i 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
80
Embed
SPEED CONTROL OF PERMANENT MAGNET …3697/datastream/OBJ/...SPEED CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR ... There are two types of three phase AC PMSMs: the surface mounted
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
i
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
ii
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.
iii
Copyright by
ABDULLAH AHMAD ALFEHAID
2015
iv
To my family
v
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.
vi
TABLE OF CONTENTS
LIST OF TABLES ....................................................................................................................... viii
LIST OF FIGURES ....................................................................................................................... ix
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 ).
6
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,
7
�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
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
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
9
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
10
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,
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
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
�# 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