SPEED CONTROL OF PERMANENT MAGNET SYNCHRONOUS …...ii ABSTRACT SPEED CONTROL OF PERMANENT MAGNET...
Transcript of SPEED CONTROL OF PERMANENT MAGNET SYNCHRONOUS …...ii ABSTRACT SPEED CONTROL OF PERMANENT MAGNET...
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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
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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
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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.
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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 )
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��� = � ( 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� �� �
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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
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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 )
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¢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
River, New Jersey: Prentice Hall, 2001.
[4] X. G. Zhang, K. Zhao, and L. Sun, “A PMSMsliding mode control system based on a
novel reaching law,” in Proc. Int. Conf. Electr. Mach. Syst., 2011, pp. 1–5.
[5] Z. Xiaoguang, S. Lizhi, Z. Ke, and S. Li, “Nonlinear speed control for PMSM system
using sliding-mode control and disturbance compensation techniques,” IEEE Trans.
Power Electron., vol. 28, no. 3, pp. 1358–1365, Mar. 2013.
[6] Li Xiaodi; Li Shihua, "Extended state observer based adaptive control scheme for PMSM
system," in Control Conference (CCC), 2014 33rd Chinese , vol., no., pp.8074-8079, 28-
30 July 2014.
[7] Jun Zhang; Zhaojun Meng; Rui Chen; Changzhi Sun; Yuejun An, "Decoupling control of
PMSM based on exact linearization," in Electronic and Mechanical Engineering and
Information Technology (EMEIT), 2011 International Conference on , vol.3, no.,
pp.1458-1461, 12-14 Aug. 2011.
[8] G. Zhu, L.-A. Dessaint, O. Akhrif and A. Kaddouri, "Speed tracking control of a
permanent-magnet synchronous motor with state and load torque observer," Industrial
Electronics, IEEE, vol. 47, no. 2, pp. 346-355, 2000.
[9] L.B. Freidovich and H.K. Khalil, “Performance recovery of feedback linearization-based
designs”, IEEE Trans. Automat. Control, 53(2008), 2324-2334.
[10] John Chiasson, Modeling and high-Performance Control of Electric Machines. Hoboken,
New Jersey: John Wiley & Sons, Inc, 2005. Ch9.
[11] Hassan K. Khalil, Nonlinear Systems. Upper Saddle River, New Jersey: Prentice Hall,
2002.
[12] Cintron-Rivera, J.G.; Babel, A.S.; Montalvo-Ortiz, E.E.; Foster, S.N.; Strangas, E.G., "A
simplified characterization method including saturation effects for permanent magnet
Machines," in Electrical Machines (ICEM), 2012 XXth International Conference on ,
vol., no., pp.837-843, 2-5 Sept. 2012.
70
[13] James W. Nilsson and Susan A. Riedel, Electric Circuits Ei. Upper Saddle River, New
Jersey: Prentice Hall, 2008.
[14] Wu, R.; Slemon, G.R., "A permanent magnet motor drive without a shaft sensor,"
in Industry Applications, IEEE Transactions on , vol.27, no.5, pp.1005-1011, Sep/Oct
1991.
[15] Ji-Hoon Jang; Seung-Ki Sul; Jung-Ik Ha; Ide, K.; Sawamura, M., "Sensorless drive of
surface-mounted permanent-magnet motor by high-frequency signal injection based on
magnetic saliency," in Industry Applications, IEEE Transactions on , vol.39, no.4,
pp.1031-1039, July-Aug. 2003.