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THESIS

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LIBRARIES
MICHIGAN STATE UNIVERSITY
EAST LANSING, MICH 48824-1048

This is to certify that the
thesis entitled

SENSORLESS SPEED CONTROL OF INDUCTION MOTORS
USING SLIDING MODE CONTROL STRATEGY

presented by

ATTAULLAH YOUSUF MEMON

has been accepted towards fulﬁllment
of the requirements for the

Masters degree in Electrical Enﬂreering

 

 

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Date

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6/01 CI/CIRC/DateDUOpBS—p. 1 5

Sensorless Speed Control of Induction Motors

Using Sliding Mode Control Strategy
By

Attaullah Yousuf Memon

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements

for the degree of
MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2004

ABSTRACT

Sensorless Speed Control of Induction Motors

Using Sliding Mode Control Strategy

By

Attaullah Yousuf Memon

The field-oriented speed control of induction motors without mechanical sensors
is considered. A sliding mode speed control algorithm is developed for the speed
control that replaces the traditional PI controller. The traditional approach for the
sensorless speed control of using ﬂux and speed observers is augmented with a sixth-
order nonlinear induction machine model that describes the motor in ﬁeld-oriented
coordinates. The model takes into consideration the errors in flux estimation. The
ﬂux regulation problem is addressed by following the traditional approach of using
PI controllers. For the speed regulation problem, the machine model is simplified by
assuming that the flux regulation takes place relatively fast and by employing a slid-
ing mode controller that presents good performance against un—modeled dynamics,
insensitivity to parameter variations, external disturbance rejection and fast dynamic
response. A performance comparison of the developed sliding mode controller with
the traditional PI controller is presented. The simulation presents the edge that the
developed sliding mode controller has over its counterpart, a traditional PI controller.
The analysis reveals the conditions under which the sliding mode controller provides
effective speed control while preserving the closed-loop system stability under uncer-

tain external load disturbances and reference speed variations.

Copyright © by
Attaullah Yousuf Memon

2004

To my father... and his fond memories

iv

ACKNOWLEDGMENTS

I am highly indebted to my advisor Dr. Hassan K. Khalil, whose keen insight
and great vision in the subject provided me with invaluable advice for the work in
this thesis. His encouragement and support, both morally and academically, made
the time span of last several months truly memorable. I can not thank him enough

for all he has done for me.

I would also like to thank Dr. Elias G. Strangas and Dr. Fang. Z. Peng for

their valuable inputs to this work and for serving on the committee.

I would like to thank National University of Sciences and Technology (NUST),

Pakistan, for the ﬁnancial support throughout the tenure of my master’s studies.

I am grateful to many friends for their help, throughout the ups and downs
during the past many months, for providing me with hundreds of memorable

moments to cherish.

And, I have no words to thank the very special person in my life, my mother,
for her love and unconditional support, for everything she has done for me. I owe
my gratitude to my best friends, my brother and my sister, for their love and

encouragement, and to my wife, for all the things she has done for me.

TABLE OF CONTENTS

LIST OF FIGURES viii
1 Introduction 1
1.1 Why Sensorless Electric Drive? ......................... 2
1.2 Speed Estimation in Induction Motor Drives ................ 3
1.3 Scope of thesis work ............................... 4

2 Literature Review 6
2.1 The Induction Motor ............................... 6
2.2 Principle of Operation .............................. 6
2.3 Reference-Frame Transformation ........................ 8
2.4 Frequency-Controlled Induction Motor Drives ................ 11
2.4.1 Static Frequency Changers ....................... 12

2.4.2 Speed Control ............................... 13

2.5 Vector Controlled Induction Motor Drives .......... ' ........ 17
2.5.1 Principle of Field Oriented Control .................. 18

3 Previous Work 20
3.1 Introduction .................................... 20

3.2 Induction Motor Model ............................. 21
3.3 Flux Observer ................................... 21
3.4 Flux Regulation .................................. 23
3.5 Speed Observer .................................. 23
3.6 Speed Controller .................................. 26

4 Speed Control Using Sliding Mode Control Strategy 32
4.1 Introduction .................................... 32
4.1.1 Sliding Mode Control .......................... 33

4.1.2 Zero Dynamics and the Relative Degree of Nonlinear Systems . 35
4.1.3 Regulation via Integral Control in Relative Degree-1 Systems . 38
4.2 Sliding Mode Controller Design For the Induction Motor ......... 39

vi

4.2.1 Zero Dynamics of the system ..................... 40

4.2.2 Speed Regulation via Integral Control ................ 45

5 Simulations 59
5.1 Performance Comparison — PI controller vs Sliding Mode Controller . . 59
5.1.1 Designing PI Controllers for the Speed Control .......... 60

5.1.2 Sliding Mode Controller - Parameters ................ 63

5.1.3 Performance Comparison ........................ 64

6 Conclusions 77
BIBLIOGRAPHY 80

vii

LIST OF FIGURES

2.1 Transformation equations as trigonometric relationships ......... 9
3.1 Sensorless control of an induction motor using traditional PI controllers 30

4.1 A typical phase portrait under sliding mode control ............ 34
4.2 Trajectory outside the boundary layer reach it in ﬁnite time and stay
inside thereafter .................................. 50

4.3 Sensorless control of an induction motor using sliding mode controller . 57

5.1 Simulation results with nominal parameters ................. 65
5.2 Simulation results with 100 percent increase in R,» and nominal Rs . . . 67
5.3 Simulation results with 100 percent increase in R, and Rs ......... 68
5.4 Simulation results for speed reversal under no load conditions with
nominal parameters ................................ 69
5.5 Simulation results for step load torque TL with nominal parameters . . 70

5.6 Simulation results for sudden step load torque increase with nominal
parameters ..................................... 71
5.7 Simulation results demonstrating loss of stability in case of PI con-
troller when under negative load torque with nominal parameters . . . 72
5.8 Simulation results for step speed command with nominal parameters . 73
5.9 Simulation results for multi-step speed reduction with nominal param-
eters ......................................... 74
5.10 Simulation results for monotonic speed reduction command with nom-
inal parameters .................................. 75

viii

CHAPTER 1

Introduction

The induction motor is the motor of choice in several industrial applications due to
its reliability, power-to—size ratio, ruggedness and relatively low cost. In the last few
decades, the induction motor has evolved from being a constant speed motor to a
variable speed, variable torque machine. DC motors had the advantage of precise
speed control at the cost of many disadvantages, including but not limited to the
maintenance requirements, complex structures and power limits. The induction
motors are robust, smaller in size, almost maintenance free and possess a wide range
of speeds when compared with the DC motors. Their mechanical dependability
is due to the reason that there is no requirement of mechanical commutation
(i.e. there are no brushes nor commutators to wear out as in the DC motors).
Another advantage is that Induction Motors can be used in many volatile environ-

ments since no sparks are produced as is the casein the commutator of the DC motors.

However, the induction motor, by itself, presents a very challenging control

problem. This is owing to the complex nature of issues that the induction motor

presents;
o The dynamic model of the induction motor is nonlinear

0 Certain state variables e.g. rotor ﬂuxes are not measurable

0 Due to the ohmic temperature rise, the rotor resistance varies considerably with

a corresponding signiﬁcant effect on the system dynamics

1.1 Why Sensorless Electric Drive?

Many variable-speed electrical drives used in general-purpose applications ranging
from simple servo systems to complex traction systems require a capability of
speed variation with a pro-deﬁned performance standard. In such applications,
it is necessary that the actual drive-speed measurements be available at every
instant in order to control the drive effectively. For this reason, many different
kinds of speed sensors have been used, including tacho generators, optical en-
coders, resolvers, etc. Elimination of such a requirement of having speed sensor on
the motor shaft represents a cost advantage, and also enhances the reliability of

the drive owing to the absence of a mechanical sensor and associated cable accessories.

The idea of developing efﬁcient sensorless electric drives has gained consider-
able. interest due to their low cost and dependability, since there is no further
requirement of having a mechanical sensor to measure speed. Instead, the intrinsic
motor electro-mechanieal properties can be utilized to estimate the rotor position
and/or speed. There are numerous methods that have been proposed by various
researchers in this ﬁeld over the last two decades. Of considerable signiﬁcance are
the collection of papers by Rajashekara et a1 [6], the tutorials by Lorenz [11] and
Holtz [4], and the monographs by Leonhard [10] and Vas [14]. The identiﬁcation
of the rotor speed is generally based on measured terminal voltages and currents.
Various dynamic models are used in order to estimate the magnitude and the spatial
orientation of the magnetic flux vector and for this purpose open loop estimators

or closed-loop observers are used, which usually differ with respect to accuracy,

robustness and sensitivity against model parameter variations.

1.2 Speed Estimation in Induction Motor Drives

There are two basic approaches for speed and position estimation in induction
motors. The ﬁrst approach uses the fundamental machine model to design model
reference adaptive systems, nonlinear observers, extended Kalman ﬁlters, or adaptive
observers. It has long been recognized that the challenging part in this approach is
keeping a load stationary at (or near) zero ﬂux frequency. The second approach uses
secondary phenomena or the parasitic effects of the machine to develop methods

that will be effective at low frequency.

In their recent work [8], Khalil and Strangas have identiﬁed some drawbacks
associated with the analysis using the ﬁrst approach that has been done so far and
subsequently published by many researchers in this ﬁeld. These are summarized

here:

0 Analysis is limited to local linear models; it is rare to ﬁnd analysis that takes

into consideration the nonlinearities of the system

0 Model uncertainty is usually ignored in the analysis, even though the presence
of such uncertainty (e.g. change of resistances with temperature) could change

the conclusions in a fundamental way

0 No analysis of the overall closed loop system. It is typical in methods based
on rotor or ﬂux position estimation that the analysis is limited to the position
estimation problem itself with no analysis of the impact of the estimation error

on the performance of the closed-100p system

The traditional field-oriented control [10, 9] is studied here, where a flux observer
is used to estimate the rotor ﬂux. The speed control problem is considered, where
the motor speed is required to track a given speed command in the presence of an

unknown load. The two key elements of the approach of [8] are:

e To keep track of the error in estimation of the rotor flux, ﬁeld orientation is
performed using the estimated flux angle and two additional state variables are

added by projecting the flux estimation error into the ﬁeld oriented coordinates

o A high-gain observer is used to estimate the Speed from current measurements

With the use of these two key elements, a nonlinear model of the induction motor
in the field-oriented coordinates was derived that formulates the ﬂux and speed
regulation problems. The ﬂux regulation problem was addressed by using the
traditional approach of using PI controllers. While addressing the speed regulation
problem, the nonlinear induction motor model was Simpliﬁed by assuming that the
ﬂux regulation takes place relatively fast and by using a high-gain PI controller to
regulate the q-axis current to its command. This results in a third-order non-linear
model in which the speed and two ﬂux estimation errors are the state variables, the
q-axis current is the control input and a speed estimate provided by the high-gain
observer is the measured output. The analysis was limited to the design of PI

controllers via linearization.

1.3 Scope of thesis work

This work is an extension of the analysis of the closed-loop system using the third-

order non-linear model [8] The goal here is to design a nonlinear feedback controller

4

for the stator voltage 123 that uses only the measurements of the stator current is,
such that the rotor speed a) asymptotically tracks a bounded time-varying reference
speed wref' Here, a nonlinear sliding-mode speed control algorithm is developed and
implemented, and the analysis of the overall closed loop system is undertaken. The
analysis provides a better insight into the speed control problem when the nonlinear-
ities of the machine model are taken into account and presents certain bounds and
conditions in which the developed speed control algorithm will work with a superior
performance. Various machine operating situations are taken into account for the
purpose of simulations using the developed speed control algorithm and the results
obtained are compared with those obtained using the traditional PI control algorithm.
The comparison simulations clearly indicate the edge that the developed speed control

algorithm has over the traditional PI control.

CHAPTER 2

Literature Review

2.1 The Induction Meter

The induction motor was invented by Nikola Tesla (1856 - 1943) in 1888. It requires
no electrical connections to the rotating member; the transfer of energy from the sta-
tionary member to the rotating member is by means of electro magnetic induction. A
rotating magnetic ﬁeld, produced by a stationary winding (called the stator), induces
an alternating emf and current in the rotor. The resultant interactionvof the induced

rotor current with the rotating ﬁeld of the stationary winding produces motor torque.

The Terque — Speed characteristic of an induction motor is directly related to
the resistance and reactance of the rotor. Hence, different Torque - Speed charac-
teristics may be obtained by designing rotor circuits with different ratios of rotor

resistance to rotor reactance.

2.2 Principle of Operation

When a set of three-phase currents displaced in time from each other by angular

intervals of 120 degrees is injected into a stator having a set of three-phase windings

displaced in space by 120 degrees electrical, a rotating magnetic ﬁeld is produced [9].
This rotating magnetic ﬁeld has a uniform strength and travels at an angular speed
equal to its stator frequency. The rotating magnetic ﬁeld in the stator induces elec-
tromagnetic forces in the rotor windings. As the rotor windings are short-circuited,
currents start circulating in them, producing a reaction. As known from Lenz’s law,
the reaction is to counter the source of the rotor currents, i.e. the induced emfs in
the rotor and, in turn, the rotating magnetic ﬁeld itself. The induced emfs will be
countered if the difference between the speed of the rotating magnetic ﬁeld and the
rotor becomes zero. The only way to achieve it is for the rotor to run in the same
direction as that of the stator magnetic ﬁeld and catch up with it eventually. When
the differential speed between the rotor and magnetic ﬁeld in the stator becomes zero,
there’s zero emf, and hence zero rotor currents resulting in zero torque production
in the motor. Depending on the shaft load, the rotor will settle down to a speed
w,» always less than the speed of the rotating magnetic ﬁeld, called the Synchronous
Speed of the machine (us. The speed differential is known as the Slip Speed “’31-

The Synchronous Speed of the machine ws is given as

(.03 = 27rf3(rad/sec) (2.1)

where f3 is the supply frequency.
The slip speed is given as

“’31 = ws — wr(rad/sec) (2.2)

The differential speed between the stator magnetic ﬁeld and rotor windings is the

slip speed, and this is responsible for the frequency of the induced emfs in the rotor

and hence the rotor currents.

The direction of rotation of an induction motor is dependant on the direction
of rotation of the stator ﬂux, which in turn is dependant on the phase sequence of

the applied voltage.

2.3 Reference-Frame Transformation

The performance of an induction machine is generally described by a set of differen-
tial equations. Some of the machine inductances appearing in these equations are
functions of the rotor speed and the coefﬁcients of the differential equations that
describe the behavior of the induction machine are generally time-varying. A change
of variables is often used in order to reduce the complexity of these differential
equations. A transformation refers the machine variables to a frame of reference that
rotates at an arbitrary angular velocity and a particular transformation can readily
be obtained from this transformation by simply assigning the speed of rotation of

the reference frame of our choice.

A change of variables that formulates a transformation of the 3 — phase variables to

the arbitrary reference frame may be expressed as [12]

fqdos = stabcs (2-3)
where

(fqdos)T = lqufdsfos] (2.4)
(fabcs)T = lfasfbsfes] (2.5)

(030 cos(0 — 27f) cos(t9 + 27r)

’ ' T '3—

2
Ks = 5 sin6 sin(0 — 2375) 3271(0 + 2w) (2.6)
i i i

_d6

In the foregoing equations, f can represent any variable like voltage, flux-linkage or
current. The frame of reference may rotate at any constant er varying angular velocity
or it may remain stationary. It is convenient to visualize the transformation equations

as the trigonometric relationships between variables as shown in Figure 2.1.

 

 

 

 

Figure 2.1. Transformation equations as'trigonometric relationships

The equations of transformation may be thought of as if qu and fds variables are
directed along the paths orthogonal to each other and rotating at an angular velocity
w,whereupon [03,be and fcs may be considered as variables directed along stationary

paths, each displaced by 120 degrees. If fag, fbs and fcs are resolved into 1213, the ﬁrst

row of equation (3.2) is obtained, and if fas, fbs and fcs are resolved into fds, the
second row is obtained. The fos variables are not associated with the reference frame,
instead, they are only related arithmetically to the abc variables, independent of 6.
Portraying the transformation as shown in Fig. 2.1 is particularly convenient when
applying it to ac machines where the direction of fas, fbs and fcs may be thought
of as the direction of the magnetic axes of the stator windings. Then, the direction
of qu and fds can be considered as the direction of the magnetic axes of the new
windings created by the change of variables. The reference frame transformation
therefore simpliﬁes the intricate equations involving 120 degree 3 — phase variables into

the 2 — phase orthogonal variables.

Commonly used reference frames

The reference frames commonly used in the analysis of electric machines and power
systems can be described as, the arbitrary, stationary, rotor and synchronous reference
frames. The arbitrary reference frame can be deﬁned as the one that can be assigned
any given angular velocity corresponding to the fundamental frequency associated
with a quantiﬁable variable like ﬂux-linkage for example. The synchronous reference
frame is the reference frame rotating at the electrical angular velocity corresponding
to the fundamental frequency of the variables associated with the stationary circuits.
In the case of ac machines, it is the electrical angular velocity of the air-gap rotating

magnetic ﬁeld established by stator currents of fundamental frequency.

Transformation between reference frames

Many times during the derivations and analyses it is convenient to relate variables in
one reference frame to another reference frame directly, without involving abc variables
in the transformation. This direct transformation takes the form of a Vector Rotator

given as

10

C05(6y ‘— 61) ‘Sin(6y "‘ 61')
Sin(6y — 01:) C03(6y "' 01-)
where 0,, and 0:1: represent the angular displacements associated with the reference

frames involved with the inter-frame transformation.

2.4 Frequency-Controlled Induction Motor Drives

The speed of an induction motor is very near to its synchronous speed. The difference
between the two being characterized by the slip speed. If the synchronous speed of
the induction motor is changed, there is a corresponding change in the speed of the
motor and this can be done by changing the supply frequency of the ac. source. The
relationship between the synchronous speed and the frequency is given by

_ 1201',

P (2.8)

Us

where as is the synchronous speed in rev/ min, f5 is the supply frequency in Hz and
P is the number of poles.

The ac. supply available for the utility purposes is of a constant frequency and
when an induction motor is operated with the utility supply, it runs at a constant
speed. For the purpose of Speed control, a frequency changer is required to change
the speed of the induction motor. The electric motor drives which use frequency

changers to achieve the speed control are referred to as Frequency- Controlled Electric

Drives.

11

2.4.1 Static Frequency Changers

The static frequency changers can be broadly classiﬁed as Direct and Indirect static
frequency changers. The direct frequency changers are some times called as Cyclo-
converters. These convert the ac. supply source frequency to a variable frequency.
The output frequency typically ranges from 0 to 0.5 f3, and for the better waveform
control of the output voltage, the frequency is limited to 0.33 f3. The smaller range

of frequency variation is suitable for low-Speed and large-power applications.

For a majority of applications, a wide frequency range is desirable due to the
requirements over the desired speed range. In such applications, the Indirect
frequency conversion methods are employed. An indirect frequency changer consists
of two power conversion stages; ﬁrst stage is Rectiﬁcation (ac to dc) and the second
stage is Inversion (dc to ac). The indirect frequency changers are broadly classiﬁed
depending on the source that supplies the input power to them and that can either
be a voltage source or a current source. In both cases, the power input is kept to a
speciﬁed constant. The output frequency becomes independent of the input supply
frequency by means of the dc link. Various conﬁgurations of the indirect frequency
changers have evolved keeping in view the diversity of applications. However,
these only differ in the way the two power conversion stages are incorporated.
The more common conﬁgurations are the so called PWM inverter fed induction
motor drive, Variable- Voltage- Variable-frequency (VVVF) induction motor drive
and Variable-Current- Variable-frequency (VCVF) induction motor drive. A detailed
description of the current and voltage source static frequency changers and the

current and voltage source inverters can be found in [9].

12

2.4.2 Speed Control

For the inverter—driven induction motor, the speed control is effectively achieved by
means of variable frequency. However, apart from the frequency, the applied voltage
also needs be varied so that the air gap ﬂux can be maintained at a constant value
without letting it to saturate. It is well known that in order to maintain the air
gap ﬂux constant, the ratio between the phase voltage and the supply frequency is
to be maintained to a constant value [9, 10]. Therefore, whenever stator frequency
is changed to obtain speed control, the stator input voltage has to be changed

accordingly to maintain the air gap ﬂux at a constant value.

The requirement of keeping the ratio between the stator voltage and the sta-
tor frequency constant, actually compounds the speed control problem in an
induction machine. This, in fact, is the difference between the speed control problem
of an induction motor and a dc motor, which requires only the voltage control for
the purpose, and the simplicity of the control problem made it preferable machine in

many applications until early 19803.

Various speed control strategies have been formulated for the induction ma-
chine, depending upon how the voltage-to-frequency ratio is implemented. The more
important and commonly employed speed control strategies are precisely revisited
here. Further details about these control strategies can be found in [9, 12]. The

commonly employed speed control strategies for the induction machines are:
1. Constant Volts/Hz Control
2. Constant Slip-speed Control

3. Vector Control or F ield-oriented Control

13

The Speed control strategy developed here is based on the Vector control or the Field
Oriented Control. Further discussion on the vector control is presented in the next

section.

Constant Volts/ Hz Control

The constant volts/ Hz control is primarily designed to accommodate variable Speed
commands by using the inverter to apply a voltage of correct magnitude and
frequency so as to approximately achieve the commanded speed without the use of
speed feedback. Therefore, it is safe to say that the simplest and the least expensive

induction motor drive strategy is constant volts / Hz control.

The speed control strategy relies on two foundations. One of them is that the
torque-speed characteristic of an induction machine suggests that the electrical rotor
speed of an induction machine is very near to the synchronous speed and hence has
a direct relationship to the electrical frequency. Thus, by controlling the frequency,
the speed can be controlled. The second foundation is based upon the phase-voltage

equation that may be expressed as [12]

dAas
dt

 

”as = 7'slas +

(2.9)

For steady-state conditions at intermediate to high speeds wherein the ﬂux-linkage
term dominates the resistive term in the voltage equation, the magnitude of the

applied voltage is related to the magnitude of the stator ﬂux-linkage by

V3 = weAs (2.10)

which suggests that in order to maintain constant ﬂux linkage without any saturation,

the stator voltage magnitude Should be proportional to the frequency.

14

The advantages of this control strategy are that it is Simple and relatively in-
expensive because of being an open loop control solution and that the speed can
be controlled to a degree without using speed feedback. This, in turn, indicates a
drawback of this control strategy; because it is open loop, some speed error will

occur, particularly at low speeds.

Constant Slip-Speed Control

In Constant Slip—speed Control, the drive system is designed so as to accept a torque
command input and hence the system demands an additional feedback loop requiring
the use of a speed sensor. The method is highly robust with respect to changes in
machine parameters and results in high efficiency of both the machine itself and
the inverter at the cost of somewhat sluggish response in closed-loop speed control

situations.

The constant slip—speed control is inherently a current source based control
strategy which offers the advantage that as the current is readily controlled and
limited, the drive becomes extremely robust. However, this comes at an expense

that the control strategy requires phase current feedback.

One of the simplest strategies for current control operation is to utilize a ﬁxed

slip-frequency, deﬁned as

wszwc—wr (2.11)

Which suggests that many different optimizations of the machine performance can
readily be obtained by appropriately selecting the slip frequency ws, including achiev-
ing the optimal torque for a given value of stator current (maximum torque per

ampere) as well as the maximum efficiency [12].

15

Field-oriented Control

Field-oriented control provides the advantages of smooth motion at slow speeds as
well as the efﬁcient operation at high speeds. In many motor drive systems, it is
desirable to make the drive act as a torque transducer wherein the electromagnetic
torque can nearly instantaneously be made equal to a torque command. In such
a system, speed control is dramatically simpliﬁed because the electrical dynamics
of the drive become irrelevant to the Speed control problem. There are a number
of permutations of this kind of control strategy, broadly known as Field Oriented
Control, and these include- stator ﬂux-oriented, rotor ﬂux—oriented, and air-gap
ﬂux-oriented control. Within these types, there are direct and indirect methods of

implementations.

In the ideal ﬁeld-oriented control, the current space vector is ﬁxed in magni-
tude and direction (in quadrature) with respect to the rotor, irrespective of its
rotation. This isolates the controllers from the time variant winding currents and
voltages, and therefore eliminates the limitation of controller frequenCy response and
phase shift on motor torque and speed. Using Field Oriented Control, the quality of

current control is largely unaffected by speed of rotation of the motor.

The motor currents and voltages are manipulated in the d-q reference frame
of the rotor. This means that measured motor currents must be mathematically
transformed from the three-phase static reference frame of the stator windings to
the two axis rotating d-q reference frame. Similarly, the voltages to be applied
to the motor are mathematically transformed from the d-q frame of the rotor to
the three phase reference frame of the stator before they can be used for PWM
output. It is these reference frame transformations, which generally require the fast

math capability of a DSP or a high performance processor, which are the heart of

16

ﬁeld-oriented control.

2.5 Vector Controlled Induction Motor Drives

Various speed control strategies employed for the induction motors generally provide
a good steady-state response. However, many of the control strategies present poor
dynamic response and the cause of such poor dynamic response is found to be that
the air-gap ﬂux linkages deviate from their set values, both in magnitude and in
phase. These variations in the ﬂux linkages have to be controlled by the magnitude
and frequency of the stator and rotor phase currents and their instantaneous phases.
The oscillations in the air-gap ﬂux linkages result in oscillations in electromagnetic
torque and generally reﬂect as speed oscillations. Further, these result in large
excursions of stator currents and present a requirement of larger peak ratings on
inverters and converters that eliminate the cost advantage that ac drives have over

their counterpart dc drives.

Induction Motor drives basically require a coordinated control of stator cur-
rents — in magnitudes, in frequencies and in phases - making it a complex and
intricate control problem. However, it is possible to have an independent control of
the flux and torque for the induction motor drives as in the dc drives. The stator
current can be resolved along the rotor ﬂux linkages and the component along the
rotor ﬂux linkages is the ﬁeld producing current. But, this calculation requires the
instantaneous position of the rotor ﬂux linkages and if this is made available, the
control problem simpliﬁes to the similar one for the separately-excited dc drives.
Since, the control is achieved in ﬁeld coordinates, therefore, it is generally referred
to as Field Oriented Control and because it relates to the control of the rotor ﬂux

linkages, it is also known as Vector Control [9].

17

2.5.1 Principle of Field Oriented Control

For the purpose of explaining principle of Field Oriented Control, it is assumed that

the position of the rotor ﬂux linkage AT is known and it is at an angle p from a

stationary frame of reference. The three stator currents can be transformed into q

and d axes currents in the synchronous reference frames by using the transformation

ias

iqsc 2 sinp sin(p — 2375) sin(p + 2371)
= ‘3- 2be

fdse 008/) 608(9 - 2,") 008(0 + 23¢)
lcs

from which the stator current is can be derived as

 

l3 2 \/(lqs(:)2 'i' (ZdSC)2
and the stator current angle is given as

i
63 = tan—1(F—3‘i)
’dse

(2.12)

(2.13)

(2.14)

where 2",“ and idsc are the q and d axes currents in the synchronous reference frames

that are obtained by projecting the stator current on the q and d axes respectively.

The current is is responsible for producing the rotor ﬁux Ar and electromag-

netic torque Te. Resolving the stator current is along Ar provides the ﬁeld producing

component 2', and the perpendicular component is the torque producing component

iT. By writing rotor ﬂux linkages and torque in terms of these components as [9]

ArCIIf

Tc (1 ATIT (I iftT

18

(2.15)

(2.16)

it can be seen that the orientation of A, in synchronous reference frames presents the
flux and torque producing components of the stator current is as do quantities and,

therefore, they are ideal for use as control variables.

The crucial thing to the implementation of the ﬁeld oriented control is the

acquiring of the instantaneous rotor ﬂux angle p, which can be written as

p=/(wr+w81)dt=/wsdt (2.17)

and the ﬁeld oriented control schemes are classiﬁed based on how p is acquired. If p is
calculated by using terminal voltages and currents, then it is known as Direct Vector
Control. On the other hand, if p is obtained by using rotor position measurements
and / or using estimators or observers with only using machine parameters, then such

a scheme is known as Indirect Vector Control.
A detailed discussion on the modeling, analysis and control schemes pertain-

ing to the vector control of induction meters is provided in [9], where a number of

direct and indirect vector control algorithms are presented for various applications.

19

CHAPTER 3

Previous Work

3.1 Introduction

The previous work [8] is reviewed in this chapter. It utilizes the fundamental induc—
tion machine model to design nonlinear observers for ﬂux and speed estimation. A
sixth-order nonlinear model of the induction machine is derived that describes the mo-
tor in ﬁeld-oriented coordinates. The model takes into consideration the error in ﬂux
estimation. The flux regulation problem is addressed by following the traditional ap-
proach of using PI controllers. For the speed regulation problem, the machine model
is simpliﬁed by assuming that the ﬂux regulation takes place relatively fast and by
using a PI controller to regulate the q-axis current to its command. A third-order
nonlinear model is derived. Using this third-order nonlinear model, the speed regu-
lation problem using a traditional PI controller is considered. The analysis presents
conditions pertaining to the design of control for sensorless operation of induction
motors. It reveals an important role played by the steady-state product of the ﬂux

frequency and the q-axis current in determining the control properties of the system.

20

3.2 Induction Motor Model

The Induction Motor is represented in stator frame of reference by the equations [9]

 

 

d _ R1~ Rr .
32A,: —— (*EI+WJ)Ar+Z;LmlsU (3'1)
d L. R R L2 .
[432223 = ——EEE(—-Z:'I+WJ)AT— (R3+ TLzrn)Is+U3 (32)
r
dw _ 3me T , l
Trina-t- — — 2Lr AT J25 — blw mTL (3.3)

where Ar 6 R2 is the rotor ﬂux, is e R2 is the stator current, vs 6 R2 is the stator
voltage, and w is the rotor speed. The parameters Lr,Ls and Lm denote the rotor,
stator, and mutual inductances, a = 1 — L3,,/Ler is the leakage parameter, 1%, and Rs
are rotor and stator resistances, m is the rotor’s moment of inertia, b1 is a friction
coefﬁcient, and p is the number of pole pairs. The resistance Rs which represents
stator resistance, R, which represents rotor resistance, the moment of inertia m, and
the friction coefﬁcient b1 are treated as uncertain parameters with Rs, 127,152 and I31 as
their nominal values, respectively. The load torque TL will be treated as a bounded

time-varying disturbance.

The 2 x 2 matrices I and J are deﬁned by

3.3 Flux Observer

For the purpose of designing the controller, the ﬁeld-orientation along the rotor ﬂux

A,- is considered. Since Ar is not measured, an open loop observer [15]

A

. Ii - R .
Ar = ("Z51 ‘1‘ WreleAr + L—erls (3.4)
r

RIQ‘

t

21

is used to estimate Ar. The ﬂux observer duplicates the flux equation (3.1) , with

the unavailable speed w replaced by its reference emf. Orienting the vectors 21,-, is, us

and e = A, — A,- along the vector 51s, and denoting the direct-axis components by Ad,

id, vd and ed, respectively, and the quadrature-axis components by Aq(:0),iq, vq and

eq, respectively, the motor can be represented by the following sixth-order nonlinear

model [8].
95% = 4.1,, + 0;..me
did _ , , , . ,. - .2
:17 _ arﬁAd — (asn + eri3Lmlld + Wref'q + aer’q/Ad
+ ’yvd - (iv-Bed - Bpweq
(27—: = '13!)de — Wrcffd — (0‘37) + ar/3Lm)iq — erm‘id’iq/dd
+ ryvq + Spiced - Ori3€q
(:4: = #l'iq(/\d — ed) + ideal — 5w " TL/m
957d = -ar€d + (Wref - PW + ermlq/Adleq
+ (03» — ar)(Lmid - Ad)
d7? = _(pwr,,f _ xx» + d,-Lm-iq//\d)cd — Grf’q
+ (Cir - Or)Lmiq + P(wrcf _ Wldd
where

For the later use, deﬁne

~ _ff‘ ~_3Lm ‘_b
a3‘f31"—2mlirib"rﬁ

In the foregoing equations, Ad, id, and iq are available for feedback, as they can be

calculated from is and A, while a), ed and (7., are not available.

22

3.4 Flux Regulation

In ﬁeld orientation, the ﬂux Ad is regulated to a reference ﬂux Am] > 0, which is
taken as a constant. By viewing (—arD8d — ,3pweq) as a disturbance input to equation
(3.6), the equations (3.5) and (3.6) can be used to design a state feedback controller
for vd to regulate Ad to Aref. There are several methods available to design such a

controller. The traditional approach of using two PI controllers [10] is considered here.

First, id is viewed as a control input to equation (3.5) and the PI controller is

designed as

K +Ki
I; : LIESS—ap‘rcf ‘- Adi

And then, the second PI controller is designed as

K K .
Vd = ( dp88+ dz)[15_ Id]

With tight feedback loops, the regulation of Ad to ’\ref can be ensured for a wide

range of variation of the term
Wrefiq + (ermig/Ad + 71rd —- arﬂed — ﬁpweq

which acts as an input on the right-hand side of equation (3.6). The design should
ensure that Ad starts at a positive value and approaches Aref monotonically so that Ad
is always positive. The initial conditions of Ad are determined by the initial conditions

of the ﬂux observer (3.4).

3.5 Speed Observer

Sensorless operation of an Induction Motor essentially requires us to use an observer
to measure the rotor speed, as no other means are available to measure the rotor

speed online. The high-gain observer [7] is a technique that works for a wide class

23

of nonlinear systems and guarantees that the output feedback controller recovers the
performance of a state feedback controller when the observer gain is sufﬁciently high.
A high—gain observer is utilized here to estimate the rotor speed. Towards that end,

rewriting equations (3.7) and (3.8) as

-5Pw/\d — f1(/\d. id. i‘I‘wrcf) + 7vq + 61 (3.11)

[1.1.qu - Du) + 52 (3.12)

Si? els-

where

is available online, and 61, 62 are uncertain terms given by

61 = [((i3 — as)" + (Cir — Or),13Lrn]lq + fjptded - CIrBCq

as = (,i — [01'qu + u(—iqed + ideq) — (b — (3)5 — TL/m

The change of variables

6
Q 2 w _ 13p,“ (3.13)
A _ l
= (if) - HEW; _ can + (dr — armLmii-q - Grier}

brings equations (3.11) and (3.12) into the form

(12'

71—; = "llpwdd ‘ fl(’\dv ids 2.(Irwrcf) '1' 7“] (3'14)
d. . .

i?” = [11'qu — be) + as (3.15)

where

_ 56 d (i A - -
53 ~ ‘52 — 35f; — FINE—1h) = f2(/\daldalq~wrcf»€d,€anL)

24

and f2 is a continuous function of its arguments. The change of variables (3.13) is
invertible provided Ad — ed 36 0. The high-gain observer, then, is represented by the

equations

diq

, * , _ a . A.
E : —,13P)\dQ _ f1(Ad12daquwrc/‘) ‘1' ’7'Uq + (‘5)(2q —- zq) (316)
d0 ~- ‘ “ 02 . 1
Et- = #74qu " 59 " (mllm - m) (3.17)

where e is a small positive parameter and 01 and 02 are positive constants that assign
the roots of 32 + 013 + 02 : 0 at desired locations in the left-half s - plane. The scaled

estimation errors

T—i, ..

"1:25 ,77220—9

satisfy the equations

€771 = 0101 - ﬁpAdﬂz (3-18)
' — _ .32. _ ' _
€772 — (HP/M1)?“ ebng €63 , (3.19)

For small 5, the closed loop system will behave as a singularly perturbed system, with
771 and 772 as the fast variables. The essence of the singular perturbation theory [7],
is that when we face a perturbation problem that is characterized by discontinuous
dependance of the system properties on the perturbation parameter 5, then the dis-
continuity of solutions caused by singular perturbations can be avoided if analyzed
in separate time scales. According to Singular perturbation theory [13], the stability

of the fast dynamics is determined by the matrix
-01 —i3P)‘d
0
Bpﬁd 0
in which Ad > 0 is treated as a constant. The characteristic equation of this matrix

is s2 + 013 + 02 = 0 ; hence, it is Hurwitz by design. From the high-gain observer

25

theory [1] , it is known that if the control input vs is bounded uniformly in c, then
the estimation error 52 — 0 will be 0(5) after a Short transient period [0, T(e)], where

lim.5_,0 T(€) =

3.6 Speed Controller

The design of speed controller can be simpliﬁed by reducing the order of the system.
First, it is assumed that the flux regulator acts fast enough to regulate Ad to its
constant reference Aref. This assumption allows us to take Ad = Arc! and, therefore,

= 5L3? leads to dropping equations (3.5) and (3.6). Also, it can be seen from

equation (3.7) that for any current command if}, we can design vq as the PI controller

vq_ WV“ _ [q]

with sufﬁciently large gains to regulate 2“,, to iq .This allows us to view 1”,, as the control

input. Thus, the speed controller can be designed using the third—order model

Eu

         

dt : —O’r€d + (Wref — pa.) + Cianij/Arcf)€q (3.20)

d6“? — ‘ L 1' A

W - —(Pwref — W + “r mlq/ refl€d — Oreo

(la) ,

a” = “(1.41”, _ eq} — bw — TL/m (3.22)

A. — c
Q = ref d + areq _ j. '2

(—Arc; )3 rm; 0., (3 3)

where Q is viewed as the measured output and,

=L09 -as)r]+( Or— Or)BLrn
t3‘1’)‘ref

 

The goal is to have 9 track “ref'

26

It is natural to use integral control to ensure zero steady-state error when
“ref and TL are constant [7]. Under the condition, 0 = wref, the equilibrium

equations are

 

 

O = ‘0‘7‘8—d + (Wrcf _ pd) + dTLmi-CI/Aref)e_q (324)
O = -(pw,.cf - pa? + erm-i-q/Arefk'd — are-q
- A
o = Maura, — Q) + if: e-q] — biz; — TL/m (3.26)
/\ , —€- (1 e. _
Q _ ref d-+ rq_h. 327
(—_)‘ref )w pAref (”q ( )

Solving equations (3.24) and (3.25) for e'd and e}, in terms of {q and J; i (:2 — wref and

substituting in equation (3.27), it can be shown that

- eryni_ .. (dr-‘(lr)L1nl‘-( (d3—0.9)17Ai-
“ “ + —r——‘ = _—13X_—J
( pa) + ref )( pw ref )wc ref

where

a‘L i.’ - ciL 2'7 '
wczpwrcf+—TX7%1 and A=a%+(—w+—IX’%1)2

To gain insight into the problem, the case is considered when d3 = as, for which the

preceding equation reduces to

(“P0 + (if; )(‘P‘I’ + LOFT—gr fLmi )Wc = 0

re

Assuming that we ;£ 0, the equation has two solutions:

.. _ (dr—Qr)L1ni‘ ~ _ 03.1},an-
It is clear that the ﬁrst solution is the one which yields zero steady—state error in the

nominal case when air = or. The equilibrium point corresponding to this solution is

27

 

—. bwref + TL/m
lq — b Cir—Or L171
#‘ATCf - p ref
_ (Cir '— Qr)LmT
w — wmf + W11 (3.28)

In order to see whether a PI controller can stabilize this equilibrium point, the equa-

tions (3.20)-(3.23) are linearized at this equilibrium point, to obtain the linear model

where
Oer; O
_C!r W3
_ GrL-y 1—.
A _ ch’; _ar —p)\"cf
. Mn:
“'0 TEL “b
0
B = (611' " Orv-ﬂan
“Aref
< _,_. 7.. 1
ref p ref 1
and

with the transfer function

C(s) .—.- C(sI — A)-IB + D = 38

in which

_ 2 wan { (i —a L1
"(3) " #Arcfls + 0T3 + J—rrjP—qlll _ W“ + (9)]

up)”,

28

a i *3. ar 2”?
(1(3) = (s + b)[(s + m)2 + (1X95???) + ”—pzﬂ—lﬂs + a. - 1113—1)

ref

It can be seen that the product wciq plays an important role in the control design.
When em}, = 0, 0(3) has a zero at the origin. Hence, it is impossible to design
any controller with the integral action. This follows from the well known theory of
servomeehanisms [3]. When Luci—q < 0, 0(3) has a real zero in the right-half s—plane;
hence, it is non-minimum phase. It is possible to design a controller with integral
action to stabilize the system, but such a controller cannot be a PI controller. This
fact can be seen by sketching the root locus of the system for different possible pole-
zero patterns. For a PI controller, the root locus will always have a branch on the
positive real axis. This leaves us with the case when Luci}, is positive. In this case, the
transfer function C(s) is minimum phase and we can design a PI controller with high-
gain feedback to stabilize the closed—loop system and achieve good tracking properties.

Such PI controller takes the form

[9 = ( u 88 wz)lwrcf _ 0l
The condition

-. -. ' L T
(41ch '2 'l-q [pwref + W] > 0
is satisfied when the motor is operated in the motoring or braking modes, but not in

the generating mode. A similar condition has also been presented in [2].

The condition wciq = 0 is satisﬁed if {q = 0 or we 2 0 [8] The case we 2 O indi-
cates operation at zero frequency, in a braking mode corresponding to certain speed
and torque. It is well known in the induction motor literature that operating the
motor at zero (or low) frequency is challenging, and that a design for such case will
have to exploit secondary phenomena of the machine, not conveyed in the machine

model (3.1)-(3.3). The case Eq = 0, regardless of the speed, indicates that the power

29

into the machine is negative.

Figure 3.1 presents a schematic for sensorless control of an induction motor

using ﬂux and speed observers. Three-phase stator currents are ﬁrst transformed

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

be!
’ ” “ W K
Hm __
H Observer 0 me ’
p Inmsc
”magma! p1 Camila Ficld_an'cnudon
0 0
l l
2 idse Vt! L. 2 yds
2 iqse V.“ T 2 '—
W Transformation FickLaticnuabn a» "1‘
4 am
e wive—
ObSCNC! g Iqs’
[’1 Controller PI Controﬂa
2/3 T
Transformation V g! e
Vas
V” 2 Vqs
Va: 3 Vds

 

 

 

 

Figure 3.1. Sensorless control of an induction motor using traditional PI controllers

30

to d — q currents. The ﬂux observer provides the ﬂux estimate Ad and angle 0, for
ﬁeld-orientation, using these d — q currents and reference speed wref. The d — q
currents are then transformed into ﬁeld-coordinates. The speed Observer utilizes
these ﬁeld currents to provide a speed estimate. Two PI controllers are used for
ﬂux regulation and regulation of the d-axis current. For the speed regulation, a PI
controller is used that regulates the q-axis current to its command. The voltage
signals provided by the PI controllers are transformed back to original coordinates

by inverse ﬁeld-orientation and d — q to three-phase transformation.

31

CHAPTER 4

Speed Control Using Sliding Mode

Control Strategy

4.1 Introduction

The complex nonlinear nature of the induction motor model together with the fact
that certain important quantities are not measured, present difficulties in designing
the high performance induction machine drive control algorithms. In addition,
the uncertainties pertaining to the imperfect knowledge of the system inputs and
disturbances together with the inaccuracies in the machine modeling contribute to
performance degradation of the feedback control system. Sliding mode control is a
popular technique in nonlinear feedback control that operates effectively over a range

of system parameter variations and disturbances.

Sliding mode control deals with robust control under the matching conditions;
that is, when uncertain terms enter the state equation at the same point as the
control input. In sliding mode control, the trajectories are forced to reach a sliding
manifold in ﬁnite time and to stay on the manifold for all the future time. Motion on

the manifold is independent of matched uncertainties. Its two main advantages are

32

o The dynamic behavior of the system may be tailored by the particular choice

of switching function

a The closed-loop response becomes totally insensitive to a particular class of

uncertainty related to the system parameter variations and disturbances.

By using a lower order model of the system, the sliding manifold is designed to achieve

the control objective .

4.1.1 Sliding Mode Control

Consider the system

a.
II

f (I) + 9(I)u

y = Mir)

The sliding mode control law for such a system takes the form
u = —B(I)sgn(8)

where ,3(1:) is bounded up and below by certain inequalities in order to satisfy the

conditions for maintaining the motion on the sliding manifold and

1, s > 0
3911(3) =
—1, s < 0.

The motion consists of a reaching phase during which the trajectories starting off
the manifold s = 0 move towards it and reach it in ﬁnite time, followed by a sliding
phase during which the motion is conﬁned to the manifold s = 0 and the dynamics of
the system are represented by the reduced order model of the system[7]. The mani-
fold s = 0 is called the sliding manifold. A sketch of the typical phase portrait for a

second-order system is shown in Figure 4.1.

33

 

l

 

T

 

 

 

 

Figure 4.1. A typical phase portrait under sliding mode control

Due to the imperfections in switching devices and delays, the sliding mode control suf-
fers from chattering. Two different approaches for reducing or eliminating chattering

are[7]

o Dividing the control into continuous and switching components so as to reduce

the amplitude of the switchin component
0 Replacing the signum sgn function by a high—slope saturation function
Using the second approach, the control law is taken as
u = —l3(x)sat(s/E)

where sat(.) is the saturation function deﬁned by

y. if lyl S 1
sait(y)=

3912(3)), if |y| > 1

34

and e is a positive constant. The slope of linear portion of sat(s/£) is (1 /5). A good
approximation requires 5 to be of a very small numerical value. In the limit, as a ——> 0,

the saturation function sat(s/€) approaches the signum function sgn(s).

4.1.2 Zero Dynamics and the Relative Degree of Nonlinear

Systems

From the view point of designing feedback control for a nonlinear system, it is
necessary to investigate certain important properties of the system. The two
important properties of a nonlinear system considered here are the relative degree

and the zero dynamics.

Relative degree of a nonlinear system is the number of derivatives of the out-
put needed to make the input explicit for the system. The same term is used for a
linear system in a context that it is the excess of poles over zeroes for the transfer

function of a given linear system.

Zero dynamics of a nonlinear system represent the internal dynamics of the
system when the output is identically zero. In linear systems, this matches nicely
with the dynamic equations whose eigenvalues are the zeroes of the transfer function
of linear system. This is critical for linear controller design because at high-gain
the closed-loop poles migrate to open-loop zeros and these zeros determine the
boundedness of the control required for linear tracking. Zero dynamics are important
for many nonlinear controller design procedures like feedback linearization, sliding
mode control and adaptive control. Details of zero dynamics matter because if
the system possess unstable zero dynamics, it can invalidate many control design
procedures. Theoretical foundation of the calculation of the relative degree and the

zero dynamics is given in [7, 5].

35

Consider a single-input single—output nonlinear system

f(r) + amu-

a.
H

y = h(I)

The calculation of the zero dynamics of this system consists of two steps

0 Bring the system in a normal form with a nonlinear invertible change of coor-

dinates z = (Na)
0 Extract. the zero dynamics equations from this form

First, calculate r components of <I> as

(MI) = MI)
(152(13) = th(a:)
em) = L;—1h(x)

where L fh(.r) = 2,31%.) (I) and L}h(r) etc. are recursively deﬁned and r is the relative

degree i.e. the smallest r for which LgLy‘thr) ¢ 0, with L9 = {El—39m). Choose the
remaining 71 — 1' new coordinates 2i: 2' = r + l,...,n so that <I>(a:) is invertible at 2: = 1:0.

Additionally, select on = r + 1, ...,n so that [5]
ngi = Lg‘l’i(1') : 0

The normal form in the new coordinates z is then

.; _. ~

~r-l — ‘1‘

221» = b(z) + a(z)u
ér-l-l = gr+l(3)
in = (111(3)
and
y = 31

where b(.r.) = L;h(;r), a(;r) = Lng—lhtr) and q,-(:r) = qu’),(:c),i = r + l, ...,n. we use the

relation
3: = (IV-1(2)

to express b(1:), a(x) and q,(.-r) as functions of z.

The above equations represent the system in the normal form. This form dc-
composes the system into an external part and an internal part. In order to obtain
y = 0, we choose the input a as —b(z)/a(z) and the initial conditions 1:0 such that
z, = 0,1 = 1, ...r. By deﬁnition, the zero dynamics of the system are then given by the

equations for z‘,.+1,...,;.'~n, with the coordinates z1,...,z,~ set to zero. With

[21, ..., ZrlT

m
H

'7 = [zr+1,---azan

37

the last (n -— r) equations of the normal form can be written as

7'1 = <1(€.n)

and the zero dynamics are obtained by setting 5 = 0:

1'7 = (1(0. 17)

4.1.3 Regulation via Integral Control in Relative Degree-1

Systems

Suppose the system

NJ?) + 9($)u

H-
II

h(1:)

=c
II

has relative degree 1 for all :2 in a domain D C R". Our goal is to design a state feedback
control law such that the output y asymptotically tracks a constant reference signal r.
When the signum function sgn(s) is approximated by the saturation function sat(s/5),
the regulation error will be ultimately bounded by a constant kg for some I: > 0. Using
an integral control provides zero steady-state error, therefore, we augment the integral
of the regulation error 3} — r with the system and design a feedback controller that
stabilizes the augmented system at an equilibrium point, say 11:33, where y = r. We

use the integrator
a = y — r = 8
with the system equations to obtain the augmented system
i = f($) + 9(Ilu

38

a = e = y — 1'
To proceed with the design of the sliding mode control, we can take,
3 = koo + e

where k0 is a positive constant. The sliding mode control law for such a system takes

the form

u = —l3(.r)sat(s/€)

4.2 Sliding Mode Controller Design For the Induc-
tion Motor

In order to proceed with designing the speed controller based on the sliding mode
control theory, the reduced-order machine model (3.20)-(3.23) is considered. To this

effect, the error functions can be deﬁned as

e = w—wmf

y = Q-wref

and the equations (3.20)—(3.23) can be rewritten as

 

 

ed = —Or€d + (—p€ + ernliq/Arcf)€q (4.1)
éq 2 —(-‘p8 + errniq/Arpf)€d - Greg + (dr -' 01‘)Lrnlq — pFAI‘Cf (4.2)
. __ . Aref .
e — p[lq(/\n,f - ed) + Lm eq] - be — bwrcf - TL/m — wrcf (4.3)
)‘ref _ ed ”ref 01'
y — ——)e — —e + e — ai 4.4
( )‘ref )‘rcf d p’\ref q q ( )

where

39

a = (ds-OQ‘U+(dr-Gr)ﬁLnl
lTpxref

 

The system is single-input-single-output where ed, eq and e are the three states, the
q-axis current is the control input and a speed estimate 9 (provided by a high-gain
observer) is the measured output. The error functions have been introduced to

conveniently proceed with the analysis.

The goal here is to address the speed regulation via integral control based on
the sliding mode control theory, such that the rotor speed 0.) asymptotically tracks a
bounded time-varying reference speed wref- Clearly, the equilibrium point of interest
pertaining to the reduced order system would be such that the flux estimation errors
ed and eq, and the speed error e are zero. The following section presents the analysis

related to the asymptotic stability of the desired equilibrium point.

4.2.1 Zero Dynamics of the system

For the purpose of stability analysis, the nominal parameters case is considered when
1?... 2 RT and R3 = R3, which makes a = 0 in equation (4.4). The resulting system has
a relative degree 1 in R3 as the control input appears in the output equation upon
calculating the ﬁrst derivative of the output. The internal (or zero) dynamics of the
system are described by equations (4.1) and (4.2) when 3; is set identically zero. The
external dynamics of the system are described by equation (4.3). Analysis of the zero
dynamics provides insight into the asymptotic stability of the origin. The system is

said to be minimum phase if the origin of the zero dynamics is asymptotically stable

[7]. When

y E 0
in equation (4.4); we obtain
=wTPfed—a—pteq (4 5)
)‘re f — 8d '

4O

Taking the time derivative of the preceding equation

a [wrefpd_ 95—6-71] , 0 [wrefed- gifeq] .
—— e + —— eq

e = ,— —
06d Arc] — ed (96d )‘rcf — ed

and substituting the values of ed and e'q from equations (4.1) and (4.2), we obtain

1

é———————awre—pew e+(aLi/z\ )w’e
(Arcf‘ed){ 7‘ rcfd refq 7‘ mq ref refq

(I2 >
72:14"?qu a2
— areed + ——-—-— +

T
—e +a eA
/\r€f P q 7' TC!

__1 2 a2 , ,
'l' (Aref _ ed)2{ — QTWTCfed + [—p' — pewref + (arL1an/Aref)wrcf] edeq

2g .
+ [a - P Lm,2q:l 2
re _ (.‘q
Aref
The second order terms in ed and eq that appear in the above relationship can be

approximated by 0(]|ef]]2), where ||ef||2. = (33 + e3. Further, substituting the value of e

from equation (4.5); we obtain

 

2
. 1 3pthiq
e — Wk“ “"wref + WM
02 . /\ , g
+ (”13C + Q’L"”"1“’ref / Aref lei + (Tnf—fﬁ larwrefed ’ £1pieql}
+ 0(1e f1?)
01‘

. _ 1 03. AW . ar

e _ m [ ” Orw'rcfed + ;€q ‘i' mkﬁwrcfed — yeq)
+ 1 0—214 L .

)‘rch‘rCf ‘ed) p med+arwr0f meq 2-q

+ Owen?) (4.6)

41

Similarly, substituting the value of e from equation (4.5) in equation (4.3) we get

 

. be)”. f “Are f b0 r
' = A. —- . —————. + +
'3 “( “’f 6‘1”" (Aref—Cdled ( Lm pow-cud“
T .
" (wref + 77% +wref) (47)

Comparing equations (4.6) and (4.7); we have

2
l a . '_
2 A
- 1 - . 2L __ .r Fl ( . _ a: )
(Aref-cdl [ arwrcfed + P eq + (Aref—ed) aer‘Cfed P eq
b M ,,[ T ,
+ (”7‘0de .— LIE—£89] — :1: eq + (bwTCf + 7,1: + “rail

01‘

 

 

 

 

AT arwr' A ()
.iq = 02f {[(b — Orlwr‘ef + (—/\-—C—f——;—§-]€d
“Aref(’\ref — edl2 " (7)1:Lmed + arwrchmEQ) ref d
._ gLA (A — e )A
0r 1 ref ” ref (1 ref
+ b — ,. - ’ —
l p ( a ) (AW — ed) Lm le‘?
T .
+ [(Aref - ed)(bwref + EL +wref)]} + 0(llefll2) (48)

The next steps involve substituting the value of ,q from (4.8) into (4.1) and (4.2)
to obtain the equations for the zero dynamics of the system. Towards that end, we

substitute the value of rig in (4.1) and after some simpliﬁcations, we obtain;

6d : —Or€d

. L a w A .
+ or meq {[(b-ar)wref+-:-—rEf—ﬂ]e

2 _ ,)
”ArefO‘ref - ed)2 “ gpLLmed " OrwrefLmeq (AMI (d)

_ a—r-A A. — )A
i _ — p ref _l‘( ref ed ref
“‘l p (b 0’) emf-ed) Lm le"

T
+ [(Arcf _ ed)(bwrcf + .r—nI-J- + u."refl]} + 0(llefll2) (49)

42

Similarly when we substitute the value of iq in equation (4.2), we obtain

 

 

 

- ' 'r r. Ar.
e'q -:_ Orgmed {[(b — arWrcf + W]€
”ArefiAref — ed)2 " EPLLmed _ arw'refLmeq ' ref ed
_ 91A A — e )A
ar _ _ p ref _l”( ref (1 ref
+l p (b 0r) (Aref ‘— ed) Lm leg
T .
+ [(Aref — ed)(bwref + ';L + Wrefﬂ}
—are — Ji—(pw ed—are )+ one u?) (4.10)
q (Aref - Cd) ref q f

A further approximation of the second-order terms in ed and eq simplifies the equations

(4.9) and (4.10) to

-L1 T _
'd = —ared + Z—gr”—(bwmf + 7,? +Wrcf)€q + 0(llqll2) (4.11)
ref
' — “er TL ' 0 2 412
eq - ’Wrcf+;’\T—(bwref+;+wref) €d+ (“Cf“) (- )
ref

Assuming that aural—(t) and TL(t) approach constant limits as t —+ 00, i.e.
limtﬁoowmfu) = pref, “mt—.oowrer) = 0, limtﬁooTLU) = TL, and neglecting the second-
order terms in ed and eq, equations (4.11) and (4.12) constitute a linear time-varying

system of the form

i‘ = [A + B(t)]:r
where
_ _ T
—Qr ﬁgmwwref + 7711‘)
A = "3f
_ L _ T
“Wref+?ﬁ7?(mrcf+7£‘) 0
re

43

. - . T
0 _r_2_m0AL [bwrcf + wref + 7%]
B(t) ___ ’1' ref

T
“Wref— :26 jib“) ref+wref+7711‘] 0
where Jimf = wrcf - Qref: TL = TL — TL, A is constant and B(t) is time-varying such

that limt__,ooB(t) = 0.

The above system can be viewed as a perturbed system where A is the sys-
tem matrix of the nominal linear system and B(t) is the perturbation term. Theory
of stability of perturbed systems [7, Example: 9.6] proves that if the origin is an

exponentially stable equilibrium point of the nominal system
gt = A11:
and if
B(t) —* O as t -—> 00

then the origin is a globally exponentially stable equilibrium point of the perturbed

system :i: = [A + B(t)]:c.

In order to investigate the stability of the origin (ed = eq = 0) as an asymptot-
ically stable equilibrium point for the system (4.11)-(4.12), we proceed with the

calculation of the eigenvalues of the matrix A.

“a? ﬁi‘g‘mwwref‘l' )
A : ref
+T
‘pwref :28 'f‘WJ ref+ 77%) 0
_ T
A+or Tigémmmﬁ -,,1;)
A] - A = ref
T"
Wref+_5'm(bw ref+7r1f>l ’\

which gives

IA] - A' = A(’\+ar) + [Wref + giéﬂ;(wref + 77%)] [%§§':(boref + 2179)]

01'
IA! —— AI = A2 + am + [pi-1,6,“ n2]

where

The eigenvalues of A

 

A12 = —92£ 2t h/a? —4[p<3,.efhf+rc2]

have negative real parts only when the product n(p§2,.ef+r<) > 0, which is the case when
the steady-state speed command emf and the steady-state load torque TL both have
the same sign, positive or negative. This condition is similar to the one obtained in
the previous work [8], referred to in Chapter 3, where the product wciq determines a
similar stability condition. The condition “pom, + K.) > 0 is satisﬁed when the motor
is operated in the motoring or braking modes only. If the motor is operated in the
generating mode the stability condition is violated as the product n(po,.cf + K.) is no

longer a positive term.

4.2.2 Speed Regulation via Integral Control

A sliding mode controller with integral action is designed in order to address the speed
regulation problem. Using integral control provides zero steady—state error. There-
fore, we augment the integral of the regulation error with the system and design
a feedback controller that stabilizes the augmented system at the desired equilib-

rium point. Consider the reduced-order machine model (3.20)-(3.23) with the error

45

frmctz’ons deﬁned as

e = w-wrcf

y = Q—wrcf

In the nominal case dr = or , d3 = as, equations (3.20)-(3.23) can be rewritten as

 

 

e'd = —a;~ed + (—pe + aerz'q/Arefkq (4.13)
e'q = —(—pe + aeriq/Aref)ed - areq — peAref (4.14)
/\
e = #l‘iinrcf — ed) + 1:: eq] — be — bwref — TL/m -— emf (4.15)
Aref _ ed wref
= —— — —. e 4.16
y ( Arcf 6 )‘rcf d p/\.,.ef q ( )

The system is single-input-single-output with ed, eq and e as the states, the q-axis cur-
rent as the control input and the speed estimate 9 (provided by a high-gain observer)

as the measured output. Augment the integrator

Q —w,.cf (4.17)

with the system (4.13)-(4.16), and take

3 = (€00 + y (4.18)

where k0 is a positive constant. The continuous sliding mode control law for the

system is

iq = —Ksat(s/e) (4.19)

46

where K should be chosen to satisfy the condition

35‘ < 0 (4.20)

outside the boundary layer {lsl g e} to ensure that the trajectories reach the boundary

layer in ﬁnite time and stay inside thereafter.

Substituting the value of y from equation (4.16) in equation (4.18) yields

 

 

 

A —e w
s — k00+(—r—i\f——d)e—A—ried+ Sr eq
ref ref p ref
Are f - Pd 6 wrc f ar
s = k06+ —— é——e' — e' + e'
( Aref ) Aref d Aref d pAref q

Substituting the values of e'd, e'q, é and 6’ from equations (4.13)-(4.15) and (4.17), we

 

 

obtain
3 = [my

Arc f - 6d . Aref .

+ ( Aref ){lllIQ(Aref - 8d) + Lm eQ] _ be — bwrcf — TL/m _ wref}
e .

_ Are f l ‘ ”red + ("Pe + Oer'q/Arefle‘d

wref '.
- ml — ..., + (-.. + Q,L,,.,/.m,)e.,}

0r .
PAref{ — (—pe + aerzq/Aref)ed — areq — peAref}

 

which simpliﬁes to

 

 

 

. _ 0,. pe(e + erf) It a2
S ‘ be, (2" + Wed + (T + we] - ...) - all”
b(’\rcf -- 6(1) + (Ir/\ref
_ ( Aref )8 + km]
A - e
_ ref (1 n
( Aref ) (bwref + TL/m + wref)}

47

 

 

Aref A2

“(Aref — ed)2 aer(6 + wrcfkq (1'ng _
+ _ — 2 Cd 'lq
ref pAref

The above expression can be written in the form

where

'11
I

 

 

0r pe(e + wmf) M
{Aref (26 + wref)ed + (T + m()\rcf — ed) _

_. (bo‘ref — ed) + air/\mf
Aref

 

)e+ koy

A , -€d .
_ (inc'f_(bwrCf+TL/m+wrcf))}

 

 

C = {11(Arcf — ed)2 _ aer(e +wrefleq _ 02L": 6d} (4.22)

2 2
’\rcf Aref pAref

The function G should satisfy the condition
G 2 GO > 0 (4.23)

for some positive constant Co, which is the case in the neighborhood of ed = eq = 0.

From (4.21),

33' = sF + .3qu (4.24)

For [5| 2 s, the sliding mode control (4.19) takes the form

iq = —ngn (s)

48

and equation (4.24) can be written as

35‘ 3F - GKlsl
sF
G

G GK|3|

|/\

F
Glsl‘al —- GKlsl
We assume that the ratio '5‘ satisﬁes the inequality
F
lgl S K - K0 (4.25)

where K0 is a positive constant. Then,

53' S G|s|(K—K0)—GK|.SI
= —-K0G'|s|
S ‘KOGOISl (4.26)

It is clear from the foregoing analysis that inequalities (4.23) and (4.25), rewritten as

>2
IV

Ko+|§|

G. 2 Go>O (4.27)

should hold over the domain of interest in order to ensure that all trajectories outside
the boundary layer {Isl _<_ e} reach it in ﬁnite time and those inside the boundary layer

cannot leave, as portrayed in Figure 4.2.

In order to analyze the system inside the boundary layer, consider

C'd : —a-r€d + (—p€ "l” Qerzq/Aref)€q (4.28)

49

 

\
\

 

 

 

 

Figure 4.2. Trajectory outside the boundary layer reach it in ﬁnite time and stay

inside thereafter

 

 

eq = —(—pe + aeriq/Aref)ed — areq — peArcf

. . ref .

e = Mllqo‘rcf _ ed) 'l' Lm te " be _ bwref - TL/m _ ”ref
(5’ = —k00 + 8

’\ref _ ed wref 0r
y z: — e— -—'e + e
( ’\ref ) Aref d pAref q
S = k00 + 3;

Inside the boundary layer, the control input (4.19) is given by

2"] = -—~K(s/e)

Substituting the value of s from equation (4.33), we obtain

iq—

 

A /\ -—e w.
(00+ (Lei—i)“ 11,. +

:— 6
)‘ref )‘ref d pAref q

 

50

(4.29)
(4.30)
(4.31)
(4.32)

(4.33)

(4.34)

(4.35)

Substituting the value of iq from equation (4.35) in equation (4.28) yields

ed 2 -ared-peeq

K , ’\re f - ed wre f a.-
- (aer/Aref): l‘”" + (77* ‘ r74 + w 6"

 

Neglecting second-order terms in ed and eq, the above equation simpliﬁes to

,-

It
e'd = —ared — peeq — (aer/Aref) -5— (1.700 + e)eq + 0(llefII2) (4.36)

Substituting the value of liq from equation (4.35) in equation (4.29), we obtain

 

K Arid—ed wref 0r
e' = pee + aL- /). —ka+ —'——e-————e + e e
q d ( r m ref) 5 l 0 ( )‘rcf ) ”\ref d PAref q d

—- areq — peArCf

which simpliﬁes to

. K
eq = peed + (arL,n/Ar(.f) —€- (koo + e)ed — areq — peAref

+ 0(an2) (437)

Similarly, substituting the value of 2", from equation (4.35) in equation (4.30) yields

 

Aref ‘ ed wref 0r
e — “(Aref_ed)?[k00+ (jg—)e- med-rpArequ
“Aref .
Lm eq — be — bwmf — TL/m - cum}-
and this simpliﬁes to
- _ BE , Aref Kar/p
e — e (A00+wrcf)ed+u(—L:———E—)eq

_ (b+ figAMf)e — (“@Aref)0 — (Wrcf + TL/m +dJref)

51

+ 0(uefn2) (4.33)

Neglecting the the second-order terms in ed and eq, the system can be characterized

by the fourth-order state model

 

 

e'd = —o,~ed — peeq — (aer/Aref) g (koa + e)eq (4.39)
(’-q : peed + (aer-/’\1'ef) if (koo + 8) Cd — areq - pCAref (4.40)
e = #TK (koa + wref>ed + u()\l:;{ — Kasr/p)eq — (b + [Ag/Md}:
— (uingref)U -- (bwref + TL/Tn + emf) (4.41)
Are '_ 2 re r
e = (if—51).. — :jed + pfrcfeq (4.42)

In order to analyze the performance of the system, inside the boundary layer, we
examine the behavior of the system in the vicinity of the equilibrium point. Towards
that end, we write the equilibrium equations under the steady-state conditions emf =

aref! TL = TL and or” = 0.

 

 

 

K
0 = —aréd — pééq “ (QTLTTl/Aref)?(k06 ‘l‘ é)éq (4.43)
K
#K . - - - Arc! Kar/p _ K _
0 = 7(‘00+wref)“d+“(2m “TM-(“524206
Kk. _ _ _
—( —Eﬂ/\ref)0 — (wrcf + TL/m) (4.45)
/\ -éd ’0 f a
0 = (r—ef—m— "’ t + " a 4.46
Aref Aref d pAref q ( )

Naturally, the desired equilibrium point for the system is when the ﬂux estimation
errors ed = eq = O and the speed error e = 0, which imply that the output w tracks the

reference input emf. Substituting these values in equation (4.45) provides the value

of a at the desired equilibrium point as

a = TIT—061’; (4..., + TL/m) (4.47)
' 7‘6

Lincarizing equations (4.39)-(4.42) about the equilibrium point (ed : eq = e = 0, o = 6),

we get the linear time-varying system

:i: = A(t).r

where
o n l T
i‘ = (ed eq 0 e)
T
1.‘ : (Pd eq 0 8)
and

( -a1- — 35$?) 0 0 \
(lanlKkné -O
5 ref r
_wret at
ref I) ref 0 1

l"T"'(koé+wmf) #(i%-M) 4445944!) 44.42%)...”

C

_p)‘rcf

 

 

Substituting the value of 6 from equation (4.47) yields

 

 

{ —0r ﬁlé:(wref + 22%) 0 0 \
- T
A(t) = _:A:§ef (bwrcf + 7711‘) —Or 0 _pA7‘(-tf
w .
“if; 13%?) 0 1
A, rf _ _ e Z ‘ , rvt
\ p :6 " Xrle-f(b“’ref + IL) ”(27,11 _ Kn; p) _#£E£Q’\T0f ’4’— M‘: 6 }

Assuming that wrcﬂt) and TL(t) approach constant limits as t -» oo, i.e.

l-im.t_.oow,.(,f(t) = emf and limtqooTLU) = TL, A(t) constitutes a linear time-varying

53

system of the form

 

 

 

[\(t) = A1+Bl(t)
where
{ a L - T
_a,. m(mref+ m) 0 0
L _ T
A1 = -2—I4VMiwref‘L7ri‘) “Or 0 “1”"?
“re a
r1 TL 0 1
ref 1) ref
pKJ) _ T ’\ K0 P #KA
l—eﬂ-xi—Amme-ﬂ 42—4) .. . ..., ——.4
K 0 0 0 0\
0 0 0 0
31(t) = Q
— re 0 O 0
ref
K”
(W 0 0 0)
where are, = Wrcf — or”, A1 is constant and Bl(t) is time-varying such that

(zimt_.oo81(t) = 0. The above system can be viewed as a perturbed system where
A1 is the system matrix of the nominal linear system and B1(t) is the perturbation
term. Theory of stability of perturbed systems [7, Example: 9.6] shows that if the

origin is an exponentially stable equilibrium point of the nominal system
:1': : A11
and if
Bl(t) —> 0 as t a 00

then the origin is an exponentially stable equilibrium point of the perturbed system

1': A(t)z.

54

 

In order to investigate the stability of (ed = eq = e = O, a = 6) as an asymp-
totically stable equilibrium point for the system (4.39)-(4.42), we proceed with the
calculation of the eigenvalues of the matrix A1, which can be represented in the

singularly perturbed form

 

 

P11 P12
P:
P21+CA1 P'z-z-l-EAz
E 5
where
K o L - T \
_Q,. erﬂ)‘ cf (beard + m) 0
._ . _ T
P11 — -Q§[fm(bwwf+7{,+) —ar 0
’1 ref
_‘:"ref_‘ at
k ref P ref 0 j
T
P12 2 (0 ’PAref 1)
P21 = ( ”Kemp —;LKar/p, —)uKk0)\r(.f )
P22 = (—/J.K/\rcf)
_ _ T ~
5A1 _ e( —X7:—f(bwrcf+7711‘), MAW/L"... 0) ) — 0(6)
5A2 = e( _b)20(e)

The eigenvalues of this matrix can be approximated, for sufﬁciently small 5, by those

of the matrices P11 — Plzngglpm and ng/e [7].

55

It can be veriﬁed that

and

_a" W (baref + 7an ) 0
— ref
P11 - P12952121 = fﬁéf‘; (Mm; + :11?) - Percy 0 pkoAref
t.
0 0 —L0
P22/5 = ‘l‘é‘Aref

and from these we see that there is a fast eigenvalue at —p{f—Aref and a slow eigenvalue

at —A:0. The remaining slow eigenvalues of A1 are the eigenvalues of the 2x2 matrix

L _ T
“07' 01%ij (bwrcf + 7711‘)
A = ref
C _
(1er - T . -
—l"\ref (bwrcf + 7711‘) — Wref 0
,\ + .... 1334mm...) + i...
AI — Ac = _ “ref
_ _ T
Wref'l'ZALwawref'l'Trlf” ’\
re

which gives

 

[AI-Ac=/\<A+ar)+ [pamﬁw boref+%)][§§§§<bbm;+%>]
01‘

'11 — AC

 

= A2 + arA + [parefrc + 3]

56

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

be!
’ “' “ 229?“ 2
Eu _..
—* 0
Observer ids: la
y venue
Hux regulator p1 Contreﬂa Field_ orientation
0 0
l I
it]! 2 id” Vt! L. 2 Vds
. -§
id: 2 tqse V... 2 F
W 3/2 . . . V
, Fcld m: qs
Transformation I _oncn on (0,4 “I”
s... 43» 20434 2 ~
Observer g [45*
Sliding Mode PI antenna
2/3 Conn-one:
Transformation un
Vas I
Vbs Vqs
V“ 3 Vds

 

 

 

Figure 4.3. Sensorless control of an induction motor using sliding mode controller

where

n = 23.514qu + 7,9)
M ref

The eigenvalues of Ac

 

A12 = —a-2L :l: %\/a$ -4[pu-J1.efﬂ+r€2]

57

 

have negative real parts only when the product ”(Wrcf + K.) > 0, which is the case
when the steady-state speed command aref and the steady-state load torque TL both
have the same sign, positive or negative. The same condition was obtained in section

4.2.1, while investigating the stability of the zero dynamics of the system.

A Schematic for sensorless control of an induction motor using sliding mode

control is presented in Figure 4.3.

58

CHAPTER 5

Simulations

The performance of the sliding mode controller is evaluated through simulations in the
following sections. These simulations provide a way to compare the performance of
the previously developed speed control algorithm using the traditional PI controllers

with the sliding mode controller developed in Chapter 4.

5.1 Performance Comparison - PI controller vs

Sliding Mode Controller

In order to examine the performance of the sliding mode controller, the simula-
tions are performed considering the operation of induction motor under different
conditions of parameter variations, load torque and reference speed variations. In
this section, we present the comparison simulations where the performance of the
sliding mode controller is compared with the traditional PI controller for the speed
control of induction motor. For the purpose of this analysis, the induction motor
is operated under similar conditions except that the speed controller is either a PI
controller or a sliding mode controller. The simulation plots for the two controllers

appear side by side representing the similar conditions being taken into consideration.

In order to obtain concrete conclusions about the performance of the two speed
control algorithms, two induction motors, with different power ratings and machine
parameters, have been simulated. However, the simulation results for only one of
the induction motors are presented in this section. Simulation results for the other

induction motor are available in the form of a compact disc.

5.1.1 Designing PI Controllers for the Speed Control

In this section, the PI controller gains are designed for two different induction motors,

nicknamed as IM-l and IM-2.

Induction Motor IM-l

The induction motor with the following machine parameters and ratings is used for

the purpose of simulations:(Taken from [7, Example 56]):

200V, 4 pole, 3-phase, 60 Hz, Y-connected, Base Power 5 hp
R3 = 0.1830, R, = 0.2779, Lm = 0.053811,Ls = 0.055311, L, = 0.056H, m = 0.0165Kg — m2.

A friction coefﬁcient of bl = 0.01K g — m2/sec has been added.

Designing PI controller gains for IM-l

The gains of the PI controllers for IM-l have been taken from [8]. These gains have

been designed while considering the nominal parameters i.e. RT 2 RT and Rs 2 R3.

The constants de and Kd, of the PI controller of vd are chosen using the

model
(1' .
73“ = ‘(0‘877 + QT.BL1n)ld + 711d + d1 = ~121.393id + 276.345’Ud + (11

where d1 is a disturbance input. The choice de = 20 and £211 = 5 assigns the
P

closed—loop poles at —5.643x103 and —0.0049x103. The magnitude of the transfer

60

function from the command input 2'; to id is almost 0 dB over the frequency band
[0,103] rad/sec, and the magnitude of transfer function from d1 to id is less than —30

dB over the frequency band [0,102] rad / sec.

The constants KI!) and K],- of the PI controller of id are chosen using the

model
d/\ - '
71211 = "aTAd + aerzd— —4.946)\d + 0.266%

. K - .
To ensure monotomc response of Ad, Fiﬁ = 5 has been taken to asagn the zero of the
p
PI controller at -5 (almost cancelation of of the pole at —4.946). The gain K fp has

been chosen as Kfp = 20, which assigns the closed-loop poles at —5.13 :t j0.48.

The constants mm and Kg,- of the PI controller of vq are chosen using the

model
(iii? = -(asn + arﬁLm + ar)iq + m, + (12 = —12l.393id + 270.3451), + .12

where (12 is a disturbance input. The choice qu = K q, = 300 assigns the closed-loop
poles at —2.776x104 and —0.995. The magnitude of the transfer function from the
command input 2'; to z’q is almost 0 dB over the frequency band [0,104] rad/sec,
and the magnitude of transfer function from d2 to rig is less than —30 dB over the

frequency band [0,900] rad /sec.

The constants Kwp and Kw,- of the PI controller of iq are chosen using the
model (3.20)-(3.23). The transfer function of the linearization at the equilibrium
point (3.28) is given by

(S) _ 52.4(32+4.9464s+0.887wcﬂl)
(s+0.6){(s+4.9464)‘-’-+(0.887{q)2]+584.416(s+4.9404—o.15912;?)

The transfer function depends on the equilibrium point. By considering a range of

possible values of wref and TL, the controller gains were chosen as K mp = K w,- = 30.

61

Induction Motor IM-2

The induction motor with the following machine parameters and ratings is used for

the purpose of simulations:(Taken from [12]):

220V, 4 pole, 3—phase, 60 Hz, Y-connected, Base Power 3 hp
Base Torque 11.9 N.m. , Base Current 5.8 Ampere, l7107‘pm

R3 = 0.4350, R,- = 0.65319, Lm = 0.0693H, L3 = 0.0694H, Lr = 0.0696H,
m. = 0.0189Kg — m2.

A friction coefﬁcient of bl = 0.01Kg — m2/sec has been added.

Designing PI controller gains for IM-2

The gains of the PI controllers have been designed while considering the nominal
parameters i.e. If, = Br and R3 = R3.

The constants de and Kd, of the PI controller of vd are chosen using the model
d7? = -(asn + cry-{313mm + vvd + d1 = -2715.,-_d + 2508.10d + d1

where d1 is a disturbance input. The choice It'd,- = de = 100 assigns the closed-loop
poles at —2.5x105 and —0.0099x102. The magnitude of the transfer function from
the command input 2'; to id is almost 0 dB over the frequency band [0,104] rad/sec,
and the magnitude of transfer function from d1 to id is less than —30 dB over the

frequency band [0,103] rad /sec.

The constants K fp and Kfi of the PI controller of id are chosen using the

model
dA . .
7,24 _—_ ——a,-/\d + 01.er Id: —9.3836Ad + 065022,,

and the choice K fp = 50 and K f,- = 100 assigns he closed-loop poles at —203.48 and

—0.981. The latter causes an almost cancelation of the zero at —1.00.

62

The constants qu and Kq, of the PI controller of vq are chosen using the

model
$21 = ‘(Osn + 0#1le + ar)iq + ’7'Uq + d2 = -2724.365id + 2508.11,, + (12

where 012 is a disturbance input. The choice mm 2 Kg, = 150 assigns the closed-loop
poles at —4.04x105 and —0.995. The magnitude of the transfer function from the
command input 2'; to iq is almost 0 dB over the frequency band [0,104] rad/sec,
and the magnitude of transfer function from (12 to iq is less than ~30 dB over the

frequency band [0,102] rad /sec.

The constants Km, and Kw,- of the PI controller of iq are chosen using the
model (3.20)-(3.23). The transfer function of the linearization at the equilibrium

point is given by

 

0(8) _ 20.1375(32+9.38363+2.16w.~i}1)
(5+0.1124)[(s+9.3836)2+(2.167612,)2]+174.35(s+9.3836—0.5i;,2)

The transfer function depends on the equilibrium point. By considering a range of

possible values of wrcf and TL, the controller gains were chosen as Kwp = K w,- = 300.

5.1.2 Sliding Mode Controller - Parameters

The PI controller gains have been designed for the two different motors; however,

the same sliding mode controller is used in both cases.

The Sliding mode controller parameter K is deﬁned based on the q-axis cur-
rent requirement for the specific induction motor drive (which is proportional to
the load torque TL). The typical value of z'q for a load torque command of 20
N.m. is approximately 25 Amperes, which suggests that the sliding mode controller

parameter K should be taken around 30 — 35 for a very smooth operation under even

63

unpredictable load conditions. The value of constant gain k0 is chosen as unity. The
signum function of the sliding mode controller is replaced with sat(§) and the value

of 5 is taken as 0.01.

5.1.3 Performance Comparison

In this section, we present the simulation plots for various cases in which the
two controllers are studied for similar situations. The simulation plots appear side

by side for the two controllers for the same situation and for the same induction motor.

The flux observer (3.4) is initiated at

, 0.1
M0) =

0
so that Ad(0) = 0.1. The speed observer (3.16)-(3.17) is implemented with a1 = 2,
a2 = 1, s = 0.0002, [1 = p and I} = b. In the simulations, the components of Us are
limited to iVmax V. The speed reference varies for different situations, however, for
a number of simulation cases, it is taken as a step input smoothed by the transfer

function l/(O.53 + 1).

Figure 5.1 shows simulation results for the two controllers for a speed com-
mand of 100 rad/sec applied at zero time with a load of 20 N .m. applied between
t = 4 and t = 8 see. This simulation is for the nominal parameters case i.e. R} = Br
and Rs = R3. Clearly, both the controllers have equivalent performance curves for

the nominal parameters case.

Figure 5.2 shows simulation results for the two controllers for a speed com-
mand of 100 rad/sec for the case when there is a 100 percent increase in the rotor

resistance R,— and the nominal stator resistance i.e. R, = R3. A load of 20 N .m. is

64

Flux Errors

Sliding Mode Controller PI Controller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.4 . ~ 0.4 . -
502/ 0.2»/
U.

0* . . . . . ‘ 0* 4 . . . . ‘

024s§1012 og§ss1p1
0
0 .
O.
U)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o . - .
0-1Lll __ ed ‘ 0.1L!3 __ ed 4
Ofrw'v eq ‘ 0f,» eq ~
‘0-1’“ . . . . . l -0.1H. , . . . . ‘
,_ 0 2 4 6 8 10 12 0 2 4 6 8 10 12
9 5 . . . . . 5 . . . - .
u“: ll
v W
3 0 [v 0 0.
8
(D _5 . l . . . -5 . 1 . a .
0 2 4 6 8 10 12 0 2 4 6 8 10 12
Time(sec)‘

Figure 5.1. Simulation results with nominal parameters

applied between t = 4 and t = 8 sec.

According to (3.28), the equilibrium values are ed 2 ed 2 0, and

where

._ dw-O L T r7
w - wref = Ly—p—XZTm—q = —O.443ozq

bwref'l'TL/m
2q

_ 60.6061(1+TL2
t) .)r

”Aref—

 

b Cir—Or L171
1) ref

65

For TL = 20, we obtain 2,, = 24.164. The simulation results conﬁrm these calculations.

The two controllers present almost equivalent performance for such parameter

variation.

Figure 5.3 repeats the simulation for the two controllers for the speed com-

 

 

 

 

 

 

 

 

 

 

 

Sliding Mode Controller PI Controller
0.4 . . . . . o_4 . . . . .
12‘ 0.2 F « 0.2 /y
LL
0» . 4 0 ' . .
0 2 4 6 8 10 12 0 2 4 6 8 10 12

 

 

Speed
0'!
o

100’..... 100*.....
M SOFN—P—f

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g 0.11~ -——ed ‘ 011»

'5 o r_\ i _ _ J 0 l . 2 ._ i.

X I: .. eq '— Jlﬁ - 1 V v

g-m - - . 4 -0.1» - - «
50 2 4 6 6 10 12 50 2 4 6 8 10 12

 

 

 

 

Speed Error
l

O" 0
[\j l
I

0'! O
[2:

 

 

Tlme (sec) Time (see)

Figure 5.2. Simulation results with 100 percent increase in Rr and nominal R3

mand of 100 rad / see, with a 100 percent increase in both the rotor resistance RT and

66

the stator resistance 12,... In this case, we expect the equilibrium point to be slightly
perturbed from the one we obtained in Figure 5.1.3. This is indeed the case. Clearly,
in this case ed and eq are no longer zero. The sliding mode controller has a slight
performance edge in this parameter variation case as it produces less speed error

when the induction motor is subjected to the load.

Figure 5.4 shows speed reversal from 50 to -50 rad/sec at no load, with nomi-

 

 

 

 

 

 

 

 

 

 

 

Sllding Mode Controller Pl Controller
0.4 . . . . r . . . .
x 0.41
5 “f 0.2K
0 . 0’ .
0 2 4 6 8 10 12 0 2 4 6 8 10 12

0
o p q D
Q
U)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flux Errors
o .
O
b ' 1
I
l
D | -4
r
r
l
T
O
I
l
l
I
I

Speed Error
I
a—b
O O
I
—-L .
O O

0 EL o

 

 

 

 

 

 

 

 

Time (sec) Time (sec)

Figure 5.3. Simulation results with 100 percent increase in RT and Rs

67

nal parameters. It can be seen that the system settles back at the equilibrium point
following a transient during speed reversal. The two controllers have almost the same
performance for speed reversal from 50 to -50 rad/sec. The sliding mode controller
has a slight edge as it produces less ﬂux estimation error during the transient. The
important factor that limits the performance is the control saturation. For both
controllers, reversing the speed from 100 to -100 rad / sec causes a control saturation.
During the control saturation the variables oscillate and the performance degrades

signiﬁcantly.

Figure 5.5 shows the simulation results for the case when the induction motor
is subjected to a step load torque TL=20 N .m. at time t = 4 seconds. This can be
viewed as a case when a sudden load torque command is applied while the induction
motor is operating under no load conditions before. Such cases can arise in practice
under different drive fault conditions. It is clear from the simulation plots that the

sliding mode controller has an edge over the PI controller in such situation.

Figure 5.6 supplements the results of Figure 5.5. This figure shows the simu-
lation results for the case when the induction motor is subjected to a sudden step
load torque TL of a higher value for a short period of time after the induction motor

has been subjected to TL=2O N.m. at t: 4 seconds.

This case represents a fault condition in that the drive load rises higher than
the nominal value for which the drive is designed. Such cases can arise in practice
when two or more induction motors operate in parallel in a drive system and one or
more of them suddenly go out of operation, resulting in a sudden load rise on the
remaining machines within the drive system. It is clear from the simulation plots

that the sliding mode controller outperforms the PI controller in such situation.

68

 

 

Sliding Mode Controller Pl Controller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.4 a 0.4 . - . - .
r
30.2f - 0.27 «
LL
0r . . - . . ‘ 0 .' - . . . ‘
1
.000 ? 4 . 9 1.0 12 .000 2 4 6 9 10 2
8
g on of—L—
(D
_100 . . 4 . . _100 . I . . 1
024681012 0????‘912
20» + 20
.310 10 4
0 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m . r
t l\ J\ A
m or: 4 - ~ eq ~ 0V;— \Y
X l
5 -0.1~ , -01. 4
0 2 4 6 8 10 12 0 2 4 6 8 10 12

 

 

 

 

 

 

 

 

Speed Error
0
l
l
O
l

 

 

Time (see)

Figure 5.4. Simulation results for speed reversal under no load conditions with nom-
inal parameters

Figure 5.7 demonstrates the importance of the condition 012ch obtained in Chapter 3
and the similar condition 14(er f + n) > 0 obtained in Chapter 4. This condition is
satisﬁed for all the cases discussed so far. In this case, we apply a speed command of
10 rad / sec at zero time. Then, at time t = 4, we apply a load of —1 N.m. For these val-
ues, {q = -—1.1564 and we 2 18.9758 ; hence, 011ch < 0. Similarly, the term M10540; + n) > 0

is violated in that for these values it turns out that 8(pu‘ircf + x) = —18.958 )6 0. It is

69

Sliding Mode Controller Pl Controller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.4 . f . . 4 . .
x J . 0.4» A , , N-
6.2 0.2K 0.2;F \JNN’N
0L . . . . . ‘ 0+ . ‘ I - . . 1
0 2 4 8 8 10 12 0 2 4 8 8 110 12
100’ .
,, «10¢ 118114
g 50. 4 O i
w 0 g -100» 1
0 2 4 6 8 110 12 0 2 4 6 8 10 12
20 ————7 20 i
1.4 10» [ + 10 ,
0 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Jo).

Flux Errors
0
o ...
i
1
2
8 a
O
O 01
__..ﬁ‘ %

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-0.1 1' ’0'? 4,2”‘2'6'1
_ O 2 4 6 8 10 12 0 2 6 8 10 12
o 5
t: 50.
w k ‘ r
g 0 ——v v o .
8 -50»
01—5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
T1me(sec)

Figure 5.5. Simulation results for step load torque TL with nominal parameters

clear from the simulation that the equilibrium has been destabilized after applying
the load. However, notice that the sliding mode controller recovers the performance
during the transient and eventually attains zero steady-state error. This simulation
shows the edge the sliding mode controller has over the PI controller for its good
performance against un-modeled dynamics, external disturbance rejection and fast.

dynamic response.

70

Sliding Mode Controller Pl Controller

 

 

 

 

 

 

 

 

 

 

 

 

0.4 e - a . . . - .
4.4 0.4’
x r , J—LUuWMJ
5 0fl 02F 4
0t . . . . . ‘ 0 . '
0 2 4 6 8 10 12 0 2 L 6 8 10 12
.8 100’ 150’ g
a 50 100t
(0 50+ .
0 ‘ 0
O 2 4 6 8 10 12 0 2 4 6 8 LO 12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 0.1 v . 1

2 l — ed

iii 0 4J- - eq ..

x .‘

.3. 1

“L -0.1 L
,_ 5O 2 4 6 8 10 12 2 4 6 8 1O 12
‘13
LL!
4:
a
(I) . . .

 

 

 

Time (sec) Time (sec)

Figure 5.6. Simulation results for sudden step load torque increase with nominal
parameters

Figure 5.8 shows the simulation results for the case when the induction motor
is subjected to a step reference speed command of 75 rad / sec at time t = 1 seconds.
The sliding mode controller obediently tracks the reference speed after a short
transient period in which the trajectories outside the boundary layer reach boundary
layer and move inside it toward the equilibrium point, giving a zero steady-state
error. The PI controller struggles through and the performance deteriorates with the

passing seconds.

71

Sliding Mode Controller PI Controller
0.4 . . 4 - . . . .

 

 

*ﬁ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flux Errors
o N

 

 

 

 

 

 

 

N-h
CO

 

 

 

 

Speed Error

 

 

 

o
N
h
a)
m r
A
o
—I
N

Time (sec) Time (see)

Figure 5.7. Simulation results demonstrating loss of stability in case of PI controller
when under negative load torque with nominal parameters

Figure 5.9 presents the case when the induction motor has already attained a
reference speed of 100 rad /sec. This represents the initial state of the system. Then,
a negative speed step command of -15 rad/sec is applied. During the transient,
the sliding mode controller recovers the performance while the PI controller loses
stability and shows an increasing deterioration in performance. Notice that, the

sliding mode controller also shows the speed error during the transient when the

72

 

 

Sliding Mode Controller Pl Controller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 . . . . . 100
'76
l!
g 50 1 50* 1
o
a.
U) 0 . 1 0 .
0 2 4 8 10 12 0 4 6 8 10 12
100 . f . . 4 100
1: -
8 50» . 50’
d.
U)
0 J .
0 2 4 6 8 10 12
'5 5° '
E 0
u r
o
o
(1
(D J .
4 6 8 10 12
E Y r ——'ed
8 _ eq‘ 1
LU 0- V1 W7 \ﬁj‘Tﬁ‘ﬁﬁﬁ—ﬂ
g \l " l "\ ’1 ’1
E .
'02 + -1 4 ‘

 

Time (sec)

 

Sliding Surface
.4;
O

 

 

 

 

-40 .
.60 . 1
'80 L . L
0 2 4 6 8 10 12
Time (sec)

Figure 5.8. Simulation results for step speed command with nominal parameters

negative speed step is applied. This is due to the presence of the PI controller in

the closed-loop that outputs the stator voltage as feed back to the induction motor.

This PI controller

Vq: (qus+k’i)§[ [1* _ [q]

regulates z'q to 2'; and outputs the q-axis stator voltage as feedback in the closed-loop.
Therefore, an error presented by this PI controller has a direct effect on the

performance of the sliding mode controller.

73

Sliding Mode Controller

 

_a
o
C

Speed Ref
:3 8

 

Pl Controller

0
N
#

4 a) l.
m
A
o
—L
N

 

 

150

' 111111111111

—5
o
O

0

Speed
0 8

 

 

O
N
4 # >
O)
on
u—I
N
N

 

 

N0)
0O

 

 

 

 

Speed Error
8
O

 

o
l
01
CL

 

 

J.
o
o
‘2
i

2 4 6 8 1O 12 0 2 4 6 8 10 12

T7

 

.0

N
4

N

 

 

 

 

 

 

Flux Errors
0
i
,
1
ill
8

 

 

 

 

 

 

 

I
P
N

2 4 6 8 10 12 0 2 4 6 8 10 12
' ' r 1 m , Time(sec)

 

O

 

Sliding Surface
1. _. N
O O O

 

 

 

o
N
b
a,
m h
—K
o
d
N

Time (see)

Figure 5.9. Simulation results for multi-step speed reduction with nominal parameters

Figure 5.10 supplements the results of the Figure. 5.9. This simulation also
presents the case when the induction motor has already attained a reference speed
of 100 rad/sec. Therefore the system is considered to have initial condition of
operating at 100 rad/sec. Then, a monotonically decreasing speed reference is
applied in such a way that the system attains the desired speed, say, 60 rad /sec at

some pre-deﬁned time. The simulation results show that the sliding mode controller

74

Sliding Mode Controller

 

v

_s
o
o

 

Speed Ref
0 8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2 4 6 8 10 PI Controller
1001 ' Y ' ' 150’ ' ' . .
f2 “WWW
8 50’ r i ll
0[ . . 0’ - , J , ‘
0 2 4 6 8 10 0 2 4 6 8 10
,_ 5 . . - . - . . .
E l 50» .
... [, l
" W ° 1 ‘1 l
il
01__5? . . . . ‘50’ . . . . l
0 2 4 6 8 10 0 2 4 6 8 10
0.1 . . - . 2 . . . 4
ii .1 ”Ed
2: 011‘“ ‘ ‘2 0 :11141141114
2 [Illil\'ll‘|)‘i1)|fl;,j,l
61L 1
_01 . . . - _2 . . . .
2 4 6 8 10 0 2 4 6 8 10
8 0'01: ' ' ' ' Time (sec)
t l
3 1
(D 0 2
or [f
g
E
a

 

 

 

0011144
0246810

Time (sec)

Figure 5.10. Simulation results for monotonic speed reduction command with nominal
parameters

smoothly tracks the reference speed in this case while the performance of the PI

controller deteriorates.

75

CHAPTER6

Conclusions

Sensorless speed control of induction motors using ﬂux and speed observers is an
emerging technology, which is pushed forward by the need to develop low-cost,
dependable drive systems. The induction motor presents a complex control problem
due to its nonlinear dynamics and dependence upon parametric variations. In order
to develop efﬁcient sensorless induction motor drive systems, it is necessary to take

into account the nonlinearities of the system.

A speed control algorithm based on sliding mode control strategy is presented
in this work. This nonlinear controller replaces the traditional PI controller used for
similar purposes. Analysis reveals the conditions under which the developed sliding
mode controller provides effective speed control, while preserving the closed-loop
system stability under uncertain external load disturbances and reference speed
variations. A performance comparison presents the edge that the developed sliding

mode controller has over the traditional PI controller.

Some aspects of the analysis of the sliding mode control has been restricted
to local regions. A more elaborate nonlocal analysis would certainly provide better

insight into the control problem, however, pursuing the same is not very clear.

76

In particular, performing nonlinear analysis requires the existence of a Lyapunov

function, which is not transparent at this point.

Human nature has a tremendous drive to seek for the best. This work is a
small step ahead of an earlier work [8], and there is a long way to go. The theoretical
analysis has been supplemented by simulation results and the next step would be

obtaining the experimental results.

77

BIBLIOGRAPHY

78

BIBLIOGRAPHY

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[2] B. Aloliwi L. Laubinger E. G. Strangas, H. K. Khalil and J. Miller. Robust track-
ing controllers for induction motors without rotor position sensor: Analysis and
experimental results. IEEE Trans. Energy Conversion, (14):1448-1458, 1999.

[3] E.J.Davison. The robust control of a servomeehanism problem for linear time-
invariant multivariable systems. IEEE Trans. Aatomat. Contra, 1(AC—21):25-—34,
1976.

[4] J. Holtz. Sensorless position control of induction motors-an emerging technology.
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[5] A. Isidori. Nonlinear Control Systems. Springer, 3rd edition, 1995.

[6] A. Kawamura K. Rajashekara and K. Matsuse, editors. Sensorless Control of
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[7] H. K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, New Jersey,
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[8] H. K. Khalil and E. G. Strangas. Sensorless speed control of induction motors.

In Proc. American Control Conference, 2004.

[9] R. Krishnan. Electric Motor Drives. Prentice Hall, Upper Saddle River, New
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[10] W. Leonhard. Control of Electrical Drives. Springer, 211d edition, 1996.

[11] R. D. Lorenz. Advances in electric drive control. International Conference on
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[12] O. Wasynczuk P. C. Krause and S. D. Sudhoff. Analysis of Electrical Machinery
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[13] H. K. Khalil P. V. Kokotovié and J. O’Reilly. Singular Perturbation Methods in
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[14] P. Vas. Sensorless Vector and Direct Torque Control. Oxford University Press,
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[15] G. C. Verghese and S. R. Sanders. Observers for flux estimation in induction
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80

   

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