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LIBRARY 7.

Michigan State
University I

 

 

This is to certify that the

thesis entitled

ENCODERLESS STATOR FIELD-ORIENTATED CONTROL OF
INDUCTION MACHINES

presented by

MOHAMMAD B I LAL MALI K

has been accepted towards fulfillment
of the requirements for

M.$. degree in EleCtrica] &
Computer Engineering

 

Aim/{A

Major pro

Date April 23, 2001

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
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6/01 cJCIRC/DateDuepes-p. 15

 

ENCODERLESS STATOR FIELD-ORIENTATED CONTROL OF

INDUCTION MACHINES

By

MOHAMMAD BILAL MALIK

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

2001

ABSTRACT
ENCODERLESS STATOR FlELD—ORIENTATED CONTROL OF
INDUCTION MACHINES
By
MOHAMMAD BILAL MALIK

Control of induction machines without using speed or position sensors is an
attractive and challenging problem. Stator flux-orientated control is one of the
schemes proposed in this context, which has its advantages and limitations. Our
work concentrates on the same and includes all the necessary ingredients,
namely, analysis, simulation and experiments. Our emphasis is on the speed
control problem, which naturally includes the torque control.

The analysis begins with a discussion on pure integrators, which play a vital role
in stator-based techniques. Next comes the analysis of steady-state errors. This
analysis is valuable in the sense that it not only gives quantitative error estimates
but also is useful in evaluating the limitations of the control scheme. There is a
handy discussion on noninteracting control i.e. the ability to control speed (or
torque) and flux independent of each other.

A series of simulations have been provided to verify and illustrate the analysis.
This kind of approach gives a better perspective and understanding of the
profound details of a problem. We have used some new and novel ideas to
develop an experimental setup. This includes the use Of a personal computer as
an embedded processor. The flexibility, power and convenience to perform new

experiments produced results that closely match analysis and simulations.

In the name of Allah, the most Merciful and the most Beneficent

To

My parents

ACKNOWLEDGEMENTS

I am grateful and indebted to my advisor Dr. Hassan Khalil for his kind guidance,
assistance, encouragement and valuable time. Working with him has been a
great learning experience in the field of nonlinear controls.

I also enjoyed working with Dr. Elias Strangas. His keen interest, support and
useful advice helped a lot in improving my understanding of machines and
constructing the experimental setup.

I appreciate and thank to all others who helped directly or indirectly in the
completion of this thesis.

I am also thankful to National University of Science and Technology, Pakistan for

supporting my study for the degree of Master of Science.

iv

TABLE OF CONTENTS

LIST OF FIGURES ............................................................................................. IX
LIST OF TABLES ............................................................................................. XIII
1 INTRODUCTION ................................................................................................ 1

1.1. Control of Induction Machine; A Historical Perspective .......................... 1

1.1.1. Control of Induction Machine using Position/Speed Measurements3

1.1.2. Control of Induction Machine without Position/Speed

Measurements ............................................................................................... 4
1.2. Field-Orientated Control and its Variants ................................................ 5
1.3. Our Contribution ..................................................................................... 6

1 .3.1 . Analysis ........................................................................................... 7

1.3.2. Experimental Setup ......................................................................... 8

2 MATHEMATICAL MODELING AND ANALYSIS ............................................ 10
2.1. Three-Phase Symmetrical Induction Machine ...................................... 10

2.1.1. Stator Based Machine Model ......................................................... 14

2.1.2. Second Order Effects .................................................................... 16
2.2. Stator Flux Estimation ........................................................................... 17

2.2.1 . Pure Integrators ............................................................................. 17

2.2.2. Current Components ..................................................................... 19

2.2.3. Open Loop and Closed Loop Integrators ....................................... 21

2.2.4. Approximation of an Integrator ...................................................... 23

2.2.5. Equivalent Errors in Polar Coordinates .......................................... 28

2.2.6. Remarks on Flux Estimation .......................................................... 33
2.3. Control of Induction Machine Using Speed Measurements .................. 34
2.3.1. The Control Variables .................................................................... 35

2.3.2. Coupled Nature of Direct and Quadrature Current Components...36

2.3.3. Flux Controller and Decoupling Through Feed FonNard ................ 36
2.3.4. Torque and Speed Controller ........................................................ 41
2.3.5. Position Measurement ................................................................... 41
2.3.6. High Gain Observers ..................................................................... 42
2.4. Speed Estimation and Error Bounds ..................................................... 44
2.5. Control of Induction Machine Using Speed Estimation ......................... 47
2.5.1. Low and High Speed Operation ..................................................... 49
2.5.2. Starting Of the induction machine .................................................. 50
2.5.3. Speed Reversal ............................................................................. 50
2.6. Lowest Possible Speeds ....................................................................... 51
2.7. Discrete-Time Implementation .............................................................. 53
2.7.1. Discrete-Time Control of Continuous-Time Nonlinear Systems ..... 53
2.7.2. Difficulties Associated with High Sampling Rates .......................... 54
2.7.3. Suggested Solutions ...................................................................... 57

3 COMPUTER SIMULATIONS ........................................................................... 58
3.1. Importance of Simulations in Design .................................................... 58
3.2. Pure Integration .................................................................................... 61
3.2.1. Equivalence of Open and Closed Loop Integration ....................... 61
3.2.2. Unknown Initial Conditions ............................................................ 66

vi

3.2.3. Transient Behavior ........................................................................ 68

3.3. Stator Flux and Speed Estimation ........................................................ 70
3.3.1. Integrator Approximation ............................................................... 72
Observations ................................................................................................... 72
3.3.2. Uncertainty in Stator and Rotor Resistances ................................. 77
Observations ................................................................................................... 77
3.4. Closed Loop System with Feedback from Measured Speed ................ 84
3.4.1. Controller Performance in Presence of Errors ............................... 86
3.4.2. Operation at different speeds and loads ........................................ 91
3.4.3. The Role of Decoupler ................................................................... 94
3.5. Closed Loop System with Feedback from Estimated Speed ................ 96
3.5.1. Controller Performance in Presence of Errors ............................... 98
Observations ................................................................................................... 98
3.5.2. Operation at Different Speeds and Loads .................................... 104

4 EXPERIMENTAL SETUP AND RESULTS .................................................... 108
4.1. Introduction ......................................................................................... 108
4.2. The Experimental Setup ..................................................................... 109
4.3. Personal Computer as Embedded Processor ..................................... 111
4.3.1. Computational Power .................................................................. 1 1 1
4.3.2. Memory Resources ...................................................................... 112
4.3.3. Available Data Types ................................................................... 112
4.3.4. Mass Data Storage ...................................................................... 112
4.3.5. Simulating Real Data ................................................................... 113

vii

4.3.6. Development Environment .......................................................... 113

4.3.7. Flexibility ...................................................................................... 113
4.3.8. Summary ..................................................................................... 114
4.4. RTLinux Philosophy ............................................................................ 114
4.5. Software Planning ............................................................................... 115
4.5.1. Hierarchy of the real time task ..................................................... 116
4.6. Digital IO Card .................................................................................... 118
4.7. Machine Control Card ......................................................................... 118
4.7.1. Analog to Digital Conversion ....................................................... 119
4.7.2. 4-Channel Digital to Analog Converter ........................................ 121
4.7.3. PWM Generation ......................................................................... 122
4.7.4. Shaft Encoder Interface ............................................................... 122
4.8. Open Loop Experiment ....................................................................... 123
4.9. Encoderless Speed Control of Induction Machine .............................. 126
Observations ................................................................................................. 127
5 CONCLUSIONS AND RECOMMENDATIONS .............................................. 130
5.1. Conclusions ........................................................................................ 130
5.2. Recommendations .............................................................................. 132
BIBLIOGRAPHY ............................................................................................... 134

viii

LIST OF FIGURES

Figure 1-1. Block diagram of control of induction machines using speed/position

measurements ............................................................................................... 4
Figure 1-2. Block diagram of encoderless control of induction machines .............. 5
Figure 2-1. Stator Flux Estimation in Polar Coordinates ..................................... 22

Figure 2-2. Frequency domain representation of disturbance with integrator
approximated by first order filter ................................................................... 25
Figure 23. Frequency domain representation of disturbance with integrator
approximated by second order filter ............................................................. 27
Figure 2-4. Error propagation from rectangular to polar coordinates .................. 31
Figure 2-5. Stator field-orientated control of induction machines using speed
measurements ............................................................................................. 40
Figure 2-6. Transfer functions of different high gain Observers ........................... 43

Figure 2-7. Encoderless stator field-orientated control of induction machines ....48

Figure 2-8. A possible scheme to handle high sampling rates ............................ 57
Figure 3-1. Comparison of open and Closed loop estimators .............................. 62
Figure 3-2. Closed loop stator flux estimation ..................................................... 62
Figure 3-3. Open loop stator flux estimation ....................................................... 63

Figure 3-4. Equivalence of closed and Open loop integrators for DC disturbances
..................................................................................................................... 64
Figure 3-5. Equivalence of Closed and open loop integrators for low frequency

disturbances ................................................................................................. 65

Figure 3-6. Stator flux estimation with pure integrator and false initial conditions
..................................................................................................................... 66
Figure 3-7. Response of a pure integrator with false initial conditions ................ 67
Figure 3-8. Investigation of transient behavior of stator flux estimator using pure
integrators in presence of parameter uncertainty ......................................... 68
Figure 3-9. Sensitivity of pure integrators to transient Offsets ............................. 69
Figure 3-10. Block diagram of stator flux and speed estimation in open loop ..... 70
Figure 3-11. Errors due to integrator approximation at high speeds (Part I) ....... 73
Figure 3-12. Errors due to integrator approximation at high speeds (Part II) ...... 74
Figure 3-13. Errors due to integrator approximation at low speeds (Part I) ......... 75
Figure 3-14. Errors due to integrator approximation at low speeds (Part II) ........ 76
Figure 3—15. Errors due to integrator approximation and uncertainty in stator
resistance at high speeds (Part I) ................................................................ 78
Figure 3-16. Errors due to integrator approximation and uncertainty in stator
resistance at high speeds (Part II) ............................................................... 79
Figure 3-17. Errors due to integrator approximation and uncertainty in stator
resistance at low speeds (Part I) .................................................................. 80
Figure 3-18. Errors due to integrator approximation and uncertainty in stator
resistance at low speeds (Part II) ................................................................. 81
Figure 3-19. Errors due to integrator approximation and uncertainty in rotor
resistance at high speeds ............................................................................ 82
Figure 3-20. Errors due to integrator approximation and uncertainty in rotor

resistance at low speeds .............................................................................. 83

Figure 3-21. Controller performance under ideal conditions using measured

speed ........................................................................................................... 85

Figure 3-22. Controller performance with integrator approximation (m, =1 rads/s )
using measured speed ................................................................................. 87

Figure 3-23. Controller performance with integrator approximation (we =1rads/s)
and Rm =1.05R, as an uncertainty using measured speed. .......................... 88

Figure 3-24. Controller performance with integrator approximation (wc =1rads/s)
and Rm =1.25R, as an uncertainty using measured speed. .......................... 89

Figure 3-25. Controller performance with integrator approximation
(we =1 rads/s ), Rm =1.05R, and Rm = 1.1R, as uncertainties using measured
speed. .......................................................................................................... 90
Figure 3-26. Controller performance at various speeds using speed
measurements ............................................................................................. 92
Figure 3-27. Controller Performance for different loads using measured speed .93
Figure 3-28. Controller performance without a decoupler ................................... 95

Figure 3-29. Controller performance under ideal conditions ............................... 97

Figure 3-30. Controller performance with integrator approximation (wt =1rads/s )

..................................................................................................................... 99

Figure 3-31. Controller performance with integrator approximation (a), = l rads/s)
and Rm =1.05R, as an uncertainty ............................................................... 100
Figure 3-32. Controller performance with integrator approximation (we =1rads/s)

and Rm = 1.25R, as an uncertainty ............................................................... 101

xi

Figure 3-33. Controller performance with error due to HGO ............................. 102

Figure 3-34. Controller performance with integrator approximation
(wt =1rads/s),Rw =1.05R, and Rm = 1.1R, as uncertainties and error due to

HGO. .......................................................................................................... 103

Figure 3-35. Controller performance at various speeds (Encoderless Case)....106

Figure 3-36. Controller performance at various loads (Encoderless Case) ....... 107
Figure 4-1. Block diagram of the experimental setup ........................................ 110
Figure 4-2. Software plan .................................................................................. 116
Figure 4-3. Hierarchy of real-time task .............................................................. 117
Figure 4-4. Block diagram of machine control card ........................................... 119
Figure 4-5. Analog to digital conversion ............................................................ 120
Figure 4-6. Anti-aliasing filter (3rd order low pass Butterworth) .......................... 121
Figure 4-7. Block diagram of digital to analog conversion ................................. 121
Figure 4-8. Block diagram of PWM generation ................................................. 122
Figure 4-9. Open Loop Estimation (Part I) ........................................................ 124
Figure 4-10. Open Loop Estimation (Part II) ..................................................... 125

Figure 4-11. Experimental results of closed loop system without using

speed/position sensors .............................................................................. 129

xii

LIST OF TABLES

Table 31. Machine parameters used in simulations ........................................... 60

Table 4-1. Parameters of machine used in actual experiment .......................... 126

xiii

CHAPTER 1

INTRODUCTION

1.1. Control of Induction Machine; A Historical Perspective

The induction machine is called the ‘workhorse’ of industry. This term carries a
deep historical background of over a century. Induction machines are known for
their simplicity, ruggedness and reliability. There are no magnets and hence no
chance of demagnetization. Their construction is suitable for volume
manufacturing. They give good efficiency whilst being compact. Besides all these
advantages, they have a lower cost. Amongst the variants of induction machine,
squirrel-cage induction machine is the most popular. One of the possible
methods to manufacture squirrel-cage induction machines is to die-cast the
aluminum rotor. Laminated steel is used as the insulator owing to its much higher
resistivity. This type of rotors can work under very harsh environments and up to
temperatures below the melting point of aluminum. We have used an induction
machine of the same construction. These features make induction machines an
attractive choice over other Options.

The most common application of induction machines is in the fixed-speed drives.
A fixed voltage and fixed frequency signal is applied. Although induction machine
is not a synchronous machine, within wide operating range the slip is small and
the machine runs near synchronism. This behavior is independent of the load

within permissible limits.

Running the machine at fixed speed naturally limits its applications and scope.
Development in the field of variable-speed drives and power electronics helped
extending the use of induction machine at variable speeds. One common
example is the use Of variable frequency variable voltage 3-phase inverters as
the power source for induction machine. This method is still popular in variable
speed drives using induction machine.

The earlier variable-speed drive techniques only provided a good steady-state
solution. The idea Of replacing DC machines with induction machines was not
feasible in applications requiring high dynamic performance. Whereas, keeping in
view the qualities of induction machine, it was highly desirable to use it in high
performance applications such as servornechanisms. This desire of industrialists
and design engineers did not come true until the concept of field orientation was
first proposed by Blaschke [2].

The dynamic model of an induction machine can be represented by a sixth-order
state space model. The non-linear dynamic structure with interactions requires
sophisticated and complex control schemes. The cost of many Of these schemes
was initially prohibitive. The developments in the field Of VLSI and micro
computing alleviated these difficulties and now a days we have the capability to
implement schemes of our choice. The contemporary development in power
electronics also reduced the cost and improved the performance and reliability Of
inverters. These inverters are one of the basic ingredients of modern control

schemes of induction machines.

1.1.1. Control of Induction Machine using Position/Speed Measurements

The first step to control the speed of induction motor is to use speed
measurements in the feedback control loop. From the point of view of machine,
we do not directly control the speed. Rather the torque is controlled whilst
maintaining the flux at a desired level. In many applications the desired flux is
constant and the problem is essentially a torque control problem. The concept of
field-orientated control makes it possible to decouple the torque control from the
flux. If we are successful in our goal, we achieve the same or at least comparable
dynamic performance as in the case of DC machines. The overall concept is
analogous to control of field and armature currents in DC machines.

Measuring flux faces some practical difficulties. It is therefore, advantageous to
employ a flux estimation scheme based on terminal voltage and current
measurements. The motor is either fed power either through a voltage source
inverter or a current source inverter. Both the methods have their own prose and
cons, but the objective remains the same. Our work concentrates on voltage-fed
induction machines.

In summary the controller uses flux estimates and speed measurements for
controlling the speed of the machine. In certain applications the requirement
could actually be to control the torque rather than speed. However, our work
concentrates on speed control problem. A simplified block diagram is illustrated

in figure 1-1.

 

Desired Speed AWE" Speed

 

 

 

 

 

 

 

 

 

 

 

Power Input to
°——’ Control the machine . Induction 9
o- _______ , Scheme Machine
DeSIred Flux Currents
& Voltages

 

 

 

 

 

 

 

Figure 1-1. Block diagram of control Of induction machines using speed/position measurements

1.1.2. Control of Induction Machine without Position/Speed Measurements

Once the basic problem of controlling induction machines was solved and many
control schemes had evolved, the attention of the design engineers and
researchers turned towards a more challenging problem. The idea is to omit the
position/speed sensor and develop a method to estimate the speed from terminal
voltages and currents. Mechanical sensors are Often undesirable because Of
space and mounting restrictions. On one hand, the system becomes costly
whereas ruggedness of the mechanical sensors is questionable as well. These
factors have a greater importance for smaller machines.

Comparing with the previous scheme, now we have to estimate both the flux and
speed. The actual speed is no longer a part of the feedback control loop. The
obvious consequence is that we have to incorporate more sophisticated
algorithms that are computationally extensive and require deeper analysis.

Another factor that is not immediately apparent is the loss of some robustness
4

properties. In this context, many schemes are available but all of them face
certain difficulties and none Of them is functional in the complete operating range
of induction machine. The hardest part is to control the machine at very low or
zero speeds. A general solution of the problem is still unknown and the quest
confinues.

A simplified block diagram of the encoderless control Of induction machines is

illustrated in figure 1-2.

 

 

 

 

 

 

 

 

 

 

 

 

De ' ed d Act al Speed
3" Spee Power Input to u
0—“ Control the machine ’ Induction e
o- ....... , Scheme Machine
DeSIred Flux Currents
& Voltages

 

 

 

 

 

Flgure 1-2. Block diagram of encoderless control of induction machines

1.2. FIeld-Orlentated Control and its Variants

Torque producing capability of machines comes from the interaction of flux with
currents. This phenomenon is governed by well-known laws Of electro-
magnetism. In an induction machine there are no permanent magnets and the
flux itself is produced by a component of current. Furthermore current is flowing

in both the stator as well as rotor windings. The two currents are coupled through

the process of electromagnetic induction. ln-between the stator and rotor is a
small air-gap. In the simplest of divisions, we have three possible types of flux
that CO-exist in induction machine. These are the stator, rotor and air-gap fluxes.
The concept of field orientation is to transform the state variables in such a way
that all the new variables are defined with respect to any one of the fluxes. We
are not redefining or altering the machine model. It is just represented with a
different set of variables. Such transformations that do no change the inherent
properties of a model are referred to as diffeomorphism in control theory. The
advantage of this procedure is that it may give us additional knowledge and
better understanding about the model. This helps in developing control schemes
effectively. Since the measurements are from the physical quantities, field-
orientated control schemes make use of both the actual and transformed
variables.

Corresponding to three different fluxes, we can have stator-, rotor- or air-gap- flux
orientated control. All of these methods have their advantages and demerits.
Depending on the application and requirements one may be superior to the
others. Details Of these methods and other possible variations can be found in

Leonhard [12], Novotony and Lipo [15] and Vas [21].

1 .3. Our Contribution

We would be working on the stator based field-orientated encoderless control of .
induction machine. The main motivation comes from a scheme proposed by
Leonhard [12]. We have modified the actual proposal to either gain certain

benefits or for the convenience of implementation. The same or similar schemes

6

have been used for many years in the research area and industry. A lot of
relevant literature is available both in textbooks as well as research papers. Our

contribution to this important field comes in two forms.

1.3.1. Analysis

A number of control systems and techniques come into existence as a result of
extensive experimentation. Some of these control schemes are actually
employed for mass production without in-depth analysis. However, this approach
prohibits getting the best out of a control scheme. Unfortunately, performing
analysis is a difficult problem in the case Of non-linear systems like induction
machine. One can find many qualitative statements in the literature that are
based on experience and skill of the design engineers. Whereas, this information
is valuable, it becomes a great help if we can deduce quantitative results in the
form of mathematical equations. Naturally it is not possible in a single effort to
resolve or analyze all the details and problems related to an elaborate control
scheme. We have tried to do analysis relating to many important issues.

TO begin with some useful work has been done on pure integrators. Integration
plays a vital role in stator-based techniques. Next comes the analysis Of steady-
state errors. These errors appear either due to parameter uncertainty or other
limitations involved in the overall scheme. This analysis is valuable in the sense
that it not only gives quantitative error estimates but also is very useful in
evaluating the limitations of the control scheme. Prior information about the valid
operating region of a control scheme is vital. There is a brief discussion on high-

gain observers, which are becoming integral part of many estimation schemes.

7

Stator-based methods require some form of decoupler to fulfill the requirements
Of noninteracting control i.e. the ability to control speed (or torque) and flux
independent of each other. There is a handy discussion on this topic. Some of
the topics have been discussed without mathematical analysis but are based on
common knowledge and logical reasoning.

A series of simulations have been provided to verify and illustrate the analysis.
This kind of approach gives a better perspective and understanding of the
profound details of a problem. Some of the results that are not obvious become
conspicuous after the combination of simulations and analysis.

A final remark is that our work is a contribution and enhancement to a well-known
control scheme. It is nothing radically new but should provide to be a useful tool
in control Of induction machines. We would say that a convenient base is ready

for a design engineer to perform actual experiments.

1 .3.2. Experimental Setup

Development of a control scheme for machines cannot be considered to be
complete without actual implementation. Most of the work in this context has
utilized power of today’s micro controllers and advanced electronics. Very recent
development in real time operating systems has added a new dimension to micro
computing. A personal computer can now be used as an embedded processor.
The computational power, system resources and flexibility Of a personal
computer can never be matched by an embedded technology. We availed this
technological breakthrough and have developed a new experimental setup with a

PC as its microcomputer. We have developed our own machine interface card

8

that has added to flexibility and convenience to perform new experiments.
Practical limitations are reduced and we have freedom to do certain tasks that
are not thought off in many other methods of implementation. We consider this

setup to be an important contribution to the field of control of induction machines.

CHAPTER 2

MATHEMATICAL MODELING AND ANALYSIS

2.1. Three-Phase Symmetrical Induction Machine

The actual dynamical model of a three-phase symmetrical induction machine is
quite complicated and nonlinear. It involves many effects that do not contribute
significantly in the normal operation of the machine. If we try tO develop a model
incorporating all the effects, it would become too complicated. The task of
designing a control would become tedious. It may actually obscure the part of the
model that plays the major role in the operation Of the machine. For these
reasons we use a simplified model. The details of the derivations and basic
assumptions can be found in Leonhard [12], Novotny [15], or Vas [21]. This
model is kind of standard in dealing with control of induction machine. It has the
added benefit that it is not particularly specific to a certain type of induction
machines. Rather it relates to the operation of an induction machine in a way that
is common to all types.

A three-phase symmetrical induction machine can be represented, without loss
of generality, by an equivalent two-phasemodel. Whereas this development is a

mathematical artifice, the physical machine does always have three phases. One

of the three currents in 15.2153 in a three-phase machine is redundant because the

10

neutral is isolated and cannot act as a current source or sink

for the terminal voltages u5,,u52,us3. Hence

is] +1.52 +1.33 =0
usl +1152 +1153 =0

We define the two-phase equivalent current vector as

i 3t

sa — 2 31

. J3 . .
lsb = 7(132 _ls3)

which can be conveniently represented by a complex variable

. 3. . 3 . .
ls :Elsl + 17(152 _ls3)

Similarly

. 3
us =5u51+ 17(1‘52 —us3)

. The same is true

(2.1)

(2.2)

(2.3)

(2.4)

Note that we have a degree of freedom over the Choice of the constants in the

above equations as discussed by Vas [21]. Different authors follow different

standards. We try to remain consistent with the above notation, which has been

used by Leonhard. The inverse relation is given by

. 2. 2 ,
lsl =31“ =§Re[ls]

1

l 2 h 2”
isZ = _§isa +7511“, =§RC L1.3.6 3
s3 =-§isa _

i

 

fish = ‘g'Re [5613—]

11

(2.5)

Similar equations hold for terminal voltages. We would like to emphasize that
from now onwards the analysis and design of the control would be carried in two-
phase variables. It will be only for the actual implementation where we will have
to shift to or form the physical three phase terminal quantities.

A final remark about the transformations we are using is that the length of the
quantities is not preserved. In fact if want to calculate the actual equivalent in

terms of physical quantities, we have to multiply the two-phase quantities by 2/3.

The result is apparent from first equation of (2.2). This may become important if
we want to make use Of the rated quantities of a machine in two-phase
representation. A typical example would be to design a controller that regulates
the flux to the rated flux of the machine.

The two-phase model of the machine is given by the following set of equations.

. di d . -
Rs'ls+Ls_d:_'+Lm-d—t(lr'ejp6)=us (t) (2.6)
. di d . --
R,-l,+L,Tt'+LmE(ts°e ”0):“, (t) (2.7)
day 2 - *
—— = — : — . . jpg —
d: Td T, 3 L,” Im[z, (1,.» ) ] T, (2.8)
dB
-— = a) 2.9
p dt ( )
Where
R, —— Stator resistance
R, — Rotor resistance
L, — Stator inductance

12

L, —— Rotor inductance

Lm — Magnetizing inductance

i, — Stator current vector

u, — Stator terminal voltage vector
i, — Rotor current vector

u, — Rotor terminal voltage vector
p — Number Of pole pairs

6 — Rotor mechanical position

T4 —— Electrical torque developed
T, —— Load torque

J — Moment of Inertia of the load and the rotor shaft
to — Mechanical speed

Besides these quantities we would be using and referring to the following

quanfifies

r, — Stator time constant (1% j

r, — Rotor time constant (% )

A, — Stator flux vector (rectangular coordinates)

xi, — Stator flux magnitude

,0 — Stator flux angle with respect to stationary frame of reference
im, — Stator based magnetizing current (rectangular coordinates)
I'm, — Magnitude of im,

13

Note: We are using bold letters for vectors or complex quantities and regular
letters for scalars. The symbol 6‘ will be used as a difference operator.

We also have the following definitions, which will be used later

L5 (1+03)Lm (210)
L . .

. (now.

where a, and a, are the stator and rotor leakage factors respectively.

The following factors are helpful in simplifying equations and are referred to in

the literature quite Often.

 

i

TS

L

Tr (2.11)
L,"

 

 

There would be some other definitions and terminology that we will be discussing

as we proceed with our analysis.

2.1.1. Stator Based Machine Model

Our main emphasis in this work is on the stator flux orientated control of a

voltage-fed Induction motor. To continue with this development we define certain

14

machine variables with respect to the stator, which is going to be our frame of
reference.

The stator flux vector is defined as
A, = L,” -i,,,, (2.12)
Where i,” is the stator based magnetizing current and is given as
im, = (1+ 03):, + gel?” (2.13)
Eliminating all the references to the inaccessible rotor currents and assuming

u, = Ofor a singly fed machine, the machine equations (2.6) to (2.9) take the form

)1, = as —R,i, (2.14)

aLsis +L,(1/r, -jwa)i, = i, +1.,(1/z', -jw) (2.15)
2 . ..

T =§Lm whim] (2.16)

The inner voltage vector is defined as

 

Rr . . o 0
vs = 1+0} 1"” + 101(Lm1ms TULs's) (2.17)
Equations (2.14) and (2.15) assume the form

v, =u, —[Rs +%R,]is -aL,i'_, (2.18)

It is easily seen that the machine model is specified completely by stator-based

quanfifies.

15

2.1.2. Second Order Effects

We will concentrate on the simplified model of the induction machine. However, it
is important for the designer to always keep in mind the unmodeled part Of the
system. Under certain circumstances any of these unmodeled effects may have
a significant role in the operation of the machine and would not be negligible any
more. The impact of this factor could render a design restricted to a specific
region of operation. Just to emphasize what we have mentioned, following is a
list Of unmodeled effects.

0 Magnetic saturation

0 Finite permeability Of the stator and rotor iron

. Iron losses

. End windings

. Slot effects including non-ideally distributed sinusoidal winding

. Sensor dynamics

0 Inverter nonlinearities include dead time

- Delays and latency less than one sampling period

a Non-ideal behavior Of anti-aliasing filters

- Non-ideal behavior of power supplies

. Approximate behavior of discretization of continuous time systems
We will discuss some of the above points during various aspects of the control

design. However, they would not be a formal part of the control schemes.

16

2.2. Stator Flux Estimation

Flux, being a physical quantity, can be measured. In fact many Of the earlier field
orientation schemes relied on measurements of the stator flux. The two most
commonly employed methods were to either insert additional sensing coils or to
make use of some sort Of magnetic field sensors like Hall effect lC’s. It was not
long before It was realized that stator flux Observers could be built using stator

voltages u, and currents i, whilst achieving somewhat similar performance as

would have been expected by inserting additional sensing coils. On the other
hand the Hall effect lC’s are not rugged and therefore not suitable for the harsh
Operating environments that exist inside a machine. In most case, there are
going to be severe thermal and mechanical stresses on the sensors. Considering
all these factors and limitations, the natural solution that appears to fit well into
the industrial requirements is to estimate flux rather than to measure it.

We will assume that the stator voltage as and current i, are available online. An
open-loop stator flux estimator is obtained by integrating equation (2.14).

1, = j (u, — Rsis )dt (2.19)
The integrand in this equation is sometimes referred to as the voltage behind the

stator resistances.

2.2.1 . Pure Integrators

Pure Integrators always face difficulties due to the following factors

17

. Low frequency drifts

There are many sources Of error that contribute to low frequency drifts.
The most significant are the DC offsets, higher internal noise inherent in
semiconductors at low frequencies and aliasing errors in sampled data
systems. The last one is very important, as it may eventually become a
major factor in determining the lowest frequency at which the encoderless
control could be employed. Small errors or spurious signals at the input of
the integrator may result in undesirably large disturbance at the output of
the integrator. The main reason is the very large gain of the integrator at
low frequencies. This prohibits us to use the pure integrator in its true
form.

0 Transient offsets
Major causes Of this problem are the unmodeled effects and inaccurate
parameters in the machine model and sensor/inverter dynamics.

0 Inappropriate initial conditions.
The lack of information about the exact conditions when the system is
powered up, forces us to make certain assumptions about the initial
conditions. These assumptions can almost never be exact.

Furthermore, we only know some nominal value of the stator resistance R”.
Hence we end up building an estimator of the form.

1, = [[(u, —R,,,i, )+d]dt (2.20)
Where d is the disturbance term. It is easy to note that if d has a DC

component then after long time the estimator would become useless. To

18

investigate the effect of R30 we neglect d and define the error in estimator as

follows
62, = .1, wt, =-[(R_, —Rso)isdt (2.21)
This expression is true for any i, but does not give us much information. On the

other hand, we are generally more interested in the steady state errors. Typical
steady state currents when expressed as complex variables would be of the form

given below.
i, = [58mm (2.22)

Putting this expression in equation (2.21) gives us

I
"61$":sz "' RSOIZUS— (2.23)

3
Although an idealization, this is an important result and tells us that the error in
the flux estimator is directly proportional to the difference in actual and nominal
stator resistances and the magnitude Of the stator current. On the other hand it is
inversely proportional to the frequency of the stator current. This Observation is
particularly important because it gives us a measure of how low we could go as

far as the speed is concerned.

2.2.2. Current Components

Equation (2.13) gives a relation of the magnetizing current to stator and rotor

currents in an induction machine. In this relation the magnetizing current in, is in
rectangular coordinates. Transforming into polar coordinates we get
im, = imefp (2.24)

19

where p is the orientation of the stator flux with respect to a stationary frame Of

reference. As we will see, this angle would in fact form the basis of the stator field
orientated control. In other words it is perhaps the most important parameter in
the design Of the control. Unfortunately in a realistic and physical situation we do

not have an exact knowledge of this stator flux angle ,0. The stator flux estimator

if transformed into polar coordinates gives us

A

im, = imem (2.25)
The principle of field orientation is primarily based on resolving the stator currents

into the direct and quadrature axes. Consider the following equation

ise-Jp = isd + jisq (2.26)
where the direct component i“, is along the direction of the moving frame of
reference defining stator flux coordinates. The quadrature component in, is

orthogonal to is].

The torque from equation (2.16) can now be expressed as

Td =-:—Im[zisisq]

2 (2.27)

= 31,, Irn[i,,,,i,q]

The actual components are unavailable and we resort to using the estimate of

the field orientation.

ise'm =1“, + jisq (2.28)

We will be extensively using these stator current components as well as the

magnetizing current in both the analysis and design of the control schemes.

20

2.2.3. Open Loop and Closed Loop Integrators

There are two types of Integrators generally discussed in literature. The
integrator we are using for the stator flux estimation is referred to as open loop
integration. Another variation of the stator flux estimator is to implement it in polar
coordinates instead of rectangular coordinates. This form is called Closed loop
integration. Differentiating equation (2.19) and transforming it into field

coordinates we get
is + 115,) = (us _ Rsis )e-J’p = (aw, — 12,1“, ) + j(u_,q -R,i,q) (2.29)
The equations for stator flux in polar coordinates are

is = (usd — R,i,d) (2.30)

. _ (“sq — Rsisq)
p — ll.

 

(2.31)

The measured quantities are u, and i, and we have to make use of equation

(2.29) in order to obtain direct and quadrature quantities in equations (2.30) and
(2.31). The transformation to field coordinates require the use of p which is in
fact the output Of the Observer. There is a feedback involved in the estimator and
hence the integration in the estimator is referred to as the Closed loop integration.

Following block diagram illustrates this fact.

21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

us _ R313 e—Jp

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2-1. Stator Flux Estimation in Polar Coordinates

One might think that the integrator drifts would be reduced for this scheme.
Leonhard [12] has said: “If the operation is performed inside a feedback loop, as
in the case when being carried out in field coordinates, the effects of integrator
drifts are reduced to normal Offsets”. However, theory and simulations do not
support this reasoning. The transformation from one estimator to the other is just
a Change of variables. It is a well-known fact in Lyapunov analysis that the
stability properties are not affected in such transformations. The only thing that
may be achieved by a change of variables is that we may be able to isolate the
unstable part of the system from the stable one. Both equations Of the estimator
in the rectangular coordinates are not asymptotically stable. The change of
variable into polar coordinates doest not even partition the system such that one
part is asymptotically stable. These issues will be illustrated through simulations
in next Chapter. According to our findings, we do not gain any benefit by
transforming into polar coordinates. In fact it may be difficult to replace the pure

integrator with a filter approximation in polar coordinates.

22

2.2.4. Approximation Of an Integrator

The arguments we have gone through so far Clearly imply that it is not possible to
employ pure integrators in a physical control system estimating the stator flux.
We have to resort to some approximation of the pure integrator. In this section
we investigate the effects of approximating pure integrators with two different
types of filters. We start with the following filter, which is in fact the most

commonly used method in this regard.

 

l = l (2.32)
s s + we
The corresponding equation for the stator flux estimator is as follows.
i, = «0,3, + u, — Rwi, +d (t) (2.33)

where d (t) is the disturbance vector. Equation (2.33) results in the following error

dynamics with the definition Of error the same as in equation (2.21).

61. = 40,51, H.261, -(R_, —R,o)i, —d (z) (2.34)
To investigate the boundedness of error we use a Lyapnov function. In the
following analysis we would assume only real variables and all the operations
would be real. This is typical in Lyapnov analysis. The rest of development

extensively uses complex variables for notational convenience.
1 T
v (6.1w ) = 3((5.1, 6.1,) (2.35)

The magnetic saturation of the machine keeps A, bounded whereas any practical

controller has saturation properties that help i, to remain bounded. Hence in a

23

physical system 1,,isandd are always bounded and we can define a bound of the

following form.

 

(0,1, —(R, —R,,,)i, —d(:)| s y (2.36)

Hence

V = 615T a), (2 37)

Vso for using"?

SI

The boundedness of error is proved for any waveforms Of 1,,isandd. This result

is, therefore a conservative estimate for the error bounds. However, if we make
use of the typical steady-state conditions given below, we get better estimates of
the steady-state errors.

i =1 ej(ws’+¢)
‘ ‘ , (2.38)
1‘s =Asel(wst+'l’)

Leaving out the disturbance term we can actually solve equation (2.33). The
natural response of this stable system decays very slowly with time. We are only
interested in the forced response.

6.13 = __1__—[wcA,e"(‘”s’+’”) — angel-(mm) _ (wcAseW -6R,I,ef¢)e‘i‘"c’] (2.39)
we + 160.

For wf >> (of which is equivalent to jw, +a2c es fa), equation (2.39) reduces to

615 z QQASej(w5HW—%) _%Isej(wst+¢_%) (2.40)
(as a):

We have left out the last term of equation (2.39), which decays after long time.

24

As far as the disturbance is concerned we do not know the exact waveforms. It

would be preferable to model it as a signal which is band limited to (ad.

 

 

 

 

 

 

 

A A A
'H(w)l . |D(a))| . |H(a))D(w)| ‘
> ‘0 - > 60 - a a)
Transfer Freq Domain Representation Disturbance
Function 0' Disturbance Amplification

 

 

 

Figure 2-2. Frequency domain representation Of disturbance with integrator approximated by first

order filter
The figure illustrates that if a), << a), then the low frequency disturbance would be
amplified 1le times. This could become a problem if a), is small. Combining

equation (2.40) with the amplification of the disturbance term we get

 

+,—|R ,,,|I, 1
||61s||<ws +;,:S“P|d(t)l
(2.41)

L —R,0|I, JCUSWI" )l

ws

It is instructive to note that we had introduced three sources of errors; namely the
approximation of the integrator with a filter, the mismatch between nominal and
exact stator resistances and the presence of the disturbance. The three terms of
the first equation correspond to these sources of errors and interestingly are
independent of each other. This is because the error dynamics equation is linear.

The error due to first term is small and does not cause much Of a difference.

25

To overcome the problem of amplification of the drift we may employ another
form of the integrator approximation given by

1 z ——s— (2.42)
S

(s+a)c)2

We would be implementing the estimator as

on
A

2, = -2w,;i, «vii, + a, — R,,i, +ai (t) (2.43)
The corresponding error dynamics are given as

51, = -2a),a.i, —a),26)., + 2wci, +0231, +12, -(R, —R,, )i‘, +d (t) (2.44)
The boundedness of 61, can be shown by Lyapnov analysis but as in the

previous case these bounds would be conservative. We therefore, try to estimate
the steady-state errors for the typical operating condition given in equation (2.38).
The forced response of equation (2.44), neglecting the disturbance term, is

52, (t) = ——l——[(w3 + 2w,w,j)A,ej(‘”s”"’) — j(R, —-R,,)w,1,ej(“’s”¢)]

. 2
(w‘ + ,m,) (2.45)

— ——l———2[(1 + wet + ja),r)A,e'j“’C‘]

(we +110.)

The last term decays though very slowly. We can carry out a similar approach to
investigate the amplification of the disturbance term. Refer to the following figure

where the disturbance is assumed to be band limited to 0),.

26

 

 

 

 

 

 

 

A A A
III (w)l |D(w)l . IH (w)D(w)|
‘/l\ > a) - b a) - b a)
(0‘. wd wd
Transfer Freq Domain Representation Disturbance
Function 0‘ Disturbance Amplification

 

 

 

Figure 2-3. Frequency domain representation of disturbance with integrator approximated by

second order filter

There are two possible ways to estimate the behavior of the filter to the

disturbance term. lfwr >> wd then the amplification of the disturbance term can be

approximated by the following expression

 

(2.46)

h(t)*d(t)= a“)

w?
where h(t) is the impulse response of the filter. It is evident from the above
expression that the DC component of the disturbance is completely blocked.
Another approximation could be as follows

h(:)*d(z)=‘”—gd(t) (2.47)

8

Furthermore If (”d < a): then the amplification of the disturbance is 0(d(t)). The

disturbance is assumed to be small itself as compared to the actual signals. Now

if a), >> we then after bringing back the disturbance term into the solution (2.45) of

equation (2.44) we get

27

zwcAs +|R5_R30|Is +

 

"at": a, a, 0(d(r))
‘ 5 (2.48)
~|R5_Rsolls
~ (I)

S

This is an interesting result when compared to equation (2.23). The conclusion is
simply that the second-order approximation of the integrator not only takes care
of the low frequency disturbances but also approximates the integrator pretty

well. It is important to note that if the condition w: >> a), is not satisfied then

equation (2.48) is not a good approximation of equation (2.45) and may cause
some undesirable results and eventually lead to an error in the speed estimate.
A slight improvement in the decaying term of equation (2.45) is possible if we use

complex conjugate poles instead of a double pole. One possible scheme could

 

 

 

be
1 s
E z 52 +5..) m2 (2'49)
C C
The corresponding error term, omitting the decaying terms, is
615 (t) = we. Asej(ws’+W) l 2 [j(Rs _ R50)wslsej(wst+¢)]
we +st ((064-1205)
IR R II (2.50)
a) A -
5 .~. C 5 + S ‘0 5 +0 d I
II a." m. 0,3 ( ())

Another advantage of this form is that it resembles more to the first order filter.

The first two terms are identical to the corresponding terms in equation (2.41).

2.2.5. Equivalent Errors in Polar Coordinates

It is sometimes convenient to carry a part of analysis in polar coordinates.

However, our stator flux estimator is in rectangular coordinates. It would be

28

useful to calculate equivalent errors in the polar coordinates. This development is

for a general case, which we would convert to our estimator later. lfy is some
complex quantity given as
y = a + jfl (2.51)
with its estimate
9=dr+jB=(a—6a)+j(fl-§fl) (2.52)
where (id and 6,6 are the errors in rectangular coordinates.
Assuming that the error is small as compared to the actual values, we can carry

out the following derivations.

IyIZ—wlz =a2+fl2—(a—6a)2-(fl—6fi)2

(2.53)
z 2a§a+ 2,6613
As |y|+|5z| z 2| yl for small errors, we get
6" yl 2 M (2.54)
lyl
Representing the quantities under consideration in polar coordinates
y = M8”
.63 (2.55)
5" = Me”
gives
y—9=|y|ej9—|5I|ej9 ==|y|(ej0—ej6) (2.56)
The error in complex exponential is therefore
5619 = M (2.57)

M

29

This equation can also be useful in finding equivalent error in rectangular
coordinates provided we have knowledge of the error in polar coordinates.

Another useful variant of equation (2.57) is

5e 16=e 16-e16=-— , =-'—.—‘ 2.58
ej(6+6) €129 ( )

 

In most of the cases we would be interested in 6e” rather than 66 itself. However,

a special case of equation (2.57) arises when 6 and error in it are small. In that

case
cos(6) == 1
cos (9) = cos(0 — 66) =1
(2.59)
sin (6) z 6
sm( ) = sin(6-60) = 6—66
resulting in
def” = [008(9) — cos(é)] + j[sin(6) —sin(é)] z j6¢9 (2.60)

We are now in a position to estimate the errors in magnitude and angle of the
magnetizing current. Equations (2.54) and (2.57) show that bounded errors in
rectangular coordinates result in bounded errors in polar coordinates. A word of
caution is that the above derivations are based on relative errors being small. If
this were not the case bounded error in rectangular coordinates may become
unbounded in polar coordinates. The following figure illustrates that the small
errors in rectangular coordinates result in bounded errors in magnitude as well as

in angle. However, if the error in rectangular coordinates becomes too large the

30

angle may no longer be bounded. The right figure illustrates this situation where
5: can be pointing in any direction. This is a matter of concern if the actual
quantities are becoming small themselves. This important fact may become one
of the factors that limit the slow speed operation of the machine where the
currents and voltages are small. The sources of errors are the same or even

worse as depicted in equation (2.48).

 

   

Error Bounds

 

 

 

 

Figure 2-4. Error propagation from rectangular to polar coordinates

The striking observation is that the philosophy of field-orientated control becomes
ineffective at the worst-case scenario of the above figure. To summarize, we say
that estimated stator flux angle is the most critical parameter for the field-
orientated control but is susceptible to extreme conditions.

Now we represent the error bounds for the steady-state condition. The errors will
be assumed to be small as compared to the actual quantities. Putting equations
(2.38) and (2.40) in equation (2.54) we get, after performing some trigonometric
manipulations

6,1, zflsinO/I—(b) (2.61)
a)

S

31

The first term of right hand side of equation (2.40) is cancelled out when put in

equation (2.54). Since l//<¢ for the motoring mode, the error in magnitude has
opposite sign as compared to 6R3. Furthermore, when the difference (rt-g) is

small, the error should be small and can be expressed as

6R I
513 =—‘ ‘ (VI-(p) (2.62)

(Us

A similar approach leads us to

56m 2 geilwsW-éi) _ 5&1. eile-SL)

a), (USA, (2 63)
z 125$ ej(w_,:+¢—§)
wSAS
From equation (2.13) 15 < i,,,, , which gives us
- 6R
|6eJP| <-|—‘I (2.64)
cast...
From equations (2.26) and (2.28)
is (ae‘fp) = 6i“, + j6isq (2.65)

Hence from equations (2.38), (2.58), (2.63) and (2.65)

6isd + jgisq : _Isej(wst+¢) [% ej(wst+’/’“12£) _élsliej(ws’+¢’%)]e-j2(wst+w)

(vs (USA,
2 (2.66)
: __0_)c_ I ej(¢-W-12£) + 2531—581'(2(¢-W)-%)
ws 3 wSAS

With the same observations that l//-¢ is small, we get

32

a) 26R I2
szS( —¢)— (01: 5( ¢)
5 5 3 2 (2.67)
w R I
6qu --;)9-I,c08(w-¢)- Rs 005(2('/”¢))
5 s s
32;, _5Rsls2
a). 3 05A.

Since (xx—(p is small the errors in isq are expected to be much higher as
compared to the errors in is). The situation is worsened by the fact that for most
of the operating region of the machine imis small. For example if I, =60,i,q =6,
a), =1and a): =20then 6i", =3, which is 50% error. It becomes important to
develop an algorithm that does not rely heavily on in). We will see that the speed

estimator developed in section 2.4 makes use of in, only.

2.2.6. Remarks on Flux Estimation

It is important to keep in mind that the above derivations are true only under
certain assumptions. Even though the assumptions are made for typical steady
state conditions, certain regions of operation of the machine may require more
stringent assumptions and approximations. A typical example is to control the
machine at very low speeds. The design parameters should therefore, be well
defined in terms of requirements and operating region. The conclusion is that
estimation schemes based on approximations and in presence of uncertainties
impose certain restrictions on the design of the control and the region of

operation.

33

For most of the cases, the error introduced by approximating the integrator with a

filter is small and can be neglected if (03 >> a)( . The only exception is the estimate

iq. However, it can be seen that the error 6i“, is not completely unknown. We

have complete information about the filter parameters and related issues. This
may help in developing some corrective measures and we could improve
estimates of the desired quantities. Though we will not pursue this idea further, it

may prove to be a useful tool in certain situations.

2.3. Control of Induction Machine Using Speed Measurements

It is natural and instructive to first investigate a control scheme using
measurement of the mechanical speed of the rotor. After the development of the
flux estimator we are in a position to implement the desired control. Once the
design and related issues are clear we can move to our ultimate goal namely the
encoderless control of the machine. Before proceeding any further it is important
to have a discussion on certain factors. There are basically two possible
configurations for the control of induction machines. One is based on current fed
induction machine where as the other one is voltage fed. As far as the model of
induction machine is concerned it appears that the current fed induction machine
is easier to control and indeed the design and analysis of the control is somewhat
simpler. This is the main reason that most of the earlier schemes of field-
orientated control of machines were developed based on current-fed induction
machine. Soon it was realized that practical difficulties involved in the design of a

current source inverter not only complicated the situation but even restricted the

34

design to be useful for only a limited region of operation. These factors become
significant in the field-weakening region as explained by Leonhard. On the other
hand voltage source PWM inverters are very popular in numerous industrial
applications. Their architecture enables reliable and high performance design of
induction machine drives. Since our emphasis is not on the theory and design of
PWM inverters, we would cut short this discussion by saying that for practical as
well as theoretical reasoning we will restrict our design to voltage-fed induction

machine where a PWM inverter will act as a three-phase voltage supply.

2.3.1. The Control Variables

One of the fundamental questions in any control problem is clear knowledge of
the state variables to be controlled. In the case of stator flux orientated control of
an induction machine, our goal is to control the direct and quadrature
components of the stator current independently of each other. If this is possible,
we would then be able to control flux and torque. The problem of speed control is
basically related to the torque control problem as would become clear in the
following discussion.

In the case of a voltage-fed induction machine, the voltage source inverter acts
as the control effort. However, it is important to realize that we are trying to
control the direct and quadrature components of the stator current. Furthermore

we lack the exact knowledge of how to resolve the stator current into desired

components. The best we can do is to use the estimates 12,1: instead of the

actual variables.

35

2.3.2. Coupled Nature of Direct and Quadrature Current Components

One of the limitations of the stator flux orientated control is that the stator current

components 5,4,1}, are not completely decoupled. If the field orientation is done

along the rotor flux, the corresponding current components are decoupled.
Leonhard has compared this phenomenon with armature reaction of a DC

machine. He has derived steady state expressions as follows

_1+o’(S’z',)2 ims

i
s 1+(0Sr,)21+0's

 

(2.68)
2.1%.
1+0,
_ (1-0)“? ims
— 21+0
1+(USTr) s (2.69)

 

i
zl— s m
( U)T'1+ S

lf there were a change in the load, a transient would result in the stator flux. A
flux controller can overcome this difficulty. However, it is known from the
multivariable control theory that without proper decoupling the transient behavior
of the system may be adversely effected. The remedy to this problem is given in

the next section.

2.3.3. Flux Controller and Decoupling Through Feed Forward

In certain cases it is possible to do away with a flux controller. However, in our
case flux control is a kind of requirement. Two natural choices are sliding mode
control and proportional integral (PI) control. The first one is expected to be

robust and has proved to be a reliable method in the rotor flux orientated control.

36

Details can be found in Strangas and Khalil [10,20]. Because of its simplicity, the
second method is more widely used. The main disadvantage is that the tuning of
proportional and integral gains could be a problem. Even if the gains appear to
work well under certain conditions, the control may deteriorate with the change in
operating conditions. In summary, the PI controller is simple to design but faces a
problem of tuning of gains.

Our case is, however, different because of the decoupling requirement. It would
become a harder problem to device a technique whilst using the sliding mode
control. We therefore, resort to Pl control. Another option to overcome the
problem is to use a scheme where decoupling is achieved by a different choice of
field orientated variables. However, that would be altogether a different scheme
and would take us away from pursuing the stator flux orientated control. If, on the
other hand, we do not use the decoupling term at all, it turns out that the transient
response of the system is slowed down to an extent that may be unacceptable as
far as high performance drive requirements are concerned. This would be
demonstrated by simulating a control scheme without a decoupler.

In the case of voltage-fed induction machines we are trying to control in, and in.

Under the assumption that our controller is capable of doing this task, we can
design a decoupling scheme in terms of a current-fed induction machine. The
main incentive is the convenience in deriving such a scheme. We make use of
the scheme proposed by Novotny and Lipo [15]. The current-fed induction

machine model in stator frame of reference is

37

Lsisq + 02;ij — 56057,. (A, — aLsisd) = 0

,1, + 7,2, = Lsisd + 61231;, - Swsz',0'Lsisq (2.70)

3p .
Td = 725154

In this model i“, and is, act as the control inputs rather than the state variables to

be controlled. A PI controller of the following form is used
isd = KP (ismef - is ) + KiJ-(A‘mref “'23 ) dt + idecoup (2.71)
where 21“,”, is the reference to be tracked and id...” is the additional decoupling

term. Substituting in second of the equations (2.70) we get

. d .. ..
213 + 1,15 = (1+ or, E) L, [KP (2.5,,4 —As ) + Kin/13,4 —/1_, )dt]
d (2.72)
+ (I + 01, 717] Lsidecoup — $0151”ij

To decouple d-axis stator flux ’15,, from the q-axis stator current, we require
d . .
(1+ 0'1, 3) ledecoup - Swsz'pLszsq = 0 (2.73)

Sws can be calculated from first of the equations (2.70)

(1+ 0'1, %] Lsisq

 

Sm = 2.74
5 1,015 —aLsisd) ( )
Eliminating Sm, from equations (2.73) and (2.74) we get
aLsiszq
id u =————.- (2.75)
“a p A, -aLsz,d

The version of the decoupler that we can implement is based on available

quanfifies.

38

’-‘2
.. alesq

idecoup = A ._.
x1 -0'L1

‘ 1;" (2.76)
aLsiSq

Linims _ 0L5 isd

 

There is a singularity in the decoupler equation when all the currents are zero.

However when the machine is running, the singularity never appears. This
follows from the fact that L," > (IL, and inu >1; >0.

After that we have developed a decoupler for the current-fed induction machine,
we now adapt it for the voltage-fed machine. With 2, converging to zero at

steady state, equation (2.30) gives

usd z Rsisd
or (2.77)

usd,ref 2" Rsoidref
The proposed flux controller works well if the gains are properly tuned. However,
it faces problems if there is considerable detuning and parameter variations.

These facts are difficult to analyze but will be demonstrated through simulations.

39

 

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2.3.4. Torque and Speed Controller
A combination of a PI speed controller followed by a PI i“, controller proves to be

a useful solution in this regard. From equation (2.16) it is clear that if the

magnetizing current in” is successfully maintained at the desired level by the flux
controller then the in, controller is essentially a torque controller. Because of the

combinations of two Pl controllers, tuning of proportional and integral gains may
become tedious and time consuming. The scheme is prone to face difficulties if
the load-torque or parameter variations are significant.

One may think to employ robust control methods like sliding mode control for the
torque and speed controllers. It has been observed by Aloliwi [19] in his
experimental work that the chattering inherent with sliding mode control becomes
undesirably large. The missing link is obviously unmodeled dynamics and
phenomena.

Under these considerations we restrict ourselves to the Pl controllers.

Simulations and experimental results with this scheme appear to be satisfactory.

2.3.5. Position Measurement

In most cases, the use of a position sensor is a preferred choice over a speed
sensor. This situation, however, requires differentiation of the position to obtain
speed. The use of high-gain observers reliably handles the problem. Strangas
and Khalil [10,20] have in fact used high-gain observers for the same purpose in

rotor flux orientated control.

41

2.3.6. High Gain Observers

The process of differentiation being non-causal is not realizable. The use of high
gain observers not only reliably calculates the derivatives in real time but is also

robust to disturbances. Consider the system

5‘1 = x2
*2 = f(X1,X2) (2.78)
y = x1

Treating f (x,,x2) as a disturbance, we can make use of the high gain observer

as follows

§l=£2+(al/5)(YI‘521)

5.92 =(“2/52)()’1‘31) (2.79)
With the change of variables
zl =(A21 — x1)/6 (2.80)
fi=h—&
we get
821 = —a1z] + 2:2 (2.81)

522 =_0221+£f(x12x’2)
It can be shown by singular perturbation theory that for bounded f(x,,x2) , 21 and
z2 are 0(a). In other words the error in differentiation is 0(8).

This system is referred to as a full-order high-gain observer. There could be other
variations such as reduced order observer and extended order observer. In terms

of transfer functions we can represent these variations as

42

A S
yro:(

 

s/€)+a
:. s
y o = (2.82)
f (S/8)2 +a1(s/£)+a2
:. s
yea =

 

(s/s)3 +al (sh?)2 +a2 (s/£)+a3
It is apparent that in all the cases the transfer function can be thought of as a

cascade of an ideal differentiator and a low pass filter. The Bode plots of the

three observers are illustrates below

 

|H(w>| |H(w>| |H(w>l

 

> a) - + a) - > a)
Reduced Order Full Order Extended Order

 

 

 

 

 

Figure 2-6. Transfer functions of different high gain observers

Whereas high gain observers appear to be sensitive to noise, they have been

used in physical control systems where the noise is always present. The

performance of high-gain observers in the presence of noise can be improved by

observing following precautions.

. Do not choose a to be very small. Increasing the bandwidth unnecessarily
will only deteriorate performance in the presence of noise.

0 Avoid using a reduced order observer. This fact is apparent from the transfer

function plot where the bandwidth is infinite.

43

2.4.

The use of extended order observer may reduce the noise but be careful
about the implementation issues of higher order filters. These issues are
explored later in the chapter. Furthermore, the higher the order the poorer is
the transient response. It is important to realize that as far as noise
performance is concerned, the transient response is as important as the
magnitude plots.

Amongst the commonly used filters, Butterworth filter has good transient
response. Bessel and Gaussian filters exhibit even better transient response
but are not very frequently used. Use of a low pass filter embedded in the
high gain observer that has the best transient behavior will slightly improve
the noise performance. For a discussion of transient behavior of filters refer to
Daniels [4] and Deliyannis [5].

Extra precaution should be taken when a high-gain observer is used for
digitized discrete-time signal. The group delay of the low pass filter should be
larger than ten times the sampling period if not more. OthenNise, at each
transition the high-gain observer is going to give sharp response. This
behavior is definitely not desirable. We observed this fact during the use of a
high-gain observer for estimating speeds from the position measurement

obtained through an optical shaft encoder.

Speed Estimation and Error Bounds

Our ultimate goal is to control the speed of the induction motor without using a

mechanical position or speed sensor. To be able to do this, we must develop a

44

reliable scheme for estimating the speed of the machine. Our emphasis would be
on a scheme proposed by Leonhard. However, we will try to go into deeper
details of the analysis and try to assess the error bounds of the speed estimate.

Transforming equation (2.17) into field coordinates we get

vse-jp -_- Vsd + jvsq = 1:; im +£00ij + jw(Lmim —0'Lsisd) (2.83)
r

 

from which we get

a) — v“, (2 84)
_ (Lmims - 0141's.: ) '

 

The idea of estimating speed appears to be simple but unfortunately we do not
have access to certain quantities in equations (2.83) and (2.84). We can only

make use of the estimators. Equation (2.18) takes the form

A

6, = u, {Rm +24%}, —aL,i, (2.85)

where R“, and Rm are the nominal values for stator and rotor resistances. 1?, is the
estimate of the derivative of is. We make use of high gain observer to find the

derivative of i, . We have the following equation
. L, .
VS - s = 6V3 =— (5R5 +2—6R, ls -O'L50(8) (2.86)
r

Note that there are no dynamics associated with this equation. The equation with

estimates as compared to equation (2.83) is

 

A —— . A IA -R ¢ ¢ 0 A ¢ ¢
vse m = .4 + 1qu = 1+: 1",, +(30'lesq + 141.sz —aL,z,d) (2.87)
r

where

45

i3,e'm =(v, —6v,)(e_jp—6e-jp)=v,e'jp—6v,e'jp—v,5e'jp—6v,5e_jp (2.88)

The first term on the right hand side is the actual expression as in equation (2.83)
whereas the last term may be neglected being a second order error term. Using

equation (2.58), the final error expression comes out to be

56,751”) = 69,, + jag, = 6v,e_jp + v,5e'ip (2 89)
_ I - .2 O _ O . - 0 I .
=6v,e ’p—v,e J pfie’p =6v,e 1” -(vd +qu)e 1'06er

Subtracting equation (2.87) from equation (2.83) we arrive at the following

important result.

6&4)... + R, aim, - 6R,6i,,,,
1 + 0,

0" . 0" (2.90)
' lms _ ' lms _ ’ _ lms _ '
1L5[w[1+0 061,dj+6w[l+a 01“,] 6w[ 0‘6.in

, , 1+0,

 

5036-17)): - + 0L, (60:1,, + (06%,, — Mg, ) +

 

 

 

Neglecting small errors, the equation simplifies to

_ (5ij + R,6i,_,_
1+ 0,

. 5i,” . im, .
1L, [64],”); -0§r,d]+5(0[1+0_3 —0z,d]]

From this equation we can now move towards our actual goal of calculating error

 

+0L, (60):; +w0‘i,q)+
(2.91)

 

 

bounds on speed estimate. We get two expressions corresponding to the real

and imaginary parts of the equation however we would use only one.

5v '
i—w[ 61”" —06i,d]
L, 1+0,

 

60) =

 

(2.92)

 

46

The dynamical behavior of the system is dependent on the controller. Therefore,
we would delay any further discussion till the actual control is designed and

analyzed.

2.5. Control of Induction Machine Using Speed Estimation

Finally we are in a position to design and implement the encoderless control of
an induction machine. We will assume that any sort of mechanical speed or
position measurement is not available. Hence we will have to rely on the estimate
of the speed.

The fundamental estimates would be of the stator flux and rotor mechanical
speed. It is interesting to note that the estimate of stator flux is independent of
the rotor speed. This is one of the major differences in the estimation of rotor and
stator flux. In rotor flux based schemes, the rotor flux estimator is a function of
rotor speed. Details could be found in Strangas and Khalil [10,20]. All the
discussion and analysis we have done in the previous section for stator flux
estimation would be valid here.

The idea would again be to estimate the stator flux angle and perform field
orientation with respect to it. In the process, we resolve the stator currents into
direct and quadrature axes components. These current components would then
be used in the estimation of the mechanical speed. The speed estimate is
therefore dependent on the stator flux estimate. Any errors in estimating stator
flux affect the accuracy of the rotor speed. Furthermore, we have already seen

that the rotor resistance comes in as another factor in the speed estimation.

47

 

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48

The point to be emphasized is that the encoderless control schemes are always
prone to different errors that may have an adverse effect. The robustness
becomes an important issue. There may be situations when we are expecting an

error in the tracking but the control may actually breakdown

2.5.1. Low and High Speed Operation

The encoderless schemes experience certain limitations. The most glaring is the
operation of the machine at low or very low speeds. It is a well-known fact that
the stator-based estimators do not behave well at low speeds. Our analysis

supports this fact as 0), appears in the denominator of error estimates. This is

one of reasons that people prefer to use other methods for low speeds. However,
the most important point to be noted in this context is that most of the factors
determining the low frequency limitations are somewhat technology dependent.
The clear implication is that by improving the methods of implementation we can
get better results. This is one of the prime objectives of our work.

The stator flux and speed estimates tend to become more and more accurate as
the speed increases. There are no special considerations involved in this regard
except that at very high speeds we have to cater for some other issues such as
field weakening.

Finally machine saturation may become an important factor for extreme
operating conditions. Unfortunately the machine model with core saturation
incorporated in it becomes complicated and we will restrict our discussion and

analysis to linear or almost linear region of the magnetic hysteresis curve.

49

2.5.2. Starting of the induction machine

Any control scheme requires certain time interval before tracking of the
references actually begins. The filter approximation can become a problem in this
context. If the time required for the control to start tracking is greater than the
group delay of the filter, the machine may not start at all. In this situation, the
stator flux estimate starts to fall down after the initial transient, whereas the
actual flux is present. The apparent solution is to decrease the bandwidth of the
filter to the required level. It turns out that this method generally requires
bandwidths that are not feasible to implement. A common technique used in this
situation is to start the machine by using some other method. The controller
takes over at some appropriate time. We have implemented the same approach,

which is working well without any problems.

2.5.3. Speed Reversal

One of the challenges of the encoderless control of the induction machine is to
achieve speed reversal. This task is in fact a difficult problem even for
synchronous machines where the situation is not as complicated. The striking
feature of the scheme we are working on is its capability to do speed reversal.
The main problem is that while the machine is in transition from one direction of
motion to the other, it passes through the generating mode. The stator flux
estimator being independent of the rotor speed is robust to this transition.
However, the controller faces some difficulties. A simple remedy to the problem
is to use a low pass filter at the output of the speed estimator. The bandwidth of

the filter should be sufficiently large so as not to disturb the stability of the

50

system. The filter can be thought of sensor dynamics. Besides, a saturation

function should be used for the speed estimate.

2.6. Lowest Possible Speeds

Now we turn our attention to the lowest speed at which the machine can be
operated using stator flux-orientated control. This problem has no definite answer
because of a wide range of factors involved in the control scheme. If the analysis
predicts failure in the steady-state conditions then the scheme is expected to
break down. However, it is possible for the scheme to fail even if steady-state
error analysis does not provide this information.

The expressions derived in section 2.2 show us that the errors in estimates
depend on different parameters and conditions. In most of the cases the error is

inversely proportional to the stator frequency 0),, which is different from the rotor

speed. The relation between the two involves the knowledge of load torque,
which is unknown itself. However, at zero speed and no load torque we have

a), = 0. This implies that it is not possible to control the motor for these conditions

even if mechanical speed is available for measurements.

Stator frequency a), is generated by the controller. However, exact value of

a), may not be obvious in certain control schemes. In those cases, we can use

a), =i[tan" [2]] (2.93)
(It i,

the relation

51

where the differentiation can be done using high-gain observers. Another useful

approximation of equation (2.93) is given below
a), = — (2.94)

where )0 is the estimate of the stator flux angle as before.

In the presence of external active loads we may achieve zero speed operation. In
this case we have to generate electrical torque that counters mechanical torque.
The consequence is non-zero frequency of terminal voltages and currents.
Considering these factors, we conclude that the limiting factor is not the rotor

speed a). Rather it is the stator frequency 0),.

In order to have an accurate estimate of the minimum possible speeds, we need
information of quantities that are generally not available. However, error in 1“,,
gives us a rule of thumb. We have seen in section 2.2.5 that 1,, is vulnerable to

extreme conditions when compared to other current components. Any control

scheme would break down if i,, and 1:, have different signs. This follows from the

fact that we are trying to maintain 1,, at a certain level based on the estimate 1;).

The above discussion gives condition for failure of the scheme. Our main interest
is to have an idea of the lowest speed at which the control scheme remains
operational. Based on our work and experience on this scheme, we would say

that a good and sound controller should able to handle the situation when

s}..-
2 5"

 

6t“,

 

 

 

(2.95)

For an example of this discussion refer to section 3.5.2.

52

2.7. Discrete-Time Implementation

Development of computers and embedded technologies has lead to better
methods of implementing sophisticated and numeric intensive controllers. The
controller has to work in real time so the hardware should have enough speed as
required. One may think that computational speed is the main issue. However
having a deeper look at the problem, it turns out that there are other matters of
significance. All of these details must be kept in mind while designing a discrete-
time digital control of a continuous time analog system. Following is a brief

discussion of these issues.

2.7.1 . Discrete-Time Control of Continuous-Time Nonlinear Systems

There can be two possible alternatives for implementing discrete-time control of a
continuous-time system. We can design the control scheme completely in
continuous time and then discretize the controllers and observers etc. The other
method is to find a discrete-time equivalent of the plant and then work in discrete-
time only. Both methods are used depending on the nature of the problem.

The situation is much complicated if we are dealing with nonlinear systems. It is
generally hard to find a discrete-time equivalent of a nonlinear system. The same
is true for the complex model of an induction machine. With this limitation, we
have to restrict ourselves to the first option i.e. perform the design in continuous
time and then find discrete-time equivalents of the controllers and observers in

our design.

53

It is important to note that for most design applications the controller for a
nonlinear system is generally linear itself. In that case, we can make use of better
discretization methods such as the bilinear transform method. This is especially
true for the PID or Pl controllers. The trapezoidal-rule integrator (which is
equivalent to bilinear transformation of a continuous-time integrator) has superior
performance as compared to a simple accumulator. This may become significant
if numerical problems like round off and truncation are causing undesirable
results.

State observers are a part of most of the control schemes. An observer is in one
way or the other a replica of a part of the plant. In many cases the observer
would therefore be nonlinear. In certain situations the controller may be nonlinear
as well. Fonivard difference methods appear to be the simplest of all if a nonlinear
system is to be discretized. The difficulties associated with fonlvard difference
method can be overcome by using sufficiently high sampling rate. As the
sampling rate is increased, the different discretization methods tend to become
more and more similar. Unfortunately for nonlinear systems it is difficult to
calculate a lower bound of the sampling rate required for satisfactory
performance. In practice, some high sampling rate is chosen based on past
experience. For the case of machines, a sampling rate of 10KHz is thought to be

sufficiently high to implement any of the common control schemes.

2.7.2. Difficulties Associated with High Sampling Rates

One of the basic problems of implementing a discrete-time control is the choice

of the sampling rate. It turns out that if this choice is too high then one may face

54

with a different class of problems. This is significant and some designers have a
tendency to overlook it. We illustrate this issue with the stator flux estimator we
have employed. Let us consider the approximation in equation (2.32). The

bilinear transform method corresponds to replacing s by

__ -l
—2— 1 Z_, (2.96)
Td l+z

 

which results in

1 d Td(1+z")
5+0)

C I (2+Tafwc)_(2_'wac)z-l

\

(2.97)

where T, is the sampling interval. For high sampling rate and lower cutoff
frequencies 7,0), is small. This results in a stringent requirement for the number

of bits we are using for our discrete memory. This effect is called coefficient
quantization and results in a displaced location of the poles from the desired. The

worst case situation would arise in cases when
trunc(2 + 72,010): trunc(2 —Tda),) = 2 (2.98)

where truncation is done for the maximum number of available bits. We would
end up in implementing the exact integrator rather than an approximation by a
low pass filter. Leaving aside this worst case, there is yet another factor that is
critical. We know that the process of quantization introduces a noise in the data
being processed. An integrator accumulates information over a period of time. If
the level of quantization noise is significant, we may get undesirably large errors.

The integration done by trapezoidal rule or other sophisticated schemes must be

55

used under these circumstances. For a detailed discussion of quantization noise
refer to Oppenheim [16] and Antoniou[1].
The approximation of equation (2.42) is even more sensitive to numerical

problems. The discrete version is

S d Td (1+z'1)(1-z‘1)

\

(2.99)
(5+wc)2 [(2+T,,w,)—(2-T,,a2,)z“]2

The denominator of the right hand side involves addition or subtraction of (La), )2

from 4. This may require prohibitive number of bits as far as implementation is
concerned. To fully appreciate the problem lets consider the situation when
sampling frequency is 10KHz and the cutoff frequency is 1 rad/s. Equation (2.99)
results in

4.9995000374975 x10'5 (1 — z"1 ) (1 + z")
1.1999800009999551+0.99980001999852:'2

 

(2.100)
One may be tempted to round this expression to 7 significant digits (more than
the accuracy of 32 bit floating point) resulting in

4.9995x10'5 (1- z-1)(1, z“)
1.1.99982"+0.9998z'2

 

(2.101)

which simplifies to

4.9995x10’5 (1+ 2“)
1099982‘1

 

(2.102)

This shoes that we have totally lost the second order filter and end up in basically

implementing the first order approximation of the integrator.

56

2.7.3. Suggested Solutions

The results in the previous section may rather be surprising that even 32-bit
floating point may not be sufficient to implement certain filters.

One obvious solution is not to go for very high sampling rate. It would be in fact a
good idea to decrease the sampling rate as low as permissible. The other

possibility is to make use of down and up samplers. The concept is illustrated as

 

 

 

 

 

 

follows.
OUT
IN Desired
—> LPF —> [D —> Filter —> (U —- LPF —>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2-8. A possible scheme to handle high sampling rates

57

CHAPTER 3

COMPUTER SIMULATIONS

3.1. Importance of Simulations in Design

Control design is an involved task. In many cases complicated analysis is
accompanied by elaborate experimental setups. Whereas analysis provides the
basis of the design, it faces certain limitations. The first step is to mathematically
model an actual physical system. It is generally wise to work on a simplified
model, which approximates the reality with reasonable accuracy. This simplifies
the control design and its analysis. However, this approach may lead to serious
difficulties if the unmodeled effects play a significant role in the operation.
Furthermore, for most of the practical and physical control design problems, it
may be difficult to carry out exact analysis. Under such circumstances, we have
to make certain assumptions or resort to approximate analysis. When we come
to actual implementation, all the unmodeled effects are physically present and
our assumptions and approximations may not be appropriate. To summarize, we
can say that there is a gap between analysis and the experiment and it is
generally not possible to fully bridge it.

The concept of simulating a control system helps reducing the distance between
analysis and experiments. We can incorporate many of the unmodeled effects in
the simulations. Naturally for a physical system, it is not always possible for all

the state variables to be available for online measurement. Measurable variable

58

get corrupted by observation noise and related issues. For certain systems, the
tuning of the gains may be critical as inappropriate values may have a damaging
effect on the system. Some experiments may be costly to conduct or there may
be certain associated hazards. Computer simulations provide a neat solution to
all of these problems.

At the first look it appears that we can overcome most of our difficulties using
computer simulations. It turns out that simulations themselves face certain
problems. The hardest part is to develop a simulation setup. There are some
intricacies of the numerical solvers and a number of parameters have to be
adjusted properly. Computational speed of the computers is of prime importance.
Sometimes it becomes exasperating to run a simulation even on very fast
computers. The computers appear to have slowed down for simulations of
moderate complexity. Sometimes we may need to modify the actual system just
for the simulations to work. These changes should have a minimal effect on the
system. Simulation of a physical system is a skill in itself. It requires experience,
patience and definitely a sound knowledge of the mathematical model and its
behavior.

On of the important points is that simulation gives an insight of the system. It is in
fact a tool to learn and understand the design of the control system. Sometimes,
the details obtained from a simulation may produce critical information that one
has missed in analysis or even in an experimental setup.

We would say that even being an idealization of the real systems, simulations

constitute an important step in the design of a control system. The extra time and

59

effort involved ultimately result in better results and reduce the overall time of the
experiment and improve its quality.

We have done extensive simulations to demonstrate most of the facts discussed
in the analysis. We have used SIMULINK as our simulator. Most of the
secondary effects such as machine saturation, inverter non-linearities, sensor
dynamics etc, have not been catered for. We want to concentrate on the core of
the design rather than spend a lot of time on effects that either do not contribute
significantly in the working of the machine or are automatically compensated for
by the closed-loop nature of the control. Whenever possible we will try to
simulate the part of the system we are interested in. This approach is more
illustrative and logical. In certain cases we will like to simulate the machine in
open loop. This would be done to demonstrate certain behavior that either
becomes unclear or may actually be compensated for by closing the loop.
Machine parameters that we have used in simulations are from a physical

induction machine. They are listed below.

 

 

 

 

 

 

Parameter Value
Stator Inductance, Ls 0.595 mH
Rotor Inductance, Lr 0.608 mH
Magnetizing Inductance, Lm 0.530 mH
Stator Resistance, Rs 0.032 £2
Rotor Resistance, Fir 0.027 9

 

 

 

 

Table 3-1. Machine parameters used in simulations

60

3.2. Pure Integration

Stator flux estimation schemes based on pure integration are sensitive to
disturbances, initial conditions and transient offsets. This unwanted behavior

associated with pure integrators is demonstrated in the following simulations.

3.2.1. Equivalence of Open and Closed Loop Integration

According to our findings, the two methods of stator flux estimation generally
referred to as open and closed-loop integration are equivalent. This fact is
contrary to the idea that closed loop integrators perform better. There are two
different notions in this regard. Leonhard [12] has suggested that by using the
closed-loop integration, drifts are reduced to normal offsets. Whereas, Vas [21]
has proposed robustness of the closed loop scheme against low frequency
disturbance. We develop simulation setup for both DC drift and low frequency
disturbance. The model of the estimator used is simplified and is driven by two-
phase sinusoidal signal that would actually be equivalent to the voltage behind
the stator resistances. Three estimators are used for each simulation. One of
them is ideal and the other two are the open and closed loop schemes with a

disturbance term. The principle is illustrated as follows

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

————~ IDEAL 0
, ESTIMATOR O
(”9

“(I -iaRs
t——' CLOSED O
(9.. + , LOOP EST. 0

“b —ibR5
* OPEN LOOP .0
DISTURBANCE —> EST'MATOR 0

21

E) Q“)

b) “5»

 

Figure 3-1. Comparison of open and closed IOOp estimators

The closed loop estimator has the following form.

 

 

 

L

 

us — R318

 

 

-.I'P

 

 

 

um

“34 —

-Ri

:sq >

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>2

 

Figure 3-2. Closed loop stator flux estimation

62

 

 

The open loop estimator has the following form.

 

 

 

 

u,—R,i, —> I .__> to
1. POLAR ——> 4.

 

 

 

 

 

 

 

 

 

Figure 3-3. Open loop stator flux estimation

The results with DC and low frequency disturbances are illustrate in figure 3-4
and figure 3-5 respectively. The important observation is that the open and
closed loop stator flux estimation schemes are behaving exactly the same way.

We have demonstrated the equivalent of open and closed loop integrators.

63

 

 

Magnitude (pu)

Actual Flux

 

.1.
N

cub
I
J

4

2

 

Magnitude (pu)
.° .0
(o d

(n

 

Phase (rads)

 

 

O

50

100
Estimation based on Closed Loop Int

 

b O)

Magnitude (pu)
"3

 

O

4

Phase (rads)
O N

-2

 

 

O

50

100

 

h

Magnitude (pu)
'5’

 

O

f

O .

-2.

 

 

 

 

\l

 

 

 

 

 

 

 

 

0
egration with DC Disturbance

50 100

 

 

v

A

 

 

 

(1......

50 100

 

Estimation based on Open Loop lntegzation with DC Disturbance

 

 

O

50

100

0

Phase (rads)

-2

 

 

 

A

 

—4

ZWWWVVVVVVV‘N
0

50 100

Difference between Open and Closgq Loop Integration Schemes

 

0.05 r

J

 

—0.05 '

 

-0.1

 

 

50
Time

100

 

.0
o
01

L

 

Phase (rads)
O

-0.05 ’

 

—0.1

 

 

50 100
Time

 

Figure 3—4. Equivalence of closed and open loop integrators for DC disturbances

64

 

 

 

Magnitude (pu)

Actual Flux

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2 4
g 1.1 . « g 2
9 S
'3 cu
0’ .c
“I . - r
2 0.9 a. 2
0.8 ‘ -4 a
0 50 100 0 50 100
1 4 Estimation based on Closed Loop Intzgration with Low Freq Dist
’5‘ A
2 4
31.2 » §
8 s,
.3 9 0
(D
: as
g 1 .c 2 .
2 “- '
0.8 * - ‘
0 50 100 0 50 100
1 4 Estimation based on Open Loop Integration with Low Freq Dist
’5 A
o. w 2 1
:12 » g
C
c» 1 2
CU _ .
2 a. 2
0.8 ‘ — -
0 50 100 0 50 100
0 1 Difference between Open 81 Closeg l1_oop Integration schmemes
0.05 J 73‘ 0.05)
S
0 T; 0
(I)
2
-0.05- I a. -0.05» J
—0.1 ‘ -0.1 ‘
0 50 100 0 50 100

Time Time

 

Figure 3-5. Equivalence of closed and open loop integrators for low frequency disturbances

65

 

3.2.2. Unknown initial Conditions

It is almost impossible to have an exact knowledge of the states of the system
when the system is first powered up. Pure integrators have no leakage and
therefore retain any information either actual or unwanted. if this information is
false, the results produced by pure integrators would certainly be wrong
throughout the operation. in other words, a pure integrator lacks the ability to
correct itself. The false information could be a result of error in initial conditions or
some unwanted transient behavior. This behavior is demonstrated in the

following setup.

 

 

 

 

 

 

 

 

 

 

 

 

(,9 1 ESTIMATOR ._____O ,1:

, (CORRECT

ua-laR. + lNiT COND) 0 p

ub-ibR, __, ESTIMATOR ____0 j
(FALSE INIT j

@ > CONDITONS) ——-—O p

 

 

 

 

 

 

Figure 3-6. Stator flux estimation with pure integrator and false initial conditions

The error in the initial conditions is 50%. We observe that the estimator results in
a 50% error. This result appears to be striking but if the zero leakage property of
the integrator is kept in mind, it is not unexpected. Both the open loop and closed
loop integration schemes face the same problem. The simulation results are

illustrated in the figure 3-7.

66

 

 

Actual Flux (Exact Initial Conditions)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2 4
A1.1 * 2
a ’a‘
‘9 S
5 cu
3 E
E 0.9 i -2 .
0.8 ‘ _4 .
0 50 100 0 50 100
Flux Estimation with 50% error in initial conditions
2 4 .
A1. . 2
.a 5 a
8 .9;
3 1 8 o .
“E m
g .c
o.
5 0.5 -2.
0 * —4 ‘
0 50 100 0 50 100
Error due to Inaccurate Initial Conditions
1 - 1 -
3 0-5 A 0.5
3 4’3
8 .9,
g 0 8 0 ’
5 -o.5 -o.5
-1 - —1 ‘
0 50 100 0 50 100
Time Time

 

Figure 3-7. Response of a pure integrator with false initial conditions

67

 

3.2.3. Transient Behavior

Transient behavior poses another problem for a pure integrator. Even if we
assume that there are no errors in the initial conditions and there are no
disturbances, the estimator based on pure integrators is vulnerable to inexact
knowledge of machine model. Let us consider the case if the nominal stator
resistance Rmdiffers by 10% from the actual stator resistance R,. The estimator
would follow a different trajectory from the ideal one. If we view the estimator at
some time tl >to with the states of the estimator at time t‘ as the initial conditions
for t>r,, then it is not difficult to correlate this situation with false initial
conditions. The estimator will not be able to settle down and there would be a
bias as well given by equation (2.23). We investigate the behavior of the
estimator using the scheme given in figure 3-8. in this case we have to simulate
the complete system to have a good idea of what is happening. The results are

given in figure 3-9 where the actual quantities are represented by dashed line.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3-Phase
Sinusoidal > (RD
Voltage Supply
ilemu'nal “terminal
32 Phase
Transform ”II _o 2!
Stator Flux
Estimator
32 Phase 1. -—° 13

 

 

 

 

Transform

 

 

 

 

 

 

Figure 3-8. investigation of transient behavior of stator flux estimator using pure integrators in
presence of parameter uncertainty

68

 

 

Estimates Errors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 0.2
2 0.15
s .—
g 0 g 0 1
Q _
-2 0.05
-4 0
0 0.5 1 0 1 2 3 4 5
Time

 

Figure 3-9. Sensitivity of pure integrators to transient offsets

69

 

3.3. Stator Flux and Speed Estimation

It may become difficult to study errors associated with an estimator if it is used in
a closed-loop control system as the controller or other components can influence
the estimator. If we want to concentrate purely on the behavior of an estimator, it
would be a better to investigate an open-loop system with no feedback from the
output of the estimator. However, since we are now investigating the behavior of

the machine as a whole, we would simulate the full system rather than a part of

it. The block diagram of the scheme is illustrated as follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3-Phase
Sinusoidal ®
Voltage Supply
iteminal ”terminal
3-2 Phase ‘
Transform u I
l
Stator Flux
__ Estimator _
‘ 3-2 Phase ‘3 I
Transform
.__ e-m
_—_: Mechanical
Speed
Estimator

 

 

 

 

 

 

b) EU)

a.

 

Figure 3-10. Block diagram of stator flux and speed estimation in open loop

70

 

Our eventual aim is to use the estimators in a closed-loop control system, where
the estimator dynamics become part of the system. In this open-loop setup,
estimator dynamics and plant dynamics are independent of each other.
Therefore, the influence of the estimator on the dynamical behavior of the closed-
loop system cannot be fully understood. However, we can infer about certain
phenomena. One of the important things in this regard is the settling down of the
estimator. The most significant information is about the steady-state errors, which
remains the same if the estimator is used either in open or closed-loop. The
analysis done in chapter 2 on the steady state errors can be verified in the
simulation of an open-loop observer with much more convenience. _

We will consider three sources of error in the operation of the machine. These
are due to the approximation of the integrator with a filter and uncertainties in
stator and rotor resistances. Stator flux estimation is independent of the last one.
The stator flux estimator is also affected by low frequency disturbances.
However, due to lack of information of the exact waveform of the disturbance we
cannot have a simulation setup that will give an idea as to how the actual
physical system will behave.

The estimates are given in the left column of a figure with the corresponding
errors in the right column. We give two graphs in each plot corresponding to two
different uncertainties in the same parameter. The estimator performance is
dependent on the speed of the machine. All the results would be given

separately for high (a), =100rads/s) and low (cos =10rads/s) speeds. We start off

with the integrator approximated by a filter.

71

3.3.1 . integrator Approximation

We had considered first and second-order filter approximations of the integrator
in section 2.2. The only difference between the two is the ability of the second-
order filter to attenuate disturbances. Since we are not simulating disturbances, it
is sufficient to restrict ourselves to the first-order filter given in equation (2.32).
Our primary objective is to study the effect of the bandwidth of the filter on the
accuracy of the estimated stator flux and the ability of the estimator to overcome
false initial conditions or transient offsets. We investigate two different cases with

cos/w, =10 and (us/w, =100, where the former is represented by the solid line
and the later by dashed line. Here a), is the supply frequency and a), is the cutoff

frequency of the filter that is approximating the integrator. We assume exact
knowledge of the machine model. The results are given in figures 3-11 to 3-14.
Observations

The filter with larger bandwidth has faster settling time but larger steady state

error as compared to the one with smaller bandwidth. The errors in in, and i“, are
small in all cases as predicted by equations (2.61) and (2.67). However the error
in is, is much larger. In fact for low speed operation, is, and 1:4 have different signs

and we have a situation similar to the one depicted in the worst-case scenario of
figure 2-4.

The error in speed estimate as given in equation (2.92), depends on certain other
quantities. The plots include all the relevant details.

The results obtained through simulations verify the derivations done in chapter 2.

72

 

 

Estimates

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p (rads)
‘3

 

 

 

 

 

 

 

0 0.5
Time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3-11. Errors due to integrator approximation at high speeds (Part I)

73

 

 

 

 

.5
C
o

co (rads/s)
8

Estimates

 

 

 

 

 

 

 

 

 

 

 

0
5
2."?
3. 0'
B m _________
>
-5 ‘
0 2 5
5

 

 

(volts)

0
l
l
l
I
I
I
I
I
I

- I

 

 

 

 

 

 

 

3
>
-5 .
0 2 5
5
4.
’0?
S.
_rn2
2.
1L
0 .
0 2 5
Time

10

o “MWAunv as —————————— 1

3

I6v,l

-O.5
O

-10

Errors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time

 

Figure 3-12. Errors due to integrator approximation at high speeds (Part II)

74

 

 

 

 

 

 

 

100

ms (amps)
5‘ 8 8

8

100

(amps) i8d (amps)
o 8

p (rads)
0

Estimates

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
0 10 20 30 40
;
0 10 20 30 40
‘mMeanu ____________
0 10 20 30 40
0 5

Time

0.15

Errors

 

 

Mun ———————————

 

 

1020304050

 

 

 

 

 

 

 

 

 

 

 

 

10 20 30 40 50
Time

 

Figure 3-13. Errors due to integrator approximation at low speeds (Part I)

75

 

 

(volts)

Estimates Errors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 . a . E 10 -
A15 5 .
$
"’ I
.310 {g 0 With —————————————
8 51 1 _5

o A A A A -10 A A A 2

0 10 20 30 40 50 0 10 20 30 40 50

5 1
A 0.5
.52
.3, 0 003 o ————————————————
>3

-0.5

-5 A A A A -1 A A - A

0 10 20 30 40 50 0 10 20 30 40 50

1 - 0.5
0.5;?

0’ 1 (3g 0 WQDOQ —————————————— 1
-0.5
—1 A A A A -o.5 A A A A

0 10 20 30 40 50 0 10 20 30 40 50

5 0.5

4.

33
3, E“ o
22

1.

0 ‘ ‘ —o.5 A -

0 20 30 0 20 30

Time Time

 

 

 

 

 

 

 

 

Figure 3-14. Errors due to integrator approximation at low speeds (Part ii)

76

 

 

3.3.2. Uncertainty in Stator and Rotor Resistances

The stator resistance turns out to be an important parameter for stator based
estimators. To investigate the errors in the flux and speed estimators due to
uncertainty in stator or rotor resistances, we have to incorporate some
approximation of the pure integrators or otherwise we will face the problem of

transient offsets. In the previous section we have seen that the ratio (us/a), = 100

causes small steady state errors but requires a longer settling time. We will
simulate the estimator with this ratio and study the effect of uncertainty in stator
and rotor resistances. We consider two cases for stator resistance with

Rm =1.1R, and Rm =1.2R, and two cases for rotor resistance with Rm =1.25R, and
Rm =1.5R,. The results are given in figures 3-15 through 3-20.

Observations

The errors in i"u and in, are small but more pronounced as compared to the case
with exact knowledge of stator and rotor resistances. The error in in, is even

larger. The steady state errors are consistent with equations (2.61), (2.67) and
(2.92). The results obtained through simulations verify the derivations done in
chapter 2.

Since the flux estimator is independent of the rotor resistance, we do not give the
results for flux errors due to uncertainty in rotor resistance.

One important observation is the dynamical behavior of the estimator, which now
exhibits an overshoot before settling down to some steady state. This

phenomenon could lead to instability of the closed loop system.

77

 

Estimates Errors
120 . . -

 

 

 

 

 

 

 

 

 

 

 

 

(amps)
3

(n
O

 

'sd

 

8

 

 

 

 

 

 

 

 

(amps)

 

 

 

D 1
-‘-——————
q

 

 

 

 

 

 

 

 

 

p (rads)
o

 

 

 

 

 

 

 

 

 

 

0 0.5 1
Time

 

 

 

Figure 3-15. Errors due to integrator approximation and uncertainty in stator resistance at high

speeds (Part I)

78

 

 

Estimates Errors
1 00 . . ~

 

 

 

a) (rads/s)
8

 

 

 

 

 

 

 

 

 

 

vsd (volts)
0

 

 

 

 

 

 

 

 

(volts)
0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3
>
-5 2 -0.5
O 1 2 3 4 5 1 2 3 4 5
5 1 1 . .
4 \
E 0.5
'5 3 '7» 1 _________________
> >
Z 2 E.
_>_"’ 0 ’ ‘
1
0 -0.5 ‘
4 5 0 1 2 3 4 5

 

 

 

Figure 3-16. Errors due to integrator approximation and uncertainty in stator resistance at high

speeds (Part ll)

79

 

 

Estimates Errors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 . - a - 10
90. l ,
E80 5
5 IE" 0» __ —
W70r 0°
3 5.
60, ‘ _
50 A A A A -10 2 A A A
o 10 20 30 4o 50 o 10 20 30 4o 50
100
13
CL
350
3
._ _5,
o A A A A -10 A A A A
o 10 20 30 4o 50 o 10 20 30 4o 50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time Time

 

 

 

Figure 3-17. Errors due to integrator approximation and uncertainty in stator resistance at low

speeds (Part I)

80

 

Estimates

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Errors
20 - . ~ . 0 " - .
l
A15’ ‘0
& Mnuao ————————————
m DU. ——————————————
E10 I" 3 ‘5
V l
3 5 +
0 ‘ ‘ ‘ ‘ -10 ‘ T ‘ ‘
0 10 20 30 40 50 0 10 20 30 40 50
1 . 1 -
A 0' 0.5 I
:‘é’
3-1 k B o.
no
>8
-2. -0.5 ‘
—3 A A A A -1 2 A A A
0 10 20 30 40 50 0 10 20 30 40 50

 

 

(volts)

 

V

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 - 0.1
.05

g 2 _ ° _________________
3. j a” 0
—. 1’
->- —o.os»

o A A 2 A —o.1 A A A A

o 10 20 so 40 50 o 10 20 30 4o 50

Time

Time

 

 

 

Figure 3-18. Errors due to integrator approximation and uncertainty in stator resistance at low

speeds (Part II)

81

 

 

 

Estimates Errors

fi"

.3
C
o
8

 

u) (rads/s)
8

O

 

 

 

 

 

 

 

 

O—L

 

oits)
l

ngw
M

 

 

 

 

 

 

 

 

   

(volts)

asqro on

V

 

 

 

O

 

 

 

 

(JD->01
3|
-L

IV | (volts)
|5v

1x)

 

 

 

o'cn

 

 

Cd

 

 

 

 

Figure 3-19. Errors due to integrator approximation and uncertainty in rotor resistance at high

speeds

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Estimates Errors
10 f . s . .
4
E If ---------------
(I)
g 5 3 2
8 \
0 . \s ________________
0 A L . . A . . .
0 10 20 30 40 50 0 10 20 30 40 50
1 - - 1.5 -
0 .
’2?
a -1
2.
3—2’ L _______________
>
-3. .
-4 A A A A
0 10 20 30 40 50
1 f - r v
0.2»
2; 0.1 k
20.5 “:53 0A \ —————————————— ~
)8 ____________ _o,1l .
0 -0.2A A
0 10 20 30 40 50 0 10 20 30 40 50
3 1
F
A f _______________
a 2~
9 ">705 _________________
I a
.2?“
0 0’ 1
0 10 20 30 40 50 0 10 20 30 40 50
Time Time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3-20. Errors due to integrator approximation and uncertainty in rotor resistance at low

speeds

83

 

3.4. Closed Loop System with Feedback from Measured Speed

Now we turn our attention to the speed control of the induction machine using
measured speed in the feedback loop. The block diagram is given in figure 2-5.
We start with a simulation of the system under ideal conditions; there are no
uncertainties or sources of errors. This simulation would serve as a standard for
other cases. Since our ultimate goal is the encoderless control of the induction
machine, we will give the plots for speed estimator even though it is not being
used in the control loop. This will help us have some visualization of the
performance of speed estimator.
The inability of the flux estimator to work at zero speed makes it necessary that
the flux and speed commands be given simultaneously or within some maximum
time. This is somewhat different from the rotor-based schemes where the flux
command precedes the speed command. The main issues that would be
demonstrated in this section would be the ability of the control system to achieve

. Starting

0 Tracking of the speed reference

0 Speed reversal

. Ability to handle changes in load torque
The load torque comes from both frictional losses and active loads. Whereas

ins and a) are the quantities we want to control, is,If and in, are fundamental to field
orientated control. The plots for u“, and um are important as they serve as the

control effort. We also give plots for I; because of the large errors in it. The

results are illustrated in figure 3-21.
84

 

 

rads/s

Volts

 

 

150 I I I I I I I I I

 

 

 

 

 

 

L. -
— i
_ms
l
ms.ref
1 l l 1 J l l J

 

 

4 6 8 10 12 14 16 18 20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I 2
_— (D -
- ' west .1
$ref -
-. L 0
l 1 l L L I l 1 l -1
0 2 4 6 10 12 14 16 18 20
2% I I I I I I r I
— I
150i- 'Sd -
T ’ sq
too—L -
50./\\ i &i t Ism‘t
./ '\ _-_, ___________ / sq SCI,GS
01 H _ . f. __________________
l /
_50 l L l l . l l l l
0 2 4 6 8 10 12 14 16 18 20
10 r T I I I I r I T
I \ usd
5L \ f ————— _ _ u 7
I .~——--- 1.x ..
0 I q
l ,_ ___________
_ _ | ’ .
5 |l
l
“'10 L l l l l l 1 I l
0 2 4 6 8 10 12 14 16 18 20

Time (Seconds)

 

Figure 3-21. Controller performance under ideal conditions using measured speed

85

 

3.4.1 . Controller Performance in Presence of Errors

The actual task and challenge to design a controller is its ability to work in an
acceptable manner in the presence of disturbances and uncertainties. In this
regard we study different cases. The following family of plots gives a good idea of
the ability of the controller to handle errors and uncertainties.

The reference speed is tracked in almost a perfect manner. The dynamic
performance of the controller however, does not remain the same as compared
to the ideal case. The tracking of current components experiences errors
corresponding to errors in the estimates. if the resistances are perturbed too
much the control scheme breaks down. We have tried to capture the effects
when the scheme is working but has started to exhibit symptoms of instability.
The results are illustrated in figures 3-22 through 3-25. The last figure combines

all the sources of errors.

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rads/s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

200 I I I I I

— i
150-L isci .
In J . .sq -
6100 fl ' 'sq.est H
< 50- \\ -

\'-~ mmmmmm r" an'l
o~ I J,,,,......,.,.,.,._...,_..._._._,=
i -"

-50 l l l l A l l l l
O 1 2 3 4 5 6 7 8 9 10

10; \ I I I I I I I I I

 

 

 

 

 

 

Volts
o 01
/
I
I
l
l
I
I
I
l
l
l
I
. I
cc
3 8

 

 

 

l
I
I
I
|
_10 l l L l L I l l J
5

Time (Seconds)

 

 

 

Figure 3-22. Controller performance with integrator approximation ((0, =lrads/s) using

measured speed

87

 

 

 

 

 

150 I I I I

 

 

 

 

 

 

 

 

 

 

 

 

_ ims

i

ms,ref
3,0,, ..........................

50 . l l J- J 1 4
0 2 4 6 8 16 18 20

8
'o
E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Volts

— — — — — — — — — 1

I N
-5 — ' I .
I in"
l l l l I L l l l

2 4 6 8 10 12 14 16 1 8 20
Time (Seconds)

 

 

 

 

 

Figure 3-23. Controller performance with integrator approximation (60, = 1 rads/s) and

Rm =1.05R3 as an uncertainty using measured speed.

88

 

 

150 I I I I I I I I r

 

 

 

 

 

 

 

 

 

2 4 6 8 10 12 14 16 18 20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q
(0
'O
S
2m I I I I I I I I
— i
150 isd .
"' " 80
In 100 w ' F
g, ,\ 'sq,est
< 50’ \ ‘
'fi-vn. —-.~..—e’~m.“~— “~~—‘
0L ! p*"'"'*““"~'*"""“'~""""
_50 l l l I I l. l l l l
0 2 4 6 8 10 12 14 16 18 20
10 " I I I I I I T I I
l — U
ad
5 ‘ -_ _____ 4
co \: _____ —1\ - - "Sq
E OI . -
I .... ———————————
_5I- I I a
II’
_10 I l l l l l l l l
0 2 4 6 8 10 12 14 16 18 20

Time (Seconds)

 

 

 

Figure 3-24. Controller performance with integrator approximation ((0, =1rads/s) and

Rm =1.25R, as an uncertainty using measured speed.

89

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rads/s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m8
FT:-
/’
I '1
I
| -
|
\
I-
|
I
I-
l
I
ll
'-
CC
88

Volts
0

III N” —————————

-5 . I [WW _
I In"
I J l l l l l

0 2 4 6 8 1 0 12 14 1 6 1 8 20
Time (Seconds)

 

b
-

 

 

 

 

 

Figure 3-25. Controller performance with integrator approximation (0), = l rads/s ), Rm =1.05R,

and Rm =1.1R, as uncertainties using measured speed.

90

3.4.2. Operation at different speeds and loads

We study the operation of the system at various loads and speeds. In the first
case we keep the active component of the load constant and vary the speed. The
reference speed starts off with some high value and then reduces to low speed
with a staircase waveform. The change in load torque is due to frictional losses. It
is easily observed that the controller is working well for different speeds and the
step changes in the speed reference is not a problem. The magnetizing current is
maintained at the desired level except for minor transients during the speed
steps.

in the other case the reference speed is kept constant and the active component
of the load torque is changed. The speed tracking is once again achieved whilst
maintaining the required magnetizing current.

One important observation in this regard is the change in i“, with load torque.

This is caused by the decoupler and is required due to the coupled nature of the
direct and quadrature currents in the stator frame of reference.

The results are illustrated in figures 3-26 and 3-27.

91

 

 

rads/s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

110~ -
100E. .................... ..
90E» — {ms .
80: lms.ref
0 5 10 15 20 25 30
II I I I T I 5
100'”3 . _______ :4
1. . .............. 3
I E
.r 2 z
I
I. ‘
0
0

 

 

 

 

 

 

 

 

30

 

 

 

 

 

 

 

 

 

 

5 1 0 1 5 20 25 30
Time (Seconds)

 

Figure 3-26. Controller performance at various speeds using speed measurements

92

 

 

 

 

120 ’ I I r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

110* ‘
2
E100... ....................................................... -
90E- _ ims ~
E ' IrrIs,ref
80' 1 I L I 1
0 10 15 20 25 30
80 ' I I f I I 5
I 5 -
mH“-_--———-_——\-___-__—. ‘4
(I) ' " V'— ‘k_ ‘3,- — (D '13
% 40E- (0 ‘
g I r _______ '" ' west ~2
L ______________ ref -
20E.-—-J'—‘ _______ r “'TL ‘1
o J 1 l I 1 O
0 5 10 15 20 25 30
..../........' ,, , isq,est
o i l 1_ l l
0 5 10 15 20 25 30
10 I I I I I
l ______
5 1‘.— _______ _ _ _ _ / - — —' — -I
£2
‘5 0 ‘
>
-5 — ”9d -
_ _ Usq
_1o 1 l J l 1
0 5 10 15 20 25 30
Time (Seconds)

Nm

 

Figure 3-27. Controller Performance for different loads using measured speed

93

 

3.4.3. The Role of Decoupler

We had developed a decoupling scheme in Section 2.3.3 and used it throughout
the simulations of the closed-loop system. One may be tempted to use

techniques with high gain to overcome the problem of interdependence of im and
isq. it turns out that these techniques can provide a different solution than direct

decoupler but at the cost of much higher control effort. The closed-loop system is
simulated without the decoupler and with higher gains for the speed and flux
controllers. Other than this, the system is working under ideal conditions without
any errors and uncertainties.

The transient response of the system is much inferior to the case in which a
decoupler is present. The settling time is longer and the flux build up is affected a
lot in the region where the machine is accelerating to reach the desired value.
The currents that flow during the transient period are large. The same is true with
the voltages that have to be applied in order to achieve the desired results.
Performance would have been worse if we had saturated the terminal voltages to
ilovolts.

As a final comment, high gain techniques are suitable for cases where we want
to overcome some uncertainty. However, if we have complete or even partial
knowledge of the uncertainty, it would be better to use this information. This is
exactly what the decoupler is doing. Though the decoupler is based on estimated
quantities but we have seen in many simulations that it is working well even in

presence of errors and uncertainties.

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

150 I r I I I
”100. ..........
Q L-
E w
50 — Ems -
lmaimef
o I L I I I
0 5 10 15 20 25 30
I 2
—— 0)
0.)
ref -1
g _. TL _
‘o
‘3
-.-._-.--.--.j°
' -1
25 30
_ fed
_ -i '
a i...
E 100’ , ' sqpst :
< ___,__.--_.-j\
0: ‘.—”-.—_. ._ !—-—._.__._._._—.—fi
\I
—100 ‘ 1 ‘ ‘
0 5 10 15 20 25 30
50 I I I I I
————— — - usq
f \\ _ U
sd
5. x _
o 0 __"‘"'"T’_—i —- ————————— 1
> | I
l ,/
l/
_50 l I I I 1
0 5 10 15 20 25 30

 

 

 

Time (Seconds)

 

Figure 3-28. Controller performance without a decoupler

95

 

3.5. Closed Loop System with Feedback from Estimated Speed

In this section we would replace the measured speed in the feedback loop with
its estimate. The scheme is illustrated in figure 2-5. It is natural to think that we
may loose the accuracy in tracking the speed reference. The goal in this section
would be to investigate the associated errors and effects. However, the dynamic
performance of the controller with feedback from the estimated speed should be
acceptable.

Once again we start with the simulation of the system under ideal circumstances
with no errors and uncertainties. For this idealization, the two schemes are
almost identical.

An important observation is the behavior of the estimator itself. There is an

improvement in the dynamical behavior specially at the starting of the machine.

96

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

150 I I I I I I I l I
8
E 1m .......... L__-
. —— i
. ms
: i
I ms,ref
50 I I I I I I I I I
0 2 4 6 8 10 12 14 16 18 20
I r I I I I I I I 2
— (D .
- - west .1
$ref _
_. L O
I I I I I I I I l -1
0 2 4 6 8 10 12 14 16 18 20
2m I I I I I I I I
150TL — :59 ‘
T " sq
r01 . . - _
CE 00 /‘ 'sq.est
< 50 3/ \ .
i \_____._J-_—_"_]
o . . /._._.._.-..-._._._._._._.1
I /
_50 I I I I l ' I I I I
0 2 4 6 8 10 12 14 16 18 20
10 ‘ I I I I I I I I I
I \ - - ”sq
m 51 \_______" —————— [\ _usd_
z | -
3 ° . ____________
_5- l,/ ..
l
_10 I 1 I I I I l I l
0 2 4 6 8 10 12 14 16 18 20

Time (Seconds)

 

Figure 3-29. Controller performance under ideal conditions

97

 

3.5.1. Controller Performance in Presence of Errors

We are basically controlling the speed of induction machine. When the measured
speed was used in the feedback loop, we could literally eliminate any steady
state errors in the speed from the desired. That problem was basically the
stability and robustness of the system in presence of errors. The situation in
which measured speed is replaced by its estimate is however different. The
accuracy and dynamics of the estimator depend on certain other factors. We
have already discussed these issues at length.

Four sources of error are considered in the following simulations. These are the
error in flux estimation due to integrator approximation, uncertainty in stator and
rotor resistances and the influence of high-gain observer. It is more illustrative to
consider one problem at a time. However, we would keep the integrator
approximation in most of the cases to be as close as possible to the physical
situation in which it is not possible to have a pure integrator. The results are
illustrated in figures 3-30 through 3-34.

Observations

The control scheme is able to achieve its objective of encoderless control. As
predicted by the analysis done in chapter 2, the actual speed settles down with a
steady-state error in it. The stator resistance is seen to be the most sensitive of
all parameters. On the other hand rotor resistance and high-gain observers have
minimal effect on both the dynamic performance as well as the steady-state
errors. Figure 3-34 combines all the effects. In this situation, the dynamic

performance is worse than other cases and the steady state error is the largest.

98

 

 

1m . . e . . , , , .-

 

 

Amps

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I I I T I I 2
Q
(I)
'O
E
200 I I I l
150- _sd .
I-I sq -1
5100 k- W ..
< 50~I \.\ .
I \-- ------- ’"“““I
I /
L I 1 I I I 1 I L I
-50
0 1 2 3 4 5 6 7 8 9 10
10' I I I I I I I I I

 

 

Ir\ ——u

 

 

 

 

 

Volts
O 01
I /
/
I
I
I
|
l
I
I
I
|
I
I
l
C
B 8

 

 

 

Time (Seconds)

 

Figure 3—30. Controller performance with integrator approximation (a), = 1 rads/s )

99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ ims
i
m ms,ref
g100i- ...........................
< 2
50. l l l l l l 1
o 2 4 6 8 14 16 13 20
I I I I I I I I I 2
50EOOMS—Oo-o—o-‘oywfi.~.-.~.‘ — (0 d
\i 0.)
$ (0
U)
B T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Volts
o

 

 

 

_10 I I I I III 4 I I I

0 2 4 6 8 10 12 1 4 1 6 1 8 20
Time (Seconds)

 

 

 

Figure 3-31. Controller performance with integrator approximation ((06 =1rads/s) and

Rm =1.05R, as an uncertainty

100

 

 

150 I I I I I I I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I
“fad .I
——I
.sq q
.I —
sqem

 

-

o
\/‘.I'H’“'H'H'H‘H'H'H'ul
c

 

l l l l

12 14 16 18 20

 

I I I I

—U

 

 

 

 

 

 

 

 

 

2 4 6 8 10 12 14 16 18 20
Time (Seconds)

 

 

 

Figure 3-32. Controller performance with integrator approximation (we =1rads/s) and

Rm =1.25R, as an uncertainty.

101

 

 

 

 

 

 

 

 

 

 

 

 

150 I I I I I I I I I
3
E100:- ........ M
< _ .
: _ Ims"
: i
: ms,ref
50 1 l I l 1 1 1 l l
O 2 4 6 8 10 12 14 16 18 20

 

 

 

 

 

rads/s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I T I I
— i
.w d
— — I
'sq -
__ I -—
SQ.6$I
fi—_---_—__'I
' f‘ ———————————————— —I
. I
I I I L I J I I

 

 

 

6 8 10 12 14 16 18 20

 

 

 

 

 

 

 

 

 

 

 

10 I I I I I I I I I
—U
8d
5 [A I\ u .
A ________________ \ _____________ _- h
3 I " I sq
‘5 o ‘
>
-5 l -1
_10 l I J l l l l

2 4 6 8 1 0 12 1 4 1 6 1 8 20
Time (Seconds)

 

Figure 3-33. Controller performance with error due to H60

102

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I I r r I I 2
50;..-m.--......c._._.-._.......‘ —(1) -
.\ __ — - west .1
fl” j '''''''' w
'8 0 -------- I I'OI ‘ E
, x ......... 0
_50_ I. . q».h_.__.._._.._......._....._.
l l l l l l l _1
O 2 4 6 8 14 16 18 20
200 I I I I I I I
In 100_ _ - Esq :
a. /\ I
E I é _ sq,93I
_1m 1 l L l l l l

 

 

 

 

14 16 18 20

 

 

 

 

 

 

 

Volts

 

 

 

 

_10 l l l l I 1 l l
0 2 4 6 8 1O 12 14 16 18 20

Time (Seconds)

 

 

 

Figure 3-34. Controller performance with integrator approximation ((1)( = l rads/s ), Rm =1.05R,

and Rm =1.er as uncertainties and error due to H60.

103

 

3.5.2. Operation at Different Speeds and Loads

Finally to investigate the operation of the machine under different conditions we
develop a couple of simulations. The format is the same as in the case when
measured speed is used in the feedback. The simulations are performed in
presence of all the errors of figure 3-34.

The key observation is the breaking down of the control scheme at low speeds.
This result is different from the case when measured speed is being used in the
feedback. Keeping in view the analysis done in chapter 2, this result is not
surprising. Assuming slip to be small we have

(as = a)

If the controller successfully tracks the references then

wzwref and 1 21m”!

"15

For low speed and low torque operation in, >> i“, and for this simulation

isd =90amps :> IS = isd =90amps

 

With I'm", = 100 amps and above assumptions equation (2.67) gives us
Jim z 300
wref

For a)", =40rads/s, we get 6i“, =7.5amps. This result is very close to the
simulation result, which gives 6i“, z7.8 amps and in, =1Samps. The extreme

conditions of in, and 2:" having different signs is avoided and the control if

functional. Refer to section 2.6 for related details. Note that this is the case of

equation (2.95).

104

If the reference speed is reduced further with a)", =20rads/swe get

6i“, =15amps. By lowering the speed it is expected that in, <15amps when

compared with the previous case. This results in in and if“, having different signs.

The obvious consequence is the breakdown of the control.
This simulation is important that it not only demonstrates limitations of operating
the machine at low speeds, but also validates the accuracy of analysis done in

chapter 2.

Figure 3-36 illustrates the robustness of the control scheme to changes in load

torques.

105

 

 

 

120

.4

 

8 8'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

30
I I I I I 5
100" :4
Q t"f"""‘“'“““'“'“‘5 43
a II
9 ‘2
‘1
0 0
0 30
150 r r x r
100 by; .
8.
50 — . d
E _______ 'l .3___ _ _ & I8d
_f -. ......1. J£.:_:..__ _ _'
0 [I if " is“ ‘
. . .. . sq'est
—50 l g 1‘ l
0 5 10 15 25 30
10 r r u u I
II"... _ _.
m 5 ”" 47/ I — —; 1' d
"5 o ' I ' J
> I I
_5 . — u d
_ _ u
-10 l 1 l l
O 5 10 15 20 25 30

Time (Seconds)

Nm

 

Figure 3-35. Controller perforrnanoe at various speeds (Encoderless Case)

106

 

 

 

Amps

Amps

Volts

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

120 . . . . .
11o -
100 ...................................................... -I
90 —5ms .
' Ims,ref
80 l J l l l

o 5 10 15 20 25 30

80 I . . . . 5

:4

«3

12

11

o

30

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 j . . . .
5 L_A__r—___~—--_f——__, _____ _
o .1
_5 _usd-

——U
sq

‘10 ' I I I 1

Time (Seconds)

Nm

 

Figure 3-36. Controller performance at various loads (Encoderless Case)

107

 

CHAPTER 4

EXPERIMENTAL SETUP AND RESULTS

4.1 . Introduction

Analysis and simulations are important stages in designing a control system. It is
natural to think that the actual target is not accomplished unless the control
scheme is physically implemented. The requirements of this final stage are
radically different from the previous steps. The idealization and approximations
assumed for the analysis and simulations are no longer there in a real system. All
the unmodeled effects, disturbances and technology dependent issues are
present. The last one of these is of prime importance in an experimental setup.
The designer has to keep in mind all the basic considerations and prerequisites
to work on the experiment.

The first step of design of the experimental setup is the selection of technology
and methods to be used. This level is a part of project management. Decisions
made at this stage play a vital role in the rest of the development. The difficultly is
that the design has not even started as yet. We have to anticipate all the
requirements and steps beforehand. Keeping in view all those factors, a design
engineer selects from the available technologies. Other technology related issues
are cost, portability and suitableness as far as personal skills are concerned. The
decisions made at this stage are long lasting. Any wrong decision could result in

high cost both in terms of expenses and time. The actual implementation follows.

108

Extra precautions have to be taken at the prototyping phase because the errors
are hard to debug. We have successfully developed an experimental setup
keeping in view all the design requirements.

To summarize, we have tried to give justice to all the ingredients of a control
design scheme from analysis to actual experimentation. The methodical

approach enabled us in completing the design with satisfactory performance.

4.2. The Experimental Setup

Our experimental setup is based on some new and novel ideas. The central
theme is the use of a personal computer as an embedded microcontroller. A PCi
digital input/output card has been incorporated for the personal computer to
communicate with the outside world. We have developed a sophisticated
machine control card. This card carries all the necessary electronics to collect
data from various sensors and issue command signals to some other devices.
The 3-phase power to induction machine comes from a PWM inverter. To
emulate load we use a DC machine that is controlled by its own controller. This
control can be manual or through the machine control card.

Voltage is directly sensed owing to the low voltage machine. Isolated current
sensors provide accurate current measurements. A 1024 grating shaft encoder
senses position. The stator resistance is a critical parameter. Its value however,
varies with temperature. A temperature sensor helps alleviating this problem. We
have also incorporated a torque sensor to accurately monitor the shaft torque.
State of the art hardware design and effective software have enabled us to

perform experiments in the most elegant of manners. We are able to easily

109

implement the controller and all its variations. The results produced match the

analysis and simulations pretty well. Block diagram of the experiment is given in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4-1.
——————————— I
I
Z} l
A A I
Microprocessor , 2 :
I
1‘ a, l :
3 . .
m L A Dlgltal '0 L I
" I v Card I I
2 | . R
I r I
l
RAM I
l
I
I
V :
Personal Computer :
DC MOTOR m
CONTROLLER Machine Control
‘ _ PWM Inverter f : Card
A K
Voltage Sensor I A A
1> I
' Current Sensor
< "1

 

 

 

      

 

Dc \
Mh'n
acreJ

  

 

 

 

 

 

 

 

 

 

 

 

Figure 4-1. Block diagram of the experimental setup

110

4.3. Personal Computer as Embedded Processor

Historically personal computers and embedded technologies have been two
different things with very different roles. Emerging technologies and systems
have changed this concept and the boundary between the two is becoming
unclear. There are many tasks that were considered to be in the domain of
personal computing that are now been done by devices that are not even
recognized as computers. Most common example is of the devices that can
access Internet. The same is true for personal computers. However, replacing
some embedded technology with personal computers is not a simple procedure.
There are certain hardware as well as software requirements for accomplishing
this task.

Once we are able to use personal computers we get certain benefits that are
either not available in the embedded technologies or are hard to achieve.

Following is a brief discussion in this regard.

4.3.1. Computational Power

The most glaring difference between commonly available micro-controllers and
personal computers is the computational power of the later. The computational
power of a personal computer may exceed its counterpart by orders of
magnitude. The difference arises from high frequency of operation, super-
pipeiining, parallel and multi-processing and other advanced features that are
common in microprocessors of today. This helped us in implementing the most

sophisticated and involved control schemes without much difficulty.

111

4.3.2. Memory Resources

In most of the controller applications, memory requirements are not stringent.
However, during the phase of development we generally require to collect data
related to state variables, controller or estimators. With high sampling rates,
these data become enormous even for a short period of operation. Large
memory of personal computers can handle this situation. This task is hard to

implement using embedded technologies.

4.3.3. Available Data Types

We have already seen in section 2.6 that 32-bit floating points may become
insufficient to handle certain common problems faced by a control scheme. On
the other hand 32-bit floating points is a standard for most microcontrollers.
Naturally there are ways to handle this situation but during the development
phase, one would like to concentrate more on the control scheme rather than
implementation issues. Personal computers use 64-bit or 80-bit floating points as
a standard. This precision exceeds the requirements for most sophisticated and
advanced control schemes. We have used 64-bit floating points for our

implementation.

4.3.4. Mass Data Storage

This is also one of the requirements that become redundant once the control
scheme is mature. However, to keep a record of development phases we require

mass data storage facility. Personal computers have large hard disks and this is

112

not a problem. On the other hand, developing a similar facility for embedded

technologies is difficult and expensive.

4.3.5. Simulating Real Date

If a design engineer is running simulations using the data collected from real
experiments, then programs like MATLAB help a lot. He is able to do all of these

tasks from the same computer.

4.3.6. Development Environment

Advanced and user-friendly environment of personal computers makes the task
of developing a control scheme much easier. High-level programming languages
like C and C++ are much more stable on personal computers as compared to
other platforms. Sophisticated debugging and related facilities are common in
personal computers. All of these factors have a big impact on the development

time of a control scheme.

4.3.7. Flexibility

A design undergoes considerable changes before it reaches a final stage. In
certain cases, we may have to change the hardware of the setup for applications
implemented through embedded technologies. This alteration is tedious and time
consuming. If the same work is being done on a personal computer, it is
expected that most of the changes would be at the level of software. The
development environment for the software on personal computers is suitable for
making change in short time. This is a very important factor that has no parallel in

embedded technologies.

113

4.3.8. Summary

Personal computers offer facilities and convenience that greatly help a design
engineer in developing a control scheme. Both the cost and development time
are considerably reduced. General awareness and user-friendliness of
computers make it easy to train new people on an experimental setup. Well-set
standards for personal computers reduce the problems of portability and
compatibility.

The requirements of final product may however, exclude the possibility of using a
personal computer. Transferring a fully developed scheme from an experimental
setup based on personal computers to some other platform is a much easier task

than developing the whole scheme on that platform.

4.4. RTLinux Philosophy

If a personal computer were so suitable for implementing real-time schemes then
what is the reason that this idea has not been utilized until recent past? A
computer is just a piece of remarkable hardware. The usefulness of the
computer, however, depends on the software that is running on it. The most
conspicuous reason that forbade the personal computer as an embedded
processor has been the unavailability of real-time operating systems (RTOS). A
realization of the powers of personal computers amongst the designers have led
to the emergence of new RTOS’s. RTLinux is one of them and inherits all the
benefits from the well-known Linux operating system.

An operating system has to take care of controlling the hardware of the computer
including all the peripherals. Some tasks are given higher priority than others.

114

The solution is not simple and the result is complex architecture of the operating
system. If the same operating system is used for handling real time tasks, it is
almost certain to miss some events that may be critical. RTLinux provides an
interesting solution to this problem. It takes control of the whole machine and
executes Linux as one of its low priority tasks. This makes it possible for the real
time tasks not to be disturbed by any other event.

The real time tasks may be interrupt-driven or time scheduled. The later is more
suitable for fixed sampling time applications and we have done the same. The

details of the operating system can be found from the RTLinux web site

http://www.rtlinux.org.

4.5. Software Planning

RTLinux provides facilities to develop real time programs. However, a real time
task has all the system level privileges. The simple implication of this status is
extreme sensitivity of the real time program to programming errors. The simplest
of errors result in a system crash, whereas more serious situations may result in
hardware damage. This calls for a very good high-level plan for the development
of the software. This issue becomes even important for our case where we have
to implement a control scheme of considerable complexity.

The real time task runs in an infinite time scheduled loop. It communicates with
the machine control card via the digital IO card and has all the necessary
software for controlling the induction machine. This software includes controller
and estimators. The relevant data is stored in a memory buffer that is shared by

a Linux task that acts as the user interface. Depending on the nature of

115

experiment this buffer has a size of tens of megabytes. A small portion of this
memory is used for commands from the user to initiate or stop the controller. The
user interface does not interfere with the real time task because of its being non-
real time. Once the experiment is done, the user has the option to store the data

to the hard disk. Figure 4-2 illustrates the software plan.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine Control Card
Via
Digital IO Card
< A?
Real Time I I Shared I, k [(2:32 32:?
Mass
Storage
(Hard Disk)

 

 

 

 

 

 

Figure 4-2. Software plan

4.5.1 . Hierarchy of the real time task

The complexity of the real time task demands for modular programming. It turns
out that we can go a step further for our software. We can also create a hierarchy
in the modules. Figure 4-3 illustrates the complete scheme. This sort of

architecture gives maximum flexibility. For example if we want to change the

116

machine control card at some stage, we only have to change one block of figure
4-3. There may be minor changes in higher levels as well. The highest level is

basically RTLinux part of the real time task.

 

 

Real Time
Supervisory
Software

 

 

 

 

 

\/

Controller and
High-Level
Software

II

Machine
Control Card
Interface

II

Digital IO
Card Interface

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4-3. Hierarchy of real-time task

117

4.6. Digital IO Card

We have selected National Instruments PCl-DlO-96 card. It has 96 digital
input/output lines. The card does not provide the fastest of the interfaces but
satisfies our requirements. In fact using a very high-speed card may become
problematic from the point of cross talk and noise immunity. The details of the

card are available from National Instruments web site mp://www.ni.com.

4.7. Machine Control Card

This card plays a fundamental role in the actual interface with the induction
machine. Figure 4-4 illustrates the facilities available on the card. There is a 24-
bit local data bus on the card. Because the number of devices is limited, we use
the concept of address lines rather than the usual address bus. A data transfer
protocol is required for the bi-directional nature of the data bus. Control lines are
dedicated for this purpose. For clarity sake, the control lines have not been
shown in the figure. This protocol is not the same for all the blocks shown in the
figure. However, it is important that there are no conflicts. A brief account of the

blocks follows.

118

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[if\
: 8-Channel ,4
<:> 12-Bit ADC N —]
é ———i
a, n, —-—V 3x PWM >
<9 ‘6 Generators
C
-- o <:>
g :1
2 h?
V
N '\
V 4-Channel >
C: DAG
J\
V Shaft Encoder (L l
S 3 Interface V
V V
Machine Control Card

 

 

 

 

 

 

Figure 4-4. Block diagram of machine control card

4.7.1. Analog to Digital Conversion

As depicted in figure 4-1, we require 7-channels of analog to digital conversion.
To improve transmission time, we use two analog to digital converter chips in
parallel. Besides, anti-aliasing filters are used which play an important role in

sampled data systems. Figure 4-5 illustrates the implementation.

119

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Gain Adjust
& Buffer
A <——~ LPF ‘
4-Channel ‘—_ """"" —°
0: 12 LS B'ts
a C: ADC IC ‘__ ________ _0
3‘3 <—— ———————— —o
g t 8 Analog Inputs
:1 <—— ———————— —o
a
& 4-Channel ‘— ------ —°
N 12 MS Bits
: ADC IC .— ________ _o
<—— ———————— —o J
V

 

 

 

Figure #5. Analog to digital conversion

We use third order Buttenovorth filter for anti-aliasing. Sampling rate of 10KHz
corresponds to Nyquist frequency of 5Khz. The transfer function of the filter is
given as

l

H(s)=
(s/wo)3+2(s/w0)2+2(s/w0)+1

(4.1)

 

The magnitude response is

 

|H(a))| = (4.2)

WW

With a cutoff frequency of 500Hz, the Nyquist frequency of 5Khz undergoes
attenuation of 1000 times which is sufficient for our design requirements. The

circuit diagram for the filter is given in figure 4-6. Refer to Millman [14] for details.

120

 

 

[ Wv ‘v‘v‘v

 

 

 

’—0 Vout

 

<>
;:
>,
Y

 

Vin

'II—iI——

 

 

 

 

 

Figure 4-6. Anti-aliasing filter (3" order low pass Butterworth)

4.7.2. 4-Channel Digital to Analog Converter

A digital to analog converter is required in transmitting signals to the DC motor

controller. This is a useful facility to have on the card. Figure 4-7 illustrates the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DAC.
A
. N 2-Channel _—° W
%
(Eu Voltage + 4 Analog
8 Reference * I Outputs
A
$ A 2-Ch I *—0
v anne
N 12 MS B‘ts
'- / DAC IC __0
v

 

 

 

Figure 4-7. Block diagram of digital to analog conversion

121

4.7.3. PWM Generation

We have used PIC16F877 microcontroller for generating PWM signals. The
microcontroller has two PWM modules built-in the chip hardware. Since for 3-
phase systems we require a third PWM signal. This signal is created in software.
The program makes use of 8-bit timer and is interrupt driven. The soft PWM can
give a duty cycle of 0-100% and is synchronized with its hardware counter part.

The interface is via 24-bit data latch, which is required to prevent data being

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

corrupted.
A
‘3 PWM a 1
g g P Module‘l a, m
Ii ‘6 a Big} [>o—o '6 g
D .1 c ._
E ::) 9 Software *— —* PWM O 55” 2
8 3 *— fi Modulez } 2 0g
-‘ ._. 4 {>o_o E ._
2': .0 .C
m - Soft ~=
a". a d PWM ° "’ 3
PIC Microcontroller ll: ‘
Address Lines
V

 

 

 

Figure 4-8. Block diagram of PWM generation

4.7.4. Shaft Encoder Interface

We have used HCTL2020 for receiving the quadrature inputs of the shaft

encoder. This IC provides 8-bit bus interface.

122

4.8. Open Loop Experiment

We run the experiment in open loop to study the estimator performance. The
basic idea is similar to the concept of figure 3-10. However, now the machine is
physically run instead of being simulated. We do not have access to physical
quantities like magnetizing current etc. The position of the rotor is measured and
speed is obtained through the use of high-gain observer. A large voltage signal is
given in the first second to facilitate the start of the motor. For the next four
seconds, a signal of reduced amplitude is given. The amplitude of the signal is
further reduced for the rest of the run. The frequency of the signal is kept
constant throughout the experiment at 25Hz. Data are collected in real time.
These data are then ported into MATLAB/SIMULINK. The estimators are run in
the simulator. The results are given in the Figure 4-9 and Figure 4-10.

The accuracy of the speed estimate is the key observation. Since we do not have
access to many of the physical quantities and the system is operating in open
loop, it is not possible to quantitatively evaluate the estimators other than the
case of speed. We can still infer that the behavior of all the estimated quantities
is not different from what is expected i.e. the waveforms are logical.

The actual speed a) is estimated by passing position measurements through a

high-gain observer. Equation 2.84 is used to get the speed estimate we”. We

have used low-pass filters to filter out high frequency noise in estimated
quantities. This noise appears primarily due to dead-time nonlinearity of PWM

inverters.

123

 

 

 

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’u?
a
E 50" k _J
__E
o I I l I J I I
O 3 4 5 6 7 8 9 10
o I I I r I I I
A -50- -
(D
g I
8 -150- - ~
-2m I I I I l I I
O 3 4 5 6 7 8 9 10
o I I I I I I I
g .50- u
g —1oo~ .
88-150- * ‘
_2m J; I I I I l I
0 3 4 5 6 7 8 9 1O
20 I I I I I I I
'o
E 0 ‘
3
-10- -
_20 l I I J I I L
0 3 4 5 6 7 8 9 10

Time (Seconds)

 

Figure 4-9. Open Loop Estimation (Part I)

 

 

 

 

100

50

isd(amps)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O I l l I I I I
o 1 2 3 4 5 6 7 8 1o
50 I I I I I I I I
a? 0' '
D.
E k
81 40V .
_3
-100- .
o 1 2 3 4 5 6 7 8 1o
6 I I I I I I I I
A 4 -
U)
E I L
>‘” 2 .
0 I I I I I I I I
o 1 2 3 4 5 6 7 8 1o
1oo-/\ _
75
O.
8
:1” 50 vi
0 l I I I I I I I
o 1 2 3 4 5 6 7 8 10

Time (Seconds)

 

Figure +10. Open Loop Estimation (Part II)

125

 

4.9. Encoderless Speed Control of Induction Machine

We now conclude our chapter by presenting our ultimate goal of controlling the
induction machine without any position or speed sensors. The complete scheme
of Figure 2-7 comes into play. The machine is started by 3-phase sinusoidal
signal. This is done to overcome the difficulties of initialization as discussed in

section 2.5.2. The control scheme takes over at time t = 23 with i

ms,ref

= 40amps
and co", =100rads/s . Command for speed reversal is given at I: 68.

We use the second-order filter of equation (2.42) with a), =1rads/s to

approximate the integrator in stator-flux estimator. The sampling rate is 10KHz.
For discrete-time implementation we have used bilinear transformation without
frequency pre-warping.

We have used the exact machine parameters as given by the manufacturer.

These parameters are summarized as follows.

 

 

 

 

 

 

Parameter Value
Stator Inductance, Ls 0.373 mH
Rotor Inductance, Lr 0.383 mH
Magnetizing Inductance, Lm 0.352 mH
Stator Resistance, Fis 0.0105 £2
Rotor Resistance, Fir 0.0138 52

 

 

 

 

Table 4-1. Parameters of machine used in actual experiment

126

The sensitivity of estimators on R, makes it important to update its value online.

Accurate temperature sensor provides stator-winding temperature. We use the

relation as given by Aloliwi [19].

T+235
(T)

R = —
‘°T0+235

S

where T is the stator-winding temperature in °C and Rm is the value of stator

resistance at temperature To.

Currents are measured using LEM isolated hall-effect current transducers.

Voltages are measured directly by the machine control card.

Observations

The results are given in figure 4-11. The most important observation is the
successful tracking of both the magnetizing current and speed references. It is
important to note that the stator and rotor resistances are now true uncertainties.
We cannot use the expressions of error bounds derived in chapter 2. We do not

have access to actual flux but the speed a) can be calculated by passing the

position measurements through a high-gain observer. The estimated speed a)“,

is obtained by using equation (2.84). We see that error in speed tracking is small.
Another observation is the successful speed reversal.

Excessive noise in estimated quantities results from dead-time non-linearity of
the inverter. This noise is improved slightly as soon as the closed loop control

becomes active at tzzs. Note that the high frequency noise appears in

127

estimated quantities only. The machine has its own time constants, which filter
out high frequency component of the input. The result is apparent in the case of
speed. We observe that there is virtually no high frequency disturbance in actual
speed a).

One final remark is that this result is achieved in the presence of all unmodeled
and certain unwanted factors that are not possible to eliminate from a physical

system.

128

 

 

 

 

150 T I I I I *I I I r
O

 

 

 

 

(amps)

 

 

 

100

 

co(rads/s)
O

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8w(rads/s)

 

 

 

L I

0 1 2 3 4 5 6 7 8 9 10
Time (Seconds)

 

 

I
A
l-
-

 

 

Figure 4-11. Experimental results of closed loop system without using speed/position sensors

129

5.1.

CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

The knowledge of capabilities and limitations is fundamental to successful

implementation of a control scheme. The stator flux-orientated control offers

faculties not available in other methods. Based on the analysis, simulations and

experiments, we are now in a position to make precise and more informative

statements. A brief account follows.

We can have a quantitative estimate of the operational range of the machine.
This requires certain information like bounds on the load torque. Section 2.6
gives a guidline in this regard.

The error analysis in chapter 2 shows that stator based schemes rely heavily
on accurate knowledge of stator resistance. This apparent demerit is actually
not as big a problem. In fact if we keep in mind that the stator quantities are
available for online measurement, it is possible to track changes in stator
resistance. Stator temperature plays an important role in this regard. We
expect minimization of errors due to uncertainty in stator resistance if some
suitable scheme updates the stator resistance in real time.

Stator flux estimation is totally independent of rotor resistance whereas the
error in estimated speed is small due to uncertainty in rotor resistance. Rotor

quantities are inaccessible. The temperature of rotor resistance undergoes

130

large variations owing to rotor losses. The obvious consequence is a greater
uncertainty in rotor resistance. Under these circumstances, stator flux-
orientated control appears to be a good choice.

Stator flux estimation is independent of rotor speed. The stator flux estimator,
therefore, remains the same for both measured and estimated schemes. It is
also reasonable to think that this independence would also contribute to
overall robustness.

The errors in estimated quantities are reduced at higher stator frequencies.
This makes stator flux orientated control to be well suited when the machine

is to be operated at considerably high speeds.

On the other hand the stator flux orientated control faces certain limitations and

difficulties.

The properties of pure integrators rule out a possibility of using them in open
loop. Therefore, stator flux estimator has to be implemented using some
approximation of the pure integrator. This affects the dynamic performance
and a steady state error is introduced as well. Section 2.2 deals with analysis
of these errors.

The stator flux estimator lacks the ability to operate at zero frequency. Under
such conditions stator flux-orientated control is impossible to use even if the
mechanical speed is available for online measurement. The simplest example
is the operation of the machine at zero speed and zero load torque. These

issues are discussed in section 2.6.

131

. Owing to increased errors at lower speeds, the stator flux orientation is not

suitable for low speed operation of the machine.

5.2. Recommendations

There are many options available for the control of induction machines. All of
these techniques work well under certain conditions and face difficulties
otherwise. Rather than making a decision that is biased towards a particular
approach, it is wise to keep in mind the actual objective of designing a control
scheme. Therefore, we would not insist that the stator flux orientated control is
the best choice under all conditions. However, there are certain advantages
associated with it. Our findings relating to stator flux-orientated control are not
different from the common understanding of this scheme. A summary of
recommendations in this context follows.

0 Stator flux orientated control is. most suited at high-speed operation. The
experimental results of figure 4-11 illustrate this important fact. Speed
estimation is accurate and the scheme has a capability of speed reversal,
which is considered a difficult proposition in encoderless control of machines.
We would suggest using this scheme under these requirements and
conditions.

. For low speed operation of induction machine, either a different scheme
should be used or stator-flux orientated control can be combined with some
other suitable scheme. The former will be the dominant controller at high

speeds and the later will take charge at low speeds. The switching between

132

the two controls can cause a problem. Leonhard [12] has proposed a method
of performing this task in a soft manner.

If a comparison based on mathematical analysis is made between different
schemes, there is a possibility of combining their advantages and getting rid

of difficulties.

133

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[3]
[9]
[10]
[1 1l

[12]
[13]

[14]

[15]

BIBLIOGRAPHY

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F. Blaschke. The principal of field-orientation as applied to the new
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A. Isidori. Nonlinear Control Systems. Springer-Verlag, 1999.
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[16]

[17]

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[20]

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Gordon R. Slemon. Electric Machines and Drives. Addison-Wesley,
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E. Strangas, H.K. Khalil and Bader Aloliwi. Comparison of Three
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E. Strangas, H.K. Khalil, Bader Aloliwi, L. Laubinger and J.M. Miller. A
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Peter Vas. Sensorless Vector and Direct Torque Control. Oxford
University Press, 1998.

135

 

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