MODELING AND CONTROL STRATEGIES FOR MULTIPROCESS ARC WELDING POWER SOURCES
by
JONATHON C. KELM
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Thesis Adviser: Prof. Kenneth A. Loparo
Department of Electrical Engineering & Computer Science
CASE WESTERN RESERVE UNIVERSITY
January, 2020 Modeling and Control Strategies for Multiprocess Arc
Welding Power Sources
Case Western Reserve University Case School of Graduate Studies
We hereby approve the thesis1 of
JONATHON C. KELM
for the degree of
Doctor of Philosophy
Prof. Kenneth Loparo 11/15/2019
Committee Chair, Adviser Date Department of Electrical Engineering & Computer Science
Prof. Vira Changkong 11/15/2019
Committee Member Date Department of Electrical Engineering & Computer Science
Prof. Robert Gao 11/15/2019
Committee Member Date Department of Mechanical & Aerospace Engineering
Prof. Wei Lin 11/15/2019
Committee Member Date Department of Electrical Engineering & Computer Science
1We certify that written approval has been obtained for any proprietary material contained therein. Dedicated to my parents, who taught me everything that really matters. Table of Contents
List of Tables vii
List of Figures viii
List of Initialisms xiv
List of Symbols xviii
Acknowledgements xxii
Abstract xxiii
Chapter 1. Introduction1
Background and Motivation1
Literature Review 14
Goals of this Work 25
Contributions of this Work 26
Thesis Organization 26
Chapter 2. Modeling 28
Welding Cables, Workpiece, and Welding Fixture 28
Electrode and Contact Tip 33
The Welding Arc 35
Summary of Load Impedances 37
Piecewise Linear Model 38
Sources of Uncertainty 47
Chapter 3. Review of the Existing System 51
iv Overview of the Existing Control System 51
Waveform Generator Reference Tracking 54
Chapter 4. Proposed Control Strategy 60
Overview of Sliding Mode Control 61
Simplified Model 62
Current Tracking 63
Voltage Tracking 87
Power Tracking 93
Regulation Mode Switching 93
Output Limiting 93
Advantages of SMC for Welding Power Sources 97
Drawbacks of Sliding Mode Control 97
Chapter 5. Hardware Implementation 99
Abortive Efforts 99
Hybrid Analog/Digital Solution 101
Examples 111
Chapter 6. Results 115
Experimental Setup 115
Current Step Command 116
Current Exponential Command 124
Chapter 7. Summary, Conclusions, and Future Work 127
Summary and Conclusions 127
Recommendations for Further Work 128
v Appendix A. Simulation 130
General Structure 130
Integration Method 135
Simulation Configuration 138
Load Change Simulation 140
Simulation Parameters 141
Complete References 142
vi List of Tables
1.1 Timeline of welding power source technology 16
2.1 Ranges of Model Parameters 50
4.1 Current error dynamics for various reference signals 80
4.2 Voltage error dynamics for various reference signals 89
A.1 Simulation parameters 141
vii List of Figures
1.1 Illustration of arc welding terminology3
1.2 “Chopper” style welding power source structure6
1.3 “Inverter” style welding power source structure7
1.4 Shielded Metal Arc Welding (SMAW) setup8
1.5 Flux-Cored Arc Welding (FCAW) setup8
1.6 Gas Metal Arc Welding(GMAW) setup9
1.7 Gas Tungsten Arc Welding(GTAW) setup 10
2.1 Circuit diagram showing sources of impedance in the welding
circuit 29
2.2 Welding cable cross-section showing electrical and geometrical
properties 30
2.3 Circuit diagram for welding cable transmission line model 30
2.4 AWG 0000 welding cable resistance versus cable length 31
2.5 AWG 0000 welding cable inductance versus cable length 32
2.6 AWG 0000 welding cable capacitance versus cable length 33
2.7 AWG 0000 welding cable resonant frequency versus cable length 34
2.8 Wire electrode fed through contact tube as in Gas Metal Arc
Welding (GMAW) and FCAW 35
2.9 V-I characteristics for GMAW arcs of various lengths 37
2.10 Circuit diagram of power section and load 40
viii 2.11 Circuit diagram of power section and load when switch is closed 40
2.12 Circuit diagram of power section and load when switch is open 41
2.13 Relationship between switch position u and choke current iL(t) 43
2.14 Current filtering effect of the welding circuit impedance 44
2.15 Inductance vs. current for the output inductor of a typical
welding power source 47
3.1 Block diagram of the existing control system 53
3.2 Example of a simplified FSM for a GMAW welding program 54
3.3 Example of a constant reference signal 55
3.4 Example of a ramp reference signal 57
3.5 Example of an exponential reference signal 57
3.6 Example of a parabolic reference signal 58
3.7 Output regulation modes in a representative sample of welding
programs 59
4.1 Circuit diagram of simplified power source and load 63
4.2 Sample current error trajectories for u = 0 for a constant reference 65
4.3 Sample current error trajectories for u = 1 for a constant reference 66
4.4 Desired current trajectory z2 = kiz1 for a constant reference 67 − 4.5 Current error trajectories with u = 0, u = 1, and sliding line
σi = 0 68
4.6 Sliding mode existence for constant current, k < Rw + Ro + Ro 70 i Lw Lw L
ix Rw Ro Ro 4.7 Sliding mode existence for constant current, ki + + 71 ≈ Lw Lw L 4.8 Sliding mode existence for constant current, k > Rw + Ro + Ro 72 i Lw Lw L
4.9 Relationship between switching function σ, control u, and
hysteresis κ 74
4.10 Relationship between load resistance Rw and hysteresis κ for
Fsw = 20 kHz 75
4.11 Relationship between load inductance Lw and hysteresis κ for
Fsw = 20 kHz 76
4.12 Current step simulation, Fsw = 20 kHz 77
4.13 Current step simulation, Fsw = 80 kHz 78
4.14 Current standard deviation vs. switching frequency 79
4.15 Current exponential simulation 82
4.16 Current ramp simulation 84
4.17 Current parabola simulation 86
4.18 Voltage exponential simulation 91
4.19 Circuit diagram for the Open Circuit Voltage (OCV) case 92
4.20 Example operating region defined by current, voltage, and power
limits 95
4.21 Example of output limiting, 20 V regulation at a 100 A minimum
current 96
5.1 Original analog SMC control board 101
5.2 Hybrid analog/digital SMC PCB 103
x 5.3 Basic structure of the existing control hardware 104
5.4 Basic structure of the modified control hardware 104
5.5 Block diagram of the SMC hardware implementation 105
5.6 SMC algorithm flowchart 107
5.7 Band-limited derivative frequency responses for various values of
τ 109
5.8 State diagram for CPLD gate drive logic 110
5.9 Oscilloscope trace showing hysteresis modulation signals 111
5.10 Oscilloscope trace showing detail of hysteresis modulation signals 112
5.11 Oscilloscope trace of 200 A current regulation at 20 kHz 112
5.12 Oscilloscope trace of 200 A current regulation at 10 kHz 113
5.13 Oscilloscope trace of current ramp regulation 113
5.14 Oscilloscope trace of current exponential regulation 114
6.1 Experimental load bank 116
6.2 84 µH coil of welding cable 116
6.3 255 µH coil of welding cable 117
6.4 LCR meter used for measuring cable inductance 117
6.5 Current step, 10 A to 100 A, with varying load resistance (existing
controls) 119
6.6 Current step, 10 A to 100 A, with varying load resistance (SMC
simulation) 119
xi 6.7 Current step, 10 A to 100 A, with varying load resistance
(hardware implementation) 119
6.8 Current step, 20 A to 200 A, with varying load resistance (existing
controls) 120
6.9 Current step, 20 A to 200 A, with varying load resistance (SMC
simulation) 120
6.10 Current step, 30 A to 300 A, with varying load resistance (existing
controls) 121
6.11 Current step, 30 A to 300 A, with varying load resistance (SMC
simulation) 121
6.12 Current step, 10 A to 100 A, with varying load inductance
(existing controls) 122
6.13 Current step, 10 A to 100 A, with varying load inductance (SMC
simulation) 122
6.14 Current step, 20 A to 200 A, with varying load inductance
(existing controls) 123
6.15 Current step, 20 A to 200 A, with varying load inductance (SMC
simulation) 123
6.16 Current exponential, 20 A to 200 A, with varying load resistance
(existing controls) 125
6.17 Current exponential, 20 A to 200 A, with varying load resistance
(SMC simulation) 125
xii 6.18 Current exponential, 20 A to 200 A, with varying load inductance
(existing controls) 126
6.19 Current exponential, 20 A to 200 A, with varying load inductance
(SMC simulation) 126
A.1 Simulator “main loop” flowchart 132
A.2 Simulated load changes from R1 = 100 mΩ to R2 = 5 mΩ 140
xiii List of Initialisms
AC Alternating Current.
ADC Analog-to-Digital Converter.
ADRC Active Disturbance Rejection Control.
ANN Artificial Neural Network.
AWG American Wire Gauge.
AWPS Arc Welding Power Source.
AWS American Welding Society.
BLS Bureau of Labor Statistics.
CAG Carbon Arc Gouging.
CC Constant Current.
CPLD Complex Programmable Logic Device.
CTWD Contact Tip to Workpiece Distance.
DAC Digital-to-Analog Converter.
DC Direct Current.
DCEN Direct Current Electrode Negative.
DCEP Direct Current Electrode Positive.
DCM Discontinuous Conduction Mode.
DSP Digital Signal Processor.
xiv EMC Electromagnetic Compatibility. emf Electromotive Force.
EMI Electromagnetic Interference.
EPDM Ethylene Propylene Diene Monomer.
FCAW Flux-Cored Arc Welding.
FLC Fuzzy Logic Control.
FOC Fractional Order Control.
FPGA Field Programmable Gate Array.
FSM Finite State Machine.
GMAW Gas Metal Arc Welding.
GTAW Gas Tungsten Arc Welding.
GTAW-P Pulsed Gas Tungsten Arc Welding.
HM Hysteresis Modulation.
I/O Input/Output.
IEC International Electrotechnical Commission.
IGBT Insulated Gate Bipolar Transistor.
LAC Linear Average Control.
LCR Inductance/Capacitance/Resistance.
LTI Linear Time-Invariant.
xv MAWPS Multiprocess Arc Welding Power Source.
MOSFET Metal-Oxide Semiconductor Field-Effect Transistor.
MPC Model Predictive Control.
OCV Open Circuit Voltage.
PC Personal Computer.
PCB Printed Circuit Board.
PI Proportional-Integral.
PID Proportional-Integral-Derivative.
PWM Pulse Width Modulation.
QSMC Quasi-Sliding Mode Control.
RK4 Classic Runge-Kutta Method.
SCR Silicon Controlled Rectifier.
SISO Single Input Single Output.
SMAW Shielded Metal Arc Welding.
SMC Sliding Mode Control.
SMPS Switched-Mode Power Supply.
SPI Serial Peripheral Interface.
xvi USA United States of America.
ZCS Zero Current Switching.
ZVS Zero Voltage Switching.
xvii List of Symbols
Fsw switching frequency.
Ga electrical conductance of welding arc.
Lw inductance of welding circuit load impedance.
L output choke inductance.
P0 electric arc cooling power.
Re electrical resistance of welding electrode.
Ro resistance of welding power source output resistor.
Rt electrical resistance of electrode contact point.
Rw resistance of welding circuit load impedance.
Tsw switching period.
V0 steady-state arc electromotive force.
Vg power source supply (input or “bus”) voltage.
Zc electrical impedance of welding cables.
Zp electrical impedance of welding fixture.
... x third time derivative of x.
xviii x¨ second time derivative of x. x˙ time derivative of x.
`e length of electrode extension.
−1 inverse Laplace transform. L {·}
κ switching function hysteresis bound.
Laplace transform. L {·}
µ0 magnetic permeability of free space.
µins magnetic permeability of insulation material.
µ magnetic permeability.
ν1 voltage tracking error.
ν2 derivative of voltage tracking error.
φ unknown disturbance function.
ψ unknown disturbance function.
ρc electrical resistivity of cable.
σi sliding surface switching function for current control.
σp sliding surface switching function for power control.
xix σv sliding surface switching function for voltage control.
τa arc time constant.
ξh comparison function, upper.
ξl comparison function, lower.
cc electrical capacitance per unit length of cable.
dins insulation thickness.
dc distance between welding cables. d duty cycle.
iL inductor current.
iw electrical current in the welding circuit.
ki switching function parameter for current regulation.
kp switching function parameter for power regulation.
kv switching function parameter for voltage regulation.
lc electrical inductance per unit length of cable.
pw power in welding circuit; pw = vwiw.
ri current reference waveform.
xx rp power reference waveform.
rv voltage reference waveform. s Laplace transform operator variable (complex frequency).
vw voltage across the welding circuit.
z1 current tracking error.
z2 derivative of current tracking error.
xxi Acknowledgements
Everyone in my life has been incredibly supportive and understanding as I have tried
to balance working full-time in industry, pursuing graduate studies, and raising twin
boys, usually feeling like I’m not doing any one of those things very well. Thanks to
all of you.
This research was sponsored by The Lincoln Electric Company of Cleveland, Ohio,
and I would be remiss if I did not mention my gratitude for the knowledge and
experience I have gained while working there. I would like to give special thanks to
Tom Matthews, Joe Daniel, Ed Hillen, Todd Kooken, Lifeng Luo, Judah Henry, and
Nick Trinnes for their support and for the insights they provided into the welding and
electronics topics in this thesis.
I am grateful to my adviser, Ken Loparo, whose wide-ranging technical expertise was
invaluable. He patiently provided guidance and wisdom, and contributed insights at
several steps along the way. I would like to thank the other Systems & Control Engi-
neering faculty and staff members at Case Western Reserve University for everything
I have learned from them.
Finally, I thank my wife Ashley for dutifully accepting her role as a “Ph.D. widow”
for the last few years. Without her love and support this work would not have
been completed. The recent arrival of our sons, Carl and Robert, gave me renewed
motivation and they are, by far, the most important project I will ever pursue.
xxii Abstract
Modeling and Control Strategies for Multiprocess Arc Welding Power Sources
Abstract
by
JONATHON C. KELM
A modern Multiprocess Arc Welding Power Source(MAWPS) is a Switched-Mode
Power Supply(SMPS) that has been designed to produce waveforms used for multiple
arc welding processes, such as Shielded Metal Arc Welding(SMAW), Gas Metal
Arc Welding(GMAW), Gas Tungsten Arc Welding(GTAW), and Flux-Cored Arc
Welding(FCAW). MAWPS control is challenging for a number of reasons, including the complex dynamics of switching power converters, transient conditions encountered in the metal transfer process, wide variations in load impedance, a need for tracking complex reference waveforms, incomplete or inaccurate models of the welding process itself, the difficulty of addressing the needs of several welding processes using a single machine, an electrically harsh environment with high levels of electromagnetic noise, and health and safety concerns.
In this work, models of the equipment in a welding setup are developed that can
be used for analysis and control system design. The models are used to develop a
simulation environment and a new control strategy for a welding power source from
Lincoln Electric, using Sliding Mode Control (SMC). While SMC has been applied
xxiii to SMPS elsewhere in the literature, this work focuses on the particular needs of the welding power source and incorporates output current, voltage, and power reference
tracking, switching frequency control, and output constraints.
A hardware implementation of the SMC strategy is described, and its performance
is compared against the existing control system and computer simulations. While
some implementation details still need to be worked out, the SMC strategy is shown
to be feasible to implement and to provide significant improvements in the current, voltage, and power tracking performance. These improvements should have a direct
impact on the welding performance of the Multiprocess Arc Welding Power Source
(MAWPS).
xxiv 1
1 Introduction
1.1 Background and Motivation
High-quality arc welding is an important part of modern manufacturing, with welding-
related expenditures of at least $34.1 billion in the United States in the year 2000,
equivalent to $325 for every household in the country [1]. Welding technology is a key
factor in industries such as automotive manufacturing, oil and gas pipelines, building
construction, mining, and defense.
The central piece of equipment in an arc welding setup is the welding power source.
A modern MAWPS is a switching power converter that has been designed to produce waveforms used for various arc welding processes. MAWPS control is challenging for
a number of reasons, including the complex dynamics of switching power converters,
transient conditions encountered in the metal transfer process, wide variations in load
impedance, a need for tracking complex reference waveforms, incomplete or inaccurate
models of the welding process itself, the difficulty of addressing the needs of several welding processes using a single machine, an electrically harsh environment with high
levels of electromagnetic noise, and health and safety concerns.
Arc welding control has been studied in the past, but most previous studies have
been done from two perspectives: that of manufacturing or process engineering, where Introduction 2 the interest lies in integrating welding into a larger manufacturing system, and that of metallurgical and materials engineering, concerned with the physics of joining the materials to be welded. Most published research in welding control treats the power source as a “black box,” and few substantial studies have been published in the open literature regarding the control of the welding power source itself. Of the literature that does exist regarding power source control, the majority has not addressed the dynamics of the welding process. Most of the suggested control strategies have been tested only in simulation, or if in an actual power source then only in contrived situations. Not much of the research has demonstrated how the principles investigated affect the welding performance of the machines.
One reason for the scarcity of research in welding power source control is that, until recently, sufficiently fast and precise data acquisition equipment was not available for studying the electrical signals involved in arc welding. Another reason is that few researchers have had access to the internals of a power source in order to study and modify it to test improvements.
The work described here attempts to model the important phenomena in the welding process as they relate to the control of the power source, and suggests a control strategy for achieving better performance than traditional controls. The new strategy is implemented by modifying a standard welding power source from Lincoln
Electric. The performance of the new strategy is compared to the performance of the existing controls.
1.1.1 Arc Welding Overview
The goal of welding is to join pieces of material by obtaining continuity between multiple elements composing a structure. Two elements to be joined are placed Introduction 3
Weld Bead
Filler Material
Base Material
Weld Pool
Spatter Joint
Workpiece
Figure 1.1. Illustration of arc welding terminology
adjacent to one another to form a joint. The material comprising the pieces to be
joined is called the base material, and the structure being welded is referred to as
the workpiece. Heat from an energy source is applied to melt the base material and
possibly a separate filler material into a volume of liquid metal along the joint called
the weld pool. The cooled and solidified weld pool is called the weld bead. Figure 1.1
illustrates these terms. The quality of a weld depends upon many factors including
the purity of the material in the weld bead, the depth at which the bead penetrates
the base material, and the presence of any solidified droplets outside of the weld bead,
referred to as spatter.
The most common energy source used for welding is an electrical arc [2], and welding processes using this energy source are called arc welding processes. This
is in contrast to other types of welding processes such as resistance welding, laser Introduction 4 welding, or oxyacetylene welding. In arc welding, a metal electrode is connected to one terminal of a welding power source, and the workpiece is connected to the other terminal. A low voltage, high current electric arc is established between the end of the electrode and the workpiece. The arc is a gaseous conductor through which current
flows to complete the welding circuit. The heat from the arc melts the base material and the filler material (if present) to form the liquid metal weld pool. The power source monitors the current and voltage in the welding circuit, and manipulates them according to logic designed to promote metal transfer in a particular way. If the weld is made properly, the materials to be joined will have become one continuous, joined piece.
Arc Welding Power Sources. Arc welding can be performed using nearly any electrical power source, including motor generators, batteries, and simple step-down transformers connected to mains power [2]. The most advanced arc welding power sources are switching power converters built with solid state power electronics. These machines can be configured using digital computers, and can be easily integrated with other industrial equipment such as factory automation and data acquisition devices.
Because they are highly configurable and typically controlled by reprogrammable soft- ware, these machines can be used to implement multiple arc welding processes using a single power source. A machine with this capability is referred to as a Multiprocess
Arc Welding Power Source(MAWPS). The high switching frequency of these power sources (in the range of 20 kHz to 200 kHz) allows high bandwidth control, smaller magnetics, and smaller size and weight for the machines compared to older technolo- gies based on generators or Silicon Controlled Rectifiers (SCRs). The efficiency of traditional arc welding power sources is typically 75 % to 85 %, translating to quite Introduction 5 large losses at high outputs [3]. Switching converters can have much higher efficiency, exceeding 95 % [4].
In the welding literature, a distinction is made between static and dynamic char-
acteristics of power sources. Static characteristics reflect changes that occur on the
order of 250 ms to 500 ms (typically related to the length of the arc), while dynamic
characteristics reflect changes at speeds measured in milliseconds or faster [5]. Since
the control of switching converters is very rapid in comparison (on the order of mi-
croseconds), both the static and dynamic properties of the power source can be con-
trolled [6].
Arc welding may be performed using Direct Current (DC) in either positive or
negative polarity, or using Alternating Current (AC). The welding polarity and hence
the direction of current flow affects the distribution of heat in the weld, with certain
polarity configurations being more desirable for particular arc welding processes. Gas
Tungsten Arc Welding (GTAW), for example, is typically done using Direct Current
Electrode Negative (DCEN) or AC, while GMAW and SMAW are typically Direct
Current Electrode Positive (DCEP) but may also be performed using AC.
Modern welding power sources based on solid state switching power converters
can be roughly placed into two categories: choppers and inverters.
Figure 1.2 shows the structure of a “chopper” welding power source. In this
topology, isolation from the AC input power is provided by a low-frequency step-down
transformer directly connected to the input voltage, at 50 Hz or 60 Hz. This input voltage is rectified and filtered to provideDC power to a digitally-controlled switching
section that runs at a rate of 20 kHz to 200 kHz. This high-frequency “chopped” signal
is smoothed by an output filter section to produce the welding output. Introduction 6
AC Input Welding Power Output
step-down capacitor output rectifier switch transformer bank filter
Figure 1.2. “Chopper” style welding power source structure
There are two chief advantages to the chopper architecture. First is that there are
fewer parts than an inverter, since the switching action is implemented using a single
switch rather than a four-switch H-bridge. Second is that the switching is done on
the output of the transformer, rather than the input, meaning that it is less likely for
the control system to cause transformer flux imbalance.
The basic structure of an inverter-style power source is shown in Figure 1.3. AC
input voltage at 50 Hz or 60 Hz is rectified and filtered to produce aDC voltage that
provides power to a full-bridge. The full-bridge switching devices on the transformer
primary side are controlled to produce the desired output waveform. The output of
the full bridge is a high-voltage, high-frequency (20 kHz to 200 kHz) AC signal that
goes through a step-down transformer, then is further rectified by an output rectifier
and smoothed by an output filter section, to produce the final welding waveform.
An advantage of the inverter topology over the chopper is that the step-down trans-
former can be physically much smaller because it operates at a higher frequency. A
disadvantage is that the switching devices are on the primary side of the transformer,
meaning that the control algorithms that produce the output waveforms for welding
must ensure that transformer flux imbalance does not cause transformer saturation.
Shielded Metal Arc Welding. In SMAW, the arc is established between a flux-
coated metal rod or “stick” electrode that is consumed during welding to provide the
filler material. Figure 1.4 shows a typical SMAW setup. SMAW is almost always Introduction 7
AC Input Welding Power Output
input capacitor step-down output output H-bridge rectifier bank transformer rectifier filter
Figure 1.3. “Inverter” style welding power source structure
performed manually. The welder manipulates the weld pool by moving the electrode
along the joint using his or her hands, and adjusts the arc length by changing the
distance from the end of the electrode to the workpiece. Since the arc length can vary widely during a weld, maintaining the desired current level in SMAW typically
demands a Constant Current (CC) control characteristic from the power source. In
addition to constant current regulation, SMAW requires well-controlled response to
short circuits (avoiding explosive rises in current upon load changes) and may require
power regulation in certain portions of the waveform in order to maintain constant
energy input while the arc length changes.
Metal transfer in SMAW was studied by Pistorius and Liu, who documented
relationships between welding voltage frequency content and the size of the metal
droplets that were transferred [7]. Blasco et al. discussed the SMAW process and the
desirable characteristics of a power source used for SMAW, concluding that a full-
bridge converter provides a good balance of size, weight, and welding performance [8].
Flux-Cored Arc Welding. FCAW uses a consumable wire electrode that is con-
tinuously fed into the weld pool using a motorized wire feeder. The FCAW wire is
hollow and contains flux that, when heated, produces gas for shielding the welding
arc. Additional shielding gas may also be used. The equipment used in a typical
FCAW setup is shown in Figure 1.5 The wire is fed at a constant rate and the arc Introduction 8
Welding Torch
Power Source
Flux-Coated Electrode
E W
Workpiece
Figure 1.4. SMAW setup
Electrode Wire Feeder wire spool
Anode and electrode wire
Power Source
Gun Gas nozzle Contact tip E W (optional) Gas Supply Weld pool (optional)
Workpiece
Figure 1.5. FCAW setup length is controlled by the wire feeding rate and the Contact Tip to Workpiece Dis- tance (CTWD). Metal is normally transferred in a spray transfer with infrequent short circuits [9]. Thus for FCAW the power source mostly sees the impedance of the arc itself, with anomalous short circuit events that must be handled without explosive changes in the current level that can result in spatter. Introduction 9
Electrode Wire Feeder wire spool
Anode, electrode wire, and gas hose
Power Source
Gun Contact tip Gas nozzle E W Gas Supply Weld pool
Workpiece
Figure 1.6. Gas Metal Arc Welding(GMAW) setup
Gas Metal Arc Welding. Like FCAW, Gas Metal Arc Welding(GMAW) uses
a consumable wire electrode. GMAW differs from FCAW in that the wire is solid
and the shielding for the arc always comes from an externally-supplied shielding gas
(commonly argon, carbon dioxide, or a mixture of these) rather than a flux inside the wire. Figure 1.6 shows a GMAW setup. GMAW is quite common because it can be
done manually but is also easily automated. GMAW processes are typically classified
according to how the metal is transferred from the end of the wire. These trans-
fer modes depend on the average current level and include short circuiting transfer,
globular transfer, spray transfer, and pulsed transfer.
Gas Tungsten Arc Welding. Gas Tungsten Arc Welding(GTAW) was developed
in the 1930s and 1940s by the aircraft industry for welding of sheet metals such as
aluminum and magnesium [10]. The GTAW electrode is a nonconsumable tungsten
rod. A separate filler material may be fed into the arc to be melted and deposited
into the weld pool. Figure 1.7 shows a GTAW setup. The arc length is controlled by Introduction 10
Welding Torch
Power Source
Tungsten Electrode
E W Filler Material
Gas Supply Workpiece
Figure 1.7. Gas Tungsten Arc Welding(GTAW) setup
moving the welding torch toward or away from the workpiece, and GTAW typically
requires a power source with aCC characteristic. To avoid tungsten contamination
in the weld pool, it is important that the electrode is not melted or fused to the workpiece. Short circuits are not expected in GTAW, and when they occur the power
source should not react as it does in other processes; the power source may reduce
the current level or turn off to prevent the tungsten “sticking” to the workpiece. In
Pulsed Gas Tungsten Arc Welding (GTAW-P), the current may be moved at a high
frequency between a high peak and a low background. In this case low overshoot and
undershoot when performing a current step are important.
Other Requirements for Welding Power Sources. Aside from welding processes
themselves, additional demands are sometimes placed on MAWPS output regulation
such as touch sensing, implementing metal cutting processes, and operating while
connected to resistive load banks for testing and calibration.
Touch sensing is used by a robot connected to an MAWPS, where the electrode is
at the end of the robot’s end effector. The power source regulates a low Open Circuit
Voltage (OCV) as the robot moves the welding torch around its work area. If the Introduction 11
torch touches the workpiece, a small current begins to flow. The power source detects
this flow of current and informs the robot that the torch has touched the workpiece.
This can be used as a method of positioning the robotic arm when making welds.
Good OCV regulation and low-end current regulation are required to perform this
effectively.
Because welding and cutting of metals are frequently done together, it is common
for a MAWPS to implement Carbon Arc Gouging (CAG), a non-welding process used
for cutting metal. The heat from an arc melts the metal and a stream of air pushes
the molten metal out of the way. Gouging is typically done at very high output levels
and causes frequent transitions between open and closed circuits as the metal is cut
away.
For calibration and testing, a MAWPS is often connected to a load bank with
a fixed impedance. This allows a technician to calibrate the voltage and current
feedback by regulating a constant output into a known load and comparing the power
source readings with external meters.
1.1.2 The Need for Improved Controls
The Bureau of Labor Statistics (BLS) indicates that in 2010 there were 337,300 welding, cutting, soldering, or brazing jobs in the United States, and this is projected
to increase 15 % by 2020. This figure is slightly above the average growth projected
for all occupations (14 %), and is much higher than the 4 % average projected increase
in production occupations [11]. The American Welding Society (AWS) found that a large majority of welding industry personnel believe a lack of skilled welders is presently a problem and will continue to be a problem in the long-term [12]. Some
in the industry see increased automation as a solution, while others believe more Introduction 12
intelligent power sources that require less training to operate will be required. Both
situations could benefit from more advanced control systems. According to Norrish, welding control improvements are needed due to a paucity of skilled welders, a need for
continuous improvement in occupational health and safety, and competitive pressures
to improve productivity and reduce cost [13].
Advances in materials and metallurgy are bringing new alloys to industry, and the
present level of knowledge about how to weld these materials is often insufficient. New welding processes are being developed to take advantage of these materials, placing
new demands on power source control systems [12]. The AWS describes development
of materials as one of the key drivers of the future of the welding industry. These new
materials are developed to have increased strength, be lighter, and be more resistant
to corrosion. From [14]:
“Improvements in the quality and reliability of joints will help over-
come the image problem of the welded joint as the weakest link
in any structure... Processing research, artificial intelligence and
robotics, advanced materials, and other developments outside the
traditional scope of welding can all be applied to making dramatic
improvements in welds.”
Paul suggested that the goal of the power source is to reduce the diversity of
the welding process by making as many factors as possible robust across parameter variations [15]. Paul proposes the concept of a “goodness factor” for welding equip-
ment, which measures how appropriate the equipment is considering settling time of
reference tracking, reduction of losses in the welding cables, high power factor, and
high efficiency. Introduction 13
1.1.3 Challenges for Control
MAWPS control presents several challenges that makes its study interesting and pro- vides avenues for research in several fields of systems and control.
Switching Power Converter Dynamics. The dynamics of switching power con- verters is nonlinear and time-varying. While the electric circuits that make up power
converters can be simple, they can exhibit complex dynamics including bifurcations,
chaos, and period doubling [16].
Metal Transfer Transients. Because the purpose of welding is to transfer metal,
there is rarely a steady state achieved during welding. Instead, transient phenomena
are the norm. It is crucial to understand the way the power source control system
handles transients in metal transfer such as current pulses to promote molten metal
ball formation and transfer and short circuit events that occur when the electrode
contacts the workpiece.
Wide Load Impedance Variations. A welding power source is required to regulate voltage into an open circuit (in reality, a very high impedance load), and is also
required to regulate current into a very low impedance load (referred to as a short
circuit). Additionally, the electric arc has a nonlinear conductance that must be
properly handled by the power source control system.
Reference Tracking. In many welding processes, it is not sufficient to regulate to
a fixed setpoint such as a constant current or a constant voltage. Instead, periodic
events occur that require specific waveform shapes such as pulses, current ramps, etc.
to be tracked by the power source.
Incomplete and Inaccurate Models. There are many variables involved in the welding process, and it is challenging to combine the models of the power source, Introduction 14 welding circuit, metal transfer process, and weld pool dynamics in a way that produces
a detailed yet tractable model of the welding process. Simplifying assumptions are
necessary to make it feasible to analyze and simulate the welding process, and these
assumptions can limit the scope of applicability of the models.
Electrically Harsh Environments. Due to the high levels of electrical current
involved in arc welding, electrical noise is a primary concern in the design of welding
power sources. Both the hardware and the control system must take precautions to
ensure that electrical noise emitted from the power source is of an acceptable level,
and that interference from the welding process itself does not affect the operation of
the welding power source or surrounding equipment.
Health and Safety Concerns. The International Electrotechnical Commission
(IEC) publishes requirements for welding power sources in terms of health and safety.
Welding can be a dangerous process that can result in electrical shock or electrocu-
tion, burns, and inhalation of fumes. All of these things can be a factor in the design
of the power source and its controls.
1.2 Literature Review
1.2.1 Historical Overview
Metal-joining processes that could be considered welding were known as far back
as the Bronze Age, but the earliest known use of an electric arc for joining metals
occurred in the 19th century [2], shortly after the electric arc was discovered inde-
pendently by Sir Humphry Davy and Vasily Vladimirovich Petrov [17]. Carbon arc welding, a process by which metals are joined by melting them with an electric arc
between a carbon electrode and the workpiece, was created in 1881 by Bernardos and Introduction 15
Olszewski, and the first US patent for arc welding with bare metal electrodes was
awarded to Charles Coffin in 1890 [18]. In 1926, the use of helium gas as a shield
for the welding arc was described by Hobart and Devers. Shielded metal arc weld-
ing, commonly referred to as “stick welding,” became available in the 1930s, as did
Submerged Arc Welding(SAW).
In 1939 and 1943 A.M. Cassie and O. Mayr, respectively, published important
papers describing the electric arc, which would become the basis for many mathe-
matical models to follow [19]. Also around this time, welding became an important
part of manufacturing for the military efforts in World War II.
The 1950s saw many developments in Gas Metal Arc Welding(GMAW), including
a patent for spray transfer arc welding and the use of carbon dioxide as a shielding
gas. Plasma arc cutting, a process for precision cutting of metals using equipment
similar to welding, was first demonstrated publicly in 1956.
In 1960 the first semiconductor power device, the Silicon Controlled Rectifier
(SCR), became available. The SCR would become the basis for many industrial welding machines designed in the decades to follow. Pulsed GMAW was developed
in the 1960s, but would not become commonly used in industry until decades later.
“Synergic” GMAW was patented in 1964 by Manz. The first robotic welding instal-
lations also occurred during the 1960s.
In 1974 the US Maritime Commission supported Jim Thommes’ first-generation
prototype welding inverter, a technology that would dominate welding power source
technology beginning in the 1990s. The 1970s in general was an important decade for
the development of power electronics, and many of the analysis and control techniques
developed then are still in use today. Introduction 16
1881 - One of the earliest carbon arc welding machines invented by De • Meritens (France). 1885 - Carbon arc welding developed by Bernardos and Olszewski (Russia). • 1889-90 - First arc welding with bare metal electrodes (C.L. Coffin, USA). • 1926 - USA patents issued to M. Hobart and P.K. Devers for helium-shielding • in arc welding. 1930s - Shielded metal arc welding and submerged arc welding become avail- • able. 1941 - GTAW developed by Meredith (USA). • 1943 - Otto Mayr published an important paper describing a model of the • electric arc, focusing on the arcing phenomenon in electric relays. 1950 - First patent for spray transfer arc welding awarded to Muller, Gibson • and Anderson. 1956 - Plasma arc cutting first displayed publicly. • Late 1950s - CO2 used as a shielding gas for GMAW. • 1960 - The Silicon Controlled Rectifier(SCR), the first power semiconductor • device, becomes available [20]. 1960s - Development of pulsed GMAW. • 1961 - First robotic welding installation. • 1964 - Synergic GMAW patented by Manz. • 1974 - US Maritime Commission supports Jim Thommes’ first-generation • prototype welding inverter. 1990s - Solid-state inverter technology begins to dominate. • 2000s - Software-controlled machines become commonplace. • Table 1.1. Timeline of welding power source technology
1.2.2 Classical Approaches to Power Electronics Control
Classical approaches to switching power converter control are based on circuit av- eraging or state-space averaging of the switching action and linearization about an operating point [21, 22, 23]. Averaging techniques have been in widespread use since the 1970s. Using averaging methods, each possible combination of switching device position is considered as a separate circuit which leads to a set of differential equations for each configuration. These equations are “averaged” together using a duty cycle to transform the variable structure system into a continuous system. This Introduction 17
transformed system can then be linearized about an operating point to allow for Lin-
ear Average Control (LAC), normally using frequency-domain techniques, i.e. Bode
plots and gain and phase margins.
A 2003 paper by Guo, Hung, and Nelms discussed the use of root locus tech-
niques to design discrete-time controllers for switching converters using averaging
methods [24]. They were able to demonstrate good performance compared with controllers designed using frequency-domain techniques.
Sun and Grotstollen pointed out that averaging methods do not work well when a
Pulse Width Modulation (PWM) converter is operating in Discontinuous Conduction
Mode (DCM), and they cannot be applied to certain architectures such as resonant
or quasi-resonant converters [25].
It is well-known that classical approaches suffer from lack of robustness to input
power variations, output load variations, and parameter variations in the converter
components. Some researchers have addressed these issues by introducing nonlinear
extensions of classical control techniques. For example, Zhu investigated the applica-
tion of nonlinear Proportional-Integral-Derivative (PID) control, in which a nonlinear
combination of the proportional, integral, and derivative error is used to produce a
control signal to aDC-DC converter [ 26]. Compared to a traditional Proportional-
Integral (PI) controller, the nonlinear PID was found to be easier to tune and more
robust for a 1 kWDC-DC converter. Other attempts to address the robustness issues
of classical techniques include adaptive methods [27, 28] and gain scheduling [29].
Higuchi developed a multi-stage digital controller for a switching power converter that
estimated the load current and switched bumplessly between different controllers to
improve performance at regulating into an inductive load [30]. Introduction 18
1.2.3 Nonlinear Approaches to Power Electronics Control
Because of the difficulties with classical methods applied to power electronics, most
research in recent decades has focused on nonlinear forms of control such as those
using artificial intelligence (e.g. Fuzzy Logic Control(FLC), Artificial Neural Net- works (ANNs)) or variable structure techniques (e.g. Sliding Mode Control(SMC)).
Sometimes these approaches are combined to obtain desirable actions from several
types of controllers. Escobar et al. compared LAC to Fuzzy Logic Control (FLC) and various forms of SMC and found LAC to perform particularly poorly [31].
Fuzzy Logic Control(FLC). Fuzzy Logic Control(FLC) is an outgrowth of the
theory of fuzzy sets, formalized in 1965 by Zadeh [32]. Instead of describing systems
using numerical variables, fuzzy logic uses linguistic variables, i.e. variables whose values are sentences in a natural language. This is formalized mathematically using
the concept of a fuzzy set. Unlike crisp sets, elements are allowed to partially belong
to one or more fuzzy sets. The advantage of FLC is that it provides a mathemati-
cal framework for specifying human preferences for control system performance that would otherwise be difficult to express using analytical formulas [33]. An example
of a rule executed by a fuzzy controller would be “If setpoint error is positive and
large and the error change is positive and small, then the actuator output should
be negative and large.” These types of rules are most commonly used as high-level
supervisory controllers.
Advantages of FLC are that it does not require an accurate model of the system,
and the controller design relies only on intuitive descriptions of desired performance
instead of analytical calculations that may be difficult to perform. Although some
see this as an advantage, fuzzy logic was famously criticized by R.E. Kalman and Introduction 19
W. Kahan for lacking mathematical rigor [34]. A disadvantage of FLC is that it is difficult to implement the complex rules without using a microcontroller.
Some researchers have successfully applied FLC to power electronics, e.g. [35, 36].
Iskender and Kaarslan showed how an arc welding power source based on a two- switch single-phase inverter could be controlled using FLC or SMC with state-space averaging. In comparing the two methods, they found FLC to outperform SMC.
Their test, performed in simulation, assumed a linear, purely resistive load of 0.2 Ω and examined a step current change from 80 A to 110 A.
Golob, Koves, and Tovornik discussed the application of fuzzy logic to GMAW process control and weld process quality monitoring, but not to the control of the power source itself [37].
A 2002 paper by Viswanathan compared the performance of a “universal” fuzzy logic controller to a universalPI controller and a set of conventionalPI controllers tuned for specific operating points (output load and input line conditions) for a boost converter. The universal controllers had a single set of tuning parameters based on a nominal operating point. The fuzzy controller was found to perform better than the universalPI controller tuned for worse-case conditions, and about the same as the set ofPI controllers tuned for specific operating points [ 38].
Another 2002 paper by Balestrino discussed the application of fuzzy logic to thePI controller for a Cuk´ converter. FLC was used as the high-level supervisor to vary the coefficients of a conventionalPI controller, changing its action based on the difference between the actual and the desired output voltage [39].
Neural Networks. ANNs have been applied to inverter control. The idea is to train the network to create binary outputs for controlling the switching devices through a Introduction 20
nonlinear mapping of the input signals [40]. The chief advantage of neural networks in
control is that they are able to easily implement complex, nonlinear transfer functions.
However, they can be difficult to implement and require offline training before they
can be used [41].
Genetic Algorithms. Schutten and Torrey described the application of genetic al-
gorithms to switching power converter control, using the technique to optimize a
performance index [42].
Model-Predictive Control. Model Predictive Control (MPC) applied to switching power converters was documented by Geyer [43] and Kouro et al. [44].
Active Disturbance Rejection Control. Gao and Sun developed a control algo- rithm based on Active Disturbance Rejection Control (ADRC) which they applied to a 1 kWDC-DC converter and found to work well in rejecting disturbances and being easy to tune [45].
Fractional-Order Control. Fractional Order Control (FOC), based on the princi- ples of the fractional calculus, was proposed for the control of a buck converter by
Calder´on,Vinagre and Feliu [46]. In a related work, the same authors suggested the use of fractional order sliding surfaces when using SMC forDC-DC converter control [47].
1.2.4 Sliding Mode Control(SMC)
The general principle of SMC is to design a sliding manifold that causes the system’s state trajectory to move toward a desired operating point. SMC operates in two phases: a reaching phase and a sliding phase. In the reaching phase, the controller forces the trajectory from an arbitrary initial condition to the sliding manifold. Once the sliding manifold has been reached, the trajectory moves along it toward the desired Introduction 21
operating point. A reaching condition ensures that regardless of the initial condition,
the trajectory is directed to reach a vicinity of the sliding manifold. The sliding
manifold is subject to an existence condition which specifies that when the trajectory
is within a small distance of the sliding manifold, it will be “pulled” toward it at which
point the controller operates in the sliding phase. Once operating in the sliding phase,
a stability condition ensures that the trajectory will always move toward the desired
operating point. The controller designer has some freedom in designing the sliding
manifold provided these conditions are satisfied.
For a switching power converter, moving along the sliding manifold can be done
by opening and closing the switching devices at a high frequency (ideally infinitely
fast). Because of the practical impossibility of implementing infinitely fast switching
due to switching losses, Electromagnetic Interference (EMI) concerns and difficulty of
magnetics design, researchers have proposed alternatives such as hysteresis modula-
tion or quasi-sliding mode fixed frequency control which exhibit many of the benefits
of SMC with somewhat degraded performance from the ideal.
SMC grew out of the theory of variable structure systems, and began to be rec-
ognized in the 1960s and 1970s as described by Utkin [48]. The specific application
of SMC to power electronics was studied in the 1980s, e.g. [49, 50, 51, 52] and
in the 1990s, e.g. [53, 54, 55]. By the late 1990s it was apparent that the the-
ory of SMC had become well understood and that, in order to see more widespread
use, practicing engineers needed to be educated about how to integrate it into their work [56]. Today, SMC is beginning to be accepted as a prominent control strategy
for power electronics due to its inherently variable structure and its ability to pro-
duce controllers that are robust across a wide range of parameter variations as seen Introduction 22
in power electronics [57, 58, 59, 60, 61]. In 2008 Tan, Lai, and Tse pointed out
that SMC applied toDC-DC converters has been heavily researched but needs more
practical applications [62]. A 2012 book by the same authors presents the theoretical
and practical aspects of SMC applied to switching power converters [41].
In 1995, Nguyen published a paper where SMC was used to control a buck con- verter to track an arbitrary voltage reference signal with adaptive hysteresis [63].
Iskender and Karaarslan compared SMC with FLC for an arc welding power source
in a 2005 work [64].
In classical control methods using fixed-frequency PWM, PWM signal design is
usually performed as a separate step from controller design. That is, the dynamics
of the converter are not considered when designing the PWM switching patterns.
Sabanovic-Behlilovic et al., in a paper describing the application of SMC to three-
phase inverters and rectifiers, noted as an advantage over classical methods that
SMC allows the switching pattern and the controller design to be performed simulta-
neously [65]. SMC applied to a single-phase inverter was described by Jezernik and
Zadravec [66].
Because SMC inherently handles the variable structure of the switching power
converter, it does not suffer from the drawbacks of classical methods when converters
are operating in DCM. SMC tends to be easy to implement in either analog or digital
domains when compared to other advanced control techniques, and it is robust across
parameter variations.
A disadvantage of SMC is that the variable frequency it requires has practical
problems for implementations including increased switching losses at high frequencies
and complicating the design of transformers and output inductors. Some authors note Introduction 23
that the variable frequency can cause problems for Electromagnetic Compatibility
(EMC)[ 41]. However, Tanaka, Ninomiya and Harada proposed a random-switching
control method for a switching converter in order to reduce the power of the switch-
ing harmonics in the output frequency spectrum [67]. In their method, a random
perturbation was deliberately added to the switching instant of the converter, and
this was found to spread the power of the noise spectrum more evenly which may
actually be beneficial for EMC.
1.2.5 Reference Tracking
Most of the literature concerning switching power supply control assumes only that
a constant output voltage is of interest. One exception is the work done by Nguyen, where SMC was used to control a buck converter to track an arbitrary voltage refer-
ence signal with adaptive hysteresis [63]. This is one of the few resources describing
the tracking problem for switching power converters.
1.2.6 The Welding Power Source
Much of the published research regardingDC-DC converters, switching inverters and
switching rectifiers applies to the design of welding power sources. Some designs
specifically for welding have been discussed in the power electronics literature. Mecke,
Fischer, and Werther presented a design for a phase-shifted, Zero Voltage Switch-
ing (ZVS)/Zero Current Switching (ZCS) full-bridge inverter intended to minimize
switching losses in arc welding applications [68]. Aigner, Dierberger and Grafham
compared different topologies for Arc Welding Power Source (AWPS) use including
series resonant converters, quasi-resonant ZCS converters, and multi-resonant ZCS
and ZVS converters in terms of their desirability for arc welding applications [69]. Introduction 24
More recently a paper by Klumpner and Corbridge discussed a two-stage design where
the inverter bridge always operates at 50 % duty cycle and a prior stage regulates the
input voltage for this inverter [70]. Wang, Wang, and Xu described the design of a
double-loop control system for an AWPS with an outer voltage control loop providing
a reference for an inner current loop [71].
1.2.7 Power Source Dynamics and Their Effect on Welding
Several researchers have commented on the effect of the power source dynamics on metal transfer in GMAW. Richardson, Bucknall, and Stares explored the shape of a pulsed GMAW waveform having a trapezoidal current shape [72]. They found that two trapezoidal pulse waveforms having the same mean current can have different wire melting rates depending on the slew rate of the current, since this affects how much time is spent in the pulse peak and thus how much current contributes to I2R heating of the wire: faster current response leads to smaller liquid metal droplets. They also compared the behavior of the idealized trapezoidal pulse with the exponential shaped pulse that is more commonly seen in actual machines. Similar results had been found earlier by Yamamoto, Harada, and Yasuda [73]. Joseph et al. compared the behavior
of the pulse waveforms of four commercially available power sources, and found the
machines to produce different results with similar settings due to the differences in wave shapes [74]. Yarlagadda et al. described a method of detecting short circuits in
pulsed GMAW by combining digital signal processing of current and voltage signals with high-speed videos of metal transfer occurring [75].
Arc length control and metal transfer have been heavily studied for GMAW. Ex-
amples of more theoretical works include those by Modenesi and Nixon, Bingul, and
Thomsen. Modenesi and Nixon’s work focused on the instability in metal transfer Introduction 25
that occurs with abrupt changes in current and/or voltage during welding. Their work also considered the effects of the shielding gas on this instability [76]. Bingul
developed a model explaining that the metal transfer mode and the V-I characteristic
of the arc both play a role in the stability of arc length regulation for GMAW[ 77].
Thomsen developed a nonlinear arc length control system for GMAW using the feed-
back linearization method [78].
1.3 Goals of this Work
From the beginning, the goals of this work have been the following:
Understand the components of a welding setup, how they interact, and how • they can be modeled, focusing on how each component affects the control of
the welding power source.
Develop a simulation environment that can be used to explore the compo- • nents of the welding setup and to test different control strategies.
Develop a new power source control strategy that improves upon the ex- • isting controls in terms of reference tracking and handling of varying load
impedance.
Implement the new strategy in hardware by retrofitting an existing welding • power source, in order to prove that it is feasible and to test its performance
against the existing controls.
Compare the new control strategy to existing controls, in scenarios that • reflect events that occur during actual weld processes.
As will be explained throughout the thesis, some of these goals have been achieved
and, for others, further work is needed. Introduction 26
1.4 Contributions of this Work
This work contributes to the existing literature on SMC applied to switching power
converters by considering output current regulation in Section 4.3 and power regula-
tion in Section 4.5 in addition to output voltage regulation in Section 4.4. Addition-
ally, the focus is placed on a non-resistive load impedance, and on transient response
and reference tracking rather than only the stability of the closed-loop system as
commonly encountered in the literature. Section 4.7 describes a method of applying
constraints to the SMC strategy, forcing the system trajectory to stay within defined
limits. This type of constrained control has been briefly described in the general
case [79] but has not been applied to switching power converters.
Chapter 2 expands upon the models of welding equipment in the literature by
collecting models for all of the components of a welding setup into one place. The
simulation environment, described in detail in Appendix A, provides a way for en-
gineers to experiment with new control strategies and to easily test performance in
different scenarios.
The hardware implementation described in Chapter 5 gives a practical industrial
application of SMC using readily-available electronic components. This implementa-
tion can be used as a guideline for other researchers looking to implement SMC in
switching power converters.
1.5 Thesis Organization
Chapter 2 presents mathematical models for the various components of a welding
system, including the electric arc, welding fixture, cables, and the power source itself. Introduction 27
Chapter 3 reviews the performance of the existing control system, and discusses ways the performance could be improved. A proposed new control strategy based on Slid- ing Mode Control(SMC) is presented in Chapter 4. This strategy is implemented in hardware, and the implementation is described in Chapter 5. Experimental results comparing the existing system with the new control strategy are described in Chap- ter 6. The thesis concludes in Chapter 7 with a review of the work completed and suggestions for further work. 28
2 Modeling
In this chapter, models of the welding circuit and power source are derived from
basic principles. The welding circuit is first considered without regard for the power
source, accounting for all of the other impedances present in the circuit. The power
source is then modeled with the welding circuit as its load, using switching power
converter analysis techniques from power electronics.
The welding circuit comprises all of the components of a welding setup through which electrical current flows during welding. The circuit diagram in Figure 2.1 shows
the power section as a current source iw(t) supplying current to the welding circuit,
and illustrates the impedances in the circuit: the cables carrying current from the
power source to the arc, having an impedance Zc; the resistance Rt of the point at which the electrode comes into contact with the current in the cable (in the case of
consumable electrode processes); the resistance Re of the electrode itself; the welding
arc with conductance Ga; and Zp, the impedance of the workpiece and welding fixture.
2.1 Welding Cables, Workpiece, and Welding Fixture
Due to the high amperage required for arc welding, large cables are used between the
power source and workpiece; American Wire Gauge (AWG) 0000 or AWG 000 are Modeling 29
Zc Rt
cables contact point Re electrode
power source iw(t)
Ga arc
Zp
workpiece
Figure 2.1. Circuit diagram showing sources of impedance in the weld- ing circuit
common sizes. After the arc itself, the welding cables are typically the most significant
contributor to the impedance of the welding circuit. This impedance can vary widely
between setups. In some cases the power source is positioned near the workpiece with
short, straight cables, in which case the cable impedance has a negligible effect. In
other situations such as at construction sites, very long cables coiled around metal
structures may separate the power source from the workpiece. Cable impedance can
have a significant effect on the quality of welds as discussed by Peters and Allgood [80].
The cable is assumed to be two parallel conductors of length `c, each with radius
rc, separated by a distance dc as shown in Figure 2.2. The impedance can be analyzed
using a simplified form of Heaviside’s transmission line model, as shown in Figure 2.3.
The model consists of a resistance Rc, capacitance Cc, and inductance Lc [81].
The resistance Rc of the cables can be computed from the conductor resistivity
ρc, radius rc, and cable length `c:
`c Rc = ρc 2 . (2.1) πrc Modeling 30
µins µins
c
rc rc dins dins ρc ρc
dc
Figure 2.2. Welding cable cross-section showing electrical and geomet- rical properties
Rc Lc
Cc
Figure 2.3. Circuit diagram for welding cable transmission line model
Welding cables are typically stranded copper (ρ 1.75 10−8 Ω m) or stranded alu- ≈ × minum (ρ 2.83 10−8 Ω m) with a thick (0.25 mm to 1.75 mm) insulator made of a ≈ × synthetic rubber such as neoprene or EPDM[ 81, 82]. Figure 2.4 shows the relation-
ship between cable length and resistance for both copper and aluminum AWG 0000
cables. Modeling 31
Figure 2.4. AWG 0000 welding cable resistance versus cable length
The approximate inductance of the transmission line in microhenries is given by [83] dc 1 dc Lc = 0.004`c ln + . (2.2) rc 4 − `c Figure 2.5 plots the inductance versus cable length for AWG 0000 welding cables separated by various distances.
The approximate capacitance in farads of the welding cables is given by [81]
π`c Cc = . (2.3) ln(dc/rc) Modeling 32
Figure 2.5. AWG 0000 welding cable inductance versus cable length
In Figure 2.6, the capacitance versus cable length is plotted for AWG 0000 welding
cables separated by various distances.
The welding cables effectively introduce a harmonic oscillator into the welding
circuit, where the relationship between the voltage at the power source Vw(s) and the voltage at the end of the cables Va(s) is given by
1/LcCc Va(s) = Vw(s). (2.4) s2 + Rc s + 1 Cc LcCc This system has a resonant frequency given by
1 fn = . (2.5) 2π√LcCC Modeling 33
Figure 2.6. AWG 0000 welding cable capacitance versus cable length
Figure 2.7 shows how the resonant frequency decreases as the cable length increases,
becoming close to typical switching frequencies as the length approaches 200 m. This
implies an inverse relationship between cable length and control system bandwidth:
as the cable length increases, the bandwidth of the control system must decrease to
avoid amplifying the resonant frequency of the welding circuit.
2.2 Electrode and Contact Tip
There is some resistance associated with the consumable SMAW electrode or the
nonconsumable tungsten electrode in GTAW. In wire-fed processes such as GMAW Modeling 34
Figure 2.7. AWG 0000 welding cable resonant frequency versus cable length
and FCAW, the wire electrode is fed through a tubular conductor that is connected to the welding cable as shown in Figure 2.8. The point where the tubular conductor comes into contact with the electrode has an associated resistance. These consumable- electrode processes typically demand a consistent Contact Tip to Workpiece Distance
(CTWD), which describes the length of electrode wire that should extend beyond the contact tip during a weld. While this length of electrode is short its cross-sectional area is small, and due to the high amperage in the welding circuit its resistance is non- negligible. Computing the resistance of the contact tip and electrode is complicated Modeling 35
gas nozzle
contact tube
workpiece electrode
Figure 2.8. Wire electrode fed through contact tube as in GMAW and FCAW
by the fact that the resistivity of metals varies as a function of temperature [84]:
`e Re = ρe(T ) 2 , (2.6) 2πre where ρe(T ) is the resistivity of the electrode material as a function of temperature
T , `e is the length of the electrode that extends beyond the contact tip, and re is the
radius of the electrode.
2.3 The Welding Arc
The welding arc is a gaseous conductor, and the relationship between voltage and
current in an arc is nonlinear [2,5, 85 ]. At low current, the total potential of
the arc falls as current increases. At some point, the potential reaches a minimum
and then begins to rise with increasing current. After this point the relationship
is essentially ohmic. In general the potential increases as the gap between the arc Modeling 36
terminals increases, and in practice welders generally consider the arc voltage to be
proportional to arc length. Several researchers have created models of the arc leading
to a nonlinear relationship between voltage, current, and arc length. Figure 2.9 shows
the relationship between voltage and current for a GMAW arc at various arc lengths
as reported by Bingul [84].
There are many factors that affect the electrical characteristics of the arc, includ-
ing the shape and material makeup of the electrodes, the ambient temperature and
pressure, surrounding electromagnetic fields, gas composition, etc. [86]. Hence, the
published models of electric arcs make many simplifying assumptions.
The most well-known electric arc models are those of Mayr and Cassie. The
Mayr model gives an expression for the conductance of an arc Ga in terms of the arc voltage va and current ia and a steady-state “cooling power” dissipated by the arc to the environment: d ln G 1 v (t)i (t) a = a a 1 (2.7) d t τa P0 − where τa is the arc time constant and P0 is the cooling power dissipated by the arc
to the environment at steady-state.
The similar Cassie model gives an expression for the conductance of an arc in
terms of the voltage and a steady-state arc Electromotive Force (emf)
2 d ln Ga 1 v = 1 2 (2.8) d t τa − V0 where V0 is the steady-state arc emf.
Tseng et al. suggested combining the Cassie and Mayr models using a smooth,
monotonically decreasing function of the current such that the conductance is given Modeling 37
60 10 mm arc length 8 mm arc length 55 4 mm arc length
50
45
40 volts
35
30
25
20 150 200 250 300 350 amperes
Figure 2.9. V-I characteristics for GMAW arcs of various lengths
by i2 vi i2 i2 d G Ga = Gmin + 1 exp 2 2 + exp 2 τa (2.9) − −It V0 −It P0 − d t which tends to the Cassie model at low currents, and the Mayr model at high currents.
This model was experimentally demonstrated by Sawicki et al. to match welding
transformer arc characteristics in several scenarios [87].
2.4 Summary of Load Impedances
Given the descriptions of the various welding circuit impedances in the previous sec-
tions, it would seem intractable to develop a model that encompasses all of them.
Indeed, it will be shown that for the purposes of developing control strategies for
a MAWPS, it is sufficient to consider the stability and tracking performance across
ranges of impedances, or under specific situations that can be modeled using simpli-
fying assumptions. Modeling 38
2.5 Piecewise Linear Model
Classical systems and control theory provides a collection of tools for analyzing and
designing Single Input Single Output (SISO), linear systems that do not change over
time (i.e. Linear Time-Invariant (LTI) systems). In order to use these tools for power
electronics systems, which are nonlinear and time-varying, it is necessary to make
simplifying assumptions about the systems and develop an approximate linear, time-
invariant model. This model typically works well enough in practice when working with a simple converter with a single operating point, but becomes cumbersome and
inaccurate when a converter must operate across a wide range of input and output
power levels.
Usually, an averaging approach can be applied to switching power converters that
are controlled by PWM. The basic strategy is to develop a model of the circuit
equivalent to each possible switch position, then combine these into a single model
by a weighted average, with the PWM duty cycle giving the weight applied to each
switch position.
The drawback of the traditional averaging approach is that it results in only a
small-signal model and does not accurately describe the large-signal response of the
circuit. Furthermore, because the linearization is performed about a specific operating
point, the controllers developed using this technique are not very robust across ranges
of output loads, input power variations, and/or variations in the parameters of the
circuit elements.
The welding power source power section and load can be modeled using the circuit
diagram in Figure 2.10, which resembles a traditional buck DC-DC converter with a Modeling 39
resistive-inductive load. This is a variable-structure system containing two subsys-
tems, one of which is active at any moment depending on the position of the switch
S.
When S is closed (u = 1), the circuit in Figure 2.11 results. This circuit is
described by the set of linear ordinary differential equations
d iL 1 1 = vw(t) + vg(t) (2.10) d t −L L d vw 1 1 1 = iL(t) vw(t) iw(t) (2.11) d t C − RoC − C
d iw 1 Rw = vw(t) iw(t). (2.12) d t Lw − Lw
This system can be written in matrix form as
˙x(t) = Ax(t) + B1vg(t) (2.13) where iL(t) 0 1/L 0 1/L − x(t) = v (t) A = 1/C 1/R C 1/C B1 = 0 . (2.14) C o − − iw(t) 0 1/Lw Rw/Lw 0 − When the switch is open (u = 0), the circuit in Figure 2.12 results. This circuit is
described by
d iL 1 = vw(t) (2.15) d t −L d vw 1 1 1 = iL(t) vw(t) iw(t) d t C − RoC − C
d iw 1 Rw = vw(t) iw(t) d t Lw − Lw Modeling 40
L iL(t) iw(t) S
Lw
+ vg(t) D C Ro vw(t) −
Rw
Figure 2.10. Circuit diagram of power section and load
L iL(t) iw(t)
Lw
+ vg(t) C Ro vw(t) −
Rw
Figure 2.11. Circuit diagram of power section and load when switch is closed
which, in matrix form, can be written
˙x(t) = Ax(t) + B2vg(t) (2.16) where x(t) and A are as in (2.14) and 0 B2 = 0 . (2.17) 0 Modeling 41
iw(t)
iL(t) Lw
L C Ro vw(t)
Rw
Figure 2.12. Circuit diagram of power section and load when switch is open
2.5.1 Power Section Waveforms
Suppose the voltages vg(t) and vw(t) are constants, i.e. vg(t) = Vg and vw(t) = Vw.
This is a valid assumption when the time interval under consideration is short, or the converter is operating in steady-state. Then (2.10) gives
d iL Vg Vw = − (2.18) d t L which has solution
Vg Vw iL(t) = iL(t0) + − (t t0). (2.19) u=1 L −
This shows that when the switch is closed, the choke current iL(t) increases linearly with a ramp rate proportional to the voltage difference Vg Vw and inversely propor- − tional to the choke inductance L. A higher input voltage tends to increase the ramp
rate, while a higher output voltage or a higher inductance tends to decrease the ramp
rate. Assuming Vg > Vw is always true, the current will always increase when the
switch is closed. Modeling 42
When the switch is open, Vg disappears from (2.18) and
d i V L = w (2.20) d t − L
results. This has solution
Vw iL(t) = iL(t0) (t t0). (2.21) u=0 − L −
Thus the current decreases linearly at a rate proportional to Vw and inversely pro-
portional to L. Figure 2.13 shows the relationship between switch position and choke
current described by equations (2.19) and (2.21).
2.5.2 Welding Current and Voltage Waveforms
Consider the Laplace transform of equation (2.12),
1 Rw siw(s) = vw(s) iw(s). (2.22) Lw − Lw
Solving for iw(s) gives
1/Rw iw(s) = vw(s). (2.23) 1 + Lw s Rw
Equation (2.23) has the form of a first-order low-pass filter with time constant Lw/Rw
and gain 1/Rw. This shows that the welding current can be considered a scaled and
filtered version of the welding voltage. The frequency response of this filtering effect
is shown in Figure 2.14 for a fixed resistance Rw = 100 mΩ.
The filter time constant increases as Lw increases, or as Rw decreases. Notice that
vw(s) lim iw(s) = , s→0 Rw which shows that at steady-state the inductance has no effect and the current is
related to the voltage through the resistance Rw of the welding circuit, as expected
from Ohm’s law. Similarly, as Lw 0, the current iw(s) vw(s)/Rw. → → Modeling 43
iL
Vg Vw m1 = − L m = Vw 2 − L
iL(t0)
t
t0
u
1
0 t
t0 t0 + dTsw t0 + Tsw
Figure 2.13. Relationship between switch position u and choke current iL(t)
2.5.3 State-Space Average Model
The two subsystems described by (2.13) and (2.16) can be combined into a single
system using the state-space averaging technique, originally described by Middlebrook
and Cuk´ [21]. Suppose the switch S is closed during the interval t [0, t0]. During ∈ this time the system (2.13) is active. The solution of this system is given by
Z t At A(t−τ) x(t) = e x(0) + e B1vg(τ) d τ t [0, t0]. (2.24) 0 ∈ Modeling 44
Figure 2.14. Current filtering effect of the welding circuit impedance
At t = t0 the switch opens, and it is open during the interval t [t0,Tsw]. During ∈ this interval the system (2.16) is active, which has solution
A(t−t0) x(t) = e x(t0) t [t0,Tsw]. (2.25) ∈
Although the system changes structure at t = t0, its state x(t) is continuous. Using
this fact, we can write the value of the state at t = Tsw in terms of the state at t = 0:
Z t0 ATsw A(Tsw−τ) x(Tsw) = e x(0) + e B1vg(τ) d τ. (2.26) 0
By defining the duty cycle d(t) [0, 1] as the fraction of the switching period Tsw ∈ in which S is conducting and d0(t) = 1 d(t) as the fraction of the time the switch is − Modeling 45
not conducting, we can write
0 ˙x(t) = Ax(t) + d(t)B1 + d (t)B2 vg(t). (2.27)
This removes the variable structure nature of the model, but it is a nonlinear equation
due to the presence of d(t)vg(t) terms. An approximate linear model can be obtained
by eliminating these terms through perturbation and linearization.
In the perturbation step, we assume that the converter is operating at a nominal
point described by the constant state vector X, with constant input voltage Vg and at
a constant duty cycle D. We consider small fluctuations around these points described
by ˜x(t),v ˜g(t), and d˜(t), respectively:
x(t) = X + ˜x(t)
vg(t) = Vg +v ˜g(t)
d(t) = D + d˜(t).
Substituting these quantities into (2.27) gives the averaged and perturbed equation
0 ˜x˙ = AX + (DB1 + D B2)Vg
0 + A˜x(t) + (DB1 + D B2)˜vg(t) (2.28)
+ (B1 B2)Vgd˜(t) + (B1 B2)d˜(t)˜vg(t). − − This equation shows that the system is described by a superposition of the constant
0 steady-state response AX + (DB1 + D B2)Vg and a time-varying response. If we
restrict our interest to small fluctuations about the steady-state response, and we
eliminate the term in d˜(t)˜vg(t) under the assumption that both d˜(t) andv ˜g(t) are
small in magnitude (and hence their product is smaller still), then the averaged, Modeling 46
perturbed, and linearized model
0 ˜x˙ (t) = A˜x(t) + (DB1 + D B2)˜vg(t) + (B1 B2)Vgd˜(t) (2.29) −
results. This is equivalent to the set of equations
1 D Vg ˜ı˙L(t) = v˜w(t) + v˜g(t) + d˜(t) (2.30) −L L L 1 1 1 v˜˙w(t) = ˜ıL(t) v˜w(t) ˜ıw(t) (2.31) C − RoC − C
1 Ro ˜ı˙w(t) = v˜w(t) ˜ıw(t). (2.32) Lw − Lw
This linear, time-invariant system that describes small fluctuations around a steady- state operating point is referred to as the small-signal model of the power converter.
2.5.4 Nonlinear Output Inductor
Another source of impedance is the output inductor in the power source, typically
8 µH to 80 µH for a MAWPS based on Switched-Mode Power Supply (SMPS). The inductor is designed to smooth the output current and prevent loss of the welding arc at low currents. Due to saturation, the inductance changes from a high value at low currents to a low value at high currents, until it remains essentially constant after a certain current level. Its inductance is therefore a nonlinear function of current, L(iL),
as shown in Figure 2.15. Modeling and simulation of nonlinear inductors has been
documented in the literature, for example by Neves and Dommel [88], Deane [89],
and Ahmedou et al. [90]. Modeling 47
40
35
30
25
20 microhenries
15
10
5 0 100 200 300 400 500 amperes
Figure 2.15. Inductance vs. current for the output inductor of a typical welding power source
2.6 Sources of Uncertainty
In the sections that follow, each component of the circuit is discussed to determine
its nominal linear component and its uncertain and/or nonlinear components. The
constants in the bounds for the unknown functions can be determined from the ranges
of tolerances and uncertainties in the model.
2.6.1 Input Power
The input Vg(t) is a rectified sinusoidal voltage. There is a range of average values over which the system can be expected to operate, i.e.
Vg(min) Vg(avg) Vg(max). ≤ ≤
High-frequency variations may occur in the input power due to noise or environmental effects. It is also possible for the amount of current drawn by the power section to Modeling 48
influence the input voltage. If the nominal average input voltage is assumed to be
1 Vg(nom) = Vg(max) Vg(min) , 2 − then the input voltage signal can be written as
Vg(x, t) = Vg(nom) + Vˆg(x, t) (2.33)
where Vˆg(x, t) is an unknown disturbance, possibly dependent upon the state of the
power converter. This unknown disturbance can describe differences of the average
input voltage from the nominal as well as high-frequency disturbances and loading
effects.
2.6.2 Semiconductor Switches
The switch is a semiconductor device such as a Metal-Oxide Semiconductor Field-
Effect Transistor (MOSFET) or Insulated Gate Bipolar Transistor (IGBT). A typical
model for a nonideal MOSFET consists of an ON resistance Ron in series with a small
uncertain voltage:
Vsw(x, u, t) = Isw(t)Ron + Vˆsw(x, u, t). (2.34)
The disturbance Vˆsw(x, t) encapsulates differences in the actual devices from the as-
sumed value of Ron as well as unmodeled nonidealities in the semiconductor switches.
The unmodeled effects may depend on the current flowing through the switches, hence
the dependence of Vˆsw on the state vector x.
2.6.3 Output Inductor
The nonideal output inductor has an inductance that is a function of the current
through the inductor. At low current levels, the inductance is high. It decreases until Modeling 49
the inductor saturates after which point it remains fairly constant. If the nominal
inductance in saturation is L(nom), then the total inductance may be written as
L(iL(t)) = L(nom) + Lˆ(iL(t))
where Lnom is a constant, nominal inductance and Lˆ(iL(t)) is a possibly time-varying
inductance that may be dependent upon the current through the inductor.
2.6.4 Output Resistor
The output resistor exists in the system to provide a path for current to flow when the
load would otherwise be an open circuit, improving the open circuit voltage regulation
of the power converter. The typical output resistor is a wirewound device having a
nominal resistance Ro(nom) with a wide tolerance of as much as 20 %. The resistance ± may change slightly with temperature, and may include some inductance due to the wirewound construction. A reasonable model of the output resistor may be written
as
Zo(x, t) = Ro(nom) + Zˆo(x, t). (2.35)
Here Ro(nom) is the rated resistance, and Zˆo(x, t) encapsulates the uncertainty asso-
ciated with the resistor tolerance, environmental effects, and the reactive component
of the impedance.
2.6.5 In the Model
Reasonable ranges of the model parameters used throughout this work are given in
Table 2.1. Modeling 50
Table 2.1. Ranges of Model Parameters
Quantity Nominal Minimum Maximum Choke Inductance, L 45 µH 8 µH 60 µH Input Voltage, Vg 80 V 75 V 100 V Output Capacitance, C 220 pF 110 pF 330 pF Output Resistance, Ro 50 Ω 40 Ω 60 Ω Welding Inductance, Lw 20 µH 10 µH 300 µH Welding Resistance, Rw 100 mΩ 5 mΩ 500 mΩ 51
3 Review of the Existing System
The machine used for the actual hardware implementation is a chopper-based
machine capable of welding all of the common welding processes (SMAW, FCAW,
GTAW, and GMAW) with welding performance that is respected in the industry.
The basic topology of the machine is shown in Figure 1.2.
This chapter serves as a brief overview of the existing power source control system, which will be used to compare the effectiveness of the proposed new strategy in
subsequent chapters.
3.1 Overview of the Existing Control System
A simplified block diagram of a welding power source control system is shown in
Figure 3.1. It is a hierarchical system that consists of an arc length control and
metal control that receive direct user inputs, and an “inner control loop” that drives
the power electronics. The subject of this thesis is the inner control loop, which is
responsible for controlling the power electronics to deliver the required output levels.
The setpoint for the inner control loop comes from the metal transfer control block, which handles the logic and waveform generation appropriate to the selected welding Review of the Existing System 52 process. The arc length control block dynamically adapts the waveform generated by the metal transfer controller to correct for deviations in arc length. Review of the Existing System 53 IGBTs on/off Loop Inner Control reference output limits regulation mode Control Waveform Generator Metal Transfer Welding scale factor Program Scale and Interpolate Control Arc Length Figure 3.1. Block diagram of the existing control system Voltage Feedback Current Feedback Deposition Control User Controls Waveform Adjustments Arc Length Adjustment Review of the Existing System 54
Figure 3.2. Example of a simplified FSM for a GMAW welding program
3.2 Waveform Generator Reference Tracking
The waveform produced by the power source is commanded by a software waveform
generator, which is a Finite State Machine (FSM) that determines a setpoint by
reacting to current and voltage feedback and time according to the particular welding
program. An example of a simplified FSM for a GMAW process is shown in Figure 3.2.
The waveform generator may command regulation of current, voltage, or power, and
may do so using step, ramp, or exponential reference signals. Review of the Existing System 55
r(t)
r0
t
Figure 3.3. Example of a constant reference signal
3.2.1 Constant Reference
A constant reference, such as the one shown in Figure 3.3, may be written
r(t) = r0 (3.1)
r˙(t) = 0 (3.2)
r¨(t) = 0, (3.3)
where r0 is a scalar constant value. Review of the Existing System 56
3.2.2 Ramp Reference
A ramp reference signal such as that shown in Figure 3.4 may be written
r(t) = βt + r0 (3.4)
r˙(t) = β (3.5)
r¨(t) = 0, (3.6)
where β is a slope (ramp rate) and r0 is an initial starting point for the ramp.
In the MAWPS application, ramp references are commonly found in two situa-
tions: ramping the voltage to a level corresponding to a desired arc length at the
start of a GMAW weld, and ramping the current to a level that will “break” a short
circuit by separating molten metal from the end of the electrode in GMAW, FCAW,
and SMAW.
3.2.3 Exponential Reference
An exponential reference signal such as the one shown in Figure 3.5 can be written
−t/τ r(t) = rf (rf r0)e (3.7) − − rf r0 r˙(t) = − e−t/τ (3.8) τ rf r0 r¨(t) = − e−t/τ , (3.9) − τ 2 where r0 is an initial starting point, rf is a destination that is approached asymptoti-
cally, and τ is a time constant specifying the rate of rise or decay for the exponential.
The exponential reference is found in several MAWPS waveforms, including the pe-
riodic increase and decrease of current in pulsed GMAW. Review of the Existing System 57
r(t)
β
r0
t
Figure 3.4. Example of a ramp reference signal
r(t)
rf
1 r (r r )e− f − f − 0
r0
t 0 τ
Figure 3.5. Example of an exponential reference signal Review of the Existing System 58
r(t)
rf
1 2 2 αt + βt + r0
r0
t
t0 tf
Figure 3.6. Example of a parabolic reference signal
3.2.4 Parabolic Reference
For a parabolic reference such as shown in Figure 3.6,
1 r(t) = αt2 + βt + r (3.10) 2 0 r˙(t) = αt + β (3.11)
r¨(t) = α. (3.12)
Welding programs are designed for a specific arc welding process, electrode, ma- terial, and/or gas type and are created and modified by an application running on a
Personal Computer (PC). The programs are compiled to a binary file programmed Review of the Existing System 59
Figure 3.7. Output regulation modes in a representative sample of welding programs
into the flash memory in the power source. This provides flexibility and means that
the power source is, effectively, a development platform for welding waveforms. The
capability of the system to implement different welding processes is limited only by
the hardware and the control system performance. A built-in high-speed data ac-
quisition system allows easy monitoring of any digital signal in the power source’s
microprocessor [91].
In a survey of 146 welding modes available in Lincoln Electric’s welding power
sources, the frequency of current, voltage, and power regulation commands along with waveform shapes were found to be as shown in Figure 3.7. The welding processes
represented were SMAW, FCAW, GMAW, and GTAW. The most common regulation
command was for current, followed by voltage and then power. An exponential shape
is overall the most common command, followed by a step and then a ramp. 60
4 Proposed Control Strategy
Sliding Mode Control(SMC) is a variable-structure, nonlinear control technique
that is simple to implement [41], is robust across parameter variations [92], and has
a wealth of supporting research [93, 56]. SMC has been applied successfully to many
areas of systems and control. Over the last 20 years, SMC has emerged as a powerful, well-accepted strategy for controlling SMPSs. Advantages of SMC over traditional
methods and other nonlinear approaches include:
SMC addresses the large-signal characteristics of the power converter re- • sponse, while traditional methods based on linearization and averaging only
address small-signal deviations about an operating point;
SMC inherently accounts for the switching action of the semiconductors used • in power electronics. In fact, it depends on this switching action in order to
achieve the control objective.
This chapter presents the proposed control strategy for a Multiprocess Arc Weld-
ing Power Source(MAWPS), based on SMC. Section 4.1 gives an overview of key
SMC concepts. Section 4.2 presents the simplified model used in the development of
the control strategy. In Section 4.3 the theory is applied to output current regulation Proposed Control Strategy 61 with ideal switching. The theory is then extended to cover switching frequency con-
trol, load impedance variations, reference tracking, output voltage and power control,
and output constraints.
4.1 Overview of Sliding Mode Control
SMC divides the control response into a reaching phase and a sliding phase. During
the reaching phase, the control forces the system trajectory from its initial condition
toward a sliding manifold that represents the desired closed-loop dynamics. Once the sliding manifold has been reached, i.e. during the sliding phase, the trajectory of the system is confined to the sliding manifold while the desired dynamics are exhibited.
In this phase, the control signal serves to maintain the system trajectory on that sliding manifold.
The general SMC approach is [94]:
(1) Define the quantity of interest and the desired dynamics during the sliding
mode. This results in the definition of a sliding manifold in the system S state space. When the system trajectory is on the manifold , the system S achieves the desired dynamics.
(2) Define a switching function σ(x, t) that serves to measure the “distance” of
the system trajectory from the sliding manifold. When σ = 0, the system
trajectory lies directly on the sliding manifold:
= x : σ(x, t) = 0 . S { }
The sign of σ informs the control whether to supply energy (i.e. u = 1) or to
allow energy to dissipate (i.e. u = 0). The switching function σ determines Proposed Control Strategy 62
the value of u(t) according to 0 σ(t) < 0 u(t) = . (4.1) 1 σ(t) > 0
Thus, σ must take values that reflect a need for energy addition when σ > 0
and dissipation when σ < 0.
4.2 Simplified Model
For the purpose of developing a Sliding Mode Control(SMC) strategy, consider the
simplified model in Figure 4.1. This model eliminates the capacitor from Figure 2.10,
and models the load impedance as a resistor in series with an inductor. Removing
the capacitor is justified by the fact that this component is usually not present or is very small in an actual welding power source. The simplified model is described by
the system
Rw 1 x˙ 1 = x1 + x2 − Lw Lw RoRw Ro Ro Ro x˙ 2 = x1 + x2 + Vgu (4.2) Lw − Lw L L where x1(t) = iw(t) is the welding current, and x2(t) = vw(t) is the welding voltage.
The control signal u(t) = 1 when the switch Q1 is closed, and u(t) = 0 when the
switch is open.
Note that the relative degree of the current x1 is two, and the relative degree of
the voltage and power is one [95]. In other words, the control u appears directly in the first derivative of the voltage and power, and in the second derivative of the Proposed Control Strategy 63
L Q1 iw(t)
+
ideal Rw
+ Vg D Ro vw(t) − ideal
Lw
−
Figure 4.1. Circuit diagram of simplified power source and load
current. This fact will be used during the development of the SMC tracking controller
to determine the form of the sliding surface to be used.
This simplified model is used to develop the SMC strategy because it is a reason-
able reflection of reality, and because it is a straightforward second-order system that
can be explored both mathematically and in simulation with relative ease.
4.3 Current Tracking
For current regulation, the control objective is for the welding current x1(t) to track
a reference signal ri(t). Define the current tracking error
z1(t) = ri(t) x1(t) (4.3) − and its derivative
z2(t) =z ˙1(t) =r ˙i(t) x˙ 1(t). (4.4) − Proposed Control Strategy 64
Substituting the system from (4.2) leads to the current tracking error system,
z˙1 = z2 RoRw Rw Ro Ro z˙2 = z1 + + z2 (4.5) − LLw − Lw Lw L Rw Ro Ro RoRw RoVg +r ¨i + + + r˙i + ri u. Lw Lw L LLw − LLw When u = 0, the system will satisfy LLw L L Lw z1 = r¨i + + + r˙i + ri (4.6) RoRw Ro Rw Rw
z2 = 0 (4.7)
and when u = 1, the system will satisfy LLw L L Lw Vg z1 = r¨i + + + r˙i + ri (4.8) RoRw Ro Rw Rw − Rw
z2 = 0. (4.9)
Figure 4.2 and Figure 4.3 show example trajectories for the u = 0 and u = 1 case,
respectively, for various initial conditions and for a constant reference ri(t) = r0, r˙i =r ¨i = 0. As t , for the u = 0 case the error settles at r0; and, for the u = 1 → ∞ case, the error settles at ri Vg/Rw. −
4.3.1 Desired Dynamics
For current tracking, it is reasonable to define the “desired dynamics” of the system as
z˙1 = kiz1 (4.10) − which has solution
−kit z1(t) = z1(0)e , Proposed Control Strategy 65
z2
ri z1
Figure 4.2. Sample current error trajectories for u = 0 for a constant reference
implying that the tracking error decays exponentially to zero as t . If the desired → ∞ dynamics can be achieved, the system trajectory will be forced to lie on the line
σi(t) , z2(t) + kiz1(t) = 0, (4.11) as shown in Figure 4.4. The line
σi(t) = 0 Proposed Control Strategy 66
z2
ri − Vg/Rw z1
Figure 4.3. Sample current error trajectories for u = 1 for a constant reference is referred to as the “sliding line,” and the goal of SMC is to drive the trajectory from an initial condition to the sliding line, and maintain it on the sliding line until equilibrium has been achieved.
4.3.2 Reaching Condition
Note that the sliding line “splits” the phase plane into two regions: a region “above” the line (i.e. σi > 0), and a region “below” the line (i.e. σi < 0). Figure 4.5 shows the sliding line along with the phase trajectories for the u = 1 and u = 0 cases. If Proposed Control Strategy 67
z2
z1
ki
σ = 0
Figure 4.4. Desired current trajectory z2 = kiz1 for a constant reference −
the control signal u is defined so that 1 if σi < 0 u(t) , (4.12) 0 if σi > 0 then the trajectory will always be directed toward the sliding line. If σi = 0, the trajectory is exactly on the sliding line. Defining the control in this way satisfies the sliding mode reaching condition, ensuring that regardless of initial condition the
trajectory will always be directed toward the sliding line under the influence of the
control u. This can be seen graphically by examining the trajectories in Figure 4.5. Proposed Control Strategy 68
z2
u = 1 u = 0
σ > 0
ri − Vg/Rw ri z1
σ˙ u=0 = 0
σ < 0 σ˙ u=1 = 0
σ = 0
Figure 4.5. Current error trajectories with u = 0, u = 1, and sliding line σi = 0
4.3.3 Existence Condition
While the reaching condition guarantees that the state trajectory will be directed
toward the sliding line, it does not, in general, guarantee that a sliding mode exists.
That is, the reaching condition alone does not guarantee that the system will “slide”
along σi = 0, satisfying the desired dynamics. For this to be the case, it must be shown that once the trajectory reaches the sliding line, it will remain upon it. Proposed Control Strategy 69
Suppose the trajectory reaches a vicinity of the sliding line σi = 0. For a sliding mode to exist, within 0 < σi < it must be the case that | |
lim σiσ˙i < 0 (4.13) σi→0 or, equivalently,
lim σ˙i < 0 and lim σ˙i > 0. (4.14) + − σi→0 σi→0 Intuitively, this sliding mode existence condition requires that the trajectory and its
derivative, under the influence of u, have opposite signs. If this is the case, then a deviation of the trajectory from σi = 0, in any direction, will be immediately corrected by the control forcing it in the opposite direction. Substituting the definition of σ
into (4.13) and (4.14) gives the sliding mode existence condition for constant current
tracking: RoRw Rw Ro Ro RoRw RoVg z1 + + ki z2 + r0 > 0 − LLw − Lw Lw L − LLw − LLw RoRw Rw Ro Ro RoRw z1 + + ki z2 + r0 < 0. (4.15) − LLw − Lw Lw L − LLw
The conditions in (4.15) can be used as guidelines for choosing an appropriate value for ki. Three conditions are shown in Figure 4.6, Figure 4.7, and Figure 4.8 for different values of ki. It can be observed that while a larger value of ki shortens the time constant of the error tracking system, it also reduces the size of the sliding line within the system phase space. If Rw, Lw, Ro, and L are known, then a good choice
for the value of ki is
Rw Ro Ro ki = + + . Lw Lw L Proposed Control Strategy 70
z2
σ > 0
r V /R r i − g w i z1
σ˙ u=0 = 0
σ < 0
σ = 0 σ˙ u=1 = 0
Figure 4.6. Sliding mode existence for constant current, k < Rw + Ro + Ro i Lw Lw L
4.3.4 Switching Frequency Control
In the ideal case, the switch Q1 commutates at an infinite frequency and the system
trajectory, while sliding, can lie entirely on the sliding line σi = 0. A hardware real-
ization of the power source controlled by the SMC strategy described in the previous
section using infinite switching frequency would be impractical, because of the lim-
itations of actual semiconductor switching frequencies. A practical implementation would limit the switching frequency to a prescribed range. This can be done by intro-
ducing a hysteresis band κ around the sliding line [41] and modifying the control ± Proposed Control Strategy 71
z2
u = 1 = 0 u = 0 = 0
> 0
r i = 0 z1 ri Vg/Rw
< 0
Rw Ro Ro Figure 4.7. Sliding mode existence for constant current, ki + + ≈ Lw Lw L
signal to be 1 if σ < κ u = − . (4.16) 0 if σ > κ
Suppose it is desired to restrict the nominal switching frequency to Fsw. One switching period Tsw = 1/Fsw consists of two events: a switch “on” event of duration ton and a switch “off” event of duration toff:
1 Tsw = = ton + toff, (4.17) Fsw Proposed Control Strategy 72
z2 σ˙ u=1 = 0
σ > 0 σ˙ u=0 = 0
ri z1 r V /R i − g w
σ < 0
σ = 0
Figure 4.8. Sliding mode existence for constant current, k > Rw + Ro + Ro i Lw Lw L
as shown on the bottom of Figure 4.9. Assuming the system is in the sliding mode,
the switching function will change by an amount
2κ = (σ ˙ u=1) ton (4.18) − when u = 1, and by an amount
2κ = (σ ˙ u=0) toff (4.19) when u = 0. Combining (4.17), (4.18), and (4.19) and rearranging,
1 2κ 2κ = − + . Fsw σ˙ u=1 σ˙ u=0 Proposed Control Strategy 73
This equation can be solved for the value of κ that corresponds to a desired Fsw:
σ˙ u=1 σ˙ u=0 κ = × . (4.20) 2 (σ ˙ u=1 σ˙ u=0) Fsw − From Equation 4.20, some observations can be made:
The hysteresis κ is inversely proportional to Fsw. The standard deviation •
(i.e. “ripple”) in the output current will increase as Fsw decreases, i.e. as the
hysteresis bands move further apart. This is illustrated in Figure 4.14 which
shows the standard deviation in current during the sliding mode, for a 250 A
constant current regulation. The simulation parameters are
Vg = 80 V Rw = 98 mΩ L = 45 µH Lw = 10 µH Ro = 50 Ω ki = 6, 120, 911.
Sinceσ ˙ i depends upon the load resistance Rw and load inductance Lw, these • parameters will affect the actual switching frequency. The SMC strategy will
adjust the switching frequency to correct for deviations in load impedance.
Depending upon the range of these parameters, it may be necessary to use a
strategy such as adaptive hysteresis [41] to modify the value of κ based on
estimates of the load resistance and/or inductance. Figure 4.10 shows the
effect of Rw on the hysteresis required for 20 kHz switching with the other
simulation parameters above held constant, and Figure 4.11 shows the effect
of Lw.
Figure 4.12 and Figure 4.13 show the current regulation step response from 20 A to
200 A and back to 20 A, for switching frequencies of 20 kHz and 80 kHz, respectively.
The simulation parameters are
Vg = 80 V Rw = 98 mΩ L = 45 µH Lw = 10 µH Ro = 50 Ω ki = 6, 120, 911. Proposed Control Strategy 74
u = 1 σ > κ
σ = +κ
σ˙ u=0
σ = 0 2κ
σ˙ u=1
σ = κ − u = 0 σ < κ −
ton toff
u
Tsw
Figure 4.9. Relationship between switching function σ, control u, and hysteresis κ Proposed Control Strategy 75
1e7
5
4
κ 3
2
1
0 25 50 75 100 125 150 175 200
Rw (milliohms)
Figure 4.10. Relationship between load resistance Rw and hysteresis κ for Fsw = 20 kHz Proposed Control Strategy 76
1e7
4
3 κ
2
1
25 50 75 100 125 150 175 200
Lw (microhenries)
Figure 4.11. Relationship between load inductance Lw and hysteresis κ for Fsw = 20 kHz Proposed Control Strategy 77 = 20 kHz sw F Figure 4.12. Current step simulation, Proposed Control Strategy 78 = 80 kHz sw F Figure 4.13. Current step simulation, Proposed Control Strategy 79
9
8
7
6
5 amperes 4
3
2
1
0 20 40 60 80 100 kilohertz
Figure 4.14. Current standard deviation vs. switching frequency
4.3.5 Time-Varying Reference Tracking
When the reference ri(t) is time-varying, the desired dynamics are also time-varying.
As shown in Table 4.1, the error dynamics depend upon the derivatives of the refer-
ence. At any instant in time, the reaching condition described in Section 4.3.2 and
the existence condition in Section 4.3.3 must be satisfied for the appropriate error
dynamics. Proposed Control Strategy 80 u g w V o LL R − t/τ − e ) 0 r u g w − V o f r LL R ( − − f r 0 r w + w R o βt LL R + + 2 u g w αt V t/τ o 2 1 − LL e R 0 r w − w ) R τ − 0 o r LL f R r + o ) + βt L ( β R w w + + R o o w αt LL R ( R L + + o L R w w o L R L u R + g w o w V + o R + L o w LL R R L + t/τ − − w w + e 0 L R r f w w r w w L 2 R R − τ + o 0 LL R r β α + + + + 2 2 2 2 z z z z o o o o L L L L R R R R + + + + o o o o w w w w R R R R L L L L + + + + w w w w w w w w L L L L R R R R − − − − 1 1 1 1 z z z z w w w w w w w w R R R R o o o o LL LL LL LL R R R R 2 2 2 2 z z z z − − − − ======1 1 1 1 2 2 2 2 ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ Current Error Dynamics z z z z z z z z t/τ − Table 4.1. Current error dynamics for various reference signals 0 e ) r 0 r + t/τ t/τ − − − βt e e f 0 0 f r + r β r r ( 2 2 τ − + + − − τ αt f 0 0 f r 2 1 r αt β α r βt r ) = ) = 0 ) = 0 ) = ) = 0 ) = ) = ) = ) = ) = ) = ) = t t t t t t t t t t t t ( ( ( ( ( ( ( ( ( ( ( ( i i i i i i i i i i i i ˙ ˙ ˙ ˙ ¨ ¨ ¨ ¨ r r r r r r r Model r r r r r Reference Constant Ramp Exponential Parabola Proposed Control Strategy 81
Exponential Reference. Figure 4.15 shows the current tracking an exponential reference
−6 −(t−t0)/250×10 ri(t) = 250 225e − starting at t0 = 1 ms. The simulation parameters are
Vg = 80 V Rw = 98 mΩ L = 45 µH Lw = 10 µH Ro = 50 Ω ki = 6, 120, 911. Proposed Control Strategy 82 Figure 4.15. Current exponential simulation Proposed Control Strategy 83
Ramp Reference. Figure 4.16 shows the current tracking a ramp reference
ri(t) = 50000(t t0) + 25 − starting at t0 = 0.5 ms. The simulation parameters are
Vg = 80 V Rw = 98 mΩ L = 45 µH Lw = 10 µH Ro = 50 Ω ki = 6, 120, 911. Proposed Control Strategy 84 Figure 4.16. Current ramp simulation Proposed Control Strategy 85
Parabolic Reference. Figure 4.17 shows the current tracking a parabolic reference
6 450 10 2 ri(t) = × (t t0) + 25 2 − starting at t0 = 1 ms. The simulation parameters are
Vg = 80 V Rw = 98 mΩ L = 45 µH Lw = 10 µH Ro = 50 Ω ki = 6, 120, 911. Proposed Control Strategy 86 Figure 4.17. Current parabola simulation Proposed Control Strategy 87
4.4 Voltage Tracking
For voltage tracking, it is desired that the average output voltage tracks the reference
signal rv(t). Intuitively, “average” suggests introducing the integral of the voltage error into the desired dynamics. It is also the case that voltage is relative degree one for the system in (4.2), as the control appears in the expression forx ˙ 2, i.e. the first derivative of the voltage (unlike current, which is relative degree two).
Consider the voltage tracking error and its integral,
Z t ν1(t) = (rv(λ) x2(λ)) d λ 0 −
ν2(t) =ν ˙1(t) = rv(t) x2(t). (4.21) −
Substituting the system dynamics from (4.2) leads to
ν˙1 = v2 Ro Ro RoRw RoVg Ro Ro ν˙2 = + ν2 x1 u +r ˙v + + rv. (4.22) − Lw L − Lw − L Lw L
Notice that the expression for the voltage error dynamics includes the current x1
explicitly. It is not possible to “decouple” the current from the voltage dynamics,
due to the presence of the inductor in the load impedance which makes the voltage
dependent upon the derivative of the current.
The switching function for voltage is defined as
σv(t) = ν2(t) + kvν1(t). (4.23) Proposed Control Strategy 88
This has the dynamics
σ˙ v =ν ˙2 + kvν2 Ro Ro RoRw RoVg Ro Ro = + k ν2 x1 u +r ˙v + + rv. (4.24) − Lw L − − Lw − L Lw L
The general concepts of the sliding mode reaching and existence conditions for
current apply also to voltage, given the voltage error dynamics in (4.24). Table 4.2
summarizes the relevant equations for various reference signals. Proposed Control Strategy 89 u g V o L R − u g V o L t/τ − R e ) − 0 r 0 − r f r + u ( g V βt − o L f + R r 2 − αt ) o 1 2 0 L r u R g + V + o L o o w βt R L R ( R L − + 0 o o r L w + R R L o + t/τ L − R o w e + R 0 L + r β o w τ − + R L + f r β αt + + + + 1 1 1 1 x x x x w w w w w w w w R R R R o o o o L L L L R R R R − − − − 2 2 2 2 ν ν ν ν o o o o L L L L R R R R + + + + o o o o w w w w R R R R L L L L 2 2 2 2 ν − ν − ν − ν − ======1 2 1 2 1 2 1 2 ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ Voltage Error Dynamics ν ν ν ν ν ν ν ν t/τ − 0 e ) r 0 r + t/τ t/τ − − − βt e e f Table 4.2. Voltage error dynamics for various reference signals 0 0 f r + r β r r ( 2 2 τ − + + − − τ αt 0 f 0 f r r 1 2 r βt β r αt α ) = 0 ) = 0 ) = 0 ) = ) = ) = ) = ) = ) = ) = ) = ) = t t t t t t t t t t t t ( ( ( ( ( ( ( ( ( ( ( ( v v v v v v v v v v v v ˙ ˙ ˙ ˙ ¨ ¨ ¨ ¨ r r r r r r r Model r r r r r Reference Constant Ramp Exponential Parabola Proposed Control Strategy 90
4.4.1 Voltage Tracking Example
Figure 4.18 shows the voltage tracking an exponential reference
−6 −(t−t0)/250×10 ri(t) = 25 20e − starting at t0 = 1 ms. The simulation parameters are
Vg = 80 V Rw = 98 mΩ L = 45 µH Lw = 10 µH Ro = 50 Ω kv = 6, 120, 911. Proposed Control Strategy 91 Figure 4.18. Voltage exponential simulation Proposed Control Strategy 92
L Q1
+
ideal
+ Vg D Ro vw(t) − ideal
−
Figure 4.19. Circuit diagram for the Open Circuit Voltage (OCV) case
4.4.2 Open Circuit Voltage
In the special case of OCV, no current flows through the welding load and the circuit is simplified to the one shown in Figure 4.19. The output voltage dynamics is described by the first-order differential equation
d vw Ro RoVg = vw + u. (4.25) d t − L L
Define the OCV tracking error and its integral
Z t ν1 = (rocv(λ) vw(λ)) d λ (4.26) 0 −
ν2 =z ˙1 = rocv(t) vw(t). (4.27) − then we have
ν˙1 = ν2 (4.28)
Ro RoVg Ro ν˙2 = ν2 u +r ˙ocv + rocv. (4.29) − L − L L Proposed Control Strategy 93
4.5 Power Tracking
For power tracking, it is desired to force the average output power
pw(t) = x1(t)x2(t)
to track a reference signal rp(t). Define the power tracking error and its integral,
Z t q1(t) = (rp(λ) x1(λ)x2(λ)) d λ 0 −
q2(t) =q ˙1(t) = rp(t) x1(t)x ˙ 2(t) x˙ 1(t)x2(t). (4.30) − −
The switching function for power regulation is defined as
σp = q2 + kpq1. (4.31)
4.6 Regulation Mode Switching
The “mode” of regulation describes the quantity being regulated (current, voltage, or
power) as well as the reference signal being commanded (constant, ramp, exponential
or parabolic).
When using SMC, switching regulation modes requires no special action. The
state of the system upon entering a new regulation mode is simply the initial condition
for the SMC control action for the new mode.
4.7 Output Limiting
Limits may be imposed on the power source output in the form of minimum and
maximum current, minimum and maximum voltage, and minimum and maximum
power values. The limits are dynamic and can change while welding. The limits Proposed Control Strategy 94
may be enforced for the safety of the welder, to protect the electronics from being
damaged, or to keep the output within an operating range defined by the welding
process. The limits create boundaries around the acceptable operating region, as
shown in Figure 4.20.
imin(t) is a minimum allowable current output;
imax(t) is a maximum allowable current output;
vmin(t) is a minimum allowable voltage output;
vmax(t) is a maximum allowable voltage output;
pmin(t) is a minimum allowable power output;
pmax(t) is a maximum allowable power output.
Minimum limits are typically commanded to prevent arc outages. For example, as the arc length grows the voltage normally increases while the current decreases.
Below a particular current level, the arc becomes unstable and can be extinguished
(referred to as a “pop out”). Setting a minimum current causes the power source to attempt to prevent the arc from extinguishing by maintaining the minimum current required to sustain the arc regardless of the value of the voltage.
Maximum voltage limits are often for safety. For example, there is an International
Electrotechnical Commission (IEC) requirement that the peak output of a welding power source does not exceed 113 VDC to minimize the chance of electrocution (IEC
60974-1). Maximum voltage may also be used to prevent the arc length from growing too large. Proposed Control Strategy 95
80
vmax 70
p 60 max
50
40 imin Operating Region Volts
30 imax
20
vmin 10
0 0 100 200 300 400 500 600 700 Amperes
Figure 4.20. Example operating region defined by current, voltage, and power limits
Output limits can be naturally incorporated into the SMC strategy by considering additional switching functions for the limits,
d σi,min = (imin x1) + ki(imin x1) d t − − d σi,max = (imax x1) + ki(imax x1) d t − − Z t σv,min = vmin x2 + kv (vmin x2) d λ − 0 − Z t σv,max = vmax x2 + kv (vmax x2) d λ − 0 − d σp,min = (pmin x1x2) + kp(pmin x1x2) d t − − d σp,max = (pmax x1x2) + kp(pmax x1x2). d t − − (4.32) Proposed Control Strategy 96
Figure 4.21. Example of output limiting, 20 V regulation at a 100 A minimum current
Then, the sliding mode control signal can be changed to 0 if σi < 0 and σi,min < 0 and σv,min < 0 and σp,min < 0 u = . (4.33) 1 if σi > 0 and σi,max > 0 and σv,max > 0 and σp,max > 0
An example of the application of output limits is shown in Figure 4.21. This sim-
ulates the waveform generator requesting a constant voltage of 25 V, with a minimum
current of 100 A at the beginning of the figure, and a minimum current of 50 A after
1 ms. Proposed Control Strategy 97
4.8 Advantages of SMC for Welding Power Sources
Traditional linear compensators allow the dynamics to be specified only at one oper-
ating point, whereas the SMC technique allows the dynamics to be specified for all
operating points during the sliding phase.
The SMC strategy can achieve order reduction. For example, for a constant
current reference signal, during the sliding phase the closed-loop dynamics is of first-
order
z˙1(t) = kiz1(t) − while the system is of second-order.
4.9 Drawbacks of Sliding Mode Control
Although SMC has advantages as a candidate control system for a welding power
source, there are some disadvantages.
While SMC has been well-accepted in academia for decades, it is not as well • known in industry as traditional methods and may be met with skepticism.
SMC can be difficult to implement using analog hardware. However, the • calculations required are typically simple enough to be easily implemented in
a microcontroller or Field Programmable Gate Array (FPGA), provided the
sample rate is high enough. Chapter 5 describes a hardware implementation
using a combination of a microcontroller, a Complex Programmable Logic
Device (CPLD), and some analog circuitry. Proposed Control Strategy 98
SMC works best when the controller can open and close the switching
device “at will.” In reality there will always be a practical limit to the possi-
ble switching frequency due to switching losses, magnetics, electromagnetic
interference issues, and other concerns. Some power electronics topologies
require synchronized switching between several switching devices, which is
most easily accomplished using a fixed PWM frequency. There are com-
promises, such as Quasi-Sliding Mode Control(QSMC), that allow a sliding
mode controller to operate at a fixed frequency, but these approaches lead
to degraded performance. 99
5 Hardware Implementation
This chapter describes the implementation in hardware of the proposed control
strategy. The system was tested by retrofitting an existing welding power source from
Lincoln Electric.
First, a description is given in Section 5.1 of initial attempts to implement SMC
that were abandoned due to being impractical. Then, the details of a workable hard- ware implementation using a hybrid analog/digital approach are given in Section 5.2.
Oscilloscope traces of examples using this implementation are shown in Section 5.3, with more complete results and discussion in Chapter 6.
5.1 Abortive Efforts
This section describes efforts to implement SMC that were tried and abandoned
before arriving at the hybrid analog/digital solution described in the remainder of
the chapter.
5.1.1 Software Solution
The first attempt to implement SMC was to keep the control board hardware un-
changed, and modify only the firmware. Because this approach would allow different Hardware Implementation 100
control strategies to be tried by simply reprogramming the firmware, it would eas-
ily be possible to realize the advantages of the new strategy in existing machines.
Furthermore, trying variations of the strategy would require only modifying code,
making it easier and faster to iterate different things.
The analog feedback circuitry uses an integrating capacitor that averages the
current and voltage signals over a switching period. This has two disadvantages: (1)
introducing an integrator into the feedback loop lowers the effective bandwidth; and
(2) the feedback processing relies on a fixed switching frequency, making Hysteresis
Modulation (HM)-based SMC impossible.
The microprocessors on the existing control board cannot feasibly implement a
high-bandwidth digital control loop.
5.1.2 Analog Hardware Solution
The second attempt to implement SMC was to design a custom Printed Circuit Board
(PCB) that would implement the SMC algorithm purely in analog hardware. This
PCB is shown in Figure 5.1. The advantage over the software solution would be
higher bandwidth than the digital implementation.
The pure analog solution was abandoned primarily for three reasons: (1) it was
impractical to “tune” the control algorithm, as the analog circuits used manual po-
tentiometers to select sliding parameters and adjust feedback filtering; (2) interfacing
the analog circuitry with the existing control board was challenging because of the
limited I/O available on the control board; and (3) high-bandwidth analog circuits
require careful attention to PCB layout and placement in order to avoid issues with
electrical noise. Hardware Implementation 101
Figure 5.1. Original analog SMC control board
5.2 Hybrid Analog/Digital Solution
The hybrid analog/digital implementation of SMC combines the abortive approaches
using the custom PCB shown in Figure 5.2. This PCB uses a TMS320F28379 quad-
core DSP from Texas Instruments. The DSP is capable of executing high-speed con-
trol loops in software, and has Analog-to-Digital Converter (ADC), Digital-to-Analog
Converter (DAC), and analog comparator peripherals on-chip. This approach allows
the SMC algorithm to be implemented in software for a high degree of flexibility, while
still having the bandwidth advantage of analog signal comparison for switching. The Hardware Implementation 102
analog comparator outputs are routed to a Xilinx XCR3064XL CPLD, used to imple-
ment the logic for theHM gate drive control. This new PCB is placed “between” the
existing control board and the existing power electronics, as shown in Figure 5.3 and
Figure 5.4. The firmware for the digital control board was modified slightly to send
a reference and regulation mode command to the new PCB using Serial Peripheral
Interface (SPI).
A block diagram of the SMC implementation in the PCB is shown in Figure 5.5.
Analog current, voltage, and power signals are read by the Digital Signal Processor
(DSP), which computes comparison values corresponding to the sliding mode switch-
ing functions and hysteresis levels, and exports them to analog comparators. These
calculations are detailed in Section 5.2.1. The feedback is compared against the ex-
ported values and the comparator output signals indicate whether the feedback is
above the “upper” hysteresis level or below the “lower” level. The comparator inputs
are routed to the CPLD, which implements the gate drive logic for the chopper section
including minimum on-time, maximum on-time, and minimum off-time requirements
as described in Section 5.2.2. Hardware Implementation 103
Figure 5.2. Hybrid analog/digital SMC PCB Hardware Implementation 104
PC Ethernet
“Chopper” Digital gate drive Power Section voltage Control Board
current
Figure 5.3. Basic structure of the existing control hardware PC Ethernet
“Chopper” Digital gate drive Power Section voltage Control Board
current
SPI
SMC Board
Figure 5.4. Basic structure of the modified control hardware Hardware Implementation 105 Gate Drive IGBT Gate Logic Drive 0 1 2 0 1 2 XCR3064XL CPLD ------+ + + + + + DAC DAC DAC DAC DAC DAC i,l ξ i,h ξ v,l ξ v,h ξ 28379 DSP p,l ξ Function Switching Calculation p,h ξ mode ADC ADC ADC Figure 5.5. Block diagram of the SMC hardware implementation w p Π Analog Multiplier w w i v Hardware Implementation 106
5.2.1 SMC Algorithm Implementation
A flowchart of the SMC algorithm implementation is shown in Figure 5.6. The ADC
sampling and control loop code runs at a rate of 1 MHz.
The analog comparators in the DSP compare an internally-generated signal to
the feedback, i.e. directly to the voltage, current, or power analog signal. In order to
implement the SMC algorithm, the switching function must be rewritten slightly to
facilitate this comparison.
Comparison Functions. Consider the switching function for current tracking,
σi(t) = z2 + kiz1.
Now, σi is compared against the upper and lower hysteresis levels:
d z1 σi < κ = + ki(ri x1) < κ − ⇒ d t − − d z1 σk > κ = + ki(ri x1) > κ. ⇒ d t −
These expressions can be rearranged as follows to change them into comparisons
against the feedback:
1 d z1 κ ξi,l , + ri + < x1 ki d t ki
1 d z1 κ ξi,h , + ri > x1. (5.1) ki d t − ki
In a similar fashion, the voltage and power switching functions can be written
Z t ξv,l , rv + kv (rv x2) d λ + κv < x2 0 − Z t ξv,h , rv + kv (rv x2) d λ κv > x2 (5.2) 0 − − Hardware Implementation 107
START
Read Current Feedback
Read Voltage Feedback
Read Power Feedback
current Regulation power Mode
voltage
Compute Error Compute Error Compute Error
Calculate σi Calculate σv Calculate σp
Compute Compute Compute ξl and ξh ξl and ξh ξl and ξh
Export to Export to Export to Comparators Comparators Comparators
STOP
Figure 5.6. SMC algorithm flowchart Hardware Implementation 108
and
1 d κ ξp,l , (ri x1x2) + rp + < x1x2 kp d t − kp 1 d κ ξp,h , (ri x1x2) + rp > x1x2, (5.3) kp d t − − kp
respectively. Hereinafter, the ξ functions are referred to as “comparison functions.”
Band-Limited Derivative. A true derivative would have frequency response
Hd(s) = s.
However, a true derivative is sensitive to noise, including quantization effects in the
representation of numbers in the microcontroller [96]. Therefore, in the implementa-
tion a band-limited derivative
s H (s) = bld 1 + τs
is used, where τ can be used to tune the frequency response of the derivative in order
to reduce the effects of noise. This is effectively a derivative followed by a first-order
low-pass filter with time constant τ. Figure 5.7 shows the magnitude and phase
responses of the band-limited derivative versus the true derivative, for a sampling
rate of T = 1 µs and various values of τ.
Discrete-Time Implementation. For the microcontroller implementation, the com- parison functions in (5.1), (5.2), and (5.3) are converted to discrete-time equivalent difference equations using the bilinear transform [96]
2 z 1 s − (5.4) → T z + 1 where T is the control loop sampling frequency, 1 MHz in this implementation. Hardware Implementation 109
Figure 5.7. Band-limited derivative frequency responses for various val- ues of τ
5.2.2 Gate Drive Signal
A FSM diagram for the gate drive logic is given in Figure 5.8. This logic creates the
signal for the gate drive given the GO HIGH and GO LOW signals for the appropriate
regulation mode. It also provides provisions for a minimum on-time, maximum on-
time, and minimum off-time:
The minimum on-time, ton,min, is the minimum amount of time the gate drive • signal can be turned on. This value exists to ensure that the IGBTs have
time to move to their switching region. Hardware Implementation 110
C: ENABLED and σ > κ A: Turn on switch; set ton = 0 INIT DISABLED ON, ton Ton,min ≤
C: DISABLED C: DISABLED A: Turn off switch C: ton > Ton,min A: Turn off switch
C: σ < κ − A: Turn off switch; set toff = 0 OFF ON, ton > Ton,min
C: σ > κ and toff > toff,min A: Turn on switch; set ton = 0
Figure 5.8. State diagram for CPLD gate drive logic
The maximum on-time, ton,max, is the maximum amount of time the gate • drive signal can be turned on. This value exists to ensure that the IGBTs
are not stressed too much while turned on.
The minimum off-time, toff,min, is the minimum amount of time the gate drive • signal can be turned off. This value exists to ensure that the IGBTs have
time to turn off completely before begin commanded on again.
Of course, in the theoretical ideal, the switches would be able to commutate at an ar-
bitrary frequency. The on-time and off-time limitations exists because of the practical
requirements of the physical IGBT switches.
An example of the gate drive logic is shown in the oscilloscope trace in Figure 5.9, with a zoomed-in view in Figure 5.10. Hardware Implementation 111
Figure 5.9. Oscilloscope trace showing hysteresis modulation signals
5.3 Examples
As examples of the implementation running in hardware, Figure 5.11 and Figure 5.12 show oscilloscope traces of a 200 A current step response with nominal switching frequencies of 20 kHz and 10 kHz, respectively. Channel 1 shows the output voltage,
Channel 2 shows the output current, and Channel 3 shows the IGBT gate drive command. A current ramp is shown in Figure 5.13, and an exponential current command is shown in Figure 5.14. Hardware Implementation 112
Figure 5.10. Oscilloscope trace showing detail of hysteresis modulation signals
Figure 5.11. Oscilloscope trace of 200 A current regulation at 20 kHz Hardware Implementation 113
Figure 5.12. Oscilloscope trace of 200 A current regulation at 10 kHz
Figure 5.13. Oscilloscope trace of current ramp regulation Hardware Implementation 114
Figure 5.14. Oscilloscope trace of current exponential regulation 115
6 Results
This chapter summarizes some experiments done to compare the existing control
system with the proposed new control strategy. For most tests, results are shown
from the existing control system and simulation. For some, results are also shown
from the hardware implementation of the SMC strategy.
6.1 Experimental Setup
For the non-welding tests, the power source is connected to an experimental load
bank as shown in Figure 6.1. This load bank allows the resistance of the load to be
changed via switches that switch in or out banks of resistors.
Most experiments have been done with no additional inductance, i.e. with 4 m welding cables connected to the power source output; with an 84 µH coil of cable as
shown in Figure 6.2 to simulate medium inductance; and with a 255 µH coil of cable
as shown in Figure 6.3 to simulate high inductance. The inductance of the cables was
measured using an LCR meter as shown in Figure 6.4. Results 116
Figure 6.1. Experimental load bank
Figure 6.2. 84 µH coil of welding cable
6.2 Current Step Command
A current step command from 10 A to 100 A is shown for the existing control system in Figure 6.5, for different values of load resistance. Compared with the simulation Results 117
Figure 6.3. 255 µH coil of welding cable
Figure 6.4. LCR meter used for measuring cable inductance results in Figure 6.6 and the results from the hardware implementation in Figure 6.7, it can be seen that the SMC strategy produces less overshoot and, generally, a faster rise time on the rising edge. There is also less undershoot on the falling edge. The fall time is dictated by the system’s open loop response due to the inductance in Results 118 the welding circuit. Similar step commands are shown in Figure 6.8 and Figure 6.9 for 20 A to 200 A steps in the existing system and in simulation. It can be observed that In the existing system, as the load resistance increases, the amount of overshoot decreases. However, with the SMC strategy, there is no noticeable overshoot at any load resistance.
The effects of inductance are shown in Figure 6.12, Figure 6.13, Figure 6.14, and
Figure 6.15. As these results indicate, the SMC strategy does a good job of reducing the overshoot and oscillation exhibited by the existing controls when the inductance increases. Results 119
Figure 6.5. Current step, 10 A to 100 A, with varying load resistance (existing controls)
Figure 6.6. Current step, 10 A to 100 A, with varying load resistance (SMC simulation)
Figure 6.7. Current step, 10 A to 100 A, with varying load resistance (hardware implementation) Results 120
Figure 6.8. Current step, 20 A to 200 A, with varying load resistance (existing controls)
Figure 6.9. Current step, 20 A to 200 A, with varying load resistance (SMC simulation) Results 121
Figure 6.10. Current step, 30 A to 300 A, with varying load resistance (existing controls)
Figure 6.11. Current step, 30 A to 300 A, with varying load resistance (SMC simulation) Results 122
Figure 6.12. Current step, 10 A to 100 A, with varying load inductance (existing controls)
Figure 6.13. Current step, 10 A to 100 A, with varying load inductance (SMC simulation) Results 123
Figure 6.14. Current step, 20 A to 200 A, with varying load inductance (existing controls)
Figure 6.15. Current step, 20 A to 200 A, with varying load inductance (SMC simulation) Results 124
6.3 Current Exponential Command
Figure 6.16 and Figure 6.17 show existing control and SMC simulation responses for a 20 A to 200 A exponential current command, with varying load resistance. The
SMC strategy can be seen to perform well in comparison to the existing controls, in simulation. The “dip” in current that occurs in the 30 mΩ and 98 mΩ cases in the existing controls, does not occur in the SMC simulation. Results 125
Figure 6.16. Current exponential, 20 A to 200 A, with varying load re- sistance (existing controls)
Figure 6.17. Current exponential, 20 A to 200 A, with varying load re- sistance (SMC simulation) Results 126
Figure 6.18. Current exponential, 20 A to 200 A, with varying load in- ductance (existing controls)
Figure 6.19. Current exponential, 20 A to 200 A, with varying load in- ductance (SMC simulation) 127
7 Summary, Conclusions, and Future Work
This chapter summarizes the work described in this thesis, and presents ideas for
further work.
7.1 Summary and Conclusions
In this thesis, models of the equipment in a welding setup were developed that can be
used for analysis and control system design. These models were described in Chap-
ter 2. The models were used to develop a simulation environment and a new control
strategy for a welding power source from Lincoln Electric, using Sliding Mode Control
(SMC). The new strategy, described in Chapter 4, was implemented in hardware as
described in Chapter 5. The performance of the existing system, SMC simulations,
and hardware implementation were compared in Chapter 6.
The original goals of this work were described in Section 1.3. Models of the weld-
ing equipment were successfully developed and described in Chapter 2. A usable
simulation environment was created, described in detail in Chapter A. The simula-
tor was used throughout this work, but could be improved further as discussed in
Section 7.2.1. Chapter 4 described the new control strategy, based on SMC. The Summary, Conclusions, and Future Work 128
hardware implementation in Chapter 5 suggests that implementing and using SMC
is feasible in an actual welding power source. Some results were given in Chapter 6,
although to achieve the original goal it would have been necessary to do more testing
in actual welding scenarios. This is now an item for future work.
7.2 Recommendations for Further Work
The work described herein is only the beginning of what can be done with SMC
applied to welding power sources. This section describes some further work and
improvements that can be done.
7.2.1 Simulation Improvements
The computer simulation could be made more user-friendly, so that it is usable by
other engineers to easily run simulations and test different designs and control sys-
tems. It would be helpful to be able to use the actual logic from the welding programs
in the simulator. It would also be useful to incorporate different power section topolo-
gies.
7.2.2 Alternative Switch Modulation Techniques
The IGBTs used as switching elements in MAWPSs are not perfect switches: there
is a nonzero transition time between their “off” and “on” states. Thus it is especially
important to understand the effective switching frequencies of the different control
strategies being proposed. Aside from the hysteresis modulation used in this work,
Pulse Width Modulation(PWM) and parabolic modulation are other switch modula-
tion techniques. Both techniques involve comparing the output signal to be regulated with one or more comparison levels set by the controller. Summary, Conclusions, and Future Work 129
PWM is a traditional method to produce a fixed-frequency switching. In this
switching strategy, the switch can only turn on at defined instants and turns off
once the output reaches the desired level or a maximum on-time has been reached, whichever comes first. This is the strategy presently employed in the Lincoln Electric
power source, which has a fixed PWM switching frequency of 20 kHz.
It has been shown that SMC can be applied to PWM switching control by con-
sidering the equivalent control ueq to be the duty cycle of the PWM.
Another potential switch modulation technique is parabolic modulation. Para-
bolic modulation is a concept proposed by Qi et al.at the 2017 Applied Power Elec-
tronics Conference [97] as a way of realizing some of the benefits of sliding mode
control while still restricting the switching frequency to a desired range.
7.2.3 Sliding Mode Parameter Selection
In this work, not much time was spent on the selection of the sliding mode parameters
ki, kv, and kp. It would be very helpful to have a way of visualizing the effect of these
parameters on the control, and to select optimal values given particular situations.
This could be included as part of the simulation environment. Appendix 130
Appendix A Simulation
In order to compare and evaluate control strategies and parameters, a simula-
tion program was designed using the C++ programming language. This appendix
describes the implementation of the simulator.
1 General Structure
A flowchart showing the general structure of the simulation “main loop” is shown in
Figure A.1. The simulation is configured using a duration tdur and a fixed step size
∆t, typically set to 1 ns. The main loop of the simulation runs a total of N times, where t N = dur . ∆t For each iteration, discrete-time equivalents of the analog feedback filters in the weld-
ing power source are executed; see the code in Listing A.1. The parameters of these
filters can be modified by changing the filter coefficients to approximate the frequency
response of the actual analog filters. The values of the hysteresis κ and the switching function σ for the active regulation mode are then computed, using the filtered and sampled values and discrete-time equivalents of the derivative and integral operations, as shown in Listing A.2 and Listing A.3. Depending upon the relationship between σ and κ, a new control u may be computed, or u may retain its previous value. At the end of the cycle, the system equations are integrated using the Classic Runge-Kutta Appendix 131
Method (RK4) algorithm, using the chosen value of u. The values of the signals dur- ing the simulation step are saved to a file for later review, and the process is repeated until all N iterations have been completed. Appendix 132
START
u = 0
n = 0
Update progress display
Update feedback filters
Compute κ
Compute σ
NO σ > κ YES
u = 1 NO σ < κ YES − Integrate system equations with u = 1 Integrate system u = 0 equations with previous u
Integrate system equations with u = 0
Save signal values
n := n + 1
NO n < N YES
STOP
Figure A.1. Simulator “main loop” flowchart Appendix 133
Listing A.1. C++ implementation of a discrete-time, 4th-order filter 1 class AnalogFilter { 2 public: 3 AnalogFilter() { 4 clear(); 5 } 6 7 void clear(void) { 8 for (int i=0; i < 4; i++) { 9 m_xnm[i] = 0; 10 m_ynm[i] = 0; 11 } 12 } 13 14 FLOAT_TYPE update(const FLOAT_TYPE& xn) { 15 FLOAT_TYPE yn; 16 yn = -m_a[1]*m_ynm[0]; 17 yn -= m_a[2]*m_ynm[1]; 18 yn -= m_a[3]*m_ynm[2]; 19 yn -= m_a[4]*m_ynm[3]; 20 yn += m_b[0]*xn; 21 yn += m_b[1]*m_xnm[0]; 22 yn += m_b[2]*m_xnm[1]; 23 yn += m_b[3]*m_xnm[2]; 24 yn += m_b[4]*m_xnm[3]; 25 m_xnm[3] = m_xnm[2]; 26 m_xnm[2] = m_xnm[1]; 27 m_xnm[1] = m_xnm[0]; 28 m_xnm[0] = xn; 29 m_ynm[3] = m_ynm[2]; 30 m_ynm[2] = m_ynm[1]; 31 m_ynm[1] = m_ynm[0]; 32 m_ynm[0] = yn; 33 return yn; 34 } 35 36 private: 37 // Coefficients for a sampling frequency of 1 MHz 38 static constexpr FLOAT_TYPE m_b[] = { 39 2.1434708483209336e-15, 8.573883393283734e-15, 40 1.28608250899256e-14, 8.573883393283734e-15, 41 2.1434708483209336e-15 42 }; 43 44 static constexpr FLOAT_TYPE m_a[] = { 45 1.0, -3.9988753153245495, 5.99662657838831, 46 -3.996627210594679, 0.9988759475309539 47 }; 48 49 FLOAT_TYPE m_xnm[4]; 50 FLOAT_TYPE m_ynm[4]; 51 }; Appendix 134
Listing A.2. C++ implementation of a discrete-time band-limited derivative 1 class Derivative { 2 public: 3 Derivative(const FLOAT_TYPE& Ts, const FLOAT_TYPE& tau) 4 : m_K(2.0/(Ts+2.0*tau)), 5 m_a1((Ts-2.0*tau)/(Ts+2.0*tau)), 6 m_xnm1(0), 7 m_ynm1(0) 8 { 9 } 10 11 void clear(void) { 12 m_xnm1 = 0; 13 m_ynm1 = 0; 14 } 15 16 FLOAT_TYPE update(const FLOAT_TYPE& xn) { 17 FLOAT_TYPE yn; 18 yn = m_K*(xn - m_xnm1) - m_a1*m_ynm1; 19 m_xnm1 = xn; 20 m_ynm1 = yn; 21 return yn; 22 } 23 24 private: 25 FLOAT_TYPE m_K; 26 FLOAT_TYPE m_a1; 27 FLOAT_TYPE m_xnm1; 28 FLOAT_TYPE m_ynm1; 29 }; Appendix 135
Listing A.3. C++ implementation of a discrete-time integrator 1 class Integrator { 2 public: 3 Integrator(const FLOAT_TYPE& Ts) 4 : m_a(Ts/2) 5 { 6 clear(); 7 } 8 9 void clear(void) { 10 m_xnm1 = 0; 11 m_ynm1 = 0; 12 } 13 14 FLOAT_TYPE update(const FLOAT_TYPE& xn) { 15 FLOAT_TYPE yn; 16 yn = m_a*(xn + m_xnm1) + m_ynm1; 17 m_xnm1 = xn; 18 m_ynm1 = yn; 19 return yn; 20 } 21 22 private: 23 FLOAT_TYPE m_a; 24 FLOAT_TYPE m_xnm1; 25 FLOAT_TYPE m_ynm1; 26 };
2 Integration Method
During each simulation iteration, once the control u has been chosen the system
equations
Rw 1 x˙ 1 = x1 + x2 − Lw Lw RoRw Ro Ro Ro x˙ 2 = x1 + x2 + Vgu Lw − Lw L L
are integrated to determine the system state value for the next iteration. The inte-
gration method used is the classic Runge-Kutta method [98], i.e. RK4. This method Appendix 136
approximates a solution to the system
˙x = f(x, t) x(0)
in an iterative fashion, using a step size of ∆t > 0.
(k + 2k + 2k + k )∆t x = x + 1 2 3 4 n+1 n 6
tn+1 = tn + ∆t where
k1 = f (xn, tn)
∆t ∆t k = f x + k , t + 2 n 2 1 n 2 ∆t ∆t k = f x + k)2, t + 3 n 2 n 2
k4 = f (xn + k3∆t, tn + ∆t) .
C++ source code for the RK4 routine is shown in Listing A.4. The rk4 function is
passed a function that computes the derivatives of the system state. There are two
such functions (for each possible value of u), as shown in Listing A.5 and Listing A.6,
respectively. Appendix 137
Listing A.4. C++ implementation of RK4 algorithm 1 State rk4(State& x, 2 FLOAT_TYPE& t, 3 const FLOAT_TYPE& h, 4 State (*f)(const State& x, const FLOAT_TYPE& t)) 5 { 6 State k1 = f(x, t); 7 State k2 = f(x + (h/2)*k1, t+h/2); 8 State k3 = f(x + (h/2)*k2, t+h/2); 9 State k4 = f(x + h*k3, t + h); 10 return x + (h/6)*(k1 + 2*k2 + 2*k3 + k4); 11 }
Listing A.5. System state derivative computation for u = 0 1 State dxdt_ueq0(const State& x, const FLOAT_TYPE& t) 2 { 3 const FLOAT_TYPE x1 = x.get_x1(); 4 const FLOAT_TYPE x2 = x.get_x2(); 5 6 const FLOAT_TYPE x1dot = -(Rw(t)/Lw(t))*x1 + (1/Lw(t))*x2; 7 const FLOAT_TYPE x2dot = (Ro(t)*Rw(t)/Lw(t))*x1 - Ro(t)*(1/Lw(t) + 1/L(t))*x2; 8 9 return State(x1dot, x2dot); 10 }
Listing A.6. System state derivative computation for u = 1 1 State dxdt_ueq1(const State& x, const FLOAT_TYPE& t) 2 { 3 const FLOAT_TYPE x1 = x.get_x1(); 4 const FLOAT_TYPE x2 = x.get_x2(); 5 6 const FLOAT_TYPE x1dot = -(Rw(t)/Lw(t))*x1 + (1/Lw(t))*x2; 7 8 const FLOAT_TYPE x2dot = 9 (Ro(t)*Rw(t)/Lw(t))*x1 - Ro(t)*(1/Lw(t) + 1/L(t))*x2 10 + (Ro(t)/L(t))*Vg(t); 11 12 return State(x1dot, x2dot); 13 } Appendix 138
3 Simulation Configuration
The simulation is flexible in allowing all system parameters to vary at any point in
time. This allows parameter changes and disturbances to be simulated easily. The
parameters are controlled by modifying a C++ header file that defines time-dependent
functions for each parameter, as shown in Listing A.7. Appendix 139
Listing A.7. Simulation parameter configuration 1 // Simulation time step, in seconds. 2 static constexpr FLOAT_TYPE k_time_step = 1.0e-9; 3 4 // Duration of the simulation, in seconds. 5 static constexpr FLOAT_TYPE k_duration = 2e-3; 6 7 // Number of "ticks" in the simulation. 8 static constexpr std::uint32_t N 9 = static_cast
Figure A.2. Simulated load changes from R1 = 100 mΩ to R2 = 5 mΩ
4 Load Change Simulation
An important event to understand is the transition from one load resistance R1 to
another R2. A sigmoid function is used in order to produce a “smooth” transition for
a load change occurring at time t0:
R1 R2 R(t) = R1 − , − 1 + e−α(t−t0) where α can be adjusted to affect the speed at which the load change occurs. Some
examples are shown in Figure A.2. Appendix 141
Table A.1. Simulation parameters
Parameter Symbol Value Time step ∆t 1 ns Control frequency Fctrl 1 MHz Derivative time constant τ 100 µs
5 Simulation Parameters
Unless otherwise noted, the simulation parameters for all results in this thesis are as given in Table A.1. Each of these parameters can be changed by modifying the source code and recompiling the simulator. Complete References 142
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