modeling and control of switched reluctance …
TRANSCRIPT
Electronics
Mechanics
Computers
The Ohio State University
Mechatronic Systems Program
MODELING AND CONTROL OF SWITCHED
RELUCTANCE MACHINES FOR ELECTRO-
MECHANICAL BRAKE SYSTEMS
By
Wenzhe Lu, PhD
Ali Keyhani, Professor
Tel: 1-614-292-4430
Mechatronics Laboratory
Department of Electrical and Computer Engineering
The Ohio State University
Columbus, Ohio 43210
August 2005
Copyright by
Mechatronics Laboratory
Department of Electrical and Computer Engineering
The Ohio State University
August 2005
ii
ABSTRACT
Modern vehicles have become more and more “electrical”. Electro-mechanical brake
(EMB) systems have been proposed to replace the conventional hydraulic brake systems.
Due to the advantages such as fault tolerant operation, robust performance, high
efficiency, and reliable position sensorless control, switched reluctance machine (SRM)
has been chosen as the servomotor of the EMB systems. This research is focused on the
modeling and control of switched reluctance machines for EMB systems. The overall
goal is to design a robust clamping force controller without position sensors for the SRM.
An accurate model and precisely estimated parameters are critical to the successful
implementation of the control system. An inductance based model for switched
reluctance machine is proposed for this research. Maximum likelihood estimation
techniques are developed to identify the SRM parameters from standstill test and online
operating data, which can overcome the effect of noise inherent in the data.
Four-quadrant operation of the SRM is necessary for the EMB system. Based on the
inductance model of SRM, algorithms for four-quadrant torque control and torque-ripple
minimization are developed and implemented.
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The control objective of the electro-mechanical brake system is to provide desired
clamping force response at the brake pads and disk. It is realized by controlling the torque,
speed, and rotor position of the SRM. A robust clamping force controller is designed
using backstepping. The backstepping design proceeds by considering lower-dimensional
subsystems and designing virtual control inputs. The virtual control inputs in the first and
second steps are rotor speed and torque, respectively. In the third step, the actual control
inputs, phase voltages, appear and can be designed. Simulation results demonstrate the
performance and robustness of the controller.
Position sensing is an integral part of SRM control. However, sensorless control is
desired to reduce system weight and cost, and increase reliability. A sliding mode
observer based sensorless controller is developed. Algorithms for sensorless control at
near zero speeds and sensorless startup are also proposed and simulated, with satisfactory
results.
Experimental testbed for the electro-mechanical brake system has been setup in the
laboratory. DSP based control system is used for SRM control. The algorithms developed
in simulation have been implemented on the testbed, with corresponding results given.
Future work is suggested to finalize the implementation of the electro-mechanical brake
system.
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ACKNOWLEDGMENTS
First I would like to express my acknowledgement to my advisor, Professor Ali
Keyhani, for his technical guidance, his constant help and support of this work, and his
kind care of my study and life in The Ohio State University. In the past six years I had
learned a lot from his rich experience in research and teaching.
During this research, I have obtained great support and help from many cooperators
and experts. I would like to express my sincere thanks to all of them. They are Mr. Harald
Klode from Delphi Automive who provided me the system model and requirements of
the electro-mechanical brake; Dr. Babak Fahimi from University of Texas-Arlington who
guided me in modeling and control of switched reluctance machines; and Prof. Farshad
Khorrami and Dr. P. Krishnamurthy from Polytechnic University who assisted me in
robust clamping force controller design.
I thank all my colleagues at The Ohio State University and especially to Min Dai,
Bogdan Proca, Geeta Athalye, Luris Higuera, Jin-Woo Jung, and Sachin Puranik. We had
a great environment and many fruitful discussions during the past several years.
Finally, I would like to express my deepest appreciation to my wife and my parents.
Without their constant support none of this would have been possible.
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TABLE OF CONTENTS
Page
ABSTRACT…………………………………………………………………………. ii
ACKNOWLEDGMENTS………..…………………………………………………. iv
TABLE OF CONTENTS……………………………………………………………. v
LIST OF TABLES………………………………………………………………… viii
LIST OF FIGURES………………………………………………………………… ix
NOMENCLATURE……………………………………………………………….. xiii
1 INTRODUCTION .................................................................................................. 1
1.1 Research Background ..................................................................................... 1
1.1.1 Brake-By-Wire........................................................................................ 2
1.1.2 Electro-Mechanical Brake Actuators ...................................................... 4
1.1.3 Switched Reluctance Machines .............................................................. 6
1.2 Research Objectives........................................................................................ 8
1.3 Dissertation Organizations............................................................................ 12
2 LITERATURE REVIEW ..................................................................................... 14
2.1 Modeling of Switched Reluctance Machines ............................................... 14
2.1.1 Flux Linkage Based SRM Model ......................................................... 15
2.1.2 Inductance Based SRM Model ............................................................. 17
2.2 Parameter Identification of Electric Machines ............................................. 18
2.2.1 Parameter Estimation in Frequency Domain and Time Domain .......... 19
2.2.2 Neural Network Based Modeling ......................................................... 19
2.2.3 Maximum Likelihood Estimation ......................................................... 20
2.3 Torque Control and Torque-ripple Minimization of SRM ........................... 21
2.4 Back-Stepping Controller Design ................................................................. 23
2.5 Position Sensorless Control of SRM ............................................................ 25
2.6 Summary....................................................................................................... 28
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3 MODELING AND PARAMETER IDENTIFICATION OF SWITCHED
RELUCTANCE MACHINES .............................................................................. 29
3.1 Introduction to Switched Reluctance Machines ........................................... 30
3.2 Inductance Based Model of SRM At Standstill ............................................ 33
3.2.1 Three-Term Inductance Model ............................................................. 36
3.2.2 Four-Term Inductance Model ............................................................... 37
3.2.3 Voltage Equation and Torque Computation ......................................... 38
3.3 Standstill Test................................................................................................ 39
3.4 Maximum Likelihood Estimation ................................................................. 40
3.4.1 Basic Principle of MLE ........................................................................ 41
3.4.2 Performance of MLE with Noise-Corrupted Data................................ 44
3.5 Parameter Estimation Results From Standstill Tests .................................... 47
3.6 Verification of Standstill Test Results .......................................................... 51
3.7 SRM Model for Online Operation ................................................................ 54
3.8 Neural Network Based Damper Winding Parameter Estimation ................. 56
3.9 Model Validation .......................................................................................... 60
3.10 Modeling and Parameter Identification Conclusions.................................... 62
4 FOUR QUADRANT TORQUE CONTROL AND TORQUE-RIPPLE
MINIMIZATION.................................................................................................. 63
4.1 Four Quadrant Operation of Switched Reluctance Machines....................... 63
4.2 Torque Control and Torque Ripple Minimization ........................................ 66
4.3 Hysteresis Current Control ........................................................................... 71
4.4 Simulation Results ........................................................................................ 72
4.5 Torque Control Conclusions ......................................................................... 73
5 DESIGN AND IMPLEMENTATION OF A ROBUST CLAMPING FORCE
CONTROLLER .................................................................................................... 75
5.1 Clamping Force Control System................................................................... 75
5.2 System Dynamics.......................................................................................... 78
5.3 Controller Design.......................................................................................... 82
5.3.1 Backstepping Controller Design – Basic Idea ...................................... 82
5.3.2 Backstepping Controller Design - Step 1.............................................. 83
5.3.3 Backstepping Controller Design - Step 2.............................................. 84
5.3.4 Backstepping Controller Design - Step 3.............................................. 85
5.3.5 Backstepping Controller Design – Generalized Control Law .............. 91
5.4 Simulation Results ........................................................................................ 93
5.4.1 Controller with Voltage Commutation Scheme.................................... 94
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5.4.2 Controller with Torque Control and Torque-Ripple Minimization ...... 97
5.4.3 Controller Robustness Test ................................................................... 98
5.5 Controller Design Conclusions ................................................................... 100
6 POSITION SENSORLESS CONTROL OF SWITCHED RELUCTANCE
MACHINES........................................................................................................ 102
6.1 Sliding Mode Observers ............................................................................. 103
6.1.1 System Differential Equations ............................................................ 103
6.1.2 Definition of Sliding Mode Observer ................................................. 105
6.1.3 Estimation Error Dynamics of Sliding Mode Observer...................... 106
6.1.4 Definition of Error Function ............................................................... 108
6.1.5 Simulation Results .............................................................................. 111
6.2 Sensorless Control At Near Zero Speeds.................................................... 114
6.2.1 Turn-On/Turn-Off Position Detection ................................................ 114
6.2.2 Simulation Results .............................................................................. 116
6.3 Sensorless Startup ....................................................................................... 117
6.4 Sensorless Control Conclusions.................................................................. 119
7 EXPERIMENTAL TESTBED AND RESULTS ............................................... 121
7.1 Experimental Testbed ................................................................................. 121
7.2 Experimental Results of Four-Quadrant Speed Control ............................. 124
7.3 Experimental Results of Clamping Force Control...................................... 127
7.4 Experimental Results Conclusions ............................................................. 131
8 CONCLUSIONS AND FUTURE WORK......................................................... 132
8.1 Conclusions................................................................................................. 132
8.2 Future Work ................................................................................................ 135
BIBLIOGRAPHY..................................................................................................... 137
APPENDIX A PARAMETERS OF SRM AND BRAKE SYSTEM..................... 149
APPENDIX B SIMULATION TESTBED............................................................. 152
APPENDIX C C PROGRAMS FOR CLAMPINP FORCE CONTROL…………152
viii
LIST OF TABLES
Table Page
Table 3.1 Estimation results with initial guesses within 90% of true values.................. 45
Table 3.2 Estimation results with initial guesses within 10% of true values.................. 46
Table 4.1 Turn-on/turn-off angles for phase A............................................................... 68
Table 6.1 Simulation results at near zero speeds .......................................................... 116
Table 6.2 Determining phases to be excited from the relationship among the peaks of the
currents in all phases............................................................................................... 119
Table A.1 Phase inductance coefficients at different rotor positions…………………150
ix
LIST OF FIGURES
Figure Page
Figure 1.1 Conventional hydraulic brake systems............................................................ 2
Figure 1.2 Brake-by-wire systems .................................................................................... 3
Figure 1.3 Sectional drawing of an electromechanically actuated disk brake.................. 5
Figure 1.4 Basic block diagram of an electro-mechanical brake...................................... 9
Figure 2.1 Nonlinear function λ(θ, i) of an SR motor .................................................... 16
Figure 2.2 Nonlinear function L(θ, i) of an SR motor .................................................... 17
Figure 3.1 Double salient structure of switched reluctance machines............................ 30
Figure 3.2 Typical drive circuit for a 4-phase SRM....................................................... 31
Figure 3.3 Soft-chopping and hard-chopping for hysteresis current control .................. 32
Figure 3.4 Inductance model of SRM at standstill ......................................................... 33
Figure 3.5 Phase inductance profile................................................................................ 34
Figure 3.6 Experimental setup for standstill test of SRM............................................... 40
Figure 3.7 Block diagram of maximum likelihood estimation ....................................... 44
Figure 3.8 Noise-corrupted signal with different signal-to-noise ratio........................... 46
Figure 3.9 Voltage and current waveforms in standstill tests......................................... 47
Figure 3.10 Standstill test results for inductance at θ =0o .............................................. 48
Figure 3.11 Standstill test results for inductance at θ =15o ............................................ 49
x
Figure 3.12 Standstill test results for inductance at θ =30o ............................................ 49
Figure 3.13 Standstill test result: nonlinear phase inductance........................................ 50
Figure 3.14 Flux linkage at different currents and different rotor positions................... 51
Figure 3.15 Cross-sectional drawing of 8/6 SRM .......................................................... 52
Figure 3.16 Flux path at aligned position ....................................................................... 52
Figure 3.17 Flux path at unaligned position ................................................................... 54
Figure 3.18 Model structure of SRM under saturation................................................... 54
Figure 3.19 Recurrent neural network structure for estimation of exciting current ....... 57
Figure 3.20 Validation of model with on-line operating data......................................... 60
Figure 3.21 Validation of model with on-line operating data (Phase A)........................ 61
Figure 4.1 Four quadrants on torque-speed plane........................................................... 64
Figure 4.2 Phase definition of an 8/6 switched reluctance motor................................... 64
Figure 4.3 Phase inductance profile and conduction angles for 4-quadrant operation... 65
Figure 4.4 Phase torque profile under fixed current ....................................................... 66
Figure 4.5 Torque factors for forward motoring operation............................................. 69
Figure 4.6 Voltage and current waveform in hysteresis current control......................... 71
Figure 4.7 Four-quadrant torque control and torque-ripple minimization...................... 73
Figure 5.1 Block diagram of clamping force control system.......................................... 76
Figure 5.2 Simulink model of electro-mechanical brake system.................................... 93
Figure 5.3 Clamping force response ............................................................................... 94
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Figure 5.4 Phase current waveforms............................................................................... 95
Figure 5.5 Rotor position, speed and torque ................................................................... 96
Figure 5.6 Four-quadrant torque speed curve ................................................................. 96
Figure 5.7 Clamping force response ............................................................................... 97
Figure 5.8 Four-quadrant torque speed curve ................................................................. 98
Figure 5.9 Clamping force response ............................................................................... 99
Figure 5.10 Phase current waveforms............................................................................. 99
Figure 5.11 Four-quadrant torque speed curve ............................................................. 100
Figure 6.1 Phase inductance profile.............................................................................. 109
Figure 6.2 Simulink model for sliding mode observer ................................................. 111
Figure 6.3 Simulation results at high speed .................................................................. 112
Figure 6.4 Simulation results at low speed ................................................................... 113
Figure 6.5 Cross section of an 8/6 switched reluctance machine ................................. 117
Figure 6.6 Peak currents at different rotor positions..................................................... 119
Figure 7.1 Block diagram of experimental testbed....................................................... 122
Figure 7.2 Components of experimental testbed .......................................................... 123
Figure 7.3 Photo of experimental testbed ..................................................................... 123
Figure 7.4 Block diagram of four-quadrant speed controller ....................................... 124
Figure 7.5 Torque and speed responses ........................................................................ 125
Figure 7.6 Four-quadrant torque-speed curve............................................................... 126
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Figure 7.7 Voltage and current waveforms................................................................... 126
Figure 7.8 Block diagram of clamping force controller ............................................... 128
Figure 7.9 Clamping force response ............................................................................. 129
Figure 7.10 Torque and speed waveforms.................................................................... 129
Figure 7.11 Four-quadrant torque-speed curve............................................................. 130
Figure 7.12 Phase voltage and current waveforms ....................................................... 130
Figure B.1 Simulink testbed of electro-mechanical brake system…………………….153
Figure B.2 Force command module…………………………………………………...154
Figure B.3 Force controller module…………………………………………………...154
Figure B.4 SRM and power converter module………………………………………..155
Figure B.5 SRM Hysteresis module (four modules for four phases)…………………156
Figure B.5 SRM back EMF module (four modules for four phases)…………………157
Figure B.6 SRM torque module (four modules for four phases)……………………...158
Figure B.7 SRM mechanic part module (four modules for four phases)……………...158
Figure B.8 Brake caliper module……………………………………………………...158
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NOMENCLATURE
LR, : Phase resistance and inductance
θL : Phase inductance at rotor position θ
mua LLL ,, : Phase inductance at aligned position, unaligned position, and a midway
between the two
dd LR , : Resistance and inductance of damper winding
ωθ , : Rotor position and speed
ωθ ˆ,ˆ : Estimates of rotor position and speed
ωθ ee , : Estimation error of rotor position and speed
iV , : Phase voltage and current
offon θθ , : Turn-on and turn-off angles
λ : Flux linkage
rN : Number of rotor poles
T : Electromagnetic torque
lT : Load torque
refT : Torque command
F : Clamping force
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refF : Force command
)(⋅ : Estimate of )(⋅
[ ]T : Transpose of [ ]
[ ]⋅E : The operation of taking the expected value of [ ]⋅
( )⋅w : Process noise sequence
( )⋅v : Measurement noise sequence
( )⋅X : State vector
( )⋅Y : Measured output vector in the presence of noise
Q : Covariance of the process noise sequence
0R : Covariance of the measurement noise sequence
( )⋅R : Covariance of the state vector
( )⋅e : Estimation error, )(ˆ)()( kYkYke −=
exp : The exponential operator
det : Determinant
)1|(ˆ −kkY : The estimated value of Y(k) at time instant k given the data up to k-1
( )⋅U : Input vector
( )⋅θ : Parameter vector
1
CHAPTER 1
1 INTRODUCTION
1.1 Research Background
Modern vehicles have become more and more “electrical”. The implementation of
by-wire systems - replacing a vehicle’s hydraulic systems with wires, microcontrollers,
and electric machines - promises better safety and handling, as well as lower
manufacturing costs and weight.
By-wire systems began to be installed well over a decade ago, first in military and
then in commercial aircraft. In these systems the control commands are not transferred in
a hydraulic/mechanical way but through electrical wires (by-wire). In this area, the
advantages against classical hydraulic/mechanical systems have proved to be so
substantial that the technique is expected to be used in other areas as well [1].
With the fast development of electric or hybrid electric vehicles (HEV), more and
more by-wire systems have been designed for vehicle components, such as throttle-by-
wire, steer-by-wire, and brake-by-wire. This dissertation is based on a brake-by-wire
project sponsored by Delphi Automotive®.
2
1.1.1 Brake-By-Wire
The braking system is one of the most important systems in vehicles. In most of the
vehicles nowadays, hydraulic actuated brake systems are used. The hydraulic brake
system is composed of the following basic components (Figure 1.1): the master cylinder
which is located under the hood, and is directly connected to the brake pedal, converts
mechanical pressure from driver’s foot into hydraulic pressure. Steel brake lines and
flexible brake hoses connect the master cylinder to the slave cylinders located at each
wheel. Brake fluid, specially designed to work in extreme conditions, fills the system.
Shoes and pads in the calipers are pushed by the slave cylinders to contact the drums
and/or rotor disks thus causing drag, which slows the car.
Figure 1.1 Conventional hydraulic brake systems
The hydraulic brake systems have proved to be very successful in automotive
industry. However, to reduce the inherent high cost and necessity of regular maintenance
3
of the hydraulic systems, people are motivated to find cheap and reliable substitute of the
brake systems, that is, electromechanically actuated brake systems, or Brake-By-Wire.
In recent years, the automotive industry and many of their suppliers have started to
develop Brake-By-Wire systems [2-4]. An electromechanical brake-by-wire system looks
deceptively simple. Wires convey the driver’s pressure from a sensor on the brake pedal
to electronic control unit that relays the signal to electromechanical brake actuators at
each wheel. In turn, the modular actuators squeeze the brake pads against the brake disk
to slow and stop the car [2].
Generally a brake-by-wire system contains the following components: four wheel
brake modules (electro-mechanical brake actuators), an electronic control unit (ECU),
and an electronic pedal module with pedal feel simulator, as shown in Figure 1.2.
Figure 1.2 Brake-by-wire systems
Electronic pedal
module with pedal
feel simulator
Electronic
control unit
(ECU)
Wheel
brake
module
4
Brake-by-wire does everything [2]: the antilock and traction control functions of
today’s antilock braking systems (ABS) plus brake power assist, vehicle stability
enhancement control, parking brake control, and tunable pedal feel, all in a single,
modular system.
The brake-by-wire technology is expected to offer increased safety and vehicle
stability to consumers and it will provide benefits to automotive vehicle manufacturers
who will be able to combine vehicle components into modular assemblies using cost
effective manufacturing processes.
1.1.2 Electro-Mechanical Brake Actuators
Among the components of brake-by-wire systems, the wheel brake module (or
electro-mechanical brake actuator, EMB) is the most important one. It receives the
electronic commands from control unit and generates the desired braking force, by means
of electric motors and corresponding mechanical systems.
A prototype of EMB developed by ITT Automotive [3] is shown in Figure 1.3. A
spindle is used to actuate the inner brake pad. A bolt at the back of the brake pad, which
fits in a recess of the pad support, prevents the spindle from rotating. The nut of the
planetary roller gear is driven by the rotor of a brushless torque motor. By integrating the
coil of the servo motor directly into the brake housing and by supporting the gear and
spindle unit with one central bearing only, the compact design is made possible. A
resolver measures the position of the rotor for electronic commutation.
5
Figure 1.3 Sectional drawing of an electromechanically actuated disk brake
To embed such functionality as ABS (Anti-lock brake system), TCS (Traction control
system) etc. in conventional hydraulic brake systems, a large number of electro-hydraulic
components are required. An electromechanically actuated brake system, however,
provides an ideal basis to convert electric quantities into clamping forces at the brakes
[3]. Standard and advanced braking functions can be realized on uniform hardware. The
software modules of the control unit and the sensor equipment determine the
functionality of the Brake-By-Wire system. The reduction of vehicle hardware and entire
system weight are not the only motivational factors contributing to the development of a
Brake-By-Wire system. The electro-mechanical brakes have the advantages in many
other aspects:
6
- Environmentally friendly due to the lack of brake fluid
- Improved crash worthiness - its decoupled brake pedal can be mounted crash
compatible for the passenger compartment
- Space saving, using less parts
- More comfort and safety due to adjustable pedals
- No pedal vibration even in ABS mode
- Simple assembly
- Can be easily networked with future traffic management systems
- Additional functions such as an electric parking brake can be integrated easily
- Reduced production and logistics costs due to the ‘plug and play’ concept with a
minimized number of parts.
1.1.3 Switched Reluctance Machines
Electric motors are used in the electro-mechanical brake systems (EMB) to drive the
brake pads. Different types of electric motors have been tried in EMB, such as DC
motors, brushless DC motors, or induction motors [2,4,5]. Some prototypes of brake-by-
wire system based on these motors have already been developed. However, due to the
importance of the brake system and the harsh working environment, a more efficient,
reliable, and fault-tolerant motor drive over wide speed range is preferred for this
application. In this research, a switched reluctance motor (SRM), is proposed to be the
servo-motor in the electro-mechanic brake system, which has the following advantages
over the other types of electric machines:
7
Fault-tolerance
The brake system requires that the motor drive system continue operation in case of
errors in phase windings, drive circuits, or sensors. The switched reluctance motors can
operate in a reduced power rating when losing one or more phases, while the other
motors may completely fail in such cases.
Fast acceleration and high speed operation
Emergency braking requires fast acceleration and high-speed operation of the motor
drive. From mechanical point of view, the phase winding or permanent magnet mounted
on the rotor prevents the motor from accelerating fast and running at super high speed.
This problem exists in all the types of motors mentioned above except switched
reluctance motors, which have neither winding nor magnet on the rotor.
Robust performance at high temperature
During braking, a lot of heat will be generated in the caliper house that may cause
high temperature. The absence of magnets or windings on the rotor of SRM suggests a
robust performance in the presence of high temperature.
Reliable position sensorless control
Position sensor in motor drives mean additional cost, space, and weight, as well as a
potential source of failure. Integration of a reliable position sensorless control turns into a
8
necessary step towards development of high-speed motor drives. Due to the limited
available computational time at high speeds, the sensorless method should be robust and
timely efficient. This has led us to develop a novel geometry based technique that
requires minimum magnetic data and computation. These, in turn, would favor SRM
drive as a superior choice for this application.
High efficiency
The inherent simplicity of the SRM geometry and control offers high efficiency and a
very long constant power speed ratio [6].
In this research, an 8/6, 42V, 50A switched reluctance motor is used in the electro-
mechanical brake system.
1.2 Research Objectives
A block diagram of the electro-mechanical brake developed in this research is shown
in Figure 1.4. The brake assembly consists of a bracket which is rigidly mounted to the
vehicle chassis and the floating caliper which is typically held on two sliding pins that are
in turn attached to the bracket.
The actuator inside the caliper housing constitutes the actual electromechanical
energy conversion device. In this design configuration, the actuator consists of a ball
screw assembly that is driven by a dual-stage planetary gear. The input gear of the first
9
planetary gear stage is connected to a 4-quadrant switched reluctance motor. This allows
for conversion of the rotary motion (torque Ta) of the servomotor into linear ball screw
action in order to create the required clamping force FCL at the wheel brake rotor.
Figure 1.4 Basic block diagram of an electro-mechanical brake
The servomotor interfaces electrically to a 4-quadrant servo-controller that controls
current and voltage (controller output u) to the motor. An encoder delivers rotor position
information to the controller for correct commutation of the motor phases.
A clamping force sensor located in the force path between the ball screw and the
caliper housing measures the clamping force FCL of the ball screw. The output signal of
this force sensor is used to close the loop on the clamping force command Fd which is
generated by the higher-level brake system controller.
The overall goal of this research is to develop and implement a low-cost drive system
consisting of a sensorless switched reluctance motor with power converter and controller.
This drive system shall be operated as part of an electromechanical clamping device for
Controller Actuator
Brake
Mechanical
Structures
Force Sensor
F d u T a F CL
10
the purpose of converting a clamping force command into a physical clamping force in a
closed-loop configuration under observation of specific dynamic requirements.
The research objectives have the following main components:
Modeling and parameter estimation of switched reluctance machines
Different model structures for switched reluctance machines can be found in literature.
An appropriate model for the electro-mechanical brake application is to be decided. Then
the parameters for the proposed model need to be estimated from test or operating data.
And the identified model and its parameters must be validated by simulation and
experiments. Noise effect on the parameter estimation is to be studied too.
Four-quadrant operation of switched reluctance machines
In electro-mechanical brake systems, the electric motor needs to be operated in all
four quadrants on the torque-speed plane to realize desired clamping force response. For
switched reluctance machines, this means to set correct turn-on/turn-off angles for each
phase and determine the proper exciting sequence. This will be combined with the torque
control described in the following part.
Torque control and torque ripple minimization of switched reluctance machines
Based on the structure and torque production principle of switched reluctance
machines, the torque generated in each phase is a highly nonlinear function of phase
11
current and rotor position. And during operation, each phase will be conducting for only a
fraction of the electric cycle. So the torque in any individual phase is discontinuous.
Since the SRM needs to be operated in four quadrants, it is even more difficult to control
the torque and minimize the torque-ripple. Based on the proposed SRM model, new
torque control algorithms will be suggested and implemented.
Robust clamping force control of electro-mechanical brake
The control objective in electro-mechanical brake system is the clamping force
between the brake pads and the brake disk. Generally the force is a nonlinear function of
the distance between the pads and the disk which corresponds to the angular movement
of the SRM rotor, and the torque seen by the motor is a nonlinear function of the force.
For SRM, the control inputs are the phase voltages. It’s hard to define a direct
relationship between the control input and the objective. A robust controller needs to be
designed to meet different requirements on the clamping force. Back-stepping technology
is to be used in this research to design the force controller.
Rotor position sensorless control of switched reluctance machines
Rotor position sensing is an integral part of switched reluctance motor control system
due to the torque-production principle of SRM. Conventionally, a shaft position sensor is
employed to detect rotor position. But this means additional cost, more space requirement
and an inherent source of unreliability. A sensorless (without direct position or speed
sensors) control system, which extracts rotor position information indirectly from
12
electrical or other signals, is expected. The sensorless algorithm needs to be stable in full
speed range. Sensorless control for motor startup (when regular electrical signals are not
available) also needs to be designed.
Implementation of control algorithms in experimental testbed
As the first step of the electro-mechanical brake system development, an offline
simulation testbed is to be setup in Matlab/Simulink®, with the parameters identified
from real motor and brake caliper. All control algorithms will be tested on the simulation
testbed first. In later steps, the experimental testbed that contains the switched reluctance
motor, power converter, and DSP will be built up. And the control algorithms will then
be implemented and tested on the experimental testbed.
Details on how these research objectives have been achieved are described in the
following chapters.
1.3 Dissertation Organizations
This dissertation is organized as follows. The research background and objectives are
introduced in Chapter 1. Literature review of related work is summarized in Chapter 2.
Chapter 3 presents the model identification and parameter estimation of switched
reluctance machines from standstill test data and operating data. In Chapter 4 the four-
quadrant torque control and torque-ripple minimization of SRM are presented. A robust
clamping force controller for the electro-mechanical brake is described in Chapter 5. In
13
Chapter 6 a sliding mode observer based sensorless control algorithm for switched
reluctance machines is developed and analyzed. Sensorless control at startup and low
speed is also given. And In Chapter 7, the experimental setup for the electro-mechanical
brake is introduced and related experimental results are shown. Overall conclusions and
future work are presented in Chapter 8.
14
CHAPTER 2
2 LITERATURE REVIEW
The idea of brake-by-wire is kind of new. But the related technologies have been
studied and applied in other areas for many years. According to the research goal and
objectives of the electro-mechanical brake systems, the related researches are focused on
electric machines (switched reluctance machines, or SRM, for this research), modeling
and parameter identification, torque control and torque-ripple minimization of SRM,
back-stepping controller design, and position sensorless control of electric machines.
Research activities in these fields can be easily found in publications. The literature
review will be based on these topics.
2.1 Modeling of Switched Reluctance Machines
To ensure the high efficiency and successful development of a complicated control
system, offline simulation is often performed first. At this stage of development, a model
of the system is designed and simulated offline. Then the parameters of the real system
will be identified and implemented into the model. An accurate model and precisely
15
estimated system parameters are critical to the successful implementation of the final
control system.
In this research, the model of brake caliper is available from cooperative company. So
the major task is to model the switched reluctance machines and power converters.
The nonlinear nature of SRM and high saturation of phase winding during operation
makes the modeling of SRM a challenging work. The flux linkage and phase inductance
of SRM vary with both the phase current and the rotor position. Therefore the nonlinear
model of SRM must be identified as a function of the phase current i and rotor position θ.
Two main types of models for SRM have been suggested in the literature – the flux
linkage based model [7-10], and the inductance based model [11-12].
2.1.1 Flux Linkage Based SRM Model
The flux linkage based model assumes the nonlinear relationship between the phase
flux linkage and phase current and rotor position. A typical flux model of SRM is as
follows [10],
)1()( if
s eθλλ −−= , (2.1)
where λ and i are the phase flux linkage and phase current, respectively, θ is the phase
angle, sλ is a constant, and
θθθθθ 2sinsin2coscos)( edcbaf ++++= . (2.2)
16
A 3-D plot in Figure 2.1 shows the λ-θ-i relationship obtained from an 8/6 switched
reluctance motor.
Figure 2.1 Nonlinear function λ(θ, i) of an SR motor
In an SRM control system, the phase winding is modeled as an inductance and a
resistance connected in series. The resistance R is assumed known and phase voltage V
and current i can be measured. Therefore, the flux linkage λ can be computed by
∫ −= dtRiV )(λ . (2.3)
From the relationship λ-θ-i, rotor position angle θ can be obtained since λ and i are
known. This is the basic operating principle of the flux linkage based model in rotor
position estimation.
17
2.1.2 Inductance Based SRM Model
In the inductance-based model, the position dependency of the phase inductance is
represented by a limited number of Fourier series terms and the nonlinear variation of the
inductance with current is expressed by means of polynomial functions [11]:
∑∞
=
+=0
)cos()(),(n
nrn nNiLiL ϕθθ , (2.4)
where rN is the number of rotor poles, nL are coefficients to be decided. In practical use,
only the first few terms of the Fourier series are used. The higher-order-term can be
ignored without significant error.
A 3-D plot in Figure 2.2 shows the L-θ-i relationship obtained from an 8/6 SRM.
Figure 2.2 Nonlinear function L(θ, i) of an SR motor
18
In electro-mechanical brake system, the switched reluctance motor will be running at
very low speed (near zero) at steady state. Only one or two phases will be continuous
conducting at such situation. Since the flux linkage based model utilizes the integration in
Eq. (2.3) to estimate the flux linkage. The errors in parameter (R) and measurements (V
and i) in the active phase may accumulate very soon, without chance to be reset (reset can
only occur when the phase is not conducting). Our simulation shows that flux linkage
model based rotor position estimation fails in long time zero speed case, which is the
steady state of an electrical brake. It reduces the output clamping force up to 25% and
creates oscillation. Same problems do not exist in inductance-based model. This makes
the inductance-based model a better choice for electro-mechanical brake application.
The inductance-based model suggested by Fahimi etc. [11] can represent the SRM at
standstill or low load condition very well. But for highly saturated condition under high
load, the model needs to be improved to include saturation effect and core losses. Also,
models with different number of terms in the Fourier series will be compared to select the
best model. This will be detailed in Chapter 3.
2.2 Parameter Identification of Electric Machines
Once a model of an electric machine is selected, how to identify the parameters in the
model becomes an important issue. Finite element analysis can provide model parameters
that will be subjected to substantial variation after the machine is constructed with
19
manufacturing tolerances. Therefore, the model parameters need to be identified from test
and/or operating data.
2.2.1 Parameter Estimation in Frequency Domain and Time Domain
Generally, the parameter estimation from test data can be done in frequency domain
or time domain. Since noise is an inherent part of the test data, the noise effect on
parameter estimation must be taken into consideration. In [13], Keyhani etc. studied the
effect of noise on parameter estimation of synchronous machines in frequency-domain,
and got the conclusion that: noise has significant impact on the synchronous machine
parameters estimated from SSFR (steady state frequency response) test data using curve-
fitting techniques. The estimated values of machine parameters are very sensitive to the
value of armature resistance used in data analysis. Even a 0.5% error in the value of
armature resistance could result in unrealistic estimation of machine parameters. Hence a
technique should be developed which provides a unique physically realizable machine
model even when the test data is noise-corrupted.
2.2.2 Neural Network Based Modeling
In [25], Karayaka etc. developed an Artificial Neural Network based modeling
technique for the rotor body parameters of a large utility generator. Disturbance operating
data collected on-line at different levels of excitation and loading conditions are utilized
for estimation. Rotor body ANN models are developed by mapping field current *
fdi and
20
power angle δ to the parameter estimates. Validation studies show that ANN models can
correctly interpolate between patterns not used in training.
Nonlinear neural network modeling of other types of electric machines have also been
reported [18]. However, for ANN based model, rich data set collected at different loading
and exciting levels are needed for training the ANN to improve the performance of the
model. This sometimes restricts the application of ANN based models.
2.2.3 Maximum Likelihood Estimation
A time-domain identification technique, which can overcome the multiple solution
sets problem encountered in the frequency response technique, is used to estimate
machine parameters. The new technique is maximum likelihood estimation, or MLE.
The MLE identification method has been applied to the parameter estimation of many
engineering problems. It has been established that the MLE algorithm has the advantage
of computing consistent parameter estimates from noise-corrupted data. This means that
the estimate will converge to the true parameter values as the number of observations
goes to infinity [22,23]. This is not the case for the least-square estimators which are
commonly used in power system applications.
Maximum likelihood estimation techniques have been successfully applied to identify
the parameters of synchronous [14-16] and induction machines [17] from noise-corrupted
data. In this research, MLE techniques will be used to identify the parameters of switched
21
reluctance machines from standstill test and operating data. The details and the results are
given in Chapter 3.
2.3 Torque Control and Torque-ripple Minimization of SRM
Due to the nature of torque production in SRM, the torque generated in any individual
phase is discontinuous. The output torque is a summation of the torque generated in all
active phases. Ripple-free torque control strategies for SRM have been studied
extensively. The most popular approach for ripple minimization has been to store the
torque-angle-current characteristics in a tabular form so that optimum phase current can
be determined from position measurements and torque requirement.
The method described in [27] is based on the estimation of the instantaneous
electromagnetic torque and rotor position from the phases terminal voltages and currents.
The flux linkage for the active phase is computed from the voltage, current and stator
resistance. Both the rotor position and torque are obtained from the third order
polynomial evaluations which coefficients are pre-computed and stored in memory
locations of the DSP used to implement the control. These coefficients are computed
from the flux linkage versus current and rotor position characteristics curve data
measured experimentally; bi-cubic spline interpolation technique is used to generate these
coefficients. The estimated torque is compared with a constant reference value and the
result of this comparison drives a current regulator to control the motor phase currents.
Simulation results have shown that the torque ripple can be reduced from a value of about
22
100% for the motor operating in open loop to about 10% when the torque ripple
minimization controller is utilized.
The method of ripple reduction by optimizing current overlapping during
commutation at all torque levels was studied in [28]. The algorithm is based on minimize
both the average and peak current hence improves the dynamic performance of the motor
and inverter. It is shown that the proposed current profiling algorithm results in the
highest possible torque/current inverter rating and an extended operating speed range
under constant torque operation.
The research by Husain etc. [29] suggests a new strategy of PWM current control for
smooth operation of the SRM drive. In this method, a current contour for constant torque
production is defined and the phase current is controlled to follow this contour. The
scheme is capable of taking into account the effects of saturation, although in some cases
more accurate modeling of the motor inductance may be required.
In [30] Le Chenadec etc. present methods for computing simple reference currents for
a current-tracking control to minimize torque ripple. In [31], nominal currents that result
in constant torque are computed for reduced current peaks and slopes, under the
constraint that at critical rotor positions each of the phases contributes half of the total
torque. In [32], the control goal, motivated by energy considerations, is to minimize the
peak phase current while requiring linear torque change in the angular range where the
two phases overlap.
23
In [33] Russa etc. propose a new commutation strategy along with a PI controller to
minimize torque ripple, where an easily invertible flux function is used in calculating
reference phase currents. Fuzzy logic and neural network based methods are proposed in
[33-34]. In [36], the algorithm proposed combines the use of a simplified model with
adaptation. Explicitly, it includes dynamic estimation of low harmonics of the combined
unknown load torque and the ripple in the produced torque (due to model simplification),
and adds appropriate terms to the commanded current to cancel these harmonics.
In this research, an inductance based model for switched reluctance machine is used.
The torque generated by the SRM can be computed directly from the phase currents and
rotor position. This provides a convenient way to control the output torque. The main
problem will be how to determine the torque generated from each phase during
overlapping to make a ripple free resultant torque according to the rotor position. This
will be detailed in Chapter 4.
2.4 Back-Stepping Controller Design
Over the past ten years, the focus in the area of control theory and engineering has
shifted from linear to nonlinear systems, providing control algorithms for systems that are
both more general and more realistic. Nonlinear control dominates control conferences
and has strong presence in academic curricula and in industry.
24
Backstepping is a design approach whose significance for nonlinear control can be
compared to root locus or Nyquist's method for linear systems. Its roots are in the theory
of feedback linearization of the 1980's. With its added flexibility to handle modeling
nonlinearities and some classes of systems that are not feedback linearizable,
backstepping is the most important ingredient in the nonlinear control advances of the
last decade.
Consider a nonlinear system [37]:
ξηηη )()( gf +=& (2.5)
u=ξ& (2.6)
where 1],[ +∈ nTT Rξη is the state and Ru∈ is the control input. The functions f and g
are smooth in a domain nRD ⊂ that contains 0=η and 0)0( =f . A state feedback
control law is to be designed to stabilize the origin ( 0=η , 0=ξ ).
This system can be viewed as a cascade connection of two components; the first
component is (2.5), with ξ as “virtual” input and η as state, and the second component is
the integrator (2.6), with u as input and ξ as state.
Suppose the component (2.5) can be stabilized by a smooth state feedback control law
)(ηφξ = , with 0)0( =φ . Suppose further that a Lyapunov function )(ηV that satisfies the
inequality
25
)()]()()([ ηηφηηη
WgfV
−≤+∂∂
, (2.7)
where )(ηW is positive definite. Integrator backstepping theory tells that the state
feedback control law
)]([)(])()([ ηφξηη
ξηηηφ
−−∂∂
−+∂∂
= kgV
gfu (2.8)
stabilizes the origin of nonlinear system (2.5-2.6), with a Lyapunov function of
2/)]([)( 2ηφξη −+V .
In this research, the electro-mechanical brake is a typical nonlinear system. There’s
no direct relationship between the control objective and the input. Iterative backstepping
control techniques will be used to design the clamping force controller, which is detailed
in Chapter 5.
2.5 Position Sensorless Control of SRM
As is mentioned in Chapter 1, rotor position sensing is an integral part of switched
reluctance machine control but a rotor position or speed sensor is not desired. This
requires position sensorless control of SRM.
A large amount of sensorless control techniques have been published in the last
decade. All these techniques come from the same main idea, that is, to recover the
encoded position information stored in the form of flux linkage, inductance, back-EMF,
26
etc. by solving the voltage equation in an active or idle phase. However, according to the
different signals and devices used, the sensorless techniques differ greatly in cost,
accuracy, and range of application. Generally the various techniques can be divided into
three categories according to the signals they use.
The first category uses external injected signal. Mehrdad Ehsani etc. propose such an
indirect SRM rotor position-sensing scheme by applying a high frequency external carrier
voltage [42]. Two modulation techniques – amplitude modulation (AM) and phase
modulation (PM), which are commonly used in communication systems, are adopted to
encode winding current that is dependent on phase inductance. Because the phase
inductance is a periodic function of rotor position, the encoded signal can be decoded to
obtain rotor position.
This scheme applies an external sinusoidal voltage to the unexcited phase winding to
generate encoding signal, so it can keep excellent track of the rotor angle continuously
and is extremely robust to switching noises. However, the fairly expensive high
frequency sinusoidal wave generator will highly increase the cost of the drive system.
And the encoding and decoding circuits will also increase the complexity, hence
unreliability, of the drive.
Instead of using external signal, the second category utilizes internal generated signal.
Iqbal Husain etc. suggest a sensorless scheme by measuring mutually induced voltage in
an inactive phase winding when adjacent phases are excited by power converter [43].
Since this voltage varies significantly as the rotor moves from its unaligned position to
27
aligned position, a simple electronic circuit can be built to capture the variation and send
it to a microprocessor to compute the rotor position.
Compared to the previous described sensing scheme, the new method directly
measures an internal signal that is available without the injection of any diagnostic pulses.
This reduces the space requirement, complexity and cost of the system. However, this
method is not suitable for high-speed applications because the speed dependent terms of
the mutually induced voltage cannot be neglected.
Despite the suitability, both these methods need to measure extra signals that are
unnecessary for operation purpose. Another method, which uses ONLY operating data, is
described by Gabriel Gallegos-Lopez and his colleagues [44]. The new method, called
CGSM (Current Gradient Sensorless Method), uses the change of the derivative of the
phase current to detect the position where a rotor pole and a stator pole starts to overlap.
It can give one position update per energy conversion without a priori knowledge of
motor parameters which is an indispensable basis of the above two methods. This makes
it applicable to most SRM topologies in a wide torque and speed range and for several
inverter topologies. Since all calculation can be done on a cheap but powerful DSP
(digital signal processor), this method has great advantages over other similar methods in
terms of low-cost, easy implementation, and robust functionality. But on the other hand,
it has the disadvantages such as needing a startup procedure, not being suitable for low
speed, and not allowing large load torque transients.
28
Ideally, it is desirable to have a sensorless scheme, which uses only operating data
measured from motor terminals while maintaining a reliable operation over the entire
speed and torque range with high resolution and accuracy. In this research, a sliding
mode observed based sensorless control algorithm is suggested and tested. Details are
given in Chapter 6.
2.6 Summary
The above literature review covers most aspects of the proposed research.
From the literature, we got the basic ideas for modeling of switched reluctance
machines. We will improve the inductance-based model for electro-mechanical brake
application. Maximum likelihood estimation based techniques, which have been
successfully utilized to other types of electric machines, will be applied to parameter
estimation of SRM from standstill and online test data.
Since the model used in this research differs from the models proposed by others, the
relative algorithms on torque control, torque-ripple minimization, and sensorless control
will be different too. This will be described in the following chapters.
29
CHAPTER 3
3 MODELING AND PARAMETER IDENTIFICATION OF
SWITCHED RELUCTANCE MACHINES
Modeling the dynamic properties of a system is an important step in analysis and
design of control systems. Modeling often results in a parametric model of the system
that contains several unknown parameters. Experimental data are needed to estimate the
unknown parameters.
Generally, the parameter estimation from test data can be done in frequency-domain
or time-domain. Since noise is an inherent part of the test data, which may cause
problems to parameters estimation, the effects of noise on different parameter estimation
techniques must be investigated. Studies on identification of synchronous machine
parameters from noise-corrupted measurements show that noise has significant effect on
frequency-domain based techniques. Sometimes unrealistic parameters are obtained from
noise-corrupted data. A time-domain technique - maximum likelihood estimation (MLE)
- can be used to remove the effect of noise from estimated parameters. The models and
the procedures to identify the parameters of switched reluctance machines using
30
experimental data will be presented in this chapter, after a brief introduction to switched
reluctance machines and their control.
3.1 Introduction to Switched Reluctance Machines
The name “switched reluctance” has now become the popular term for this class of
electric machine. A switched reluctance motor (SRM) is a rotating electric machine
where both stator and rotor have salient poles, as shown in Figure 3.1. SR motors differ
in the number of phases wound on the stator. Each of them has a certain number of
suitable combinations of stator and rotor poles (for example, 6/4, 8/6, …).
Figure 3.1 Double salient structure of switched reluctance machines
Phase windings are mounted around diametrically opposite stator poles (A-A, B-
B, …). There is neither phase winding nor magnet on the rotor of SRM. The motor
31
operates on the well-known minimum reluctance law - excitation of a phase will lead to
the rotor moving into alignment with that stator pole, so as to minimize the reluctance of
the magnetic path. The motor is excited by a sequence of voltage/current pulses applied
at each phase. The exciting sequence (instead of the current direction) determines the
rotating direction of the SRM. For example, an exciting sequence of A-B-C-D for the
motor shown in Figure 3.1 will result in counterclockwise rotation; while an exciting
sequence of C-B-A-D will result in clockwise rotation of the motor.
Also note that the voltage/current pulses need to be applied to the respective phase at
the exact rotor position relative to the excited phase, so rotor position sensing is an
integral part of SRM control.
A typical drive circuit for SRM is shown in Figure 3.2.
Figure 3.2 Typical drive circuit for a 4-phase SRM
According to this configuration, two types of voltage chopping are available, as
shown in Figure 3.3:
32
− Soft chopping, as shown in Figure 3.3 (a), only one switch (S2) is switching
during conducting period (with S1 being kept on). A positive voltage is applied to
conducting phase when S2 is on, and a zero voltage is applied when S2 is off.
− Hard chopping, as shown in Figure 3.3 (b), both switches (S1 and S2) are
switching during conducting period. So a positive voltage is applied to conducting
phase when S1 and S2 are on, and a negative voltage is applied when S1 and S2
are off (before the current drops to zero).
(a) soft-chopping (b) hard-chopping (c) switching circuit
Figure 3.3 Soft-chopping and hard-chopping for hysteresis current control
In this research, soft-chopping with 20kHz switching frequency is used.
SR machines have a significant torque ripple, especially when operated in single-
pulse voltage control mode. This is the price to pay for high efficiency. Algorithms must
33
be developed to control the voltage pulses applied to the phase windings to minimize the
torque ripple. This will be described in Chapter 4.
3.2 Inductance Based Model of SRM At Standstill
There are two main types of SRM model: flux linkage based model and inductance-
based model. Both models have been designed and simulated with the electro-mechanical
brake system. After detailed study and comparison of the two models, the inductance-
based model has been chosen for our system, which completely fulfills the requirements
of the brake system and provides a flexible method for indirect rotor position sensing.
For a switched reluctance motor, the dynamic system refers to a phase winding which
can be represented by an inductor in series with a resistor, as shown in Figure 3.4. The
corresponding resistance and the inductance are the system parameters.
Figure 3.4 Inductance model of SRM at standstill
34
Since the phase inductance changes periodically with the rotor position θ, it can be
expressed as a Fourier series with respect to θ: To simplify the inductance expression, we
select the origin of the rotor position θ to be at the aligned position, as shown in Figure
3.5. Under this definition, the phase inductance L reaches its maximum value aL at
0=θ (aligned position) and its minimum value uL at rN/πθ = (unaligned position).
So the phase inductance can be expresses as
∑=
=m
k
rk NkiCiL0
cos)(),( θθ , (3.1)
where rN is the number of rotor poles, i is phase current, θ is rotor position, and m is the
number of terms included in the Fourier series.
Figure 3.5 Phase inductance profile
θ
),( iL θ
)(iLθ
θrN/π0
uL
aL
35
To determine the coefficients )(iCk in the Fourier series, the inductances at several
specific positions need to be known. Use )(iLθ to represent the inductance at position θ,
which is a function of phase current i and can be approximated by a polynomial:
∑=
=p
n
n
niaiL0
,)( θθ , (3.2)
where p is the order of the polynomial and na ,θ are the coefficients of polynomial. In our
research, p = 5 is chosen after we compare the fitting results of different p values (p = 3,
4, 5, and 6 have been tried and compared.)
For an 8/6 SR machine, we have 6=rN . When o0=θ is chosen at the aligned
position of phase A, then o30=θ is the unaligned position of phase A. Usually the
inductance at unaligned position can be treated as a constant [11]:
constL =030. (3.3)
In [11], The authors suggest using the first three terms of the Fourier series, but more
terms can be added to meet accuracy requirements.
36
3.2.1 Three-Term Inductance Model
If three terms (m = 2) are used in the Fourier series, then we can compute the three
coefficients 0C , 1C , and 2C from inductances at three positions: 00
L (aligned position),
030
L (unaligned position), and 015
L (a midway between the above two positions). Since
=
2
1
0
00
00
30
15
0
)30*12cos()30*6cos(1
)15*12cos()15*6cos(1
111
0
0
0
C
C
C
L
L
L
, (3.4)
we have
−
−=
0
0
0
30
15
0
2
1
0
4/12/14/1
2/102/1
4/12/14/1
L
L
L
C
C
C
. (3.5)
Or in separate form, we have
++= 000 153000 )(2
1
2
1LLLC ,
)(2
100 3001 LLC −= , (3.6)
−+= 000 153002 )(2
1
2
1LLLC .
Note that 00
L and 015
L are curve-fitted as a polynomial of phase current i.
37
3.2.2 Four-Term Inductance Model
If Four terms are used in the Fourier series, then we can compute the four coefficients
0C , 1C , 2C , and 3C from phase inductance at four different positions: 00
L (aligned
position), 010
L , 020L , and 0
30L (unaligned position). Since
=
3
2
1
0
000
000
000
30
20
10
0
)540cos()360cos()180cos(1
)360cos()240cos()120cos(1
)180cos()120cos()60cos(1
1111
0
0
0
0
C
C
C
C
L
L
L
L
, (3.7)
we have
−−
−−
−−=
0
0
0
0
30
20
10
0
3
2
1
0
6/13/13/16/1
3/13/13/13/1
3/13/13/13/1
6/13/13/16/1
L
L
L
L
C
C
C
C
. (3.8)
Or in separate form, we have
+++= )()(2
1
3
10000 20103000 LLLLC ,
)(3
10000 30201001 LLLLC −−+= , (3.9)
)(3
10000 30201002 LLLLC +−−=
−−−= )()(2
1
3
10000 20103003 LLLLC .
38
3.2.3 Voltage Equation and Torque Computation
Based on the inductance model described above, the phase voltage equations can be
formed and the electromagnetic torque can be computed from the partial derivative of
magnetic co-energy with respect to rotor angle θ. They are listed here:
)(
)(
dt
di
i
LLi
dt
diLiR
dt
LidiR
dt
diRV
∂∂
+∂∂
++⋅=
+⋅=+⋅=
ωθ
λ
, (3.10)
where
dt
dθ=ω , (3.11)
∑= ∂
∂=
∂∂ m
k
rk kNi
iC
i
L
0
cos)(
θ , (3.12)
∑=
−=∂∂ m
k
rrk kNkNiCL
1
sin)( θθ
. (3.13)
And
])([)sin(
])cos()([
]),([),(
1
0
∑ ∫
∫∑
∫
=
=
−=
∂
∂=
∂
∂=
∂
∂=
m
k
krr
m
k
rk
c
diiiCkNkN
diikNiC
diiiLiWT
θ
θ
θ
θ
θ
θθ
. (3.14)
39
Eq. (3.14) represents the torque generated in one phase of the switched reluctance
machine. The total electromagnetic torque is a summation of the torque generated in all
active phases. Since the phase inductance L has an explicit expression with respect to θ
and i, the above equations can be computed analytically.
3.3 Standstill Test
There are two parameters (R and L) in the model that need to be estimated. Test data
can be obtained from standstill tests.
The basic idea of standstill test is to apply a short voltage pulse to the phase winding
with the rotor blocked at specific positions, record the current generated in the winding,
and then use maximum likelihood estimation to estimate the resistances and inductances
of the winding. By performing this test at different current level, the relationship between
inductance and current can be curve-fitted with polynomials.
The experimental setup is shown in Figure 3.6.
40
Figure 3.6 Experimental setup for standstill test of SRM
Before testing, the rotor of SRM is blocked at a specific position (with the phase to be
tested at aligned, unaligned, or other positions). A DSP system (dSPACE DS1103
controller board) is used to generate the gating signal to a power converter to apply
appropriate voltage pulses to that winding. The voltage and current at the winding is
measured and recorded. Later on, the test data is used to identify the winding parameters.
3.4 Maximum Likelihood Estimation
To minimize the effects of noise caused by the converter harmonics and the
measurement, maximum likelihood estimation (MLE) technique is applied to estimate the
parameters.
41
3.4.1 Basic Principle of MLE
Suppose the system dynamic response is represented by
+=
++=+
)()()()(
)()()()()()1(
kvkxCky
kwkuBkxAkx
θθθ
, (3.15)
where θ represents the system parameters,
x(k) represents system states,
y(k) represents the system output,
u(k) is the system input,
w(k) is the process noise, and
v(k) is the measurement noise.
To apply the maximum likelihood estimation method, the first step is to specify the
likelihood function [19-23]. The likelihood function )(θL , is defined as
∏=
−
−=N
k
T
mkekRke
kRL
1
1 )()()(2
1exp
))(det()2(
1)(
πθ , (3.16)
where ( )⋅e , ( )⋅R , N and m denotes the estimation error, the covariance of the estimation
error, the number of data points, and the dimension of y, respectively.
Maximizing )(θL is equivalent to minimizing its negative log function, which is
defined as:
[ ] )2log(2
1))(det(log
2
1)()()(
2
1)(
)(log)(
11
1 πθ
θθ
mNkRkekRkeV
LV
N
k
N
k
T ++=
−=
∑∑==
− . (3.17)
42
The parameter vector θ can be computed iteratively using Newton’s approach
[20,24]:
θθθθ
∆+=
=+∆
oldnew
GH 0, (3.18)
where H and G are the Hessian matrix and the gradient vector of )(θV . They are defined
by:
θθ
θθ
∂∂
=∂
∂=
)()(2
2 VG
VH . (3.19)
The H and G matrices are calculated using the numerical finite difference method as
described in [17,23].
To start iterative approximation of θ , the covariance of estimation error )(kR is
obtained using the Kalman filter theory [18-22]. The steps are as follows:
1) Initial conditions: The initial value of the state is set equal to zero. The initial
covariance state matrix 0P is assumed to be a diagonal matrix with large positive
numbers. Furthermore, assume an initial set of parameter vector θ .
2) Using the initial values of the parameters vector θ , compute the matrices A, B,
and C.
3) Compute estimate )1|(ˆ −kky from )1|(ˆ −kkx :
)1|(ˆ)1|(ˆ −=− kkxCkky . (3.20)
4) Compute the estimation error of )(kY :
43
)1|(ˆ)()( −−= kkykyke . (3.21)
5) Compute the estimation error covariance matrix )(kR :
TCkkPCRkR ⋅−⋅+= )1|()( 0 . (3.22)
6) Compute the Kalman gain matrix:
1)()1|()( −⋅⋅−= kRCkkPkK T . (3.23)
7) Compute the state estimation covariance matrix at instant k and k+1:
QAkkPAkkP
kkPCkKkkPkkP
T +⋅⋅=+
−⋅⋅−−=
)()|()()|1(
)1|()()1|()|(
θθ . (3.24)
8) Compute the state at instant k and k+1:
)()()|(ˆ)()|1(ˆ
)()()1|(ˆ)|(ˆ
kuBkkxAkkx
kekKkkxkkx
⋅+⋅=+
⋅+−=
θθ. (3.25)
9) Solve Eq. (3.18) for θ∆ and compute the new θ such that
maxmin θθθ ≤≤ new . (3.26)
10) Repeat steps (2) through (9) until )(θV is minimized.
The above mechanism for maximum likelihood estimation is illustrated in Figure 3.7.
44
Figure 3.7 Block diagram of maximum likelihood estimation
A model of the system is excited with the same input as the real system. The error
between the estimated output and the measured output is used to adjust the model
parameters to minimize the cost function )(θV . This process is repeated till the cost
function is minimized.
3.4.2 Performance of MLE with Noise-Corrupted Data
To test the performance of the above algorithm, some simulations have been done in
Matlab. The circuit shown in Figure 3.4 is used for this simulation. A step input voltage
is applied to the circuits, different noise is added to the output currents to get noise
“corrupted” data with different signal-to-noise ratio (S/N). Then MLE techniques are
45
used to estimate the parameters (R and L) with different initial guesses. The estimated
results are shown in the following tables.
The true value of R is 0.157 Ω and that of L is 0.290 mH.
In Table 3.1, the initial guesses are within 90% of true values. The relative estimation
errors are also shown in the table.
In Table 3.2, the initial guesses are within 10% of true values.
Some noise-corrupted signals with different signal-to-noise ratio are shown in Figure
3.8 below.
S/N initR (Ω) estR (Ω) Re (%)
initL (mH) estL (mH) Le (%)
∞ 0.9R 0.157000 0.0000 0.9L 0.290000 0.0000
2000:1 0.9R 0.157000 0.0001 0.9L 0.289993 -0.0023
1000:1 0.9R 0.156998 -0.0013 0.9L 0.290033 0.0115
500:1 0.9R 0.156999 -0.0003 0.9L 0.289980 -0.0068
200:1 0.9R 0.156998 -0.0077 0.9L 0.290143 0.0492
100:1 0.9R 0.157014 0.0086 0.9L 0.289838 -0.0558
50:1 0.9R 0.156935 -0.0416 0.9L 0.290753 0.2595
20:1 0.9R 0.156984 -0.0101 0.9L 0.290242 0.0833
10:1 0.9R 0.156836 -0.1046 0.9L 0.293724 1.2842
5:1 0.9R 0.156972 -0.0175 0.9L 0.291014 0.3496
Table 3.1 Estimation results with initial guesses within 90% of true values
46
S/N initR (Ω) estR (Ω) Re (%) initL (mH) estL (mH) Le (%)
∞ 0.1R 0.157000 0.0000 0.1L 0.289995 -0.0015
2000:1 0.1R 0.157002 0.0015 0.1L 0.289956 -0.0151
1000:1 0.1R 0.156998 -0.0014 0.1L 0.290034 0.0118
500:1 0.1R 0.156997 -0.0016 0.1L 0.290059 0.0203
200:1 0.1R 0.156993 -0.0046 0.1L 0.290221 0.0763
100:1 0.1R 0.157049 0.0314 0.1L 0.289092 -0.3129
50:1 0.1R 0.156984 -0.0101 0.1L 0.290154 0.0531
20:1 0.1R 0.156971 -0.0185 0.1L 0.289974 -0.0090
10:1 0.1R 0.157013 0.0084 0.1L 0.289203 -0.2748
5:1 0.1R 0.157110 0.0701 0.1L 0.286158 -1.3248
Table 3.2 Estimation results with initial guesses within 10% of true values
Figure 3.8 Noise-corrupted signal with different signal-to-noise ratio
From the simulation results we can see that, the estimation algorithm can accurately
identify the system parameters from very ‘noisy’ data (S/N > 20:1), even with poor initial
guesses.
47
Normally the signal-to-noise ratio of voltage and current measurements is above
100:1~200:1. So MLE can assure the correct estimation of the system parameters.
3.5 Parameter Estimation Results From Standstill Tests
The motor used in this research is a 42V/50A 8/6 SRM. Tests are performed at
several specific positions for current between 0~50 ampere. Maximum likelihood
estimation is used to get the parameters of the model shown in Figure 3.4.
In Figure 3.9, the voltage and current waveforms used in standstill tests are shown.
The blue dotted curve shows the current calculated from the estimated parameters (R and
L). It matches the measurement very well.
Figure 3.9 Voltage and current waveforms in standstill tests
0 1 2 3 4 5
x 10-3
-1
0
1
2
3I (A)
estimatedactual
0 1 2 3 4 5
x 10-3
0
5
10
15
Voltage (V)
time (s)
48
The inductance estimation and curve-fitting results at aligned, midway, and unaligned
position are shown in Figure 3.10 - Figure 3.12.
The results show that the inductance at unaligned position doesn’t change much with
the phase current and can be treated as a constant. The inductances at midway and
aligned position decrease when current increases due to saturation.
0 5 10 15 20 25 30 0
1
2
3
4
5
6 x 10
-3 inductance at theta = 0 deg test data curve-fitting
(A)
(H)
Figure 3.10 Standstill test results for inductance at θ =0o
49
0 5 10 15 20 25 30 0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8 x 10
-3 inductance vs current at theta = 15 deg test data curve-fitting
(A)
(H)
Figure 3.11 Standstill test results for inductance at θ =15o
0 5 10 15 20 25 30 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 x 10
-3 inductance vs currentat theta = 30 deg test data curve-fitting
(A)
(H)
Figure 3.12 Standstill test results for inductance at θ =30o
50
A 3-D plot of inductance shown in Figure 3.13 depicts the profile of inductance
versus rotor position and phase current. At theta = 0 and 60 degrees, phase A is at its
aligned positions and has the highest value of inductance. It decreases when the phase
current increases. At theta = 30 degrees, phase A is at its unaligned position and has
lowest value of inductance. The inductance here keeps nearly constant when the phase
current changes.
Figure 3.13 Standstill test result: nonlinear phase inductance
In Figure 3.14, the flux linkage versus rotor position and phase current based on the
estimated inductance model is shown. The saturation of phase winding at high currents is
clearly represented. At aligned position, the winding is highly saturated at rated current.
51
Figure 3.14 Flux linkage at different currents and different rotor positions
3.6 Verification of Standstill Test Results
The results obtained from standstill test can be verified by comparing them with the
inductances calculated from the physical dimension of the switched reluctance machine,
or from finite element analysis results.
A cross-section of the 8/6 SRM used in this research is shown in Figure 3.15. Phase A
is now at its aligned position, and phase C is at its unaligned position.
When phase A at aligned position is excited, the flux generated by the two phase-
windings will mainly cross the two airgaps between the two pairs of stator poles and rotor
poles, and the stator and rotor iron (as shown in Figure 3.16).
52
Figure 3.15 Cross-sectional drawing of 8/6 SRM
Figure 3.16 Flux path at aligned position
Ignoring the contribution from the machine iron and the fringing in the airgap, the
two identical inductances can be computed as
g
ANLL c0
2
21
µ== (3.27)
where
53
N = 10 is the number of turns of the phase winding,
µ0 = 4π x 10-7 is the permeability of air,
Ac = 6.68 x 45 x 10-6 is cross-sectional area, and
g = (34.95-34.925)/2 x 10-3 is the airgap length.
The phase inductance aL is the equivalent of 1L and 2L in series. So
HLLLa
3
3
672
21 1004.6100125.0
104568.6104102 −
−
−−
×=×
××××××=+=
π (3.28)
According to the standstill test result, the inductance at aligned position at low current
(non-saturated) is about 5.942 x 10-3 H, which is very close to the calculation result.
When phase C at unaligned position is excited, the flux generated by the two phase-
windings will NOT cross the two major airgap between the stator poles and rotor yoke,
because of the high reluctance of the airgap. Instead, the flux will cross the shorter airgap
between the stator pole and two adjacent rotor poles, as shown in Figure 3.17.
Under such cases, it is very hard to estimate the equivalent airgap length. Finite
element analysis (FEA) is needed to get a good estimate of the inductance here. The
inductance at unaligned position got from FEA is about 4.2 x 10-4 H, which is close to the
standstill test result 4.9 x 10-4 H, as shown in Figure 3.12.
54
Figure 3.17 Flux path at unaligned position
3.7 SRM Model for Online Operation
For online operation case, especially under high load, the losses become significant.
There are no windings on the rotor of SRM. But similar as synchronous machines, there
will be circulating currents flowing in the rotor body and makes it work as a damper
winding. Considering this, the model structure may be modified as shown in Figure 3.18,
with dR and dL added to represent the losses on the rotor.
Figure 3.18 Model structure of SRM under saturation
55
The phase voltage equations can be written as:
Vi
i
RRR
R
i
i
L
LL
d
d
d
d
+
⋅
−−−=
⋅
−
1
00
0 2
1
2
1
&
&
, (3.29)
where i1 and i2 are the magnetizing current and damper current.
It can be re-written in state space form as:
DuCxy
BuAxx
+=
+=&, (3.30)
where
][ 21 iix = , ][ 21 iiy += , ][Vu = ,
−−−⋅
−=
−
d
d
d
d
RRR
R
L
LLA
0
0
1
,
⋅
−=
−
1
0
0
1
d
d
L
LLB ,
]11[=C , and
0=D .
The torque can be computed as follows (notice that L is the magnetizing winding):
])([)sin(
])cos()([
]),([),(
1
111
11
0
1
1111
∑ ∫
∫∑
∫
=
=
−=
∂
∂=
∂
∂=
∂
∂=
m
k
krr
m
k
rk
c
diiiCkNkN
diikNiC
diiiLiWT
θ
θ
θ
θ
θ
θθ
. (3.31)
56
During on-site operation, we can easily measure phase voltage V and phase current
21 iii += . But the magnetizing current ( 1i ) and the damper winding current ( 2i ) are not
measurable. Let’s assume that the phase parameters R and L obtained from standstill test
data are accurate enough for light load case. And we want to attribute all the errors at
high load case to damper parameters. If we can estimate the exciting current 1i during
online operation, then it will be very easy to estimate the damper parameters. A two-layer
recurrent neural network is formed and trained here for such purpose.
3.8 Neural Network Based Damper Winding Parameter Estimation
During online operation, there will be motional back EMF in the phase winding. So
the exciting current 1i will be affected by:
− Phase voltage V,
− Phase current i,
− Rotor position θ, and
− Rotor speed ω.
To map the relation ship between 1i and V, i, θ, ω, different neural network structures
(feed forward or recurrent), with different number of layers, different number of neurons
in each layer, and different transfer functions for each neuron, are tried. Finally the one
shown in Figure 3.19 is used. It is a two-layer recurrent neural network. The feeding-back
of the output i to input makes it better in fitting and faster in convergence.
57
The first layer is the input layer. The inputs of the network are V, i, θ, ω (with
possible delays). One of the outputs, the current i, is also fed back to the input layer to
form a recurrent neural network.
The second layer is the output layer. The outputs are phase current i (used as training
objective) and magnetizing current 1i .
Figure 3.19 Recurrent neural network structure for estimation of exciting current
A hyperbolic tangent sigmoid transfer function – “tansig()” is chosen to be the
activation function of the input layer, which gives the following relationship between its
inputs and outputs:
58
112,1
4
1
,11 byLWpIWni
ii +⋅+⋅=∑=
, (3.32)
11
2)(
1211 −+
==− ne
ntansiga . (3.33)
A pure linear function is chosen to be the activation of the output layers, which gives:
211,22 baLWn +⋅= , (3.34)
2221 )( nnpurelinay === ; (3.35)
311,33 baLWn +⋅= , (3.36)
3332 )( nnpurelinay === . (3.37)
After the neural network is trained with simulation data (using parameters obtained
from standstill test). It can be used to estimate exciting current during on-line operation.
When 1i is estimated, the damper current can be computed as
12 iii −= , (3.38)
and the damper voltage can be computed as
RiVV ⋅−=2 . (3.39)
The damper resistance dR and inductance dL can then be identified using maximum
likelihood estimation.
59
The data used for training is generated from simulation of SRM model obtained from
standstill test. The model is simulated at different DC voltages, different reference
currents, and different speed. The total size of the sample data is 13,351,800 data points.
The training procedure is detailed as follows:
First, from standstill test result, we can estimate the winding parameters (R and L) and
guess initial damper parameters ( dR and dL ). The guesses of dR and dL may not be
accurate enough for online model. It will be improved later through iteration.
Second, build an SRM model with above parameters and simulate the motor with
hysteresis current control and speed control. The operating data under different reference
currents and different rotor speeds are collected and sent to neural network for training.
Third, when training is done, use the trained ANN model to estimate the magnetizing
current ( 1i ) from online operating data. Compute damper voltage and current according to
equations (3.38) and (3.39). And then estimate dR and dL from the computed 2V and 2i
using output error estimation. This dR and dL can be treated as improved values of
standstill test results.
Repeat above procedures until dR and dL are accurate enough to represent online
operation (it means that the simulation data matches the measurements well).
60
In our research, the neural network can map the exciting current from and V, i, θ, ω
very well after training of 200 epochs.
3.9 Model Validation
To test the validity of the parameters obtained from above test, a simple on-line test
has been performed. In this test, the motor is accelerated with a fixed reference current of
20 ampere. All the operating data such as phase voltages, currents, rotor position, and
rotor speed are measured. Then the phase voltages are fed to an SRM model running in
Simulink, which has the same rotor position and speed as the real motor. All the phase
currents are estimated from the Simulink mode and compared with the measured currents.
The results are shown in Figure 3.20 and Figure 3.21.
0.05 0.1 0.15-20
-10
0
10
20
Current Response in Phase A - cov = 1.1962
time (s)
measured current (A)estimated current (A)measured voltage (V)
0.05 0.1 0.15-20
-10
0
10
20
Current Response in Phase B - cov = 0.7887
time (s)
measured current (A)estimated current (A)measured voltage (V)
0.05 0.1 0.15-20
-10
0
10
20
Current Response in Phase C - cov = 0.5873
time (s)
measured current (A)estimated current (A)measured voltage (V)
0.05 0.1 0.15-20
-10
0
10
20
Current Response in Phase D - cov = 0.6806
time (s)
measured current (A)estimated current (A)measured voltage (V)
Figure 3.20 Validation of model with on-line operating data
61
In Figure 3.20, the phase current responses are shown. The dashed curve is the
voltage applied to phase winding; the solid curve is the measured current; and the dotted
curve is the estimated current. An enlarged view of the curves for phase A is shown in
Figure 3.21. It is clear that the estimation approximates to the measurement quite well.
Figure 3.21 Validation of model with on-line operating data (Phase A)
To compare online model with standstill one, we compute the covariance of the errors
between the estimated phase currents and the measured currents. The average covariance
for standstill model is 0.9127, while that for online model is 0.6885. It means that the
online model gives much better estimation of operating phase currents.
62
3.10 Modeling and Parameter Identification Conclusions
An inductance-based model has been selected for the switched reluctance motor used
in this research. To represent the losses at heavy load condition, a damper winding is
added into the model structure. Model validation results prove that the addition of the
damper winding can represent the switched reluctance machine at on-line operation better
than the simple R-L model.
Noise, which is inherently presented in the field test data, has significant impact on
the machine parameters estimated from frequency-domain based techniques. Maximum
likelihood estimation (MLE) based techniques can identify the model parameters from
very noisy test data. They are proposed in this chapter to identify the SRM parameters
from standstill test and operating data.
For the improved SRM model, the voltage and current for the damper winding are not
available from measurements. A two-layer recurrent neural network is proposed here to
estimate the damper winding current, so the parameters of the damper winding can be
identified.
Model validation by comparing the actual and estimated phase current under the same
voltage pulses proves that the proposed model and the identified parameters can represent
the dynamics of the SRM quite well.
63
CHAPTER 4
4 FOUR QUADRANT TORQUE CONTROL AND TORQUE-RIPPLE
MINIMIZATION
To get desired clamping force response, the servomotor in electro-mechanical brake
system needs to be operated in all four quadrants of the torque-speed plane. At the same
time, the ripple in the torque generated by the motor must be minimized to reduce
acoustic noise and mechanical vibration.
4.1 Four Quadrant Operation of Switched Reluctance Machines
In this context the four quadrants mean the four sections on the torque-speed plane.
As shown in Figure 4.1, the torque (T) – speed (ω) plane is divided into four quadrants.
In the 1st quadrant, both torque and speed are positive. This corresponds to the forward
motoring region of the motor. In the 2nd quadrant, speed is positive while torque is
negative. This corresponds to the forward braking (or, generating) region. Accordingly,
the 3rd quadrant corresponds to the backward motoring region; and the 4
th quadrant
corresponds to the backward braking (generating) region of the motor.
64
Figure 4.1 Four quadrants on torque-speed plane
For switched reluctance motor (SRM) used in this research, the rotating direction (or
the sign of the speed ω) is determined by the exciting sequence. For example, using the
phase definition in Figure 4.2, if the exciting sequence is phase A-B-C-D-A-…, then the
rotor will rotate in counter clockwise (CCW) direction; and if the exciting sequence is
phase C-B-A-D-C-…, the rotor will rotate in clockwise (CW) direction.
Figure 4.2 Phase definition of an 8/6 switched reluctance motor
T
ω
I
ω > 0 T > 0
II
ω > 0 T < 0
III
ω < 0 T < 0
IV
ω < 0 T > 0
stator rotor
65
The working mode of the SRM (motoring or generating, or the sign of the torque T )
is determined by the conduction angle for each phase. For the 8/6 SRM used in this
application, the phase inductance has a profile shown in Figure 4.3. According to the
assumed forward direction, the phase inductance increases in region [–30o, 0
o], which
corresponds to forward motoring (quadrant I) operation; and the phase inductance
decreases in region [0o, 30
o], which corresponds to forward generating/braking (quadrant
II) operation. Similarly, the backward motoring (quadrant III) operation happens in
region [30o, 0
o], and backward generating/braking (quadrant IV) operation happens in
region [0o, -30
o].
Figure 4.3 Phase inductance profile and conduction angles for 4-quadrant operation
So for four-quadrant operation of the SRM, we just need to set the corresponding
exciting sequence and conduction angles for each phase according to the desired
30o
Forward direction
Lu
-30o
L(θ, i)
La
Forward motoring
Backward braking
Backward motoring
Forward braking
0o
66
operating quadrant. There’s no need to consider the polarity of the voltage applied to
each phase (or in other words, the direction of the current in each phase winding).
4.2 Torque Control and Torque Ripple Minimization
At any time, the resultant output torque of SRM is the summation of the torque in all
phases:
∑=
=N
j
jTT1
, (4.1)
where N is the number of phases, and jT is the torque generated in phase j, which is a
nonlinear function of phase current i and rotor position θ. If the phase current i is fixed,
the torque (of an 8/6 SRM) will have a profile as shown in Figure 4.4.
-30 -20 -10 0 10 20 30-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Theta (o)
Torque (N)
Figure 4.4 Phase torque profile under fixed current
67
It is clear that high torque is not available near aligned/unaligned position even when
high phase current is presented. To generate a ripple-free output torque, there must be
overlapping between phases.
During phase overlapping, the current in one phase is decreasing, and that in the other
phase is increasing. To obtain a constant torque, the summation of the torque generated
by these currents must equal to the torque generated in non-overlapping period. To
determine the desired torque produced by each phase, torque-factors are introduced here,
which are defined as
∑∑==
==N
j
refj
N
j
j TfTT11
)(θ , (4.2)
where )(θjf is the torque factor for phase j at rotor position θ, and Tref is the expected
output torque.
The motor used in this research is an 8/6switched reluctance motor. To generate
desired torque (T = Tref), the torque factor must meet the following requirements (N = 4):
• 1)(4
1
=∑=j
jf θ , (4.3)
• )3/()( πθθ += jj ff , (4.4)
• )12/)(()( πθθ ×−−= kjff kj . (4.5)
68
There are many different choices that can meet the above requirements. Appropriately
chosen torque factors will facilitate the current control and produce satisfactory torque
output. In our simulation, the following torque factors are adopted.
For 8/6 SRM, the inductance increasing/decreasing period for each phase is π/6. We
choose a conduction angle of π/8. This means,
8/πθθ =− onoff . (4.6)
So the phase overlapping for each two adjacent phases is π /8 - π /12 = π /24. During
phase overlapping, the total torque is distributed in the two phases according to a sine
function of the rotor position. The torque factor for phase j is expressed as follows,
<≤−−−+
−<≤+
+<≤−−
=
other
fjonjoffjoff
joffjon
jonjonjon
j
0
24/)24/(24cos5.05.0
24/24/1
24/)(24cos5.05.0
)(,,,
,,
,,,
θθπθπθθπθθπθ
πθθθθθ
θ . (4.7)
The turn-on/turn-off angles ( jon,θ and joff ,θ ) for phase A in different operating
quadrants are chosen as follows,
Quadrant I Quadrant II Quadrant III Quadrant IV
Aon,θ -30o 5
o 7.5
o -27.5
o
Aoff ,θ -7.5 o 27.5
o 30
o -5
o
Table 4.1 Turn-on/turn-off angles for phase A
69
The turn-on/turn-off angles for the other phases can be obtained by phase shifting of a
multiple of π /12.
The torque factors for all four phases in forward motoring operation are shown in
Figure 4.5 below.
-30 -20 -10 0 10 20 300
0.2
0.4
0.6
0.8
1
1.2
theta
Torque Factor
A B C D
Figure 4.5 Torque factors for forward motoring operation
In the above figure, the thick dotted line is the summation of all the four torque
factors. It’s obvious that it is 1 at any rotor position.
70
Once the torque factors are chosen, the reference current for different phases at
specified rotor position can be determined through the following steps.
1) Compute the torque factors for all phases according to current rotor position and
turn-on/turn-off angles (Eq. 4.7);
2) Compute the desired torque for each phase by multiplying the Tref with the
corresponding torque factor;
3) Assume 01 =ji , max2 ii j = (maximum phase current allowed), j=1,2,3,4;
4) Compute 2/)( 21 jjj iii += ;
5) Calculate jT generated by ji according to Eq. (3.14);
6) Compare jT with the desired torque for phase j calculated in step 2).
If the difference between the computed jT and the designed torque is small
enough, let jjref ii =, and stop.
Else if jT is less than desired torque, let jj ii =1 , and go to step 4.
Else if jT is greater than desired torque, let jj ii =2 , and go to step 4.
Iterate the steps until satisfactory results are obtained.
In DSP implementation of this algorithm, table-lookup and interpolation can be used
to avoid the massive computation needed.
71
4.3 Hysteresis Current Control
From previous section, the reference current for each reference jrefi , is determined by
the desired torque and current rotor position. Then hysteresis current control technique
will be applied to maintain the phase current within an acceptable range around the
reference current. The principle of hysteresis current control is shown in Figure 4.6.
Figure 4.6 Voltage and current waveform in hysteresis current control
As shown in Figure 4.6, when the reference current jrefi , is given, a lower limit and
an upper limit for the phase current can be obtained, according to the control
requirements and the switching frequency of the power converter. Then the power
switches are turned on and off by comparing the actual phase current with the two limits.
jrefi ,
72
4.4 Simulation Results
Based on the above torque control technique, a Simulink© model is built to test the
performance of the algorithm. Here we use a torque ripple coefficient to measure the
ripple in the torque. It is defined as follows,
[ ]%100
)(1 0
0
2
×−
=∫
+
ave
t
tave
T
dtTtT
TR
τ
τ, (4.8)
where
∫+
=τ
τ0
0
)(1 t
tave dttTT , (4.9)
and 0t is the time when a phase starts to conduct, τ+0t is the time when the next phase
starts to conduct. For the 8/6 SRM used in this research, we have
12/)()( 00 πθτθ =−+ tt .
The simulation results for four-quadrant torque control and torque ripple
minimization are shown in Figure 4.7. The upper curves show the torque command (solid
red) and actual torque generated by the switched reluctance motor (dashed blue). The
lower curve shows the rotor speed. It’s easy to find that the motor operated in all four
quadrants. The torque ripple in all operating region is less than 4%, which is very
satisfactory.
73
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.2
0
0.2
0.4
0.6
0.8
Torque vs Time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-200
-100
0
100
200
Speed vs Time
Time(s)
actual
desired
Figure 4.7 Four-quadrant torque control and torque-ripple minimization
4.5 Torque Control Conclusions
Switched reluctance machines have a significant torque ripple, especially when
operated in single-pulse voltage control mode. This is not desired for electro-mechanical
brake systems. To minimize the torque ripple, phase overlapping is necessary and the
determination of torque generation in overlapping phases is very important in torque
ripple minimization. To get a ripple-free torque, the summation of the torque generated in
all active phases must always be the same. In this research, a torque factor is defined to
determine the torque required in each phase according to the rotor position and the
specified turn-on/turn-off angles. Then the torque command is converted into current
command according to the inductance based SRM model. Finally voltage-chopping
74
technique is used to generate voltage pulses for each phase to maintain the phase current
around the desired value.
Since four-quadrant operation of the SRM is necessary for EMB system, the above
torque control algorithms need to be implemented in all four quadrants, with different
turn-on/turn-off angles and exciting sequence for different quadrants. Simulations show
satisfactory results of the proposed algorithms.
75
CHAPTER 5
5 DESIGN AND IMPLEMENTATION OF A ROBUST CLAMPING
FORCE CONTROLLER
The control objective of the electro-mechanical brake (EMB) system is to provide
desired clamping force response at the brake pads and disk. It is realized by controlling
the torque, speed, and rotor position of the servomotor in the EMB system. In this
research, an 8/6 switched reluctance motor (SRM) is used as the servomotor.
In this chapter, a robust clamping force controller will be designed and implemented.
5.1 Clamping Force Control System
The electro-mechanical brake system developed in this research contains the
following components:
- An 8/6 switched reluctance motor, with corresponding power converter and
sensors
76
- A dual-stage planetary gear and a ball screw, which allows for conversion of the
rotary motion of the SR motor into linear ball screw action in order to create the
required clamping force on the brake disk
- A clamping force sensor located in the force path between the ball screw and the
caliper housing measures the clamping force
- A controller, which receives brake commands and clamping force feedback, and
generates control output – the switching signal for power converter to provide
voltage pulses to the SRM
A block diagram of the clamping force control system is shown in Figure 5.1.
Figure 5.1 Block diagram of clamping force control system
Generally, the clamping force F applied at the brake disk is a nonlinear function of
rotor position θ, and the load torque LT seen by the SR motor is a nonlinear function of
the clamping force F. The dependence of the load torque on the clamping force and the
Controller Power
converter SRM Gear and
ball screw
Brake
pads / disk
Sensors and
encoder
Clamping
force sensor
V, i, θ
V T G F
F
refF
77
dependence of the clamping force on the rotor position are governed by the mechanical
coupling between the motor and the brake disk. In general, F and LT are given by
)(θµFF = , (5.1)
)(θµLLT = , (5.2)
with Fµ and Lµ being nonlinear functions.
Physically, it is meaningful to assume that Fµ is monotonic and that the slope of the
function Fµ is positive and bounded, i.e.
max,min, FF
F σθµ
σ ≤∂
∂≤ . (5.3)
Furthermore, typically, LT is proportional to F with the proportionality constant being
essentially the gear ratio of the coupling between the motor shaft and the disk pads. For
generality, however, we only require that
LLL FT τσ +≤ (5.4)
with Lσ and Lτ being nonnegative constants.
The control objective is to make the clamping force F track a given reference
trajectory refF . Practically, it is reasonable to consider refF and its first derivative to be
bounded almost everywhere.
78
5.2 System Dynamics
The mechanical dynamics of the SR motor are given by
LTDTJ −−= ωθ&& , (5.5)
where
J is the rotational inertia of the motor,
D is the viscous friction coefficient,
θ is the rotor position,
ω is the rotor speed, ωθ =& ,
LT is the load torque, and
T is the electromagnetic torque generated by the SRM.
The total electromagnetic torque T is given by
∑=
=N
j
jj iTT1
),(θ , (5.6)
where N is the number of phases, jT is the torque contribution of the phase j, and ji is
the torque contribution of the phase j. The torque jT can be computed as
θ
θ
θθ
∂
∂
=∂
∂=
∫ji
jjjjc
jj
diiiLWiT
0,),(
),( , (5.7)
where ),( jiL θ is the inductance associated with phase j at rotor position θ under phase
current ji . The detailed inductance model of SRM is given in Chapter 3.
79
For design simplicity, we use the three-term Fourier series model here:
−−+
−−+=
)2
)1((2cos)(
)2
)1((cos)()(),(
2
10
r
rj
r
rjjj
NNjNiC
NNjNiCiCiL
πθ
πθθ
, (5.8)
where rN is the number of rotor poles.
To determine the three coefficients 0C , 1C , and 2C , we use the inductances at three
positions: )( ja iL (aligned position), uL (unaligned position), and )( jm iL (a midway
between the above two positions). Note that uL can be treated as a constant but )( ja iL
and )( jm iL are functions of the phase current ji and can be approximated by the
polynomials
∑
∑
=
=
=
=
p
n
n
jnjm
p
n
n
jnja
ibiL
iaiL
0
0
)(
)(
, (5.9)
where p is the order of the polynomials and na , nb are the coefficients.
Using Eq. (3-6), the three coefficients for the Fourier series can be computed as
++= )())((2
1
2
1)(0 jmujaj iLLiLiC ,
[ ]ujaj LiLiC −= )(2
1)(1 , (5.10)
−+= )())((2
1
2
1)(2 jmujaj iLLiLiC .
80
With the above definition, the torque expression given in Eq. (5.7) reduces to
))]2
)1((2sin())(2)((
))2
)1((sin())([(4
),(
****
**2
r
rjmuja
r
rujajr
jj
NNjNiLLiL
NNjNLiLi
NiT
πθ
πθθ
−−−++
−−−−= (5.11)
where
∑
∑
=
=
+=
+=
p
n
n
jnjm
p
n
n
jnja
ibn
iL
ian
iL
0
**
0
**
2
2)(
2
2)(
. (5.12)
The electrical dynamics of the switched reluctance motor are given by
( )
∂
∂+
∂
∂++⋅=
+⋅=+⋅=
dt
di
i
iLiLi
dt
diiLiR
dt
iiLdiR
dt
diRV
j
j
jj
j
j
jj
jj
j
j
jj
),(),(),(
),(
θω
θ
θθ
θλ
(5.13)
where jV is the voltage applied on phase j, and R is the resistance of phase winding.
Using “standard” from, Eq. (5.13) can be written as
∂
∂−⋅−
∂
∂+
= ωθ
θ
θθ
),(
),(),(
1 j
jjj
j
j
jj
j iLiiRV
i
iLiiL
dt
di. (5.14)
From physical considerations, it is meaningful to assume that the denominator in Eq.
(5.14) is positive within the range of operation of the SR motor, i.e.,
81
0),(
),( >>∂
∂+ ε
θθ
j
j
jji
iLiiL . (5.15)
Using the inductance model in Eq. (5.8), we have
))]2
)1((2sin())(2)((
))2
)1((sin())([(2
),(
r
rjmuja
r
rujarj
NNjNiLLiL
NNjNLiL
NiL
πθ
πθ
θ
θ
−−−++
−−−−=∂
∂
, (5.16)
and
)2
)1((2cos()())((2
1
2
1
))2
)1((cos())((2
1
)())((2
1
2
1),(),(
**
*
**
r
rjmuja
r
ruja
jmuja
j
j
jj
NNjNiLLiL
NNjNLiL
iLLiLi
iLiiL
πθ
πθ
θθ
−−
−++
−−−+
++=∂
∂+
,
(5.17)
where
∑
∑
=
=
+=
+=
p
n
n
jnjm
p
n
n
jnja
ibniL
ianiL
0
*
0
*
)1()(
)1()(
. (5.18)
Equations (5.5)-(5.18) summarize the dynamics of the SRM based clamping force
control system. A robust force controller will be designed according to these equations.
82
5.3 Controller Design
In this section, the clamping force controller for the switched reluctance motor based
brake system is designed using robust backstepping. The backstepping design proceeds
by considering lower-dimensional subsystems and designing virtual control inputs (or
equivalently, state transformations).
5.3.1 Backstepping Controller Design – Basic Idea
The virtual control inputs in the first and second steps are rotor speed ω and torque T,
respectively. In the third step, the actual control inputs jV , j = 1, …, N, appear and can be
designed.
In the first step of backstepping, the one-dimensional system
ωθµ∂
∂= FF& (5.19)
is considered and ω is regarded as the virtual control input. The objective is to make the
clamping force F track the designed force response refF .
However, since ω is not the actual control input, the error between ω and the desired
ω is formulated as
*
2 ωω −=z . (5.20)
83
In the second step of backstepping, the torque T will be regarded as the virtual control
input and will be designed to make 2z small, i.e., ω converges to the virtual control law
*ω designed for ω, and hence, F converges to refF .
Since T is also not the actual control input, the process is repeated by introducing an
error 3z which is the difference between T and the desired T
*
3 TTz −= . (5.21)
At the third step of backstepping, the control inputs jV , j = 1, …, N, appear and the
control law is designed at that step can actually be implemented.
5.3.2 Backstepping Controller Design - Step 1
For the first step, the force tracking error is defined as
refFFz −=1 . (5.22)
A Lyapunov function
2
112
1zV = (5.23)
is used here.
Differentiating Eq. (5.23), we obtain
84
refFF
refF
ref
Fzzzzk
Fz
FFzzzV
&
&
&&&&
121
2
11
1
1111 )(
−∂
∂+
∂
∂−=
−
∂
∂=
−==
θµ
θµ
ωθµ
, (5.24)
where
*
2 ωω −=z ,
11
* zk−=ω , (5.25)
and 01 >k is a design freedom.
Choosing )(111 refFFkzk −−=−=ω would result in the practical tracking of
clamping force F in the one-dimensional system described in Eq. (5.19).
Using Eq. (5.3), Eq. (5.24) reduces to
refFF FzzzzkV &&121max,
2
1min,11 ++−≤ σσ (5.26)
5.3.3 Backstepping Controller Design - Step 2
A new Lyapunov function is defined as
2
2
2
122
1z
cVV += (5.27)
where 02 >c is a design freedom. Differentiating Eq. (5.27),
85
[ ]
( )[ ] [ ] refFLrefL
refFF
refF
L
refFF
refF
L
refFF
Fzc
kzkzz
c
kFzz
Jc
zkzzJc
Dzz
Jcz
Jc
k
Fzzzzk
Fzc
kz
c
k
TzJc
zJc
Dzz
Jcz
Jc
k
Fzzzzk
FkkTDTJ
zc
FzzzzkV
&
&
&
&
&
&&
2
2
11122max,
2
112
2
1122
2
32
2
2
2
2
2
121max,
2
1min,1
2
2
12
2
1
2
2
2
2
32
2
2
2
2
2
121max,
2
1min,1
112
2
121max,
2
1min,12
1
1
11
)(11
++++++
+++−
++−≤
−∂
∂+
−−+−
++−≤
−
∂
∂+−−+
++−≤
στσ
σσ
ωθµ
ω
σσ
ωθµ
ω
σσ
(5.28)
where
*
3 TTz −= ,
)]([ 1222
*
refFFkkzkT −+−=−= ω , (5.29)
with 02 >k being a controller gain to be picked by the designer.
5.3.4 Backstepping Controller Design - Step 3
For the third and final step of the backstepping, we use the Lyapunov function
86
2
3
3
232
1z
cVV += (5.30)
where 03 >c is a design freedom.
To differentiate Eq. (5.30), the dynamics of the torque will be required. The torque
dynamics can be derived from Eq. (5.6) and (5.11) as
∑=
=N
j
jj iTT1
),(θ&& , (5.31)
∂
∂+
∂
∂−⋅−
∂
∂+
∂
∂=
∂
∂+
∂
∂=
j
j
jj
j
jjj
j
jj
j
j
jj
jj
i
iLiiL
iLiiRV
i
TT
dt
di
i
TTiT
),(),(
),(
),(
θθ
ωθ
θ
ωθ
ωθ
θ&
, (5.32)
where
))]2
)1((2cos())(2)((2
))2
)1((cos())([(4
****
**22
r
rjmuja
r
rujajrj
NNjNiLLiL
NNjNLiLi
NT
πθ
πθ
θ
−−−++
−−−−=∂
∂
, (5.33)
))]2
)1((2sin())(
2)(
(
))2
)1((sin()(
[4
))]2
)1((2sin())(2)((
))2
)1((sin())([(2
****
**
2
****
**
r
r
j
jm
j
ja
r
r
j
ja
jr
r
rjmuja
r
rujajr
j
j
NNjN
i
iL
i
iL
NNjN
i
iLi
N
NNjNiLLiL
NNjNLiLi
N
i
T
πθ
πθ
πθ
πθ
−−∂
∂−
∂
∂+
−−∂
∂−
−−−++
−−−−=∂
∂
. (5.34)
87
Differentiating Eq. (5.30) and using Eq. (5.28), we have
[ ]
( )[ ]
[ ]
−
∂
∂+
−−++
+++
+++
+++−
++−≤
refF
L
refF
LrefL
refFF
FkkTDTJ
kTzc
Fzc
kzkzz
c
k
FzzJc
zkzzJc
Dzz
Jcz
Jc
k
FzzzzkV
&
&
&&
1123
3
2
2
11122max,
2
1
12
2
1122
2
32
2
2
2
2
2
121max,
2
1min,13
)(1ˆ1
1
1
ωθµ
ω
σ
τσ
σσ
(5.35)
where, for notational convenience, we have introduced TT &=ˆ . The control law will first
be designed in terms of T . The control laws in terms of the input voltages jV will then
be obtained through a commutation scheme.
Designing
)]([ˆ122333 refFFkkkTkzkT −++−=−= ω (5.36)
with 03 >k being a design freedom, Eq. (5.35) reduces to
[ ]
( )[ ]
[ ]
[ ]2233
3
22
3
3
3
2
2
11122max,
2
1
12
2
1122
2
32
2
2
2
2
2
121max,
2
1min,13
1
1
zkzzJc
kz
c
k
Fzc
kzkzz
c
k
FzzJc
zkzzJc
Dzz
Jcz
Jc
k
FzzzzkV
refF
LrefL
refFF
++−
+++
+++
+++−
++−≤
&
&&
σ
τσ
σσ
88
[ ]
[ ]
[ ] refF
LrefL
Fzc
kkzkzz
c
kk
FzzJc
k
zkzzJc
Dk
&3
3
211123max,
3
21
13
3
2
1123
3
2
)(
+++
+++
++
σ
τσ
(5.37)
It can be rearranged as
refL
refL
refL
refL
ref
F
F
FL
FLF
F
Fzc
kkz
Jc
kFz
Jc
k
Fzc
kz
JcFz
JcFz
Jc
kz
c
k
Jc
Dz
c
kk
Jc
Dk
Jc
k
Jczz
c
kk
Jc
k
Jc
Dkkzz
c
k
JcJc
Dkzz
zc
kz
Jc
kzk
V
&
&&
&
3
3
213
3
23
3
2
2
2
12
2
2
2
1
3
22
3
2
max,1
2
2
2
3
max,21
3
2
3
2
2
2
32
3
max,2
2
1
3
2
3
2131
2
max,
2
1
22
1max,21
2
3
3
32
2
2
22
1min,13
1
+++
++++
+
++
++++
+++
++++
−−−≤
τσ
τσ
σ
σ
σσ
σσσ
σ
(5.38)
From Eq. (5.38), it is seen that by picking 1k , 2k , 3k , 2c , and 3c appropriately, 3V
satisfies
[ ]222
33 Lrefref FFVV τχγ +++−≤ && (5.39)
89
where
= 3
2max,1 ,,min k
J
kk Fσγ (5.40)
and χ is a constant independent of the controller gains 1k , 2k , and 3k . The proof of Eq.
(5.39) follows by quadratic over-bounding which is standard in the backstepping
literature [50].
From Eq. (5.39), it follows that practical tracking is achieved, i.e., the force tracking
error refFFz −=1 can be regulated to an arbitrarily small compact set by picking 1k , 2k ,
and 3k large enough.
To obtain control laws in terms of the input voltages jV , j = 1, …, N, two alternative
methods can be used.
Method one: torque control and torque-ripple minimization
Since the desired torque *T can be obtained from Eq. (5.36), the torque control
algorithm described in Chapter 4 can be applied here to determine the reference current
jrefi , for each phase. Then hysteresis current control will be used to maintain the phase
current around the reference. This is the most commonly used technique in controller
implementation.
90
Method two: voltage commutation and PWM
The following commutation scheme is used to get the control laws in terms of phase
voltages:
ωθ
θθθ
∂
∂+⋅+
∂
∂+=
),(~),(),(
j
jjj
j
j
jjj
iLiiRV
i
iLiiLV , (5.41)
∑
∑
=
=
∂
∂
∂
∂−
∂
∂
=N
j j
j
N
j
j
j
j
j
i
T
TT
i
T
V
1
2
1
ˆ
~ω
θ . (5.42)
From Eq. (5.42), we have
∑∑== ∂
∂−=
∂
∂ N
j
jN
j j
j
j
TT
i
TV
11
ˆ~ω
θ, (5.43)
which satisfies Eq. (5.32), and jV~ approximates
dt
di j.
The input voltages obtained from Eq. (5.41-5.42) are continuous voltages. In SRM
control, the phase voltage can only be switched among DCV± and 0. So PWM techniques
need to be applied here to approximate the desired phase voltage waveforms.
In controller implementation, method one will be used, which provides desired torque
with minimal torque ripple.
91
5.3.5 Backstepping Controller Design – Generalized Control Law
The overall controller is given by
refFFz −=1 , (5.22)
11
* zk−=ω , (5.25)
*
2 ωω −=z , (5.20)
)]([ 1222
*
refFFkkzkT −+−=−= ω , (5.29)
*
3 TTz −= , (5.21)
)]([ˆ122333 refFFkkkTkzkT −++−=−= ω , (5.36)
and
ωθ
θθθ
∂
∂+⋅+
∂
∂+=
),(~),(),(
j
jjj
j
j
jjj
iLiiRV
i
iLiiLV , (5.41)
∑
∑
=
=
∂
∂
∂
∂−
∂
∂
=N
j j
j
N
j
j
j
j
j
i
T
TT
i
T
V
1
2
1
ˆ
~ω
θ. (5.42)
In implementation, Eq. (5.41) and (5.42) can be replaced by torque control algorithms
given in Chapter 4.
In above control laws, the controller design freedoms are 1k , 2k , 3k , 2c , and 3c .
Noting the structure of the controller, the control law can be slightly generalized to
92
provide additional damping in the electrical dynamics and to provide more design
freedom in shaping the force dynamics. The generalized control law is given by
( )ωωKTK
dttFtFKFFKFFKT
T
t
refirefdrefp
−−
−−−−−−= ∫0 111 )()()()(ˆ && (5.44)
∑
∑
=
=
+
∂
∂
∂
∂−
∂
∂
=N
j
T
j
j
N
j
j
j
j
j
i
T
TT
i
T
V
1
2
1
ˆ
~
ε
ωθ
(5.45)
jcur
j
jj
j
j
jjj iKiL
iVi
iLiiLV ⋅−
∂
∂+
∂
∂+= ω
θ
θθθ
),(~),(),( (5.46)
with pK , dK , iK , TK , ωK , and curK being controller gains free to be picked by the
designer. The term Tε is added to avoid the singularity of the commutation scheme when
all currents are zero. Eq. (5.45) and (5.46) can be replaced by the torque control
algorithm given in Chapter 4.
Note that the controller design does not require the knowledge of the mechanical
parameters D and J of the motor. Also, the functional form of )(θµFF = and
)(θµLLT = are not required to be known. Magnitude bounds on D, J, Fµ , and Lµ are
sufficient for picking the controller gains. While the controller does require knowledge of
the electrical parameters of the motor, the design does provide considerable robustness to
uncertainty in the electrical parameters also. Simulation results are given in the following
section.
93
5.4 Simulation Results
The performance of the controller is verified through simulation studies in Matlab/
Simulink® environment. The Simulink model of the electro-mechanical brake system is
shown in Figure 5.2. The parameters of the switched reluctance motor and the brake
system are given in Appendix A.
Figure 5.2 Simulink model of electro-mechanical brake system
The force reference trajectory is specified to be 2500 N till the actual clamping force
reaches 2000 N within 0.1 s, and 1600 N after that. The phase currents are constrained to
be maintained in the Range [0, 50] A. The parameters of the controller were chosen to be
20=pK , 002.0=dK ,
2=iK , 3500=TK ,
85=ωK , 1=curK . (5.47)
94
As mentioned in previous section, there are two methods to get the control law in
terms of voltages after we get the torque reference. Both methods are simulated and the
results are shown in the following sections.
5.4.1 Controller with Voltage Commutation Scheme and PWM Implementation
Using the voltage commutation scheme described in Eq. (5.45-5.46), the results are
shown in Figure 5.3 through Figure 5.6.
In Figure 5.3, the reference clamping force (dotted red curve) and the actual clamping
force response (solid blue curve) are shown. The force reference is set to 2500 N initially.
At 0.06 second, the actual clamping force reaches 2000 N and the force reference drops
to 1600 N. At 0.1 second, the clamping force reaches steady state and stays at 1600 N.
0 0.05 0.1 0.15 0.2 0.250
500
1000
1500
2000
2500
Time (sec)
Force (N)
Clamping Force Response
Actual Force
Force Reference
Figure 5.3 Clamping force response
95
The phase current waveforms are shown in Figure 5.4. At steady state, the speed of
the switched reluctance motor is nearly zero. Only phase A and D are conducting and
they share the desired load torque. The currents in phase B and C are zero.
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i1 (A)
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)i2 (A)
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i3 (A)
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i4 (A)
Phase Currents
Figure 5.4 Phase current waveforms
In Figure 5.5, the rotor position, speed, and torque during the control process are
shown. At first, the switched reluctance motor is accelerated to 1500 RPM in a very short
time. When the force reference drops at 0.06 second, the SRM is decelerated and
reserved rotation direction very quickly to meet the decreased clamping force
requirements. Four-quadrant operation of the SRM is necessary here to provide good
clamping force response. The four-quadrant torque speed curve is shown in Figure 5.6.
96
0 0.05 0.1 0.15 0.2 0.250
500
1000
Theta (o)
0 0.05 0.1 0.15 0.2 0.25-1000
0
1000
2000
Speed (RPM)
0 0.05 0.1 0.15 0.2 0.25-5
0
5
time (sec)
Torque (Nm)
Rotor Position, Speed, and Torque
Figure 5.5 Rotor position, speed and torque
-3 -2 -1 0 1 2 3 4-1000
-500
0
500
1000
1500
2000
Speed (RPM)
Torque (Nm)
Torque-Speed curve
Figure 5.6 Four-quadrant torque speed curve
97
5.4.2 Controller with Torque Control and Torque-Ripple Minimization
Using the torque control and torque-ripple minimization algorithm developed in
Chapter 4, the results are shown in Figure 5.7 and Figure 5.8.
In Figure 5.7, the clamping force response under the new controller is shown.
Comparing it with Figure 5.3, we can see that almost identical clamping force responses
are obtained.
0 0.05 0.1 0.15 0.2 0.250
500
1000
1500
2000
2500
Time (sec)
Force (N)
Clamping Force Response
Actual Force
Force Reference
Figure 5.7 Clamping force response
In Figure 5.8, the four-quadrant torque speed curve with new controller is shown. It is
obvious that the torque-ripple in this case is much smaller than that in the previous
controller implementation.
98
-4 -3 -2 -1 0 1 2 3 4-1000
-500
0
500
1000
1500
Torque (Nm)
Speed (RPM)
Torque-Speed curve
Figure 5.8 Four-quadrant torque speed curve
5.4.3 Controller Robustness Test
To demonstrate the robustness of the controller, we consider the following case:
− Only the constant terms in the inductance expression are available for use in
the controller
− Errors in the parameters of resistance R and unaligned inductance uL up to
50%
− Furthermore, the load is changed to include a cascaded first-order linear block
with gain 1.1 and time constant 2 ms.
99
The simulation results are shown in Figure 5.9 through Figure 5.11.
0 0.05 0.1 0.15 0.2 0.250
500
1000
1500
2000
2500
Time (sec)
Force (N)
Clamping Force Response
Actual Force
Force Reference
Figure 5.9 Clamping force response
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i1 (A)
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i2 (A)
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i3 (A)
0 0.05 0.1 0.15 0.2
0
20
40
60
time (sec)
i4 (A)
Phase Currents
Figure 5.10 Phase current waveforms
100
-3 -2 -1 0 1 2 3 4-1000
-500
0
500
1000
1500
Torque (Nm)
Speed (RPM)
Clamping Force Response
Figure 5.11 Four-quadrant torque speed curve
It is seen that the performance of the force controller is retained in spite of the errors
in electrical parameters and the change of load. Though the switched reluctance has a
different trajectory in torque speed plane (as shown in Figure 5.11) , the clamping force
response is almost identical to previous cases.
5.5 Controller Design Conclusions
A robust nonlinear clamping force controller for an SRM based electro-mechanical
brake system has been design and simulated in this chapter. The controller is designed via
a backstepping procedure. It does not require knowledge of the mechanical parameters of
the motor and the functional forms of the relationships among the motor position,
101
clamping force, and the motor load torque. Moreover, the controller provides significant
robustness to uncertainty in the SRM parameters. The proposed controller design can also
be extended to electro-mechanical brake systems using other type of electric machines.
102
CHAPTER 6
6 POSITION SENSORLESS CONTROL OF SWITCHED
RELUCTANCE MACHINES
Sensorless (without direct position or speed sensors) control system, which extracts
rotor position information indirectly from electric signals, is important for automotive
application due to the need for minimum package size, high reliability, and low cost for
electric motor driven actuators.
Ideally, it is desirable to have a sensorless scheme, which uses only operating data
measured from motor terminals while maintaining a reliable operation over the entire
speed and torque range with high resolution and accuracy.
Various methods of sensorless control for switched reluctance machines have been
investigated and reported in literature, each with its advantages and disadvantages. With
the rapid progress in digital signal processor (DSP), observer-based sensorless algorithms
have attracted more and more attention. Generally only operating signals are needed for
this kind of sensorless algorithms. And a model of the SRM will be running in the
103
memory (of DSP) which has the same input as the real motor. The difference between the
estimated output signals and the measured ones is used to extract the position information.
Sliding mode observer, with its advantages of inherent robustness of parameters
uncertainty, computational simplicity, and high stability, provides a powerful approach to
implement sensorless schemes.
However, sliding mode observers often fail at very low (near zero) speed or ultra high
speed. Also for SRM at standstill, no operating signals are available for the observer to
estimate the rotor position. So methods for sensorless startup and low-speed operation of
the SRM must be provided.
6.1 Sliding Mode Observers
To define a sliding mode observer for SRM drive system, we first need to setup a
model of the system and build the system differential equations. The sliding mode
observer will be defined based on these equations.
6.1.1 System Differential Equations
The SRM drive system differential equations include the electromagnetic equations,
the electromechanical equations, and the mechanical equations, which are given in
Chapter 5. They are listed here again for reference:
104
The phase voltage equations can be expressed as
)(dt
di
i
LLi
dt
diLiR
dt
diRV
j
j
jj
j
j
jj
j
jj ∂
∂+
∂
∂++⋅=+⋅= ω
θ
λ, (6.1)
where ),( jj iLL θ= .
By converting to “standard” format of differential equations, Eq. (6.1) can be
represented as
⋅
∂
∂−⋅−
∂
∂+
= j
j
jj
j
j
jj
j iL
iRV
i
LiL
i ωθ
1& . (6.2)
The voltage equations (6.2 with j = 1…N, where N is the number of phases) are
nonlinear differential equations.
The torque-speed equation can be expressed as
[ ]
−=−= ∑
=L
N
j
jL TTJ
TTJ 1
11ω& , (6.3)
where J is the moment of inertia of the rotor, TL is the load torque, and Tj is the
electromagnetic torque generated in phase j. Tj can be computed as follows,
θ
θ
θ ∂
∂=
∂
∂= ∫ jjjjjc
j
diiiLWT
),(,
. (6.4)
105
The mechanical equation can be expressed as
ωθ =& , (6.5)
where ω is the rotor speed.
Equations (6.2), (6.3), and (6.5) form the system differential equations of the
switched reluctance machine and brake system, which are used to define the sliding mode
observer.
6.1.2 Definition of Sliding Mode Observer
According to the system differential equations derived from the inductance model of
SRM, a second-order sliding mode observer for rotor position and speed estimation can
be defined as follows,
)sgn(ˆˆfekθωθ +=
&, (6.6)
)sgn(ˆˆ1ˆ
1
fL
N
j
j ekTTJ
ωω +
−= ∑
=
& , (6.7)
where
ωθ ˆ,ˆ,ˆ i are the estimations of ωθ ,, i ,
jT and LT are phase torque and load torque estimated from ωθ ˆ,ˆ,ˆ i , and
106
fe is an error function based on measured and estimated variables. The details about
fe will be given later in this section.
Note that if the estimate of load torque LT is not available, Eq. (6.7) can be simplified
as
)sgn(ˆfekωω =& . (6.8)
Generally the gain ωk will be chosen large enough; the first term in Eq. (6.7) can be
neglected without any problem.
6.1.3 Estimation Error Dynamics of Sliding Mode Observer
To describe the observer error dynamics, the following estimation errors are defined:
θθθˆ−=e , (6.9)
ωωω ˆ−=e . (6.10)
Differentiating Eq. (6.8) yields
( ))sgn(
)sgn(ˆ
ˆ
f
f
eke
ek
e
θω
θ
θ
ωωθθ
−=
+−=
−=&&& (6.11)
107
Differentiating Eq. (6.8) yields
ωωω&&& ˆ−=e (6.12)
Substituting Eq. (6.3) and (6.8) into (6.12), we have
)sgn(1
1
f
N
j
Lj ekTTJ
e ωω −
−= ∑
=
& . (6.13)
Since the gain ωk may be selected to be large enough such that the first term in Eq.
(6.13) may be neglected, the speed error dynamics become
)sgn( feke ωω −=& . (6.14)
Eq. (6.11) and (6.14) represent the estimation errors dynamics of the sliding mode
observer.
From Eq. (6.11) we can find that if the error function fe is chosen to carry the same
sign as θe , and the observer gain θk is chosen to satisfy
ωθ ek > , (6.15)
then θe& will always have different sign than θe . The sliding surface 0=θe will be
reached in finite time.
Once the sliding surface is reached, the estimation error dynamics become
0=θe& , (6.16)
108
ωθ
ωω e
k
ke −=& . (6.17)
So the speed error ωe will decrease to zero exponentially.
6.1.4 Definition of Error Function
From above analysis we get the conclusion that the error function fe , which
compares measured electrical variables with their corresponding estimated values, need
to carry the same sign as θe .
There are many possible error functions that can meet this requirement, hence
stabilize the error dynamics. In this research, the following error function is used:
∑=
−⋅
−−=
N
j
jj
r
rf iiNN
jNe1
)ˆ(2
)1(ˆsinπ
θ . (6.18)
Now let’s analyze why fe has the same sign as θe . The motor has two operating
modes: motoring and generating. The analysis will be done in both modes.
Motoring operation mode
As shown in Figure 6.1, when the rotor position θ lies between [ rN/π− ,0], the phase
inductance increases with θ, and the SR motor works in motoring mode. If the estimated
rotor position 1θ is greater than the actual rotor position 1θ , then we have
109
0ˆ11 <−= θθθe . (6.19)
Since the phase inductance ),(),ˆ( 11 jj iLiL θθ > , the estimated current ji must be less
than the actual current ji with the same phase voltage. So
0ˆ >− jj ii . (6.20)
During the motoring region [ rN/π− ,0], we always have
02
)1(ˆsin <
−−
r
rNN
jNπ
θ . (6.21)
So we have the conclusion that 0<fe , or in other words, fe has the same sign as θe .
Same conclusion can be drawn if the estimated rotor position 1θ is less than the actual
rotor position 1θ .
Figure 6.1 Phase inductance profile
),( iL θ
2θrN/π0 2θ
1θ1θrN/π−
110
Generating operation mode
As shown in Figure 6.1, when the rotor position θ lies between [0, rN/π ], the phase
inductance decreases with θ, and the SR motor works in generating mode. If the
estimated rotor position 2θ is greater than the actual rotor position 2θ , then we have
0ˆ22 <−= θθθe . (6.22)
Since the phase inductance ),(),ˆ( 22 jj iLiL θθ < , the estimated current ji must be
greater than the actual current ji with the same phase voltage. So
0ˆ <− jj ii . (6.23)
During the motoring region [0, rN/π ], we always have
02
)1(ˆsin >
−−
r
rNN
jNπ
θ . (6.24)
So we have the conclusion that 0<fe , or in other words, fe has the same sign as θe .
Same conclusion can be drawn if the estimated rotor position 2θ is less than the actual
rotor position 2θ .
In summary, the error function defined in Eq. (6.18) always carries the same sign as
that of the rotor position error θe . It can stabilize the error dynamics of the sliding mode
observer defined in Eq. (6.6) and (6.8). When the sliding surface 0=θe is reached, the
111
rotor position estimate θ equals to the actual rotor position θ , and the speed estimate ω
converges exponentially to the actual speed ω .
6.1.5 Simulation Results
The sliding mode observer defined above has been implemented in Simulink model.
In actual system, the voltage and current measurements are often noise-corrupted. To test
the validity of the sliding mode observer under noise effect, noise signals have been
added to the voltage and current signals in Simulink, as shown in Figure 6.2.
Figure 6.2 Simulink model for sliding mode observer
The simulation results are shown in Figure 6.3 and Figure 6.4. In Figure 6.3, the SR
motor starts to run at 500 RPM, and the speed jumps to 1000 RPM after 1 second. In
Figure 6.4, the SR motor starts to run at 90 RPM, and the speed drops to 40 RPM after 1
112
second. In both figures, the solid red curves represent the actual rotor position and speed,
and the dotted blue curves represent the estimated rotor position and speed from the
sliding mode observer.
It is demonstrated that the sliding mode observer can get satisfactory estimates of
rotor position and speed in despite of the noise in measurements. It uses only operating
signals, so no additional sensors or external circuitry are needed. The computational
simplicity and robust stability properties of sliding mode observers make it a good choice
for sensorless control of SRM.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
100
200
300
400
time (s)
theta (o)
actual theta
estimated theta
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
200
400
600
800
1000
time (s)
speed (RPM) actual speed
estimated speed
Figure 6.3 Simulation results at high speed
113
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
100
200
300
400
time (s)
theta (o)
actual theta
estimated theta
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
20
40
60
80
100
time (s)
speed (RPM)
actual speed
estimated speed
Figure 6.4 Simulation results at low speed
However, the sliding mode observer may fail at very low speed and ultra high speed.
Also it cannot be used for sensorless startup because of the absence of electric signals at
standstill.
In this research, the switched reluctance motor needs to be run at near zero speed at
steady state. So sensorless control algorithms for such cases must be studied and
developed.
114
6.2 Sensorless Control At Near Zero Speeds
When the rotor speed is low (near zero speeds), the sliding mode observer based
controller cannot give satisfactory results. Other methods must be used for this speed
region.
6.2.1 Turn-On/Turn-Off Position Detection
Re-organize the terms in phase voltage equation (6.1), we have
ωθ∂
∂+
∂
∂++⋅= j
j
j
j
j
jjjj
Li
dt
di)
i
LiL(iRV . (6.25)
The last item in Eq. (6.25) represents the motional back-EMF, which can be ignored
at near zero speeds. So we have
dt
di/)iRV(
i
LiL
j
jj
j
j
jj ⋅−≈∂
∂+ . (6.26)
Let’s observe Eq. (6.26). The right side of Eq. (6.26) can be computed directly from
terminal measurements Vj and ij; and the left side is determined by rotor position and
phase current. At different rotor position θ, the left side of Eq. (6.26) will have a different
value (with ij known). So if at any moment, the value of the right side matches the value
of the left side at a given position θ, we can know that the rotor reaches the specified
position. Based on this observation, we can easily find a way to determine the turn-
on/turn-off moments for the SRM.
115
Let’s define
j
j
jjji
LiL)i,(F
∂
∂+=θ , (6.27)
and
j
jjjji
t)iRV()i,V(G∆∆⋅−= . (6.28)
Then when phase j is conducting and if
),(),( , jjjjoff iVGiF =θ , (6.29)
then the turn-off position of phase j is arrived and phase j should be turned off.
Similarly, when phase j is conducting and if
),(),( 1, jjjjon iVGiF =+θ , (6.30)
then the turn-on position of phase j+1 is arrived and phase j+1 should be turned on.
By the way, F(θ, ij) can be computed from the inductance model as
( )
( )
( ) θ
θ
θ
rmua
rua
mua
NLLL
NLL
LLLiF
2cos2
1
2
1
cos2
1
2
1
2
1),(
**
*
**
+++
−+
++= (6.31)
116
where
( )∑=
+=p
n
n
na ianiL0
* 1)( , (6.32)
and
( )∑=
+=p
n
n
nm ibniL0
* 1)( , (6.33)
6.2.2 Simulation Results
The above algorithm is simulated in Simulink. The desired turn-on/turn-off angles
and the actual turn-on/turn-off angles (obtained from above algorithm) are listed in Table
6.1. The position estimation errors are also listed in the table.
Angle Desired (o) Actual (o) Error (o)
θon,A -30 -29.49 0.51
θoff,A -7 -6.14 0.86
θon,B -15 -14.49 0.51
θoff,B 8 8.86 0.86
θon,C 0 0.51 0.51
θoff,C 23 23.86 0.86
θon,D 15 15.51 0.51
θoff,D 38 38.86 0.86
Table 6.1 Simulation results at near zero speeds
117
It is clear that the errors are negligible. Satisfactory results at near zero speeds (less
than 20 RPM in our case) can be obtained.
6.3 Sensorless Startup
When the rotor is at standstill, there are no operating voltage and current signals
available for position detection with the above algorithms. So some detecting voltage
pulses need to be injected into phase windings for position sensing. The voltage pulses
must be short enough so the torque generated in the phases can be neglected. The
detecting voltage signals can be injected into only one or multiple phase windings. The
easiest way is to inject an identical voltage pulse into each phase one by one, and
compare the peaks of the currents generated in them. This is valid because at any rotor
position, different phases have different airgap between the stator and the rotor pole
(hence different inductances), as shown in Figure 6.5.
Figure 6.5 Cross section of an 8/6 switched reluctance machine
118
At the rotor position shown in Figure 6.5, the phase inductances have the following
relationship
BACD LLLL >>> . (6.34)
If an identical voltage pulse is injected, the peaks of the currents generated in these
phases will have the relationship
BpeakApeakCpeakDpeak iiii ,,,, <<< . (6.35)
At this moment, if the motor is to rotate clockwise, then phase C should be excited;
and if the motor is to rotate counterclockwise, then phases D and A should be excited.
For rotor at different positions, the peaks of the currents have different but definite
relationships. So the controller can decide which phases to be turned on according to this
information.
The relationship among the peaks of currents for rotor at different positions is shown
in Figure 6.6. The phase conduction cycle can be divided into 8 regions. In each region,
the peaks of the currents have a unique relationship, and the phases to be excited can be
determined by this relationship. This is summarized in Table 6.2.
Once the SR motor is started, operating voltage and current signals will be available.
The sensorless algorithm proposed above can be applied, according to the range of the
speed.
119
Figure 6.6 Peak currents at different rotor positions
Peak currents Rotor position Phase to be excited
Ic>Id>Ib>Ia [0o, 7.5o] B, C
Id>Ic>Ia>Ib [7.5o, 15o] C
Id>Ia>Ic>Ib [15o, 22.5o] C, D
Ia>Id>Ib>Ic [22.5o, 30o] D
Ia>Ib>Id>Ic [30o, 37.5o] D, A
Ib>Ia>Ic>Id [37.5o, 45o] A
Ib>Ic>Ia>Id [45o, 52.5o] A, B
Ic>Ib>Id>Ia [52.5o, 60o] B
Table 6.2 Determining phases to be excited from the relationship among the peaks of the
currents in all phases
120
6.4 Sensorless Control Conclusions
In this chapter, a sliding-mode observer based sensorless control algorithm has been
designed and simulated. Satisfactory results are obtained for medium and high speed
operation of the switched reluctance machine. For very low speed operation, which is the
steady state of the electro-mechanical brake system, a method by matching a position-
related variable with its calculated value from voltage and current measurements is
suggested. The turn-off position of the conducting phase and the turn-on position of the
next phase can be easily determined through this algorithm, which is enough for control
of the SRM. Also, voltage probing at standstill is suggested to start the SRM without
position sensors. It is realized through comparing the relationship of the peaks of the
currents generated in all phases when probed by the same short voltage pulse.
By combining the above algorithms, a sensorless scheme that yields satisfactory
results over full speed range can be obtained.
121
CHAPTER 7
7 EXPERIMENTAL TESTBED AND RESULTS
In previous chapters, a model of the electro-mechanical brake system has been setup
in Simulink. The clamping force controller and position sensorless control algorithms for
switched reluctance machines have been designed and implemented in simulation. The
next step is to setup an experimental testbed, and test and implement the controllers in the
real system.
7.1 Experimental Testbed
A DSP based experimental testbed for the electro-mechanical brake system has been
setup in Mechatronics laboratory of the Ohio State University. A block diagram of the
testbed is shown in Figure 7.1. The testbed contained two sets of electric motor drive: one
is for a switched reluctance motor, which is the servomotor of the EMB system; the other
is for an induction machine, which serves as the load to the SRM. Since the caliper is not
available, the induction machine will be used to simulate the brake load. A torque sensor
122
is shaft coupled in between the two electric machines. Both motor drives are DSP based,
with its power converter, position encoder, and voltage and current sensors.
Figure 7.1 Block diagram of experimental testbed
The details of the components in the experimental testbed are shown in Figure 7.2.
The DSP system used for SRM drive is dSPACE DS1103 controller board system. It
contains dual CPU – Motorola PowerPC 604e and TI TMS320F240 DSP. DS1103
provides interfaces in Matlab/Simulink to access the hardware and software system of
controller board, which is very convenient tool for control development. The DSP system
used for induction motor drive is dSPACE DS1102 control system.
A picture of the testbed is shown in Figure 7.3.
123
Figure 7.2 Components of experimental testbed
Figure 7.3 Photo of experimental testbed
124
7.2 Experimental Results of Four-Quadrant Speed Control and Torque Ripple
Minimization
In this experiment, a PID speed controller for the switched reluctance motor is
implemented. The controller takes the speed command and the actual speed as input, and
output the torque command. Torque control and torque ripple minimization algorithm
mention in Chapter 4 is then applied to convert the torque command into phase current
command. Finally hysteresis current control is used to decide the appropriate voltage
pulses for each active phase. A block diagram of the controller is shown in Figure 7.4.
Figure 7.4 Block diagram of four-quadrant speed controller
To test the speed controller, the speed command is first set to be 500 RPM, then
switched to –500 RPM after 2 seconds, and finally reset to zero. During this process, the
motor is operating in all four quadrants of the torque speed plane. The experimental
results are shown in Figure 7.5 through Figure 7.7.
Power
Converter
PID
Controller
Torque Control and Torque
Ripple Minimization
Hysteresis
Current Control
SRM
refω refT refi
Gating
Signal V
i
θω,
125
In Figure 7.5, the torque and speed curves during the whole process are shown. The
SR motor is first accelerated in forward direction, then slowed down and reversed.
Finally it is stabilized at steady state. From the torque waveform we can see that the
torque ripple is quite small. This proves the performance of the ripple minimization
algorithm.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1
0
1
2
Torque (Nm)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1000
-500
0
500
1000
Speed (RPM)
Time (s)
Figure 7.5 Torque and speed responses
The four-quadrant torque-speed curve is shown in Figure 7.6. The motor starts from
the 1st quadrant, then passes the 2
nd, 3
rd, and 4
th quadrant, and finally returns to the origin.
In Figure 7.7, the voltage and current waveforms for phase A and B are shown. It is
seen that phase overlapping exists to minimize torque ripple.
126
-1.5 -1 -0.5 0 0.5 1 1.5 2-600
-400
-200
0
200
400
600
Speed (RPM)
Torque (Nm)
Torque-speed curve
Figure 7.6 Four-quadrant torque-speed curve
0.26 0.28 0.3-10
0
10
20
30
IA (A)
0.26 0.28 0.3-30
-20
-10
0
10
20
30
VA (V)
Time (s)
0.26 0.28 0.3-10
0
10
20
30
IB (A)
0.26 0.28 0.3-30
-20
-10
0
10
20
30
VB (V)
Time (s)
Figure 7.7 Voltage and current waveforms
127
The experimental results verify the effectiveness of the torque control and torque
ripple minimization algorithms developed in Chapter 4.
7.3 Experimental Results of Clamping Force Control
In this experiment, the clamping force controller developed in Chapter 5 is
implemented and tested.
Since a brake caliper and a clamping force sensor are not currently available in the
experimental testbed, an induction motor is used to simulate the caliper load and a
“virtual” force sensor is implemented in DSP to simulate the force sensor. The virtual
force sensor takes the angular movement of the rotor as input and computes the clamping
force according to the following equations:
θθθθ~
1043.1~]10904.5
~)10235.4
~1019.1[(5.2 6101316 ×+×+×−×=F , (7.1)
=π
θθ
00125.0
28
~, (7.2)
where θ is the angular movement of the motor, θ~ is the linear movement of the brake
pad, and F is the clamping force between the brake pad and brake disk.
Four-quadrant torque control and torque-ripple minimization algorithms have been
implemented in the DSP to convert the torque command obtained from force controller to
phase current command.
128
A block diagram of the controller is shown in Figure 7.8.
Figure 7.8 Block diagram of clamping force controller
The clamping force command is set as follows: the initial force command is 2500N;
when the actual clamping force reaches 2000N, the force command drops to and stays at
1600N.
The experiment results are shown in Figure 7.9 through Figure 7.12.
In Figure 7.9, the clamping force response is shown. Comparing with the simulation
result in Figure 5.7, the experimental response is not as good but still quite satisfactory.
Considering that the induction motor cannot fully represent the characteristics of the
brake load, this result is quite acceptable.
The SR motor torque and speed responses are shown in Figure 7.10, with very small
torque ripple. The four-quadrant torque-speed curve is shown in Figure 7.11, and the
phase voltages/currents waveforms are shown in Figure 7.12.
Power
Converter
Force
Controller
Torque Control and Torque
Ripple Minimization
Hysteresis
Current Control
SRM
refF refT refi
Gating
Signal V
i
θForce
Sensor
F
129
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
500
1000
1500
2000
2500
3000
Time (s)
Clamping Force (N)
Clamping Force Response
Clamping Force
Force Command
Figure 7.9 Clamping force response
0 0.5 1 1.5 2 2.5 3 3.5 4-1
-0.5
0
0.5
1
Torque (Nm)
0 0.5 1 1.5 2 2.5 3 3.5 4-100
0
100
200
300
400
Speed (RPM)
Time (s)
Figure 7.10 Torque and speed waveforms
130
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-100
-50
0
50
100
150
200
250
300
350
Speed (RPM)
Torque (Nm)
Torque-speed curve
Figure 7.11 Four-quadrant torque-speed curve
0.25 0.3 0.35 0.4-10
0
10
20
30
IA (A)
0.25 0.3 0.35 0.4-30
-20
-10
0
10
20
30
VA (V)
Time (s)
0.25 0.3 0.35 0.4-10
0
10
20
30
IB (A)
0.25 0.3 0.35 0.4-30
-20
-10
0
10
20
30
VB (V)
Time (s)
Figure 7.12 Phase voltage and current waveforms
131
7.4 Experimental Results Conclusions
In Chapters 4 and 5, algorithms for four-quadrant torque control, torque-ripple
minimization, and clamping force control are developed and simulated. The validity of
these algorithms is to be verified by experimental test results.
In this chapter, the experimental testbed is introduced and tests for four-quadrant
speed control and clamping force control are performed. The test results prove that the
performance of the controllers developed in previous chapters meet the requirements of
electro-mechanical brake system.
The sensorless control algorithms proposed in Chapter 6 haven’t been tested in this
research. It will be included in future work.
132
CHAPTER 8
8 CONCLUSIONS AND FUTURE WORK
8.1 Conclusions
The field of vehicle braking is on the verge of a revolution. The impact of new Brake-
By-Wire technology is heralding a rate of change in brake systems technology not seen
since the disc brake came into common use. This change will be driven forward by
systems development and hold advantages for the vehicle manufacturer in terms of
vehicle dynamics and design packaging.
In Brake-By-Wire systems, the conventional hydraulically actuated brakes are
replaced by electro-mechanical brakes (EMB). Electric motors are used in the EMB to
drive the brake pads onto the brake disks. An efficient, reliable, and fault-tolerant
operation of the electric motor in the EMB system is highly desired. Switched reluctance
motors, which operate on the principle of minimum reluctance, have the inherent
advantages of fault-tolerance and ability to work with high speed and high temperature
133
due to the lack of phase windings or magnets on the rotor. These characteristics make
SRM a good choice for electro-mechanical brake systems.
Modeling the dynamical properties of a system is an important step in analysis and
design of the control system. Generally, the parameter estimation from test data can be
done in frequency-domain or time-domain. Since noise is an inherent part of test data and
it has great impact on frequency-domain based methods, time-domain based methods,
such as maximum likelihood estimation (MLE), are proposed to identify the parameters
of the switched reluctance machines. An inductance-based model for the SRM is detailed
in this dissertation, and the MLE techniques are successfully applied to identify the
parameters of the SRM model from standstill test and onsite operating data.
Based on the inductance model of SRM, phase voltage equations and torque
equations can be formed, which provides a way to compute the electromagnetic torque
generated by the SRM from input phase voltages. On the other hand, when a desired
torque is given, the corresponding phase currents/voltages needed to generate this torque
can be determined if the motor speed is known and the turn-on/turn-off angles for each
phase is specified. This is the basis of the four-quadrant torque control and torque-ripple
minimization algorithms proposed in Chapter 4.
The control objective of the EMB system is to generate desired clamping force at the
brake pads and disk. It is realized by controlling the torque, speed, and rotor position of
the SRM in the EMB system. However, there’s no direct relationship between the control
134
objective and the control inputs, which are the phase voltages for the switched reluctance
motor. In this dissertation, the clamping force controller is designed using robust
backstepping, which proceeds by considering lower-dimensional subsystems and
designing virtual control inputs. The virtual control inputs in the first and second steps
are rotor speed ω and torque T, respectively. In the third step, the actual control inputs –
phase voltages, appear and can be designed. The proposed clamping force controller does
not require knowledge of the mechanical parameters of the motor and the functional
forms of the relationships among the motor position, clamping force, and the motor load
torque. Moreover, the controller provides significant robustness to uncertainty in the
SRM parameters. Simulation results have proved this.
According to the torque generation principle of the switched reluctance machines,
rotor position sensing is an integral part of SRM control. At the same time, the position
sensor/encoder is a source of additional space, extra cost, and potential cause of failure.
Rotor position sensorless control of SRM is desired for the electro-mechanical brake
systems. Sliding mode observer (SMO), with its advantages of inherent robustness of
parameters uncertainty, computational simplicity, and high stability, provides a powerful
approach to implement sensorless schemes. A SMO-based sensorless control algorithm is
developed and simulated in this research, with excellent performance in wide speed range.
However, SMO based controller doesn’t work well at near-zero speeds, which is the
steady state of the electro-mechanical brake system. A method by matching a position-
related variable with its calculated value from voltage and current measurements is
suggested to detect the turn-off position of the conducting phase and the turn-on position
135
of the next phase, which is enough for control of the SRM at low speeds. Also, by
probing the phase windings of SRM at standstill and comparing the relationship of the
peaks of the currents generated in all phases, sensorless startup of the SRM is realized.
To test the algorithms developed above, an experimental testbed has been setup in the
lab. Experiments on four-quadrant speed control and clamping force control are
performed. The test results prove the validity of the SRM model and its parameters, as
well as the desired performance of the torque and clamping force controllers.
8.2 Future Work
This dissertation covers most of the issues related to the implementation of a switched
reluctance motor based electro-mechanical brake system. Within the scope of this work,
there still remain a few aspects to be addressed to improve:
− The sensorless control algorithms developed in Chapter 6 need to be implemented
and tested on experimental testbed. Sensorless control in full speed range of the
SRM in EMB needs to be verified.
− The integration of the sensorless algorithms with the torque and clamping force
control algorithms needs to be completed. Currently the torque and force
controllers use the rotor position signal sensed by an encoder. Later when
sensorless control is tested and implemented, the estimated rotor position will be
used instead. The effect of the errors in position estimation on the torque and
136
force control needs to be investigated. A robust force controller is to be
implemented.
− The induction motor on the testbed needs to be replaced by brake caliper and all
tests performed above need to be repeated and modified for the “real” brake
system. A clamping force sensor is necessary in the first stage, but methods to
eliminate the force sensor need to be investigated. The final goal is to realize
position-sensorless and force-sensorless control of the EMB system.
− Currently all control functions are implemented on dSPACE DS1103 system
which is perfect for rapid control prototyping. When all functions are verified on
the development system, a low-cost implementation system needs to be
recommended for mass production of manufacturer.
With the successful accomplishments of the above work, the SRM based EMB can
finally be implemented in future hybrid/electric vehicles, which will hopefully appear
in market around year 2010.
137
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Modulation Methods for Sensorless SRM Drives,” Proceedings of the 24th Annual
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149
10 APPENDIX A
PARAMETERS OF SWITCHED RELUCTANCE MOTOR AND
BRAKE CALIPER SYSTEM
A.1 Parameters of Switched Reluctance Motor
− Number of phases: N = 4
− Number of stator poles: sN = 8
− Number of rotor poles: rN = 6
− Rated Voltage: V = 42 V DC
− Rated Current: I = 50 A
− Phase winding resistance: R = 0.2996 Ω
− Phase winding inductance: 50304.0=uL mH
150
− Phase winding inductance at other rotor positions: ∑=
=p
n
n
niaiL0
,)( θθ , where p is
the order of the polynomial and na ,θ are the coefficients of polynomial. They are
listed in Table A.1.
θ p paa ,0, θθ →
0o 5 5.1892e-003
6.4641e-005
-7.7645e-005
6.5907e-006
-2.1974e-007
2.6214e-009
10 o 5 4.2396e-003
-1.9346e-004
-2.8209e-006
4.9546e-007
-1.3845e-008
1.1391e-010
15 o 5 2.6605e-003
-4.7806e-005
-2.9271e-006
4.3566e-008
4.2784e-009
-9.6186e-011
20 o 5 1.1875e-003
-1.9716e-005
3.8457e-006
-2.3203e-007
4.3266e-009
-1.9516e-011
Table A.1 Phase inductance coefficients at different rotor positions
− Damper winding resistance: 32i0.0041i0.2613i5.4105-37.9659 ×−×+×=dR Ω
− Damper winding inductance: mH 5678.0=dL
151
A.2 Parameters of Brake Caliper
θθθθ~
1043.1~]10904.5
~)10235.4
~1019.1[(5.2 6101316 ×+×+×−×=F
=π
θθ
00125.0
28
~
=π00125.0
28
1
5.2
FTl
where θ is the angular movement of the motor, θ~ is the linear movement of the brake
pad, F is the clamping force between the brake pad and brake disk, and lT is the load
torque connected to the SR motor.
152
11 APPENDIX B
SIMULATION TESTBED OF SWITCHED RELUCTANCE MOTOR
BASED ELECTRO-MECHANICAL BRAKE SYSTEM
The simulation testbed for switched reluctance motor based electro-mechanical brake
system is implemented in Matlab/Simulink®. As shown in Figure B.1, it has four
modules:
− Force Command (Figure B.2)
− Force Controller (Figure B.3)
− Switched Reluctance Motor and Power Converter (Figure B.4 through Figure B.7)
− Brake Caliper (Figure B.8)
153
Figure B.1 Simulink testbed of electro-mechanical brake system
154
Figure B.2 Force command module
Figure B.3 Force controller module
155
Figure B.4 SRM and power converter module
156
Figure B.5 SRM Hysteresis module (four modules for four phases)
157
Figure B.5 SRM back EMF module (four modules for four phases)
158
Figure B.6 SRM torque module (four modules for four phases)
Figure B.7 SRM mechanic part module (four modules for four phases)
Figure B.8 Brake caliper module
159
12 APPENDIX C
C PROGRAM FOR FOUR-QUADRANT CLAMPING FORCE
CONTROL
This is the C program implemented on dSPACE DS1103 control system for four-
quadrant clamping force control of the EMB testbed.
/************************************************************ * * FILE: * torquecontrol.c * * RELATED FILES: * Brtenv.h - Basic Real Time Environment * * DESCRIPTION: * Four-Quadrant Clamping Force control of SRM * using DS1103 controller board. * * AUTHOR: * Wenzhe Lu * Mechatronics Laboratory, The Ohio State University * March 20, 2005 * *************************************************************/ #include <Brtenv.h> /*----------------------------------------------------------*/ #define PI 3.1415926 #define DT 5e-5 /* 50 us simulation step size */ #define NPHASES 4 /* number of phases */ #define I_MAX_SET 52 /* peak current rating */ #define I_RMS_SET 1600 /* square of rms current rating */ #define SCALECURRENT 100 /* 24.2 current transformation ratio */ #define SCALEVOLTAGE 200 /* 19.9 voltage transformation ratio */ #define THETA_A_ALIGNED 37.4 /*47.7148 aligned position of phase A */ #define I_UPPER_LIMIT 49.0 /* Upper limit of phase current */ /* variables for TRACE */ Float64 u[NPHASES] = 0.0, 0.0, 0.0, 0.0; Float64 i[NPHASES] = 0.0, 0.0, 0.0, 0.0; Float64 i_rms[NPHASES] = 0.0, 0.0, 0.0, 0.0; Float64 i_oc_rms = 0.0, i_oc_max = 0.0; Float64 Iref[NPHASES] = 0.0, 0.0, 0.0, 0.0; UInt32 phase_oc = 0; UInt32 count[NPHASES] = 0, 0, 0, 0; UInt32 bConduct[NPHASES] = 0, 0, 0, 0;
160
UInt32 bGrayCode; Float64 fRotorPosition; Float64 fAbsRotorPosition = -1e6; Float64 fPrevPosition; /*Float64 fPrevAbsPosition[10];*/ Float64 fSpeeds[6]; Float64 fPrevAbsPosition; UInt32 nSpeedCount = 0; UInt32 OCcount[NPHASES] = 0, 0, 0, 0; UInt32 iSample = 0; UInt32 bOverCurrent = 0; /* over current flag */ UInt32 bRotate = 0; /* starting flag */ UInt32 bDemo = 0; /* four-quadrant demo flag */ Float64 fSpeedCmd = 100; /* RPM */ Float64 fInt = 0; Float64 th_on = 7.5, th_off = 30; /* turn-on/off angle */ UInt32 gatingsignal = 0x00000000; /* gating signal */ Float64 exec_time; /* execution time */ Int16 task_id = 0; /* communication channel */ Int16 index = -1; /* communication table index */ UInt8 m[NPHASES] = 1, 1, 1, 1; Float64 ts = 0, fSpeed = 0, tspeedctrl = 0; Float64 PID_P = .01, PID_I = 1e-6; /* PID controller coefficients */ /* functions defined in this program */ void isr_timerA(void); /* ISR of Timer A */ void rms_current_calculation(void); void over_current_protection(void); void rotor_position_detection(void); UInt32 Graycode2Binary(UInt32 bGrayCode); void speed_control(void); void torque2iref(Float64 Tref, Float64 theta, Float64 w, Float64 *Iref, Float64 *theta_on, Float64 *theta_off); /*----------------------------------------------------------*/ void main(void) int k; /* init ds1103 */ ds1103_init(); /* init communication with slave_dsp */ ds1103_slave_dsp_communication_init(); /* Initialize Bit 0~7 of group 2 for Bit In: Encoder bit01~08 */ ds1103_slave_dsp_bit_io_init(task_id, 2, SLVDSP1103_BIT_IO_BIT0_MSK | SLVDSP1103_BIT_IO_BIT1_MSK | SLVDSP1103_BIT_IO_BIT2_MSK | SLVDSP1103_BIT_IO_BIT3_MSK | SLVDSP1103_BIT_IO_BIT4_MSK | SLVDSP1103_BIT_IO_BIT5_MSK | SLVDSP1103_BIT_IO_BIT6_MSK | SLVDSP1103_BIT_IO_BIT7_MSK, SLVDSP1103_BIT_IO_BIT0_IN | SLVDSP1103_BIT_IO_BIT1_IN | SLVDSP1103_BIT_IO_BIT2_IN | SLVDSP1103_BIT_IO_BIT3_IN | SLVDSP1103_BIT_IO_BIT4_IN | SLVDSP1103_BIT_IO_BIT5_IN | SLVDSP1103_BIT_IO_BIT6_IN | SLVDSP1103_BIT_IO_BIT7_IN); /* register read function in the command table for group 2 */ ds1103_slave_dsp_bit_io_read_register(task_id, &index, 2); /* init digital I/O ports: set I/O port1 to output set I/O port2 to output set I/O port3 to input set I/O port4 to input */ ds1103_bit_io_init(DS1103_DIO1_OUT | DS1103_DIO2_IN | DS1103_DIO3_OUT | DS1103_DIO4_IN);
161
/* Gating signal: IO17-SA1, IO19-SA2, IO21-SB1, IO23-SB2, IO16-SC1, IO18-SC2, IO20-SD1, IO22-SD2 */ /* Encoder: bit01-IO09 bit02-IO11 bit03-IO13 bit04-IO15 */ /* Encoder: bit05-IO08 bit06-IO10 bit07-IO12 bit08-IO14 */ /* Encoder: bit09-IO24 bit10-IO26 bit11-IO28 bit12-IO30 */ /* Joystick: B1-IO25, B2-IO27 */ /* turn off all switches */ ds1103_bit_io_write(0x00000000); /* adjust mux 1 (converter 1) to channel 2: Ia, mux 2 (converter 2) to channel 6: Ib, mux 3 (converter 3) to channel 10:Ic, mux 4 (converter 4) to channel 14:Id, */ ds1103_adc_mux_all(2, 6, 10, 14); /* start AD convertion */ ds1103_adc_delayed_start(DS1103_ADC1 | DS1103_ADC2 | DS1103_ADC3 | DS1103_ADC4); /* wait 5 us for AD convertion*/ ds1103_tic_delay(5.0e-6); /*for (k=0; k<10; k++) fPrevAbsPosition[k] = 0.0;*/ for (k=0; k<6; k++) fSpeeds[k] = 0.0; /* periodic event in ISR */ ds1103_start_isr_timerA(DT, isr_timerA); /* Background task */ while(1) master_cmd_server(); host_service(0, 0); /* COCKPIT service */ /*----------------------------------------------------------*/ void isr_timerA(void) ds1103_begin_isr_timerA(); /* overload check */ host_service(1, 0); /* TRACE service */ ds1103_tic_start(); /* start time measurement */ ts = ts + DT; /* samples current when iSample = 0; else samples voltage */ if (iSample == 0) iSample = 1; /* read results of last convertion */ ds1103_adc_read2(1, &i[0], &i[1]); ds1103_adc_read2(3, &i[2], &i[3]); i[0] = i[0] * SCALECURRENT + .192; /* zero shifting */ i[1] = i[1] * SCALECURRENT + .192; i[2] = i[2] * SCALECURRENT + .192; i[3] = i[3] * SCALECURRENT + .192; /* adjust mux 1 (converter 1) to channel 4: Va, mux 2 (converter 2) to channel 8: Vb, mux 3 (converter 3) to channel 12:Vc, mux 4 (converter 4) to channel 16:Vd, */ ds1103_adc_mux_all(4, 8, 12, 16);
162
else iSample = 0; /* read results of last convertion */ ds1103_adc_read2(1, &u[0], &u[1]); ds1103_adc_read2(3, &u[2], &u[3]); u[0] = u[0] * SCALEVOLTAGE + .374; /* zero shifting */ u[1] = u[1] * SCALEVOLTAGE + .374; u[2] = u[2] * SCALEVOLTAGE + .374; u[3] = u[3] * SCALEVOLTAGE + .374; /* adjust mux 1 (converter 1) to channel 2: Ia, mux 2 (converter 2) to channel 6: Ib, mux 3 (converter 3) to channel 10:Ic, mux 4 (converter 4) to channel 14:Id, */ ds1103_adc_mux_all(2, 6, 10, 14); /* start AD convertion */ ds1103_adc_delayed_start(DS1103_ADC1 | DS1103_ADC2 | DS1103_ADC3 | DS1103_ADC4); if (iSample == 1) rms_current_calculation(); else over_current_protection(); rotor_position_detection(); if (bRotate == 1) speed_control(); else gatingsignal = 0x00000000; ds1103_bit_io_write(gatingsignal); fInt = 0; tspeedctrl = 0; exec_time = ds1103_tic_read(); ds1103_end_isr_timerA(); /*----------------------------------------------------------*/ void rms_current_calculation(void) /* Calculate the square of RMS current */ UInt8 j; for (j=0; j<NPHASES; j++) i_rms[j] = (i_rms[j] * (Float64)count[j] + i[j] * i[j]) / (Float64)(count[j] + 1); count[j]++; /*----------------------------------------------------------*/ void over_current_protection(void) UInt8 j; phase_oc = 0; for (j=0; j<NPHASES; j++)
163
if (i[j] > I_MAX_SET) OCcount[j]++; else if (OCcount[j] > 0) OCcount[j]--; if ((OCcount[j] > 3) || (i_rms[j] > I_RMS_SET)) phase_oc = j+1; if (phase_oc != 0) /* turn off all switches */ gatingsignal = 0x00000000; ds1103_bit_io_write(gatingsignal); /* record fault current */ i_oc_rms = sqrt(i_rms[phase_oc-1]); i_oc_max = i[phase_oc-1]; /* set over current flag */ bOverCurrent = 1; bRotate = 0; OCcount[0] = 0; OCcount[1] = 0; OCcount[2] = 0; OCcount[3] = 0; /*----------------------------------------------------------*/ UInt32 Graycode2Binary(UInt32 bGrayCode) UInt32 bin; bin = bGrayCode ^ (bGrayCode >> 8); bin = bin ^ (bin >> 4); bin = bin ^ (bin >> 2); bin = bin ^ (bin >> 1); return bin; /*----------------------------------------------------------*/ void rotor_position_detection(void) UInt8 bTemp8; UInt16 slave_err_read; UInt32 bTemp32; UInt8 j; Float64 fAngle; bTemp32 = ds1103_bit_io_read(); ds1103_slave_dsp_bit_io_read_request(task_id, index); /* read bitmap from the specified I/O group */ slave_err_read = ds1103_slave_dsp_bit_io_read(task_id, index, &bTemp8); if (slave_err_read != SLVDSP1103_NO_DATA) bGrayCode = 0; if ((bTemp8 & 0x02) != 0) bGrayCode = bGrayCode | 0x800; if ((bTemp8 & 0x80) != 0)
164
bGrayCode = bGrayCode | 0x400; if ((bTemp8 & 0x10) != 0) bGrayCode = bGrayCode | 0x200; if ((bTemp8 & 0x04) != 0) bGrayCode = bGrayCode | 0x100; if ((bTemp8 & 0x01) != 0) bGrayCode = bGrayCode | 0x80; if ((bTemp8 & 0x40) != 0) bGrayCode = bGrayCode | 0x40; if ((bTemp8 & 0x08) != 0) bGrayCode = bGrayCode | 0x20; if ((bTemp8 & 0x20) != 0) bGrayCode = bGrayCode | 0x10; if ((bTemp32 & 0x01000000) != 0) bGrayCode = bGrayCode | 0x8; if ((bTemp32 & 0x04000000) != 0) bGrayCode = bGrayCode | 0x4; if ((bTemp32 & 0x10000000) != 0) bGrayCode = bGrayCode | 0x2; if ((bTemp32 & 0x40000000) != 0) bGrayCode = bGrayCode | 0x1; fRotorPosition = Graycode2Binary(bGrayCode) * 360.0 / 4096.0; if (fAbsRotorPosition == -1e6) fAbsRotorPosition = (fRotorPosition - THETA_A_ALIGNED); while (fAbsRotorPosition < -30.0) fAbsRotorPosition += 60.0; while (fAbsRotorPosition >= 30.0) fAbsRotorPosition -= 60.0; fPrevAbsPosition = fAbsRotorPosition; nSpeedCount = 0; else fAngle = fRotorPosition - fPrevPosition; if (fAngle > 180.0) fAngle -= 360.0; else if (fAngle < -180.0) fAngle += 360.0; fAbsRotorPosition += fAngle; nSpeedCount++; /*for (j=0; j<9; j++) fPrevAbsPosition[j] = fPrevAbsPosition[j+1]; fPrevAbsPosition[9] = fAbsRotorPosition;*/ fPrevPosition = fRotorPosition; fAngle = (fRotorPosition - THETA_A_ALIGNED); while (fAngle < th_on) fAngle += 60.0; while (fAngle >= th_on+60.0) fAngle -= 60.0; if ((fAngle >= th_on) && (fAngle < th_off)) bConduct[0] = 1; count[2] = 0; else bConduct[0] = 0; if ((fAngle >= th_on+15) && (fAngle < th_off+15)) bConduct[1] = 1; count[3] = 0;
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else bConduct[1] = 0; if ((fAngle >= th_on+30) && (fAngle < th_off+30)) bConduct[2] = 1; count[0] = 0; else bConduct[2] = 0; if ((fAngle >= th_on+45) || (fAngle < th_off-15)) bConduct[3] = 1; count[1] = 0; else bConduct[3] = 0; /*if ((fabs(fAbsRotorPosition - fPrevAbsPosition) >= 3.0) || (nSpeedCount >= 20))*/ if (nSpeedCount >= 10) /*fAngle = fAbsRotorPosition - fPrevAbsPosition[0];*/ fAngle = fAbsRotorPosition - fPrevAbsPosition; fSpeed = fAngle / 360.0 / (DT * nSpeedCount) * 60; /*fSpeed = fSpeeds[5] + (fSpeed - fSpeeds[0]) / 6;*/ for (j=0; j<5; j++) fSpeeds[j] = fSpeeds[j+1]; fSpeeds[5] = fSpeed; nSpeedCount = 0; fPrevAbsPosition = fAbsRotorPosition; void speed_control(void) UInt32 bPhase; Float64 Tref; int k; if (bDemo == 1) tspeedctrl += DT; if (tspeedctrl < 2.0) fSpeedCmd = 400; else if (tspeedctrl < 4.5) fSpeedCmd = -400; else if (tspeedctrl < 5.0) fSpeedCmd = 200; else fSpeedCmd = 0; bDemo = 0; bRotate = 0; tspeedctrl = 0; return; fInt += (fSpeedCmd - fSpeed); Tref = PID_P * (fSpeedCmd - fSpeed) + PID_I * fInt; if (fSpeed == 0.0) if (fSpeedCmd > 0) fSpeed = 1e-3; else fSpeed = -1e-3;
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torque2iref(Tref, fRotorPosition-THETA_A_ALIGNED, fSpeed, Iref, &th_on, &th_off); for (k=0; k<NPHASES; k++) if (Iref[k] > I_UPPER_LIMIT) Iref[k] = I_UPPER_LIMIT; if (bOverCurrent == 0) for (bPhase = 0; bPhase<=3; bPhase++) if ((i[bPhase] < .9*Iref[bPhase]) && (m[bPhase] == 1) && (bConduct[bPhase] == 1)) switch (bPhase) case 0: gatingsignal |= 0x000a0000; break; case 1: gatingsignal |= 0x00a00000; break; case 2: gatingsignal |= 0x00050000; break; case 3: gatingsignal |= 0x00500000; break; m[bPhase] = 0; else if (((i[bPhase] > Iref[bPhase]) && (bConduct[bPhase] == 1)) && (m[bPhase] == 0)) switch (bPhase) case 0: gatingsignal &= 0xfffdffff; break; case 1: gatingsignal &= 0xffdfffff; break; case 2: gatingsignal &= 0xfffeffff; break; case 3: gatingsignal &= 0xffefffff; break; m[bPhase] = 1; else if (bConduct[bPhase] == 0) switch (bPhase) case 0: gatingsignal &= 0xfff5ffff; break; case 1: gatingsignal &= 0xff5fffff; break; case 2: gatingsignal &= 0xfffaffff; break; case 3: gatingsignal &= 0xffafffff; break; m[bPhase] = 1;
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ds1103_bit_io_write(gatingsignal); else gatingsignal = 0x00000000; ds1103_bit_io_write(gatingsignal); Float64 torque_factor(Float64 theta, Float64 theta_on, Float64 theta_off) Float64 tf; if (theta >= 0.0) theta = fmod(theta, PI/3); else theta = PI/3 - fmod(-theta, PI/3); if (theta > PI/6) theta = theta - PI/3; theta_on = theta_on * PI / 180; theta_off = theta_off * PI / 180; if ((theta >= theta_on) && (theta < (theta_on + PI/24))) tf = 0.5 - 0.5 * cos(24 * (theta - theta_on)); else if ((theta >= (theta_on + PI/24)) && (theta <= (theta_off - PI/24))) tf = 1.0; else if ((theta > (theta_off - PI/24)) && (theta <= theta_off)) tf = 0.5 + 0.5 * cos(24 * (theta - theta_off - PI/24)); else tf = 0.0; return tf; void torque2iref(Float64 Tref, Float64 theta, Float64 w, Float64 *Iref, Float64 *theta_on, Float64 *theta_off) /* % Get current reference from torque reference, rotor position, and speed. % % Input: % Tref : Torque reference (Nm) % theta : rotor position (rad) % w : rotor speed (rad/s) % Output: % Iref[0] : Current reference for phase A (amp) % Iref[1] : Current reference for phase B (amp) % Iref[2] : Current reference for phase C (amp) % Iref[3] : Current reference for phase D (amp) % theta_on : turn-on angle (deg) % theta_off : turn-off angle (deg) */ Float64 Torque[21][11] = 0.0, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0, 0.0, 0.0035, 0.0063, 0.0081, 0.0085, 0.0077, 0.0062, 0.0044, 0.0027, 0.0013, 0.0, 0.0, 0.0140, 0.0253, 0.0321, 0.0337, 0.0308, 0.0248, 0.0177, 0.0109, 0.0051, 0.0, 0.0, 0.0315, 0.0570, 0.0723, 0.0759, 0.0693, 0.0558, 0.0398, 0.0244, 0.0113, 0.0, 0.0, 0.0561, 0.1016, 0.1289, 0.1353, 0.1233, 0.0993, 0.0707, 0.0434, 0.0201, 0.0, 0.0, 0.0880, 0.1594, 0.2021, 0.2121, 0.1932, 0.1555, 0.1105, 0.0678, 0.0314, 0.0,
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0.0, 0.1273, 0.2304, 0.2922, 0.3064, 0.2791, 0.2244, 0.1594, 0.0977, 0.0452, 0.0, 0.0, 0.1738, 0.3147, 0.3989, 0.4183, 0.3809, 0.3061, 0.2173, 0.1331, 0.0616, 0.0, 0.0, 0.2275, 0.4118, 0.5220, 0.5474, 0.4984, 0.4005, 0.2843, 0.1741, 0.0806, 0.0, 0.0, 0.2879, 0.5213, 0.6609, 0.6932, 0.6312, 0.5075, 0.3604, 0.2208, 0.1022, 0.0, 0.0, 0.3547, 0.6423, 0.8146, 0.8547, 0.7787, 0.6265, 0.4454, 0.2731, 0.1266, 0.0, 0.0, 0.4273, 0.7738, 0.9818, 1.0308, 0.9400, 0.7572, 0.5392, 0.3312, 0.1537, 0.0, 0.0, 0.5048, 0.9146, 1.1610, 1.2200, 1.1140, 0.8989, 0.6415, 0.3950, 0.1837, 0.0, 0.0, 0.5865, 1.0630, 1.3505, 1.4208, 1.2994, 1.0509, 0.7520, 0.4645, 0.2166, 0.0, 0.0, 0.6714, 1.2176, 1.5483, 1.6312, 1.4948, 1.2122, 0.8704, 0.5397, 0.2525, 0.0, 0.0, 0.7586, 1.3766, 1.7524, 1.8493, 1.6988, 1.3820, 0.9963, 0.6205, 0.2913, 0.0, 0.0, 0.8471, 1.5383, 1.9608, 2.0732, 1.9097, 1.5592, 1.1291, 0.7066, 0.3331, 0.0, 0.0, 0.9360, 1.7010, 2.1713, 2.3009, 2.1259, 1.7428, 1.2684, 0.7981, 0.3779, 0.0, 0.0, 1.0243, 1.8632, 2.3822, 2.5305, 2.3460, 1.9319, 1.4137, 0.8947, 0.4257, 0.0, 0.0, 1.1112, 2.0235, 2.5917, 2.7604, 2.5687, 2.1255, 1.5645, 0.9962, 0.4763, 0.0, 0.0, 1.1961, 2.1805, 2.7983, 2.9890, 2.7925, 2.3227, 1.7201, 1.1023, 0.5298, 0.0 ; Float64 f[4], Te[4], T[21], th; int j, k, m; /* Determine theta_on and theta_off according operating quadrant */ if ((Tref*w) >= 0) /* 1st or 3rd quadrant */ if (w >= 0) /* 1st quadrant */ theta_on[0] = -30; theta_off[0] = -7.5; else /* 3rd quadrant */ theta_on[0] = 7.5; theta_off[0] = 30; else /* 2nd or 4th quadrant */ if (w >= 0) /* 2st quadrant */ theta_on[0] = 5; theta_off[0] = 27.5; else /* 4th quadrant */ theta_on[0] = -27.5; theta_off[0] = -5; /* calculate torque distribution factors among phases and calcualte current reference for each phase */ for (k=0; k<4; k++) if (theta-k*PI/12 >= 0) th = fmod(theta-k*PI/12, PI/3);
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else th = PI/3 - fmod(-theta+k*PI/12, PI/3); if (th > PI/6) th -= PI/3; j = (int)(fabs(th) / (PI/60)); if (j > 9) j = 9; for (m=0; m<21; m++) T[m] = Torque[m][j]+(Torque[m][j+1]-Torque[m][j])/(PI/60)*(fabs(th)-PI/60*j); f[k] = torque_factor(theta-k*PI/12, theta_on[0], theta_off[0]); Te[k] = fabs(Tref) * f[k]; if (Te[k] == 0.0) Iref[k] = 0.0; else for (j=0; j<21; j++) if (Te[k] < T[j]) break; if (j <= 0) Iref[k] = 0.0; else if (Te[k] >= T[20]) Iref[k] = 50.0; else Iref[k] = 2.5*sqrt(j*j-2*j+1 + (2*j-1)*(Te[k]-T[j-1])/(T[j]-T[j-1]));