iterative learning control for smooth operation of

School of Engineering and Information Technology,

Charles Darwin University,

Australia

A Thesis for the degree of Doctor of Philosophy

Iterative Learning Control for Smooth Operation of Permanent Magnet

Synchronous Motors

Kheng Cher Yeo

Submitted on the 16th of March, 2017

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Declaration

I hereby declare that the work herein, now submitted as a thesis for the degree of

Doctor of Philosophy at the Charles Darwin University, is the result of my own

investigations, and all references to ideas and work of other researchers have been

specifically acknowledged.

I hereby certify that the work embodied in this thesis has not already been accepted in

substance for any degree, and is not being currently submitted in candidature for any

other degree.

____________________________________

Kheng Cher Yeo

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Page III

Abstract

Permanent Magnet Synchronous Motors (PMSM) have many advantages over other types

of motors and are used in many applications. However, undesirable torque ripples are

associated with these motors, caused by design, manufacturing imperfections or

measurement inaccuracies. These torque ripples occur as periodic functions of the rotor

position. While there are several control schemes to minimise torque ripple, such as using

observers or pre-compensation, Iterative Learning Control (ILC), an adaptive control

method capable of reducing torque ripples that are periodic in nature, has not been

extensively investigated to date and may be a suitable method to significantly reduce

torque ripple for PMSMs.

Various methods of ILC are described in literature, such as the Single Channel First Order

ILC (SCFO-ILC), which uses information from the previous cycle within a certain frequency

range for the iterative learning process; Multi-Channel ILC (MC-ILC) which uses multiple

channels; Higher Order ILC (HO-ILC), which uses information from more than one previous

cycle; and adaptive ILC in which the learning gains vary with the error. Most ILC schemes

used in PMSM control are the Proportional type ILC (P-ILC) or its variations. Although other

types of ILC schemes are available, the effectiveness of these schemes in minimising torque

ripple for PMSM are not described in detail in literature.

Simulations and experimental results of this thesis showed that the various ILC schemes

were able to suppress the major torque ripple harmonics of PMSM. Of the four categories

of ILC, P-type ILC with a forgetting factor has the widest learnable band while D-type ILC

and MC-ILC has the narrowest. MC-ILC has the lowest Torque Ripple Factor (TRF), fastest

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convergence and is relatively robust to parameter variations. Together with HO-ILC and

adaptive ILC, they have lower TRF and faster convergence compared to SCFO-ILC.

Two new ILC schemes were proposed in this work: Multi-Channel Higher Order ILC and

Multi-Channel Adaptive ILC. Compared to SCFO-ILC, Multi-Channel Higher Order ILC

converges faster and has a lower TRF. However, it is not robust to parameter variations.

Multi-Channel Adaptive ILC on the other hand is robust, has a low TRF and converges the

fastest among other ILC schemes. It can therefore be concluded that the Multi-Channel

Adaptive ILC, developed in this thesis, is a suitable ILC scheme for PMSMs to minimise

torque ripple.

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List of Publications

Results from this research contributed to the following publications:

K. C. Yeo, G. Heins and F. De Boer, "Comparison of torque estimators for PMSM," Power

Engineering Conference, 2008. AUPEC '08. Australasian Universities, Sydney, NSW, 2008, pp.

1-6.

K. C. Yeo, G. Heins and F. De Boer, "Indirect adaptive feedforward control for Permanent

Magnet motors," Power Engineering Conference, 2009. AUPEC 2009. Australasian

Universities, Adelaide, SA, 2009, pp. 1-5.

K. C. Yeo, G. Heins and F. De Boer, "Indirect adaptive feedforward control in compensating

cogging torque and current measurement inaccuracies for Permanent Magnet

motors," 2009 IEEE International Conference on Control and Automation, Christchurch,

2009, pp. 2136-2142.

K. C. Yeo, G. Heins, F. De Boer and B. Saunders, "Adaptive feedforward control to

compensate cogging torque and current measurement errors for PMSMs," 2011 IEEE

International Electric Machines & Drives Conference (IEMDC), Niagara Falls, ON, 2011, pp.

942-947.

Kheng Cher Yeo, G. Heins and F. De Boer, "Position based iterative learning control to

minimise torque ripple for PMSMs," IECON 2011 - 37th Annual Conference on IEEE

Industrial Electronics Society, Melbourne, VIC, 2011, pp. 4727-4732.

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Acknowledgements

This thesis would not have finished without the patience and guidance of Friso De Boer and

Greg Heins, assistance in the experimental testing from Ben Saunders and contributions

from the past researchers at Charles Darwin University for the work they have done in this

area.

I am also truly grateful to my family for their support and encouragement, my parents and

friends for always believing in me.

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Contents

Declaration ................................................................................................................................ I

Abstract ................................................................................................................................... III

List of Publications ................................................................................................................... V

Acknowledgements ................................................................................................................ VII

Contents .................................................................................................................................. IX

Table of Figures ..................................................................................................................... XIII

Glossary of Terms................................................................................................................. XVII

List of Acronyms .................................................................................................................... XXI

Chapter 1 Introduction ............................................................................................................ 1

1.1 Background .................................................................................................................. 2

1.1.1 Overview of Permanent Magnet Motors .............................................................. 2

1.1.2 Factors Contributing to Torque Ripple of PMSMs ................................................ 4

1.1.3 Control Methods for Smooth Operation ............................................................... 5

1.1.4 Iterative Learning Control ..................................................................................... 9

1.2 Aim of Research ......................................................................................................... 10

1.3 Structure of Thesis ..................................................................................................... 10

Chapter 2 Modelling PMSMs Control .................................................................................... 11

2.1 Dynamic Model of a Permanent Magnet Motor ........................................................ 11

2.2 Field Oriented Control ................................................................................................ 14

2.3 Current Controllers for PMSM ................................................................................... 18

2.3.1 PID Current Control ............................................................................................. 18

2.3.2 Hysteresis Current Control .................................................................................. 20

2.4 Torque Ripple of PMSMs ............................................................................................ 21

2.4.1 Manufacturing Imperfections ............................................................................. 21

2.4.2 Measurement Inaccuracies ................................................................................. 27

2.4.3 Total Torque Ripple in a PMSM ........................................................................... 31

2.5 Discussion ................................................................................................................... 33

Chapter 3 ILC for Smooth Operation of PMSMs .................................................................... 35

3.1 Feedforward Control .................................................................................................. 35

3.2 Pre-Compensation Techniques .................................................................................. 38

3.2.1 Direct Pre-Compensation Technique .................................................................. 38

3.2.2 Indirect Pre-compensation Technique ................................................................ 39

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3.3 ILC Schemes ................................................................................................................ 41

3.3.1 Single Channel First Order ILC ............................................................................. 45

3.3.2 Multi-Channel ILC ................................................................................................ 60

3.3.5 Higher Orders ILC ................................................................................................ 63

3.3.4 Adaptive ILC ......................................................................................................... 66

3.3.5 Comparison of ILC schemes ................................................................................ 68

3.4 ILC for PMSM .............................................................................................................. 69

3.4.1 Domain of Operation: Time, Frequency and Position ......................................... 71

3.4.2 Multi-Channel Higher Order ILC .......................................................................... 74

3.4.3 Multi-Channel Adaptive ILC ................................................................................. 75

3.5 Discussion ................................................................................................................... 76

Chapter 4 Simulation of Control Methods for PMSMs .......................................................... 79

4.1 Simulation Scenario .................................................................................................... 79

4.2 Field Oriented Control ................................................................................................ 82

4.3 Using Pre-compensation Techniques ......................................................................... 86

4.4 Iterative Learning Control .......................................................................................... 92

4.4.1 Single Channel First Order ILC ............................................................................. 93

4.4.2 Multi-Channel ILC .............................................................................................. 104

4.4.3 Higher Order ILC ................................................................................................ 109

4.4.4 Adaptive ILC ....................................................................................................... 114

4.4.5 Multi-Channel Higher Order ILC ........................................................................ 121

4.4.6 Multi-Channel Adaptive ILC ............................................................................... 123

4.5 Discussion ................................................................................................................. 126

Chapter 5 Experimental Setup ............................................................................................. 129

5.1 Hardware and Software Specifications .................................................................... 129

35.1.1 Motor .............................................................................................................. 130

5.1.2 Mechanical Design ............................................................................................ 130

5.1.3 Eddy Current Brake ........................................................................................... 131

5.1.4 Sensors .............................................................................................................. 131

5.1.5 DSP .................................................................................................................... 132

5.1.6 Data Acquisition using Labview ......................................................................... 133

5.1.7 Matlab/Simulink ................................................................................................ 133

5.2 Determining the Motor Parameters ........................................................................ 133

5.2.1 BEMF Shapes ..................................................................................................... 133

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5.2.2 Cogging Torque ................................................................................................. 136

5.2.3 Electrical Subsystem .......................................................................................... 137

5.2.4 Mechanical Subsystem ...................................................................................... 137

5.3 Design of the Current Controller .............................................................................. 137

5.4 Discussion ................................................................................................................. 138

Chapter 6 Experimental Results ........................................................................................... 141

6.1 Compensation Scheme Setup .................................................................................. 141

6.2 Torque Ripple Factor of Control Schemes ............................................................... 147

6.2.1 Field Oriented Control ....................................................................................... 147

6.2.2 Pre-compensation Technique ........................................................................... 148

6.2.3 Single Channel First Order Iterative Learning Control ...................................... 153

6.2.4 Multi-Channel Iterative Learning Control ......................................................... 164

6.2.5 Higher Order Iterative Learning Control ........................................................... 166

6.2.6 Adaptive Iterative Learning Control .................................................................. 168

6.2.7 Multi-Channel Higher Order Iterative Learning Control ................................... 175

6.2.8 Multi-Channel Adaptive Iterative Learning Control .......................................... 176

6.2.9 Comparison of ILC Schemes .............................................................................. 177

6.3 Variations to motor parameters J and b .................................................................. 178

6.4 Discussion ................................................................................................................. 186

Chapter 7 Conclusion ........................................................................................................... 191

7.1 Further Work ............................................................................................................ 192

Appendix A ........................................................................................................................... 195

A.1 Cogging Torque Variation with Temperature .......................................................... 195

A.2 BEMF Variation with Temperature .......................................................................... 198

A.3 Current Gain and Offset Errors Variation ................................................................ 200

References ........................................................................................................................... 203

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Page XIII

Table of Figures

FIGURE 1-1: SURFACE MOUNT PERMANENT MAGNETS MOTORS (COILS NOT SHOWN) [19] .............. 2

FIGURE 1-2: BEMF SHAPES ..................................................................................................................... 3

FIGURE 2-1: MODEL OF PM MOTOR .................................................................................................... 13

FIGURE 2-2: PMSM MODEL .................................................................................................................. 13

FIGURE 2-3: REFERENCE FRAMES FOR FOC [54] .................................................................................. 14

FIGURE 2-4: FOC ON A PMSM .............................................................................................................. 17

FIGURE 2-5: SIMPLIFIED MODEL USING FOC ON A PMSM ................................................................... 17

FIGURE 2-6: PI CURRENT CONTROLLERS FOR PMSM (SIMPLIFIED) ..................................................... 19

FIGURE 2-7: HYSTERESIS CONTROLLERS [67] ....................................................................................... 20

FIGURE 2-8: PMSM CONTROL WITH TORQUE RIPPLE .......................................................................... 32

FIGURE 3-1: FEEDFORWARD-FEEDBACK CONTROL .............................................................................. 36

FIGURE 3-2: DIRECT PRE-COMPENSATION TECHNIQUE ....................................................................... 39

FIGURE 3-3: INDIRECT PRE-COMPENSATION TECHNIQUE ................................................................... 40

FIGURE 3-4: BASIC ILC CONFIGURATION .............................................................................................. 42

FIGURE 3-5: P-ILC ................................................................................................................................. 46

FIGURE 3-6: BODE PLOT OF A SECOND ORDER SYSTEM ...................................................................... 47

FIGURE 3-7: CONVERGENCE FOR P-ILC ................................................................................................ 48

FIGURE 3-8: PF-ILC ................................................................................................................................ 49

FIGURE 3-9: CONVERGENCE FOR PF-ILC .............................................................................................. 50

FIGURE 3-10: D-ILC ............................................................................................................................... 51

FIGURE 3-11: CONVERGENCE FOR D-ILC (NO FILTER) .......................................................................... 52

FIGURE 3-12: CONVERGENCE FOR D-ILC (LPF - 25HZ) .......................................................................... 53

FIGURE 3-13: CONVERGENCE FOR D-ILC (LPF - 50HZ) .......................................................................... 53

FIGURE 3-14: CONVERGENCE FOR D-ILC (LPF - 100HZ) ........................................................................ 54

FIGURE 3-15: PD-ILC ............................................................................................................................. 55

FIGURE 3-16: CONVERGENCE FOR PD-ILC (KD VARIES) ........................................................................ 56

FIGURE 3-17: CONVERGENCE FOR PD-ILC (KP VARIES) ......................................................................... 56

FIGURE 3-18: PI-ILC .............................................................................................................................. 58

FIGURE 3-19: CONVERGENCE FOR PI-ILC (KI VARIES) ........................................................................... 59

FIGURE 3-20: MC-ILC ............................................................................................................................ 61

FIGURE 3-21: CONVERGENCE CONDITION FOR MC-ILC ....................................................................... 62

FIGURE 3-22: CONVERGENCE CONDITION FOR MC-ILC ....................................................................... 63

FIGURE 3-23: HO-ILC ............................................................................................................................ 64

FIGURE 3-24: CONVERGENCE CONDITION FOR HO-ILC ....................................................................... 65

FIGURE 3-25: ADAPTIVE P-ILC .............................................................................................................. 68

FIGURE 3-26: ILC FOR SPEED RIPPLE MINIMISATION ........................................................................... 70

FIGURE 3-27: ILC FOR TORQUE RIPPLE MINIMISATION ....................................................................... 70

FIGURE 3-28: MCHO-ILC ....................................................................................................................... 75

FIGURE 3-29: MULTI-CHANNEL ADAPTIVE ILC ..................................................................................... 76

FIGURE 4-1: SIMULATION SETUP ......................................................................................................... 80

FIGURE 4-2: ELECTRICAL AND MECHANICAL SUBSYSTEM OF A PMSM ............................................... 81

FIGURE 4-3: MECHANICAL BLOCK ........................................................................................................ 81

FIGURE 4-4: IDEAL SCENARIO ............................................................................................................... 83

FIGURE 4-5: CASE 1: NON-IDEAL SINUSOIDAL BEMF ........................................................................... 84

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FIGURE 4-6: CASE 2: CURRENT MEASUREMENT ERRORS .................................................................... 84

FIGURE 4-7: CASE 3: COGGING TORQUE .............................................................................................. 85

FIGURE 4-8: OUTPUT TORQUE (ALL CASES) ......................................................................................... 86

FIGURE 4-9: DIRECT PRE-COMPENSATION CONTROL SETUP ............................................................... 87

FIGURE 4-10: OUTPUT TORQUE USING DIRECT PRE-COMPENSATION CONTROL ............................... 88

FIGURE 4-11: INDIRECT PRE-COMPENSATION CONTROL ..................................................................... 88

FIGURE 4-12: CASE 1: NON-IDEAL BEMF COMPENSATED .................................................................... 89

FIGURE 4-13: CASE 2: COGGING TORQUE COMPENSATED .................................................................. 89

FIGURE 4-14: CURRENT MEASUREMENT ERRORS COMPENSATED ..................................................... 90

FIGURE 4-15: OUTPUT TORQUE (ALL CASES) ....................................................................................... 90

FIGURE 4-16: ITERATIVE LEARNING CONTROL ..................................................................................... 92

FIGURE 4-17: PLOT OF P-ILC FOR DIFFERENT KP VALUES ..................................................................... 94

FIGURE 4-18: PLOT OF P-ILC ................................................................................................................. 95

FIGURE 4-19: PLOT OF PF-ILC (KP = 0.4) WITH DIFFERENT FORGETTING FACTOR ............................... 96

FIGURE 4-20: PLOT OF D-ILC WITH DIFFERENT CUTOFF FREQUENCIES ............................................... 97

FIGURE 4-21: PLOT OF D-ILC WITH DIFFERENT KD VALUES .................................................................. 98

FIGURE 4-22: PLOT OF D-ILC ................................................................................................................ 98

FIGURE 4-23: PLOT OF PD-ILC .............................................................................................................. 99

FIGURE 4-24: COMPARING PD-ILC WITH DIFFERENT VALUES ............................................................ 100

FIGURE 4-25: PLOT OF PD-ILC ............................................................................................................ 101

FIGURE 4-26: PI-ILC WITH VARYING KI VALUES .................................................................................. 102

FIGURE 4-27: PI-ILC ............................................................................................................................ 102

FIGURE 4-28: COMPARISON OF MULTI-CHANNEL ILC WITH SINGLE CHANNEL ILC ........................... 105

FIGURE 4-29: PLOT OF 2 CHANNELS ILC FOR DIFFERENT KP,LOW VALUES ........................................... 106

FIGURE 4-30: PLOT OF 2 CHANNELS ILC ............................................................................................. 106

FIGURE 4-31: PLOT OF 3 CHANNELS ILC FOR DIFFERENT LEARNING GAINS ...................................... 107

FIGURE 4-32: PLOT OF 3 CHANNELS ILC ............................................................................................. 108

FIGURE 4-33: COMPARING MC-ILC METHODS ................................................................................... 108

FIGURE 4-34: SECOND ORDER ILC WITH DIFFERENT KP VALUES ........................................................ 110

FIGURE 4-35: 2ND

ORDER ILC WITH VARYING KP VALUES ................................................................... 110

FIGURE 4-36: PLOT OF HO-ILC (2ND

ORDER) ....................................................................................... 111

FIGURE 4-37: 3RD

ORDER ILC WITH VARYING KP VALUES (1) .............................................................. 111

FIGURE 4-38: 3RD

ORDER ILC WITH VARYING KP VALUES (2) .............................................................. 112

FIGURE 4-39: PLOT OF HO-ILC (3RD

ORDER) ....................................................................................... 112

FIGURE 4-40: COMPARING HO-ILC METHODS ................................................................................... 113

FIGURE 4-41: PLOT OF ADAPTIVE P-ILC FOR Α = 0.1 .......................................................................... 114



FIGURE 4-44: PLOT OF ADAPTIVE P-ILC (OUTPUT TORQUE) .............................................................. 116

FIGURE 4-45: PLOT OF ADAPTIVE P-ILC (TRF) ..................................................................................... 116

FIGURE 4-46: PLOT OF ADAPTIVE PD-ILC FOR DIFFERENT KD VALUES ............................................... 117

FIGURE 4-47: PLOT OF ADAPTIVE PD-ILC (OUTPUT TORQUE) ............................................................ 118

FIGURE 4-48: PLOT OF ADAPTIVE PD-ILC (TRF) .................................................................................. 118

FIGURE 4-49: COMPARISON OF ADAPTIVE ILC ................................................................................... 119

FIGURE 4-50: COMPARISON OF PD, MC, HO AND ADAPTIVE ILC ....................................................... 120

FIGURE 4-51: PLOT OF MCHO-ILC ...................................................................................................... 122

FIGURE 4-52: COMPARISON OF MC, HO AND MCHO ILC ................................................................... 122

FIGURE 4-53: PLOT OF MCA-ILC (OUTPUT TORQUE) ......................................................................... 124

FIGURE 4-54: PLOT OF MCA-ILC (TRF) ................................................................................................ 124

FIGURE 4-55: COMPARISON OF VARIOUS ILC SCHEMES .................................................................... 125

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FIGURE 5-1: EXPERIMENTAL MOTOR USED FOR RESEARCH .............................................................. 130

FIGURE 5-2: BEMF SHAPES OF THE EXPERIMENTAL MOTOR ............................................................. 134

FIGURE 5-3: BEMF SHAPES OF THE EXPERIMENTAL MOTOR (CLOSE-UP).......................................... 135

FIGURE 5-4: BEMF IMBALANCES ........................................................................................................ 136

FIGURE 5-5: COGGING TORQUE OF THE EXPERIMENTAL MOTOR ..................................................... 136

FIGURE 5-6: TORQUE AND SPEED WAVEFORMS ................................................................................ 137

FIGURE 5-7: BODE PLOT OF PI CURRENT CONTROLLER ..................................................................... 138

FIGURE 6-1: SCHEMATIC OF COMPENSATION SCHEME FOR PMSM CONTROL ................................. 142

FIGURE 6-2: TORQUE ESTIMATION WITHOUT A FILTER ..................................................................... 143

FIGURE 6-3: TORQUE ESTIMATION WITH LPF .................................................................................... 144

FIGURE 6-4: IMPLEMENTING DSP BASED ZPF .................................................................................... 145

FIGURE 6-5: TORQUE ESTIMATION WITH ZPF (LUT OF SIZE 4096) .................................................... 145

FIGURE 6-6 TORQUE ESTIMATION WITH ZPF (LUT OF SIZE 256)........................................................ 146

FIGURE 6-7: TORQUE RIPPLE USING FOC ........................................................................................... 147

FIGURE 6-8: TRF FOR DIRECT FF CONTROL (USING SPEED INFORMATION) ....................................... 148

FIGURE 6-9: TRF FOR DIRECT FF CONTROL (USING A TORQUE TRANSDUCER) .................................. 149

FIGURE 6-10: TRF FOR INDIRECT PRE-COMPENSATION - TΔΛ ............................................................. 150

FIGURE 6-11: TRF FOR INDIRECT PRE-COMPENSATION - TΔI .............................................................. 151

FIGURE 6-12: TRF FOR INDIRECT PRE-COMPENSATION - TCOG ........................................................... 151

FIGURE 6-13: TRF FOR INDIRECT PRE-COMPENSATION - ALL ............................................................ 152

FIGURE 6-14: EXPERIMENTAL PLOT OF P-ILC FOR DIFFERENT KP VALUES ......................................... 154

FIGURE 6-15: PLOT OF P-ILC ............................................................................................................... 155

FIGURE 6-16: COMPARING LPF WITH ZPF IN TORQUE ESTIMATION ................................................. 156

FIGURE 6-17: PLOT OF PF-ILC WITH VARYING FORGETTING FACTORS .............................................. 157

FIGURE 6-18: PLOT OF PF-ILC ............................................................................................................. 157

FIGURE 6-19: PLOT OF D-ILC WITH DIFFERENT CUTOFF FREQUENCIES ............................................. 158

FIGURE 6-20: PLOT OF D-ILC WITH DIFFERENT KD VALUES ................................................................ 158

FIGURE 6-21: PLOT OF D-ILC .............................................................................................................. 159

FIGURE 6-22: PLOT OF PD-ILC FOR DIFFERENT KP AND KD VALUES .................................................... 160

FIGURE 6-23: PLOT OF PD-ILC ............................................................................................................ 161

FIGURE 6-24: PI-ILC ............................................................................................................................ 161

FIGURE 6-25: PLOT OF PI-ILC .............................................................................................................. 162

FIGURE 6-26: COMPARISON OF SINGLE CHANNEL FIRST ORDER ILC SCHEMES ................................. 163

FIGURE 6-27: COMPARING P-ILC AND MC-ILC ................................................................................... 164

FIGURE 6-28: PLOT OF MC-ILC FOR DIFFERENT KP,HIGH VALUES .......................................................... 165

FIGURE 6-29: PLOT OF MC-ILC ........................................................................................................... 166

FIGURE 6-30: HO-ILC WITH VARYING LEARNING GAINS .................................................................... 166

FIGURE 6-31: HO-ILC WITH DIFFERENT LEARNING GAINS ................................................................. 167

FIGURE 6-32: PLOT OF HO-ILC ............................................................................................................ 167




FIGURE 6-36: PLOT OF ADAPTIVE P-ILC (OUTPUT TORQUE) .............................................................. 170

FIGURE 6-37: PLOT OF ADAPTIVE P-ILC (TRF) ..................................................................................... 170

FIGURE 6-38: PLOT OF ADAPTIVE PD-ILC FOR DIFFERENT KD VALUES ............................................... 171

FIGURE 6-39: PLOT OF ADAPTIVE PD-ILC (OUTPUT TORQUE) ............................................................ 172

FIGURE 6-40: PLOT OF ADAPTIVE PD-ILC (TRF) .................................................................................. 172

FIGURE 6-41: COMPARING ADAPTIVE P-ILC AND ADAPTIVE PD-ILC .................................................. 173

FIGURE 6-42: COMPARING ADAPTIVE AND NON-ADAPTIVE ILC ........................................................ 173

FIGURE 6-43: COMPARING BETWEEN DIFFERENT CATEGORIES OF ILC SCHEMES ............................. 174

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FIGURE 6-44: PLOT OF MCHO-ILC ...................................................................................................... 176

FIGURE 6-45: PLOT OF MCA-ILC (OUTPUT TORQUE) ......................................................................... 176

FIGURE 6-46: PLOT OF MCA-ILC (TRF) ................................................................................................ 177

FIGURE 6-47: COMPARISON OF PROPOSED ILC SCHEMES WITH EXISTING ILC SCHEMES ................. 177

FIGURE 6-48: ROBUSTNESS OF DIRECT PRE-COMPENSATION TECHNIQUE ....................................... 179

FIGURE 6-49: ROBUSTNESS OF P-ILC .................................................................................................. 179

FIGURE 6-50: ROBUSTNESS OF PF-ILC ................................................................................................ 180

FIGURE 6-51: ROBUSTNESS OF D-ILC ................................................................................................. 180

FIGURE 6-52: ROBUSTNESS OF PD-ILC ............................................................................................... 181

FIGURE 6-53: ROBUSTNESS OF PI-ILC ................................................................................................. 181

FIGURE 6-54: ROBUSTNESS OF MC-ILC .............................................................................................. 182

FIGURE 6-55: ROBUSTNESS OF HO-ILC ............................................................................................... 182

FIGURE 6-56: ROBUSTNESS OF ADAPTIVE P-ILC ................................................................................. 183

FIGURE 6-57: ROBUSTNESS OF ADAPTIVE PD-ILC .............................................................................. 183

FIGURE 6-58: ROBUSTNESS OF MCHO-ILC ......................................................................................... 184

FIGURE 6-59: ROBUSTNESS OF MCA-ILC ............................................................................................ 184

FIGURE A.0-1: COGGING TORQUE VARIATION WITH TEMPERATURE ................................................ 195

FIGURE A.0-2: MAXIMUM AMPLITUDE OF COGGING TORQUE VARIATION WITH TEMPERATURE ... 196

FIGURE A.0-3: PLOT OF COGGING TORQUE VARIATIONS WITH TRF .................................................. 197

FIGURE A.0-4: VARIATION OF BEMF WITH TEMPERATURE ............................................................... 198

FIGURE A.0-5: VARIATION OF BEMF AMPLITUDE WITH TEMPERATURE ........................................... 199

FIGURE A.0-6: PLOT OF TORQUE CONSTANT VARIATION WITH TEMPERATURE ............................... 199

FIGURE A.0-7: PLOT OF CURRENT GAIN VS TRF ................................................................................. 201

FIGURE A.0-8: PLOT OF CURRENT OFFSET VS TRF .............................................................................. 201

Page XVII

Glossary of Terms

𝑖 actual current (A)

𝜇 adaptive learning gain

𝑇𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 additional harmonic components of cogging torque (Nm)

𝐼𝑎 armature current (A)

C Clarke transform matrix

𝐺𝑝 closed loop transfer function of a system

Tcog cogging torque (Nm)

ia current for phase a (A)

ib current for phase b (A)

ic current for phase c (A)

𝜖𝑎 current scaling error for phase a

𝜖𝑏 current scaling error for phase b

∆𝑖 current offset error (A)

∆𝑖𝑎 current offset error for phase a (A)

∆𝑖𝑏 current offset error for phase b (A)

Id* current command for the d-axis (A)

Iq* current command for the q-axis (A)

u control signal

Td derivative time for PID controller (s)

𝑦𝑑 desired signal

Tem electromagnetic torque (Nm)

𝑘𝑇 electromagnetic torque constant (NmA-1)

e error between the output and reference

Page XVIII

�̂�𝑟𝑖𝑝 estimated torque ripple (Nm)

𝜙 flux per pole (Wb)

λs flux linkage due to the permanent magnets (Wb)

λd flux linkage for d-axis (Wb)

λq flux linkage for q-axis (Wb)

α forgetting factor

Ls inductance (H)

Ld inductance for d-axis (H)

Lq inductance for q-axis (H)

Ti integral time for PID controller (s)

kp iterative learning gain for P-type iterative learning

kd iterative learning gain for D-type iterative learning

ki iterative learning gain for I-type iterative learning

Φ learning gains of higher order iterative learning

TL load torque (Nm)

𝑖𝑎,𝑚𝑒𝑎𝑠 measured current for phase a (A)

𝑖𝑏,𝑚𝑒𝑎𝑠 measured current for phase b (A)

J mass moment of inertia (kgm2)

𝑇𝑛𝑎𝑡𝑖𝑣𝑒 native harmonic components of cogging torque (Nm)

∆𝜆𝑠 non-ideal component of flux density distribution

J number of iterations

𝑦 output signal

T output torque (Nm)

P Park transform matrix

K proportional gain for PID controller

p pole pairs

Page XIX

𝑟 reference signal

Tref reference torque (Nm)

θe rotor electrical angular position (rad)

θm rotor mechanical angular position (rad)

ωe rotor electrical angular velocity (rad/s)

ωm rotor mechanical angular velocity (rad/s)

𝜖 scaling error

Rs stator resistance (Ω)

is stator current (A)

vs stator voltage (V)

vs* stator voltage command (V)

𝐻𝑓𝑓 transfer function of feedforward systems

𝐻𝑓𝑏 transfer function of feedback systems

𝐾𝑎 torque constant (NmA-1)

Trip torque ripple (Nm)

𝑇Δλ torque ripple from non-ideal flux density distribution (Nm)

𝑇Δλ𝑛𝑠 torque ripple caused by non-sinusoidal flux density distribution (Nm)

𝑇Δλ𝑎𝑠𝑦 torque ripple caused by asymmetry flux density distribution (Nm)

𝑇Δλ𝑖𝑚 torque ripple caused by imbalanced flux density distribution (Nm)

𝑇∆𝑖 torque ripple due to current measurement errors (Nm)

𝑇∆𝑖,𝑜𝑠 torque ripple due to current offset errors (Nm)

𝑇∆𝑖,𝑠𝑐 torque ripple due to current scaling errors (Nm)

b viscous friction coefficient (Nms)

vd* voltage command for the d-axis (A)

vq* voltage command for the q-axis (A)

Page XX

Page XXI

List of Acronyms

AC Alternating Current

A/D Analog/Digital

BEMF Back Electromotive Force

BLDCM Brushless Direct Current Motor

CAD Computer Aided Design

D-ILC Differential Type Iterative Learning Control

DFT Discrete Fourier Transform

DSP Digital Signal Processor

DC Direct Current

FOC Field Oriented Control

HO-ILC Higher Orders Iterative Learning Control

IDFT Inverse Discrete Fourier Transform

LUT Lookup Table

LPF Low Pass Filter

ILC Iterative Learning Control

MRAS Model Reference Adaptive System

MC-ILC Multi-Channel Iterative Learning Control

Page XXII

MCA-ILC Multi-Channel Adaptive Iterative Learning Control

MCHO-ILC Multi-Channel Higher Order Iterative Learning Control

PM Permanent Magnet

PMSMs Permanent Magnet Synchronous Motors

PPWT Pre-Programmed Waveform Techniques

PID Proportional-Integral-Derivative

P-ILC Proportional Type Iterative Learning Control

PD-ILC Proportional-Differential Type Iterative Learning Control

Pf-ILC Proportional Type Iterative Learning Control with forgetting factor

PWM Pulse Width Modulation

RMS Root Mean Square

SCFO-ILC Single Channel First Order Iterative Learning Control

SVPWM Space Vector Pulse Width Modulation

TE Torque Estimation

IT Texas Instruments

TRF Torque Ripple Factor

ZPF Zero Phase Filter

Chapter 1 Introduction

Permanent Magnet Synchronous Motors (PMSMs) are used in many applications such as

electric cars, pool pumps, servo applications, etc. This is due to the ease of control, the

declining cost of permanent magnets and a higher speed range for this type of motors [1, 2].

Compared to induction motors, PMSMs have a higher torque to inertia ratio, improved

efficiency, and smaller size. Induction motors on the other hand have lower cost and higher

operating temperatures [3].

While PMSMs offer many advantages in comparison to other types of motors, undesirable

torque ripple is also associated with these motors. Torque ripple can arise due to

manufacturing imperfections or measurement inaccuracies [4]. These torque ripples are

periodic in nature [5], and if not compensated by the controller will have a negative impact

on the motor’s performance. Research in PMSM [6-11] has shown torque ripple between

the range of 2% to 4% of rated torque.

A range of control methodologies have been described in literature to minimise torque

ripple of PMSMs [5, 7, 10-12]. Iterative Learning Control (ILC) is however of particular

interest due to their inherent capability to deal with disturbances that are repetitive in

nature [5, 13, 14].

The aim of this research is therefore to investigate whether ILC methods can effectively

deal with the periodic nature of torque ripple of PMSMs, and, if so, which ILC method may

be the preferred approach for this application.

1.1 Background

This section gives an overview of Permanent Magnet (PM) motors, the factors that cause

torque ripple in PMSMs, control methods that are used to achieve smooth operation and

lastly, ILC methods.

1.1.1 Overview of Permanent Magnet Motors

PM motors have many advantages that have led to their widespread use. These advantages

include quick dynamic speed response, high power factor, improved efficiency and high

power density compared to induction motors [15-17].

PM motors are also used in many applications including high performance servo and

robotics applications where rapid dynamic response and high reliability are required. In

applications where efficiency and size are the main concern such as in electric vehicles, PM

motors are also the preferred choice [18].

Figure 1-1: Surface Mount Permanent Magnets Motors (coils not shown) [19]

Torque is produced in a PM motor due to the interaction of the magnetic fields caused by

the permanent magnet and the stator coils, as shown in Figure 1-1. The interaction of the

magnet and the stator windings also gives rise to different shapes of the Back Electromotive

Force (BEMF) as shown in Figure 1-2. For a full revolution of the motor, the BEMF can be

modelled as either sinusoidal or trapezoidal depending on how the magnet and stator

windings are placed.

Figure 1-2: BEMF Shapes

A review of these two types of motors can be found in [2, 20]. There are generally two main

types of PM motors depending on the shape of their BEMF. The first type of PM motor is

known as Brushless Direct Current Motor (BLDCM) which has a trapezoidal shape BEMF. To

produce a constant torque, the phase currents have to be rectangular in shape. Assuming

instantaneous current commutation, only two of the three current phases are conducting.

The advantages of BLDCM are that only a single current sensor and a low resolution Hall

effect position sensor are needed for control, thereby reducing cost. More information on

how BLDCMs function can be found in [1, 2]. As the control of the inner loop of a BLDCM

system is only to control the input current, the control scheme is relatively simple. However,

the drawback of BLDCM is the inherently high torque ripple [2].

The second type of PM motor is the PMSM where its BEMF varies sinusoidally with the

rotor position. The stator requires sinusoidally shaped currents to produce a constant

torque. At least two current sensors are needed for current control as three phases are

0 pi/2 pi 3*pi/2 2*pi

-1

-0.5

0

0.5

1

Sinusoidal BEMF

Radians

Magnitude

0 pi/2 pi 3*pi/2 2*pi

-1

-0.5

0

0.5

1

Trapezidal BEMF

Radians

Magnitude

conducting at the same time [2]. More details about PMSMs function will be discussed in

section 2.1. Compared to control of BLDCMs, effective control of PMSMs requires a high

resolution position sensor. Alternatively, a complex estimation scheme for the rotor

position is required. However, because PMSMs have much better tracking abilities, smaller

torque ripples and a higher speed range, it is generally the preferred type of motor for high

performance applications where smooth operation and precise tracking are important [20].

1.1.2 Factors Contributing to Torque Ripple of PMSMs

Although there are many advantages of PMSMs, one significant drawback is torque ripple.

This can lead to undesirable vibrations, acoustic emissions, reduction in efficiencies and

speed oscillations. This is undesirable in applications that require smooth and steady

operations [13]. Although at high frequencies the undesirable effects of torque ripples are

filtered by the motor system inertia, these effects remain significant at low speeds [2].

Moreover, in applications such as electric cars, torque ripples can result in unwanted noise

that can distract driver or passengers. In machine tool applications, mechanical oscillations

induced by torque ripples can potentially leave visible marks on machined surfaces.

Overall, torque ripple reduces efficiency, creates unnecessary vibration and acoustic

emissions and unwanted position and speed oscillations. There is therefore a need for

smooth operation of PMSMs, i.e. the production of torque without the unwanted torque

ripples.

Torque ripples can arise due to manufacturing imperfections and measurement

inaccuracies. The two main problems associated with manufacturing imperfections are

cogging torque and non-ideal sinusoidal flux density distributions [13, 21, 22]. Cogging

torque can be defined as the pulsating torque components generated by the interaction of

the rotor magnetic flux and angular variations in the stator magnetic reluctance [4].

Cogging torque is one of the inherent problems associated with PMSMs and may not be

eliminated completely even with well-known methods for cogging torque reduction.

Furthermore, motors from the same batch may have different cogging torque waveforms

[23, 24]. Cogging torque can be 3% of rated motor torque and as high as 25% for poorly

designed machines [2, 25].

For PMSM drives, the flux density distribution is assumed to be sinusoidal and sinusoidal

currents are used to drive the motors. This assumes that the flux density distribution is

ideal in terms of its shape, symmetry and balance. However, an imbalance, asymmetry or

non-sinusoidal flux density distribution shape will result in unwanted torque ripples in the

output torque [26]. More details about manufacturing imperfections are covered in section

2.4.1.

Sensors are used to measure important attributes of a PMSM system, including the

currents and the position of the rotor. However, sensor signals may not accurately

represent these variables due to noise, nonlinear behaviour due to environmental

conditions such as temperature, and quantization effects. Using these inaccurate

measurements as inputs to the controller will result in torque ripple in the output torque

[27]. The main issues associated with current measurements are inaccuracies due to scaling

and offset errors [13, 27]. These factors are discussed in more detail in section 2.4.

Variations of current scaling and offset errors up to ±20% would lead to torque ripple of

0.24% of the rated torque. This is discussed in more detail in Appendix A.3.

1.1.3 Control Methods for Smooth Operation

In view of the many causes of torque ripple, there is a need for an efficient and effective

method to achieve torque ripple minimisation. In general, there are two ways to achieve

this.

One approach to reduce torque ripple is to improve the design of the motor to optimise the

interaction between the stator currents and the BEMF waveforms. However, this may not

be practical. Techniques to achieve high accuracy of motor construction may be impractical

for mass produced motors as they are too costly [4]. In addition, there are design trade-offs

as optimising a motor design for reduced cogging torque will reduce the maximum torque

of a motor [28, 29].

An alternative approach is active control of the stator currents. Jahns and Soong [4], divided

the control methods to minimize torque ripple into the following five categories:

1. Commutation torque minimisation – this method is only relevant for BLDCM

2. Speed loop disturbance rejection – this method is only relevant for low speed

operation

3. High speed current regulator saturation – this method is only relevant for high

speed operation

4. Estimators and observers – this methods requires a high resolution encoder for low

speed operation

5. Programmed current waveform control – this method is dependent on an accurate

predefined (off-line) model of the system.

Of all the methods suggested, only estimators or observers will be able to cope with the

factors that cause torque ripples and their varying nature with temperature and operating

setpoint. Due to the high cost associated with the torque sensor and the additional space

needed to mount the sensor, the use of a torque sensor is not considered for many control

schemes.

Feedback Control

Feedback control is an error driven type of control, which in most applications tries to

match the plant output with the reference input [30]. The most common types of feedback

control methods are Proportional-Integral-Derivative (PID). Feedback control can be used

to minimise torque ripples. Feedback control for PMSMs is further discussed in section

2.3.1.

Pre-compensation Techniques

Pre-compensation techniques can be used as a supplement to feedback control. The

compensation is based on prior knowledge of the plant and process disturbances. Ideal pre-

compensation can result in zero error between the reference and the output with the

possibility of no torque ripple in the case of PMSMs. Pre-compensation techniques also

have the ability to achieve a much faster transient response compared to using a feedback

control structure only. Using pre-compensation, all possible causes of torque ripple can be

pre-compensated for to achieve torque ripple minimisation if they are known beforehand

[31, 32]. This is somewhat similar to a feedforward control but is not limited to the use of

reference signals only. Both terms have been used interchangeably by some researchers.

Pre-compensation techniques will require system output signals for the compensation to be

accurate.

However, ideal pre-compensation is limited by many factors of real world implementations

such as inaccurate modelling of non-linearities of the system, parameters that can vary with

time or operating conditions as well as noise. Pre-compensation techniques also assume

that all variables of the system are known beforehand through measurements or system

modelling [31, 32]. In practice, it can be difficult to achieve sufficient accuracy for an

application. Hence, a combination of both feedback control and pre-compensation can be

utilised to achieve the desired performance. More details on pre-compensation techniques

are covered in section 2.4.

Adaptive Control

Adaptive control can deal with modelling inaccuracies, noises or the variations of

parameters due to changes in temperature or operating conditions. Thus, an adaptive

controller should be able to modify the control outputs to cope with changes in the

dynamics of the process or due to other disturbances.

Some proposed schemes in literature assume the cogging torque to be negligible [5, 13, 33,

34]. However for mass produced motors, cogging torque can be a major cause of torque

ripple [24]. There are techniques to reduce cogging torque such as stator slot skewing or

improving rotor magnet design techniques that includes varying the magnet arc length,

varying the magnet strength, shifting the magnet poles or varying the radial shoe depth

[35]. However, a cost penalty is usually associated with this approach which may not be

appropriate for mass produced motors.

According to Jahns [20], there is no known method that can effectively minimise torque

ripples under all conditions. Most schemes are either too computational intensive, have

complicated motor designs or require exact information of motor parameters. Furthermore,

some methods proposed for torque estimation extract information from the electrical

subsystem. These methods have the drawbacks of high reliance on accurate current

measurements and are therefore limited in their ability to minimise cogging torque.

Therefore, an adaptive approach to minimise torque ripple in PMSMs may be the solution

to cope with the inherent modelling inaccuracies and noise.

1.1.4 Iterative Learning Control

ILC is an adaptive control method capable of reducing the influence of periodic

disturbances. In a repetitive loop, the controller attempts to reduce the error to zero based

on the information from the previous iteration [36].

Since most of the causes of torque ripples (mentioned in section 1.1.2) are periodic with

respect to the position (angle) of the motor, ILC may be a suitable method to minimise

these torque ripples. ILC will require significant memory storage requirements in real time

for the necessary information of the previous cycles. This is not an issue if the control

system is connected with a computer which is used to store the information. However, the

transfer time lag may be an issue if there is significant exchange of information required

between the controller and the computer. Moreover, if an online DSP based control is

required, the requirement of large memory spaces may be an issue. External memory may

have to be used, which will add to the total cost for production.

ILC has been researched extensively in robotics. The most commonly used ILC schemes are

the proportional type ILC (P–ILC). Other types of ILC, such as differential type (D-ILC),

variable learning ILC or multi-channel ILC schemes have shown an improved performance

for robotic control [37-45]. However, the use of ILC to minimise torque ripple of PMSMs has

been limited. Many researchers used only the P-ILC and some have shown only simulated

results. Nevertheless, P-ILC has been shown to be able to suppress torque ripple for a

PMSM [46-52].

1.2 Aim of Research

In view of the limited experimental verification of the different ILC methods to minimise

torque ripples of PMSMs, the goal of this research can now be formulated as follows:

1. Can Iterative Learning Control schemes effectively be used to minimise torque

ripple of Permanent Magnet Synchronous Machines without using a torque

transducer, and if so,

2. Which Iterative Learning Control schemes are suitable for torque ripple

minimisation, given the typical properties of PMSMs and their real time control

systems?

1.3 Structure of Thesis

The layout of the remaining thesis is as follows: Chapter 2 covers modelling and control of

PMSMs. Chapter 3 discusses in depth the different ILC schemes described in literature that

may be useful for torque ripple minimisation of PMSMs. Chapter 4 shows a theoretical

comparison of the reviewed methods, using simulation results. Chapter 5 describes the

experimental setup and the design of experiments. Chapter 6 describes the experimental

results and compares them to the theoretical simulation results of chapter 4. Finally,

chapter 7 summarises the research outcomes and outlines future work.

Chapter 2 Modelling PMSMs Control

In order to minimise torque ripple of PMSMs, various methods have been proposed and

evaluated by other researchers. As discussed in section 1.1.3, minimising torque ripple

through improved motor design may not be economically feasible due to the high cost that

can be associated with this. Thus, researchers have investigated how torque ripple can be

minimised using control of the stator currents instead.

A control method which is simple to design and yet able to minimise torque ripple given the

practical constraints of mass produced motors is highly desirable. As the conventional PID

controller is a widely used feedback method [30], this method of control will be used as a

baseline of comparison for evaluation of the literature, as well as simulations and

experiments in later chapters.

As mentioned in section 1.1.4, Iterative Learning Control (ILC) has the potential to reduce

periodic disturbances. Since torque ripples of PMSMs are periodic in nature, ILC may be a

suitable control method to reduce the torque ripples of PMSMs.

This chapter covers the dynamic model of a PMSM and a common method of controlling

PMSMs using field oriented control. The factors contributing to torque ripple of PMSMs will

also be discussed in depth. ILC methods will be covered in section 3.3.

2.1 Dynamic Model of a Permanent Magnet Motor

The dynamic model of a permanent magnet (PM) motor is derived based on the following

assumptions:

1. The PM motor is unsaturated [53]

2. Eddy currents and hysteresis losses are negligible [53]

3. BEMF is proportional to angular velocity and independent of current [12]

4. Torque produced is proportional to phase currents [12]

With these assumptions, the equations for the motor and electrical dynamics of a PM

motor are as follows:

ee

dt

d

(2.1)

mLm bTT

Jdt

d

1 (2.2)

essssss idt

dLiRv (2.3)

where 𝜃𝑒 is the rotor electrical angular position, 𝜔𝑒 and 𝜔𝑚 are the rotor electrical and

mechanical angular velocity respectively, 𝐽 is the mass moment of inertia, 𝑏 is the viscous

friction coefficient, 𝑇 and 𝑇𝐿 are the output torque and load torque respectively, 𝑣𝑠 is the

stator voltage, 𝑖𝑠 is the stator current, 𝐿𝑠 is the inductance, 𝑅𝑠 is the stator resistance and

𝜆𝑠 are the flux linkage due to the permanent magnets.

For a three phase system, the equation for the electromagnetic torque is:

𝑇𝑒𝑚 = 𝜆𝑠 ∙ 𝑖𝑠 (2.4)

where 𝑇𝑒𝑚 is the electromagnetic torque and is the dot product of the flux linkages and

stator currents.

In additional, some methods require electrical position or angular velocity and it is useful to

note that in a complete mechanical revolution, there will be an electrical revolution for

each pair of poles of the PM motors. Thus,

𝜃𝑒 = 𝑝𝜃𝑚 (2.5)

𝜔𝑒 = 𝑝𝜔𝑚 (2.6)

where 𝜃𝑚 is the rotor mechanical angular position and p is the number of pole pairs.

The output torque, 𝑇 is thus:

𝑇 = 𝑇𝑒𝑚 − 𝑇𝐿 (2.7)

where 𝑇𝐿 is the load torque. Figure 2.1 shows the model of the PM motor based on the

equations above.

Where there is no external load torque and the BEMF is sinusoidal, the whole model can be

simplified to Figure 2.2.

This PMSM model will be used throughout this thesis where 𝑣𝑠 is the input to the PMSM, 𝑖𝑠,

𝑇, 𝜔𝑚, 𝜔𝑒 , 𝜃𝑚, 𝜃𝑒 are the outputs. 𝑖𝑠 and 𝑇 can be measured using current sensors and

torque transducer respectively. 𝜃𝑚 can be measured using an encoder and 𝜃𝑒 can be found

using equation 2.5. 𝜔𝑚 is the derivate of 𝜃𝑚 and 𝜔𝑒 can be found using equation 2.6.

Possible practical applications where such model can be applied include pumps, fans,

vibrators and etc.

𝜔𝑒

𝜔𝑚 𝑇𝑒𝑚 +

_ 𝜔𝑚 𝜃𝑚 1

𝐿𝑠𝑠 + 𝑅𝑠

+ _ ∙

𝑇𝐿

𝑣𝑠

𝑝

𝜆𝑠

×

𝑖𝑠 𝑇

𝑇 PMSM

𝑖𝑠

𝜔𝑚 /𝜔𝑒

𝜃𝑚/𝜃𝑒

𝑣𝑠

1

𝑠

1

𝐽𝑠 + 𝑏

Figure 2-1: Model of PM Motor

Figure 2-2: PMSM Model

2.2 Field Oriented Control

Field Oriented Control (FOC) allows controlling an Alternating Current (AC) machine, such as

PMSM, as if it is a Direct Current (DC) motor. In a DC motor, the flux and torque can be

controlled independently since the currents that are produced are orthogonal to one

another [2].

The electromagnetic torque produced is thus:

𝑇𝑒𝑚 = 𝐾𝑎𝜙𝐼𝑎 (2.8)

where 𝐾𝑎 is a constant for a particular machine, 𝐼𝑎 is the armature current and 𝜙 is the flux

per pole. If the flux is unchanged, then the torque can be controlled solely by the armature

current. In an AC machine, the field of the stator and rotor are not orthogonal and only the

stator current can be controlled. FOC can then be used to transform the stationary

reference frame of the currents to the rotating reference frames consisting of torque and

flux. This then allows an AC motor to be controlled just like a DC motor [2].

Figure 2-3: Reference Frames for FOC [54]

Figure 2-3 shows the transformation using FOC techniques. This enables the control of 3

phase currents Ia, Ib and Ic using just two constant values of Id and Iq in the rotating

reference frame. The transformation from the 3 phase stationary reference frame to a 2

phase reference frame is known as the Clarke transformation. The Park transformation is a

further transformation required to change the 2 phase reference frame to the rotating

reference frame [55].

In a balanced system, where 𝐼𝑎 + 𝐼𝑏 + 𝐼𝑐 = 0, the Clarke transform matrix, C is:

𝐶 =2

3[1 −

1

2−

1

2

0√3

2−

√3

2

] (2.9)

and the inverse C-1:

𝐶−1 =3

2

[

2

30

−1

3

√3

3

−1

3−

√3

3 ]

(2.10)

The Park transform matrix, P is:

𝑃 =2

3

[ cos (𝜃) cos (𝜃 −

2𝜋

3) cos (𝜃 +

2𝜋

3)

sin(𝜃) sin (𝜃 −2𝜋

3) sin (𝜃 +

2𝜋

3)

1

2

1

2

1

2 ]

(2.11)

and the inverse P-1:

𝑃−1 =

[

cos (𝜃) sin(𝜃) 1

cos (𝜃 −2𝜋

3) sin (𝜃 −

2𝜋

3) 1

cos (𝜃 +2𝜋

3) sin (𝜃 +

2𝜋

3) 1]

(2.12)

More details about both transformations can be found in [55]. Due to these

transformations, the equation 2.3 can be rewritten in the rotating reference frame as:

eqdddsd idt

dLiRv (2.13)

edqqqsq idt

dLiRv (2.14)

The relationship between the flux linkages with the inductances are:

𝜆𝑞 = 𝐿𝑞𝑖𝑞 (2.15)

𝜆𝑑 = 𝐿𝑑𝑖𝑑 + 𝜆𝑚 (2.16)

where 𝐿𝑑 and 𝐿𝑞 are the stator inductance in the d and q axis.

The equation for the electromagnetic torque becomes:

dqqdem iipT 2

3

dqqqmdd iiLiiLp

2

3

qdqdqm iiLLip 2

3 (2.17)

where Tem is the electromagnetic torque and p is the number of motor pole pairs. Assuming

a surface mounted PMSM is non-salient, 𝐿𝑑 = 𝐿𝑞 and will be represented by 𝐿. The

electromagnetic torque can be re-written as

𝑇𝑒𝑚 =3

2𝑝𝜆𝑚𝑖𝑞 = 𝑘𝑇𝑖𝑞 (2.18)

where 𝑘𝑇 =3

2𝑝𝜆𝑚𝑖𝑞 is the electromagnetic torque constant.

To achieve the maximum torque output, the d-axis current is controlled to be zero [56].

Therefore, the controlling of the PMSM using FOC can now be done using Iq alone. For a

reference torque Tref, the current command iq* becomes:

T

ref

qk

Ti * (2.19)

Figure 2-4 shows how FOC can be used on a PMSM.

Figure 2-4: FOC on a PMSM

The superscript star is used to denote a command. id* and iq* are the current commands for

the d and q axis respectively. vd* and vq* are the voltage commands for the d and q axis

respectively. vs* is the stator voltage command. The block C is the current controller and

the dq-abc block is the transformation from the dq-frame to the abc-frame or from the abc-

frame to the dq-frame. 𝜃𝑒 is required for these transformations to occur.Since the aim of

the thesis is about torque ripple minimisation, the whole control scheme can be simplified

to Figure 2-5 with the torque as a reference.

Figure 2-5: Simplified Model using FOC on a PMSM

In the case where the external load torque is zero, the output torque from equation 2.7

becomes:

ref

qtem TikTT (2.20)

The output torque is rarely used in the control scheme due to the high cost associated with

torque measurement. Controlling the torque can instead be done through controlling the q-

𝑣𝑞∗

𝑣𝑑∗

𝑣𝑠∗

𝜃𝑒

𝑖𝑠 PMSM + _

+ _

C

C

𝑖𝑞∗

abc

dq

dq

abc

FOC

𝑇 𝑇𝑟𝑒𝑓 1

𝑘𝑇

𝑖𝑞∗

𝑖𝑑∗ = 0

𝑣𝑠∗

𝜃𝑒

𝑖𝑠

FOC PMSM

𝑖𝑑∗ = 0

axis current, 𝑖𝑞. The FOC is one of the most popular control schemes and the setup in Figure

2.5 is being used throughout this thesis as a baseline comparison. The common current

controllers used in the above diagram are either the PID or hysteresis controllers.

2.3 Current Controllers for PMSM

In Figure 2.5, where FOC is being used for PMSM control, the current controllers play an

important part in ensuring a smooth torque output. The bandwidth of the current

controllers must be wide enough to ensure torque ripple within a certain frequency range

can be controlled and minimised. Two main types of current controllers are being discussed

in the next two sections, PID current control and hysteresis current control.

2.3.1 PID Current Control

PID controller is a type of feedback control which is an error driven type of control where

the controller tries to match the system output to a reference. Thus, a difference between

the reference and output, the error, must first occur before any actions will be taken to

minimise this error [30].

The PID controller is useful in many applications as it be tuned even though a mathematical

model of the system is not known. A PID controller is therefore used in many industrial

applications [30, 57]. The controller consists of a proportional action, and integral action

and differential action in parallel. The output from the controller is thus given by:

𝑢(𝑡) = 𝐾 (𝑒(𝑡) +1

𝑇𝑖∫ 𝑒(𝑡)𝑑(𝑡) + 𝑇𝑑

𝑑

𝑑𝑡𝑒(𝑡)

𝑡

0) (2.21)

where u is the control signal, K is the proportional gain, e is the error between the output

and reference, Ti is the integral time and Td is the derivative time.

A larger proportional gain K results in faster response. However, a large K might lead to

instability and oscillation. A smaller Ti would result in eliminating the steady state error

more quickly. However, the downside is the large overshoot that may be undesirable in

some applications. Finally, a larger Td results in smaller overshoot but may lead to

instability due to noise amplification caused by the high gain of the differentiating

operation [30].

In motor industries, the PI controller is often used instead of PID to prevent any

amplification of noise caused by the D action.

The Ziegler Nichols tuning methods can be used to tune a PID controller. However, in most

cases these tuning methods still require further fine tuning using manual adjustments [58].

Self-tuning PID methodologies are also available, however they generally require significant

time to self-tune and are limited in the type of systems where they can be applied [59].

Figure 2-6 shows how the outputs of the current controllers provide the voltage inputs to

the motor system.

Figure 2-6: PI Current Controllers for PMSM (Simplified)

While the PID controller is relatively easy to implement and functions acceptably on many

systems, performance is often limited compared to more advanced control methodologies.

In some cases, PID controllers perform poorly or cannot be applied at all due to the

complexity of the control system [60]. Since PID control utilises constant parameters, it will

not be able to take variations of system parameters into account for torque ripple

minimisation [60].

PMSM + _ PI

𝑣𝑠∗

𝑖∗

𝑖𝑚𝑒𝑎𝑠

𝑇

𝑖𝑠

sensors

2.3.2 Hysteresis Current Control

Similar to the PI control, hysteresis control is also based on a feedback loop with

comparators to modulate the output. The output is modulated within a certain range

known as the hysteresis band, h. This scheme is easy to implement and has fast response

time but it has the problem of a variable switching frequency which induces electrical losses

and adds of high frequency harmonics to the system [61]. The uses of hysteresis controllers

in PMSMs are discussed in [62-64]. Although hysteresis controllers are easy to implement,

the switching frequency of the converter depends on the load parameters which vary with

the AC voltage. The randomness of the limit cycle makes the control operation unsmooth

[65].

Hysteresis controllers have rectangular error fields (refer to Figure 2.7) in the rotating

frame and different hysteresis values can be chosen to control the d and q current

components [66].

Figure 2-7: Hysteresis Controllers [67]

2.4 Torque Ripple of PMSMs

In an ideal scenario, the output torque is exactly the same as the reference torque.

However in an actual implementation of a PMSM motor control system, torque ripple is

present due to the factors mentioned in section 1.1.2. The two main factors, manufacturing

imperfections of the motor (which causes cogging torque and non-ideal sinusoidal flux

density distribution) and measurement errors resulting from the sensors (which causes

error in current and position/speed measurements) will be discussed in detail in this section.

In most PMSM control systems, current sensors and an encoder are used to measure the

currents and position of the rotor. Speed of the rotor is estimated by differentiating the

position signal and torque can be estimated from either the currents and/or the speed

depending on the control scheme used. Accurate measurements of the currents and

position are critical to deliver the desired output. Inaccurate measurements, which are fed

back into the controller, will result in greater torque ripple. This is further complicated by

manufacturing imperfections whereby the motors have cogging torque and non-ideal

sinusoidal flux density distributions. Unless taken into account by the control scheme, a

greater torque ripple will result if cogging torque and a non-sinusoidal flux density

distribution are present.

Other factors not discussed in this thesis and may have effect on the torque ripple relates

to the design of the motor – placement of the magnets, variation of the skew and

magnetisation and the types of magnets used (either rare-earth or ferrite) [19, 68, 69].

2.4.1 Manufacturing Imperfections

To remain cost effective, mass produced motors have manufacturing tolerances. These

manufacturing imperfections result in cogging torque and a non-ideal sinusoidal flux

density distribution. For the former, the methods used for cogging torque reduction such as

stator slot skewing or magnetic arc length variation may not produce the desirable effect.

To make matters worse, PM motors from the same batch can have different values of

cogging torque [23, 24]. Similarly for the latter, the flux density distribution of a PMSM may

not be perfectly sinusoidal and this can produce unwanted torque ripples in the output

torque. However, these imperfections in the manufacturing process result in torque ripples

and should therefore be minimised by the controller to produce a smooth output torque.

Cogging Torque

Cogging torque, 𝑇𝑐𝑜𝑔, can be defined as the pulsating torque components generated by the

interaction of the rotor magnetic flux and angular variations in the stator magnetic

reluctance [4]. It is always present, even when the motor is not powered. According to

Grcar [25], cogging torque can be 3% of rated motor torque. However, for PMSMs that are

poorly designed or manufactured, cogging torque can even be as high as 25% of the rated

motor torque [2]. The cogging torque of the test motor used in this research is 6% of the

rated motor torque. This motor is part of a pool pump system. More details can be found in

Chapter 5.

Cogging torque has a mean value of zero and is a periodic function of rotor position. Its

harmonics appear at frequencies that are multiples of the position based fundamental

frequencies [18, 23]. The harmonic components of cogging torque are made up of the

native and additional components. Native harmonic components of cogging torque are

caused by the particular design of the motor while additional harmonic components of

cogging torque are due to manufacturing inaccuracies [23].

Since cogging torque is a periodic function of the rotor position, 𝜃𝑚, the cogging torque

equation, as defined by L. Gašparin [23], is thus:

𝑇𝑐𝑜𝑔(𝜃𝑚) = 𝑇𝑛𝑎𝑡𝑖𝑣𝑒(𝜃𝑚) + 𝑇𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙(𝜃𝑚) (2.22)

where 𝑇𝑛𝑎𝑡𝑖𝑣𝑒 is the native harmonic components of cogging torque and 𝑇𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 is the

additional harmonic components of cogging torque.

Analytic modelling of cogging torque is a challenging task and techniques are available to

reduce cogging torque. Most techniques relate to machine design with the aid of various

numerical tools. Techniques such as stator slot skewing, varying the magnet arc length,

varying the magnet strength, varying the radial shoe depth or shifting the magnet poles

have been reported as successful [35]. However, cogging torque is not included in some

literatures, as cogging torque is assumed to be negligible. Moreover, many of these

methods require specialised tools and the high cost associated with it may not be practical

for mass produced motors. Cogging torque may however be neglected for specially made

motors designed for a particular application. Islam [70] raises the issues in reducing cogging

torque of mass produced motors. In all the experiments tested using the above-mentioned

techniques, cogging torque cannot be eliminated completely due to imperfection and

irregularities in the magnet dimensions. In general, cogging torque cannot be neglected or

assumed to be negligible for mass produced motors with the current state of art of

manufacturing techniques.

There are a variety of control methods developed to minimise cogging torque instead.

Jahns [71] uses a graphical optimisation process to determine the best combination of

current and BEMF shape to output a constant electromagnetic torque. This technique is

computational intensive. Heins [72] compared different pre-programmed waveform

techniques (PPWT) to minimise cogging torque. A torque transducer is used to measure

the torque and a decoupling method is used to determine the cogging torque. Besides

using an expensive torque transducer, this method is also not able to adapt to any

variations to cogging torque changes that can be caused by temperature changes. Seguritan

[73] compensates for cogging torque by estimating the amplitude and phase of the torque

pulsation through filtering of the system signals. The frequencies that compose the first

harmonics are taken to be known quantities. Least square estimation is then carried out to

find the required amplitudes and phases. Grcar [29] uses a similar technique of estimating

the magnitudes and phases of the torque component. Ruderman [74] detects the torque

harmonics via a FFT and tunes the parameters of the feedforward compensator using a

recursive estimation technique. The above three mentioned methods require significant

computational resources in order to suppress cogging torque. Jia uses current harmonic

injection technique to produce torque harmonics that can compensate for cogging torque.

Compensation for the 1st and 2nd harmonics of the cogging torque are done and higher

orders are ignored [75]. This may not be effective if the cogging torque has higher order

harmonics. Favre uses an iterative method for a given BEMF shape and modified the

amplitude and phase angle of the different current harmonics one at a time to gradually

reduce existing torque harmonics [76]. This technique may not be suitable for rapid

changing input signals.

In summary, cogging torque can be a major cause of torque ripple and the control methods

used to control a PMSM must be able to minimise cogging torque to achieve torque ripple

minimisation.

Non-Ideal sinusoidal flux density distribution

For PMSM drives, the flux density distribution is assumed to be sinusoidal and sinusoidal

currents are used to drive the motors. This assumes that the flux density distribution is

ideal in terms of its shape, symmetry and balance. However, there could be an imbalance,

asymmetry or non-sinusoidal flux density distribution resulting in torque ripples [26].

The flux density distribution 𝜆𝑠 can thus be considered to be the summation of the ideal

sinusoidal flux density distribution (λs,ideal) and any error (Δ𝜆𝑠) caused by non-sinusoidal

shape, asymmetry or imbalances.

𝜆𝑠 = 𝜆𝑠,𝑖𝑑𝑒𝑎𝑙 + Δ𝜆𝑠 (2.23)

where ∆𝜆𝑠, the non-ideal component of 𝜆𝑠 is the sum of the flux density distribution error

caused by non-sinusoidal shape, asymmetry and imbalances.

A non-ideal sinusoidal flux density distribution interacting with purely sinusoidal stator

currents can give rise to periodic torque ripples. The resultant flux linkage between the

permanent magnets and the stator currents contains harmonics of the order 5, 7, 11 … in

the 3-phase 120o reference frame. However, in the synchronous rotating reference frame,

these harmonics appear in the 6th, 12th and other multiples of the sixth harmonics as shown

in the equation below [77].

𝜆𝑚 = 𝜆𝑑0 + 𝜆𝑑6𝑐𝑜𝑠𝑐𝑜𝑠6𝜃𝑒 + 𝜆𝑑12𝑐𝑜𝑠12𝜃𝑒 + ⋯ (2.24)

where λd0, λd6 and λd12 are the dc, 6th and 12th harmonic of the d-axis flux linkage

respectively and θe is the electrical angle. Combining equations 2.18 and 2.24, we have

𝑇𝑒𝑚 =3

2𝑝𝜆𝑚𝑖𝑞

=3

2𝑝𝑖𝑞(𝜆𝑑0 + 𝜆𝑑6𝑐𝑜𝑠𝑐𝑜𝑠6𝜃𝑒 + 𝜆𝑑12𝑐𝑜𝑠12𝜃𝑒 + ⋯)

=3

2𝑝𝜆𝑑0𝑖𝑞 +

3

2𝑝𝜆𝑑6𝑐𝑜𝑠𝑖𝑞𝑐𝑜𝑠6𝜃𝑒 +

3

2𝑝𝜆𝑑12𝑖𝑞𝑐𝑜𝑠12𝜃𝑒 + ⋯

= 𝑇0 + 𝑇6𝑐𝑜𝑠6𝜃𝑒 + 𝑇12𝑐𝑜𝑠12𝜃𝑒 + ⋯

= 𝑇0 + 𝑇Δλ𝑛𝑠 (2.25)

where 𝑇0, 𝑇6 and 𝑇12 are the dc, 6th and 12th torque harmonics respectively. 𝑇Δλ𝑛𝑠is the sum

of the torque harmonics caused by non-sinusoidal flux density distribution. Therefore, the

6th and 12th and other multiples of the sixth torque harmonics are caused largely by the

non-ideal flux density distribution shapes which are not fully sinusoidal in nature.

Asymmetry and imbalances in the flux density distribution will also affect the torque output.

Similarly, if they are not compensated for by the control scheme, torque ripples may also

result. The torque ripple caused by asymmetry and an imbalanced flux density distribution

can be represented by 𝑇Δλ𝑎𝑠𝑦 and 𝑇Δλ𝑖𝑚

respectively.

Therefore, the torque ripple resulted from the non-ideal sinusoidal flux density distribution

(due to non-sinusoidal shape, asymmetry or imbalance flux density distribution) is

represented by the notation 𝑇Δλ whereby

𝑇Δλ = 𝑇Δλns+ 𝑇Δλ𝑎𝑠𝑦

+ 𝑇Δλ𝑖𝑚 (2.26)

where 𝑇Δλns, 𝑇Δλ𝑎𝑠𝑦

and 𝑇Δλ𝑖𝑚 are the torque ripple due to the non-sinusoidal shape,

asymmetry and an imbalance in the flux density distribution respectively. While FOC offers

an easier way of controlling PMSM, it can bring about torque ripple if the flux density

distribution is not sinusoidal or symmetric.

Literature about removing or minimising torque ripples due to a non-ideal flux density

distribution is quite extensive. The most common techniques range from pre-programmed

waveform techniques [72, 78], injecting of current harmonics [79] or using least square

methods to find the required torque harmonic components [17, 29, 74]. Most of the

techniques are similar to the techniques to suppress cogging torque. Since the torque

harmonics caused by a non-ideal sinusoidal flux density distribution can be detected from

the electromagnetic torque, different torque estimation techniques using the measured

stator currents can also be used [80].

2.4.2 Measurement Inaccuracies

Hardware components are needed for real time implementation of the motor drive system.

These include current sensors, Analog/Digital (A/D) converter, encoder, rectifier, inverter,

etc. Measurements are subjected to noise and the degree of accuracy of the measurement.

High frequency measurement noise exists in all electronics and is inevitable. An average

filter or low pass filter is often used to remove these high frequency noises. The accuracy of

these sensors is another issue as any inaccuracies in these sensors can also lead to torque

ripples. The two main types of hardware measurement inaccuracies are current scaling

error and current offset error.

By assuming that the current error is linear, the measurement error can be simplified to an

offset and a gain as shown in the equation below [27].

𝑖𝑚𝑒𝑎𝑠 = 𝜖𝑖 + ∆𝑖 (2.27)

where 𝑖𝑚𝑒𝑎𝑠 is the measured current, 𝑖 is the actual current, 𝜖 is the scaling error and ∆𝑖 is

the offset error for the stator currents.

Current Offset Error

In a balanced system where the sum of the stator currents is zero, only two currents need

to be measured for FOC. This also reduce the number of current sensors needed for control

and thus save cost. Direct Current (DC) offset can give rise to torque ripple [27]. The actual

currents are measured by Hall-effect sensors and converted into voltage signals, which are

analogue signals which have to be converted to digital signals for many modern digital

control systems. If there are any inherent offsets in these devices, it will results in a DC

offset. Similarly, any unbalanced DC supply voltage in the sensors will also result in a DC

offset. Thus, the measured current is the sum of the actual current and the current offset

error as shown below:

𝑖𝑎,𝑚𝑒𝑎𝑠 = 𝑖𝑎 + ∆𝑖𝑎 (2.28)

𝑖𝑏,𝑚𝑒𝑎𝑠 = 𝑖𝑏 + ∆𝑖𝑏 (2.29)

where 𝑖𝑎and 𝑖𝑏 are the actual stator currents for phase a and phase b, ∆𝑖𝑎 and ∆𝑖𝑏 are the

current offset errors, 𝑖𝑎,𝑚𝑒𝑎𝑠 and 𝑖𝑏,𝑚𝑒𝑎𝑠 are the measured stator currents for phase a and

b of the stator currents respectively. In the rotating frame, this becomes

𝑖𝑑,𝑚𝑒𝑎𝑠 = 𝑖𝑑 + ∆𝑖𝑑 (2.30)

𝑖𝑞,𝑚𝑒𝑎𝑠 = 𝑖𝑞 + ∆𝑖𝑞 (2.31)

where ∆𝑖𝑑 and ∆𝑖𝑞 are the results of Park’s transformation for ∆𝑖𝑎 and ∆𝑖𝑏

∆𝑖𝑑 =2

3[∆𝑖𝑎𝑠𝑖𝑛𝜃𝑒 + ∆𝑖𝑏𝑠𝑖𝑛 (𝜃𝑒 −

2𝜋

3) + (−∆𝑖𝑎 − ∆𝑖𝑏)𝑠𝑖𝑛 (𝜃𝑒 +

2𝜋

3)]

=2

√3√∆𝑖𝑎

2 + ∆𝑖𝑎∆𝑖𝑏 + ∆𝑖𝑏2𝑐𝑜𝑠(𝜃𝑒 + 𝜑) (2.32)

∆𝑖𝑞 =2

3[∆𝑖𝑎𝑐𝑜𝑠𝜃𝑒 + ∆𝑖𝑏𝑐𝑜𝑠 (𝜃𝑒 −

2𝜋

3) + (−∆𝑖𝑎 − ∆𝑖𝑏)𝑐𝑜𝑠 (𝜃𝑒 +

2𝜋

3)]

=2

√3√∆𝑖𝑎

2 + ∆𝑖𝑎∆𝑖𝑏 + ∆𝑖𝑏2𝑠𝑖𝑛(𝜃𝑒 + 𝜑) (2.33)

where 𝜑 = tan−1 (√3∆𝑖𝑎

∆𝑖𝑎+2∆𝑖𝑏)

The effect of ∆𝑖𝑑 is negligible compared to ∆𝑖𝑞 which directly affects the output torque.

From equation 2.18,

𝑇𝑒𝑚 = 𝑘𝑇𝑖𝑞

= 𝑘𝑇(𝑖𝑞,𝑚𝑒𝑎𝑠 − ∆𝑖𝑞)

= 𝑘𝑇(𝑖𝑞∗ − ∆𝑖𝑞)

= 𝑇𝑟𝑒𝑓 − 𝑇∆𝑖,𝑜𝑠 (2.34)

where 𝑇∆𝑖,𝑜𝑠 is the torque ripple due to current offset error and

𝑇∆𝑖,𝑜𝑠 = 𝑘𝑇2

√3√∆𝑖𝑎

2 + ∆𝑖𝑎∆𝑖𝑏 + ∆𝑖𝑏2𝑠𝑖𝑛(𝜃𝑒 + 𝜑) (2.35)

It can be seen from equation 2.35 that offset errors in current measurements give rise to

torque ripples at the fundamental frequency.

There are a limited number of papers in literature about removing current offset errors for

PMSM control. Qian [13] used iterative learning control to remove this periodic torque

harmonic. However, an accurate torque estimator is needed before compensation is

possible. The torque estimator will require an accurate model of the plant in order to give

satisfiable torque estimation [13]. Heins [32] used a pulsating torque decoupling technique

to determine the values of the offsets and compensate for them separately. However, a

torque transducer is used to give accurate torque measurement. The offline method is to

turn off the current supply and measure the output current separately. The offset currents

can thus be determined with this method [32]. This is a onetime pre-compensation and the

current offsets are from then onwards assumed to be constant. However, this might not be

the case and any changes in the offsets are not compensated for using this method. The

method used by Qian can be considered a collective approach whereby torque ripple is

compensated regardless of their sources. Heins used a distributive approach whereby

cogging torque and current measurement errors were compensated separately.

Current Scaling Error

Scaling is necessary to convert the output of the sensor, such as a shunt or a Hall effect

sensor, from volt [V] to current [A]. Inevitably, scaling error will be introduced by these

conversions [27]. Furthermore, a digital system will result in additional scaling errors. This is

particularly the case if a Digital Signal Processor (DSP) which can only handle integers is

applied, as was the case for this research.

If the current signals are well within the bandwidth of the current control system, it can be

assumed that the measured phase currents follow the command phase current 𝑖∗:

𝑖𝑎,𝑚𝑒𝑎𝑠 = 𝑖𝑎∗ = 𝜖𝑎𝑖𝑎 = 𝜖𝑎𝑖𝑐𝑜𝑠(𝜃𝑒) (2.36)

𝑖𝑏,𝑚𝑒𝑎𝑠 = 𝑖𝑏∗ = 𝜖𝑏𝑖𝑏 = 𝜖𝑏𝑖𝑐𝑜𝑠 (𝜃𝑒 −

2𝜋

3) (2.37)

where 𝜖𝑎 and 𝜖𝑏 are the scaling errors for phase a and b respectively.

Similarly,

∆𝑖𝑞 = 𝑖𝑞,𝑚𝑒𝑎𝑠 − 𝑖𝑞

=2

3[𝑖𝑎

∗𝑐𝑜𝑠𝜃𝑒 + 𝑖𝑏∗𝑐𝑜𝑠 (𝜃𝑒 −

2𝜋

3) + (−𝑖𝑎

∗ − 𝑖𝑏∗)𝑐𝑜𝑠 (𝜃𝑒 +

2𝜋

3)]

−2

3[𝑖𝑎∗

𝜖𝑎𝑐𝑜𝑠𝜃𝑒 +

𝑖𝑏∗

𝜖𝑏𝑐𝑜𝑠 (𝜃𝑒 −

2𝜋

3) + (−

𝑖𝑎∗

𝜖𝑎−

𝑖𝑏∗

𝜖𝑏) 𝑐𝑜𝑠 (𝜃𝑒 +

2𝜋

3)]

=2

√3[𝑖𝑎

∗ (1 −1

𝜖𝑎) 𝑠𝑖𝑛 (𝜃𝑒 +

𝜋

3) + 𝑖𝑏

∗ (1 −1

𝜖𝑏) 𝑠𝑖𝑛𝜃𝑒]

= 𝐼 [𝜖𝑎−𝜖𝑏

√3𝜖𝑎𝜖𝑏𝑠𝑖𝑛 (2𝜃𝑒 +

𝜋

3) −

𝜖𝑎+𝜖𝑏

2𝜖𝑎𝜖𝑏+ 1] (2.38)

Therefore, the resulting torque ripple due to scaling error from equation 2.18 and 2.38 is

𝑇∆𝑖,𝑠𝑐 = 𝑘𝑇𝐼 [𝜖𝑎−𝜖𝑏

√3𝜖𝑎𝜖𝑏𝑠𝑖𝑛 (2𝜃𝑒 +

𝜋

3) −

𝜖𝑎+𝜖𝑏

2𝜖𝑎𝜖𝑏+ 1] (2.39)

where 𝑇∆𝑖,𝑠𝑐 is the torque ripple due to scaling error. Thus, the scaling errors in current

measurement give rise to a torque ripple at twice the fundamental frequency.

There are a limited number of papers in literature about removing current scaling errors.

Current scaling error can also be removed using the same approach to remove current

offset errors as suggested by Qian and Heins [13, 32].

The resulting torque ripple resulting from both current offset and scaling errors is

represented by 𝑇∆𝑖 where

𝑇∆𝑖 = 𝑇∆𝑖,𝑜𝑠 + 𝑇∆𝑖,𝑠𝑐 (2.40)

Encoder inaccuracy and the misplacement of encoder on the shaft will also have an impact

on the output torque resulting in additional torque ripple. A calibration process can be

carried out to negate their corresponding effects on the output torque [81]. These two

factors are unlikely to vary and once calibrated, their effects on torque ripple will be

minimal. Unlike cogging torque and flux density distribution which varies with temperature,

current offset error may drift with time and current scaling error also vary with reference

torque.

2.4.3 Total Torque Ripple in a PMSM

The above-mentioned factors can lead to the production of torque ripple. The amount of

torque ripple produced depends on the method of control, the types of sensors used as

well as the motor design. Research in PMSM [6-11] has shown torque ripple between the

range of 2% to 4% of rated torque. Figure 2-8 shows the sources of torque ripple (as shaded)

mentioned in earlier sections for a PMSM system (load torque is assumed to be zero).

Figure 2-8: PMSM Control with Torque Ripple

The non-ideal sinusoidal flux density distribution ∆𝜆𝑠 results in torque ripple in the output

torque and can be represented by 𝑇∆𝜆. The measured currents due to measurement errors

result in torque ripple in the output torque and can be represented by 𝑇∆𝑖. 𝑇𝑐𝑜𝑔 is produced

due to the interaction between the rotor magnetic flux and the angular variations in the

stator magnetic reluctance. The total torque ripple due to these three factors is thus,

TTTT icogrip (2.41)

Assuming no load torque, the output torque from equation 2.16 is now:

ripem TTT (2.42)

Thus, a good control method should be able to minimise these factors to obtain the desired

smooth output from the motor system.

The current controllers mentioned in section 2.3 are not capable of minimising torque

ripple caused by cogging torque, non-ideal sinusoidal BEMF, current scaling and offset error

as they occur outside the current loop. Current scaling and offset errors on the other hand

bring inaccurate measurements into the current controllers.

∆𝜆𝑠

+ _

+ _ 𝑣𝑠∗

𝜃𝑒

𝑖𝑠 C

C

𝑖𝑞∗

𝑖𝑑∗

Tref

1

𝑘𝑇 1

𝐿𝑠𝑠 + 𝑅𝑠

1

𝐽𝑠 + 𝑏

1

𝑠 ∙

dq

abc

+ _ + + T

𝑇𝑐𝑜𝑔

𝑇𝑒𝑚

abc

dq Sensor

s

𝜃𝑚 𝜔𝑚

X 𝑝

𝜆𝒔 +

𝑖𝑑,𝑚𝑒𝑎𝑠 𝑖𝑞,𝑚𝑒𝑎𝑠

𝑖𝑠,𝑚𝑒𝑎𝑠

2.5 Discussion

This chapter has covered the various causes of torque ripple for a PMSM and how PMSM

can be controlled using field oriented control. FOC is widely used in PMSM due to the ease

of controlling a PMSM. Hysteresis current controllers were used in the 1980s to 1990s,

however recent researchers used PI controllers instead.

There are many different types of PMSM such as interior PMSMs or surface mounted

PMSMs. Both types have their advantages and disadvantages and characteristics suitable

for different applications. The winding layout can also affect the motor performance. If

windings are distributed over multiple slots, there are more copper losses and thus

efficiency is reduced [82, 83]. In comparison, fractional pitch machines with concentrated

windings have lesser copper losses and thus better efficiency [83].

The control scheme in the rest of this thesis now focusses on the torque control system,

assuming that the current control loop is used within its specified operation region. An

additional speed loop is not considered for this thesis as the main aim of this research is to

minimise torque ripple in the output torque without the use of a torque transducer.

The next chapter discusses Iterative Learning Control (ILC) methodologies. ILC has shown to

be effective in reducing disturbances that are periodic in nature, discussed in section 2.4,

and may therefore be suitable to minimise torque ripples of PMSMs.

Chapter 3 ILC for Smooth Operation of PMSMs

Chapter 2 covered the modelling of PMSMs as well as the contributing factors to torque

ripple for PMSMs. This chapter will discuss feedforward control, pre-compensation

techniques, the various ILC methods and how ILC are used for PMSM control.

There are many control theories to improve the dynamic system response but to achieve

the desired response may not always be possible. This can be due to unmodelled system

dynamics or parameter changes that can happen during the system operation [84]. ILC can

possibly overcome the limitations of these methods especially for a system that operates in

a repetitive manner. This can be speed, position or torque control for a PMSM system. In

this case, ILC is able to achieve good tracking even when there are uncertainties in the

model or when the system structure is unknown [37]. However, learning controllers such as

ILC may have a problem with maintaining stability for a long period whereby the error

reduces for a number of iterations before it starts to increase again [85].

3.1 Feedforward Control

Feedforward control has the ability to achieve operation without torque ripple if a perfect

model of the system were available. Feedforward is different from a feedback control in

that a pre-determined control signal is generated without any output response from the

plant. Pre-knowledge about the plant and any disturbances are needed for this type of

control.

Figure 3-1 shows how feedforward control injects the control signals (usually based on an

approximation of the inverse plant model) and is not influenced by the feedback loop. The

FF block is designed to be 𝐺−1 and the output will then be the same as the reference.

Feedforward control is normally used together with a feedback control. The idea is to

minimise the control signal from the feedback loop by using the feedforward controller.

However, if there are inaccuracies in the inverse model, the feedback loop will be able to

minimise the resulting error.

Figure 3-1: Feedforward-Feedback Control

The control input, 𝑢 to the plant is thus:

𝑢 = 𝑢𝑓𝑓 + 𝑢𝑓𝑏

= 𝑟(𝐻𝑓𝑓) + (𝑟 − 𝑦)𝐻𝑓𝑏

= 𝑟(𝐻𝑓𝑓+𝐻𝑓𝑏) − 𝑦(𝐻𝑓𝑏) (3.1)

Where 𝐻𝑓𝑓 and 𝐻𝑓𝑏 are the transfer functions of the feedforward and feedback systems

respectively, 𝑟 is the reference signal and 𝑦 is the output.

If feedforward control is implemented with FOC, from equation 2.13,

𝑣𝑑 = 𝑅𝑠𝑖𝑑 + 𝐿𝑑𝑑

𝑑𝑡𝑖𝑑 − 𝜆𝑞𝜔𝑒 (3.2)

Since 𝑖𝑑 is controlled to be zero and assuming perfect current tracking,

𝑣𝑑 = −𝜆𝑞𝜔𝑒 (3.3)

G + _ FB 𝑢 𝑟 𝑦 + +

FF 𝑢𝑓𝑓

𝑢𝑓𝑏

From equation 2.6,

𝑣𝑑 = −𝜆𝑞𝑝𝜔𝑚 (3.4)

From equation 2.2, assuming load torque to be zero,

𝑣𝑑 = −𝜆𝑞𝑝 (𝑇

𝐽𝑠+𝑏) (3.5)

In ideal system whereby 𝑇 = 𝑇𝑟𝑒𝑓 , the reference torque, the feedforward voltage

command for the d-axis is

𝑣𝑑∗ = (

−𝜆𝑞𝑝

(𝐽𝑠+𝑏))𝑇𝑟𝑒𝑓 (3.6)

Similarly from equation 2.14,

𝑣𝑞 = 𝑅𝑠𝑖𝑞 + 𝐿𝑞𝑑

𝑑𝑡𝑖𝑞 + 𝜆𝑑𝜔𝑒 (3.7)

From equation 2.6 and 2.18,

𝑣𝑞 = 𝑅𝑠 (𝑇𝑟𝑒𝑓

𝑘𝑇) + 𝐿𝑞

𝑑

𝑑𝑡(𝑇𝑟𝑒𝑓

𝑘𝑇) + 𝜆𝑑𝑝𝜔𝑚 (3.8)

From equation 2.2, assuming load torque to be zero,

𝑣𝑞 = (𝑅𝑠

𝑘𝑇)𝑇𝑟𝑒𝑓 + (

𝐿𝑞

𝑘𝑇)

𝑑

𝑑𝑡𝑇𝑟𝑒𝑓 + 𝜆𝑑𝑝 (

𝑇

𝐽𝑠+𝑏) (3.9)

In ideal system whereby 𝑇 = 𝑇𝑟𝑒𝑓, the feedforward voltage command for the q-axis is

𝑣𝑞∗ = (

𝑅𝑠

𝑘𝑇+

𝜆𝑑𝑝

(𝐽𝑠+𝑏))𝑇𝑟𝑒𝑓 + (

𝐿𝑞

𝑘𝑇)

𝑑

𝑑𝑡𝑇𝑟𝑒𝑓 (3.10)

𝜆𝑞 , 𝜆𝑑 , 𝐽, 𝑏, 𝑝, 𝑅𝑠, 𝑘𝑇 and 𝐿𝑞 are motor parameters that can be pre-determined. Thus, the

commands 𝑣𝑑∗ and 𝑣𝑞

∗ can be used as feedforward inputs to the PMSM. However this

feedforward method is unable to minimise torque ripple caused by cogging torque, current

measurement errors and non-ideal sinusoidal flux density distribution as equation 3.6 and

3.10 do not take them into account. However, this can be dealt with using pre-

compensation techniques as discussed in the next section.

3.2 Pre-Compensation Techniques

The contributing factors to the torque ripple of a PMSM as shown in Figure 2-8 can be pre-

compensated. This then allows torque ripple caused by cogging torque, current

measurement errors and non-ideal sinusoidal flux density distribution to be removed from

the output torque. There are two ways to implement pre-compensation with the aim of

minimising torque ripple in the output torque. The first method is the direct or collective

approach whereby torque ripple is being compensated as a whole, disregarding how it

came about. The second way is the indirect or distributive approach whereby torque ripple

is being compensated separately according to their contributing factors – cogging torque,

non-ideal sinusoidal flux density distribution, current scaling and offset errors.

3.2.1 Direct Pre-Compensation Technique

In the direct pre-compensation technique, first the torque has to be measured or estimated

using any information that can correlate to the output torque. These can come from the

speed information, sound or vibrations. Next, the torque ripple can be found and

subtracted from the reference torque. The command current from equation 2.19 for the q-

axis is now:

𝑖𝑞∗ =

1

𝑘𝑇(𝑇𝑟𝑒𝑓 − �̂�𝑟𝑖𝑝) (3.11)

where �̂�𝑟𝑖𝑝 is the estimated torque ripple.

This can be considered a collective method whereby the effects of the individual factors on

the output torque are not considered. The main focus is solely on the final output and

trying to minimise it to achieve the desired reference signal. Figure 3-2 shows how the

estimated torque ripple compensation �̂�𝑟𝑖𝑝 can be implemented using a position driven

lookup table (LUT).

Figure 3-2: Direct Pre-Compensation Technique

This method assumes all causes of torque ripple are accounted for and are known. Any

inaccuracies or unaccounted factors may result in a worse outcome i.e. higher torque ripple.

Otherwise, this is a simple and effective pre-compensation method to minimise torque

ripple. Simulation results can be found in section 4.3 4.3 Using Pre-compensation

Techniquesand experimental results in section 6.2.2.

3.2.2 Indirect Pre-compensation Technique

The second approach is to identify the individual factors contributing to torque ripple and

eliminate them separately. This can be considered a distributive method whereby each

factor has to be minimised or eliminated before error minimisation can be achieved. If

cogging torque (𝑇𝑐𝑜𝑔) and the torque ripple due to non-ideal sinusoidal density flux

distribution (𝑇∆𝜆) compensation can be pre-determined using offline measurement, they

can be removed from the system using this technique. The current offset (∆𝑖) and scaling

errors (𝜀) of the measured current can also be pre-determined and corrected before the


𝑘𝑇

𝑖𝑞∗

𝑖𝑑∗

𝑣𝑠∗

𝜃𝑒

𝑖𝑠

FOC PMSM + _

�̂�𝑟𝑖𝑝 LUT

𝜃𝑚

Park transformation. Figure 3-3 shows the bock diagram for indirect pre-compensation

control scheme using two position driven lookup tables (LUTs).

Figure 3-3: Indirect Pre-Compensation Technique

The pre-compensation for cogging torque and torque ripple due to non-ideal sinusoidal flux

density distributions can be implemented using position driven lookup tables. The

amplitude of 𝑇∆𝜆 depends on the 𝑇𝑟𝑒𝑓 and will be discussed further in section 5.2.1.

Similarly, this method of control assumes all the three factors are accurately measured and

compensated. The shape of the cogging torque, flux density distribution and current

measurement errors have to be determined beforehand. Any inaccuracies will also result in

torque ripple in the output torque. This method can also be easily implemented to remove

known torque ripple.

Both approaches have their advantages and disadvantages. The indirect approach allows

changes to be made only to either one of the three compensated factors and enables one

to understand how the factors may change under different conditions. This however

requires more effort than the direct method if all three parameters have to be changed.

The disadvantage to both methods is the reliance on the accuracies of the pre-

compensation compensation.


𝑘𝑇

𝑖𝑞∗

𝑖𝑑∗

𝑣𝑠∗

𝜃𝑒

FOC PMSM + _

𝑇𝑐𝑜𝑔

_

𝑇∆𝜆

+ _ 𝑖𝑠𝑐

∆𝑖

1

𝜀

LUT

𝑖𝑠

𝜃𝑚

LUT

𝜃𝑚

Heins decouples the torque ripple and provides a pre-compensation for each identified

cause of the torque ripple [32]. This paper assumed the main causes of torque ripple

comes from cogging torque and current measurement errors. Offline measurement was

first carried out to identify cogging torque and current measurement errors before they are

compensated in the control scheme [32].

3.3 ILC Schemes

ILC has been used in many areas, ranging from robotics, batch processing, rotary systems,

actuators, power electronics, etc [37]. Arimoto [86] was considered the first person who

proposed this control method for improving the control input based on previous

information for systems that are repetitive in nature.

The aim of ILC is to find a term in a repetitive manner such that when j ,

𝑦𝑗(𝑡) → 𝑦𝑑(𝑡) (3.12)

where j is the number of iterations, 𝑦𝑗 is the output signal and 𝑦𝑑 is the desired signal.

Thus, ILC repetitively finds an input signal until the desired output is achieved. Consider a

continuous time, non-linear dynamic systems with the state and output equations below:

�̇�𝑗(𝑡) = 𝑓(𝑥𝑗(𝑡), 𝑡) + 𝐵(𝑥𝑗(𝑡), 𝑡)𝑢𝑗(𝑡) (3.13)

𝑦𝑗(𝑡) = 𝑔(𝑥𝑗(𝑡), 𝑡) (3.14)

where j is the number of cycle, 𝑥(𝑡) ∈ 𝑅𝑛 is the state vector, 𝑦(𝑡) ∈ 𝑅𝑝 is the output

vector and 𝑢(𝑡) ∈ 𝑅𝑟 is the input vector. The matrix functions f, g and B are known to have

certain properties. Thus, for an output of 𝑦𝑑(𝑡), the goal is to find a signal 𝑢𝑑(𝑡) in a

repeated number of cycles such that as 𝑗 → ∞,

)()( tutu dj

and so

𝑦𝑗(𝑡) → 𝑦𝑑(𝑡)

The control signal 𝑢(𝑡) is updated based on the summation of the previous input and the

actions taken:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝐿(. , 𝑒𝑗−1(𝑡)) (3.15)

where 𝐿(. ) is any chosen function and 𝑒𝑗−1 = 𝑦𝑑(𝑡) − 𝑦𝑗−1(𝑡). Figure 3-4 shows the basic

configuration for ILC.

Figure 3-4: Basic ILC Configuration

ILC schemes have the following assumptions [37]:

Each iteration has a fixed duration.

The system always starts from the same initial condition in each iteration.

Invariance of the system dynamics is maintained throughout the process.

The output 𝑦𝑗(𝑡) can be measured in a deterministic way.

The dynamics of the system are deterministic.

Considering the setup from Figure 3.4,

𝑌𝑗 = 𝑃𝑈𝑗

System

Memory

𝑦𝑗

𝑢𝑗

ILC 𝑦𝑑

𝑢𝑗−1 _ +

Memory

𝑒𝑗−1

𝑦𝑗−1

𝐸𝑗 = 𝑌𝑑 − 𝑌𝑗

𝑈𝑗 = 𝑈𝑗−1 + 𝐿𝐸𝑗−1 (3.16)

where 𝑃 denotes the transfer function for the system and 𝐿 is the transfer function of the

ILC scheme. The convergence condition can be derived as follows:

𝐸𝑗+1 = 𝑌𝑑 − 𝑌𝑗+1

= 𝑌𝑑 − 𝑃𝑈𝑗+1

= 𝑌𝑑 − 𝑃(𝑈𝑗 + 𝐿𝐸𝑗)

= 𝐸𝑗 − 𝐿𝑃𝐸𝑗

= (1 − 𝐿𝑃)𝐸𝑗

𝐸𝑗+1

𝐸𝑗= 1 − 𝐿𝑃

‖𝐸𝑗+1

𝐸𝑗‖ = ‖1 − 𝐿𝑃‖ < 1 (3.17)

Therefore, with the above condition in which 𝐿 must fulfil for a given transfer function 𝑃,

when 𝑗 → ∞, the error will go to 0.

A survey of ILC schemes was done by Ahn and et al. [37] in which ILC schemes were being

put into ten categories that include general structure, general update rules, typical ILC

problems, etc. There are further sub-categories within the ten categories as the authors

tried to organise the different variations of ILC and how it can be used. To systematically

present how ILC can be categorised, the following will be used to identify the different ILC

schemes for this research.

Categories of ILC

1. Single Channel First Order ILC (SCFO-ILC)

2. Multi-Channel ILC (MC-ILC)

3. Higher Orders ILC (HO-ILC)

4. Adaptive ILC

SCFO-ILC is the simplest type of ILC schemes and many different updating rules can be

applied to this scheme. These are the rules to determine how iterative learning is being

carried out for each iteration. Different updating rules will result in different convergence

rates, stability and learnable bands. Convergence rates determine how fast it takes for the

ILC schemes to reach the desired outcome. Generally, the faster the convergence rate, the

more desirable it is. However, the trade-off to fast convergence is the stability of the

system. The stability of the system is important because if the learning occurs too quickly,

there is a possibility for the system to become unstable. This is probably due to the control

system trying the compensate the noise instead of the internal system parameters [87]. The

learnable band is defined as the frequency range within which the convergence conditions

hold [88]. Outside this learnable band, the ILC schemes may not be able to control the

system to achieve the desired outcome. There are different updating rules for ILC such as

the P-type ILC (P-ILC), D-type ILC (D-ILC), PD-type ILC (PD-ILC), PI-type ILC (PI-ILC) and PID-

type ILC (PID-ILC) [36, 37, 44, 45, 48, 89-92]. P-ILC or its variations are the most commonly

used ILC schemes for PMSM [13, 46, 48-52]. This will be covered in detail in section 3.3.1

for the different updating rules.

MC-ILC uses multiple channels in the updating process. For example, in a single channel P-

ILC, only a single learning gain is used to compensate for all frequencies whereas in multi-

channel P-ILC, multiple learning gains are used instead. A learning gain is used for each

channel and these channels work together simultaneously and provide different ILC

schemes for different frequencies. A Zero Phase Filter (ZPF) or discrete Fourier

transform/inverse discrete Fourier transform pair can be used to separate the different

channels. This allows variation to the learning gains for each channel to achieve better

tracking performance than single channel ILC [93].

First order ILC only uses information from the previous cycle. Higher Order ILC on the other

hand is able to use information from n previous cycles. Where n > 1, it is known as HO-ILC

schemes [37, 39, 94]. It was demonstrated that the higher order ILC schemes can possibly

have faster convergence compared to first order ILC. In particular, good performance can

also be achieved in the presence of disturbances [39]. Simulations have been done to show

that higher order ILC was able to reduce the effects of noise [95].

Adaptive ILC uses a variable learning gain for ILC. For non-adaptive ILC, the learning gain is

fixed and usually chosen to be small to ensure stability. If a high gain is chosen, it will result

in a small stability margin and can cause severe impairment to the robustness of the system

to uncertainty [87]. However, to achieve fast convergence the learning gain should be

relatively high. This becomes a design trade-off between convergence and stability.

Adaptive ILC is able to overcome this problem whereby the learning gain can vary so as to

achieve fast convergence and maintain stability. The learning gain can be high initially when

the error is high and can gradually decrease when the error reduces [37, 51, 96-100]. The

learning gains can also be found using fuzzy or neural network control methods [84, 101,

102].

3.3.1 Single Channel First Order ILC

Figure 3-4 shows the basic configuration of ILC whereby the control signal is updated based

on the previous control signal and the error between the desired and actual output. In

SCFO-ILC, the learning only takes place within one frequency range. The width of the

learnable frequency range will depend on the value of the learning gain and the mechanics

of the system.

P-type ILC

P-ILC uses a portion of the previous error and updates its input for the next cycle. The

learning gain will determine the rate of convergence and the stability of the system. In

general, the error will be compensated over time and the ideal control signal is generated

to achieve the desired output. The general form of P-ILC is:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝐿(∙)(𝑦𝑑(𝑡) − 𝑦𝑗−1(𝑡)) (3.18)

When the input signal 𝑢𝑗(𝑡) is in action, the results that it produced cannot be seen at the

same time as that can only happen later. Thus, the learning law of a P type ILC does not

have the ability to predict the direction of error. However, P-ILC can be easily implemented

as only the measurements of state variables are needed. It does not require derivative

signals which can be very noisy [103].

A simplified form of P-ILC is:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝑘𝑝(𝑦𝑑(𝑡) − 𝑦𝑗−1(𝑡)) (3.19)

where 𝑘𝑝 is a scalar number chosen as the learning gain. Figure 3-5 shows the block

diagram for P-ILC.

Figure 3-5: P-ILC

+ +

M 𝑢𝑗−1

𝑘𝑝 𝑦𝑑(𝑡) − 𝑦𝑗−1(𝑡) 𝑢𝑗

The block M in the figure above is the memory block to store the signal from the previous

cycle. The convergence condition as derived in Equation 3.17 can be expressed as:

|1 − 𝑘𝑝𝐺𝑝(𝑗𝜔)| < 1 (3.20)

where 𝐺𝑝 is the closed loop transfer function of the system.

The test motor described in chapter 5, using the values of the variables – kT, J, b, L and R,

with stator voltages as inputs and torque as output had been approximated to a first order

system. This assumes that the BEMF is sinusoidal and cogging torque is not present. The

transfer function of the closed loop system is then derived based on the parameters of the

PI current controller (refer to section 5.3). The whole setup can be approximated to a

second order transfer function with

𝐺𝑝 =0.8249

7.72×10−8𝑠2+2.78×10−4𝑠+0.8249.

The bode plot of 𝐺𝑝 can be seen in Figure 3-6. This is the approximate closed loop second

order transfer function of the experimental setup.

Figure 3-6: Bode Plot of a Second Order System

-60

-40

-20

0

20

Magnitude (

dB

)

102

103

104

105

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/s)

Figure 3-7 shows the convergence condition for a range of 𝑘𝑝 values on a closed loop

system with the transfer. From equation 3.20, the system is able to converge when the

curves are between the values of 0 to 1.

Figure 3-7: Convergence for P-ILC

The corresponding frequency axis will therefore identify the range of frequency that the ILC

schemes can operate and learn. Therefore, the learnable band for P-ILC is from 0 to 366 Hz

when 𝑘𝑝 = 1.0 and from 0 to 494 Hz when 𝑘𝑝 = 0.2. Table 3.1 shows the learnable band

for P-ILC for different values of 𝑘𝑝.

Table 3.1: Learnable Band for P-ILC

𝑘𝑝 Learnable Band (Hz)

0.2 0 – 494

0.4 0 – 467

0.6 0 – 436

0.8 0 – 404

1.0 0 – 366

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

Frequency (Hz)

Converg

ence C

onditio

n

Convergence condition for a range of kp values

kp=0.2

kp=0.4

kp=0.6

kp=0.8

kp=1.0

The choice of the learning gain determines the speed of convergence as the learning

process becomes quicker when |1 − 𝑘𝑝𝐺𝑝(𝑗𝜔)| becomes smaller. The error tends to go to

zero when |1 − 𝑘𝑝𝐺𝑝(𝑗𝜔)| tends to zero [104]. However, the learnable band becomes

smaller when the learning gain increases.

P-type ILC with forgetting factor (Pf-ILC)

The P-ILC is not robust to uncertainties in the system or measurement errors. A forgetting

factor can be introduced in the P-ILC to improve the robustness of the learning scheme

[105]. Figure 3-8 shows the block diagram of Pf-ILC with the forgetting factor, α. With the

forgetting factor, the equation 3.19 becomes

𝑢𝑗(𝑡) = (1−∝)𝑢𝑗−1(𝑡) + 𝑘𝑝(𝑒𝑗−1(𝑡)) (3.21)

Figure 3-8: Pf-ILC

A recommended range of 0.01 to 0.05 is given as a suitable value of α as the error bound

can be 20 to 100 times higher. The tracking errors bounded are inversely proportional to

the forgetting factor i.e. the bigger the forgetting factor, the smaller are the tracking error

bound. Therefore, there is a conflict between the tracking error bound and the forgetting

factor [103, 104].

The convergence condition is now

|1 − 𝛼 − 𝑘𝑝𝐺𝑝(𝑗𝜔)| < 1 (3.22)

𝑒𝑗−1 𝑘𝑝 +

+

M

𝑢𝑗

𝑢𝑗−1

1 − 𝛼

Figure 3-9: Convergence for Pf-ILC

Figure 3-9 shows that with the addition of the forgetting factor, the system is now able to

converge for a wider learnable frequency range. Table 3.2 shows the learnable band for Pf-

ILC for different values of α. For the same value of pk = 0.4, the learnable frequency

increases from 467 Hz (α = 0) to 559 Hz (α = 0.1).

Table 3.2: Learnable Band for Pf-ILC (kp = 0.4)

𝛼 Learnable Band (Hz)

0 0 – 467

0.05 0 – 501

0.10 0 – 559

D-type ILC

The D-ILC form suggested by Arimoto is [106]:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝐿(∙)(�̇�𝑑(𝑡) − �̇�𝑗−1(𝑡)) (3.23)

0 100 200 300 400 500 600 700 800 900 10000.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Frequency (Hz)

Converg

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n

Convergence condition for a range of and kp = 0.4

= 0

= 0.05

= 0.10

In the absence of measurement noise and uncertainties, D-ILC is able to achieve zero

tracking errors [106]. However, it requires the derivation of the signal. Implementation of

D-ILC becomes harder compared to P-ILC as some derivatives may not be measurable or it

becomes noisy after numerical differentiation. For a PMSM with an encoder, differentiation

has to be carried out twice to get the acceleration signal needed to estimate the torque. As

a result, this estimated torque can be very noisy. Since the tracking errors bounds are

proportional to the noise, this reduces the effect of D-ILC in practice [107].

Similarly, a simplified form of the D-ILC can be represented as:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝑘𝑑�̇�𝑗−1(𝑡) (3.24)

where kd is the scalar learning gain of the D-ILC. Figure 3-10 shows the block diagram for D-

ILC.

Figure 3-10: D-ILC

The convergence condition is now:

|1 − 𝑘𝑑𝑗𝜔𝐺𝑝(𝑗𝜔)| < 1 (3.25)

Figure 3-11 shows that the convergence condition for a range of 𝑘𝑑 values are all above the

value of 1. This means that the example system does not converge.

𝑒𝑗−1 𝑘𝑑 + +

M

𝑢𝑗

𝑢𝑗−1

𝑑

𝑑𝑡

Figure 3-11: Convergence for D-ILC (no filter)

This is the case whereby the error is not filtered after differentiation. With the filter, the

convergence condition becomes:

|1 − 𝑘𝑑𝑗𝜔𝐻(𝑗𝑤)𝐺𝑝(𝑗𝜔)| < 1 (3.26)

where H is the transfer function of the filter.

To implement D-ILC, the error has to be differentiated. Since differentiation results in

amplification of noise, a filter is needed. The following figures show the convergence

conditions when a LPF is used with different cut-off frequencies.

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

12

14

16

18

Frequency (Hz)

Converg

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Convergence condition for a range of kd values - no filter

kd=0.002

kd=0.003

kd=0.004

kd=0.005

kd=0.006

Figure 3-12: Convergence for D-ILC (LPF - 25Hz)

It can be seen from figure 3.8 that there is a region of stability with the addition of the Low

Pass Filter (LPF). The learnable band ranges from 105 Hz (when 𝑘𝑑 = 0.006) to 121 Hz (when

𝑘𝑑 = 0.002).


Similarly, when a LFP with cutoff frequency at 50 Hz, the learnable band ranges from 110 Hz

(𝑘𝑑 = 0.006) to 160 Hz (𝑘𝑑 = 0.002).

0 100 200 300 400 500 600 700 800 900 1000

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Frequency (Hz)

Converg

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Convergence condition for a range of kd values - LPF (25Hz)

kd=0.002

kd=0.003

kd=0.004

kd=0.005

kd=0.006

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (Hz)

Converg

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kd=0.002

kd=0.003

kd=0.004

kd=0.005

kd=0.006


Likewise for a LPF of 100 Hz, the learnable frequency ranges from 44 Hz (𝑘𝑑 = 0.005) to 193

Hz (𝑘𝑑 = 0.006). Using the value of 𝑘𝑑 > 0.006, the graphs are greater than 1 for all

frequencies. This means that the system does not converge and there are no learnable

bands. It can be seen that the design of the filter plays an important role in ensuring the

convergence of the D-ILC system.

Table 3.3: Learnable Band for D-ILC (Hz)

Learnable Band (Hz) 𝑘𝑑

0.002 0.003 0.004 0.005 0.006

Cutoff

frequency

of LPF (Hz)

25 0 – 121 0 – 117 0 – 113 0 – 109 0 – 105

50 0 – 160 0 – 148 0 – 137 0 – 124 0 – 110

100 0 – 193 0 – 160 0 – 118 0 – 44 0

Table 3.3 shows that the lower the cutoff frequency, the narrower is the learnable band.

While a larger 𝑘𝑑 may imply faster learning, the learnable frequency decreases with

increasing 𝑘𝑑.

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

Converg

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kd=0.002

kd=0.003

kd=0.004

kd=0.005

kd=0.006

PD-type ILC

The PD-ILC is a combination of both P-ILC and D-ILC. It has been shown that PD-ILC is able

to achieve faster convergence than P-ILC [92]. PD-ILC can also be robust to initialisation

errors that are non-zero [38]. A combination of the traditional PID as a feedback controller

was used with PD-ILC as additional compensation to the control signal. This was used in the

control of a quadruped robot and the trunk attitude improved by three times [44].

From equation 3.6 and 3.11, the equation for PD-ILC can be expressed as:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝑘𝑝𝑒𝑗−1(𝑡) + 𝑘𝑑�̇�𝑗−1(𝑡) (3.27)

Figure 3-15: PD-ILC

and the convergence condition is:

|1 − (𝑘𝑝 + 𝑘𝑑𝑗𝜔)𝐺𝑝(𝑗𝜔)| < 1 (3.28)

The next two figures show the convergence condition for a range of 𝑘𝑝 and 𝑘𝑑 values.

𝑒𝑗−1 𝑘𝑑 +

+

M

𝑢𝑗

𝑢𝑗−1

𝑑

𝑑𝑡

+

𝑘𝑝

Figure 3-16: Convergence for PD-ILC (kd varies)

The learnable band is very narrow for 𝑘𝑑 values greater than of 0.004. There are two

learnable bands when 𝑘𝑑 = 0.003 (0 to 37 Hz and 85 to 223 Hz). For 𝑘𝑑 = 0.002, the

learnable band ranges from 0 to 248 Hz.

Figure 3-17: Convergence for PD-ILC (kp varies)

Similarly, from figure 3.13, when 𝑘𝑝 ≥ 0.6, the graphs are greater than 1 for all frequencies.

This means that there are no learnable bands for PD-ILC with 𝑘𝑝 ≥ 0.6 and 𝑘𝑑 = 0.003.

0 100 200 300 400 500 600 700 800 900 10000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

Converg

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Convergence condition for a range of kd values with LPF of 50Hz, k

p = 0.4

kd=0.002

kd=0.003

kd=0.004

kd=0.005

kd=0.006

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

Converg

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Frequency (Hz)

Convergence condition for a range of kp values, k

d = 0.003 with LPF of 50Hz

kp=0.2

kp=0.4

kp=0.6

kp=0.8

kp=1.0

There are two learnable bands when 𝑘𝑝 = 0.4 (0 to 37 Hz and 85 to 223 Hz). For 𝑘𝑝 = 0.2,

the learnable band ranges from 0 to 368 Hz.

The right combination of 𝑘𝑝 and 𝑘𝑑 values has to be chosen to achieve a fast and stable

response. Table 3.4 shows the learnable bands for different values of 𝑘𝑝 and 𝑘𝑑.

Table 3.4: Learnable Band for PD-ILC

Learnable

frequency

Range (Hz)

𝑘𝑑

0.002 0.003 0.004 0.005 0.006

𝑘𝑝

0.2 0 – 381 0 – 368 0 – 343 0 – 319 0 – 42

86 – 282

0.4 0 – 248 0 – 37

85 – 223

0 – 28 0 – 23 0 – 20

0.6 0 0 0 0 0

0.8 0 0 0 0 0

1.0 0 0 0 0 0

When PD-ILC is used, a high 𝑘𝑝 should not be used as it may go beyond the convergence

region and has no learnable band.

PI-type ILC

ILC can also use the integral of the error as the compensation. Authors in [91] has found

that the I-component can help in the convergence of ILC schemes and PI-type ILC can

perform better than P-ILC in terms of convergence speed. They have also shown that the I-

component is of little use if the number of time instants in an iteration is large. Therefore

the authors do not recommend the use of the I-term in ILC in practice since the time

instants is large in an iteration.

A method to find the optimisal values of the PID coefficients was proposed. Simulation

results were used to show the effectiveness of the proposed scheme and guaranteed

monotonic convergence to zero has been proven [89]. PID-type ILC scheme is able to

converge and was proven with linear operator theory. Simulations were shown to justify

the effectiveness of the proposed scheme as the convergence speed has increased with

optimal parameters [108]. Simulation results were used to show that PID-ILC has fast

convergence speed. However, no comparison has been made to other types of ILC schemes

[90]. P-component is the stabiliser in ILC scheme and it can bring about monotonic

convergence. The I-component increases the rate of convergence and the effect of non-

zero initial errors are minimised. The effect of disturbances in the inputs is minimised by

the D-component [42].

The equation for PI-ILC is [42]:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡) + 𝑘𝑝𝑒𝑗−1(𝑡) + 𝑘𝑖 ∫ 𝑒𝑗−1(𝑡)𝑡

𝑜𝑑𝑡 (3.29)

The convergence condition can be written as:

|1 − (𝑘𝑝 +𝑘𝑖

𝑗𝜔)𝐺𝑝(𝑗𝜔)| < 1 (3.30)

𝑒𝑗−1 +

M

𝑢𝑗

𝑢𝑗−1

𝑘𝑝

𝑘𝑖 ∫

Figure 3-18: PI-ILC

Figure 3-19 shows that the I-term has no impact on the learnable band as the convergence

condition is the same as the P-ILC.

Figure 3-19: Convergence for PI-ILC (ki varies)

As mentioned by Douglas, ILC learns from one iteration to another and is similar to the

integrator effect of the I-component. Therefore, it is not commonly used in the learning

process [40]. It can be seen that the additional I-component does not really have an impact

on the learnable band.

Other Updating Rules

There are many other updating rules, and it is not possible to include all of them here. A

selection of relevant update rules for this application can however be presented.

An additional current cycle error was used and results showed convergence can be

achieved faster compared to using only the previous cycle error [13, 109].

An A-type ILC that has the benefits of both P (simplicity) and D (predictive) type ILC was

proposed. The proposed scheme was able to converge in the presence of uncertainties and

noises [103].

0 100 200 300 400 500 600 700 800 900 10000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Frequency (Hz)

Converg

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Convergence condition for a range of ki values, k

p = 0.4

ki = 0

ki = 0.05

ki = 0.10

ki = 0.15

ki = 0.20

An optimised approach through minimising a quadratic criterion can also be incorporated in

ILC whereby the learning gain is found using this optimised approach [41, 97, 110].

Based on literature, it is difficult to determine which updating rules of the Single Channel

First Order ILCs should be considered. This depends on the conditions imposed on the

learning control, whether a wide learnable band is desired, how fast the convergence can

be in relation to stability and finally the robustness of the system. Chapter 4 compares the

simulated results of the different ILC schemes for a PMSM and chapter 6 discusses the

experimental results using the experimental setup.

3.3.2 Multi-Channel ILC

For the P-ILC schemes covered in section 3.3.1, the learning control has only one learning

gain or two in the case of PD-ILC. Another approach is to deal with each harmonic

component individually so that different learning gains are used for each harmonic in the

learning scheme. Thus, for a system with n harmonic components, there should be n

learning gains to compensate for all frequencies. A Discrete Fourier Transform (DFT) of the

previous input is necessary and Inverse Discrete Fourier Transform (IDFT) for the current

input signal. The ILC is therefore implemented in the frequency domain instead of the time

domain [93].

However, in Multi-Channel ILC (MC-ILC) several harmonic components can be lumped

together. Thus, less learning gains are needed than harmonic components. In addition, the

learning gains can still be updated in the time domain instead of the frequency domain [93].

MC-ILC uses multiple learning gains in parallel, as shown in Figure 3.20. The frequency band

of the i-channel is defined by the filter Fi(s) and ki is the scalar learning gain for the ith

channel.

The update of the input signal for MC-ILC still takes place in the time domain. The total

input update is thus the summation of all the updates from the channels. A Zero Phase

Filter (ZPF) approach can be implemented to produce the different channels that have to be

learned [93].

The overall learning law for MC-ILC is:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡)+(∑ 𝑘𝑖𝐹𝑖𝑛𝑖=1 )𝑒𝑗−1(𝑡) (3.31)

The convergence condition is [93]:

|1 − 𝐺𝑝(𝑗𝜔)∑ 𝑘𝑖𝐹𝑖𝑛𝑖=1 (𝑗𝜔)| < 1 (3.32)

For a 2 channel system, the equation can be simplified into:

𝑢𝑗(𝑡) = 𝑢𝑗−1(𝑡)+𝑘𝑝,𝑙𝑜𝑤𝑒𝑗−1,𝑙𝑜𝑤(𝑡)+𝑘𝑝,ℎ𝑖𝑔ℎ𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝑡) (3.33)

Where the subscript low refers to the lower frequency range and subscript high refers to

the higher frequency range. Figure 3-21 shows an example where a ZPF filter is used with

the cutoff frequency at the 50 Hz.

𝑒𝑗−1

𝐹1

𝐹𝑖

𝐹𝑛

𝜁1

𝜁𝑖

𝜁𝑛

.

.

.

𝑘𝑝,𝑖

M

𝑢𝑗

𝑢𝑗−1

𝑘𝑝,1

𝑘𝑝,𝑛

.

.

.

.

.

.

.

.

.

+

Figure 3-20: MC-ILC

Figure 3-21: Convergence condition for MC-ILC

Figure 3-21 shows a comparison between 2-channel P-ILC and a single channel P-ILC where

𝑘𝑝,𝑙𝑜𝑤 = 𝑘𝑝,ℎ𝑖𝑔ℎ = 𝑘𝑝 = 0.4. The learnable band for the 2-channel P-ILC is now 0 to 46 Hz

and 58 Hz to 211 Hz. This is only useful if there are no significant error harmonics between

47 Hz to 57 Hz. Compared to using a single channel P-ILC where the learning gain is 0.4 (0 to

467 Hz), it can be observed that the learnable band has reduced significantly. Although the

learnable band is smaller for multi-channel ILC, one advantage with MC-ILC is that different

learning gains can be used for the different channels.

Figure 3-22 shows different learning gains being used for the lower channel. As a result, the

channels will have different convergence speeds. This is particularly useful for a system

where there are many error harmonics in the lower channel and only a few in the higher

channel. In this case, a higher learning gain can be used in the lower channel and a lower

learning gain to be used in the higher frequency. This can also avoid the harmonics in the

higher frequency channel becoming unstable.

0 100 200 300 400 500 600 700 800 900 10000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Frequency (Hz)

Converg

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Convergence condition for MC-ILC and P-ILC

P-ILC

MC-ILC

Figure 3-22: Convergence Condition for MC-ILC

Table 3.5 shows the learnable band where the learning gain for the lower channel is varied

and learning gain for the higher channel remains constant. Similarly when a higher learning

gain for the lower channel is used, the learnable band becomes smaller.

Table 3.5: Learnable Bands for MC-ILC (2 channels)

kp,low Learnable Band (Hz)

0.4 0 – 46, 58 – 211

0.6 0 – 46, 63 – 207

0.8 0 – 43, 70 – 204

1.0 0 – 39, 77 - 201

3.3.5 Higher Orders ILC

Higher Order ILC (HO-ILC) schemes use information from n previous cycles where n > 1 [37,

39, 94]. It has been found that HO-ILC is robust and in the absence of measurement errors

is able to achieve zero error between the reference and output [94]. However, it has been

found that higher orders of ILC did not have much impact on performance and robustness

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (Hz)

Converg

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n

Convergence condition for a range of kp,low

values, kp,high

= 0.4

kp,low

= 0.4

kp,low

= 0.6

kp,low

= 0.8

kp,low

= 1.0

P-ILC, kp = 0.4

[43]. The learning rule for ILC can either be of first order or higher. The equation for HO-ILC

is:

𝑢𝑗(𝑡) = ∑ 𝜓𝑘𝑢𝑗−𝑘(𝑡)𝑀𝑘=1 + ∑ Φ𝑘 [

𝑒1,𝑗−𝑘(𝑡)

⋮𝑒𝑘,𝑗−𝑘(𝑡)

]𝑀𝑘=1 (3.34)

where k is the number of cycles, M is the order of the updating law, 𝜓 and Φ are the

learning gains. From the equation, it can be seen that information from previous cycles

such as the control inputs and output errors are used in computing the current control

input. This gives a higher degree of freedom in selecting the learning gains for each of the

cycles.

The convergence condition for HO-ILC is [39]:

∑ 𝜓𝑘𝑀𝑘=1 = 1 (3.35)

⌊∑ 𝜓𝑘 − Φ𝑘𝐺𝑝(𝑗𝜔)𝑀𝑘=1 ⌋ < 1 (3.36)

For a second order ILC, the learning rule becomes:

𝑢𝑗(𝑡) = 𝜓1𝑢𝑗−1(𝑡) + Φ1𝑒𝑗−1(𝑡) + 𝜓2𝑢𝑗−2(𝑡) + Φ2𝑒𝑗−2(𝑡) (3.37)

where Φ1 and Φ2 are learning gains for the first and second order respectively.

Figure 3-23: HO-ILC

𝑒𝑗−1 Φ1 +

M

𝑢𝑗

𝑢𝑗−1

M

𝑒𝑗−2 Φ2 +

+ M

𝑢𝑗−2

𝜓1

𝜓2

The convergence conditions become:

𝜓1 + 𝜓2 = 1 (3.38)

|𝜓1 − Φ1𝐺𝑝(𝑗𝜔)| + |𝜓2 − Φ2𝐺𝑝(𝑗𝜔)| < 1 (3.39)

Figure 3-24: Convergence Condition for HO-ILC

Figure 3-24 shows the comparison between HO-ILC and P-ILC where 𝜓1 = 𝜓2 = 0.5,

Φ1 = Φ1 = 𝑘𝑝 = 0.4. The learnable band for 2nd Order ILC is 0 to 405 Hz compared to 0 to

467 Hz for single order P-ILC. As the order increase from the first to second, the learnable

band decreases.

Table 3.6: Learnable Bands for HO-ILC

No. of Orders Learnable Band (Hz)

1 0 – 467

2 0 – 405

Tan tried to answer the questions as to whether higher order ILC is better than a lower

order ILC [111]. A comparison is made between first and second order ILC schemes. It was

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (Hz)

Converg

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Convergence condition for HO-ILC

1st Order ILC

2nd Order ILC

found that first order ILC has faster convergence speed compared to the second order. In

fact, it was deduced that first order ILC is the fastest compared to any higher order ILC

schemes for an infinite number of iterations. Higher order ILC may work better for finite

number of iterations [111]. However, it was shown that second order ILC may be able to

converge faster that first order ILC [39]. A comparative study on first and second order ILC

schemes was conducted and it was found that second order design is not better than first

order ILC in terms of performance or robustness. It was further pointed out that second

order ILC schemes needed double the amount of memory that first order ILC schemes. As

second order ILC makes use of error information from more than one iteration, it may be

able to smooth the behaviour of the system better than first order ILC. However, more

memory space is needed for this additional information. This can be a problem if the ILC

scheme has to be implemented on a DSP [43].

3.3.4 Adaptive ILC

The aim of adaptive ILC is to find a learning gain that is able to adjust itself based on the

magnitude of the error. The learning gain of an A-ILC should be large if the error is large and

becomes smaller when the error decreases. The ILC schemes discussed in previous sections

used a fixed step parameter as the learning gain. However, for faster convergence, it is

better to have an adaptive learning gain whereby the step increase for each iteration is

controlled [51].

The equation for adaptive P-type ILC can be expressed as:

𝑢𝑟,𝑗(𝑡) = 𝑢𝑟,𝑗−1(𝑡)+𝜇(𝑡)𝑒𝑗−1(𝑡) (3.40)

where 𝜇 is the adaptive P learning gain. Automatic tuning of learning gains of ILC was

discussed and Longman suggested ways in which they can be implemented in real time [96].

The first step is to stop the learning when the root mean square error reaches a threshold.

As noise comes into play when the error gets smaller, care has to be taken care that the

learning is not affected by noise. A ZPF is used for stabilisation. Learning only takes place

within the frequency range where the system is known. Beyond that, it is disregarded. The

second step is to adjust the learning gains based on the frequency and the error. The third

step is to use linear phase lead compensation where the cutoff frequency can be adjusted

and the final step is phase cancellation whereby FFTs are used in the learning process. In

this case, the variable learning gain, 𝜇 can be updated based on Model Reference Adaptive

System (MRAS). A recursive least square algorithm or any adaptive schemes whereby

convergence is satisfied based on Lyapunov function can be used [96]. Model reference

adaptive system can also be used with ILC for nonlinear system with uncertainties or with

unknown parameters [98]. A recursive least square algorithm to adjust the learning gain for

ILC schemes was used. Simulations were then used to demonstrate the effectiveness of the

proposed scheme [100]. The robust adaptive method in which convergence is satisfied

based on Lyapunov function has also been proposed [99]. A variable step-size scheme that

changes the learning gain to minimise the conflict between mean square error and

convergence speed was also proposed. A faster convergence speed and lower error was

obtained using the proposed method [51]. The learning gains of ILC schemes can also be

adjusted based on fuzzy control or neural network. Neural network based P-ILC was used

whereby the learning gain is varied according to the neural network [101]. Offline training is

first done to the neural network system using input-output sampled data. The robot

dynamic system is also estimated offline. Two fuzzy systems to compensate for any

uncertainties due to unknown nonlinearities have also been proposed [112].

Figure 3-25 shows an adaptive P-type ILC where the block A represents any of the adaptive

schemes mentioned earlier.

Figure 3-25: Adaptive P-ILC

The rate of convergence will depend on the adaptive scheme that is chosen.

3.3.5 Comparison of ILC schemes

Table 3.7 shows a comparison of the learnable bands between the ILC schemes for PMSM

systems. As a comparison, 𝑘𝑝 was chosen to be 0.4 for all learning schemes. On the whole,

P-ILC with forgetting factor has the widest range of learnable band whereas D-ILC has the

narrowest range. MC-ILC has learnable band with gaps thus making the overall learnable

band smaller than other ILC schemes.

Table 3.7: Comparing Learnable Bands of ILC Schemes

ILC Schemes Learnable Band (Hz)

SCFO-ILC

P 0 – 467

Pf 0 – 501

D 0 – 160

PD 0 – 248

PI 0 – 467

MC-ILC 2 Channels 0 – 46, 58 – 211

HO-ILC 2nd Order 0 – 405

Adaptive Varies

𝑒𝑗−1 + +

M

𝑢𝑗

𝑢𝑗−1

𝜇𝑗−1

A

The learnable band of adaptive ILC will vary with the learning gain. Generally, when the

learning gain increases, the learnable band decreases. This concludes the various categories

of ILC schemes and the next section will discuss the implementation of ILC schemes on a

PMSM.

3.4 ILC for PMSM

Section 3.3 discussed different types of ILC schemes. This section discusses how ILC is used

on a PMSM.

ILC has been used on PMSM systems due to the periodic nature of the torque ripple. The

ILC schemes considered in this section are the same as what have been discussed in section

3.3.1. SCFO-ILC is the most commonly used ILC schemes used for PMSM control [13, 46, 48-

50, 52]. The next 2 diagrams show how ILC can be implemented on PMSM control. It is

referred as the cascade ILC whereby ILC schemes are added to existing systems without

significant changes to the existing control system. This is to avoid additional cost that may

be incurred due to reconfiguration or replacement of the controller [104]. The learning

process of ILC may be hindered by noise or non-repeating disturbances. To minimise the

effect of these factors, a feedback controller is used in combination with ILC [40].

For both setups in Figure 3.26 and Figure 3.27, the output of the ILC block provides

additional compensation to the q-axis command current [104].

For a P-ILC, the equation can be written as:

∆𝑖𝑞,𝑗(𝑡) = ∆𝑖𝑞,𝑗−1(𝑡) + 𝑘𝑝(𝑒𝑗−1(𝑡)) (3.41)

where the error

𝑒(𝑡) = 𝜔𝑟𝑒𝑓(𝑡) − 𝜔(𝑡) (3.42)

Figure 3-26 shows how ILC can be used to achieve speed ripple minimisation [48].

Figure 3-26: ILC for Speed Ripple Minimisation

Output speed information can be derived from the derivation of 𝜃 which is measured using

an encoder. Figure 3-27 shows another setup on how ILC can be used to achieve torque

ripple minimisation [13].

Figure 3-27: ILC for Torque Ripple Minimisation

In this setup, the error

𝑒(𝑡) = 𝑇𝑟𝑒𝑓(𝑡) − 𝑇(𝑡) (3.43)

This scheme requires the output torque to be measured or estimated. In both cases, ILC is

used to provide an additional compensation to the command current of the q-axis.

Simulated results were used to show that the proposed ILC by Zheng and Qiao with a

passive filter works better than without the filter [52]. A P-type ILC was shown in the torque

loop to minimise torque ripple. However, it is not clear how much improvement was made

over the proposed method. A robust negative high gain ILC was proposed by Suja and

Amuthan for the current loop [47]. Only simulation results were shown with the P-type ILC

𝜔 𝜔𝑟𝑒𝑓 𝑖𝑞∗

𝑣𝑠∗

𝜃𝑒

𝑖𝑠

FOC PMSM _ + +

+ PI

ILC

𝑖𝑑∗ = 0


𝑘𝑇

𝑖𝑞∗

𝑣𝑠∗

𝑒

FOC PMSM +

+ 𝑖𝑑∗ = 0

_ +

ILC

where the researchers showed an improvement using ILC with the negative high gain

current controller compared to the conventional ILC with PI controller [47]. A P-type ILC

was used in conjunction with a PI Controller to minimise torque ripple. The previous cycle

of the reference Iq was used with the output of a PI controller to produce the current cycle

of the reference Iq. Compared to other methods, the proposed method did not need the

feedback of the output torque [49].

In real time applications of ILC schemes, the size of the memory is an important factor to

consider [104]. Filters are also used in many systems with ILC schemes due to measurement

noise or possible disturbances to the system. In general, most ILC schemes implemented for

control of PMSMs use the P-type ILC schemes. A forgetting factor is introduced in some of

the schemes to improve the robustness. Implementations of ILC can also been done in the

time domain, frequency domain and position domain.

3.4.1 Domain of Operation: Time, Frequency and Position

Other than the time domain, ILC has also been used in other domains for PMSM control. ILC

in the frequency domain with current cycle feedback was proposed where convergence can

be improved through this method [113]. ILC was used as a replacement for a PI torque

controller to produce the reference current Iq. Using this method, torque ripple was

reduced from 48% with a PI torque controller compared to only 2% when ILC controller was

used instead. It is not clear what type of ILC that was being used as only a generic ILC

scheme was shown [46]. The ILC scheme was implemented in both the time and frequency

domains. Both schemes are able to minimise torque ripple. Torque Ripple Factor (TRF) is

used in this research to justify the improvement of the proposed schemes. It is the ratio of

the peak to peak torque ripple to the rated torque. The TRF dropped from 14.7% to 3.9%

when the frequency domain scheme was used compared to 4.3% when the time domain

scheme was used. This is due to the forgetting factor used in the time domain to improve

the robustness against changes to system parameters and noise whereas forgetting factor

was not used in the frequency domain. Only the P-type ILC was used together with a

forgetting factor to increase robustness. [13].

In the frequency domain, a P-ILC can be represented as:

𝑢𝑗(𝐹𝑖) = 𝑢𝑗−1(𝐹𝑖) + 𝑘𝑝(𝑒𝑗−1(𝐹𝑖)) (3.44)

where 𝐹𝑖 refers to the ith frequency harmonics

Similarly a P-ILC in the position domain can be represented as:

𝑢𝑗(𝜃) = 𝑢𝑗−1(𝜃) + 𝑘𝑝(𝑒𝑗−1(𝜃)) (3.45)

where 𝜃 can refer to either the electrical position or the mechanical position of the rotor.

Time based ILC can only compensate torque ripple at only one frequency whereas

frequency based ILC needed multiple FFT in the control schemes if there are multiple

torque ripple harmonics that needed to be minimised. Since the torque ripples discussed in

section 2.4 are periodic and are the same for each position of the motor, a position based

ILC can be used to minimise these torque ripples. An angle based ILC method in contrast to

a time based ILC was proposed by Yuan and et al [50]. This is similar to implementing ILC in

the position domain. The results showed that the proposed angle-based ILC is better at

reducing torque ripple than a time based ILC at any speed but only the P-type ILC was

shown [50]. Position based ILC was also proposed by Qian and et al [13] but a

computational intensive Fourier series expansion was needed for the iterations.

The application of other types of ILC schemes for controlling a PMSM discussed in section

3.3 have been limited to date, which will be further investigated in this thesis.

In view of the limited research of the various types of ILC schemes for PMSM control, this

thesis seeks to achieve the following:

1. Compare different SCFO iterative learning schemes as discussed in section 3.1.1. To

the author’s knowledge, comparative analysis of different ILC schemes with regards

to their effectiveness to minimise torque ripple for PMSM has not been done or has

been limited to date.

2. Investigate the effectiveness of MC-ILC for PMSM control. To the author’s

knowledge, the practicality and usefulness of implementing MC-ILC in minimising

torque ripple for PMSM control has not been done or has been limited to date.

3. Investigate the effectiveness of HO-ILC. To the author’s knowledge, the practicality

and usefulness of implementing HO-ILC in minimising torque ripple for PMSM

control has not been done or has been limited to date.

4. Investigate the effectiveness of adaptive ILC. To the author’s knowledge, the

practicality and usefulness of implementing adaptive ILC in minimising torque ripple

for PMSM control is limited to date.

5. Compare and analyse the above four categories of ILC schemes in terms of their

learnable band, ability to minimise torque ripple, rate of convergence and

robustness to parameter changes.

6. Implement DSP based ILCs – to implement ILC on a DSP without the use of

additional resources such as a computer. ILC schemes that required computational

intensive calculations may not be practical for most industrial applications.

Two not previously investigated ILC schemes are also proposed by the author for further

investigation. The first scheme is the Multi-Channel Higher Order ILC (MCHO-ILC) in

contrast to the Single Channel First Order ILC (SCFO-ILC). MC-ILC has the advantage of using

different learning gains for different channels and should be able to achieve faster

convergence while maintaining stability. HO-ILC on the other hand may be able to smooth

the behaviour of the system better than first order ILC. The downside to this scheme is the

memory size required which may be a problem for DSP based ILC implementation. The

second scheme is the Multi-Channel Adaptive Iterative Learning Control (MCA-ILC),

combining the advantage of fast convergence of adaptive ILC and the flexibility of MC-ILC.

These two schemes are further discussed in the next section.

3.4.2 Multi-Channel Higher Order ILC

MCHO-ILC combines features of both MC-ILC and HO-ILC. Combining equation 3.31 and

equation 3.34, the learning law for a 2-channel and 2nd order ILC, the equation can be

expressed as:

𝑇𝑟,𝑗(𝜃𝑚) = 𝜓1𝑇𝑟,𝑗−1(𝜃𝑚)+𝑘𝑝1,𝑙𝑜𝑤𝑒𝑗−1,𝑙𝑜𝑤(𝜃𝑚) +

𝑘𝑝1,ℎ𝑖𝑔ℎ𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝜃𝑚)+

𝜓2𝑇𝑟,𝑗−2(𝜃𝑚)+𝑘𝑝2,𝑙𝑜𝑤𝑒𝑗−2,𝑙𝑜𝑤(𝜃𝑚) +

𝑘𝑝2,ℎ𝑖𝑔ℎ𝑒𝑗−2,ℎ𝑖𝑔ℎ(𝜃𝑚) (3.46)

The convergence condition from equation 3.19, 3.25 and 3.26 becomes:

𝜓1 + 𝜓2 = 1 (3.47)

|𝜓1 − 𝑘𝑝1𝐹1(𝑗𝜔)𝐺𝑝(𝑗𝜔)| + |𝜓2 − 𝑘𝑝2𝐹2(𝑗𝜔)𝐺𝑝(𝑗𝜔)| < 1 (3.48)

MCHO-ILC has the benefits of both HO-ILC and MC-ILC. However the downside is the high

usage of memory space to implement this scheme for DSP based ILC control. Second order

ILC requires 4 LUTs and 2 Channels ILC requires 3 LUTs and an additional 2 LUTs to

implement DSP zero phase filtering. This will be further discussed in section 6.1. MCHO-ILC

on the other hand would require 6 LUTs and an additional of 2 LUTs totalling 8 LUTs.

Figure 3.28 shows how MCHO-ILC can be implemented.

3.4.3 Multi-Channel Adaptive ILC

MCA-ILC is basically MC-ILC with variable learning gains for the different channels. For a 2

channel adaptive P type ILC, there are two learning gains associated with the lower and

higher frequency channels. The equation for MCA-ILC can be expressed as:

𝑇𝑟,𝑗(𝜃𝑚) = 𝑇𝑟,𝑗−1(𝜃𝑚)+𝜇𝑗−1,𝑙𝑜𝑤(𝜃𝑚)𝑒𝑗−1,𝑙𝑜𝑤(𝜃𝑚)

+𝜇𝑗−1,ℎ𝑖𝑔ℎ(𝜃𝑚)𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝜃𝑚) (3.49)

where

𝜇𝑗(𝜃𝑚) = 𝜇𝑚𝑎𝑥‖𝑝𝑗(𝜃𝑚)‖

2

‖𝑝𝑗(𝜃𝑚)‖2+𝐶

(3.50)

𝑝𝑗(𝜃𝑚) = 𝛼𝑝𝑗−1(𝜃𝑚) +1−𝛼

𝑘𝑡𝑒𝑗−1(𝜃𝑚) (3.51)

𝑝𝑗 is updated per revolution, where j is the number of iterations, where α (0 ≤ α ≤ 1) is a

smoothing factor and C is a positive constant. When pj becomes large, µj tends to µmax.

When pj is small, µj is small. Thus, µj (0 ≤ µj ≤ µmax) varies according to pj. To ensure

Figure 3-28: MCHO-ILC

+

𝑒𝑗−1𝑒𝑗−1

𝑘𝑝1,𝑙𝑜𝑤𝑘𝑝 𝑇𝑟,𝑗𝑢𝑗

M

𝑒𝑗−2 +

ZPFlow

+

M

M

𝑒𝑗−2

ZPFhigh

𝑘𝑝2,𝑙𝑜𝑤

𝑘𝑝1,ℎ𝑖𝑔ℎ

𝑘𝑝2,ℎ𝑖𝑔ℎ

𝜓1

𝜓2

M

stability, µmax should be less than 2 [114]. This is the variable step size concept where the

learning gain changes are controlled to meet the requirements of low mean square error

and fast convergence [114]. The values chosen for c and α will have an impact on the

learning gain(s), µ.

Figure 3.29 shows how MCA-ILC can be implemented. The learning gains for both low and

high channels vary accordingly to the adaptive algorithm. Zero Phase Filtering (ZPF) is used

to filter the signals into the 2 required channels. 5 LUTs are needed to implement this

scheme on a DSP based ILC method. 2 LUTs to carry out zero phase filtering (to be

discussed in section 6.1) and 3 LUTs to implement first order ILC.

3.5 Discussion

ILC has the potential to achieve minimal torque ripple in PMSM as the torque ripple is

periodic with respect to the rotor position. The focus of this thesis will therefore be on ILC

for PMSMs as the use of various ILC schemes have not been comprehensively investigated.

Field oriented control with PI current feedback will be used as a baseline comparison and

pre-compensation techniques will be included as it has the best potential of achieving zero

torque ripples for a time invariant system that is well known and can be accurately

modelled.

𝑒𝑗−1 +

+

M

𝑇𝑟,𝑗

𝑇𝑟,𝑗−1

𝜇𝑗−1,𝑙𝑜𝑤

A

ZPFlow

𝜇𝑗−1,ℎ𝑖𝑔ℎ ZPFhigh

Figure 3-29: Multi-Channel Adaptive ILC

P-ILC is the most commonly used ILC methods to minimise torque ripple for PMSM control.

Although other types of ILC exist in literature, they are not commonly used in PMSM

control and thus the effectiveness of these control methods is not clear.

The challenges of implementing a DSP-based ILC for PMSMs are as follows:

1. Limited memory space – due to the nature of ILC to store information from the

previous cycle, memory space of the DSP becomes an issue. Although additional

memory can be added, this adds on to the total cost of the hardware. To keep cost

at a minimum, the implementation of ILC will be confined to the memory space of

the DSP i.e. no additional external memory will be used.

2. Accurate online torque estimation – to achieve the aim of torque ripple

minimisation, torque estimation is needed. Although there are many ways to

estimate the torque from the literature, most of them assumed cogging torque to

be negligible. If cogging torque is to be included, speed information which is

derived from position (using encoder) can be used. This however will require the

motor to run at relatively low speeds during the learning process.

3. Filtering process – this is needed to remove the unwanted noise due to the double

differentiation process from position to torque information. Using a LPF with high

cutoff has the problem of noisy torque estimation. A low cutoff on the other hand

produces a clean signal but becomes inaccurate due to the phase shift. Ideally, a

zero phase filter should be used as it has the potential of producing a clean and

accurate signal. Two LUTs will be needed to implement ZPF in a DSP based ILC. This

will be further discussed in section 6.1.

This concludes the discussion of various ILC schemes and how they can be used on PMSM

control. Chapter 4 will show the simulated comparison of ILC methods on PMSM to achieve

the aim of minimal torque ripple.

Chapter 4 Simulation of Control Methods for PMSMs

Chapter 2 discussed field oriented control of PMSMs and the various causes of torque

ripple. Chapter 3 gave an overview of pre-compensation techniques and the different types

of Iterative Learning Control methods for PMSMs. In this chapter, a simulation study will be

presented evaluating the effectiveness of various ILC methods to minimising torque ripple.

The simulation scenario will first be discussed followed by a comparison of the ILC methods

considered. A Torque Ripple Factor (TRF) will be used to quantify how these control

methods perform in minimising torque ripple. Since torque output information is readily

available during simulation, it will be used as one of the inputs to the iterative learning

process.

4.1 Simulation Scenario

Simulations were done in Matlab/Simulink using the embedded coder of the Texas

Instruments (TI) C2000 DSP, a fixed point processor that will also be used for the

experimental setup. As fixed-point processors only supports integer mathematical

operations, so called IQ-math blocks were used to allow computation of floating-point

numbers. These are blocks provided by TI to enable the processor to calculate 32 bit fixed-

point numbers efficiently.

There are several parts to the simulation scenario as shown in Figure 4.1:

1. Motor – to simulate the electrical and mechanical subsystems of a PMSM.

2. Pulse Width Modulation (PWM) – to simulate the mechanics of a real motor

whereby Space Vector Pulse Width Modulation (SVPWM) is used to provide the

switching sequence of a three-phase voltage source inverter to the motor.

3. Field Oriented Control (FOC) – Two PI current controllers are used to control the 𝑖𝑑

and 𝑖𝑞 currents. Park and Inverse Park Transformation blocks are then used to

convert signals from the 𝑑𝑞, rotating reference frame to 𝛼𝛽, 2 phase reference

frame or 𝛼𝛽 back to 𝑎𝑏𝑐, 3 phase reference frame respectively.

4. I Sensor – to simulate the current measurement errors made up of current scaling

and offset errors

5. mech2elec – to convert 𝜃𝑚 to 𝜃𝑒 where 𝜃𝑚 = 𝑝 × 𝜃𝑒 (𝑝 is the number of pole pairs)

6. Offset – to align the flux density distribution of the motor with the current

waveforms

7. Scope – to observe the motor outputs (speed, encoder position, actual currents and

torque)

8. Reference – 𝐼𝑞∗ =

𝑇𝑟𝑒𝑓

𝑘𝑇

Figure 4-1 below shows the entire simulation scenario. The control setup using field

oriented control can be seen in the figure below where the reference current Id is set to 0

to optimise torque output.

Figure 4-1: Simulation Setup

Figure 4-2 shows that three phase currents (Iabc) and voltages (Vabc) are also being used to

drive the motor during simulation. The simulation is designed to be as close as possible to

the experimental setup. Therefore, the simulation is closely modelled after the test motor

using the same parameters whenever possible. These include the flux density distribution

and the cogging torque, which were measured off the test motor and the values inserted

into a Lookup Table (LUT). These LUTs were then placed in the mechanical block as shown

in figure 4.3.

Figure 4-2: Electrical and Mechanical subsystem of a PMSM

The mechanical subsystem block above consists of the following:

Figure 4-3: Mechanical Block

The Tc block allows the selection of either the cogging torque of the test motor or no

cogging torque and the BEMF block, allows the selection of the BEMF of the test motor or

an ideal sinusoidal BEMF.

Table 4.1 shows other parameters used in the simulation.

Table 4.1: Motor Parameters

Parameters Values

J 0.0042 kgm2

b 0.1457 Nms

L 0.002954 H

R 0.62 Ω

kt 0.537

Pole pair 10

Max. current 6 A

Sampling time, Ts 1

16384 s

Encoder resolution 4096

Torque ripple can be defined in many ways. The calculation of torque ripple suggested by

Gieras and Wing [115] will be used to determine how effective the various control methods

are in reducing the torque ripple. For this research, it is termed as Torque Ripple Factor

(TRF) and expressed as a percentage:

𝑇𝑅𝐹 =𝑇𝑟,𝑟𝑚𝑠

𝑇𝑎𝑣𝑒𝑟𝑎𝑔𝑒× 100% (4.1)

4.2 Field Oriented Control

The PI current controller uses the same parameters in the simulation as the experimental

setup, which was tuned using Ziegler Nichols open loop method (to be discussed in chapter

5). As the experimental motor has 20 poles and 24 slots, the lowest common multiple is

thus 120. Torque harmonics up to the 120th order will therefore be of interest, which will be

shown on all plots.

In an ideal situation, whereby the sources of torque ripple are not present, the output

torque follows the reference torque with no torque ripple. Figure 4-4 shows that the

simulated setup is able to follow the reference torque without any error.

Figure 4-4: Ideal Scenario

Case 1: Using the flux density distribution of the test motor

The ideal scenario shown above used an ideal sinusoidal flux density distribution. In this

case, an experimentally determined flux density distribution of the test motor (to be

discussed in chapter 5) is used instead with a sinusoidal BEMF with imbalances between the

three phases. The resulting output torque is shown in Figure 4.5.

It can be seen that with a torque reference of 1 Nm, there is a small TRF, in this instance

0.8%. Although the BEMF of the test motor is sinusoidal in shape, there are imbalances

among the three phases and these can result in torque ripple as shown in Figure 4.5.

Although the effect is small, nonetheless it adds to the torque ripple in the output torque.

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0.8

1

1.2

Output Torque

Time (s)

Torq

ue (

Nm

)

0 20 40 60 80 100 1200

0.05

0.1FFT of Output Torque

Orders

Magnitude

TRF = 0.0%

Figure 4-5: Case 1: Non-ideal sinusoidal BEMF

Case 2: With current measurement errors (current scaling and offset errors)

In this case, current measurements errors are simulated using the current (I) sensor block.

A resulting 10th order and the 20th order on the output torque harmonics appear as shown

in Figure 4-6.

Figure 4-6: Case 2: Current Measurement Errors

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0.8

1

1.2

Output Torque

Time (s)

Torq

ue (

Nm

)

0 20 40 60 80 100 1200

0.05


Orders

Magnitude

TRF = 0.8%

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0.8

1

1.2

Output Torque

Time (s)

Torq

ue (

Nm

)

TRF = 1.1%

0 20 40 60 80 100 1200

0.05


Orders

Magnitude

They are caused by current offset errors and current scaling errors respectively. As

explained in section 2.4.2, current offset and scaling errors in current measurements give

rise to a torque ripple at the fundamental frequency and twice the fundamental frequency

respectively. Due to the Park transformation, these translate to the 10th and 20th order for a

10 pole pair PMSM. Ia has a scaling of 0.98 and an offset of 0.018 A while Ib has no scaling

and offset errors. Figure 4-6 shows a TRF of 1.1% when only current measurement errors

are present.

Case 3: Cogging Torque

Lastly, the actual cogging torque of the test motor was used in the simulation. Figure 4-7

shows the output torque with a TRF of 7.6%. More details about how this cogging torque

was derived will be discussed in chapter 5. For the test motor used, cogging torque is the

largest contributor to torque ripple. The two peaks at 20th and 24th order are the

fundamental frequencies of the cogging torque induced by the stator and rotor. At

multiples of 20 and 24, more peaks can also be seen. The peak at the 120th order is the

native harmonic of the motor which is the lowest common multiple of 20 and 24.

Figure 4-7: Case 3: Cogging Torque

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0.8

1

1.2

Output Torque

Time (s)

Torq

ue (

Nm

)

TRF = 7.6%

0 20 40 60 80 100 1200

0.05


Orders

Magnitude

Figure 4-8 shows the output torque whereby all 3 cases were considered. This setup

exhibited a TRF of 8.1%. This setup that used FOC control will be used as a baseline

comparison for all other methods.

Figure 4-8: Output Torque (All Cases)

It can be seen that the torque ripple caused by non-ideal sinusoidal flux density distribution

imbalance (𝑇∆𝜆), cogging torque (𝑇𝑐𝑜𝑔) and torque ripple caused by current measurement

errors, (𝑇∆𝑖) cannot be minimised using the current controller as they occurred outside the

current control loop. The final torque ripple is the summation of all the above three factors.

Therefore, to minimise torque ripple, pre-compensation can be done to remove all these

causes. This will be shown in the next two sections.

4.3 Using Pre-compensation Techniques

As discussed in section 3.2, pre-compensation techniques have the potential to eliminate all

torque ripples if all the sources of the torque ripple can be identified, modelled and

compensated. Two variations of pre-compensation control will be discussed: the direct and

indirect pre-compensation techniques.

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0.8

1

1.2

Output Torque

Time (s)

Torq

ue (

Nm

)

TRF = 8.1%

0 20 40 60 80 100 1200

0.05


Orders

Magnitude

Direct Pre-compensation Control

Figure 4-9 shows how direct pre-compensation control can be incorporated with FOC of

PMSM.

Figure 4-9: Direct Pre-compensation Control Setup

Before the pre-compensation, the output torque has to be determined first. The torque

ripple attained thereafter can then be put into a LUT (Tr^) which is driven by the encoder.

This is then fed to the reference torque whereby the torque ripple will be subtracted. This

becomes a direct pre-compensation method whereby all sources of torque ripple are being

compensated.

Figure 4-10 shows that the majority of the torque ripple is being suppressed and a low TRF

of 0.6% is being achieved (93% reduction in TRF). The reason why TRF is not 0% is due to

the resolution of the LUT used. The resolution used has an impact on the accuracy of the

torque waveform and thus a higher TRF if the resolution is smaller. Another reason could

be due to the current controller that can only operate within a certain bandwidth.

This pre-compensation method is very effective in reducing torque ripple if there are no

variations to the motor parameters or the reference torque.

Figure 4-10: Output Torque Using Direct Pre-compensation Control

Indirect Pre-compensation Technique

Figure 4-11 shows how the individual causes of torque ripple were being compensated

separately in indirect pre-compensation. Two encoder-driven LUTs (Tcog and Tk) were being

used to compensate for cogging torque and non-ideal sinusoidal flux density distribution

respectively.

Figure 4-11: Indirect Pre-compensation Control

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Output Torque

TRF (FOC) = 8.1%

TRF (FF - Direct) = 0.6%

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude

FFT of Output Torque

FOC

FF - Direct

The magnitude of the flux density distribution is dependent on the reference torque.

Therefore a gain was needed to adjust the output of that LUT accordingly. The current

scaling and offset errors can be determined offline and compensated respectively. The

impact of each factor can be seen clearly using this method as shown in the next three

figures.

Figure 4-12: Case 1: Non-ideal BEMF Compensated

When only 𝑇∆𝜆 was compensated, the TRF dropped by a small amount to 7.9%.

Figure 4-13: Case 2: Cogging Torque Compensated

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Output Torque - Indirect Pre-Compensation: T

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


TRF (FOC) = 8.1%

TRF (T

compensated) = 7.9%

FOC

T

compensated

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Output Torque - Indirect Pre-Compensation: Tcog

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


TRF (FOC) = 8.1%

TRF (Tcog

compensated) = 1.2%

FOC

Tcog

compensated

When 𝑇𝑐𝑜𝑔 was compensated, TRF of the output torque reduced to 1.2%

Figure 4-14: Current Measurement Errors Compensated

Finally, when only current measurement errors were compensated, the TRF dropped to

7.7%. If all three factors were compensated, a low TRF of 0.4% was achieved (95%

reduction in TRF) as shown in figure 4.15.

Figure 4-15: Output Torque (All Cases)

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Output Torque - Indirect Pre-Compensation: Ti

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


TRF (FOC) = 8.1%

TRF (Ti

compensated) = 7.7%

FOC

Ti

compensated

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Output Torque - Indirect Pre-Compensation: T

Ti

Tcog

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


TRF (FOC) = 8.1%

TRF (T

Ti

Tcog

compensated) = 0.4%

FOC

T

Ti

Tcog

compensated

The LUTs used to generate the cogging torque and the flux density distribution in the motor

model and the LUTs used in the pre-compensation were identical and thus the lower TRF

compared to using the direct approach. The remaining TRF was probably due to the

limitation of the current controller that operated within a certain bandwidth.

Table 4.2 shows a summary of the TRF when different combinations of 𝑇∆𝜆, 𝑇𝑐𝑜𝑔 and 𝑇∆𝑖

were being compensated.

Table 4.2: TRF for Indirect Pre-compensation Control

Compensated Factors TRF (%)

None 8.1

𝑇∆𝜆 7.9

𝑇𝑐𝑜𝑔 1.2

𝑇∆𝑖 7.7

𝑇∆𝜆, 𝑇∆𝑖 7.5

𝑇∆𝜆, 𝑇𝑐𝑜𝑔 0.9

𝑇∆𝑖, 𝑇𝑐𝑜𝑔 0.8

𝑇∆𝜆, 𝑇∆𝑖, 𝑇𝑐𝑜𝑔 0.4

Using indirect FF method enables one to study the individual impact of these factors on the

TRF. In some cases, it may be appropriate to just monitor and adaptively compensate for

cogging torque only. The adaptation can be an adjustment to the gain of the cogging torque

due to its variation with temperature. Overall, pre-compensation control was effective if

the necessary parameters are known and compensated for. Since pre-compensation

methods require the compensated parameters to be determined and known, any variations

to these parameters would result in inaccurate compensation. This could then results in

torque ripple in the output torque.

4.4 Iterative Learning Control

As discussed in chapter 3, ILC is effective in minimising torque ripple that is periodic in

nature. A learning gain is used to determine the amount of learning for each iteration. In

the learning process, the system has to be stable and yet it must be able to converge

quickly. There is a trade-off between fast convergence and stability [87]. Figure 4-16 shows

how ILC schemes can be incorporated with FOC for PMSM. The error between the output

torque and reference torque is also put into an encoder driven LUT which serves as an input

to the ILC scheme. In practice, if it is not feasible to measure the torque directly, it will have

to be estimated using a torque estimation scheme. This will be further discussed in section

6.1.

Figure 4-16: Iterative Learning Control

Other than the learnable band discussed in chapter 3, convergence is another important

criteria for choosing the type of ILC methods and the corresponding learning gain(s). In this

thesis, convergence 𝛽𝑐 is defined as follows:

The number of iterations taken for the Torque Ripple Factor (TRF) to reach and

remain bounded within ±𝑛% of the steady state TRF.

Steady state TRF, TRFss is the average TRF of the last 𝑚 iterations

For the purpose of comparing the various ILC methods in simulated and experimental cases,

𝑛 = 25%

𝑚 = 5, where the total number of iterations = 20

4.4.1 Single Channel First Order ILC

In Single Channel First Order (SCFO) ILC the iterative learning only takes place within a

frequency range. There are different ways in which the learning gain of the ILC schemes can

be applied. The P-ILC, Pf-ILC, D-ILC, PD-ILC and PI-ILC will be discussed.

P-ILC

For this setup, ILC is used to estimate the torque ripple which is then used as a pre-

compensation signal to be subtracted from the reference torque. Since the torque

harmonics discussed in section 2.4 are periodic to the rotor position, the equation is

expressed in the position domain and not in the time domain.

It can be expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = �̂�𝑟,𝑗−1(𝜃𝑚) + 𝑘𝑝𝑒𝑗−1(𝜃𝑚) (4.2)

A range of 𝑘𝑝 values were used in the simulation to show how the TRF varies for each

iteration.

Figure 4-17: Plot of P-ILC for different kp values

Figure 4-17 shows that with higher 𝑘𝑝, the system is able to converge faster. Using a 𝑘𝑝

value of 1.0, it is able to achieve the lowest possible TRF in the 1st iteration. There are two

explanations as to why TRF did not go to 0% on the 1st iteration when 𝑘𝑝 = 1.0:

1. The estimated torque is inserted in a LUT that is driven by the encoder. The LUT has

a resolution of 4096 (as in the encoder) and this represents a digital form of the

estimated torque ripple.

2. Torque ripple with harmonic orders greater than its learnable band will not be

compensated. This will result in the error outside the learnable band adding up and

leading to the eventual instability.

Figure 4-18 shows the output torque after 20 iterations when 𝑘𝑝 = 0.4. It can be seen that

the major harmonics are now suppressed and a low TRF of 1.0% (88% reduction in TRF) is

achieved at the end of the 20th iteration. It takes about 4 iterations for the TRF to remain

steady. On the whole, P-ILC is effective in reducing torque ripple although it may take more

iterations if a lower learning gain is chosen.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of P-ILC for different kp values

kp = 0.2

kp = 0.4

kp = 0.6

kp = 0.8

kp = 1.0

Figure 4-18: Plot of P-ILC

Although 𝑘𝑝 = 1.0 may seem to be an ideal learning gain to achieve the faster convergence

possible, it may not be a wise choice as a learning gain in practice. An accurate estimation

of the torque ripple is assumed and if the estimated torque is not accurate, it may result in

higher torque ripple instead of near perfect compensation. If the iterations continue, the

TRF will increases thus leading to instability eventually. The learning has to stop after an

acceptable TRF is achieved.

P-ILC with forgetting factor (Pf-ILC)

A forgetting factor, α can be used together with the P-ILC to improve the robustness of

system as discussed in chapter 3. The equation can be expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = (1 − 𝛼)�̂�𝑟,𝑗−1(𝜃𝑚) + 𝑘𝑝𝑒𝑗−1(𝜃𝑚) (4.3)

Figure 4-19 shows the TRF over 20 iterations when different values of α are chosen.

Although, the robustness can be improved, the downside is the higher TRF.

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

P-ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude

FFT of Output Torque after 20 iterations

Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of P-ILC for kp = 0.4

Average TRF (last 5) = 1.0%

Figure 4-19: Plot of Pf-ILC (kp = 0.4) with different forgetting factor

It can be seen that when α increases, TRF at steady state increases too. This is because not

100% of the previous estimated torque ripple is used in the current compensation. Similar

to P-ILC, the torque ripple harmonics are suppressed and a TRF of 1.3% is achieved (84%

reduction in TRF) at the end of the 20th iterations for 𝛼 = 0.05. It takes about 5 iterations

for this control scheme to converge.

D-ILC

For D-ILC, the error is differentiated and the learning scheme can be expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = �̂�𝑟,𝑗−1(𝜃𝑚) + 𝑘𝑑�̇�𝑗−1(𝜃𝑚) (4.4)

A low pass filter is used after the differentiation. Figure 4-20 shows the plot of D-ILC for 20

iterations using the three different cut-off frequencies can be seen in the next figure.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of Pf-ILC with kp = 0.4 and varying

= 0

= 0.05

= 0.10

Figure 4-20: Plot of D-ILC with different cutoff frequencies

Figure 4-20 shows that by using a LPF with a cutoff frequency of 100 Hz, the system is still

very noisy and thus the high TRF. On the other hand, a LPF with a lower cutoff frequency

results in a more stable system but the learning range is reduced (as discussed in chapter 3).

Moreover, some of the higher harmonics may not be compensated if the cutoff frequency

is too low. Using a LPF with cutoff frequency of 50 Hz seems to be the best choice for the

setup.

Figure 4-21 shows the plot of D-ILC for different kd values using a LPF of 50 Hz:

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of D-ILC (kd = 0.004) with different cutoff frequencies

25Hz

50Hz

100Hz

Figure 4-21: Plot of D-ILC with different kd values

Figure 4-22 shows the output torque after 20 iterations when 𝑘𝑑 = 0.004. The torque

ripple harmonics are suppressed and a low TRF of 0.7% is achieved (91% reduction in TRF)

at the end of the 20th iterations. However, it takes about 9 iterations for this control

scheme to converge.

Figure 4-22: Plot of D-ILC

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of D-ILC for different kd values

kd = 0.002

kd = 0.003

kd = 0.004

kd = 0.005

kd = 0.006

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0.5

1

1.5

Samples

Torq

ue (

Nm

)

D-ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of D-ILC for kd = 0.004


PD-ILC

PD-ILC uses a combination of both the error and the differentiated error and can be

expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = �̂�𝑟,𝑗−1(𝜃𝑚) + 𝑘𝑝𝑒𝑗−1(𝜃𝑚) + 𝑘𝑑�̇�𝑗−1(𝜃𝑚) (4.5)

The right combination of 𝑘𝑝 and 𝑘𝑑 values has to be chosen to achieve a fast and stable

response. Figure 4-23 shows the plot of PD-ILC for 20 iterations using kp = 0.4 and varying kd

values.

Figure 4-23: Plot of PD-ILC

When 𝑘𝑝 and 𝑘𝑑 values are too large, the system becomes unsteady. Table 4.3 shows the

TRF for a range of 𝑘𝑝 and 𝑘𝑑 values at the end of 20 iterations. The yellow and red regions

shows the combination of kp and kd whereby it causes the whole system to become

unstable within the first 20 iterations. The green region shows the possible combinations

whereby TRF decreases and remains stable within the first 20 iterations.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of PD-ILC for different kd values, k

p = 0.4

kd=0.002

kd=0.003

kd=0.004

kd=0.005

kd=0.006

Table 4.3: TRF (20th

iteration) for PD-ILC with varying kp and kd values (%)

kp \ kd 0.002 0.003 0.004 0.005 0.006

0.2 0.78 0.60 0.51 0.44 0.44

0.4 0.74 0.73 0.81 0.95 1.31

0.6 3.65 4.59 5.52 7.75 9.70

0.8 >10 >10 >10 >10 >10

1 >10 >10 >10 >10 >10

Although some combinations of kp = 0.2 with kd results in a lower TRF at the end of the 20

iterations, it converges slower compared to using kp = 0.4 as shown in Figure 4.24. There is

a compromise between a lower TRF and a faster convergence.

Figure 4-24: Comparing PD-ILC with different values

Figure 4-25 shows the output torque after 20 iterations when 𝑘𝑝 = 0.4 and 𝑘𝑑 = 0.003.

The major harmonics have been suppressed and a low TRF of 0.6% is achieved. However, it

takes about 8 iterations for the TRF to converge.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparing PD-ILC

kp=0.2, k

d=0.005

kp=0.4, k

d=0.003


PI-ILC

PI-ILC uses a combination of both the error and the integral of the error and can be

expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = �̂�𝑗−1(𝜃𝑚) + 𝑘𝑝𝑒𝑗−1(𝜃𝑚) + 𝑘𝑖 ∫ 𝑒𝑗−1(𝜃𝑚)𝑡

𝑜𝑑𝑡 (4.6)

Figure 4-26 shows the output torque when 𝑘𝑝 = 0.4 and with varying 𝑘𝑖 values. The result

is the same as P-ILC when 𝑘𝑝 = 0.4.

For a fixed 𝑘𝑝and within a certain range, the Integral part of the scheme did not seem to

have any effect on the TRF nor the rate of convergence. However, when 𝑘𝑖 becomes too

large, the system will result in instability.

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

PD-ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of PD-ILC for kp = 0.4, k

d = 0.003


Figure 4-26: PI-ILC with varying ki values

Figure 4-27 shows the output torque when 𝑘𝑝 = 0.4 and 𝑘𝑖 = 0.1. The result is almost the

same as P-ILC when 𝑘𝑝 = 0.4. It can be seen that the major harmonics are now suppressed

and a low TRF of 1.0% is achieved at the end of the 20th iteration. However, it takes about 5

iterations for the TRF to converge.

Figure 4-27: PI-ILC

Table 4.4 shows the TRF for a range of 𝑘𝑝 and 𝑘𝑖 values.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of PI-ILC for different ki values, k

p = 0.4

ki = 0.1

ki = 0.2

ki = 0.3

ki = 0.4

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

PI-ILC: Output Torque after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of PD-ILC for kp = 0.4, k

i = 0.1

Initial

Final

Initial

Final


Table 4.4: TRF for PI-ILC

kp \ ki 0.1 0.2 0.3 0.4 0.5

0.2 1.80 1.78 1.78 2.41 >10

0.4 1.03 1.03 1.03 >10 >10

0.6 2.69 2.57 2.48 >10 >10

0.8 >10 >10 >10 >10 >10

Summary

It is not easy to determine which type of learning scheme is better at simulation level. The

learning has to be stopped at some point once a minimum threshold is reached otherwise

noise will just add up indefinitely [96]. In the simulated comparison, it is assumed that the

learning will stop after 20 iterations and the chosen learning gain has to be relatively stable

up to that point. Since only torque ripple up to the 120th harmonics are of interest and the

learning process whereby ILC is enabled has to be done in low speed operation (less than 1

Hz), a learnable range of 0 to 150 Hz will suffice and the convergence condition has to be

satisfied within this range. Three criteria will be used to select the appropriate learning

gains for each scheme as a comparison with other schemes.

Criteria:

1. Has a learnable band with frequency ranges from 0 to 150 Hz

2. TRF has to be relatively stable for at least 20 iterations.

3. For learning gains that satisfy the first two criteria, the one with the quicker

convergence will be chosen

Table 4.5 shows the comparison of the different learning schemes for SCFO-ILC. Depending

on the values of the learning gains, the TRF and the rate of convergence will be affected.

Choosing a large learning gain may result in a faster convergence but it may result in

instability.

Table 4.5: Comparing SCFO-ILC

SCFO-ILC Learning Gain(s) TRFss (%) Convergence

P-ILC 𝑘𝑝 = 0.4 1.0 4

Pf-ILC 𝑘𝑝 = 0.4, α = 0.05 1.3 5

D-ILC 𝑘𝑑 = 0.004 0.7 9

PD-ILC 𝑘𝑝 = 0.4, 𝑘𝑑 = 0.003 0.6 8

PI-ILC 𝑘𝑝 = 0.4, 𝑘𝑖 = 0.1 1.0 5

D-ILC had the slowest convergence among SCFO-ILC. This effect can also be seen in PD-ILC

when compared to P-ILC. However, the additional learning gain in PD-ILC resulted in a lower

TRFss when compared to P-ILC. For SCFO-ILC, PD-ILC has the lowest TRFss while P-ILC has the

fastest convergence. Both P-ILC and PD-ILC will be used to represent SCFO-ILC as a

comparison with other categories of ILC methods.

4.4.2 Multi-Channel ILC

Multi-Channel ILC (MC-ILC) uses multiple channels in the updating process. Figure 4-28

shows that for the same value of learning gain of 𝑘𝑝 = 0.4 used for all the channels, MC-ILC

has similar results as P-ILC.

Figure 4-28: Comparison of multi-channel ILC with single channel ILC

However, the advantage of MC-ILC is that the learning gains can be different for the

different channels.

2 Channels ILC

For a 2 channel system, the equation can be simplified into:

�̂�𝑟,𝑗(𝜃𝑚) = �̂�𝑟,𝑗−1(𝜃𝑚)+𝑘𝑝,𝑙𝑜𝑤𝑒𝑗−1,𝑙𝑜𝑤(𝜃𝑚)+𝑘𝑝,ℎ𝑖𝑔ℎ𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝜃𝑚) (4.7)

where the subscript low represents the lower frequency channel and the subscript high

represents the higher frequency channel for 2 Channels ILC.

In this case, where the lower frequencies has much higher torque ripple harmonics, the

learning gain for the lower channel can be higher. Figure 4-29 shows a 2 channel ILC

scheme whereby the learning gain for the lower channel is varied. A zero phase filter is

used to separate the learning band in two channels: one from 0 to 85 Hz and the other

from 85 Hz onwards.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of Multi-Channel ILC

Single Channel ILC: Average TRF (last 5) = 1.0%

2 Channels ILC: Average TRF (last 5) = 1.0%


Figure 4-29: Plot of 2 channels ILC for different kp,low values

Figure 4-29 shows that the fastest convergence can be achieved using a kp,low value of 1.0.

MC-ILC gives the flexibility of choosing different learning gains for different regions of the

torque harmonics needed to be compensated. In the lower frequency where there are

more major peaks, choosing a value of 1.0 can quickly suppress them.

Figure 4-30: Plot of 2 Channels ILC

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of 2 Channels ILC for different kp,low

values, kp,high

= 0.4

kp,low

= 0.4

kp,low

= 0.6

kp,low

= 0.8

kp,low

= 1.0

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

2 Channels ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of 2 Channels ILC for kp,low

= 1.0, kp,high

= 0.4


The output torque of a 2 channels ILC whereby 𝑘𝑝,𝑙𝑜𝑤 = 1.0 and 𝑘𝑝,ℎ𝑖𝑔ℎ = 0.4 can be seen

in Figure 4.30. A low TRF of 0.5% is achieved and convergence is reached in just two

iterations.

3 Channels ILC

For a 3 channels system, the equation can be expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = �̂�𝑟,𝑗−1(𝜃𝑚)+𝑘𝑝,𝑙𝑜𝑤𝑒𝑗−1,𝑙𝑜𝑤(𝜃𝑚)

+𝑘𝑝,𝑚𝑒𝑑𝑖𝑢𝑚𝑒𝑗−1,𝑚𝑒𝑑𝑖𝑢𝑚(𝜃𝑚)

+𝑘𝑝,ℎ𝑖𝑔ℎ𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝜃𝑚) (4.8)

where the subscript low, medium and high represents the low, medium and high frequency

channels for a 3 Channels ILC method. A zero phase filter is used to separate the learning

band into three channels:

Low channel: from 0 to 36 Hz

Medium channel: from 36 Hz to 85 Hz

High channel: from 85 Hz onwards.

Figure 4-31: Plot of 3 Channels ILC for different learning gains

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of 3 Channels ILC for different kp,low

and kp,medium

values, kp,high

= 0.4

kp,low

= kp,medium

= 0.4

kp,low

= kp,medium

= 0.6

kp,low

= kp,medium

= 0.8

kp,low

= kp,medium

= 1.0

Figure 4-32: Plot of 3 Channels ILC

The output torque of 3 Channels ILC whereby 𝑘𝑝,𝑙𝑜𝑤 = 𝑘𝑝,𝑚𝑒𝑑𝑖𝑢𝑚 = 1.0 and 𝑘𝑝,ℎ𝑖𝑔ℎ = 0.4

can be seen in Figure 4-32. A low TRF of 0.5% is also achieved and convergence is also

reached in two iterations.

Summary

Figure 4-33 shows a comparison between ILC with different number of channels.

Figure 4-33: Comparing MC-ILC Methods

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

SamplesT

orq

ue (

Nm

)

3 Channels ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of 3 Channels ILC for kp,low

= kp,medium

= 1.0, kp,high

= 0.4


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of Multi-Channel ILC

Single Channel ILC: Average TRF (last 5) = 1.0%



Higher channels ILC shows quicker convergence and lower TRFss compared to a single

channel ILC scheme. Although, a 2 channels ILC and 3 channels ILC methods have the same

TRFss and the same convergence, a 2 channels ILC is preferred for the following reasons:

2 channels ILC requires a zero phase filter to separate a channel into two while 3

channels ILC would require 2 zero phase filter. In experimental setup with DSP-

based zero phase filtering technique (discussed in section 6.1), this would mean

more LUTs (more memory space) are needed for higher channel ILC.

Some attenuation of the original waveform may be inevitable depending on the

location of the torque ripple harmonics. This would correspond to lower accuracy

of higher channel ILC compared to lower channel ILC.

Table 4.6: Comparing MC-ILC Methods

Multi-Channel ILC TRFss (%) Convergence

Single 1.0 4

2 Channels 0.5 2

3 Channels 0.5 2

Due to the above mentioned advantages of 2 channels ILC over 3 channels ILC, the 2

channels ILC method will be used as a comparison to the other categories of ILC methods.

4.4.3 Higher Order ILC

Single order ILC uses only the previous error and the previous input. HO-ILC on the other

hand uses information from multiple cycles of previous errors and previous inputs.

For a 2nd Order ILC, the equation can be expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = 𝜓1�̂�𝑟,𝑗−1(𝜃𝑚) + 𝑘𝑝1𝑒𝑗−1(𝜃𝑚) + 𝜓2�̂�𝑟,𝑗−2(𝜃𝑚) + 𝑘𝑝2𝑒𝑗−2(𝜃𝑚)

(4.9)

Figure 4-34 shows that when the learning gains increases, a faster and lower TRF can be

achieved. However, for the case when 𝑘𝑝1 = 𝑘𝑝2 = 0.8, the system gradually becomes

unstable. 𝜓1 and 𝜓2 are both set to 0.5.

Figure 4-34: Second Order ILC with different kp values

Figure 4-35 shows another example with different values of 𝑘𝑝1 and 𝑘𝑝2.

Figure 4-35: 2nd

Order ILC with varying kp values

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of 2nd Order ILC for different kp values

kp1

= kp2

= 0.2

kp1

= kp2

= 0.4

kp1

= kp2

= 0.6

kp1

= kp2

= 0.8

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of 2nd Order ILC for different kp values

kp1

= 0.2, kp2

= 0.8

kp1

= 0.4, kp2

= 0.6

kp1

= 0.6, kp2

= 0.4

kp1

= 0.8, kp2

= 0.2

Figure 4-36 shows the plot for HO-ILC where kp1 = 0.6 and kp2 = 0.4. It can be seen that

TRF reduces to 0.4% at TRF becomes stable in 3 iterations.

Figure 4-36: Plot of HO-ILC (2

nd Order)

For a 3rd Order ILC, the equation can be expressed as:

�̂�𝑟,𝑗(𝜃𝑚) = 𝜓1�̂�𝑟,𝑗−1(𝜃𝑚) + 𝑘𝑝1𝑒𝑗−1(𝜃𝑚) + 𝜓2�̂�𝑟,𝑗−2(𝜃𝑚) + 𝑘𝑝2𝑒𝑗−2(𝜃𝑚)

+𝜓3�̂�𝑟,𝑗−3(𝜃𝑚) + 𝑘𝑝3𝑒𝑗−3(𝜃𝑚) (4.10)

In Figure 4.37, the learning gains, 𝑘𝑝1 = 𝑘𝑝2 = 𝑘𝑝3 = 𝑘𝑝 and 𝜓1, 𝜓2 and 𝜓3 are set to

0.333.

Figure 4-37: 3rd

Order ILC with varying kp values (1)

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

) 2nd Order ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of 2nd Order ILC for kp1

= 0.6, kp2

= 0.4


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of 3rd Order ILC for different kp values

kp = 0.2

kp = 0.4

kp = 0.6

kp = 0.8

Figure 4-38: 3rd

Order ILC with varying kp values (2)

Figure 4.39 shows the plot for 3rd Order ILC where kp1 = 0.4, kp2 = 0.4 and kp3 = 0.2. It

can be seen that TRF reduces to 0.4% at TRF becomes stable in 5 iterations.

Figure 4-39: Plot of HO-ILC (3

rd Order)

Summary

Figure 4.40 and Table 4.7 show the comparison between ILC with different number of

orders. The kp1 value of the 2nd order ILC was higher than that of the 3rd order ILC and thus

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of 3rd Order ILC for different kp values

kp1

= 0.2, kp2

= 0.2, kp3

= 0.6

kp1

= 0.2, kp2

= 0.4, kp3

= 0.4

kp1

= 0.2, kp2

= 0.6, kp3

= 0.2

kp1

= 0.4, kp2

= 0.2, kp3

= 0.4

kp1

= 0.4, kp2

= 0.4, kp3

= 0.2

kp1

= 0.6, kp2

= 0.2, kp3

= 0.2

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

) 3rd Order ILC: Output Torque after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of 3rd Order ILC for kp1

= 0.4, kp2

= 0.4, kp3

= 0.2

Initial

Final

Initial

Final


a larger reduction in TRF in the first iteration. The kp1 value of the 3rd order ILC was the

same as the kp value of the 1st order ILC. Thus both schemes had the same reduction in TRF

in the first iteration. The effects of kp2 and kp3 in the 3rd order ILC can then be seen in

subsequent iterations whereby the reduction in TRF was more than that of the 1st order ILC.

As it took more iteration for the higher order learning to take place, higher order ILC may

not have faster convergence compared to lower order ILC. It can be seen that 2nd Order ILC

has the faster convergence and will be used to compare with other categories of ILC

schemes.

Figure 4-40: Comparing HO-ILC Methods

Table 4.7: Comparing HO-ILC Methods

Higher Orders ILC TRFss (%) Convergence

1st Order 1.0 4

2nd Order 0.4 3

3rd Order 0.4 5

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of Higher Orders ILC

1st Order ILC: Average TRF (last 5) = 1.0%

2nd Order ILC: Average TRF (last 5) = 0.4%

3rd Order ILC: Average TRF (last 5) = 0.4%

4.4.4 Adaptive ILC

Adaptive ILC allows the learning gain to be adjustable based on the error. For an adaptive P-

ILC using variable step-size [51], the equation is:

𝑇𝑟,𝑗(𝜃𝑚) = 𝑇𝑟,𝑗−1(𝜃𝑚)+𝜇𝑗−1(𝜃𝑚)𝑒𝑗−1(𝜃𝑚) (4.11)

where


2


(4.12)


𝑘𝑡𝑒𝑗(𝜃𝑚) (4.13)

The values chosen for c and α are going to have an impact of the learning gain, µ. It can be

seen from the next thee figures how these parameters have an impact on the convergence

and the TRF.

Figure 4-41: Plot of Adaptive P-ILC for α = 0.1

For α = 0.1, a higher c value result in a higher learning gain and lower TRF at the end of the

20th iterations.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Iterations

TR

F(%

)

Plot of Adaptive P-ILC for different c values, = 0.1

c = 0.005

c = 0.01

c = 0.02

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations

Variable Learning Gain

c = 0.005

c = 0.01

c = 0.02


Similar trend is observed for α = 0.5.


Lastly, for α = 0.9, the system seems to be unstable after some time. c = 0.01 gives the best

trade-off between convergence rate and a low TRF. Using a value of α = 0.1, the system is

able to converge the fastest. With these two values chosen, the plot of adaptive P-ILC can

be seen in the next figure.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

IterationsT

RF

(%)


c = 0.005

c = 0.01

c = 0.02

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


c = 0.005

c = 0.01

c = 0.02

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Iterations

TR

F(%

)


c = 0.005

c = 0.01

c = 0.02

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


c = 0.005

c = 0.01

c = 0.02

Figure 4-44: Plot of Adaptive P-ILC (Output Torque)

The major hjarmonics are being suppresed quickly bringing it to a low TRF of 0.8% at the 1st

iteration and converge to a TRF of 1.3% after 3 iterations.

Figure 4-45: Plot of Adaptive P-ILC (TRF)

The learning gain, µ goes to approximately 0.91 at the first instance and remains steady at

around 0.32 after 8 iterations. On the whole, adaptive P-ILC is able to achieve a low TRF at

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Adaptive P-ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of Adaptive P-ILC ( = 0.1, c = 0.01)


0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


the first iteration and TRF remains relatively stable at about 1.3% and the end of the 20th

iteration.

Adaptive PD-ILC

Since PD-ILC has a lower TRF compared to P-ILC, an additional D-ILC can also be used in

conjunction with the adaptive P-ILC.

The equation for adaptive PD-ILC becomes:

𝑇𝑟,𝑗(𝜃𝑚) = 𝑇𝑟,𝑗−1(𝜃𝑚)+𝜇𝑗−1(𝜃𝑚)𝑒𝑗−1(𝜃𝑚) + 𝑘𝑑�̇�𝑗−1(𝜃𝑚) (4.14)

Figure 4-46 shows that using the same values of a and c but with the addition of 𝑘𝑑, the

variable learning gain and TRF for 20 iterations.

Figure 4-46: Plot of Adaptive PD-ILC for different kd values

The additional D-ILC has an impact on the TRF and the variable learning gain µ. A higher kd

value results in a lower µ value at the end of the 20th iteration and converges the slowest.

The learning gain kd = 0.001 is chosen and the output torque is shown below:

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Iterations

TR

F(%

)

Plot of Adaptive PD-ILC for different kd values, = 0.1, c = 0.01

kd = 0.001

kd = 0.002

kd = 0.003

kd = 0.004

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


kd = 0.001

kd = 0.002

kd = 0.003

kd = 0.004

Figure 4-47: Plot of Adaptive PD-ILC (Output Torque)

Figure 4-48 shows that all major harmonics are suppressed quickly after 2 iterations and

TRF remains low at about 1.0% throughout thereafter.

Figure 4-48: Plot of Adaptive PD-ILC (TRF)

Similarly, the learning gain, µ increases to approximately 0.91 at the 1st iteration, 0.45 at

the 2nd iteration and stabilises at around 0.26 thereafter. On the whole, adaptive PD-ILC is

able to converge in 2 iterations and have a low TRFss of 1.0%.

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

Adaptive PD-ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of Adaptive PD-ILC ( = 0.1, c = 0.01, kd = 0.001)


0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


Summary

Figure 4.49 shows the comparison between the non-adaptive and adaptive P-ILC and PD-ILC.

Figure 4-49: Comparison of Adaptive ILC

Although adaptive P-ILC has the lowest TRF in the 1st iteration, its TRFss is the highest among

the 4 schemes compared.

Table 4.8: Comparing Adaptive ILC

Adaptive ILC TRFss (%) Convergence

P 1.0 4

PD 0.7 9

Adaptive P 1.3 3

Adaptive PD 1.0 2

PD-ILC has lower TRFss compared to P-ILC and similarly adaptive PD-ILC has lower TRFss

compared to adaptive P-ILC. In addition, adaptive PD-ILC has the fastest convergence of 2

iterations. Adaptive ILC is able to converge faster due to the adaptive learning gain.

However, the TRFss for the adaptive schemes are higher than the non-adaptive

counterparts. There is a tradeoff for tuning c and α in the adaptive scheme in which faster

convergence may result in a higher TRFss.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of P, PD, Adaptive P and Adaptive PD ILC

P-ILC: Average TRF (last 5) = 1.0%

PD-ILC: Average TRF (last 5) = 0.6%

Adaptive P-ILC: Average TRF (last 5) = 1.3%

Adaptive PD-ILC: Average TRF (last 5) = 1.0%

Summary of comparison

Figure 4-50 shows the plot comparing SCFO-ILC (P-ILC and PD-ILC), MC-ILC (2 channels), HO-

ILC (2nd order) and adaptive ILC (PD) schemes. MC-ILC shows the best results achieving

stable TRF in 2 iterations and has a relatively low TRF of 0.5%. HO-ILC has the lowest TRF of

0.4%. Adaptive PD-ILC also achieves stable TRF in 2 iterations but TRF remains at

approximately 1.0%. This is similar to the results observed from PD-ILC, the additional D-ILC

is able to achieve lower TRF. The adaptive P learning makes the convergence faster.

Figure 4-50: Comparison of PD, MC, HO and Adaptive ILC

HO-ILC has the lowest TRFss while MC-ILC and adaptive PD-ILC has the faster convergence.

MC-ILC and HO-ILC outperform SCFO-ILC by having a lower TRFss and a faster convergence.

The results for the different categories of ILC can be summarised in the table 4.9.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of P, PD, Multi-Channel, Higher Order and Adaptive ILC



MC-ILC: Average TRF (last 5) = 0.5%

HO-ILC: Average TRF (last 5) = 0.4%


Table 4.9: Summary of Comparison for Different ILC Schemes

ILC Schemes TRFss (%) Convergence

SCFO-ILC (P-ILC) 1.0 4

SCFO-ILC (PD-ILC) 0.6 8

MC-ILC (2 Channels) 0.5 2

HO-ILC (2nd Order) 0.4 3

Adaptive ILC (PD) 1.0 2

4.4.5 Multi-Channel Higher Order ILC

In MCHO-ILC, multiple channels together with higher orders are used in the iterative

learning schemes.

The equation for MCHO-ILC is:

𝑇𝑟,𝑗(𝜃) = 𝜓1𝑇𝑟,𝑗−1(𝜃)+𝑘𝑝1,𝑙𝑜𝑤𝑒𝑗−1,𝑙𝑜𝑤(𝜃)+𝑘𝑝1,ℎ𝑖𝑔ℎ𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝜃)

+𝜓2𝑇𝑟,𝑗−2(𝜃)+𝑘𝑝2,𝑙𝑜𝑤𝑒𝑗−2,𝑙𝑜𝑤(𝜃)+𝑘𝑝2,ℎ𝑖𝑔ℎ𝑒𝑗−2,ℎ𝑖𝑔ℎ(𝜃) (4.15)

Figure 4-51 shows the simplest form of MCHO-ILC with 2 channels and 2nd order ILC.

Convergence is reached in just 3 iterations and with a low TRFss of 0.4%.

Figure 4-51: Plot of MCHO-ILC

Figure 4-52 and Table 4.10 shows a comparison between MC-ILC, HO-ILC and MCHO-ILC.

Simulation results show that MCHO-ILC is unable to converge faster than MC-ILC and HO-

ILC but it still retain the lower TRFss of 0.4%.

Figure 4-52: Comparison of MC, HO and MCHO ILC

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

SamplesT

orq

ue (

Nm

)

MCHO-ILC: Output Torque after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of MCHO-ILC for kp,low 1

= 0.6; kp,high1

= 0.9; kp,low 2

= 0.6; kp,high2

= 0.9

Initial

Final

Initial

Final


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of Multi-Channel, Higher Order and MCHO ILC



MCHO-ILC: Average TRF (last 5) = 0.4%

Table 4.10: Comparing MC-ILC, HO-ILC and MCHO-ILC Schemes




MCHO-ILC 0.4 3

4.4.6 Multi-Channel Adaptive ILC

In Multi-Channel Adaptive ILC (MCA-ILC), the learning gains for both channels are adaptive

and thus able to adapt to the error. The equation for MCA-ILC is:

𝑇𝑟,𝑗(𝜃) = 𝑇𝑟,𝑗−1(𝜃)+𝜇𝑗−1,𝑙𝑜𝑤(𝜃)𝑒𝑗−1,𝑙𝑜𝑤(𝜃)+𝜇𝑗−1,ℎ𝑖𝑔ℎ(𝜃)𝑒𝑗−1,ℎ𝑖𝑔ℎ(𝜃)

(4.16)

where


2


(4.17)


𝑘𝑡𝑒𝑗(𝜃𝑚) (4.18)

MCA-ILC is able to supress all torque harmonics and reach a low TRF of 0.5% after 1

iteration as shown in the next two figures.

Figure 4-53: Plot of MCA-ILC (Output Torque)

Figure 4-54: Plot of MCA-ILC (TRF)

The learning gain for the lower frequency is still steadily decreasing but the learning gain

for the higher frequency has already stabilised at about 0.7.

Figure 4-55 shows the comparison between 7 ILC schemes. MCA-ILC has the fastest

convergence while PD-ILC is the slowest.

0.5 1 1.5 2 2.5 3 3.5

x 104

0.8

1

1.2

Samples

Torq

ue (

Nm

)

MCA-ILC: Output Torque after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of MCA-ILC for low

= 0.9, high

= 0.1, clow

= 0.001, chigh

= 0.0001


0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


low

high

Figure 4-55: Comparison of Various ILC Schemes

Table 4.11 shows the average TRF and the number of iterations needed for the ILC schemes

to achieve stable TRF.

Table 4.11: Comparison of Various ILC Schemes


SCFO-ILC (P-ILC) 1.0 4

SCFO-ILC (PD-ILC) 0.6 8



Adaptive ILC (PD) 1.0 2

MCHO-ILC 0.4 3

MCA-ILC 0.5 1

HO-ILC and MCHO-ILC has the lowest TRF while adaptive PD-ILC has the highest. On the

whole MCHO-ILC is still able to perform better than SCFO-ILC by having a lower TRF and

achieving stable TRF in a lesser number of iterations. MCA-ILC on the other hand is able to

achieve stable TRF in just one iteration and has relatively low TRF. It can be considered the

best performing ILC schemes among the schemes compared.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of various ILC Schemes






MCHO-ILC: Average TRF (last 5) = 0.4%

MCA-ILC: Average TRF (last 5) = 0.5%

4.5 Discussion

In this chapter, the simulation setup was first discussed with the aim of modelling it as

realistic as possible. The actual characteristics and parameters of the experimental motor

are being used in the simulation. These include the BEMF shapes for the three phases,

cogging torque, current measurement errors.

For PMSM control, P-ILC and Pf-ILC are the two most commonly used ILC schemes.

Experiments have shown that torque ripple harmonics can be suppressed using these two

schemes [5, 13, 49, 52]. Adaptive P-ILC using variable step size has been tested using

simulation and was also able to suppress torque ripple [51]. No literature was found that

reviews other ILC methods for PMSM control. A comprehensive comparison between the

different ILC schemes and their effectiveness in minimising torque ripple for PMSM control

was also not found in literature.

Among all schemes, D-ILC took the most number of iterations for the system to converge

compared to other ILC schemes. MC-ILC, HO-ILC and adaptive ILC schemes were better than

SCFO-ILC schemes. They had relatively lower TRF and can achieve convergence faster. The

proposed MCHO-ILC was not able to converge quickly but it had the lowest TRF of 0.4%.

The other proposed MCA-ILC had the fastest convergence and a relatively low TRF of 0.5%.

They were able to achieve TRFss comparable to using pre-compensation schemes and were

adaptive in nature.

The simulation results have been summarised in Table 4.12:

Table 4.12: Comparison of all Control Schemes

Schemes Categories TRFss (%) Convergence

FOC - 8.1 -

Pre-Compensation

Technique

Direct 0.6 -

Indirect 0.4 -

SCFO-ILC

P 1.0 4

Pf 1.3 5

D 0.7 9

PD 0.6 8

PI 1.0 5

MC-ILC 2 Channels 0.5 2

3 Channels 0.5 2

HO-ILC 2nd Order 0.4 3

3rd Order 0.4 5

Adaptive ILC P 1.3 3

PD 1.0 2

Proposed ILC MCHO 0.4 3

MCA 0.5 1

This chapter completes the simulation analysis of the various control methods. Chapter 5

will discuss about the experimental setup follow by the experimental results in chapter 6.

Chapter 5 Experimental Setup

Chapter 4 presented the simulated results of various control schemes used to achieve

torque ripple minimisation of a PMSM. This chapter describes the experimental setup of

PMSM to validate the results from the simulation. Depending on the type of control

method, different parameters are needed to be determined. Generally, parameters include

the motor inertia (J), the viscous friction (b), the inductance (L) and the resistance (R),

representing the mechanical and electrical subsystems of the motor.

For field oriented control using Parks’ transformation, the torque constant (𝑘𝑇) and the

offset have to be determined as well. There are a few parameters such as the number of

pole pairs (𝑝), the number of stator slots and rotor slots which can be determined quite

easily. Other parameters that can affect the torque ripple are the cogging torque, non-

sinusoidal BEMF or imbalances, current scaling and current offset errors. They can be

determined experimentally and used for the control scheme if required.

The type of hardware and software used in the experimental control setup will first be

discussed followed by the determination of the motor parameters and characteristics of

the current controller.

5.1 Hardware and Software Specifications

This section will discuss the inherited experimental PMSM setup that was used to conduct

the experiments.

35.1.1 Motor

The test motor as shown in Figure 5.1 is a single sided, three phase axial flux motor used in

pool pump system with a rated power of 750 W and a rated torque of 3 Nm. It has 20 poles

and 24 slots whereby the poles are skewed at approximately 9°. The test motor has a

sinusoidal flux density distribution and relatively high cogging torque as shown in later

sections.

Figure 5-1: Experimental motor used for research

5.1.2 Mechanical Design

The transfer function of the experimental setup should not change over the range of

operation. Measurements should also not be affected by resonant frequencies, external or

internal drive forces and bearing loads. The twin shaft layout was used in the design of the

test rig whereby the stator and rotor are mounted on different shafts [116]. The equation

of motion is:

𝐽𝑠�̈�𝑠 + 𝑏𝑏𝑒𝑎𝑟𝜃�̇� + 𝑘𝑠𝑒𝑛𝑠𝑜𝑟(𝜃𝑠) = 𝑇𝑚 (5.1)

where 𝐽𝑠is the stator angular moment of inertia, 𝜃𝑠 is the stator angular position and

𝑘𝑠𝑒𝑛𝑠𝑜𝑟 is the measurement output from the torque sensor. Assuming a non-moving stator,

𝜃𝑠 ≈ 𝜃�̇� ≈ �̈�𝑠 ≈ 0, therefore

𝑘𝑠𝑒𝑛𝑠𝑜𝑟(𝜃𝑠) = 𝑇𝑚 (5.2)

By having a two shaft layout, the measured torque is the actual motor torque and is not

affected by other parameters. However, the two shafts must be carefully aligned otherwise

the axial concentricity is not guaranteed. The air gap between the rotor and stator is kept at

1 mm.

5.1.3 Eddy Current Brake

The eddy current brake is chosen to provide the braking force needed to allow variation to

the velocity of the motor.

5.1.4 Sensors

Sensors are needed in order to measure the output torque, stator current and position of

the rotor.

Torque Sensor

A Kistler 9339A piezoelectric sensor was used. This model was chosen due to the stiffness

that it provides. The torque sensor was placed so that the only force it measured was

coming from the motor torque and to minimise cross talk forces [117]. This torque sensor

was used to independently verify the effectiveness of the control schemes discussed. It was

however not used for real time control.

The crosstalk between the shear, axial and bending moments for Kistler 9333A are as

follows [117]:

Shear force (FX,Y) on torque (MZ): < 0.3 mNm/N (1 in 3333)

Axial force (FZ) on torque (MZ): ±0.05 mNm/N (1 in 20,000)

Bending moment (MX,Y) on torque (MZ): < 8 mNm/N (1 in 125)

This torque sensor is able to measure up to ±10Nm when used together with a Kistler

charge amp type 5073 which has an error of less than ±0.18 nm [118].

Current Sensors

The current sensors chosen were the closed loop type (LEM LTS-25 NP) and have an

accuracy of 0.7% over a range of ±25 A. The bandwidth of the sensors is 100 kHz. To avoid

potential interferences to the current sensors, a separate board was used to mount them

so that they were at a distance from the current inverter.

Position Sensor

The position sensor used was a 12 bit (4096 count), BEI Model HS35 incremental encoder

that can provide the position of the encoder from count of 0 to 4095 (2𝜋 rad).

5.1.5 DSP

The Texas InstrumentsTM DSP TMS320F2812 was chosen together with a Spectrum Digital

eZdspF2812 board. Programming of the DSP can be done using the block diagrams in

Simulink.

5.1.6 Data Acquisition using Labview

In order to capture data from the torque transducer, encoder and current sensors, a data

acquisition program using Labview was developed. A National Instruments PCI6259 D/A

data acquisition card was used together with a data acquisition computer using Labview.

The data can be captured in a text batch file. A number of revolutions of data can be

recorded with this setup.

5.1.7 Matlab/Simulink

Matlab/Simulink was used to design the control schemes. Matlab was also used to process

the captured data.

5.2 Determining the Motor Parameters

A number of motor parameters were determined offline. They are the BEMF shapes, the

torque constant 𝑘𝑇, cogging torque, J, b, L and R.

5.2.1 BEMF Shapes

To determine the BEMF shapes, an external DC motor was used to drive the rotor. The

voltages for the three phases were then recorded using Labview and Figure 5-2 shows the

speed normalised BEMF for the 3 phases.

Figure 5-2: BEMF Shapes of the Experimental Motor

From Figure 5-2, it can be seen that the BEMF is sinusoidal in shape and thus the Park

transformation can be used to simplify the control scheme. To utilise Park transformation,

another two parameters have to be determined: the BEMF offset (227

4096 𝑟𝑎𝑑) and the torque

constant (𝑘𝑇 = 0.537 𝑁𝑚𝐴−1).

A close-up view of the BEMF shapes as shown in Figure 5-3 show that the BEMF for the

three phases are not equal in amplitude and these corresponds to harmonics appearing at

orders other than the 10th.

0 500 1000 1500 2000 2500 3000 3500 4000-0.4

-0.2

0

0.2

0.4

Encoder

Voltage (

V)

Shape of BEMF

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

Harmonics

Magnitude

FFT of BEMF

Figure 5-3: BEMF Shapes of the Experimental Motor (close-up)

To investigate the impact of BEMF imbalances among the 3 waveforms on the output

torque, perfect sinusoidal currents at maximum amplitude for the three phases were

simulated and passed through the measured BEMF. The electromagnetic torque can then

be found using equation 2.4. Figure 5-4 shows the maximum torque ripple as a result of

BEMF imbalances among the three phases which can cause a maximum TRF of 0.9%. These

imbalances may be due to the unequal magnetic field strengths from the individual

magnets.

0 500 1000 1500 2000 2500 3000 3500 4000

0.32

0.34

0.36Shape of BEMF (close-up)

Voltage (

V)

Encoder

0 20 40 60 80 100 1200

1

2

3

4

5x 10

-3

X: 70

Y: 0.0007936

Harmonics

Magnitude

FFT of BEMF (close-up)

0 500 1000 1500 2000 2500 3000 3500 4000-0.1

-0.05

0

0.05

0.1

Encoder Position

Torq

ue R

ipple

(N

m)

Max Torque Ripple caused by BEMF imbalance

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

Harmonics

Magnitude

FFT of Torque Ripple caused by BEMF imbalance

Figure 5-4: BEMF Imbalances

5.2.2 Cogging Torque

An external drive is used to measure the cogging torque, while the motor is not in

operation. In this way, the rotor is not affected by the stator and accurate torque

measurements are possible. A rubber ‘O’ ring drive belt is used together with an external

DC motor to connect to the rotor. Figure 5-5 shows the cogging torque waveform of the

experimental motor. The peak at the 20th and 24th order was caused by the stator and rotor

respectively. These two peaks are the fundamental frequencies of the cogging torque

induced by the stator and rotor. More peaks can then be observed at multiples of 20 and 24.

The peak at the 120th order is the native harmonic of the motor which is the lowest

common multiple of 20 and 24. The cogging torque is approximately 6% of the rated torque.

Figure 5-5: Cogging Torque of the Experimental Motor

It is assumed that the effect of the bearings on the measured cogging torque is negligible.

0 500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder

Tc (

Nm

)

Cogging Torque Waveform

0 20 40 60 80 100 1200

0.05

0.1

Harmonics

Magnitude

FFT of Cogging Torque

5.2.3 Electrical Subsystem

The inductance, L and resistance, R were measured offline and the values were:

𝐿 = 2.954 𝑚𝐻

𝑅 = 0.62 Ω

5.2.4 Mechanical Subsystem

To determine the mechanical subsystem, the output torque and speed were measured as

shown in Figure 5-6. Least square minimisation using Matlab command “arx” was then used

to determine the system. The motor inertia, J was found to be 0.0042 kgm2 and the viscous

friction, b was 0.1457 Nms.

Figure 5-6: Torque and Speed Waveforms

5.3 Design of the Current Controller

The current controller plays a very important part in the ability of any PMSM control

scheme to minimise torque ripples. If the current controller is not fast enough or has a

limited bandwidth, achieving torque ripple minimisation will be restricted to very low

0 500 1000 1500 2000 2500 3000 3500 40000.8

0.9

1

1.1

Theta

Torq

ue (

Nm

)

Torque

0 500 1000 1500 2000 2500 3000 3500 4000

6.4

6.6

6.8

7

7.2

7.4

Theta

Speed r

ad/s

)

Speed

harmonics. The current controller was tuned using the Ziegler Nichols open loop method

[119]. Maximum proportional gain, Ku is found to be 0.4 and the corresponding period, Tu is

11.6 s.

For a PI controller:

𝐾𝑝,𝑐𝑐 = 0.4𝐾𝑢 = 0.4 × 6.1875 = 2.475

𝐾𝑖,𝑐𝑐 =1

𝑇𝑖=

1

0.8𝑇𝑢=

1

0.8×11.66= 0.1072𝑠−1

Figure 5-7 shows the bode plot for the closed loop response of the current controller. The

bandwidth of the PI current controller was found to be 1313.5 Hz which is wide enough for

the required operation.

Figure 5-7: Bode Plot of PI Current Controller

5.4 Discussion

The sampling frequency should at least be five times lower than the lowest resonant

frequency so that the error of system linearity is lower than 5%, otherwise torque

measurements can be affected [120]. The test rig has been designed so that the resonance

-50

-40

-30

-20

-10

0

10

Magnitu

de (

dB

)

100

101

102

103

104

-900

-720

-540

-360

-180

0

Phase (

deg)

Bode Diagram

Frequency (Hz)

Experimental Data

Simulated Plot

frequency is as high as possible. Through experimental analysis of the setup, it was found

that the lowest resonant frequency was 518 Hz [116]. Finite element analysis and modal

analysis using 3D Computer Aided Design (CAD) were also carried out on the same

experimental setup and the lowest resonant frequency was found to be 490 Hz and 498 Hz

respectively [19]. Since the native harmonic of the experimental motor was at the 120

orders, the minimum sampling frequency 𝑓𝑠 is thus:

𝑓𝑠 =1

5(490

120) = 0.82 Hz

Therefore, torque measurements at speeds below 0.82 Hz will most likely not be affected

by the resonant frequencies.

This concludes the description of the experimental setup used for this research. Chapter 6

will provide the experimental results of the ILC methods discussed.

Chapter 6 Experimental Results

Chapter 4 presents the simulated comparison between the various ILC methods and

chapter 5 describes the experimental setup. However, since no torque sensor will be used

for control, Chapter 6 will first discuss the method of torque estimation used. This is then

followed by the experimental results of the control schemes described in Chapter 3 and

simulated in Chapter 4. Lastly, robustness to parameter variations in the torque estimator

used for the various ILC schemes will also be discussed.

6.1 Compensation Scheme Setup

For the experimental implementation, there are several steps involved in the entire

iterative learning process.

1. Torque estimation – to effectively minimise torque ripple, the output torque has to

be known. Since it is not cost effective to use a torque transducer to measure the

output torque in practical applications, the torque has to be estimated.

2. Iterative learning is used to estimate the torque ripple.

3. Pre-compensation – The estimated torque ripple after iterative learning acts as a

pre-compensation signal and is subtracted from the reference torque.

Figure 6.1 shows the schematic of the compensation scheme to control PMSM. The TE

block is the torque estimation block and the error between the estimated torque and the

reference torque is fed into the ILC block. The estimated torque ripple after iterative

learning is then subtracted from the reference torque. Since the torque ripple as discussed

in section 2.4 are periodic to the encoder position, the proposed ILC schemes will be

implemented in the position domain where 𝜃𝑚 is the mechanical position of the rotor.

The Torque Estimation (TE) block is further subdivided into two parts:

1. Estimate the torque from the speed

2. Filtering the estimated torque

Estimate the torque from the speed

In order to capture cogging torque information in the torque estimation, speed information

is needed. However due to the low pass filtering effect of the mechanical system, higher

torque harmonics are not observed in the speed waveform if the motor is at high speed. It

was recommended that the upper speed threshold should be 1

6𝑝 of the speed loop

bandwidth where p is the number of machine pole pairs for this method to be effective.

Thus, torque estimation that includes cogging torque estimation can only be done at low

speeds [4].

Re-arranging equation 2.2, assuming 𝑇𝐿 = 0, the output torque is

𝑇 = 𝐽�̇� + 𝑏𝜔 (6.1)

𝜔𝑚 𝑇𝑟𝑒𝑓 1

𝑘𝑇

𝑖𝑞∗

𝑣𝑠∗

𝑒

FOC PMSM

𝑖𝑑∗ = 0

_ + ILC TE

_ +

�̂� �̂�𝑟𝑖𝑝,𝑗−1 �̂�𝑟𝑖𝑝,𝑗

Compensation scheme

Figure 6-1: Schematic of compensation scheme for PMSM control

Thus, with J and b of the mechanical system known, the output torque can be estimated

using speed information. The position, 𝜃 of the rotor can be measured using an encoder. To

get speed, 𝜔 information, 𝜃 needs to be differentiated. To get �̇� information, 𝜔 has to be

differentiated again. The torque can then be estimated from the above equation. This

result in noise due to the double differentiation that are carried out in the estimated torque

and thus the need for it to be filtered before the estimated torque can be used back on the

system.

If 𝑇𝐿 ≠ 0, the load torque will have to be pre-modelled and added to the above estimated

torque.

Filtering the estimated torque

Very often, filtering of estimated signals is necessary due to the amplification of noise by

differentiating the variables. Feeding a noisy signal back into the system is undesirable and

this makes filtering a very important process. Without a filter, the estimated torque is very

noisy and cannot be used. Figure 6-2 shows the estimated torque using equation 6.2 but

without using a filter. The Root Mean Square (RMS) error of the estimated torque is 9.5%.

Figure 6-2: Torque Estimation without a filter

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5Estimated Torque Without Filter

Encoder Position

Torq

ue (

Nm

)

0 500 1000 1500 2000 2500 3000 3500 4000-0.4

-0.2

0

0.2

0.4

Encoder Position

Torq

ue E

rror

(Nm

)

Error of Estimated Torque Without Filter

RMS Error = 9.5%

A Low Pass Filter (LPF) can be used to remove the high frequency noise. Employing the use

of a simple second order LPF can significantly improve the accuracy of the estimated torque

with a lower RMS error of 4.3% as shown in Figure 6.3. the cutoff frequency is set at 125 Hz

with a damping factor of 0.707.

Figure 6-3: Torque Estimation with LPF

The estimated torque is now cleaner but due to the phase lag, the estimated torque is still

not accurate. Zero Phase Filtering (ZPF) can be used to get a more accurate estimated

torque. The only problem is implementing it real time in the DSP. Figure 6.4 shows how ZPF

can be done real time in the DSP [121]. The output is a filtered version of the input but

without any phase shift. The phase shift caused by the first LPF is negated by the first time

reversal block and the second LPF. Considering a component of the input signal with a

phase of x° and the first LPF causes a phase shift of -α°. The first time reversal block will

then conjugate the phase and adds another phase shift of -δ°. This results in the phase

being (–x + α – δ)° at point A. The second LPF will also result in a phase shift of -α°. Thus

the phase is now (–x – δ)° which then undergoes another time reversal and an additional

phase shift of -δ°. The final output is now x + δ – δ = x°. Therefore, by implementing this

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5Estimated Torque using LPF

Encoder Position

Torq

ue (

Nm

)

0 500 1000 1500 2000 2500 3000 3500 4000-0.4

-0.2

0

0.2

0.4

Encoder Position

Torq

ue E

rror

(Nm

)

Error of Estimated Torque using LPF

RMS Error = 4.3%

filter, we will now have the output with the same phase as the input over the whole

frequency range [121].

To do the time reversal block in real time, a lookup table (LUT) can be used to store all the

time samples. After the LPF, the signal can be inserted into the LUT in a reverse order. The

reverse signal is then passed through the LPF again and gone through the time reversal

block a second time. With this, the DSP-based ZPF can be implemented. Due to the time

reversal block, the estimated torque is not the instantaneous torque as it always lags

behind by two revolutions. Figure 6-5 shows the estimated torque using ZPF with LUT of

size 4096 which has the same resolution as the encoder used in the experimental setup.

The accuracy is now improved with a lower RMS error of 1.9% (56% improvement

compared to just using a LPF)

Figure 6-5: Torque Estimation with ZPF (LUT of size 4096)

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5Estimated Torque using ZPF4096

Encoder Position

Torq

ue (

Nm

)

0 500 1000 1500 2000 2500 3000 3500 4000-0.4

-0.2

0

0.2

0.4

Encoder Position

Torq

ue E

rror

(Nm

)

Error of Estimated Torque using ZPF4096

RMS Error = 1.9%

LPF Time

Reversal

LPF Time

Reversal

Input Output A

x° (x-α) ° (–x + α – δ)° (–x – δ)° x °

Figure 6-4: Implementing DSP based ZPF

The size of the lookup table can be scaled according to the memory available in the DSP. A

larger LUT is able to have more accurate torque estimation but requires more memory

space. Figure 6-6 shows that by using a LUT of size 256, it show a similar RMS error to just

using a LPF. As it can be observed, the filtered signal is of a much lower resolution

(resolution is 16 times smaller).

Figure 6-6 Torque Estimation with ZPF (LUT of size 256)

Using a LUT of different size yields different results as can be seen in table 3.8.

Table 6.1: Comparing size of LUT on torque estimation accuracy

Filters Size of LUT RMS error (%)

No Filter - 9.5

LPF - 4.3

ZPF

4096 1.9

2048 1.9

1024 2.0

512 2.5

256 4.5

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5Estimated Torque using ZPF256

Encoder Position

Torq

ue (

Nm

)

0 500 1000 1500 2000 2500 3000 3500 4000-0.4

-0.2

0

0.2

0.4

Encoder Position

Torq

ue E

rror

(Nm

)

Error of Estimated Torque using ZPF256

RMS Error = 4.5%

Therefore by using a much lower resolution encoder of 512 will still give better result than

just a simple LPF.

6.2 Torque Ripple Factor of Control Schemes

The two aims of this thesis are to see if ILC schemes can minimise torque ripple of a PMSM

without a torque transducer and which ILC schemes that have been discussed are more

effective. Torque Ripple Factor (TRF) will be used to determine the effectiveness of the

control schemes in achieving this aim. A good control method will be able to minimise all

the major torque harmonics to achieve minimal torque ripple.

6.2.1 Field Oriented Control

Figure 6-7 shows the torque ripple when Field Oriented Control (FOC) scheme is being

implemented. The TRF is 8.1%, with the major torque harmonics shown below.

Figure 6-7: Torque Ripple using FOC

0 500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Torque Ripple

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude

FFT of Torque Ripple

TRF = 8.1%

This is similar to the simulation results in which the torque harmonics are caused by cogging

torque (𝑇𝑐𝑜𝑔), non-ideal sinusoidal density flux distribution (𝑇∆𝜆) and current measurement

errors (𝑇∆𝑖).

6.2.2 Pre-compensation Technique

Pre-compensation Control has the potential to achieve the lowest TRF if the contributing

factors of torque ripple can be compensated accurately and completely. The direct and

indirect pre-compensation control results are shown in the next three figures.

Direct Pre-compensation Technique

Figure 6-8 shows the direct pre-compensation control in which speed information (from

equation 6.1) is used to estimate the output torque. Double differentiation of the position

information is needed to obtain the output torque and this result in noise amplification.

The DSP based ZPF as discussed earlier was used to remove the high frequency noise.

Figure 6-8: TRF for Direct FF Control (Using Speed Information)

500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Torque Ripple of Direct Pre-Compensation using Speed Information

TRF (FOC) = 8.1%

TRF (Direct Pre-Compensation) = 1.4%

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


FOC

Direct Pre-Compensation

TRF is a low 1.4% (83% reduction) and most of the peaks have been suppressed. Even if ZPF

has been applied to the estimated torque, there is still a small RMS error of 1.9% remaining.

This result in a higher TRF compared to the Figure 6.9. For comparison, if a torque sensor is

used to measure the output torque, the TRF becomes even lower, 0.8% (90% reduction) as

shown in Figure 6.9. This can be viewed as the best possible result (lowest TRF) that can be

attained experimentally.

The measured torque is inserted into a mechanical rotor position, 𝜃𝑚 , driving the LUT that

has a resolution of 4096. The size of the LUT used affects the degree of accuracy of the

estimated torque and thus affects the TRF. Furthermore, as mentioned in chapter 5 the

torque transducer has an error of ±0.18 Nm or less. These contribute to why the TRF is 0.8%

and not lower.

Figure 6-9: TRF for Direct FF Control (Using a Torque Transducer)

Indirect Pre-compensation Technique

In indirect pre-compensation control, the compensations for cogging torque (𝑇𝑐𝑜𝑔), non-

ideal sinusoidal density flux distribution (𝑇∆𝜆) and current measurement errors (𝑇∆𝑖) are

500 1000 1500 2000 2500 3000 3500 4000

0.8

1

1.2

Encoder Position

Torq

ue (

Nm

)

Torque Ripple of Direct Pre-Compensation using Torque Transducer

TRF (FOC) = 8.1%

TRF (Direct Pre-Compensation) = 0.8%

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


FOC

Direct Pre-Compensation

done separately. The next three figures show the torque ripple if each of these three

contributing factors are compensated individually.

Figure 6-10 shows the output torque when non ideal sinusoidal flux density distribution

(𝑇∆𝜆) is being compensated. There is only a slight improvement of 0.2% as the experimental

motor has a sinusoidal BEMF and there is only a slight imbalance between the three BEMF

waveforms. From Figure 5.3, the maximum TRF due to 𝑇∆𝜆 is approximately 0.9%. For the

case whereby the reference torque is 1 Nm, which is one-third of the maximum torque, the

torque ripple caused by 𝑇∆𝜆 is approximately 1

3× 0.9 = 0.3%. This 0.3% is not fully

compensated could be due to the limited resolution of the LUT used.

Figure 6-10: TRF for Indirect Pre-Compensation - TΔλ

Figure 6-11 shows the TRF as 7.6% when current measurement errors are compensated. It

can be seen that the 10th order (due to current offset errors) is now suppressed. The bulk of

the 20th order is made up mainly by the cogging torque and thus compensation to the

current scaling errors (20th order) is not very effective.

500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Torque Ripple of Indirect Pre-Compensation with T

compensated

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


TRF (FOC) = 8.1%

TRF (Indirect Pre-Compensation) = 7.9%

FOC

Indirect Pre-Compensation

Figure 6-11: TRF for Indirect Pre-Compensation - TΔi

Finally in Figure 6-12, it can be seen that by compensating the cogging torque, it yields the

best result due to its dominance in the torque ripple. A low TRF of 1.4% is achieved just by

compensating the cogging torque. The torque harmonics left are due mainly to the other

two remaining contributing factors.

Figure 6-12: TRF for Indirect Pre-Compensation - Tcog

If all three factors are compensated, a low TRF of 0.9% is achieved as shown in Figure 6-13.

500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder PositionT

orq

ue (

Nm

)

Torque Ripple of Indirect Pre-Compensation with Ti

compensated

TRF (FOC) = 8.1%

TRF (Indirect Pre-Compensation) =7.6%

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


FOC


500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Torque Ripple of Indirect Pre-Compensation with Tcog

compensated

TRF (FOC) = 8.1%


0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


FOC


Figure 6-13: TRF for Indirect Pre-Compensation - All

A breakdown of the TRF when different factors are being compensated can be seen in the

Table 6.2. Table 6.2 shows that when a torque ripple contributing factor has been

compensated, the TRF will decrease.

Table 6.2: Indirect FF Control

Factors compensated TRF (%)

None 8.1

𝑇∆𝜆 7.9

𝑇𝑐𝑜𝑔 1.4

𝑇∆𝑖 7.6

𝑇∆𝜆, 𝑇∆𝑖 7.5

𝑇∆𝜆, 𝑇𝑐𝑜𝑔 1.2

𝑇∆𝑖, 𝑇𝑐𝑜𝑔 1.0

𝑇∆𝜆, 𝑇∆𝑖, 𝑇𝑐𝑜𝑔 0.9

500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Torque Ripple of Indirect Pre-Compensation with T

Tcog

Ti

compensated

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


TRF (FOC) = 8.1%


FOC


How much the TRF decreases depends on its contribution to the torque ripple. Since

cogging torque is the largest contributor to the torque ripple for the experimental motor,

by compensating for cogging torque, the TRF is able to decrease by the largest amount.

However, to achieve torque ripple minimisation, the other two factors have to be

compensated as well. The reason why the indirect FF control method has slightly higher TRF

than the direct FF method as the former methods uses 2 lookup tables for cogging torque

and non-ideal sinusoidal flux density distribution compensation. This result in additional

quantisation errors for the indirect FF method compared to the direct FF method, which

uses only 1 lookup table. This is different from the simulation results in which indirect pre-

compensation (TRF = 0.4%) is lower than using direct pre-compensation (TRF = 0.6%). In

simulation, the cogging torque and BEMF are simulated using LUTs in the PMSM model.

Thus, when indirect pre-compensation is used, near perfect compensation can be achieved

for cogging torque and non-ideal sinusoidal flux density.

6.2.3 Single Channel First Order Iterative Learning Control

This section will show the experimental results for the five ILC schemes that belong to the

category of single channel first order ILC: P-ILC, P-ILC with forgetting factor (Pf-ILC), D-ILC,

PD-ILC and PI-ILC.

P-ILC

Figure 6-14 shows the plot of P-ILC with varying 𝑘𝑝 values.

Figure 6-14: Experimental Plot of P-ILC for different kp values

It can be observed that when 𝑘𝑝 increases, the TRF decreases more rapidly. It can be seen

that when 𝑘𝑝 values goes higher than 0.8, it starts to have a slower convergence instead.

Since torque estimation is not perfect, it begins to have the reverse effect of having higher

TRF when higher 𝑘𝑝 values are used. This is different from the simulation results where the

𝑘𝑝 value of 1.0 can be used as the learning gain to achieve the lowest TRF in the 1st

iteration.

Table 6.3: Experimental Results for P-ILC

𝑘𝑝 TRFss (%) Convergence

0.2 1.3 19 (8)

0.4 1.3 4

0.6 1.4 > 20 (2)

0.8 1.5 5

0.9 1.5 4

1 1.6 3

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of P-ILC for different kp values

kp = 0.2

kp = 0.4

kp = 0.6

kp = 0.8

kp = 0.9

kp = 1.0

Table 6.3 shows the TRFss and convergence for the various learning gains used for P-ILC.

TRFss increases with increasing 𝑘𝑝. This is due to the accumulation of noise in the system. If

the sudden bump in the 19th iteration of the TRFss for 𝑘𝑝 = 0.2 is neglected, the

convergence will happen in 8 iterations. Similarly for the case of 𝑘𝑝 = 0.6. The presence of

non-repeatable causes in the system affects the learning process and may result in the TRF

decreasing and then increases again [104]. The higher the learning gain, the amount of

noise accumulated over time will also be higher. Figure 6-15 shows the output torque for P-

ILC when the learning gain of 𝑘𝑝 = 0.4 was used. It has the lowest TRFss of 1.3% and takes

4 iterations to converge.

Figure 6-15: Plot of P-ILC

Figure 6.16 shows the benefit of using the DSP-based Zero Phase Filter (ZPF) in comparison

to a LPF in the torque estimation process. Due to the more accurate estimated torque

ripple when ZPF is used, a lower TRF was achieved.

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder Position

Torq

ue (

Nm

)

P-ILC: Torque Ripple after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of P-ILC, kp = 0.4


Figure 6-16: Comparing LPF with ZPF in torque estimation

The downside to using DSP based ZPF filter are the additional memory required and the

estimated torque ripple is not the current torque ripple, as discussed in section 6.1.

Pf-ILC

Figure 6-17 shows the impact of different forgetting factors, α on the TRF. Similar to the

simulated results, when α is higher, the TRFss is higher. This is because not 100% of the

previous information is being used in the learning process when α is greater than zero.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of P-ILC (kp = 0.2) using different filters for torque estimation

Using LPF

Using ZPF

Figure 6-17: Plot of Pf-ILC with varying forgetting factors

Figure 6-18 shows the output torque for Pf-ILC.

Figure 6-18: Plot of Pf-ILC

When the learning gain of 𝑘𝑝 = 0.4 and α = 0.05 were used for Pf-ILC, it takes 3 iterations

for the system to converge and a TRFss of 1.3% is achieved.

D-ILC

In D-ILC, filtering of the signal is needed due to differentiation of the error signal. Similar to

the simulation results, figure 6.12 shows that using a LPF with a cutoff frequency of 50 Hz

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F

Pf-ILC (kp = 0.4) comparing forgetting factor

= 0

= 0.05

= 0.10

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder Position

Torq

ue (

Nm

)

Pf-ILC: Torque Ripple after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of Pf-ILC, kp = 0.4, = 0.05


has the best result. Using a lower cutoff frequency of 25 Hz, the error caused by the

additional phase shift results in a slightly higher TRF. Conversely, a higher cutoff frequency

of 100 Hz is not useful as it is unable to remove the noise introduced by the differentiation.

Figure 6-19: Plot of D-ILC with different cutoff frequencies

Figure 6-20 shows the TRF of D-ILC over a range of 𝑘𝑑 values for 20 iterations.

Figure 6-20: Plot of D-ILC with different kd values

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of D-ILC (kd = 0.004) with different cutoff frequencies

25Hz

50Hz

100Hz

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of D-ILC for different kd values

kd = 0.002

kd = 0.003

kd = 0.004

kd = 0.005

kd = 0.006

Table 6.4 shows the TRFss and convergence for the various learning gains used for D-ILC.

Similarly, when the learning gain increases, TRFss increases. It can be seen that the fastest

convergence can be achieved when 𝑘𝑑 = 0.004 with a relatively low TRFss of 1.4%.

Table 6.4: Experimental Results for D-ILC

𝑘𝑑 TRFss (%) Convergence

0.002 1.2 9

0.003 1.3 7

0.004 1.4 5

0.005 1.6 5

0.006 1.9 6

Figure 6-21 shows the output torque when a learning gain of 𝑘𝑑 = 0.004 was used for D-

ILC. TRFss of 1.4% is achieved and it takes about 5 iterations for the system to converge.

Figure 6-21: Plot of D-ILC

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder Position

Torq

ue (

Nm

)

D-ILC: Torque Ripple after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of D-ILC, kd = 0.004

Initial

Final

Initial

Final


PD-ILC

Figure 6-22 shows the plots for different combination of 𝑘𝑝 and 𝑘𝑑 values.

Figure 6-22: Plot of PD-ILC for different kp and kd values

From Table 6.5, it can be observed that PD-ILC with a 𝑘𝑝 value of 0.8 and 𝑘𝑑 value of 0.001

gives the fastest convergence and the lowest TRFss.

Table 6.5: Experimental Results for PD-ILC

𝑘𝑝, 𝑘𝑑 TRFss (%) Convergence

0.4, 0.004 1.6 3

0.6, 0.004 2.7 4

0.8, 0.001 1.4 2

0.8, 0.002 1.6 4

0.8, 0.004 4.9 10

0.9, 0.001 1.8 4

The output torque using this set of values can be seen in Figure 6-23. A TRFss of 1.4% is

achieved and converges after 2 iterations.

0 5 10 15 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of PD-ILC for different kp and k

d values

kp = 0.4, k

d = 0.004

kp = 0.6, k

d = 0.004

kp = 0.8, k

d = 0.001

kp = 0.8, k

d = 0.002

kp = 0.8, k

d = 0.004

kp = 0.9, k

d = 0.001


PI-ILC

Figure 6-24 shows the plots for different combination of 𝑘𝑝 and 𝑘𝑖 values.

Figure 6-24: PI-ILC

From Figure 6-24 and Table 6.6, PI-ILC does not perform better than the P-ILC. This is similar

to the simulation results. However, if 𝑘𝑖 becomes too big, the TRF will begin to grow and

the system becomes unstable.

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder Position

Torq

ue (

Nm

)

PD-ILC: Torque Ripple after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of PD-ILC, kp = 0.8, k

d = 0.001

Initial

Final

Initial

Final


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F

PI-ILC (kp = 0.4) comparing different k

i values

ki = 0

ki = 0.05

ki = 0.10

ki = 0.20

Table 6.6: Experimental Results for PI-ILC

𝑘𝑖 TRFss (%) Convergence

0 1.3 4

0.05 1.3 4

0.10 1.3 4

0.20 1.6 >20

Figure 6.25 shows that by using PI-ILC, TRFss was reduced to 1.3% and convergence to

steady TRF is achieved in 4 iterations.

Figure 6-25: Plot of PI-ILC

Comparison of SCFO-ILC Schemes

Figure 6-26 and Table 6.7 show the comparison between the 5 SCFO-ILC schemes. It can be

seen that among the 5 SCFO-ILC schemes, PD-ILC takes the least number of iterations to

converge. D-ILC on the other hand is the slowest, taking 6 iterations to converge.

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder Position

Torq

ue (

Nm

)

PI-ILC: Torque Ripple after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of PI-ILC, kp = 0.4, k

i = 0.10


Figure 6-26: Comparison of Single Channel First Order ILC Schemes

In terms of TRFss, D-ILC and PD-ILC have slightly higher TRF compared to the other 3 ILC

schemes. If a lower TRF is desired, Pf-ILC may be the best choice but if faster convergence is

required, PD-ILC may be more suitable.

Table 6.7: Comparing different Single Channel First Order ILC Schemes

ILC Methods TRFss (%) Convergence

P 1.3 4

Pf 1.3 3

D 1.4 5

PD 1.4 2

PI 1.3 4

Therefore, Pf-ILC and PD-ILC can be said to be the best among these 5 SCFO-ILC schemes

and will be used as comparisons for the other categories of ILC schemes in the next sections.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F

Comparison of Single Channel First Order ILC

P-ILC

Pf-ILC

D-ILC

PD-ILC

PI-ILc

6.2.4 Multi-Channel Iterative Learning Control

In MC-ILC, the torque harmonics can be separated into different regions and being

compensated separately. Depending on the motors to be tested and its characteristics, it

may not be necessary to employ more learning channels due to the following reasons:

1. Memory space limitation – Filtering is required to separate the torque harmonics

into different portions. If DSP-based ZPF is used, there will be a need for more

memory spaces. Thus, the more channels used, the more memory is required.

2. The gap between the torque harmonics – the torque harmonics have to be

reasonably far apart for the filtering process to be effective.

Thus, for the experimental setup, two channel iterative learning will be used. It can be seen

from Figure 6-27 that if the same learning gains are chosen for multi-channel ILC and the

single channel ILC, the graphs exhibit similar reduction in TRF. The only difference between

them is the noise that is left after all the major torque harmonics are removed.

Figure 6-27: Comparing P-ILC and MC-ILC

To gain the most benefit out of multi-channel learning, different learning gains can be used

for different channels instead. The learning gain for the lower frequencies, 𝑘𝑝,𝑙𝑜𝑤 can be set

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparing P-ILC and MC-ILC

P-ILC: kp = 0.4

MC-ILC: kp,low

= kp,high

= 0.4

P-ILC: kp = 0.8

MC-ILC: kp,low

= kp,high

= 0.8

higher and the learning gain for the higher frequencies, 𝑘𝑝,ℎ𝑖𝑔ℎ can be lower. Figure 6-28

shows the plot of MC-ILC with 𝑘𝑝,𝑙𝑜𝑤 = 0.9 for a range of 𝑘𝑝,ℎ𝑖𝑔ℎ values.

Figure 6-28: Plot of MC-ILC for different kp,high values

From Table 6.8, for 𝑘𝑝,ℎ𝑖𝑔ℎ = 0.3, MC-ILC has the lowest TRFss and the fastest convergence.

Table 6.8: Experimental Results for MC-ILC

𝑘𝑝,ℎ𝑖𝑔ℎ TRFss (%) convergence

0.2 1.3 3

0.3 1.2 2

0.4 1.3 8

0.5 1.2 15

0.6 1.3 3

Figure 6.29 shows the plot for 𝒌𝒑,𝒍𝒐𝒘 = 𝟎. 𝟗 and 𝒌𝒑,𝒉𝒊𝒈𝒉 = 𝟎. 𝟑. A TRFss of 1.2% is achieved

and converges after 2 iterations.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Plot of MC-ILC, kp,low

= 0.9 and different kp,high

values

kp,high

= 0.2

kp,high

= 0.3

kp,high

= 0.4

kp,high

= 0.5

kp,high

= 0.6

Figure 6-29: Plot of MC-ILC

6.2.5 Higher Order Iterative Learning Control

Figure 6-30 and Figure 6-31 show the TRF of HO-ILC using varying learning gains of 𝑘𝑝1and

𝑘𝑝2.

Figure 6-30: HO-ILC with varying learning gains

Some instability in the TRF can be seen when 𝑘𝑝1 ,𝑘𝑝2 ≥ 0.6

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder PositionT

orq

ue (

Nm

)

MC-ILC: Torque Ripple after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of MC-ILC, kp,low

= 0.9, kp,high

= 0.3


0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F

HO-ILC

kp1

= kp2

= 0.2

kp1

= kp2

= 0.4

kp1

= kp2

= 0.6

kp1

= kp2

= 0.8

Figure 6-31: HO-ILC with different learning gains

Figure 6-32 shows the torque ripple when 𝑘𝑝1 = 0.4 and 𝑘𝑝2 = 0.4. A low TRFss of 1.1%

was achieved and it takes 5 iterations for the system to converge.

Figure 6-32: Plot of HO-ILC

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F

HO-ILC

kp1

= 0.2, kp2

= 0.8

kp1

= 0.4, kp2

= 0.6

kp1

= 0.6, kp2

= 0.4

kp1

= 0.8, kp2

= 0.2

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder Position

Torq

ue (

Nm

)

HO-ILC: Torque Ripple after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of HO-ILC, kp1

= kp2

= 0.4

Initial

Final

Initial

Final


6.2.6 Adaptive Iterative Learning Control

Adaptive ILC uses a variable learning gain that changes with the error. The adaptive scheme

chosen will determine the rate of convergence.

Adaptive P-ILC

The next 3 figures shows the TRF and the changes to the variable learning gain, µ for

different values of α and c. As a recap, α (0 ≤ α ≤ 1) is a smoothing factor and c is a positive

constant. When error is big, µ becomes big and when error becomes smaller, µ decreases

as well.


Figure 6-33 shows that by using a lower c value, µ becomes larger in the first iteration and

has higher learning gain on the whole.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Iterations

TR

F(%

)


0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


c = 0.005

c = 0.01

c = 0.02

c = 0.005

c = 0.01

c = 0.02


Similar trends were observed in Figure 6-34 and Figure 6-35.


The values of α = 0.1 and c = 0.01 are chosen due as they gives the best trade-off between

convergence rate and a low stable TRF. The output torque can be seen in Figure 6.36 for

this pair of chosen parameters.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

IterationsT

RF

(%)


c = 0.005

c = 0.01

c = 0.02

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


c = 0.005

c = 0.01

c = 0.02

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Iterations

TR

F(%

)


0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


c = 0.005

c = 0.01

c = 0.02

c = 0.005

c = 0.01

c = 0.02

Figure 6-36: Plot of Adaptive P-ILC (Output Torque)

A TRFss of 1.3% is achieved and the system converges in about 2 iterations. The variable

learning gain, µ increases from 0 to about 0.82 in the first iteration and drops steadily to 0.1

in about 4 iterations as shown in Figure 6.37.

Figure 6-37: Plot of Adaptive P-ILC (TRF)

500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Adaptive P-ILC: Torque Ripple after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of Adaptive P-ILC, = 0.1, c = 0.01

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations



Adaptive PD-ILC

An additional D-ILC can be used together with the adaptive P-ILC. The range of different 𝑘𝑑

values used together with adaptive P-ILC can be seen in Figure 6-38.

Figure 6-38: Plot of Adaptive PD-ILC for different kd values

Choosing a 𝑘𝑑 value of 0.001 has the lowest TRF. The variable learning gain, µ increases

from 0 to 0.8 and drops steadily to 0.1. Higher values of 𝑘𝑑 results in µ retaining the high

learning gain even though TRF has drop. Figure 6-39 shows the output torque using

adaptive P-ILC and 𝑘𝑑 = 0.001.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Iterations

TR

F(%

)Plot of VPD-ILC for different k

d values, a = 0.1, c = 0.01

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations

miu


kd = 0.001

kd = 0.002

kd = 0.004

Figure 6-39: Plot of Adaptive PD-ILC (Output Torque)

A TRFss of 1.2% is achieved and converges in 2 iterations.

Figure 6-40: Plot of Adaptive PD-ILC (TRF)

Figure 6-41 compares adaptive P-ILC with adaptive PD-ILC using the same values of c and α.

500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Adaptive PD-ILC: Torque Ripple after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of Adaptive PD-ILC, = 0.1, c = 0.01, kd = 0.001


0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


Figure 6-41: Comparing Adaptive P-ILC and Adaptive PD-ILC

Adaptive PD-ILC has a TRFss of 1.2% and converges in 2 iterations while adaptive P-ILC has a

TRFss of 1.3% and also converges in 2 iterations. Adaptive PD-ILC is able to perform slightly

better than adaptive P-ILC in terms of TRFss. As a comparison to the non-adaptive P-ILC and

PD-ILC, Figure 6-42 shows the TRF of these four schemes over 20 iterations.

Figure 6-42: Comparing adaptive and non-adaptive ILC

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

IterationsT

RF

(%)

Comparing Adaptive P and Adaptive PD ILC

Adaptive PD-ILC

Adaptive P-ILC

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations


c = 0.01, = 0.1, kd = 0.001

c = 0.01, = 0.1

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)

Comparison of ILC Methods

P-ILC

PD-ILC

Adaptive P-ILC

Adaptive PD-ILC

It can be seen Table 6.9 that adaptive PD-ILC had lower TRF than PD-ILC and the adaptive P-

ILC was able to converge faster than P-ILC.

Table 6.9: Comparison between adaptive and non-adaptive ILC


P 1.3 4

PD 1.4 2

Adaptive P 1.3 2

Adaptive PD 1.2 2

Adaptive ILC is able to have either faster convergence or lower TRFss compared to the non-

adaptive ILC. Adaptive PD-ILC due to its fast convergence and low TRFss will be used to

represent adaptive ILC to compare with other categories of ILC schemes.

Comparison between different categories of ILC schemes

Figure 6-43 shows the TRF over 20 iterations for the 4 categories of ILC schemes.

Figure 6-43: Comparing between different categories of ILC schemes

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)


Pf-ILC

PD-ILC

MC-ILC

HO-ILC

Adaptive PD-ILC

It can be seen from Table 6.9 that HO-ILC has the lowest TRFss while PD-ILC, MC-ILC and

adaptive PD-ILC took the least number of iterations to converge. This is quite similar to the

simulation results in which HO-ILC has the least TRFss and MC-ILC and adaptive PD-ILC have

the fastest convergence. On the whole, MC-ILC, HO-ILC and adaptive ILC seemed to

perform better than SCFO-ILC in terms of TRF and rate of convergence.

Table 6.10: Comparison between different categories of ILC schemes


Pf 1.3 3

PD 1.4 2

MC 1.2 2

HO 1.1 5

Adaptive PD 1.2 2

6.2.7 Multi-Channel Higher Order Iterative Learning Control

MCHO-ILC combines both MC-ILC and HO-ILC. Figure 6-44 shows the torque ripple when

MCHO-ILC was used. A low TRF of 1.2% is achieved and it takes 3 iterations for this scheme

to converge.

Figure 6-44: Plot of MCHO-ILC

6.2.8 Multi-Channel Adaptive Iterative Learning Control

As discussed in chapter 4, there are benefits to both variable learning and multi-channel

learning. The Multi-Channel Adaptive ILC (MCA-ILC) scheme has the benefits of adaptability,

a low TRF and fast convergence. In this control scheme, the learning gains for the higher

and lower frequencies are adapting according to the error.

Figure 6-45: Plot of MCA-ILC (Output Torque)

500 1000 1500 2000 2500 3000 3500 4000

-0.2

0

0.2

Encoder PositionT

orq

ue (

Nm

)

MCHO-ILC: Torque Ripple after 20 iterations

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of MCHO-ILC, kp1,low

= 0.9, kp2,low

= 0.1, kp1,high

= 0.9, kp2,high

= 0.1

Initial

Final

Initial

Final


500 1000 1500 2000 2500 3000 3500 4000

-0.2

-0.1

0

0.1

0.2

Encoder Position

Torq

ue (

Nm

)

Multi-Channel Adaptive P-ILC: Torque Ripple after 20 iterations

Initial

Final

0 20 40 60 80 100 1200

0.05

0.1

Orders

Magnitude


Initial

Final

From Figure 6-45 and Figure 6-46, a TRFss of 1.2% is achieved in 2 iteration and learning

gains for both the low and high channels stabilized at about 0.2 and 0.1 respectively.

Figure 6-46: Plot of MCA-ILC (TRF)

6.2.9 Comparison of ILC Schemes

Figure 6-47 and Table 6.11 show the comparison of the two new ILC schemes to other

categories of ILC schemes.

Figure 6-47: Comparison of proposed ILC schemes with existing ILC schemes

0 2 4 6 8 10 12 14 16 18 200

5

10

Iterations

TR

F(%

)

Plot of Multi-Channel Adaptive P-ILC, a = 0.1, clow

= chigh

= 0.005

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Iterations



low

high

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Iterations

TR

F(%

)


Pf-ILC

PD-ILC

MC-ILC

HO-ILC

Adaptive PD-ILC

MCHO-ILC

MCA-ILC

MCHO-ILC is able to perform reasonably well with a low TRFss of 1.2% and converges in 3

iterations. The downside to this ILC scheme is the large number of LUTs needed. MCA-ILC

on the other hand does not required large numbers of LUTs, able to adjust the learning

gains according to the error, has a low TRFss and is able to converge in 2 iterations.

Table 6.11: Comparison of proposed ILC schemes with other ILC schemes


(no. of iterations)

Pf 1.3 3

PD 1.4 2

MC 1.2 2

HO 1.1 5

Adaptive PD 1.2 2

MCHO 1.2 3

MCA 1.2 2

6.3 Variations to motor parameters J and b

Using equation 6.2 to estimate torque from speed information assumes that J and b are

constant and do not vary with time. Moreover, it assumes these values are accurate. If

there are any inaccuracies or if these parameters changes over time, the estimated torque

will not be accurate. This section will discuss about the robustness of the control schemes

to variations in J and b.

Figure 6-48 shows the robustness of direct pre-compensation technique to variations in J

and b. There is a 47.0% variation in TRF for the J variation of ±10% and 22.0% variation of

TRF for a variation of b of ±5%.

Figure 6-48: Robustness of Direct Pre-Compensation Technique

Figure 6-49 shows the robustness of P-ILC to variations in J and b. There is a 14.8% variation

of TRF for a J variation of ±10% and 22.0% variation of TRF for a b variation of ±5%.

Figure 6-49: Robustness of P-ILC

Figure 6-50 shows the robustness of P-ILC with forgetting factor to variations in J and b.

There is a 11.3% variation of TRF for a J variation of ±10% and 4.7% variation of TRF for a b

-10% -5% 0 +5% +10%-20

0

20

40

Variation of J%

Change in T

RF

Robustness of Direct Pre-Compensation Technique

-5% -2.5% 0 +2.5% +5%0

10

20

30

Variation of b

% C

hange in T

RF

-10% -5% 0 +5% +10%-15

-10

-5

0

Variation of J

% C

hange in T

RF

Robustness of P-ILC, kp = 0.4

-5% -2.5% 0 +2.5% +5%-20

-10

0

10

20

Variation of b

% C

hange in T

RF

variation of ±5%. By adding a forgetting factor, the robustness of the scheme has indeed

improved.

Figure 6-50: Robustness of Pf-ILC

Figure 6-51 shows the robustness of D-ILC to variations in J and b. There is a low 7.1%

variation of TRF for a J variation of ±10% and 8.2% variation of TRF for a b variation of ±5%.

Figure 6-51: Robustness of D-ILC

-10% -5% 0 +5% +10%-10

-5

0

5

Variation of J

% C

hange in T

RF

Robustness of Pf-ILC, kp = 0.4, = 0.05

-5% 0 +5%-4

-2

0

2

Variation of b

% C

hange in T

RF

-10% -5% 0 +5% +10%-5

0

5

Variation of J

% C

hange in T

RF

Robustness of D-ILC, kd = 0.004

-5% -2.5% 0 +2.5% +5%0

5

10

Variation of b

% C

hange in T

RF

Figure 6-52 shows the robustness of PD-ILC to variations in J and b. There is a 12.5%


Figure 6-52: Robustness of PD-ILC

Figure 6-53 shows the robustness of PI-ILC to variations in J and b. There is a 17.8%

variation of TRF for a J variation of ±10% and a large 63.0% variation of TRF for a b variation

of ±5%.

Figure 6-53: Robustness of PI-ILC

-10% -5% 0 +5% +10%-10

-5

0

5

Variation of J

% C

hange in T

RF

Robustness of PD-ILC, kp = 0.8, k

d = 0.001

-5% -2.5% 0 +2.5% +5%-10

0

10

20

30

Variation of b

% C

hange in T

RF

-10% -5% 0 +5% +10%-10

-5

0

5

10

Variation of J

% C

hange in T

RF

Robustness of PI-ILC, kp = 0.4, k

i = 0.1

-5% -2.5% 0 +2.5% +5%0

20

40

60

80

Variation of b

% C

hange in T

RF

Figure 6-54 shows the robustness of MC-ILC to variations in J and b. There is a low 5.1%


Therefore, MC-ILC is robust to variations to both J and b.

Figure 6-54: Robustness of MC-ILC

Figure 6-55 shows the robustness of HO-ILC to variations in J and b. There is a 24.9%


Figure 6-55: Robustness of HO-ILC

-10% -5% 0 +5% +10%0

2

4

6

Variation of J

% C

hange in T

RF

Robustness of MC-ILC

-5% -2.5% 0 +2.5% +5%-10

-5

0

5

Variation of b

% C

hange in T

RF

-10% -5% 0 +5% +10%-20

-10

0

10

20

Variation of J

% C

hange in T

RF

Robustness of HO-ILC

-5% -2.5% 0 +2.5% +5%-20

-10

0

10

20

Variation of b

% C

hange in T

RF

Figure 6-56 shows the robustness of adaptive P-ILC to variations in J and b. There is a 15.9%


Figure 6-56: Robustness of Adaptive P-ILC

Figure 6-57 shows the robustness of adaptive PD-ILC to variations in J and b. There is a

11.7% variation of TRF for a J variation of ±10% and 16.9% variation of TRF for a b variation

of ±5%.

Figure 6-57: Robustness of Adaptive PD-ILC

-10% -5% 0 +5% +10%-20

-15

-10

-5

0

Variation of J

% C

hange in T

RF

Robustness of Adaptive P-ILC

-5% -2.5% 0 +2.5% +5%-15

-10

-5

0

Variation of b

% C

hange in T

RF

-10% -5% 0 +5% +10%-10

-5

0

5

10

Variation of J

% C

hange in T

RF

Robustness of Adaptive PD-ILC

-5% -2.5% 0 +2.5% +5%0

5

10

15

20

Variation of b

% C

hange in T

RF

Figure 6-58 shows the robustness of MCHO-ILC to variations in J and b. There is a 24.5%


MCHO-ILC is not very robust to variations in J and b.

Figure 6-58: Robustness of MCHO-ILC

Figure 6-59 shows the robustness of MCA-ILC to variations in J and b. There is a 7.2%


Figure 6-59: Robustness of MCA-ILC

-10% -5% 0 +5% +10%-10

0

10

20

Variation of J

% C

hange in T

RF

Robustness of MCHO-ILC

-5% -2.5% 0 +2.5% +5%-20

0

20

40

Variation of b

% C

hange in T

RF

-10% -5% 0 +5% +10%-10

-5

0

5

Variation of J

% C

hange in T

RF

Robustness of MCA-ILC

-5% -2.5% 0 +2.5% +5%-10

-5

0

5

Variation of b

% C

hange in T

RF

To easily compare the robustness of the different control schemes, Table 6.12 will be used.

Table 6.12: Robustness to J and b variations

Average between torque gain

and offset variations

Robustness

< 10% Very Good

10% - 15% Good

15% - 20% Average

20% - 25% Bad

> 25% Very Bad

Table 6.13: Comparison of robustness of control schemes

Control Methods

% Variation in

TRF for ±10%

variation in J (%)

% Variation in

TRF for ±5%

variation in b (%)

Robustness

Pre-Compensation

(Direct) 47.0 22.0

Very Bad

P-ILC 14.8 22.0 Average

Pf-ILC 11.3 4.7 Very Good

D-ILC 7.1 8.2 Very Good

PD-ILC 12.5 28.6 Bad

PI-ILC 17.8 63.0 Very Bad

MC-ILC 5.1 7.7 Very Good

HO-ILC 24.9 21.2 Bad

Adaptive P-ILC 15.9 11.6 Good

Adaptive PD-ILC 11.7 16.9 Good

MCHO-ILC 24.5 37.4 Very Bad

MCA-ILC 7.2 9.8 Very Good

Table 6.13 shows the comparison of robustness to variations in J and b for the various

control schemes that used the torque estimator discussed in section 6.1. Indirect pre-

compensation technique and FOC do not use torque estimator in the control scheme and

thus excluded in this comparison. It can be seen that direct pre-compensation technique

has one of the worst performance in terms of robustness compared to other control

schemes. Appendix A shows the effect of how various parameters changes with

temperature. For pre-compensation techniques that are not adaptive, these variations will

have a great impact on the TRF. On the other hand, D-ILC, MC-ILC, Pf-ILC and MCA-ILC are

among the best in terms of robustness compared to other ILC schemes. As mentioned in

literature, the use of the forgetting factor (Pf-ILC) improves the robustness of P-ILC and thus

less likely to be impacted by the variations in J and b.

6.4 Discussion

The TRF for FOC and pre-compensation control methodologies have been included as

comparison in Table 6.14. It can be seen that pre-compensation methods have the lowest

TRFss. As expected, the downside of the pre-compensation methods is its robustness. It also

assumes that the mechanics of the system to be controlled can be accurately modelled. Any

inaccuracies may result in a higher TRF. Although the pre-compensation methods discussed

are not robust to parameter variations, they can serve as a benchmark for evaluating other

controllers in terms of torque ripple.

SCFO-ILC, PD-ILC and D-ILC had the lowest TRFss. PI-ILC had similar performance as P-ILC.

This showed that the I-term in ILC was not really useful in minimising torque ripple. On the

whole, MC-ILC, HO-ILC and adaptive ILC were able to perform better than SCFO-ILC in terms

of TRFss and convergence. MC-ILC, adaptive PD-ILC and the proposed MCA-ILC can be said

to be the best ILC schemes compared in terms of their overall performance in convergence,

TRFss and robustness. MCA-ILC has adaptable learning gains while MC-ILC does not. This

gives MCA-ILC the ability to cope with other types of parameter changes that may occur

due to temperature changes (for further information on temperature variability, refer to

Appendix A). Although adaptive PD-ILC also has the ability to adapt due to the variable

learning gain, the use of differentiation in the “D” learning may not be effective in a noisy

environment.

Table 6.14: Comparison of all control methods

Control Methods

Categories

TRFss

(%)

Convergence

(no. of iterations)

Robustness

FOC - 8.1 - -

Pre-compensation Direct 0.8 - Very Bad

Indirect 0.9 - -

SCFO-ILC

P-ILC 1.3 4 Average

Pf-ILC 1.3 3 Very Good

D-ILC 1.4 5 Very Good

PD-ILC 1.4 2 Bad

PI-ILC 1.3 4 Very Bad

MC-ILC 2 Channels 1.2 2 Very Good

HO-ILC 2nd Order 1.1 5 Bad

Adaptive ILC P 1.3 2

Good

PD 1.2 2 Good

Proposed ILC Schemes MCHO 1.2 3

Very Bad

MCA 1.2 2 Very Good

Table 6.15 shows the results from both simulations and experiments for the various

categories of ILC schemes.

6.15: Table of Comparison between ILC Schemes (1)

Control Schemes Simulations Experimental

TRFss

(%)

Convergence

(no. of iterations)

TRFss

(%)

Convergence

(no. of iterations

SCFO-ILC (P) 1.0 4 1.3 4

MC-ILC (2) 0.5 2 1.2 2

HO-ILC (2nd) 0.4 3 1.1 5

Adaptive ILC (PD) 1.0 2 1.2 2

MCHO-ILC 0.4 3 1.2 3

MCA-ILC 0.5 1 1.2 2

From Table 6.15, similar trends between simulated and experimental results can be

observed:

HO-ILC has the lowest TRFss

P-ILC has the highest TRFss

MCA-ILC has one of the fastest convergence

The gaps between the various ILC schemes in experimental results are not as wide as in

simulations. The experimental results depend largely on the accuracy of the estimated

torque. This is not a problem for the simulated results as the simulated torque is used in

the iterative learning process. However, the simulation results showed that MCA-ILC has

the potential to achieve one step convergence.

Table 6.16 shows the comparison between the different categories of ILC schemes in terms

of robustness, adaptability, learnable band and the number of LUTs needed. Of the four

categories of ILC, SCFO-ILC and HO-ILC have the widest learnable band. HO-ILC is not robust

to parameter variations of J and b whereas MC-ILC is very robust. However, MC-ILC needs

the most number of LUTs among the four categories. Adaptive ILC has the benefit of a

variable learning gain and is also relatively robust to parameter variations of J and b.

6.16: Table of Comparison between ILC Schemes (2)

Control

Schemes

Robustness Adaptable

learning

gain

Learnable

band

Number of

LUT needed

SCFO-ILC

(P)

Average No Wide 2

MC-ILC (2) Very Good No Narrow 5

HO-ILC

(2nd)

Bad No Wide 4

Adaptive

ILC (PD)

Good Yes Varies 3

MCHO-ILC Very Bad No Narrow 8

MCA-ILC Very Good Yes Varies 5

The two proposed ILC schemes allow a higher degree of freedom in tuning the learning

process. This gives the two schemes more flexibility in the learning process and thus has the

potential to achieve better performance. MCHO-ILC in comparison to SCFO-ILC is able to

converge faster and has lower TRF (from Table 6.15). However, it is not robust to parameter

variations. MCA-ILC on the other hand, is very robust, has very low TRF and converges the

fastest. This makes MCA-ILC an ideal ILC schemes to be applied in industrial applications as

the proposed scheme was carried out without the need of an additional computer.

If adaptable learning gain is needed, choosing adaptive ILC will suffice as it does not require

too much memory space and is robust to parameter variations. If memory space is not an

issue, MCA-ILC will be the better choice due to the potential of lower TRF and faster

convergence as simulated. If adaptable learning gain is not needed and memory space is

limited, P-ILC is a good choice and forgetting factor can be included to improve robustness.

MC-ILC requires more memory space and is ideal for cluster of torque ripple harmonics in

which the filtering process can work ideally. HO-ILC may be used if a low torque ripple is

required. MCHO-ILC in theory works fine but may not be ideal in real world applications due

to the large number of memory space needed and the narrow learnable band despite its

low TRFss and fast convergence.

This summarise the results from chapter 6, chapter 7 will present conclusions and

recommendations for future work.

Chapter 7 Conclusion

In this research, various Iterative Learning Control (ILC) schemes were investigated and

compared against their effectiveness in minimising torque ripple for Permanent Magnet

Synchronous Machines (PMSMs). Robustness for parameter changes were evaluated as

well.

For PMSMs, the contributors of torque ripple are periodic to the rotor position and this

makes ILC an attractive method of control used in conjunction with Field Oriented Control

(FOC) of PMSMs. However, only the Proportional type ILC (P-ILC) and its variations have

been explored by researchers to minimise torque ripple for PMSMs. Although other ILC

schemes are described in literature, their effectiveness to minimise torque ripple for a

PMSM had not yet been investigated.

This thesis investigated first the Single Channel First Order ILC (SCFO-ILC) such as D-ILC, PD-

ILC and PI-ILC. These schemes proved to be equally effective in minimising torque ripple

compared to P-ILC. P-ILC with the optimal learning gain of 1 has the potential to achieve

convergence in a single iteration. However, this assumes that the estimated torque is equal

to the actual torque which is not possible in practice. In this case, other ILC methods can

work in conjunction with P-ILC such as PD-ILC that utilises both P-type and D-type learning

rules. On the whole, SCFO-ILC is straightforward to implement, requires minimal memory

storage and has a wide learnable band.

Other categories of ILC were also investigated. MC-ILC has fast convergence, low Torque

Ripple Factor (TRF), high robustness to parameter variations in the torque estimator, but it

requires more memory space and has a much narrower learnable band. HO-ILC also has low

TRF and a wide learnable band. However, it also requires a significant amount of memory

space and is not robust to parameter variations in the torque estimator. Lastly, adaptive ILC

has fast convergence with an adaptive learning gain and is relatively robust to parameter

variations in the torque estimator.

Table 6.14 compares all the various methods discussed in terms of steady state TRF, steps

to convergence and robustness. ILC schemes have shown to be very effective in their

overall performance to achieve a low TRF and remain fairly robust to certain parameter

changes. Lastly, Table 6.16 compares the various ILC schemes in terms of their robustness,

range of learnable band, memory space needed and whether the learning gain is adaptive.

The newly proposed ILC scheme, MCHO-ILC, has a low TRF but is not robust and requires a

lot of memory space. The also proposed MCA-ILC on the other hand has fast convergence,

low TRF, is very robust, making it a suitable ILC scheme for many PMSMs applications.

7.1 Further Work

This thesis covered a selection of the most widely described ILC schemes. There are other

variations of ILC schemes which can be investigated in the future.

There are three types of SCFO-ILC schemes that were not investigated in this thesis that

may also be suitable: ILC in which the data of the current cycle is used together with the

data of the previous cycle; optimal ILC whereby the learning gains are calculated to achieve

optimal convergence; and adaptive ILC which uses other adaptive schemes such as fuzzy

logic or neural network.

Frequency based ILC could also be further investigated. A more powerful DSP would be

required as a frequency domain based ILC will require significantly more computing

resources.

Although ILCs were used as a cascade structure to existing control structures of the PMSM

(refer to section 3.4), other structures consisting of only ILC may also be beneficial, in

particular to overcome the limitations of the bandwidth of the PID controlled current loop.

This is another area that could be further investigated.

Lastly, it is assumed that once the torque ripple is compensated at low speed, the motor

can then run at much higher speed without the associated torque ripple. The performance

of the proposed methods can be tested over a range of speeds to investigate whether

similar amount of torque reduction is possible.

Appendix A

Variation of parameters with temperature will have an impact on the output torque. The

impact of cogging torque, BEMF and current scaling and offset errors with temperature

changes were investigated with the results shown below.

A.1 Cogging Torque Variation with Temperature

The cogging torque waveform for the test motor in CDU is taken at different temperature

ranging from 25°C to 55°C as shown in Figure A.0-1.

Figure A.0-1: Cogging Torque Variation with Temperature

It can be seen that there is little to no phase shift for the cogging torque waveforms at

different temperature. However, the amplitude decreases with increasing temperature.

There is a 5.4% drop (from 0.2521 Nm to 0.2384 Nm) in cogging torque when temperature

increases from 25°C to 55°C. This is due to the widening of the air gap and the changes in

the magnetic strength with increasing temperature. If there is no pre-compensation for

0 500 1000 1500 2000 2500 3000 3500 4000

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Encoder Position

Coggin

g T

orq

ue (

Nm

)

Cogging Torque Variation with Temperature (C)

25C

30C

35C

40C

45C

50C

55C

cogging torque, an increase in temperature, resulting in a drop in cogging torque, would

thus result in a drop in torque ripple (neglecting the effects on other parameters).

Figure A.0-2: Maximum Amplitude of Cogging Torque Variation with Temperature

From Figure A.0-2, it can be observed that the maximum amplitude of the cogging torque

waveforms decreases with increasing temperature. If there is pre-compensation for cogging

torque without the ability to adapt to temperature changes, this would have a reverse

effect of an increase torque ripple. This is due to the incorrect pre-compensation being put

into the system. To investigate the effect of cogging torque changes on the TRF, the actual

cogging torque is first measured with a torque transducer and placed in a lookup table (LUT)

using the indirect pre-compensation technique. All other factors that can contribute to

torque ripple are also being compensated. The TRF is 1.17% with no obvious harmonics till

the 120th order. This remaining 1.17% TRF is mainly due to noise. This TRF is different from

the experimental results in chapter 6 for indirect pre-compensation technique as a different

rotor was used.

Assuming a linear relationship between temperature and the amplitude of the cogging

torque, an increase of 30°C result in a drop of 5.4% in cogging torque amplitude, then an

25 30 35 40 45 50 550.238

0.24

0.242

0.244

0.246

0.248

0.25

0.252

0.254

Temperature (C)

Coggin

g T

orq

ue (

Nm

)Maximum Amplitude of Cogging Torque Variation with Temperature

increase of 120°C (assuming operating temperature of about 145°C) would cause a drop of

22.4% in cogging torque amplitude.

To investigate the impact of cogging torque amplitude changes on TRF, the gain for the pre-

compensation for cogging torque was varied from 0.8 to 1.2 and the output torque was

measured.

Figure A.0-3: Plot of Cogging Torque Variations with TRF

Figure A.3 shows how the gain of the pre-compensated cogging torque would affect the

TRF. TRF increases from 1.17% to 2.52% when the gain of the cogging torque decreases

from 1 (at 25°C) to 0.8 (at 134°C). When pre-compensation of cogging torque is not

correctly adjusted to temperature changes, there could be an increase of 2.9% in TRF.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.21

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3Changes in Cogging Torque Amplitude

Gain

TR

F(%

)

A.2 BEMF Variation with Temperature

The BEMF waveforms for the test motor in CDU is taken at different temperature ranging

from 25°C to 55°C as shown in Figure A.0-4. The amplitude of the BEMF changes with

temperature and can be seen in Figure A.0-5.

Figure A.0-4: Variation of BEMF with Temperature

Similar trends are seen for the BEMF for phase b and c. the variation of the maximum

amplitude of the BEMF for the three phases can be seen in Figure A.0-5.

0 500 1000 1500 2000 2500 3000 3500 4000-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Encoder Position

Am

plit

ude (

V/r

ad)

Back EMFa Variation with Temperature

25C

30C

35C

40C

45C

50C

55C

Figure A.0-5: Variation of BEMF Amplitude with Temperature

The changes in the amplitude of the BEMF for the three phases will have an impact on the

torque constant. Figure A.0-6 shows the variation of the torque constant with temperature.

Figure A.0-6: Plot of Torque Constant Variation with Temperature

Table A.1 shows the percentage changes of the amplitude of the BEMFs and the torque

constant.

25 30 35 40 45 50 550.18

0.185

0.19

Temperature (C)

Am

plit

ude (

V/r

ad) Variation of Maximum Amplitude of BEMF

a with Temperature

25 30 35 40 45 50 550.18

0.185

0.19

Temperature (C)

Am

plit

ude (

V/r


b with Temperature

25 30 35 40 45 50 550.18

0.185

0.19

Temperature (C)

Am

plit

ude (

V/r


c with Temperature

25 30 35 40 45 50 550.274

0.275

0.276

0.277

0.278

0.279

0.28

0.281

Temperature (C)

Am

plit

ude (

V/r

ad)

Torque Constant Variation with Temperature

Table A.1: Variation of BEMF with Temperature

Parameters 25°C 55°C Percentage change

Amplitude of BEMFA 0.1854 0.181 -2.37%

Amplitude of BEMFB 0.1903 0.1857 -2.41%

Amplitude of BEMFC 0.1866 0.1822 -2.36%

Torque constant 0.2811 0.2744 -2.38%

A temperature change from 25°C to 55°C will result in 2.38% decrease in the torque

constant. This means that for a constant torque reference, the output torque will now be

2.4% lesser than what it should be if the torque constant value in the control scheme

remains unchanged with temperature changes.

A.3 Current Gain and Offset Errors Variation

The gain and offset of ia were varied from 80% to 120% to investigate the impact of these

changes on the TRF when indirect pre-compensation technique was used. The next two

figures showed how these changes will affect the TRF.

Figure A.0-7: Plot of Current Gain vs TRF

Figure A.4 shows how the gain of the measured current would affect the TRF. TRF increases

from 1.17% to 9.5% when the gain of the measured current decreases from 1 to 0.8.

Likewise from Figure A.8, TRF increases from 1.17% to 22.2% when the offset of the

measured current decreases from 1 to 0.8.

Figure A.0-8: Plot of Current Offset vs TRF

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.21

2

3

4

5

6

7

8

9

10Plot of Current Gain and TRF

Gain

TR

F(%

)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

5

10

15

20

25Plot of Current offset and TRF

Gain

TR

F(%

)

There was a variation of current scaling error of less than 0.5% over time and this

contributes to about 0.1% TRF from Figure A.7. Current offset on the other hand varied by

about 0.02A over time and this would contribute to about 0.73% increase in TRF according

to Figure A.8. From these results, variations of current scaling and offset errors up to ±20%

would lead to torque ripple of 0.24% of the rated torque.

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