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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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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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 4-42: PLOT OF ADAPTIVE P-ILC FOR Α = 0.5 .......................................................................... 115
FIGURE 4-43: PLOT OF ADAPTIVE P-ILC FOR Α = 0.9 .......................................................................... 115
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 6-33: PLOT OF ADAPTIVE P-ILC FOR Α = 0.1 .......................................................................... 168
FIGURE 6-34: PLOT OF ADAPTIVE P-ILC FOR Α = 0.5 .......................................................................... 169
FIGURE 6-35: PLOT OF ADAPTIVE P-ILC FOR Α = 0.9 .......................................................................... 169
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
Page 1
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.
Page 2
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
Page 3
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
Page 4
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].
Page 5
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.
Page 6
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.
Page 7
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
Page 8
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.
Page 9
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].
Page 10
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.
Page 11
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]
Page 12
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)
Page 13
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
Page 14
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
Page 15
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)
Page 16
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.
Page 17
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
Page 18
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.
Page 19
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
Page 20
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]
Page 21
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
Page 22
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:
Page 23
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
Page 24
[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].
Page 25
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)
Page 26
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].
Page 27
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:
Page 28
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)
Page 29
= 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
Page 30
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.
Page 31
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).
Page 32
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
Page 33
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.
Page 34
Page 35
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
Page 36
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
Page 37
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
Page 38
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.
Page 39
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
𝑇 𝑇𝑟𝑒𝑓 1
𝑘𝑇
𝑖𝑞∗
𝑖𝑑∗
𝑣𝑠∗
𝜃𝑒
𝑖𝑠
FOC PMSM + _
�̂�𝑟𝑖𝑝 LUT
𝜃𝑚
Page 40
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.
𝑇 𝑇𝑟𝑒𝑓 1
𝑘𝑇
𝑖𝑞∗
𝑖𝑑∗
𝑣𝑠∗
𝜃𝑒
FOC PMSM + _
𝑇𝑐𝑜𝑔
_
𝑇∆𝜆
+ _ 𝑖𝑠𝑐
∆𝑖
1
𝜀
LUT
𝑖𝑠
𝜃𝑚
LUT
𝜃𝑚
Page 41
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 → ∞,
Page 42
)()( 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
Page 43
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.
Page 44
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
Page 45
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
Page 46
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
Page 47
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)
Page 48
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
Page 49
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
Page 50
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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Convergence condition for a range of and kp = 0.4
= 0
= 0.05
= 0.10
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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
Page 52
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
Page 53
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).
Figure 3-13: Convergence for D-ILC (LPF - 50Hz)
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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Convergence condition for a range of kd values - LPF (50Hz)
kd=0.002
kd=0.003
kd=0.004
kd=0.005
kd=0.006
Page 54
Figure 3-14: Convergence for D-ILC (LPF - 100Hz)
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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Convergence condition for a range of kd values - LPF (100Hz)
kd=0.002
kd=0.003
kd=0.004
kd=0.005
kd=0.006
Page 55
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
Page 56
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
ence C
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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
Page 57
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
Page 58
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
Page 59
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
Page 60
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
Page 61
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
Page 62
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
Page 63
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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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
Page 64
[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
Page 65
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
Page 66
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].
Page 67
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.
Page 68
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
Page 69
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)
Page 70
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
𝑇 𝑇𝑟𝑒𝑓 1
𝑘𝑇
𝑖𝑞∗
𝑣𝑠∗
𝑒
FOC PMSM +
+ 𝑖𝑑∗ = 0
_ +
ILC
Page 71
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
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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.
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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
Page 74
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.
Page 75
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
Page 76
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
Page 77
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.
Page 78
Page 79
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.
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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
Page 81
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.
Page 82
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.
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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%
Page 84
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
0.1FFT of Output Torque
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
0.1FFT of Output Torque
Orders
Magnitude
Page 85
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
0.1FFT of Output Torque
Orders
Magnitude
Page 86
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
0.1FFT of Output Torque
Orders
Magnitude
Page 87
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.
Page 88
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
Page 89
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
FFT of Output Torque
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
FFT of Output Torque
TRF (FOC) = 8.1%
TRF (Tcog
compensated) = 1.2%
FOC
Tcog
compensated
Page 90
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
FFT of Output Torque
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
FFT of Output Torque
TRF (FOC) = 8.1%
TRF (T
Ti
Tcog
compensated) = 0.4%
FOC
T
Ti
Tcog
compensated
Page 91
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.
Page 92
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
Page 93
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.
Page 94
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
Page 95
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%
Page 96
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
Page 97
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
Page 98
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
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 D-ILC for kd = 0.004
Average TRF (last 5) = 0.7%
Page 99
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
Page 100
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
Page 101
Figure 4-25: Plot of PD-ILC
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
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 PD-ILC for kp = 0.4, k
d = 0.003
Average TRF (last 5) = 0.6%
Page 102
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 1.0%
Page 103
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.
Page 104
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.
Page 105
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%
3 Channels ILC: Average TRF (last 5) = 0.9%
Page 106
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
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 2 Channels ILC for kp,low
= 1.0, kp,high
= 0.4
Average TRF (last 5) = 0.5%
Page 107
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
Page 108
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
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 3 Channels ILC for kp,low
= kp,medium
= 1.0, kp,high
= 0.4
Average TRF (last 5) = 0.5%
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 Multi-Channel ILC
Single Channel ILC: Average TRF (last 5) = 1.0%
2 Channels ILC: Average TRF (last 5) = 0.5%
3 Channels ILC: Average TRF (last 5) = 0.5%
Page 109
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)
Page 110
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
Page 111
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
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 2nd Order ILC for kp1
= 0.6, kp2
= 0.4
Average TRF (last 5) = 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
Page 112
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 0.4%
Page 113
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%
Page 114
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
𝜇𝑗(𝜃𝑚) = 𝜇𝑚𝑎𝑥‖𝑝𝑗(𝜃𝑚)‖
2
‖𝑝𝑗(𝜃𝑚)‖2+𝐶
(4.12)
𝑝𝑗(𝜃𝑚) = 𝛼𝑝𝑗−1(𝜃𝑚) +1−𝛼
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
Page 115
Figure 4-42: Plot of Adaptive P-ILC for α = 0.5
Similar trend is observed for α = 0.5.
Figure 4-43: Plot of Adaptive P-ILC for α = 0.9
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
(%)
Plot of Adaptive P-ILC for different c values, = 0.5
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
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, = 0.9
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
Page 116
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
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 Adaptive P-ILC ( = 0.1, c = 0.01)
Average TRF (last 5) = 1.3%
0 2 4 6 8 10 12 14 16 18 200
0.5
1
Iterations
Variable Learning Gain
Page 117
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
Variable Learning Gain
kd = 0.001
kd = 0.002
kd = 0.003
kd = 0.004
Page 118
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
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 Adaptive PD-ILC ( = 0.1, c = 0.01, kd = 0.001)
Average TRF (last 5) = 1.0%
0 2 4 6 8 10 12 14 16 18 200
0.5
1
Iterations
Variable Learning Gain
Page 119
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%
Page 120
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
P-ILC: Average TRF (last 5) = 1.0%
PD-ILC: Average TRF (last 5) = 0.6%
MC-ILC: Average TRF (last 5) = 0.5%
HO-ILC: Average TRF (last 5) = 0.4%
Adaptive PD-ILC: Average TRF (last 5) = 1.0%
Page 121
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%.
Page 122
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 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, Higher Order and MCHO ILC
MC-ILC: Average TRF (last 5) = 0.5%
HO-ILC: Average TRF (last 5) = 0.4%
MCHO-ILC: Average TRF (last 5) = 0.4%
Page 123
Table 4.10: Comparing MC-ILC, HO-ILC and MCHO-ILC Schemes
ILC Schemes TRFss (%) Convergence
MC-ILC (2 Channels) 0.5 2
HO-ILC (2nd Order) 0.4 3
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
𝜇𝑗(𝜃𝑚) = 𝜇𝑚𝑎𝑥‖𝑝𝑗(𝜃𝑚)‖
2
‖𝑝𝑗(𝜃𝑚)‖2+𝐶
(4.17)
𝑝𝑗(𝜃𝑚) = 𝛼𝑝𝑗−1(𝜃𝑚) +1−𝛼
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.
Page 124
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
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 MCA-ILC for low
= 0.9, high
= 0.1, clow
= 0.001, chigh
= 0.0001
Average TRF (last 5) = 0.5%
0 2 4 6 8 10 12 14 16 18 200
0.5
1
Iterations
Variable Learning Gain
low
high
Page 125
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
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
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
P-ILC: Average TRF (last 5) = 1.0%
PD-ILC: Average TRF (last 5) = 0.6%
MC-ILC: Average TRF (last 5) = 0.5%
HO-ILC: Average TRF (last 5) = 0.4%
Adaptive PD-ILC: Average TRF (last 5) = 1.0%
MCHO-ILC: Average TRF (last 5) = 0.4%
MCA-ILC: Average TRF (last 5) = 0.5%
Page 126
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.
Page 127
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.
Page 128
Page 129
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.
Page 130
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)
Page 131
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.
Page 132
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.
Page 133
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.
Page 134
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
Page 135
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
Page 136
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
Page 137
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
Page 138
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
Page 139
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.
Page 140
Page 141
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
Page 142
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
Page 143
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%
Page 144
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%
Page 145
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
Page 146
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%
Page 147
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%
Page 148
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
FFT of Torque Ripple
FOC
Direct Pre-Compensation
Page 149
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
FFT of Torque Ripple
FOC
Direct Pre-Compensation
Page 150
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
FFT of Torque Ripple
TRF (FOC) = 8.1%
TRF (Indirect Pre-Compensation) = 7.9%
FOC
Indirect Pre-Compensation
Page 151
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
FFT of Torque Ripple
FOC
Indirect Pre-Compensation
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%
TRF (Indirect Pre-Compensation) = 1.4%
0 20 40 60 80 100 1200
0.05
0.1
Orders
Magnitude
FFT of Torque Ripple
FOC
Indirect Pre-Compensation
Page 152
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
FFT of Torque Ripple
TRF (FOC) = 8.1%
TRF (Indirect Pre-Compensation) = 0.9%
FOC
Indirect Pre-Compensation
Page 153
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.
Page 154
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
Page 155
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
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, kp = 0.4
Average TRF (last 5) = 1.3%
Page 156
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
Page 157
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
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 Pf-ILC, kp = 0.4, = 0.05
Average TRF (last 5) = 1.3%
Page 158
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
Page 159
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 1.4%
Page 160
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
Page 161
Figure 6-23: Plot of PD-ILC
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 1.4%
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
Page 162
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
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 PI-ILC, kp = 0.4, k
i = 0.10
Average TRF (last 5) = 1.3%
Page 163
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
Page 164
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
Page 165
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
Page 166
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
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 MC-ILC, kp,low
= 0.9, kp,high
= 0.3
Average TRF (last 5) = 1.2%
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
Page 167
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 1.1%
Page 168
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: Plot of Adaptive P-ILC for α = 0.1
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(%
)
Plot of Adaptive P-ILC for different c values, = 0.1
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
c = 0.005
c = 0.01
c = 0.02
Page 169
Figure 6-34: Plot of Adaptive P-ILC for α = 0.5
Similar trends were observed in Figure 6-34 and Figure 6-35.
Figure 6-35: Plot of Adaptive P-ILC for α = 0.9
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
(%)
Plot of Adaptive P-ILC for different c values, = 0.5
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
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, = 0.9
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
c = 0.005
c = 0.01
c = 0.02
Page 170
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
FFT of Output Torque after 20 iterations
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
Variable Learning Gain
Average TRF (last 5) = 1.3%
Page 171
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
Variable Learning Gain
kd = 0.001
kd = 0.002
kd = 0.004
Page 172
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
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 Adaptive PD-ILC, = 0.1, c = 0.01, kd = 0.001
Average TRF (last 5) = 1.2%
0 2 4 6 8 10 12 14 16 18 200
0.5
1
Iterations
Variable Learning Gain
Page 173
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
Variable Learning Gain
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
Page 174
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
ILC Methods TRFss (%) Convergence
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(%
)
Comparison of ILC Methods
Pf-ILC
PD-ILC
MC-ILC
HO-ILC
Adaptive PD-ILC
Page 175
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
ILC Methods TRFss (%) Convergence
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.
Page 176
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
FFT of Output Torque after 20 iterations
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
Average TRF (last 5) = 1.2%
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
FFT of Output Torque after 20 iterations
Initial
Final
Page 177
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
Variable Learning Gain
Average TRF (last 5) = 1.2%
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(%
)
Comparison of ILC Methods
Pf-ILC
PD-ILC
MC-ILC
HO-ILC
Adaptive PD-ILC
MCHO-ILC
MCA-ILC
Page 178
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
ILC Methods TRFss (%) Convergence
(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%.
Page 179
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
Page 180
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
Page 181
Figure 6-52 shows the robustness of PD-ILC to variations in J and b. There is a 12.5%
variation of TRF for a J variation of ±10% and 28.6% variation of TRF for a b variation of ±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
Page 182
Figure 6-54 shows the robustness of MC-ILC to variations in J and b. There is a low 5.1%
variation of TRF for a J variation of ±10% and 7.7% variation of TRF for a b variation of ±5%.
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%
variation of TRF for a J variation of ±10% and 21.2% variation of TRF for a b variation of ±5%.
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
Page 183
Figure 6-56 shows the robustness of adaptive P-ILC to variations in J and b. There is a 15.9%
variation of TRF for a J variation of ±10% and 11.6% variation of TRF for a b variation of ±5%.
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
Page 184
Figure 6-58 shows the robustness of MCHO-ILC to variations in J and b. There is a 24.5%
variation of TRF for a J variation of ±10% and 37.4% variation of TRF for a b variation of ±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%
variation of TRF for a J variation of ±10% and 9.8% variation of TRF for a b variation of ±5%.
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
Page 185
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
Page 186
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,
Page 187
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
Page 188
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.
Page 189
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
Page 190
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.
Page 191
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
Page 192
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.
Page 193
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.
Page 194
Page 195
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
Page 196
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
Page 197
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(%
)
Page 198
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
Page 199
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
ad) Variation of Maximum Amplitude of BEMF
b with Temperature
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
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
Page 200
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.
Page 201
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(%
)
Page 202
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.
Page 203
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