which is an illegal state since

channel A and B are 90 degrees apart. If this should happen, then it is considered noise and

{Ò� = 0. The Boolean expression for {Ò� can be extracted from the truth table and put in sum of

products,

{Ò� = E̅LÕ=ÕG + E̅LÕ=GÕ + E̅L=ÕGÕ + E̅L=G + ELÕ=ÕGÕ + ELÕ=G + EL=ÕG + EL=GÕ

which simplifies to,

{Ò� = E̅aLÕ�=ÕG + =GÕ� + L�=ÕGÕ + =G�d + EaLÕ�=ÕGÕ + =G� + L�=ÕG + =GÕ�d ( 4.2 )

40

the term =ÕG + =GÕ = =⨁G and the term =ÕGÕ + =G = =⨁GÕÕÕÕÕÕÕ. Equation ( 4.2 ) can be rewritten to

obtain the following:

{Ò� = E̅aLÕH=⨁GK + L�=⨁GÕÕÕÕÕÕÕ�d + EaLÕ�=⨁GÕÕÕÕÕÕÕ� + LH=⨁GKd the term LÕH=⨁GK + L�=⨁GÕÕÕÕÕÕÕ� = L⨁=⨁G and the term LÕ�=⨁GÕÕÕÕÕÕÕ� + LH=⨁GK = L⨁=⨁GÕÕÕÕÕÕÕÕÕÕÕÕ. With

this substitution,

{Ò� = E̅[L⨁=⨁G] + EaL⨁=⨁GÕÕÕÕÕÕÕÕÕÕÕÕd Which simplifies to,

{Ò� = E⨁L⨁=⨁G

A a B b clk

0 0 0 0 0

0 0 0 1 1

0 0 1 0 1

0 0 1 1 0

0 1 0 0 1

0 1 0 1 0

0 1 1 0 0

0 1 1 1 1

1 0 0 0 1

1 0 0 1 0

1 0 1 0 0

1 0 1 1 1

1 1 0 0 0

1 1 0 1 1

1 1 1 0 1

1 1 1 1 0

Table 4.1. Truth table for the driving clock clk of the edges counter.

41

Figure 4.4. State diagram for channel A and B and the direction of rotation.

The direction of rotation can be obtained by sequence detection. Figure 4.4 shows the

state diagram for channel A and B and the direction of rotation. Let ×H�K = × and ×H� − 1K =� be the present and past state of the direction of rotation, respectively. The convention here is

that the direction × is set to zero when the rotation is counterclockwise, and set to one when the

rotation is clockwise. Table 4.2 shows the state transition table of the state diagram for channel A

and B and the direction of rotation. The symbol X in Table 4.2 represents a “do not care state”

and it is used whenever an illegal sequence occur. The Boolean expression for the direction × is

found using the following 5-Variables Karnaugh map,

LÕ L E̅=Õ E̅= E= E=Õ E̅=Õ E̅= E= E=Õ GÕ�̅ 0 1 X 0 GÕ�̅ 1 X 0 0 GÕ� 1 1 X 0 GÕ� 1 X 0 1 G� 0 1 1 X G� X 0 1 1 G�̅ 0 0 1 X G�̅ X 0 0 1

The simplified Boolean expression is then,

× = LÕ=GÕ + ELÕ= + EØGÕ� + LÕ=� + E=ÕG + EG� + E̅LGÕ + L=Õ�

42

Past State Present State

a b d A B D

0 0 0 0 0 0

0 0 0 0 1 1

0 0 0 1 0 0

0 0 0 1 1 X

0 0 1 0 0 1

0 0 1 0 1 1

0 0 1 1 0 0

0 0 1 1 1 X

0 1 0 0 0 0

0 1 0 0 1 0

0 1 0 1 0 X

0 1 0 1 1 1

0 1 1 0 0 0

0 1 1 0 1 1

0 1 1 1 0 X

0 1 1 1 1 1

1 0 0 0 0 1

1 0 0 0 1 X

1 0 0 1 0 0

1 0 0 1 1 0

1 0 1 0 0 1

1 0 1 0 1 X

1 0 1 1 0 1

1 0 1 1 1 0

1 1 0 0 0 X

1 1 0 0 1 0

1 1 0 1 0 1

1 1 0 1 1 0

1 1 1 0 0 X

1 1 1 0 1 0

1 1 1 1 0 1

1 1 1 1 1 1

Table 4.2. State transition table for the state diagram of channel A and B and the direction of

rotation.

The interface circuit of the incremental encoder is implemented using the FPGA module in the

NI PCIe-7852R. Figure 4.5 shows the complete x4 circuit drawn in LabView 2014.

43

Figure 4.5. Incremental encoder interface circuit.

Figure 4.6. High frequency corruption of Channel A and B.

44

4.1.3 Incremental Encoder’s Digital Filter

The x4 circuit alone is not immune to noise. Figure 4.6 shows Channel A and Channel B

of the incremental encoder which are corrupted with high frequency noise. The x4 circuit would

count these high frequency switching resulting in an incorrect position measurement. To add

robustness to the incremental encoder’s interface circuit, a digital filter is designed and

implemented. The digital filter only allows signals of a predetermined time length to be qualified

as a valid signal. The digital filter starts counting at the system’s clock once it detects an edge.

Then, it waits for a period of time set by the user before passing the detected edge. While it is

waiting, it is also continuously checking the signal at the system’s clock. If an edge is detected

before the time period is over, the counter is reset and the signal does not pass through. On the

other hand, the signal is passed to the x4 circuit when the period is over and the signal had not

changed. Figure 4.7 shows the implementation of the digital filter circuit in LabView 2014. The

filter order is where the user sets the counts; hence, setting the waiting time.

4.1.4 PWM Controller Circuit

A PWM controller circuit is a digital circuit that provides switching signals for the three

phase inverter. The input of the PWM controller circuit is the pulse width for each phase pole of

the three phase inverter represented in count values which are calculated by the control algorithm

loop. Then, the PWM controller circuit outputs three pairs of switching signals; one pair for each

phase pole, making a total of six signals. The PWM controller circuit also applies a dead-time

operation where both switches in a phase pole are turned off for a predetermined time period

45

(appropriately chosen to suit the electronic switches used in the three phase inverter) to avoid

shoot through.

Figure 4.7. Implementation of the digital filter circuit in LabView.

The PWM controller circuit uses an up-down counter to realize the switching frequency

�́. The up-down counter starts from zero and increments by one at the system’s clock for one-

half of the switching period. Once the counter reaches one-half of the switching period, it starts

decreasing at the same rate until the counter reaches zero where it starts the same cycle again

creating a triangular wave shown in Figure 4.8. The switching period is governed by:

�� = |OÙO

46

where �� is the switching period, |O is the switching period count which represents the number

of counts the up-down counter must go through to realize the switching period, and ÙO is the

system’s clock. The low-side and the high-side switching signals of each phase pole are created

by comparing the up-down counter’s current value with a modified pulse width count received

from the control algorithm loop. The modification to the pulse width count is done by the PWM

controller to accommodate for the dead-time. Let LPWC and HPWC be the low-side and high-

side pulse width count; respectively. Then,

�|ÚO = |ÚO + 12×�O

Ð|ÚO = |ÚO − 12×�O

where PWC is the pulse width count which represents the number of counts needed to realize the

high-side pulse width and it is calculated in the control algorithm loop, and �O is the dead-time

count which represents the number of counts needed to realize the dead-time. For example, a

switching frequency of 10�Ðm at the system’s clock is realized using the following:

�� = 1́� = 110�Ðm = 100­h

|O = ÙO ∗ �� = 40iÐm ∗ 100­h = 4000{Û��h

which means that the up-down counter must increment by one from zero to 2000 and then

decrement by one until it reaches zero at 40iÐm to realize the switch frequency of 10�Ðm. As a

continuation of the example, Let |Ú=60­h be the required pulse width for the high-side switch

and let � = 100�h be the required dead-time, then

47

|ÚO = |Ú�� ∗ |O2 = 60­h100­h ∗ 4000{Û��h2 = 1200{Û��h

×�O = ×��� ∗ |O2 = 100�h100­h ∗ 4000{Û��h2 = 4{Û��h

Ð|ÚO = |ÚO − 12×�O = 1200 + 12 ∗ 4 = 1202{Û��h

�|ÚO = |ÚO + 12×�O = 1200 − 12 ∗ 4 = 1198{Û��h

which means that at one point |O = Ð|ÚO and at 4{Û��h later |O = �|ÚO while the up-

down counter is incrementing. In addition, |O = �|ÚO and at 4{Û��h later |O = Ð|ÚO

while the up-down counter is incrementing. These specific events will be used to create the low-

side and the high-side switching signals.

Figure 4.8. Signals generation by the PWM controller circuit.

48

Figure 4.8 shows the up-down counter and the switching signals for a phase pole verses

time. Let the events H|O = Ð|ÚOK = Ü and H|O = �|ÚOK = Ú. Furthermore, Let ÐH�K = Ð

and ÐH� − 1K = ℎ be the present and past state of the high-side switching signal, respectively.

Additionally, Let �H�K = � and �H� − 1K = Ò be the present and past state of the low-side

switching signal, respectively. Also, Let Y be false when the up-down counter is incrementing

and true when the up-down counter is decrementing. Table 4.3 shows the state transition table

for the high-side switching signal and Table 4.4 shows the state transition table for the low-side

switching signal. The Boolean expression for the high-side and low-side switching signals Ð and

�, respectively, are found using the following Karnaugh maps,

High-side Low-side

ÞÕℎÕ ÞÕℎ Þℎ ÞℎÕ ÞÕÒ ̅ ÞÕÒ ÞÒ ÞÒ ̅ÜØ 0 1 1 0 ÚØ 0 1 1 0 Ü 0 0 1 1 Ú 1 1 0 0

The simplified Boolean expressions are,

Ð = ÜØℎ + Þℎ + ÜÞ

� = ÚØ Ò + ÞÕÒ +ÚÞÕ

Input Past State Present State

U Y h H

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 1

Table 4.3. State transition table for the high-side switching signal.

49

Input Past State Present State

W Y l L

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 0

Table 4.4. State transition table for the Low-side switching signal.

Figure 4.9 shows the implementation of the PWM controller circuit for one phase pole in

LabView 2014. To protect the three phase inverter, the output of the PWM controller circuit is

ANDed with an active-low fault signal to force the output of the PWM controller to shut-down.

Fault signals are produced in the three phase inverter to signal a problem such as an over current

detection.

4.1.5 Control Algorithm Loop

The control algorithm loop collects the measured quantities and then calculates the

appropriate control signal to drive the motor. The control algorithm loop is built in LabView to

be deployed in the real-time operating system installed on the target PC. The control algorithm

loop is divided into blocks, each of which is built as a SubVI to simplify debugging and to make

it easier to read and follow.

50

4.1.5.1 Three Phase to α-β Transformation

The currents �6 and �7 cannot directly be measured and only the three phase currents are

accessible for measurement. Therefore, the measured three phase currents �0. �1, and �2 are

mapped first to the alpha-beta coordinates using the following transformation [10]:

3������# 4 = %23()))))* 1 − 12 −120 √32 −√321√2 1√2 1√2 -..

.../"�0�1�2$

Figure 4.9. Implementation of the PWM controller circuit for one phase pole.

51

where �� and ��� are the two-phase equivalent currents and �# is the zero-sequence current

which is identically zero for a balanced three phase system. Figure 4.10 shows the

implementation of the transformation in LabView.

Figure 4.10. Implementation of the three phase to α-β transformation in LabView.

4.1.5.2 α-β to d-q Transformation

�� and ��� are used together with the measured position of the rotor � to obtain the

currents �6 and �7. The relationship is governed by [10]:

;�6�7< = 9 cos����� sin�����− sin����� cos�����: ;�����< Figure 4.11 shows the implementation of the transformation in LabView.

52

4.1.5.3 Extended High-Gain Observer

The extended high-gain observer ( 2.30 )-( 2.32 ) is discretized using Euler’s method and

it is given by,

SubVI

SubVI Block

Figure 4.11. Implementation of α-β to d-q transformation in LabView.

�cH��RK = �� 5��H��R½\K + e\� §�H��R½\K − �cH��R½\K¨8 + �cH��R½\K ��H��RK = �� 5�̀�7£¤¥H��R½\K − ¬̀��H��R½\K + ­̂C7H��R½\K + b̀H��R½\K + e>�> §�H��R½\K − �cH��R½\K¨8+ ��H��R½\K b̀H��RK = �� 5e��� ��H��R½\K − �c�8 + b̀H��R½\K where � = 1,2,3, …., ��R = ���, and ��R½\ = ��H� − 1K. Figure 4.12 shows the implementation of

the discrete extended high-gain observer in LabView.

53

Figure 4.12. implementation of the discrete extended high-gain observer in LabView.

54

4.1.5.4 Feedback Linearization

Figure 4.13 shows the implementation of the saturated speed controller ( 2.37 ) in

LabView.

Figure 4.13. Implementation of the speed controller ( 2.37 ) with saturation in LabView.

55

4.1.5.5 PI Controller

The PI current controllers are built with anti-windup. The integration part of the PI

controller is calculated using the trapezoidal method. Let ¢HK be the tracking error and NHK be

the output of the PI controller, then the PI controller’s mathematical model in continuous time is,

NHK = ��¢HK + �� ¦¢H�K����à

Figure 4.14. Block diagram of the PI controller.

where �� and �� are the proportional and integral gains of the PI controller, respectively.

Figure 4.14 shows the block diagram of the PI controller. Applying the trapezoidal method to the

continuous model of the PI controller yields the following discrete model of the PI controller,

NH��RK = ��¢H��RK + 12 �����¢H��R½\K + ¢H��RK� To apply anti-windup operation, the output of the integral part of the PI controller is passed

through a logical process. If the integral output is larger than a predetermined saturation limit

á > 0, the integrator is reset with an initial condition of á. On the other hand, if the integral

output is less than −á, the integrator is reset with an initial condition of −á. This way the

integration process is stopped once the output of the integral reaches the saturation limit

56

preventing the integrator from winding-up. Figure 4.15 shows the implementation of the discrete

PI controller with anti-windup in LabView.

Figure 4.15. Implementation of the discrete PI controller with anti-winding in LabView.

4.1.5.6 d-q to α-β Transformation

The voltages �6 and �7 are used together with the measured position of the rotor � to

obtain �� and ���. The relationship is governed by [10]:

5�����8 = 9cos����� − sin�����sin����� cos����� : 5�6�78

57

Figure 4.16 shows the implementation of the transformation in LabView.

Figure 4.16. Implementation of d-q to α-β transformation in LabView.

4.1.5.7 α-β to Three Phase Transformation

The three phase output voltages of the control algorithm loop �0. �1, and �2 are

calculated using the two-phase components ��, ���, and the zero-sequence voltage �# via the

following transformation [10]:

"�0�1�2$ = %32()))))* 23 0 √23−13 1√3 √23−13 − 1√3 √23 -.

..

../"������# $

58

�# is identically zero for a balanced three phase system. Figure 4.17 shows the implementation

of the transformation in LabView.

Figure 4.17. Implementation of α-β to three phase transformation in LabView.

4.2 Experimental Results

To further validate the performance of the proposed control method, three experiments

were conducted. Experiment I is performed using the nominal parameters of the PMSM,

experiment II is carried out with 20% change in the nominal parameters, and in experiment III

an external load is applied while the PMSM is running at constant speed.

The nominal parameters of the used surface mount PMSM are shown in Table 3.1. The

proportional and integral gains of the PI current controllers are: �� = 20, and �� = 1200,

59

respectively; and they are chosen such that the current response is fast and has a minimal

overshoot to protect the motor from overcurrent. The constant �¯ = 25. The parameters of the

extended high-gain observer are: e\ = 3, e> = 3, and e� = 1, and they are chosen such that the

conditions ( 2.47 )-( 2.49 ) are satisfied with � < 9. With � = 2.145 ∗ 10½�, the parameter � of

the extended high-gain observer is chosen to be 0.01 so that the assumption � ≪ � ≪ 1 holds.

All experiments are performed with a 10kHz sampling frequency and a 2500PPR incremental

encoder.

Figure 4.18. (a) Simulation and experimental speed of PMSM using nominal parameters, (b)

simulation and experimental speed deviation from target speed.

60

4.2.1 Experiment I

In this case, the motor is at standstill when a step command of 100rad/s is applied at

= 0.1h. The nominal parameters in Table 3.1 are used in the controller and the motor in this

case is not externally loaded. Figure 4.18.a shows the commanded speed, the speed of the motor

from simulation and the estimated speed obtained from the experiment. Figure 4.18.a also shows

the trajectory of the desired target system. It can be seen that the control law was able to regulate

the speed and shape the transient response of the speed to the desired trajectory. Figure 4.18.b

shows the error between the target speed and the speed of the simulated motor, and the error

between the target system and the estimated speed of the motor obtained from the experiment.

Furthermore, Figure 4.18 shows that there is a small difference between the simulation and the

experimental speed trajectories and that is expected because the simulation was performed in

continuous-time and without taking into account the inverter, the incremental encoder, and noise.

4.2.2 Experiment II

In this case, the nominal parameters of the PMSM are increased by 20% and then used in

the controller. Then, a step command of 100rad/s is applied at = 0.1h while the PMSM is at

standstill. Figure 4.19.a shows the commanded speed, the speed of the motor from simulation

and the estimated speed obtained from the experiment. Figure 4.19.a also shows the trajectory of

the desired target system. Even though there is a 20% increase in the nominal parameters, the

control law was capable of regulating the speed, and to a great extent, shaping the transient

response of the speed to the desired trajectory. Figure 4.19.b shows the error between the target

speed and the speed of the simulated motor, and the error between the target system and the

61

estimated speed of the motor obtained from the experiment. Furthermore, Figure 4.19 shows that

there is a small difference between the simulation and the experimental speed trajectories.

Figure 4.19. (a) Simulation and experimental speed of PMSM when the nominal parameters are

increased by 20%, (b) simulation and experimental speed deviation from target

speed when the nominal parameters are increased by 20%.

4.2.3 Experiment III

In this case, the motor is rotating at a constant speed of 100rad/s. Then, a step of

external load of about 2N.m is applied at about t = 0.1s and removed at about t = 0.7s. Figure 4.20 shows the estimated speed of the motor before and after the external load was

applied. At the moment the external load was applied the speed of the motor dropped to about

Time (s)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Speed (rad/s)

0

50

100

Commanded Speed

Simulation Speed

Target Speed

Experimental Speed

Time (s)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

e*- e (rad/s)

-10

0

10

Target Speed-Simulation Speed

Target Speed-Experimental Speed

62

90rad/s but recovered quickly. The speed of the motor increased to about 109rad/s when the

external load was removed.

Figure 4.20. Speed of PMSM before and after the external load was applied.

Time (s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Speed (rad/s)

80

85

90

95

100

105

110

115

120

63

CHAPTER 5

Conclusion and Future Work

5.1 Conclusion

An effective control method that not only regulates the speed of a PMSM but also shapes

the transient response of the speed was presented. The measured rotor position is used to drive an

extended high-gain observer that estimates both the speed and the disturbance. The speed is

shaped and regulated using feedback linearization method. Two methods were presented to

design the extended high-gain observer. The first method is based on the full model of the

PMSM and produced a fourth order extended high-gain observer. The second method is based on

a reduced model of the PMSM and produced a third order extended high-gain observer. The

reduction of the model was made possible by creating fast current loops that allowed us to utilize

singular perturbation theory to replace the current variables by their quasi-steady-state

equivalent. The second method is preferred over the first method because of advantages such as

lower observer gain and lower noise amplification. The design of the speed controller and the

extended high-gain observer is based on the nominal parameters of the PMSM. The closed loop

system was shown to have exponential stability and also the trajectory of the speed approaches

that of the target system.

In addition, simulation and experimental results under different operating conditions were

performed and shown which confirmed performance and robustness of the control method. The

64

results were satisfactory in terms of accomplishing the objective which is to regulate and shape

the speed of the PMSM.

5.2 Future Work

5.2.1 Field Weakening

The design of the proposed speed controller is intended to regulate the speed of a PMSM

below or at the rated speed. However, there are many applications that require speeds higher than

the rated speed; such applications would be machine-tool spindles, and electric vehicles.

Therefore, an improvement over the proposed control method would be extending the design of

the speed controller to include speeds beyond the rated speed. Realization of such speeds

requires the use of Field Weakening.

For a given drive system, the rated speed is set by the rated torque of the machine and the

available DC-link voltage which determines the machine’s maximum input voltage. The induced

back Electro Motive Force (EMF) above the rated speed is larger than the machine’s maximum

input voltage which restricts the current flow and thus torque production and speed gain. This

problem can be solved by weakening the air gab flux linkage which reduces the induced back-

EMF [3]. For surface mounted PMSMs, the air gab flux linkage is weakened by the direct

current �6. Therefore, optimization techniques are used to create reference current signals for �6

and �7 to keep the air gab flux linkage constant above the rated speed.

65

5.2.2 Sensorless Control

The proposed control method requires both current and position measurements. The

current measurement is usually performed by measuring the voltage across a resistor that is

connected in series with each phase or using hall-effect sensors. Both solutions are readily

available and they are relatively cheap. On the other hand, position sensors such as optical

encoders and resolvers are expensive and contain moving parts making them undesirable. It is

actually possible to control motors without position sensors, hence the name sensorless control.

There are two popular sensorless control techniques, sensorless control based on the induced

back-EMF [14] and sensorless control based on high frequency signal injection [15]. The back-

EMF based control technique uses voltage and current measurements to estimate the back-EMF

and then determines the position of the rotor. A drawback of such technique is that it cannot be

used in low speeds because the measured signals are dominated by noise. The high frequency

signal injection based sensorless control technique uses high frequency phase voltages to inject

current into the machine. The current is then measured and the position is estimated based on a

high-frequency model of the PMSM. This control method estimates the position in a wide range

of speeds including stand still which is an advantage over the back-EMF based control method.

However, it suffers from losses, increased acoustic noise, and vibration. Improvements over

these control methods have taken place such as using different state observers, and combining

the two techniques together to estimate the rotor position. The topic of sensorless control is still

under development. It is a promising control technique that is worth pursuing as a research area

in the future.

66

APPENDIX

67

APPENDIX

Definition of Functions çèH∙Kto çéH∙K \́ = −²� − �̀�̀ ³ �>�~���> + ;L� H¬ − ¬̀K − ¬̀�̀ H� − �̀K< ��~��� − §G − L¬� ¨ H¬ − ¬̀K�~��

+ ;��̀̄ H� − �̀KH�¯ − ¬̀K + 5L� H�¯ − ¬K + G8 H¬ − ¬̀K< ¢¯+ H�¯ − ¬̀K ;��̀̄ H� − �̀K + L� H¬ − ¬̀K< �±> + ²� − �̀�̀ ³ H�¯ − ¬̀Ke>±\+ ;²� − �̀�̀ ³ ­̂�� − H¬ − ¬̀KL − H­ − ­̂K��< ¢7 − L­� H¬ − ¬̀KC7 + L� H¬ − ¬̀K±�− ²� − L�� ³ H¬ − ¬̀K� + 1� �� �

>́ = ��6£¤¥� − ����~�� − ¢¯� ;1� ��~��� + ¬� �~�� + 1� H�¯ − ¬K¢¯ + 1� H�¯ − ¬̀K�±> − ­� C7+ 1� ±� + 1�� � < + ����~�� − ¢¯�¢7

�́ = 1�̀ �>�~���> + ¬̀�̀ ��~��� − ��̀̄ H�¯ − ¬̀K¢¯ − ��̀̄ H�¯ − ¬̀K�±> − e>�̀ H�¯ − ¬̀K±\ − ­̂���̀ ¢7+ ����~�� − ¢¯� §�6£¤¥ − ¢6¨

�́ = −²� − �̀�̀ ³ �>�~���> + ;L� H¬ − ¬̀K − ¬̀�̀ H� − �̀K< ��~��� − §G − L¬� ¨ H¬ − ¬̀K�~��+ ;��̀̄ H� − �̀KH�¯ − ¬̀K + 5L� H�¯ − ¬K + G8 H¬ − ¬̀K< ¢¯+ H�¯ − ¬̀K ;��̀̄ H� − �̀K + L� H¬ − ¬̀K< �±> + ²� − �̀�̀ ³ H�¯ − ¬̀Ke>±\+ + �� ;²� − �̀�̀ ³ ­̂�� − H¬ − ¬̀KL − H­ − ­̂K��< ;1� ��~��� + ²¬� + �� ³�~��+ ²1� H�¯ − ¬K − �� ³ ¢¯ − �� H�¯ − ¬̀K±> − ²­� + 1 ³ C7 + 1� ±� + 1�� � <− L­� H¬ − ¬̀KC7 + L� H¬ − ¬̀K±� − ²� − L�� ³ H¬ − ¬̀K� + 1� �� �

68

BIBLIOGRAPHY

69

BIBLIOGRAPHY

[1] Werner Leonhard, “Control of Electrical Drive,” 3rd

Edition. Springer 2001.

[2] Rik De Doncker, Duco W.J. Pulle, and Andre Veltman, “Advanced Electric Drives:

analysis, modeling, and control.” Springer Dordrecht Heidelberg London New York

2011.

[3] R. Krishnan, “Electric Motor Drives: modeling, analysis, and control.” Upper Saddle

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