Adaptive Sliding Mode Observer

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ADAPTIVE SLIDING MODE OBSERVER AND LOSS

MINIMIZATION FOR SENSORLESS FIELD ORIENTATION

CONTROL OF INDUCTION MACHINE

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor

of Philosophy in the Graduate School of The Ohio State University

By

Jingchuan Li, M.S.E.E

* * * * *

The Ohio State University

2005

Dissertation Committee:

Professor Longya Xu, Advisor

Professor Donald G. Kasten

Professor Vadim I. Utkin

Approved by

______________________________

Adviser

Graduate Program in Electrical and Computer

Engineering



UMI Number: 3197781

3197781

2006

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Copyright

All rights reserved. This microform edition is protected againstunauthorized copying under Title 17, United States Code.

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P.O. Box 1346Ann Arbor, MI 48106-1346

by ProQuest Information and Learning Company.





ii

ABSTRACT

Induction machines are widely used in industry application. Consequently more

and more attention has been given to design and development of induction machine

control. High performance of induction machine control is achieved by so called field

orientation control (FOC). Speed sensorless technology has also been proposed for

decades to overcome the disadvantages of cost and fragility of a mechanical speed sensor.

However, due to the high order, multiple variables and nonlinearity of induction machine

dynamics, the development of advanced induction machine control is still a challenging

task.

In this research, a sliding mode based flux and speed estimation technique for speed

sensorless control of field oriented induction machine is first investigated. The parameter

sensitivity of the control method is also analyzed. A robust sliding mode speed controller

is also presented, which has the advantage of disturbance rejection and avoiding re-tuning

gains comparing to traditional PI controller

Then an adaptive sliding mode observer is proposed and the stability is verified by

Lyapunov theory. Two sliding mode current observers are utilized to compensate the

effects of parameter variation on the rotor flux estimation, which make flux estimation

more accurate and insensitive to parameter variation. The convergence of the estimated

flux to actual rotor flux is proved by the Lyapunov stability theory.



iii

Finally, an efficiency optimization method which does not require extra hardware and

is insensitive to motor parameters is presented. The relationship between stator current

minimization and motor loss minimization in the induction motor vector control system is

investigated. A fuzzy logic based search method is simulated and implemented. It is

determined that the motor loss minimization can be achieved by minimizing stator

current in practice.

An experimental setup is presented in the appendix to verify the proposed approaches.

The simulation and experimental results are presented to demonstrate the potential and

practicality of the presented approaches.



iv

Dedicated to my wife and my parents



v

ACKNOWLEDGMENTS

I would like to express my first acknowledgement to my advisor, Professor Longya

Xu, for his academic guidance, his constant help and support of my research. His

supervision has broadened my knowledge in power electronics and drive system. I had

learned a lot from his rich experience.

I would like to thank Professor Donald Kasten and Professor Vadim Utkin for being

my dissertation committee. They give me many insightful comments and constructive

suggestions in review of my research proposal and dissertation.

I would like to express my appreciation to Professor Stephen Sebo, Professor Giorgio

Rizzoni and Dr. Zheng Zhang for their kindly help during my research work.

I thank all my colleagues of the Power Electronics and Electric Machines (PEEM)

group at The Ohio State University and especially to Dr. Mongkol Konghirun, Dr. Jingbo

liu, Mr. Song Chi, Mr. Jiangang Hu, Dr. Mihai Comanescu, Mr. Reza Esmaili and Ms.

Debosmita Das. We had a very good corporation and had many fruitful discussions

during the past several years.

Finally, I would like to express my deepest appreciation to my wife, Yan, who has

been sharing hardships and happiness over years, and my parents and my brothers. Their

caring, understanding, and encouragement have motivated my research and study.

Without their constant support none of this would have been possible.



vi

VITA

June1993………………………………...………..…………...B.S. Electrical Engineering

Xi’an Jiaotong University , Xi’an, China

January 1995………………………….…….….…..…………M.S. Electrical Engineering

Xi’an Jiaotong University , Xi’an, China

May 1999………………………………………………….Puchuan Power Electronics Ltd.

Shenzhen, China

September 2000……………………………………………..Graduate Research Associate

The Ohio State University, Columbus, OH

PUBLICATIONS

Research Publication

Jingchuan Li, Longya Xu, Zheng Zhang, “An Adaptive Sliding Mode Observer for Induction Motor Sensorless Speed Control,” IEEE Trans. Industry Applications, Vol. 41,

No. 4, pp.1039 - 1046, 2005.

Jingchuan Li, Longya Xu, Zheng Zhang , “A New Efficiency Optimization Method onVector Control of Induction Motors,” Electric Machines and drives Conference, IEMDC

2005.

Jingchuan Li, Longya Xu, Zheng Zhang, “An Adaptive Sliding Mode Observer for Induction Motor Sensorless Speed Control,” IEEE Industry Applications Conference,

IAS 2004. Volume 2, pp1329 – 1334, 2004.



vii

Jingchuan Li, Longya Xu, “Investigation of cross-saturation and deep bar effects of

induction motors by augmented d-q modeling method”, IEEE Industry ApplicationsConference, IAS 2001, Volume 2, pp 745 – 750, 2001.

Codrin-Gruie Cantemir, Gabriel Ursescu , Jingchuan Li , Chris Hubert, Giorgio Rizzoni,

“An 1800 HP, Street Legal Corvette: An Introduction to the AWD Electrically-Variable

Transmission,” SAE 2005 World Congress.

Codrin G. Cantemir, David Mikesell, Nicholas Dembski, Jingchuan Li, Giorgio Rizzoni,

“Hybrid Electric Refuse Vehicle,” IEEE Vehicular Power and Propulsion – IEEE VPP2004, Paris.

FIELDS OF STUFY

Major Field: Electrical and Computer Engineering

Studies in:

Power Electronics and Electrical Machine Control Prof. Longya Xu

Control Prof. Vadim Utkin



viii

TABLE OF CONTENTS

Page

Abstract……………………………………………….…………….…..…...ii

Dedication………………………………………………………….……….iv

Acknowledgments……………………………………….……….………….v

Vita……………………………………….……….……………………..….vi

List of Tables………………………………………….…………...………..xi

List of Figures……………………………………………………..……….xii

Chapters

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

2. Backgroud and literature review................................................................4

2.1. Field Orientation Control of Induction Motors ..................................................... 4

2.1.1. Co-ordinate transformation............................................................................. 5

2.1.2. Induction motor dynamic model..................................................................... 82.1.3. Basic scheme of Field Orientation control.................................................... 12

2.1.4. Direct Field Orientation control (DFO) ........................................................ 142.1.5. Indirect Field Orientation Control (IFO) ...................................................... 16

2.1.6. Variable speed control of induction machines.............................................. 172.2. Speed Sensorless Control Technology of Induction Machines ........................... 19

2.3. Efficiency optimization of induction machine control ........................................ 23



ix

3. Sliding Mode Flux Observer for DFO sensorless Control ......................26

3.1. Introduction ......................................................................................................... 26

3.2. Induction motor model ........................................................................................ 26

3.3. Sliding mode current observer............................................................................. 293.4. Rotor flux and speed estimation .......................................................................... 31

3.5. Simulation by Matlab .......................................................................................... 32

3.6. Simulation by HIL(hardware-in-the-loop) .......................................................... 363.7. Experimental results ............................................................................................ 39

3.8. Parameter sensitivity analysis.............................................................................. 443.9. Conclusion........................................................................................................... 51

4. Robust sliding mode speed controller......................................................52

4.1. Introduction ......................................................................................................... 52

4.2. Sliding mode controller design............................................................................ 534.3. Continuous sliding mode controller .................................................................... 534.4. Implementation in the induction motor drive system.......................................... 56

4.5. Simulation results ................................................................................................ 57

4.6. Experimental results ............................................................................................ 594.7. Conclusion........................................................................................................... 64

5. Adaptive Sliding Mode Rotor Flux and Speed Observers ......................65

5.1. Introduction ......................................................................................................... 65

5.2. Sliding mode Current and flux observer design .................................................. 655.2.1. Current observer I ......................................................................................... 67

5.2.2. Current observer II........................................................................................ 68

5.2.3. Rotor flux observer design............................................................................ 695.3. Adaptive speed estimation................................................................................... 70

5.4. Stability analysis.................................................................................................. 71

5.5. Simulation results ................................................................................................ 72

5.5.1. Simulation results by MATLAB................................................................... 725.5.2. HIL Evaluation results by TI 2812 DSP....................................................... 78

5.6. Experimental results ............................................................................................ 80

5.7. Conclusion........................................................................................................... 84

6. Efficiency Optimization on Vector Control of Induction Motors ...........86

6.1. Introduction ......................................................................................................... 86

6.2. Principle of Fuzzy logic controller ...................................................................... 886.2.1. Fuzzifier ........................................................................................................ 89

6.2.2. Rule base....................................................................................................... 91



x

6.2.3. Inference engine............................................................................................ 92

6.2.4. Defuzzifier .................................................................................................... 926.2.5. Implementation procedure for fuzzy logic controller ................................... 93

6.3. Motor losses determination ................................................................................. 94

6.4. Comparison of minimum losses point and minimum stator current point .......... 97

6.5. Fuzzy controller for efficiency optimization..................................................... 1026.6. Simulation results .............................................................................................. 107

6.7. Experimental results .......................................................................................... 108

6.8. Conclusion......................................................................................................... 116

7. Summary and future work .....................................................................117

7.1. Summary............................................................................................................ 117

7.2. Future work........................................................................................................ 119

Bibliography ................................................................................................120

Appendix A Experimental Setup................................................................127



xi

LIST OF TABLES

Table Page

6.1 Rule base…………………………………………………………..……………….106



xii

LIST OF FIGURES

Figures Pages

2.1 Stator current space vector............................................................................................ 6

2.2 Park transformation....................................................................................................... 8

2.3 Phasor diagram of a field oriented induction motor ................................................... 12

2.4 General block diagram for a field orientation control system .................................... 13

2.5 The scheme of direct field orientation ...................................................................... 15

2.6 The scheme of indirect field orientation................................................................... 16

3.1 Conrol system block ................................................................................................... 33

3.2 Step speed command at start up.................................................................................. 34

3.3 Real and estimated stator currents .............................................................................. 34

3.4 Real and estimated rotor fluxes................................................................................... 35

3.5 Motor response for trapezoid speed command ........................................................... 35

3.6 Real and estimated rotor flux...................................................................................... 36

3.7 HIL simulation results for 1 hp motor ........................................................................ 37

3.8 HIL simulation results for 1.1kw motor ..................................................................... 37

3.9 HIL simulation results for 5hp motor ......................................................................... 38

3.10 HIL simulation results for 50hp motor ..................................................................... 38

3.11 Experimental setup.................................................................................................... 40



xiii

3.12 Estimated and measured current (current siα ; curve 2: phase current si α ) ............. 41

3.13 Estimated rotor flux and flux angle .......................................................................... 41

3.14 Estimated rotor flux trajectory.................................................................................. 42

3.15 Four quadrate operation (±1200 rpm) (curve1: speed command*

r ω ; curve 2:

estimated speed r ω ˆ ; curve 3: torque current qi ; curve 4: phase current ai ) ............. 42

3.16 Motor response for step speed change from standstill to 950 rpm........................... 43

3.17 Phase current response for step speed change .......................................................... 43

3.18 Coefficient1k in the observer is increased by 20% ................................................... 46

3.19 Coefficient β in the observer is increased by 20%................................................... 47

3.20 Speed response with Rr unchage under 0.5 pu torque disturbance............................ 48

3.21 Speed response with Rr decreasing 20% and 0.5 pu torque disturbance ................... 48

3.22 Speed response with Rr increasing 20% and 0.5 pu torque disturbance.................... 49

3.23 Experimental result with T r unchage under 2.0 N.m torque disturbance.................. 49

3.24 Experimental result with T r

increase 22% under 2.0 N.m torque disturbance.......... 50

3.25 Experimental result with T r decrease 22% under 2.0 N.m torque disturbance......... 50

4.1 Comparison of PI controller and the sliding mode controller ...................................... 58

4.2 External torque rejection performance ....................................................................... 58

4.3 Speed tracking simulation with a triangle speed command...................................... 59

4.4 Speed response atl T =1.0 Nm (curve1: real speed r ω ; curve 2: estimated speed r ω ˆ ;

curve 3: flux current d i ; curve 4: torque current qi ; curve 5: phase current ai ) ........ 60

4.5 Speed response atl T =2.0 Nm (curve1: real speed r ω ; curve 2: estimated speed r ω ˆ ;

curve 3: flux current d i ; curve 4: torque current qi ; curve 5: phase current ai ) ........ 61

4.6 Speed response with external torque step changel T =0.5 Nm ..................................... 61



xiv

4.7 Speed response with external torque step changel T =1.0 Nm ..................................... 62

4.8 Speed response with external torque step changel T =2.0 Nm .................................... 62

4.9 Four quadrant operation with trapezoidal speed command (between 300rpm and1400rpm) ................................................................................................................... 63

4.10 Four quadrant operation with trapezoidal speed command (between ±1200rpm).................................................................................................................................... 63

5.1 Configuration of the proposed flux and speed observer. ............................................ 67

5.2 Real and estimated speed at a step speed command................................................... 73

5.3 Real and estimated current.......................................................................................... 73

5.4 Real and estimated rotor flux...................................................................................... 74

5.5 Coefficient 1k in the observer is increased by 20% .................................................... 76

5.6 Coefficient β in the observer is increased by 20%. .................................................. 77

5.7 Speed step response from –0.5pu to 0.5pu (curve 1: speed command *

r ω ; curve 2: real

speed r ω ; curve 3: estimated speed r ω ~ ) .................................................................... 78

5.8 Rotor flux estimation. (curve 1: real flux r α λ ; curve 2: estimated flux r α λ ~ ; curve 3:

estimated flux angle r θ ~

) .............................................................................................. 79

5.9 Trapezoidal speed at ±0.5pu. (curve 1: phase current ai ; curve 2: torque current qi ;

curve 3: estimated speed r ω ~ ) ..................................................................................... 79

5.10 Transient response to speed step command ±900rpm at no load (curve1: speed

command *

r ω ; curve 2: estimated speed r ω ~ ; curve 3: torque current qi ; curve 4:

phase current ai ) ........................................................................................................ 81

5.11 Real and estimated currents. (curve 1: measured current siα ; curve 2: observed

current siα ˆ ; curve 3: observed current siα

~) ................................................................. 82



xv

5.12 Transient response due to trapezoidal speed command (±900rpm) at no load (curve

1: speed command *

r ω ; curve 2: estimated speed r ω ~ ; curve 3: torque current qi ,;

curve 4: phase current ai ) .......................................................................................... 82

5.13 Transient response due to trapezoidal speed command (±900rpm) at puT l 5.0=

(curve

1: speed command *

r ω ; curve 2: estimated speed r ω ~ ; curve 3: torque current qi ;

curve 4: phase current ai ) .......................................................................................... 83

5.14 Speed response due to step change command from 360rpm to 1260rpm at

puT l

5.0= . (curve1: real speed; curve 2: estimated speed r ω ~ ; curve 3: torque current

qi ; curve 4: phase current ai ) ..................................................................................... 83

5.15 Transient response for step disturbance torque (curve1: real speedr ω ; curve 2:

estimated speed r ω ~ ; curve 3: torque current qi ) ........................................................ 84

6.1 Explaination of efficiency improvement .................................................................... 87

6.2 Block diagram of a fuzzy control system ................................................................... 89

6.3 Explaination of membership function ...................................................................... 90

6.4 Different shapes of membership functions .......................................................... 91

6.5 Motor losses with respect to dsi at different load torque ............................................ 96

6.6 Input power with respect to dsi ................................................................................... 98

6.7 Stator current variation with respect to dsi .................................................................. 98

6.8 Minimum input power point and input power point corresponding to minimum stator

current......................................................................................................................... 99

6.9 Measured input power vs flux level at different torques ............................................ 99

6.10 Measured stator current vs flux level at different torques ...................................... 100

6.11 The ratio ipk with frequency for different motors ................................................... 101

6.12 Principle of efficiency optimization control by stator current .................................. 103

6.13 Fuzzy controller ...................................................................................................... 103



xvi

6.14 Membership for fuzzy logic controller ................................................................... 105

6.15 d-axis reference current and q-axis current............................................................. 107

6.16 Stator peak current .................................................................................................. 108

6.17 Motor losses............................................................................................................ 108

6.18 Fuzzy search at rpmn NmT l 900,2.0 == ........................................................................ 110

6.19 Fuzzy search at rpmn NmT l 900,5.0 == ........................................................................ 110

6.20 Fuzzy search at rpmn NmT l 600,2.0 == ..................................................................... 111

6.21 Fuzzy search at rpmn NmT l 1200,2.0 == ................................................................... 111

6.22 Current, power and efficiency variation at rpmn NmT l 900,2.0 == ............................. 112

6.23 Current, power and efficiency variation at rpmn NmT l 900,5.0 == ............................ 113

6.24 Current, power and efficiency variation at rpmn NmT l 1200,2.0 == ......................... 114

6.25 Current, power and efficiency variation at rpmn NmT l 600,2.0 == .......................... 115

6.26 Comparison of efficiency curves at different operation points with and without fuzzy

optimization control. (solid line is with fuzzy optimization, dash line is without fuzzyoptimization) ............................................................................................................ 116

A.1 Experimental setup………………………………….……………………………………127

A.2 Hysteresis dynamometer ……………………………………..…………………………128

A.3 Dynamometer controller and power analyzer ………….………..……………………128



xvii

NOMENCLATURE

• r s R R , stator and rotor resistance

• lr ls L L , stator and rotor leakage inductances

• m L magnetizing inductance

• mls s L L L += total stator inductance

• mlr r L L L += total rotor inductance

• σ leakage coefficient,r s

m

L L

L2

1−=σ

• r T rotor time constant,r

r

r R

LT =

• r ω , eω , sω motor speed, synchronous speed and slip speed

• l e T T , electromagnetic and load torques

• P number of pole pairs

• J motor inertia constant

• B f coefficient of friction

• p differential operator

• qsds vv , d and q components of the stator voltages

• qsds ii , d and q components of the stator currents



xviii

• qr dr ii , d and q components of the rotor currents

• qsds λ λ , d and q components of the stator flux linkages

•qr dr

λ λ , d and q components of the rotor flux linkages

• s s vv β α , α and β components of the stator voltages

• s s ii β α , α and β components of the stator currents

• r r ii β α , α and β components of the rotor currents

• s s β α λ λ , α and β components of the stator flux linkages

• r r β α λ λ , α and β components of the rotor flux linkages

• s s ii β α ˆ,ˆ estimated stator currents

• r r β α λ λ ˆ,ˆ estimated rotor flux linkages

• r ω ˆ estimated rotor speed

• s s ii β α

~,

~ estimated stator currents by second current observer

• r r β α λ λ ~

,~

estimated rotor flux linkages by adaptive sliding mode

observer

• r ω ˆ estimated rotor speed by adaptive sliding mode observer

• ai phase a current

• si phase peak current

List of Abbreviations

• FOC field orientation control



xix

• DFO direct field orientation

• IFO indirect field orientation

• PID proportional integral differential

• FLC fuzzy logic control

• SLM sliding mode control

• FSMC fuzzy sliding mode control

• MRAS model reference adaptive scheme

• LMC loss model controller

• SC search controller



1

CHAPTER 1

1. INTRODUCTION

Induction motors are relatively inexpensive and rugged machines because they can be

built without slip rings or commutators. They are widely used in industry application.

Consequently much attention has been given to induction motor control for starting,

braking, four-quadrant operation, etc. Open loop control of the machine with variable

frequency may provide a satisfactory variable speed drive when the motor has to

operate at steady torque without stringent requirements on speed regulation. When the

drive requirements include fast dynamic response and accurate speed or torque

control, an open loop control is unsatisfactory. Hence it is necessary to operate the motor

in a closed loop mode. The dynamic operation of the induction machine drive system has

an important effect on the overall performance of the system. The control of induction

motors is a challenging problem since it has a nonlinear model, rotor variables are rarely

measurable and its parameters vary with operating conditions.

Several techniques are used to control the induction motor. These schemes can be

classified into two main categories: 1) Scalar control, One of the first ways of controlling

induction machines was the volts/hertz speed control also known as scalar method in

which the machine was excited with constant voltage to frequency ratio in order to

maintain a constant air gap flux and hence provide maximum torque sensitivity. This

method is relatively simple but does not yield satisfactory results for high

INTRODUCTION



2

performance applications. This is due to the fact that in the scalar method, an

inherent coupling exists between the torque and air gap flux, which leads to a

sluggish response of the induction machine. 2) Field oriented control (vector control). To

overcome the limitation of the scalar control method, field oriented methods were

developed. In field oriented control methods the variables are transformed into a

reference frame in which the dynamics behave like dc quantities. The decoupling

control between the flux and torque allows the induction machine to achieve fast

transient response. The field oriented induction machine drive therefore, can be used for

high performance applications where traditionally dc machines have been used.

The above traditional control schemes require a speed sensor for closed loop

operation. The speed sensor has several disadvantages from the standpoint view of drive

cost, reliability, and noise immunity. Various speed sensorless approaches have been

proposed in the literature recently. However, due to the high order, multiple variables and

nonlinearity of induction motor dynamics, estimation of the rotor speed and flux

without the measurement of mechanical variables is still very challenging. Another

problem for classic vector control is the efficiency improvement. It has been reported that

65% of the electric energy in US is consumed by electric motors [1]. In industrial sector

alone, 76% is consumed by motors and over 90% of these are induction motors.

Induction motors have a high efficiency at rated speed and load. However, at light loads,

iron losses increase, reducing the efficiency considerably. For a 500 hp motor with proper

control, it is reported that the reduction in losses translates into annual savings of $7000

at the energy cost of $0.05/kW.Hr [2] . Obviously, there is a clear motivation for

efficiency improvement.



3

This dissertation is organized as follows. The background and principle of field

orientation control of induction motor are summarized in Chapter 2. In addition, the

existing work up to now about induction motor sensorless control and efficiency

improvement are reviewed. An effective sliding mode flux and speed observer for direct

field orientation control is presented and the parameter sensitivity is analyzed in Chapter

3. The simulation and experimental results are also presented. In Chapter 4, a robust

chattering free sliding mode speed controller is presented and analyzed. The results are

compared with conventional PI controller. An adaptive sliding mode observer is proposed

in Chapter5. The stability is derived using Lyapunov theory and the speed estimation is

based on adaptive mechanism. A loss minimization algorithm by fuzzy logic for

induction motor control is presented in Chapter 6. This new method has the advantage

that it is insensitive to motor parameters and does not require extra hardware, at the same

time it can improve motor efficiency dramatically especially in light load. The simulation

and experimental results show the effectiveness and validation of the method. The

conclusion and suggested future work are discussed in Chapter 7.



4

CHAPTER 2

2. BACKGROUD AND LITERATURE REVIEW

As stated above, induction motor drive system is becoming more and more a

competitive system in many high performance motion drive application. There are three

main major components in an induction motor drive system: an induction motor, a power

electronic device and a controller. The field orientation control (FOC), integrating modern

control theory, power electronic and DSP/micro-processor technology, has made possible

the development of high performance induction motor drive systems. In this chapter, the

basis of field orientation control of induction motors is summarized. The state of art of

speed sensorless control and motor loss minimization is reviewed. The existing problems in

the implementation of induction motor drive system are outlined.

2.1. Field Orientation Control of Induction Motors

An electric motor can be thought of as a controlled source of torque. The torque

developed in the electric motor is a result of the interaction between current in the

armature and the magnetic filed produce by motor. Independent control of the field and

armature current is feasible in separately-excited DC motors where the current in the

stator winding determines the magnetic field of the motor, while the current in the rotor

armature winding can be used as a direct means of torque control. In a similar manner to

that in DC motors, the induction motor control can be accomplished by a decoupled

BACKGROUD AND LITERATURE REVIEW



5

control of flux and torque. The action of holding a fixed, orthogonal spatial angle

between the field flux and the armature MMF is emulated in induction machines by

orienting the stator current with respect to the rotor flux so as to attain decoupled

controlled flux and torque. Such controllers are called field orientation controllers (also

referred to as vector controllers).

A basic understanding of the decoupled flux and torque control resulting from field

orientation can be obtained from the d-q axis model of an induction machine with the

reference axes rotating at synchronous speed. This control is based on projections that

transform a three-phase time and speed dependent system into a two co-ordinate (d- and

q- axis) time invariant system. These projections lead to a structure similar to that of a

DC machine control.

2.1.1. Co-ordinate transformation

The three-phase voltages, currents and fluxes of induction motors can be analyzed in

terms of complex space vectors [3-6]. With regard to the currents, the space vector can be

defined as follows. Assuming that cba iii ,, are the instantaneous currents in the stator

phases, 0=++ cba iii , the complex stator current vector siv

is defined by:

)( 2

cba s iiik i α α ++=v

( 2.1)

whereπ

α 3

2 j

e= ,π

α 3

4

2 j

e= represent the spatial operators,3

2=k . The following

diagram shows the stator current complex space vector.



6

a

i s

b

c

α

β

siα

si β

Figure 2.1 Stator current space vector

where (a, b, c) are the three-phase system axes. This current space vector depicts the three

phase sinusoidal system. The space vector can also be plotted in another reference frame

with only two orthogonal β α − axis. The real part of the space vector is equal to the

instantaneous value of the direct-axis stator current component siα . The imaginary part is

equal to the quadrature axis stator current component si β . Thus, the stator current space

vector in the stationary reference frame attached to the stator can be expressed as:

s s s jiii β α +=v

( 2.2)

The space vectors of other motor quantities (voltages, rotor currents, magnetic fluxes,

etc.) can be defined in the same way as the stator current space vector.

1) Clarke transformation

In symmetrical three-phase machines, the direct and quadrature axis stator currents

(as shown in Figure 2.1) are fictitious two-phase current components. Assuming α -axis

is in the same direction with a-axis, we have following relations with respect to the actual

3-phase stator currents



7

)ii( k i

iiik i

csbs s

csbsas s

−=

−−=

2

3

2

1

2

1

β

α

( 2.3)

The constant k (3

2= ) for the non-power-invariant transformation. In this case, the

quantities asi and siα are equal. If it’s assumed that 0=++ cba iii , the quadrature-phase

components can be expressed utilizing only two phases of the three-phase system:

csbs s

as s

iii

ii

3

2

3

1+=

=

β

α

(2.4)

2) Park and inverse park transformation

The components siα and si β , calculated with a Clarke transformation, are attached to

the stator reference frame β α − system. In vector control, all quantities must be

expressed in the same reference frame. The stator reference frame is not suitable for the

control process. The space vector s

i is rotating at a rate equal to the angular frequency of

the phase currents. The components siα and si β change with time and speed. These

components can be transformed from the stator reference frame to the d-q reference

frame rotating at the same speed as the angular frequency of the phase currents. The dsi

and qsi components do not then depend on time and speed. If the d-axis is aligned with

the rotor flux, the transformation is illustrated in Figure 2.2



8

d

α

β

q

is

r

θ

α si

β si

sd i

sqi

Figure 2.2 Park transformation

The components dsi and qsi of the current space vector in the d-q reference frame are

determined by the following equations:

−

=

s

s

qs

ds

i

i

cos sin

sincos

i

i

β

α

θ θ

θ θ (2.5)

The inverse Park transformation from the d-q to the β α − coordinate system is found

by the following equations:

−=

qs

ds

s

s

i

i

cos sin

sincos

i

i

θ θ

θ θ

β

α (2.6)

2.1.2. Induction motor dynamic model

The system model defined in the stationary β α − coordinate system attached to the

stator is expressed by the following equations.



9

s s s sdt

d i Rv α α α λ += (2.7)

s s s sdt

d i Rv β β β λ += ( 2.8)

r r r r r dt

d i Rv β α α α ωλ λ ++= ( 2.9

r r r r r dt

d i Rv α β β β ωλ λ −+= ( 2.10)

where

r m s s s i Li L α α α λ += ( 2.11)

r m s s s i Li L β β β λ += ( 2.12)

smr r r i Li L α α α λ += ( 2.13)

smr r r i Li L β β β λ += ( 2.14)

Besides the stationary reference frame, induction motor model can be formulated in a

general d-q reference frame, which rotates at a general speed eω . The motor model

voltage equations in the general reference frame can be expressed by using the

transformations of the motor quantities from one reference frame to the general reference

frame. The two phase d-q model of an induction machine rotating at the synchronous

speed will help to carry over this decoupled control concept. This model can be described

by the following set of differentia equations

qsedsds sdsdt d i Rv λ ω λ −+= ( 2.15)

dseqsqs sqsdt

d i Rv λ ω λ ++= ( 2.16)



10

qr r edr dr r dr )( dt

d i Rv λ ω ω λ −++= ( 2.17)

dr r eqr qr r qr )( dt

d i Rv λ ω ω λ −−+= ( 2.18)

dr mds sds i Li L +=λ ( 2.19)

qs sqr mqs i Li L +=λ ( 2.20)

dr r dsmdr i Li L +=λ ( 2.21)

qsmqr r qr i Li L +=λ ( 2.22)

)(2

3dsqr qsds

r

me ii L

PLT λ λ −= ( 2.23)

r r l e P

B p

P

J T T ω ω +=− ( 2.24)

This induction motor model is often used in field orientation control (vector control)

algorithms. To achieve this, the reference frames may be aligned with the stator flux-

linkage space vector, the rotor flux-linkage space vector or the magnetizing space vector.

The most popular reference frame is the reference frame attached to the rotor flux

linkage. This can be accomplished be choosing eω to be the instantaneous speed of rotor

flux and locking the phase of reference system such that the rotor flux is entirely in the d-

axis, resulting in

0=qr λ ( 2.25)

This expresses the field orientation concept in d-q variables. Assuming the machine is

supplied from a current regulated source so the stator equation can be omitted, the d-q

equations in a rotor flux oriented frame becomes [7]:



11

dr dr r pi R λ +=0 ( 2.26)

dr r eqr r i R λ ω ω )(0 −−= ( 2.27)

0=+=qr mqs sqr

i Li Lλ ( 2.28)

qsdr

r

me i

L

PLT λ

2

3= ( 2.29)

Equation (2.29) demonstrates the desired torque control properties in terms of the

current components and the rotor . If the rotor can be kept constant just as it is in the

D.C. machine, then the instantaneous torque control can be achieved by controlling

the current component. From these equations, the following relations can be

obtained:

qs

r

mqr i

L

Li −= ( 2.30)

ds

r

mdr i

pT

L

+=

1λ ( 2.31)

dr r

qsm

sT

i L

λ ω = ( 2.32)

Where slip speed is denoted by r e s ω ω ω −= and r r r R LT = is the rotor time constant. In

the steady state, dsmdr i L=λ and 0=dr i . The phasor diagram of the field oriented

induction machine is illustrated in Figure 2.3.



12

dqsi

e

qsiqs

r

me

qr i L

Li −=

e

dsi

dqr dr λ λ rr

=

q- axis

d- axis

Figure 2.3 Phasor diagram of a field oriented induction motor

Equation (2.31) shows that the machine flux can be determined by controlling the current

component dsi . Therefore, in the steady state, the constant flux can be obtained by

constant dsi . As a result, the torque control can be easily obtained by controlling dsi as

seen in (2.29). Equation (2.32) is the most important expression for the practical

implementation of the induction machine in indirect field control which will be discussed

later.

2.1.3. Basic scheme of Field Orientation control

Field orientated controlled machines need two constants as input references: the

torque component (aligned with the q-axis) and the flux component (aligned with d-axis).

Since the field orientation control is simply depended on projections, the control structure

can handle instantaneous electrical quantities. This makes the control accurate in every

working operation and independent of the limited bandwidth mathematical model. The

field orientation control thus has advantages in the following ways:



13

1) The ease of reaching constant reference (torque component and flux component of

the stator current)

2) The ease of applying direct torque control because in the d-q reference frame the

expression of the torque is iT ⋅∝ λ

By maintaining the amplitude of the rotor flux at a fixed value we have a linear

relationship between torque and torque component current. We can then control the

torque by controlling the torque component of stator current vector. The general block

diagram of a field orientation control system for an induction motor is shown in Figure

2.4.

Inverter

Induction

Motor

θ

*

qi

*

d iPI

PI

)(1 θ −T PWM

a,b,c

to

β α ,

)(θ T

qi

d i

*

α U

*

β U

ai

bi

d i

qiα i

β i

Figure 2.4 General block diagram for a field orientation control system

There are many variations of field orientation control of induction machine.

Depending on the reference frame transformation used, two types of field orientation

control are mostly used: the rotor flux orientation (RFO) [8-10] and the stator flux



14

orientation (SFO) [11-13]. In the rotor flux orientation vector control, the reference frame

rotates synchronously with the rotor flux, while in the stator flux orientation the reference

frame rotates with the stator flux. In both these reference frames, the dynamics of an

induction machine appear similar to a dc machine allowing it to be controlled like a dc

machine. The rotor field orientation control of induction machine can also be classified as

a direct field orientation control [14-17] or an indirect field orientation control [18-20]

depending on how the flux information necessary to perform the reference frame

transformation is obtained.

2.1.4. Direct Field Orientation control (DFO)

Knowledge of the instantaneous position of the flux vector, with which the revolving

reference frame is aligned, constitutes the necessary requirement for proper field

orientation. Usually, the identification of flux position can be based on direct

measurement or estimation from other measurable quantities. Such an approach is what

so called direct field orientation (DFO). Only the air gap flux can be measured directly. A

simple scheme for estimation of rotor flux vector is based on measurement of air-gap flux

and stator current. The disadvantage of direct measurement method is that a flux sensor is

expensive and needs special installation and maintenance, thus, spoil the ruggedness of

the induction motor. In practice, the rotor flux is usually computed from the stator voltage

and current. This technique requires the knowledge of the stator resistance along with

the leakage and magnetizing inductance. This method is commonly known as the voltage

model observer [21]. The scheme of a direct field orientation is shown in Figure 2.5.



15

Flux

observer

dqr λ siαβ

svαβ

eθ

)/(1r r tg α β λ λ −

Figure 2.5 The scheme of direct field orientation

The stator flux along the α and β axes, in the stationary frame of reference, can

be estimated by the equations:

s s s s i Rvˆ p α α α λ −= (2.33)

s s s s i Rvˆ p β β β λ −= (2.34)

The rotor flux can be calculated from above

)i Lˆ ( L

Lˆ s s

m

r r α σ α α λ λ −= (2.35)

)i Lˆ ( L

Lˆ s s

m

r r β σ β β λ λ −= (2.36)

where )(2

r

m s

L

L L L −=σ is leakage induction. This method depends on parameters

such as the stator resistance and the leakage inductance. The study of parameter

sensitivity [22,23] shows that the leakage inductance can significantly effect system

performance such as stability, dynamic response, and utilization of the machine and the

inverter. The major difficulty in this case is the need for three motor parameters . The

stator resistance is a significant problem because of temperature dependable, the two

inductance parameters are only moderately affected by saturation. There are also problems

with integrating low frequency signals and with the fact that the stator resistance



16

voltage drop becomes dominant at low speed. These limitations preclude use of this

scheme at low speed. However, it is practical over a reasonable speed range and is used in

many implementations.

2.1.5. Indirect Field Orientation Control (IFO)

Indirect field orientation is based on the slip relation as shown in Equation (2.32). The

control algorithm for calculation of the rotor flux angle using IFO control is shown in

the Figure 2.6. This algorithm is based on the assumption that the flux along the q-

axis is zero which imposes a condition on the command slip that isdr r

qsm

sT

i L

λ ω = , a

necessary and sufficient condition to guarantee that all the flux are aligned along d -

axis and the flux along q-axis is zero. The angle can be then calculated by adding the

slip angle and the rotor angle. The slip angle includes the necessary and sufficient

condition for decoupled control of the flux and torque.

ds

qs

r i

i

T *

*1

sω

r ω eω

dsi*

qsi*

∫

IM

eθ

1−T

Figure 2.6 The scheme of indirect field orientation

The IFOC is an open loop feed forward control in which the slip frequency is fed

forward, guaranteeing the field orientation. This feed forward control is very sensitive to



17

the rotor open circuit time constant T r . Therefore, T r must be known in order to achieve

a decoupled control of torque and flux by controlling iqs and ids respectively. When T r is

not set correctly the motor will be detuned, and the controller performance will become

sluggish due to loss of decoupled control of the torque and flux .

2.1.6. Variable speed control of induction machines

A typical variable speed induction machine drive system consists of an induction

machine, a power inverter and a DSP/microprocessor based controller. Generally, there

are two feedback loops typically used to implement field orientation control and speed

control. The field orientation control is implemented in the inner current loop, the

decoupled control of flux and torque can be obtained by d - and q-axis current

regulator. In the area of controlled electric drives, the drive inertia and load

characteristics change widely. Although current control is important to torque

performance, the speed controller has directly impacts the system performance. It is

desirable to have a drive system that can provide fast dynamic response, a parameter-

insensitive control feature, and rapid recovery from speed drop caused by impact loads. A

satisfactory speed controller is extremely important for achieving desired response.

Traditionally, a proportional-integral (PI) controller is often used in the outer

speed regulation loop [24-26]. The PI controller offers fair performance in a stable

and robust manner if it is well tuned. The PI controller is usually designed in a linear

region ignoring the saturation-type non-linearity. At some working area, the behavior of

such controller could be satisfied. When the controller is applied to variable speed motor

drives, the performance deterioration is referred to a windup phenomenon, which causes

large overshoot, slow setting time, and, sometimes, even instability. So the parameters of



18

the PI controller should be modified according to the various operating condition of

induction motor. This would add difficulty to the on-line debugging and cause several

drawbacks in using PI as the speed controller. 1) PI controller has a set of fixed PI

gains cannot satisfy the requirements for different speed commands. 2) PI speed

controller lacks the ability to handle detuning problems when parameters of the

machine vary. 3) Tuning PI gains is very time consuming. The limitations of the PI

controller have motivated research into alternative control techniques such as fuzzy logic,

sliding mode control, etc.

Fuzzy Logic Control (FLC) provides a systematic method to incorporate human

experience in the controller [27-37]. FLC can perform better with high nonlinearities

and overcome parameters variation emphasizes the importance of exploring control

techniques other than a conventional PI controller. Recent literature has explored the

potential of fuzzy control for machine drive applications It has been shown that a

properly designed direct fuzzy controller can outperform conventional PID controllers.

However, the performance will still degrade when the machine is severely detuned.

Another approach proposed and widely studied is the discontinuous sliding mode

control [38-43]. SLM control offers attractive features such as insensitivity to parameter

variations (as long as the bounds of the parameter variations are known) and

computationally simple to implement. However, it is reported by many authors that

sliding mode in motion control exhibits chattering imposed by the discontinuity of the

control action. The essential of sliding mode control is that the discontinuous feedback

control switches on one or more manifolds in the state space. Ideally, the switching of

control occurs at an infinitely high frequency to eliminate deviations from sliding



19

manifolds. In practice, the frequency is not infinitely high due to the finite switching time

and with effects of un-modeled dynamics. This chattering is quite undesirable for most

applications. Different schemes have been suggested to eliminate the chattering such as

sliding-mode controller with boundary layer [44], fuzzy sliding-mode controller (FSMC)

[45]. Bartolini [46,47] proposes to introduce an integrator into the controller and design a

discontinuous control as the derivative of the actual control signal. The chattering

problem is addressed in [48] by analyzing the saturation function as an approximation to

the discontinuous switching element in the presence of singularly perturbed actuator

dynamics. Among the alternatives to eliminate chattering, the most promising is a

chattering free sliding mode technique which produces a continuous signal to the system

[49-52].

2.2. Speed Sensorless Control Technology of Induction Machines

The approach of speed senseless control of induction motor has been receiving more

and more attention in industry application since it can reduce cost and avoid fragility of a

mechanical speed sensor, and eliminate the difficulty of installing the sensor in some

applications. Different techniques for obtaining the rotor speed, estimating the rotor flux

of an induction machine for sensorless control have been extensively studied in the past

two decades and can be broadly classified as:

1) Magnetic-saliency-based methods [53-55];

2) Voltage model and current model flux and speed estimations [56-58];

3) Model reference adaptive schemes [59-61];

4) Adaptive observer based approaches [62-64];



20

5) Extended Kalman filters [65,66];

6) Sliding mode observers [67-70];

7) Artificial neural network and artificial intelligence based sensorless control [71,72].

Magnetic-saliency-based methods are proposed in [53,54], which allow standstill and

low speed operation. These approaches rely on the motor response to the injection of

relatively high-frequency test signals, which investigate the motor saliency due to

saturation or geometric construction. They need high precision in the measurement and

increase the hardware and software complexity with respect to a standard vector control

scheme. When applied with the high frequency signal injection [55], the method may cause

torque ripples, vibration, and audible noise. Moreover, motors having a low saliency

content do not give an appreciable response, whereas enhancing the saliency requires a

proper machine design, therefore the saliency based technique is machine specific and

can not be applied to a standard machine.

The problems when using non-magnetic-saliency-based methods are flux integration,

unavailability of the signal at low speed and parameter sensitivity. Different schemes to

overcome these problems and to improve the sensorless control have been proposed in the

literature.

The voltage model flux estimations [56,57] have problems at low frequency regions,

because the signal to noise ratio of the stator voltage measurement is very poor, and

voltage drop on the stator resistance is dominant. The voltage model is also sensitive to

the leakage inductance. The current model flux observer is considered to have better

performance at low speeds. Also, its accuracy is relatively unaffected by the leakage

inductance for any operating condition. However, it does not work well at high speed due



21

to rotor resistance variation. To get better performance, it has been suggested to use the

current model observer at low speed and voltage model observer at high speed [56, 57].

To further improve the observer performance, close loop rotor flux observers are proposed

[58].

Model reference adaptive schemes (MRAS) are proposed in [59-61], where one of the

flux estimators acts as a reference model, and the other acts as the adaptive estimator. The

estimation is based on the comparison between the outputs of two estimates, and the

output errors are then used to drive a suitable adaptation mechanism that generates the

estimated speed. These schemes require integration and. To overcome the integration

problem, Peng [60] suggested the use of back-EMF and instantaneous reactive power as

alternative ways to estimate the velocity in the adaptive controller. However, the

performances are still limited by parameter variations and the accurate flux estimation

problem still remains.

Adaptive observer based approaches [62,63] can have preferred performance using

the derived adaptive laws with relatively simple computation. However, their robustness

to parametric uncertainties is never guaranteed. Reduced order observers are designed in

[64], in which only the rotor flux, not the stator current is estimated. The correction is

then applied by using the error between the actual stator voltage vector and an estimate

ones. However, this requires adding voltage sensors to the system, which is not desirable.

Extended Kalman filters have been proposed in [65, 66] as a potential solution for

better flux estimation. Unfortunately, this approach contains some inherent disadvantages

such as computational expense and having no specific design and tuning criteria.



22

Sliding mode has been documented to have the advantages of robustness and

parameters insensitivity [67] and been recognized as the prospective control methodology

for electric motors. Flux observers have been designed [67-69] using the sliding mode

technique for sensorless speed control of induction machines. These algorithms use a

current model flux observer and apply a correction term based on the current estimation

error. The observers require the rotor speed and rotor time constant for the current and

flux estimations. Therefore, an error in the estimated speed or rotor time constant will

affect the current and flux estimations, and thus degrade the observer accuracy.

Other algorithms for speed sensorless vector control, such as artificial neural network

[71] and artificial intelligence (AI) [72], can achieve high performance, but are relatively

complicated and require large calculation time.

An effective sliding mode based flux and speed estimation technique for sensorless

control of field-oriented induction machine is presented. The flux observer model is

decoupled by the proposed sliding function, which makes the observed rotor flux

independent of rotor speed. But the observed flux calculation is still sensitive to

parameter variation. To overcome this sensitivity and make flux and speed estimation

robust to parameter variations, an adaptive sliding mode flux and speed observer is

proposed. Two sliding mode current observers are used in the method. The effects of

parameter deviation in the rotor flux observer can be alleviated by these two current

sliding mode observers. The stability of the method is proven by Lyapunov theory. An

adaptive speed estimation is also derived from the stability theory.



23

2.3. Efficiency optimization of induction machine control

Induction machines consume most of the world’s electrical energy every year.

Improving efficiency of electrical drives is important not only for energy saving, but also

for environmental protection. Induction motors normally operate at rated flux in a

variable frequency drive to get a best transient response. However, most of the time, the

drive system operates with light loads. In this case the core losses become excessive

causing poor efficiency. To improve the motor efficiency, the flux must be reduced,

obtaining a banlance between the copper and iron losses.

A number of methods for efficiency improvement through flux control have been

proposed in the literature. They can be classified into three basic types. The simple pre-

computed flux program as a function of torque is widely used for light load efficiency

improvement. This method, however, yields only a partial improvement in the system

efficiency. The second approach is based on the modeling of the motor and the losses to

derive an objective function. The objective function is optimized (either minimized or

maximized) to yield the maximum efficiency. Thus, this method treats the situation

analytically by properly modeling the losses and is called Loss Model Controller

(LMC)[73-75]. The third method is on-line efficiency optimization control on the basis of

search, and has a feedback nature that finds the maximum efficiency and is called Search

Controller (SC) [76-80].

The LMC method has the advantage that it is fast, however, the accuracy depends on

correct modeling of the motor drive and the losses. Garcia [73] proposed a simple loss

model consisting of computation of iron loss, rotor and stator losses in function of stator

current and in the frame. For a given speed and torque, the solution of the loss model



24

yields the flux current for which the total loss is minimal. Lorenz and Yang [74] pointed

that major loss saving is possible by considering the system as a whole and employing

dynamic programming to select the operating flux. Kioskeridis and Margaris [75]

calculated the total of iron loss, copper losses, and stray loss and found an optimal flux

level that minimizes the total loss. Thus, LMC method is to develop controllers for

different drive systems by building the loss models and including different applications.

The LMC method consists in the real time computation of losses and corresponding

selection of flux level that results in minimum losses. As the loss computation is based

on a machine model, parameter variations caused by temperature and saturation effects

tend to yield suboptimal efficiency operation.

SC method on the other hand offers optimum efficiency based on the exact

measurement of input power (or DC bus power). Sul and Park [76] proposed a method that

maximizes the efficiency by means of finding optimal slip. The technique can be

considered as an in direct way to minimize the input power. For the vector drive, Kirschen

et al. [77] reduced the flux in small steps to reach to the optimum condition. Kim et al.

[78] adjusted the squared rotor flux according to a minimum power algorithm using

search method. Sousa et al. [79] reduced the reference flux current by minimizing input

DC bus power using fuzzy logic, where the torque pulsation is overcome by using feed

forward pulsating torque compensation. Ta and Hori [80] improved the convergence rate

by a golden-section-based search algorithm. However, problems still exist in selecting

the upper and lower limit of the flux-producing current before the algorithm starts. Thus,

the object in the SC is to reduce the search time and torque pulsations. Moreover, The SC



25

method does not require the knowledge of machine parameters and completely

insensitive to parameter changes.

To make a comparison, we can conclude that the SC is always slow comparing to

LMC and the LMC works on the model and not on the actual drives. In LMC, the loss

minimization optimum flux is calculated analytically. The main advantage is the

simplicity of the method and not requiring extra hardware. However, it is sensitive to

motor parameters which change considerably with temperature and load condition.

Performance of the LMC method deteriorates when parameters change, the online

estimation of the parameters makes the method far more complicated. On the other

hand, SC method measures input power to searches the flux where the motor runs at

maximum efficiency. This approach is insensitive to motor parameters and operating

condition. However, it does require extra hardware to measure DC bus current and does

not be used in the classical vector control system where additional sensor is not available.

To take the advantages of both LMC and SC, an efficiency optimization method by

minimizing the stator current is presented. This approach does not require extra hardware

and is insensitive to motor parameters. The relationship between stator current

minimization and motor losses minimization in the induction motor vector control system

has been investigated. It is pointed that minimum stator current point is very close to

minimum losses point in most cases and the losses minimization can be achieved by

minimizing stator current in practice. A fuzzy logic based search method is simulated and

implemented. Simulation and experimental results are given in the paper to verify the

proposed method.



26

CHAPTER 3

3. SLIDING MODE FLUX OBSERVER FOR DFO SENSORLESS

CONTROL

3.1. Introduction

In the past decade, a wide range of nonlinear methods for feedback control, state

estimation, and parameter identification has merged. Among them, sliding mode control

gained wide acceptance because sliding mode method can offer many good properties, such as

insensitivity to parameter variations, external disturbance rejection, and fast dynamic response.

In this chapter, based on the concept of equivalent control of sliding mode, a speed

observation system, which comprises a current observer, a rotor flux observer and a rotor

speed observer, is presented for a direct rotor flux oriented induction motor drive.

3.2. Induction motor model

Induction motors can be modeled in various reference frames. Commonly used

reference frames include stationary reference frame, which is fixed to the stator, and

synchronous frame, which is rotating at the synchronous speed. In this Chapter, by

defining stator currents and rotor fluxes as the state variables, we can rewrite the

induction motor model in the stationary frame as

s s s sdt

d i Rv α α α λ += ( 3.1)

SLIDING MODE FLUX OBSERVER FOR DFO SENSORLESS

CONTROL



27

s s s sdt

d i Rv β β β λ += (3.2)

r r r r r dt

d i Rv β α α α ωλ λ ++= (3.3)

r r r r r dt

d i Rv α β β β ωλ λ −+= (3.4)

r m s s s i Li L α α α λ += (3.5)

r m s s s i Li L β β β λ += (3.6)

smr r r i Li L α α α λ += (3.7)

smr r r i Li L β β β λ += (3.8)

For squirrel rotor, 0,0 == r r vv β α . Eliminating the rotor currents and stator fluxes from

Equations (3.5)-(3.8), we have

s sr

r

m s i L

L

Lα α α σ λ λ += (3.9)

s sr

r

m s i L

L

L β β β σ λ λ += (3.10)

)(1

smr

r

r i L L

i α α α λ −= (3.11)

)(1

smr

r

r i L L

i β β β λ −= (3.12)

Substituting Equations (3.9)-(3.12) into Equations (3.1)-(3.4), yield the induction motor

model in β α − coordinate system

s

s

s

r r

m s

s

r r

r

m

s

r

r r

m

s

s v L

iT L

L R L L

L LT L

L L

idt d

α α β α α σ σ

λ ω σ

λ σ

1)(11112

++−+= (3.13)

s

s

s

r r

m s

s

r r

r

m

s

r

r r

m

s

s v L

iT L

L R

L L

L

LT L

L

Li

dt

d β β α α β

σ σ λ ω

σ λ

σ

1)(

11112

++−−= (3.14)



28

s

r

mr r r

r

r iT

L

T dt

d α β α α λ ω λ λ +−−=

1(3.15)

s

r

mr r r

r

r iT

L

T dt

d β α β β λ ω λ λ ++−=

1(3.16)

Letdt

d p = ,

r

r r

R

LT = ,

s Lk

σ

12 = ,

r s

m

L L

L2

1−=σ ,

r

m

L

Lk 2= β , )(2

21

r r

m s

T L

L Rk k += ,

r

m

T

Lk =3 ,

the above equations become:

s sr r r

r

s vk ik T

pi α α β α α λ βω λ β

21 +−+= (3.17)

s sr r r

r

s vk ik T

pi β β α β β λ βω λ β

21 +−−= (3.18)

sr r r

r

r ik T

p α β α α λ ω λ λ 3

1+−−= (3.19)

sr r r

r

r ik T

p β α β β λ ω λ λ 3

1++−= (3.20)

Equation (3.17) - (3.20) can be written in matrix form as

VIAΛI 21 k k p +−= β (3.21)

IAΛΛ 3k p +−= (3.22)

where ],[ T

s s ii β α =I ,T

r r ],[ β α λ λ =Λ ,

−=

r

r

r

r

T

T

1

1

ω

ω

A

It can be seen that the term AΛ appears in both current and flux equations of the

machine. So this model has the advantage that the coupling terms between α and β axes



29

are exactly same and the coupling terms can be replaced by the same sliding function

both in current and flux equations.

3.3. Sliding mode current observer

As noted above, the current and flux observer can be designed by replacing the term

AΛ with sliding function U . The sliding mode current observer is defined as:

+

−

=

s

s

s

s

r

r

s

s

v

vk

i

ik

U

U

i

i p

β

α

β

α

β

α

β

α β 21 ˆ

ˆ

ˆ

ˆ( 3.23)

and the rotor flux observer can be written as

+

−=

s

s

r

r

r

r

i

ik

U

U p

β

α

β

α

β

α

λ

λ

ˆˆ

3 (3.24)

or in matrix form, we have

VIUI 21 k ˆ k ˆ p +−= β (3.25)

IUΛ ˆ k ˆ p 3+−= (3.26)

where

][ T

s s i ,i ˆ β α =I ,

T

r r ˆ ,ˆ ˆ ][ β α λ λ =Λ , T][ r r U ,U β α =U

) s( signuU sr α α 0−= , ) s( signuU sr β β 0−=

and

s s s s iii s α α α α −== ˆ , s s s s ii i s β β β β −==

<−>=

01

01

s

s

s sif

sif ) s( sign

α

α

α ,

<−>= 01

01

s

s

s sif

sif ) s( sign

β

β

β

Two independent sliding functions r U α and r U β are designed for the α and β

axes of the current observer, respectively. It is noted that the sliding functions r U α



30

and r U β are only dependent on the error between the measured and estimated phase

currents. The α and β axes rotor fluxes are only intergration of these sliding

functions and their own currents. So, the designed current and flux observers for

the α and β axes have no coupling between them, making the current and the flux

observer models completely decoupled.

The stability of the observre can be proved by Lyapunov stability theory. Let us select

the Lyapunov function as

n

T

n s sV

2

1= (3.27)

where s sn s s s β α = . The Lyapunov function V is positive definite, which satisfies the

Lyapunov stability first condition. The derivative of V is

n

T

n s sV && = (3.28)

To satisfy the Lyapunov stability, second condition must satisfy 0<V & . From Equation

(3.13) and (3.17), we have

IAΛUI 1k )( sn −−== β &&

01 <−−= IIAΛUIT T k )( V β &

Thus

010 <−−− IIAΛIIT T

k ) )( signu( β

s s

T T

ii

k

u β α

β +

−>

IIAΛI 1

0 (3.29)

If 0u is large enough, found by existence condition, the sliding mode ( 0=n s ) will

occur. The system trajectories reach the sliding manifold.



31

Let )( qr r

r

dr ds

T i λ ω

λ +=Γ , )( dr r

r

qr

qsT

i λ ω λ

+=Φ , )(221

qsds iik

+−=Ω β

qsds

slid

ii

Bu

+

Ω+Φ+Γ=>

||||||0

where slid B is the boundary of sliding function. When the system reach the sliding

surface 0=n s , that means the observed currents converge to the actual ones, then the

flux estimation just an integration of sliding mode function without need of other

information related motor parameter or speed. The resulting equivalent control depends

on machine parameters and is difficult to implement. It is reasonable to assume that the

equivalent control is the slow component of real control that can be obtained by using a

low-pass filter.

β r

eq

β r r α

eq

αr U 1 µs

1U ,U

1 µs

1U

+=

+= ( 3.30)

From Equations (3.2) and (3.3), (3.4), we have

−=

r

r

r

r

r

r eq

r

eq

r

λ

λ

T ˆ

1ω

ω T ˆ 1

U

U

β

α

β

α ( 3.31)

3.4. Rotor flux and speed estimation

Based on the equivalent control concept, if the observered currents converge to

measured ones, the rotor flux can be calculated from Equation (3.26)

dt )i k U ( ˆ s

eq

r r α α α λ 3+−= ∫ (3.32)

dt )i k U ( ˆ s

eq

r r β β β λ 3+−= ∫ (3.33)



32

Moreover, the angle of the rotor flux can be calculated by the following equation:

)ˆ

ˆ ( tanˆ

r

r

α

β

λ

λ θ 1−= (3.34)

This rotor flux angle is used in the field orientation control. Noted that the rotor fluxes

are estimated only by sliding function and the their own currents. It requires no speed

information, making control system very easy to implement.

From Equation (3.31), we have the form

−=

eq

r

eq

r

r r

r r

r

r

r U

U

λ λ

λ λ

λ

1

ω T ˆ

1

β

α

α β

β α ( 3.35)

where22

r r r ˆ ˆ ˆ β α λ λ λ +=

The rotor speed and rotor time constant can be calculated as:

( )eq

r r

eq

r r

r

r U λ U λ

λ

1ω

β α α β −= (3.36)

It is important to notice that the observer structure is decoupled in the sense that the

estimation process for α and β axis fluxes are independent because of the choice of

sliding mode function.

3.5. Simulation by Matlab

Figure 3.1 shows the diagram of system block using sliding mode observer for direct

filed orientation control of induction motors. Two motor phase currents are measured.

These measured currents are used for the Clarke transformation module. The outputs of

this projection are designated by siα and si β . These two components of the current are the

inputs of the Park transformation that gives the current in the d, q rotating reference



33

frame. The dsi and qsi components are compared to the references*

dsi (which is the

output of speed regulation) and*

qsi (the torque reference).

3-phase

Inverter

Recitifier

Induction

Motor

θ

*r ω

r ω ∆*

qsi

*

dsir ω

d,q

to

β α ,

)(1 θ −T

Space

Vector

PWM

a,b,c

to

β α ,

SlidingMode

FluxObserver

to

d,q

β α ,

)(θ T

PWM1~6

qsi

dsi

*

qU

*

d U

*

α U

*

β U

ai

bi

siα

si β

Speed

observer

Flux

weakening

Speed

regulator

PI

regulator

PI

regulator

Figure 3.1 Conrol system block

The presented flux and speed observers have been simulated by Matlab for direct

field orientation control. The simulation is based on per unit system. The motor

parameters are as following:

1 HP 4 poles

R s = 6.3 ohms Rr = 8.2 ohms

Lls = Llr = 19.8 mh Lm = 335 mh

Figure 3.2 through 3.6 show the simulation results. The real motor speed and

estimated speed are shown in Figure 3.2 for a step speed command at start up. Figure 3.3



34

shows the real and estimated stator currents. Once the estimated currents converge to real

currents, the rotor fluxes can be calculated from the equivalent control as shown in Figure

3.4, where the estimated rotor fluxes converge to real rotor fluxes. The transient speed

response for trapezoid command is shown in Figure 3.5. Figure 3.5(b) shows the speed

track error and Figure 3.6 shows the real and estimated rotor flux under this condition.

Figure 3.2 Step speed command at start up

Figure 3.3 Real and estimated stator currents



35

Figure 3.4 Real and estimated rotor fluxes

Figure 3.5 Motor response for trapezoid speed command



36

Figure 3.6 Real and estimated rotor flux

3.6. Simulation by HIL(hardware-in-the-loop)

The algorithm is also evaluated by Hardware-in-the-loop (HIL) using TMS320F2812

digital signal processor. HIL evaluation is to use a computer model of the process as the

real target hardware, and on the other hand, the control and estimation algorithm are

implemented in real time. The purpose of HIL is to make evaluation of the proposed

algorithm as closely as possible to those that would be encountered in the real time

implementation. The dynamics of electric machine is modeled by five differential

equations. The control and estimated algorithms are implemented in 32-bit Q-math

approach, interacting with the motor model rather than the real targeted physical system.

The main advantages of this evaluation are: 1) the control software are implemented

and evaluated in real time and can be debugged very easily in the absence of motor; 2)

The control software can be easily transferred to the real drive system with only minor

changes. Figures 3.7 through 3.10 show the HIL simulation for induction motors at



37

different power rates. (from 1 hp motor to 50hp motor). The results show that the sliding

mode algorithm can be implemented in TMS320F2812 fixed-point DSP and can be

simulated for different motors. The results are very similar to the results of Matlab

simulation. The results also prove that the sliding mode algorithm is stable and can be

successfully implemented by DSP hardware.

Figure 3.7 HIL simulation results for 1 hp motor

Figure 3.8 HIL simulation results for 1.1kw motor



38

Figure 3.9 HIL simulation results for 5hp motor

Figure 3.10 HIL simulation results for 50hp motor



39

3.7. Experimental results

A conventional voltage-source power inverter is used to drive the motor. The

TMS320x24xx DSP generates six pulse width modulated (PWM) signals, which control

the six power devices in the inverter. Two of the motor phase currents (ia and ib) are

sensed using the inverter leg resistors and measured by the two analog-to-digital

converters (ADCs) in TMS320x24xx. The advantages of this method are the low cost and

ability to eliminate the common-mode voltage as long as the measurement circuit is

referenced to the DC bus common. The measured current of this method is no longer

motor phase current, but half-bridge current. If the low side switch is conducting (through

either the transistor or freewheeling diode) then the current is equal to that motor phase

current. This certainly occurs periodically throughout the PWM cycle, so a reconstruction

circuit including a sample and hold amplifier is required. In addition, the DC bus voltage

is also measured by an ADC channel. This information is used to calculate the three

phase voltages of the motor.

The experimental system consists of the following hardware components:

• Power drive board (include rectifier circuit and IGBT module);

• TMS320F2407 EVM platform;

• Three-phase induction motor with a (optional) sprocket;

• IBM compatible PC with Code Composer (CC) installed and emulator;

• Additional instruments such as oscilloscope, digital multi-meter, current sensing

probe and function generator.

The experimental setup and connection is illustrated in Figure 3.11. The experimental

results are shown in Figures 3.12 through 3.17. The estimated stator current and



40

measured current are compared in Figure 3.12. It shows these currents are very close to

each other and eventually the estimated one will converge to the measured one. Figure

3.13 shows the estimated rotor flux and flux angle, which is used for the Park and

inverse-Park transformation. The rotor flux trajectory estimated by the sliding mode

observer is shown in Figure 3.14. Figure 3.15 shows the transient response of drive

system to a trapezoidal speed command. The performance of motor speed step change

from zero speed to 950rpm is shown in Figures 3.16 and 3.17.

AC~

Power drive board

DSP board

Figure 3.11 Experimental setup



41

20ms/div

Figure 3.12 Estimated and measured current (current siα ; curve 2: phase current si α )

20ms/div

Figure 3.13 Estimated rotor flux and flux angle

ia

r ˆ θ

r ˆ α λ

siα

si α

1A/div

1A/div

5A/div





43

Speed step response

0

200

400

600

800

1000

1200

0 1 2 3 4 5

Time (s)

S p e e d ( r p m )

Speed Command

Actual speed

Estimated speed

Figure 3.16 Motor response for step speed change from standstill to 950 rpm

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0 1 2 3 4 5

Time (s)

P h a s e c u r r e n t ( A )

Figure 3.17 Phase current response for step speed change



44

3.8. Parameter sensitivity analysis

The flux observer model presented above has accurate flux estimation when motor

parameters are exactly known. However, when the motor parameters are changed due to

temperature or different from preset values, the estimated flux and speed will deviate from the

real values. To investigate the influence of parameter variation, we change the coefficients

1k , 2k , and β in observers. There will be errors1k ∆ ,

2k ∆ , β ∆ exist if these parameter are

changed. The current observer will be in the form of

VIUI )(ˆ)()(ˆ2211 k k k k p ∆++∆+−∆+= β β ( 3.37)

The observed current error is

[ ] VIIAΛUI 211ˆ)( k k k p ∆+∆−−−∆+= β β β ( 3.38)

By selecting0u large enough, we have 0=I . Then the equivalent control becomes

VIAΛU β β β β β β

β

∆+

∆−

∆+

∆+

∆+= 21 ˆ k k

eq( 3.39)

The error is

VIAΛAΛUU β β β β β β

β

∆+∆

−∆+

∆+

∆+∆

−=−=∆ 21 ˆ k k eq

( 3.40)

This error will cause incorrect flux and speed estimation. Figures 3.18 and 3.19 show

the simulation results when 1k and β are change by 20%. It can be seen from the

simulation that the estimated flux is deviated from actual value and the observed speed

fluctuate around real speed. The influence of rotor resistance change is shown in Figures

3.20 - 3.22. In Figure 3.20 the parameters in observers have the same value as the

induction motor model does. In Figures 3.21 and 3.22, the rotor resistance in flux

observers is increased and decreased by 20% separately. As can be easily observed, the



45

changes of rotor resistance will produce substantial speed error in steady state. This

phenomenon is also observed in experimental results as shown in Figures 3.23 - 3.25. In

Figure 3.23, the rotor time constant in flux observers is exactly the same as motor actual

value in the experiments. In Figures 3.24 and 3.25, the rotor time constant value in

observers is changed on purpose by ±22%. When T r varies, speed varies ±5.5%

(Approximately ∆n=±30rpm @550rpm). The simulation and experiments indicate that

the system is insensitive to rotor parameter under no load condition. When load increases,

the motor currents increase, the error of equivalent control will increase, thus the speed

error is more dependent on rotor parameter. In low speed, system performance is also

sensitive to parameter variation.



46

(a) r α λ real rotor flux, obsλ estimated rotor flux

(b) r ω real rotor speed, robsω estimated rotor speed

Figure 3.18 Coefficient1k in the observer is increased by 20%



47

(a) r α λ real rotor flux, obsλ estimated rotor flux

(b) r ω real rotor speed, robsω estimated rotor speed

Figure 3.19 Coefficient β in the observer is increased by 20%



48

Figure 3.20 Speed response with Rr unchage under 0.5 pu torque disturbance

Figure 3.21 Speed response with Rr decreasing 20% and 0.5 pu torque disturbance



49

Figure 3.22 Speed response with Rr increasing 20% and 0.5 pu torque disturbance

0

100

200

300

400

500

600

700

0 2 4 6 8 10

Time (s)

S

p e e d ( r p m )

Real Speed

Estimated Speed

Figure 3.23 Experimental result with T r unchage under 2.0 N.m torque disturbance



50

0

100

200

300

400

500

600

700

0 2 4 6 8 10

Time (s)

S p e e d ( r p m )

Real Speed

Estimated Speed

Figure 3.24 Experimental result with T r increase 22% under 2.0 N.m torque disturbance

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Time (s)

S p e e d ( r p m )

Real Speed

Estimated speed

Figure 3.25 Experimental result with T r decrease 22% under 2.0 N.m torque disturbance



51

3.9. Conclusion

A sliding mode flux and speed observer is presented and implemented. The terms

containing fluxes, which are common in both current and flux equation, are estimated by

a sliding function, which makes d- and q- axis flux equations decoupled in the stationary

frame. The flux estimation is easy to calculate and merely an integration of known terms,

which make the algorithm simple to implement. However, the parameter sensitivity

analysis shows that the equivalent control will detune if the parameters in observer is

incorrect, causing flux and speed estimation incorrect, which is confirmed by simulation

and experiments.



52

CHAPTER 4

4. ROBUST SLIDING MODE SPEED CONTROLLER

4.1. Introduction

A high performance speed motor drive should possess good command tracking and

load regulation dynamic responses, and these responses should be insensitive to the

operating condition. The uncertainties usually are composed of plant variations, external

load disturbance, and nonlinear dynamics of the plant. Many researches have been

reported for the robust speed control of an induction motor. The proportional (P),

proportional plus integral (PI), proportional plus integral plus derivative (PID)

conventional controllers are very easy to design and implement. The proposed continuous

sliding mode controller is robust to load changes and system disturbances. Also this

method overcomes the chattering problem which is the main concern when using

discontinuous sliding mode controller. It can prevent the performance degradation and

avoid tedious tuning process comparing the conventional PI controller. The

simulations and experimental results prove that the proposed controller is robust to

external disturbance and can also follow speed command trajectories very well without

re-tuning the controller

ROBUST SLIDING MODE SPEED CONTROLLER





54

)t ( u ) x , x( B ) x ,(x f x 212122 +=& (4.6)

mmmn u , x , x ℜ∈ℜ∈ℜ∈ −21

B is a nonsingular matrix, m(B)rank = . The aim is to drive the state of the system to

manifold defined by

)t , x( ) x( )t ( : x S 0==−= σ ξ ϕ (4.7)

where x is the state vector obtained by augmenting 1 x and 2 x . )t ( ϕ is the time dependent

part of the sliding function and contains reference inputs to be applied to the controlled

plant. ) x( ξ denotes the state dependent part of )t , x( σ .

The stability conditions for selected control must be examined first. This selection

should ensure the stability of the system’s motion in the origin of the subspace, whose

coordinates are distances from the sliding mode manifold. For the selected manifold (4.6)

the first choice is the Lyapunov function in a quadratic form of control error as in Equation

(4.3). The solution 0= )t , x( σ will he stable if the time derivate of the Lyapunov

function can be expressed as a negative definite function:

σ σ DV T −=& (4.8)

where D is a positive definite matrix. Thus, the derivative of the Lyapunov function will

be negative definite, and this will ensure stability. From (4.4) and (4.8), we have

0=+ ) D( T σ σ σ & (4.9)

A solution for the equation above is

0=+ ) D( σ σ & (4.10)

The expression for the derivative for the sliding function is



55

) BuG f G f G( 22211 ++−=

−=

ϕ

ξ ϕ σ

&

&&&(4.11)

where

2211 xG xG +=ξ and 2211 xG xG &&& +=ξ

The equivalent control can be found by 0=σ & , so

0=−= ξ ϕ σ &&& (4.12)

Then

) f G f G( ) BG( ueq 22111

2 −−= − ϕ & (4.13)

From (4.10) and (4.11), we have

σ

ϕ σ

D

) BuG f G f G(

−=

++−= 22211&&

(4.14)

then

σ

σ ϕ

D ) BG( u

D ) BG( ) f G f G( ) BG( u

eq

1

2

1

22211

1

2

−

−−

+=

+−−= &(4.15)

Multiplying 1

2

− ) BG( to both sides in Equation (4.11)

u f G f G BG BG −−−= −− )()()( 2211

1

2

1

2 ϕ σ && (4.16)

Replacing the first term on the right by equ , the above equation becomes

uu BG eq −=− σ &12 )( (4.17)

That is

σ &1

2

−+= ) BG( uueq (4.18)



56

Which indicates that the equivalent control is the sum of current control value and

σ &1

2

− ) BG( . The current value of control input is not available to use. An approximation

of previous value is used instead.

σ &12

−− += ) BG( )t ( u )t ( ueq (4.19)

where 0 , +→∆∆−=− t t

Substituting (4.18) into (4.14), we obtain

−=−− ++=

t t | ) D( ) BG( )t ( u )t ( u σ σ &1

2 (4.20)

The term −=− +

t t | ) D( ) BG( σ σ &1

2 is used in updating a recursive formula for the control

input. On the sliding manifold, )t ( u− becomes the same as the equivalent control.

Although Equation (4.19) is an approximation of (4.14) in discrete time, it can be used to

push σ toward zero, so that (4.10) holds and stability is reached [49].

4.4. Implementation in the induction motor drive system

The induction machine torque and mechanical equation can be expressed by

r dt

d ω

θ = (4.21)

) BT T ( J dt

d r f l e

r ω ω

−−=1

(4.22)

where J is motor inertia, l T is load torque, f B is friction coefficient.

If the induction motor is in field oriented control, for a fixed rotor flux, the motor torque

can be written as

qst qsdr

r

me ik i

L

PLT == λ

3

2(4.23)



57

where

dr

r

mt

L

PLk λ

3

2=

Equation (4.22) becomes

qst

r f l r i

J

k ) BT (

J dt

d −−−= ω

ω 1(4.24)

Let 21 x , x r == ω θ , the regular form for motor control can be rearranged as

21 x

dt

dx= (4.25)

KuT J

x B J dt

dxl f +−−= 11 2

2 (4.26)

where J

k K t =

4.5. Simulation results

A 1 HP cage-rotor induction machine using continuous sliding mode control is

simulated. Figures 4.1 through 4.3 show the simulation results. The simulation

results of PI controller and the proposed sliding mode controller at induction motor

start-up are compared in Figure 4.1. In the figure, the curves PI-1, PI-2, PI-3

correspond to different PI gains. It shows that continuous sliding mode has

much better transient performance, and even more important, it overcomes the

performance degradation with speed and avoids tedious tuning process. The

torque rejection performance of the proposed controller is shown in Figure 4.2,

where a 0.5 pu step load is applied to the machine between 0.6 and 1 second and

then released. The sliding mode controller rejects this disturbance very well.



58

Figure 4.3 shows the speed tracking simulation results for four-quadrant

operation with a triangle speed command.

PI-1

PI-2

PI-3

Slide mode

Figure 4.1 Comparison of PI controller and the sliding mode controller

Figure 4.2 External torque rejection performance



59

Figure 4.3 Speed tracking simulation with a triangle speed command


The proposed chattering free sliding mode controller has been implemented on

the prototype 1 HP induction machine. Figure 4.4 and 4.5 show the motor transient

response to step load changes at 1.0 Nm and 2.0 Nm respectively, where r ω is

measured motor speed,r ω ˆ is estimated motor speed, id is d-axis current, iq is q-axis

current, ia is phase current. The results show that the controller has good dynamic

performance and speed rejection to load change. To further demonstrate the speed

robustness, the test data have been collected through the data acquisition system as

shown in Appendix A.1. Figures 4.6 - 4.8 show the motor speed response and

external step torque applied to the motor atl T =0.5 Nm, 1.0 Nm and 2.0 Nm

respectively. It can be seen from the results that even at high load torque (l T =2.0

Nm) step change, the motor speed change is within 10 rpm, about 2%, which shows





61

500ms/div

Figure 4.5 Speed response atl T =2.0 Nm (curve1: real speed r ω ; curve 2: estimated speed

r ω ˆ ; curve 3: flux current d i ; curve 4: torque current qi ; curve 5: phase current ai )

Speed response under torque step disturbance (0.5 Nm)

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10

Time (s)

S p e e d ( r p

m )

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T o r q u e ( N

m )

Speed

Torque

Figure 4.6 Speed response with external torque step changel

T =0.5 Nm

id

iq

ia

r ω ˆr ω

5A/div

909rpm/div





63

0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

Time (s)

S p e e d ( r p m )

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

T o r q u e ( N m )

Figure 4.9 Four quadrant operation with trapezoidal speed command (between

300rpm and 1400rpm)

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10

Time (s)

S p e e d ( r p m )

-5

-4

-3

-2

-1

0

1

2

3

4

5

T o r q u e ( N m )

Figure 4.10 Four quadrant operation with trapezoidal speed command (between

±1200rpm)



64

4.7. Conclusion

A robust and chattering free continues sliding mode controller is presented and

implemented in this Chapter. The controller has been tested for various command

speeds through the simulation and experimental results. These results prove that the

proposed continues sliding mode controller is robust and have good rejection to external

disturbance. The simulation and experimental results show that its dynamic performance

as well as steady state performance is much better than conventional PI controller and

free of re-tuning.



65

CHAPTER 5

5. ADAPTIVE SLIDING MODE ROTOR FLUX AND SPEED

OBSERVERS

5.1. Introduction

In the sensorless speed control of induction motors with direct field orientation, the rotor

flux and speed information are dependent on the observers. However, the exact values of the

parameters that construct the observers are difficult to measure and changeable with respect to

the operating conditions. When the motor parameters are changed and thus different from the

preset values, the estimated flux and speed will deviate from the real values. To make flux and

speed estimation robust to parameter variations, an adaptive sliding mode flux and speed

observer is proposed in the Chapter. Two sliding mode current observers are used in the

proposed method. The effects of parameter deviations on the rotor flux observer can be

alleviated by the interaction of these two current sliding mode observers. The stability of

the method is proven by Lyapunov theory. An adaptive speed estimation is also derived

from the stability theory.

5.2. Sliding mode Current and flux observer design

As defined in Chapter 3, the induction motor model can be expressed in the stationary

frame as:

ADAPTIVE SLIDING MODE ROTOR FLUX AND SPEED

OBSERVERS



66

VIAΛI 21 k k p +−= β ( 5.1)

IAΛΛ 3k p +−= ( 5.2)

where

],[ T

s s ii β α =I stator currents

T

r r ],[ β α λ λ =Λ rotor fluxes

],[ s s vv β α =V stator voltages

The configuration of the proposed flux and speed estimators is shown in Figure 5.1.

The adaptive sliding mode observer consists of two sliding mode current observers and one

rotor flux observer. The rotor flux observer is based on the current estimation from the

two current observers. The rotor speed observer takes the outputs from the second current

observer and the rotor flux observer as its inputs and generates the estimated rotor speed

as the output. The estimated speed is then fed back to the second current observer for its

adaptation. The estimation of the motor speed is derived from a Lyapunov function,

which guarantees the system convergence and stability. Once the sliding functions of the

current observers reach the sliding surfaces, the rotor flux will converge to the real value

asymptotically. Each sub-observer of the overall adaptive sliding mode observer is

discussed in the following sections.



67

Current sliding mode observer I

Current sliding mode observer II

Rotor Flux observer

Speed estimation

VIUI 211ˆˆ k k p +−= β

221

~~~UVIΛAI ++−= k k p β

IULUΛ~

)(~

321 k p eqeq +−−= β

( ) ( )∫ −+−= r r I r r P r U U K U U K β α α β β α α β λ λ λ λ ω ~~~~~

2222

r ω ~r ω ~

r r β α λ λ ~

,~

eq1U

eq2U

r r β α λ λ

~

,

~

Figure 5.1 Configuration of the proposed flux and speed observer.

5.2.1. Current observer I

The first sliding mode current observer is defined as [69]:

VIUI 211 k ˆ k ˆ p +−= β ( 5.3)

whereT

s sii ]ˆ,ˆ[ˆ β α =I the first observer currents.

T U U ],[ 111 β α =U the first sliding functions.

)(

)(

1011

1011

β β

α α

s signuU

s signuU

−=

−=

, s s

s s

ii s

ii s

β β β

α α α

−=

−=

ˆ

ˆ

1

1

The sliding mode surface is defined as:

111 , β α s s sn = (5.4)

According to the above formulae, the current error equation is

1111 )( I I k p eAΛUe −−= β ( 5.5)

where IIe −= ˆ1 I

By selecting 01u large enough, the sliding mode will occur ( 01 =n s ), and then it

follows that



68

011 == I I p ee (5.6)

From the equivalent control concept [67], if the current trajectories reach the sliding

manifold, we have

AΛU =1eq(5.7)

Equation (5.7) indicates that the equivalent control equals to the rotor flux multiplied

by the A matrix, which is the common part in (5.2). The rotor flux can be obtained by

integrating this equivalent control without speed information as discussed in Chapter 3.

The flux estimation is accurate when the motor parameters 1k , 2k , and β are known.

However, if the parameters in observers are different from the real values, there will be

some errors 1k ∆ , 2k ∆ , β ∆ in the coefficients of the observers. Then the estimated flux

and speed will be incorrect. In order to compensate this divergence, a second sliding

mode current observer is used for the flux estimation.

5.2.2. Current observer II

The second sliding mode current observer is designed differently from (5.3) as

221

~~~UVIΛAI ++−= k k p β (5.8)

where T

s s ii ]~

,~

[~

β α =I , the second observer currents

T

s s]

~,

~[

~ β α λ λ =Λ , the observed rotor fluxes

T U U ],[ 222 β α =U , the second sliding function

)(

)(

2022

2022

β β

α α

s signuU

s signuU

−=

−=,

s s

s s

ii s

ii s

β β β

α α α

−=

−=~

~

2

2

The sliding mode surface is:



69

222 , β α s s sn = (5.9)

By subtracting (5.1) from (5.8), the error equation becomes:

2212 )~

( UeΛΛAe +−−= I I k p β (5.10)

where IIe −=~

2 I

From equivalent control point of view, we have

Λ−=−−= AeΛΛAU β β )~

(2eq(5.11)

where ΛΛe −=Λ

~ . The second equivalent control equals to the negative multiplication of

the estimated rotor flux error and the A matrix. It is noticed that the second current

observer needs the rotor speed as the input.

5.2.3. Rotor flux observer design

Combining the results from (5.7) and (5.11), the rotor flux observer can be

constructed as

IULUΛ~

)(~

321 k p eqeq +−−= β (5.12)

where L is the observer gain matrix to be decided such that the observer is asymptotically

stable.

From (5.3) and (5.8), the equivalent controls obtained individually by the two current

observers will deviate from their real values if the motor parameters are incorrect.

Consequently the rotor flux estimation based on each individual control will also be

inaccurate. To reduce this deviation on rotor flux estimation, the rotor flux observer is

designed from the combination of two equivalent controls, where the effects of parameter

variations are largely cancelled. From (5.7) and (5.11), the error equation for the rotor

flux is



70

ΛΛ

Λ

−=+−=

++−−=

LAeeLAe

eAΛULUe

23

2321

I

I eqeq

k

k )( p β (5.13)

5.3. Adaptive speed estimation

In order to derive the adaptive scheme, Lyapunov’s stability theorem is utilized. If we

consider the rotor speed as a variable parameter, the error equation of flux observer is

described by the following equation:

ΛALLAee~

∆−−= ΛΛ p (5.14)

where

∆−

∆=∆

0

0

ω

ω A , r r

~ω −=∆

r ω ~is the estimated rotor speed.

The candidate Lyapunov function is defined as

λ ω /2∆+= ΛΛ eeT

V (5.15)

where λ is a positive constant. We know that V is positive definite. The time derivative of

V becomes

λ ω ω β

β

λ ω ω

/~2)(~

~)()(

/~2~

~)(

r r

T

T T

T

r r

T

T T

dt

d

dt

d

pV

∆+−

−+−=

∆+−

−+−=

−

−

ΛΛ

Λ

ΛΛΛ

eq2

1

T

1

eq2TT

T

TT

UA∆ALΛ

ΛAL∆AU

eLALAe

e∆ALΛ

ΛAL∆eeLALAe

(5.16)

LetT

AL γ = , γ is an arbitrary positive constant. With this assumption, the equation

above becomes:



71

λ ω ω β

γ γ /~2

~~)(

r r

T T

T

dt

d

pV

∆+−

−

+−= ΛΛ

eq2eq2

TT

∆AUΛΛ∆AU

eLALAe

(5.17)

Let the second term equal to the third term in (5.17), we can find the following adaptive

scheme for rotor speed identification:

)~~

(~22 r r r U U

dt

d β α α β λ λ

β

γλ ω −= (5.18)

where T U U ],[ 22 β α =eq2U

In practice, the speed can be found by the following proportional and integral adaptive

scheme:

( ) ( )∫ −+−= r r I r r P r U U K U U K β α α β β α α β λ λ λ λ ω ~~~~~

2222 (5.19)

where K P and K I are the positive gains.

5.4. Stability analysis

Since the second term equal to the third term in (5.17), the time derivative of V

becomes

01

20

01

2

2

2

2

2

2

<

+

+

−=

+−=

ΛΛ

ΛΛ

ee

)eLALAe TT

r

r

r

r T

T

T

T

( pV

ω γ

ω γ ( 5.20)

It is apparent that (5.20) is negative definite. From Lyapunov stability theory, the flux

observer is asymptotically stable, guaranteeing the observed flux to converge to the real

rotor flux.



72


To evaluate the proposed algorithm for the rotor flux and speed estimation, computer

simulations have been conducted by MATLAB. To further investigate the implemental

feasibility, the estimation and control algorithm are evaluated by HIL (hardware-in-the-

loop) testing. A 1 HP induction motor was used in the simulation and also in the

experiments.

5.5.1. Simulation results by MATLAB

Figures 5.2 and 5.3 show the induction motor response to a step speed command of

± 0.5pu ( ± 900rpm) where the motor parameters are exactly known. The actual machine

model is used to calculate the current, flux and speed of the motor. The observer model as

described above is used to estimate the rotor flux and speed. Figure 5.2 shows the speed

command, real speed, estimated speed and the speed estimation error. Figure 5.3 shows

the real and estimated rotor flux and the flux estimation error. It can be seen that the

estimated speed and flux converge to the real values very quickly.



73

Figure 5.2 Real and estimated speed at a step speed command.

Figure 5.3 Real and estimated current



74

Figure 5.4 Real and estimated rotor flux.

To study the effects of parameter variation on the speed and flux observers, the

parameters in the observers are changed on purpose in the simulation. Figure 5.5 shows

the simulation results when the coefficient 1k in the observers is changed by 20% from

its actual value, where the flux obsr _ α λ and speed robsω are estimated by the proposed

method, and 1 _ obsr α λ and1robs

ω are estimated by the previous method using only one

current sliding mode observer as in Chapter 3. It is noticed that even 1k is incorrect, the

estimated rotor flux and speed by the new observer still converge to the real values, but in

previous model, there is an offset in the rotor flux estimation and fluctuation in the rotor

speed estimation. The dc offset of flux estimation by previous method is caused by the

incorrect equivalent control 1eqU . If 1k changes, the equivalent control 1eqU will



75

detune. The integration of incorrect 1eqU causes dc offset on the flux estimation.

Whereas in the new flux observer, this dc offset is cancelled by using two current

observers. The effects of coefficient β variation on the flux and speed estimation are

shown in Figure 5.6. We can also observe obvious fluctuations in speed estimation. There

is still an error on the rotor flux estimation by the proposed method as shown in Figure

5.6(a), but the new method eliminates the dc offset caused by the parameter variation,

which can be observed in results simulated by the previous model.



76

(a) r α λ : real rotor flux, obsr _ α λ : estimated by the proposed method,1 _ obsr α λ : estimated by

previous method.

(b) r ω : real rotor speed, robs : estimated by the proposed method, 1robsω : estimated by

previous method.

Figure 5.5 Coefficient 1k in the observer is increased by 20% .



77

(a) r α λ : real rotor flux, obsr _ α λ : estimated by the proposed method, 1 _ obsr α λ : estimated by

previous method.

(b) r ω : real rotor flux, robsω : estimated by the proposed method, 1robsω : estimated by

previous method.Figure 5.6 Coefficient β in the observer is increased by 20%.



78

5.5.2. HIL Evaluation results by TI 2812 DSP

The results evaluated by HIL are shown in Figures 5.7 through 5.9. Figure 5.7 shows

the motor step response to a speed command at ±0.5pu ( ±900rpm). Figure 5.8 shows the

real and estimated rotor flux and the estimated flux angle. Figure 5.9 shows the motor

response to a trapezoidal speed command. The results show that the method can be

successfully implemented by the fixed-point DSP.

Figure 5.7 Speed step response from –0.5pu to 0.5pu (curve 1: speed command *

r ω ; curve

2: real speed r ω ; curve 3: estimated speed r ω ~ )



79

r α λ

r α λ ~

r θ ~

100ms/div

Figure 5.8 Rotor flux estimation. (curve 1: real flux r α λ ; curve 2: estimated flux r α λ ~

;

curve 3: estimated flux angle r θ ~

)

Figure 5.9 Trapezoidal speed at ±0.5pu. (curve 1: phase current ai ; curve 2: torque

current qi ; curve 3: estimated speed r ω ~ )

0.6pu/div

1.2126pu/div



80


In order to evaluate the performance of the proposed algorithm experimentally, an

induction motor drive system was set up. The setup consists of a 1 HP induction motor, a

power drive board and a DSP controller board. The external load is imposed by a

hysteresis dynamometer. The experimental setup is shown in Appendix A.1. The control

algorithm is implemented by Texas Instruments TMS320F2812 32-bit fixed-point DSP.

It has following characteristics:

• High-Performance Static CMOS Technology, 150 MHz (6.67-ns Cycle Time)

• High-Performance 32-Bit CPU

• Flash Devices: Up to 128K x 16 Flash

• 12-Bit ADC, 16 Channels

The test was first performed on the motor in four-quadrant operations. Figure 5.10

shows the motor response to a commanded step change speed at ±900rpm. Figure 5.11

shows the measured current and two sliding mode observer currents. It is seen that the

sliding mode functions enforce the two observed currents to the measured ones very

closely. Once these two observer currents converge to the measured ones, the estimated

rotor flux converges to the real rotor flux. The motor responses to a trapezoidal speed

command when the motor runs at no load are shown in Figure 5.12. To further

investigate the motor transient performance at load conditions, an external torque

puT l

5.0= is applied when the motor runs at the same trapezoidal speed command as in

Figure 5.12. The waveform of speed command *

r ω , estimated speed r ω ~ , torque current

qi , and phase current ai are shown in Figure 5.13. The estimated rotor speed response to a



81

step change of command from 360rpm to 1260rpm with a load torque of puT l

5.0= is

shown in Figure 5.14. To investigate the speed robustness, a step disturbance torque

( puT l

5.0= ) is applied and then removed at motor speed n=900rpm. Figure 5.15 shows the

estimated rotor speed response and the torque current response. As evidenced by the

testing results, the induction motor drive functions very well by the proposed algorithm.

Figure 5.10 Transient response to speed step command ±900rpm at no load (curve1:

speed command *

r ω ; curve 2: estimated speed r ω ~ ; curve 3: torque current qi ; curve 4:

phase current ai )

1091rpm/div

5A/div



82

siα

siα ˆ

siα ~

50ms/div

Figure 5.11 Real and estimated currents. (curve 1: measured current siα ; curve 2:

observed current siα ˆ ; curve 3: observed current siα

~)

Figure 5.12 Transient response due to trapezoidal speed command (±900rpm) at no

load (curve 1: speed command *

r ω ; curve 2: estimated speed r ω ~ ; curve 3: torque current

qi ,; curve 4: phase current ai )

2182rpm/div

5A/div

2182rpm/div

3A/div

3A/div

3A/div



83

Figure 5.13 Transient response due to trapezoidal speed command (±900rpm) at

puT l

5.0= (curve 1: speed command *

r ω ; curve 2: estimated speed r ω ~ ; curve 3: torque

current qi ; curve 4: phase current ai )

Figure 5.14 Speed response due to step change command from 360rpm to 1260rpm at

puT l

5.0= . (curve1: real speed; curve 2: estimated speed r ω ~ ; curve 3: torque current qi ;

curve 4: phase current ai )

5A/div

1091rpm/div

1091rpm/div

5A/div

1091rpm/div



84

Figure 5.15 Transient response for step disturbance torque (curve1: real speedr ω ;

curve 2: estimated speed r ω ~ ; curve 3: torque current qi )

5.7. Conclusion

An adaptive sliding mode observer for sensorless speed control of induction motor is

presented in this Chapter. The proposed algorithm consists of two current observers and

one rotor flux observer. The two sliding mode current observers are utilized to

compensate the effects of parameter variations on the rotor flux estimation. When the

motor parameters are deviated from initial value by temperature or operation conditions,

the errors of two equivalent controls from current observers will be largely cancelled,

which make the flux estimation more accurate and insensitive to parameter variations.

Although additional sliding mode current observer is used, the complexity of the method

is not increased too much. The stability and convergence of the estimated flux to real

rotor flux are proved by the Lyapunov stability theory. Digital simulation and

1091rpm/div

1091rpm/div

1.5A/div



85

experiments have been performed in the paper. The effectiveness of the approach is

demonstrated by the results.



86

CHAPTER 6

6. EFFICIENCY OPTIMIZATION ON VECTOR CONTROL OF

INDUCTION MOTORS

6.1. Introduction

Electrical machines consume most of the world’s electrical energy every year.

Improving efficiency of electrical drives is important not only for energy saving, but also

for environmental protection. In an induction motor drive system, to get a best transient

response the induction motor normally operates at rated flux. However, when the drive

system operates with light loads, the core losses become excessive, causing drive system

poor efficiency. To improve the motor efficiency, the flux must be decreased, obtaining a

banlance between the copper and iron losses. This phenomenon can be illustrated in

Figure 6.1. T 1 and T 2 are motor torque-speed curves at different frequencies f 1 , f 2, where

T 1 is at rated frequency and voltage, T 2 is at reduced flux level because the applied

frequency is increased ( f 2>f 1) and voltage is reduced. 1η and 2η are efficiency curves

corresponding to T 1 and T 2 respectively. The operating point ‘a’ in the Figure 6.1 can be

achieved either by curve T 1 or T 2, but the efficiencies are quit different. It is seen that the

efficiency (point ‘c’) corresponding to curve T 2, which nearly reaches maximum point, is

much higher than the efficiency (point ‘b’) at which the motor operates at rated flux level.

EFFICIENCY OPTIMIZATION ON VECTOR CONTROL OF

INDUCTION MOTORS



87

In the vector control drive system, the flux optimum strategy is to find the maximum

efficiency at each operating point.

A number of methods for efficiency improvement through flux control have been

proposed in the literature. They can be classified into two main categories. The first

category is called Loss Model Controller (LMC) [73-75]. This method is based on the

loss model of the induction motor. The flux level is selected according to the computation

of minimum motor losses. The second method is Search Controller (SC) [29-33]. This

method searches the maximum motor efficiency by measuring the input power or DC bus

power.

Figure 6.1 Explaination of efficiency improvement

The LMC method has the advantage that it is simple and fast. However, the accuracy

depends on correct modeling of the motor drive and the losses. On the other hand, SC

method measures input power or DC bus power to searches the flux where the motor runs

at maximum efficiency. This approach is insensitive to motor parameters, but it does



88

require extra hardware to measure DC bus current and can not be used in the classical

vector control system where additional sensor is not available.

An efficiency optimization method by minimizing the stator current which does not

require extra hardware and insensitive to motor parameters is presented. Minimizing

stator current method has been used in induction motor scalar control by I. Kioskeridis

[75]. In this Chapter, the relationship between the stator current minimization and the

motor loss minimization in an induction motor vector control system is investigated. It is

pointed that minimum stator current point is very close to minimum loss point in most

cases and the loss minimization can be achieved by minimizing stator current in practice.

A fuzzy logic based search method is simulated and implemented. Simulation and

experimental results are given in the paper to verify the proposed method.

6.2. Principle of Fuzzy logic controller

The fuzzy logic control is based on fuzzy logic or fuzzy inference system that is able

to simultaneously handle numerical data and linguistic knowledge. It is a nonlinear

mapping of a given input data set into an output data set. A block diagram of a fuzzy

control system is shown in Figure 6.2. The fuzzy controller is composed of the

following four elements: fuzzifier, rule base, inference engine, and defuzzifier.



89

Inference

engine

F u z z i f i e r

D

e f u z z i f i e r

Rule Base

Process

Crisp input r(t)

Fuzzy controller

Crisp output

u(t) Output

y(t)

Figure 6.2 Block diagram of a fuzzy control system

6.2.1. Fuzzifier

A fuzzifier maps crisp numbers into fuzzy sets and converts controller inputs into

information that the inference engine can easily use to activate and apply rules. It is

needed in order to activate rules which are in terms of linguistic variables. The

fuzzifier includes two parts: choice of membership function and choice of scaling

factor.

A fuzzy variable has values that are expressed by the natural language. This

meaning of the linguistic values can be quantified by membership function. For

example, the stator current of a motor can be defined by the qualifying linguistic

variables: Small, Medium, or Large, where each is represented by a triangular or

straight line segment membership function. A membership function is a curve that

defines how the values of a fuzzy variable in a certain region are mapped to a membership

value between 0 and 1. The fuzzy sets can have more subdivisions such as Zero, Very small,

Medium small, Medium large, Very large for a more precise description of the fuzzy

variable.



90

For example, in Figure 6.3, if the current is below 20A, it belongs completely to the set

Small, that is, the MF value is 1; whereas for 35A, it is in the set Small by 25% (MF = 0.25)

and to the set Medium by 50% (MF = 0.5). At current 40A , it belongs completely to the set

Medium (MF = 1) and not in the set Small and Big (MF = 0). If the current is above 70A,

it belongs completely to the set Big (MF = 1), where MF = 0 for Small and Medium.

A membership function can have different shapes such as triangular,

trapezoidal, or Gaussian membership function which are shown in Figure 6.4 (a),

(b), and (c). The simplest and most commonly used membership function is the

triangular type, which can be symmetrical or asymmetrical in shape.

Small Medium Big

Current (A)

µ ( c u r r e n t )

0 10 20 30 40 50 60 70

Figure 6.3 Explaination of membership function



91

(a) Triangular (b) Trapezoidal

(c) Gaussian

Figure 6.4 Different shapes of membership functions

6.2.2. Rule base

A rule base (a set of If-Then rules), which contains a fuzzy logic quantification

of the expert’s linguistic description of how to achieve good control. Once the rules

have been established, a fuzzy logic system can be viewed as a mapping from inputs

to outputs.

Rules may be provided by experts or can be extracted from numerical data. The

general form of the linguistic rules is

If premise Then consequent

As it can be seen, the premises (which are sometimes called “antecedents”) are

associated with the fuzzy controller inputs and are on the left-hand-side of the rules.

The consequents (sometimes called “actions”) are associated with the fuzzy

controller outputs and are on the right-hand-side of the rules. Notice that each



92

premise (or consequent) can be composed of the conjunction of several terms, e.g., ‘IF t l is

very cold AND v1 is quite low, THEN voltage u must be very big.’ This one rule

reveals that we will need an understanding of:

1) Linguistic variables versus numerical values of a variable (e.g., very cold versus

-5°C);

2) Quantifying linguistic variables (e.g., t l may have a finite number of linguistic

terms associated with it, ranging from extremely hot to extremely cold), which

is done using fuzzy membership functions;

3) Logical connections for linguistic variables (e.g., "AND," "OR ," etc.);

4) Implications, i.e., "IF A THEN B."

Using the above approach, we could write down rules for all possible cases. In practical,

since only a finite number of linguistic variables and linguistic values are specified, the

number of possible rules is also finite.

6.2.3. Inference engine

Inference engine, also called fuzzy inference, which emulates the expert’s

decision making in interpreting and applying knowledge about how best to control

the plant. It handles the way in which rules are combined. Just as humans use many

different types of inferential procedures to help us understand things or to make decisions,

there are many different fuzzy logic inferential procedures. Only a very small number of

them are actually being used in engineering applications of fuzzy logic system.

6.2.4. Defuzzifier

The defuzzifier maps output sets into crisp numbers. It converts the conclusions

of the inference mechanism into actual inputs for the process. The most common



93

used defuzzification method is Center of Gravity (COG). COG defuzzification

method is often used in spite of some amount of complexity in the calculation. In

the COG method, the crisp output of is obtained by using the center of gravity, in

which the crisp or variable is taken to be the geometric center of the output fuzzy

variable value area. The general expression for COG is

∑∫ ∑ ∫ =

i

i icrisp

)i(

)i( b

µ

µ δ (6.1)

where ib is the center of membership function, ∫ iµ denote the area under membership

function.

6.2.5. Implementation procedure for fuzzy logic controller

In order to design a fuzzy logic based algorithm, the following steps need to be

performed.

1) Analyze whether the problem has sufficient elements to warrant a fuzzy

logic application. Get all the information from the operator of the plant to be

control led.

2) Selection of input/output variables and fuzzy sets. Define the universe of

discourse of the variables and convert to corresponding per unit variables as

necessary.

3) Definition of membership functions. Formulate the fuzzy sets and select the

corresponding MF shape of each. For a sensitive variable, more fuzzy sets

are needed. If a variable requires more precision near steady state, use more

crowding of membership functions near the origin.





95

2

r mm k P ω = (6.5)

where mk is mechanical loss coefficient.

When the motor is running under the rotor flux field orientation, we have relationship

0=dr i (6.6)

and

qs

r

m

qr i L

Li −= ( 6.7)

in steady state. From (6.1) to (6.4), we can get the expression for total motor losses:

( ) ( )

22

2

22222

2222

])[(

2

3

r mqs

r

mr

dsmeeeh

qr dr r qsds sloss

k i L

L Li Lk k

ii Rii R P

ω ω ω σ +++

++++=

(6.8)

The motor torque can be expressed by

qsdsqsds

r

m

e i Kiii L

L pT ==

2

22

3 (6.9)

wherer

m

L

L p K

2

22

3= .

Substituting (6.9) into (6.8), we have

2

22

2

2)

1(

2

3r m

ds

e

dsloss k i K

BT Ai P ω ++= (6.10)

where

22)( meeeh s Lk k R A ω ω ++= ,

2

222

2

2

)(r

mr eeeh

r

mr s

L

L Lk k

L

L R R B σ ω ω +++= .



96

Equation (6.10) gives an expression of induction motor losses in a vector control

system. It shows the relationship of motor losses with rotor flux ( dsi ), motor torque and

speed. It is obvious that if speed and torque are constant, the loss just is a function of dsi .

Figure 6.5 shows the curves of total losses with respect to dsi at different load torque

(speed is 900rpm). It is seen from Figure 6.5 that if speed is fixed, then the d-axis current

dsi corresponding to minimum loss point is different at different load torque. So we can

find the dsi value that corresponds to minimum loss at each operating point.

Figure 6.5 Motor losses with respect to dsi at different load torque

Minimum loss point



97

6.4. Comparison of minimum losses point and minimum stator current

point

As state above, assuming the motor parameters are constant and independent on rotor

flux, it can be seen from the loss expression (6.10) that at each operating point ( r ω and

eT fixed), the motor loss is just a function of d-axis current component dsi . In general,

minimum loss corresponds to minimum input power if output power is constant. Figure

6.6 shows the curves of input power with respect to dsi (output power is fixed). This plot

gives us the appropriate *dsi corresponding to the point of the optimum efficiency. To

investigate the relationship between stator current minimization and motor loss

minimization, the stator current variation with respect to dsi is plotted in Figure 6.7. It is

interesting to note that the minimum stator current points are very close to minimum loss

(or input power) points at each load condition. To illustrate this more explicitly, the

minimum input power points and the input power points corresponding to minimum

stator currents are plotted in same figure as shown in Figure 6.8. This phenomenon is also

demonstrated by experimental results. Figures 6.9 and 6.10 show the experimental input

power and stator current with motor flux level (v/f value).



98

Figure 6.6 Input power with respect to dsi

Figure 6.7 Stator current variation with respect to dsi



99

Figure 6.8 Minimum input power point and input power point corresponding to minimum

stator current

Figure 6.9 Measured input power vs flux level at different torques



100

Figure 6.10 Measured stator current vs flux level at different torques

To further investigate the relationship between minimum losses and minimum stator

current, a mathematical expression is derived. By taking the derivative of motor losses

expression (6.10) with respect to dsi , the magnetizing current pdsi _ * corresponding to the

point of the minimum loss can be obtained as

K T

A Bi e

pds 4 _ * = (6.11)

The stator current can be written as

22

2222 1

ds

edsqsds s

i K

T iiii +=+= (6.12)

By setting the derivative of (6.12) with respect to dsi to zero, it yields that

K

T i e

ids = _ *

(6.13)

Let ipk denote the ratio of idsi _ * with respect to pdsi _

* , we have



101

4

_ *

_ *

B

A

i

ik

pds

ids

ip == (6.14)

The plot of the ratio ipk with respect to frequency for different motors is shown in

Figure 6.11. It can be seen that the ratio ipk is dependent on motor parameters and

operating frequency. The value of ipk increases as the frequency increases. But the value

is still in the range from 0.8 to 1.4. As illustrated in Figures 6.6 and 6.9, the motor input

power (or loss) curves are quit flat around the minimum loss points. This means that even

the minimum stator current points are somewhat away from the minimum loss points, the

motor losses by finding minimum stator current are still very close to minimum losses.

Therefore, the minimum stator current can be used to minimize the motor losses in

practice.

Figure 6.11 The ratio ipk with frequency for different motors

50hp

10hp

1hp



102

6.5. Fuzzy controller for efficiency optimization

Several fuzzy logic based loss minimum algorithms have been reported previously

[78,79]. The advantage of minimizing stator current to minimize motor losses is that no

extra hardware is needed and can be implemented in a classical structure of induction

motor speed sensorless vector control.

The principle of efficiency optimization control by controlling stator current can be

explained in Figure 6.12. The program searches the minimum stator current by adjusting

the magnetizing current. If the magnetizing current is decreased, then the rotor flux is

reduced, causing a corresponding increase in the torque current to keep the developed

torque constant. As the rotor flux is decreased, the iron loss and copper loss decreases at the

same time, resulting in a decrease of stator current. After the flux level reaches to some

level, the iron loss will continue reduce while the copper loss will increase. However, the

total system loss and stator current will still decrease. Since the minimum stator current

point is very close to minimum losses point, the search continues until the system settles

down at the minimum stator current point. Any excursion beyond the minimum point will

force the controller to return.



103

dsi∗

qsi

loss P

nT

si

Minimum point

t

Figure 6.12 Principle of efficiency optimization control by stator current

Based on the principle above, the fuzzy controller is designed as in Figure 6.13. The

fuzzy controller uses stator peak current change si∆ and d-axis current change dsi∆ as its

inputs. )1()()( −−=∆ k ik ik i s s s

, )1()()( −−=∆ k ik ik i dsdsds. The output is d-axis reference

increment ref dsi _ ∆ . )1()()( _ _ _ −−=∆ k ik ik i ref dsref dsref ds .

Z-1

Z-1

Fuzzy

Inference and

Defuzzification

)(k i s∆)(k i s

)(k ids

+

-

+

- )(k ids∆

)( _ k i ref ds∆

)1( _ −k i ref ds

)( _ k i ref ds+

+

Figure 6.13 Fuzzy controller



104

The membership functions for fuzzy controller are shown in Figure 6.14. All

membership functions are triangular for simplicity. The fuzzy output is calculated using

COG (center of gravity) defuzzification approach.

The rule base for fuzzy control is given in Table 6.1. The basic idea is that if the last

control action indicated a decrease of stator current, the search proceeds in the same

direction. In case the last control action resulted in an increase of stator current, the

search direction is reversed. For example, IF NS i s =∆ AND N ids =∆ , THEN NS i ref ds =∆ _ .

This rule means that IF the stator current increment si∆ is negative small (NS) and the

last d-axis current dsi∆ is negative (N), THEN the new excitation current increment

ref dsi _ ∆ is negative small (NS).



105

-1 -0.5 0 0.5 1 si∆

NS ZE PS PBNB

(a) Input membership function si∆

-1 0 1dsi∆

PN

(b) Input membership function dsi∆

-1 -0.5 0 0.5 1ref d i _ ∆

NB NS ZE PS PB

(c) Output membership function ref dsi _ ∆

Figure 6.14 Membership for fuzzy logic controller



106

Table 1 Rule base

dsi∆ N P

si∆ ref dsi _ ∆

PB PB NB

PS PS NS

ZE ZE ZE

NS NS PS

NB NB PB

P = Positive

N = Negative

PB = Positive Big

PS = Positive Small

ZE = Zero

NS = Negative Small

NB = Negative Big



107

The disadvantage of this efficiency control is that the transient response is relatively

slow. To overcome this, for any change in load or speed command, the fuzzy efficiency

controller is turned off and the controller’s attention is directed to the system

performance, in this case the rated flux current is used instead.


The system is first simulated by MATLAB. The simulation results are shown in

Figures 6.15 through 6.17. Figure 6.15 shows the search process of d-axis current

command change and q-axis current variation at a load toque of 1.0=l T pu. Figures 6.16

and 6.17 show that the stator current and motor losses decrease until the controller

reaches steady state (almost minimum point). The core losses and copper losses are also

shown in the process. The results show that the motor loss is greatly reduced by the

proposed method.

Figure 6.15 d-axis reference current and q-axis current

ref dsi _

qsi

Fuzzy begins



108

Figure 6.16 Stator peak current

Figure 6.17 Motor losses



induction motor drive system was set up. The external load is imposed by a hysteresis

dynamometer. The fuzzy search algorithm is implemented using TI TMS320F2812 32-bit

fixed-point DSP.

lossestotal P

fe P

cu P

Fuzzy begins

siFuzzy begins



109

The test was implemented at different motor speeds and load torques. Figures 6.18

through 6.26 show the experimental results. The fuzzy logic search processes at 900rpm

with load torque NmT l 2.0= and NmT l 5.0= are shown in Figures 6.18 and 6.19

respectively, where si is stator current magnitude, ref dsi _ d-is axis current command,

qsi q-axis torque current, and ai phase current. Figure 6.20 and 6.21 show the test

results at 600rpm and 1200rpm respectively. It can be seen from the results that at light

load, the stator current is greatly reduced, which will cause motor efficiency increase.

Figures 22 - 25 show the input power and motor efficiency variation with stator current

reduction during fuzzy search. The results are shown at rpmn NmT l 900,2.0 == ,

rpmn NmT l 900,5.0 == , rpmn NmT l 1200,2.0 == , and rpmn NmT l 600,2.0 == respectively.

After fuzzy logic is switched on, the stator current is deceased from 1.25A to 0.65A in

Figure 6.22, while the input power is deceased from 62W to 39W and efficiency is

increased from 30% to 52%. In Figure 6.23, when the load is increased, the input power

is reduced from 89W to 78W. At the same time the efficiency is increased from 49% to

61%. The same results are expected in Figures 6.24 and 6.25 for motor speed at 1200rpm

and 600rpm. The experimental results show the motor efficiency is greatly improved at

light load. The comparison of efficiency curves at different operation points with and

without fuzzy optimization control is shown in Figure 6.26.



110

500ms/div

Figure 6.18 Fuzzy search at rpmn NmT l

900,2.0 ==

500ms/div

Figure 6.19 Fuzzy search at rpmn NmT l 900,5.0 ==

Fuzzy begins

Fuzzy begins

si

qsi

ai

si

ref dsi _

qsi

ai

ref dsi _

2A/div

2A/div



111

500ms/div


500ms/div


Fuzzy begins

si

ref dsi _

qsi

ai

Fuzzy begins

si

ref dsi _

qsi

ai

2A/div

2A/div



112

0

0.4

0.8

1.2

1.6

6 7 8 9 10 11 12

Time (s)

I n p

u t c u r r e n t ( A )

(a) Input current (A)

0

40

80

120

160

6 7 8 9 10 11 12

Time (s)

P o w

e r ( W )

(b) Input power (W)

0

0.2

0.4

0.6

0.8

1

6 7 8 9 10 11 12

Time (s)

E f f i c i e n c y

(c) Efficiency

Figure 6.22 Current, power and efficiency variation at rpmn NmT l 900,2.0 ==



113

0

0.4

0.8

1.2

1.6

4 6 8 10 12 14

Time (s)

I n o

u t c u r r e n t ( A )


0

40

80

120

160

200

4 6 8 10 12 14

Time (s)

I n p u t

p o w e r ( W )

(b) Input power (W)

0

0.2

0.4

0.6

0.8

1

4 6 8 10 12 14

Time (s)

E f f i c i e n c y

(c) Efficiency




114

0

0.4

0.8

1.2

1.6

0 2 4 6 8 10 12 14

Time (s)

I n p u t c u

r r e n t ( A )


0

40

80

120

0 2 4 6 8 10 12 14

Time (s)

I n p u t p o w e r ( W )

(b) Input power (W)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

Time (s)

E f f i c i e n c y

(c) Efficiency




115

0

0.4

0.8

1.2

1.6

0 5 10 15 20

Time (s)

I n o u t c u r r e n t ( A )


0

20

40

60

0 5 10 15 20

Time (s)

I n p u t p o w e r ( W )

(b) Input power (W)

0

0.2

0.4

0.6

0 5 10 15 20

Time (s)

E f f i c i e n c y

(c) Efficiency




116

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5Torque (Nm)

E f f i c i e n c y

Figure 6.26 Comparison of efficiency curves at different operation points with andwithout fuzzy optimization control. (solid line is with fuzzy optimization, dash line iswithout fuzzy optimization)

6.8. Conclusion

An efficiency optimization method which does not require extra hardware and

insensitive to motor parameters is presented. The relationship between stator current

minimization and motor losses minimization in the induction motor vector control system

is investigated. A fuzzy logic based search method is simulated and implemented. It is

pointed that the motor loss minimization can be achieved by minimizing stator current in

practice. The simulation and experimental results demonstrate the effectiveness of the

approach.

600rpm

1200rpm900rpm



117

CHAPTER 7

7. SUMMARY AND FUTURE WORK

7.1. Summary

A large variety of induction machine control schemes are used in industrial

applications. Open loop control systems maintain the stator v/f ratio at a predetermined

level to establish the desired machine flux. The ratio is satisfied only at low or moderate

dynamic requirements. Field orientation technology can provide high performance control

of induction machine by aligning a revolving reference frame with a space vector of

selected flux and allowing the induction motor to emulate a separately excited dc machine.

The speed sensorless control and loss minimization of induction drive have gained more

and more attention because the fragile speed sensor and energy crisis, which are also the

main focus of this research.

There are two key issues related to a direct field oriented drive system: flux estimation

and speed estimation. In this research, a flux and speed observer using the sliding

mode technique is presented and investigated in Chapter 3. To overcome the

parameter sensitive problem, a robust adaptive sliding mode observer is proposed in

Chapter 5 and the stability is verified by Lyapunov theory. A continuous sliding mode

speed controller is presented in Chapter 4 to avoid re-tuning PI gains. A Fuzzy logic

based loss minimization method is proposed in Chapter 6. The simulation and

SUMMARY AND FUTURE WORK



118

experimental results are given to demonstrate the effectiveness and practicality of the

approach.

The major accomplishment of the research can be summarized as following:

• An effective sliding mode flux and speed observer is presented and

investigated. The parameter sensitivity is analyzed using the equivalent control

of sliding mode method. It shows that the deviation of motor parameters will

cause error in the equivalent control of sliding mode, and the system

performance will detune by this parameter deviation. This phenomenon is

conformed with simulation and experimental results.

• To overcome this parameter sensitivity problem, an adaptive sliding mode

observer is proposed and the stability is verified by Lyapunov theory. Two

sliding mode current observers are utilized to compensate the effects of

parameter variation on the rotor flux estimation, which make flux estimation

more accurate and insensitive to parameter variation. The speed information is

estimated by adaptive mechanism. The convergence of the estimated flux to

actual rotor flux is proved by the Lyapunov stability theory.

• In conventional speed PI controller, the PI gains are very sensitive to

operating condition and tuning the gains is a very time consuming job. To

achieve better performance and avoid tuning, a continuous sliding mode speed

controller is presented. The comparison of this presented controller with PI

controller is also presented.



119

• An efficiency optimization method is presented which has the advantage of no

requiring extra hardware and insensitive to motor parameters. The relationship

between stator current minimization and motor losses minimization in the

induction motor vector control system is compared and investigated. A fuzzy

logic based search method is simulated and implemented using TI 2812 DSP.

The simulation and experimental results show that this approach has greatly

improved the motor efficiency especially at light load.

7.2. Future work

Simulations and experimental results for the flux estimation, speed estimation

and loss minimization show the great promise of the methods proposed in this

dissertation. However, the robustness of the proposed adaptive sliding mode observer

will be further investigated in the future work. To reduce the search time and torque

pulsations, the fuzzy logic search method will be improved in practical implementation.



120

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127

9. APPENDIX A EXPERIMENTAL SETUP


induction motor drive system was set up. The setup consists of a induction motor, a

power drive board and a DSP controller board. The experimental setup is shown in

Figure A.1

Drive board

DSP board

Dynamometer

controller

`

AC

Power

Motor

under

test

Power analyzer

Figure A.1 Experimental setup

The external load is modeled by a MAGTROL hysteresis dynamometer as shown in

Figure A.2. The dynamometer controller DSP 6500 (Figure A.3 (lower one)) can provide

superior motor testing capabilities by using state-of-the-art digital signal processing

technology. Precise torque loading can be provided independent of shaft speed. The

motor input power and efficiency are measured through MAGTROL 6530 power

analyzer as shown in Figure A.3 (upper one).

APPENDIX A EXPERIMENTAL SETUP



Figure A.2 Hysteresis dynamometer

Documents

Adaptive Sliding Mode Observer