AN APPLICATION OF OBSERVER FOR POSITION SENSORLESS STEPPER
MOTOR DRIVES
NOR ARYMASWATI BINTI ABDULLAH
UNIVERSITI TEKNOLOGI MALAYSIA
AN APPLICATION OF OBSERVER FOR POSITION SENSORLESS STEPPER
MOTOR DRIVES
NOR ARYMASWATI BINTI ABDULLAH
This project report submitted in partial fulfilment of the requirements
for the award of the degree of Master of Engineering
(Electrical - Mechatronics and Automatic Control)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
NOVEMBER 2009
iii
Dedicated with deepest love to:
My family, parents and siblings for their support, guidance and love.
My dearest friends for being there whenever I needed them.
iv
ACKNOWLEDGEMENT
In the name of ALLAH, the Most Beneficent, The Most Gracious and the Most
Merciful who has given me persistence in completing the project.
First of all, I would like to express my sincere gratitude to my project
supervisor, Dr Salinda Buyamin, for the commitment, encouragement and valuable
advice she provided me along the entire course of this project and report. Her
rigorous views to do the project and inspire the idea to solve problems are precious
for my future career.
I wish to thank my lab mates at Process Control Laboratory , Mr Alifa, Mr
Rozaimi and Mr Azuni from Advanced Machine laboratory for the intellectual input
and technical assistance .
I would like to thank to my colleague members, from MEM class of 2008/09
session for their support and advice.
A special thanks to my department, Malaysian Nuclear Agency who gives me
the opportunity to continue my study.
v
ABSTRACT
A control method for stepper motor drives system can be made in open-loop
circumstance which mean the system control did not require any feedback input
signal in order to run the system. By applying the right sequences of pulses, the
stepper motor capable to operate as other motion control. However, the performance
of such system cannot be achieved to high level condition and demanded a feedback
signal input to compensate the error produced while running the drive system.
Therefore, a physical sensor or an encoder is placed in the motor system to obtain the
feedback and form a close-loop system for error compensation. Nevertheless, the
prices of these instruments are expensive, bulky and also may degrade the system
performance. As a result this project presents a sensorless system in stepper motor
drive system as an alternative to develop a close-loop system where the input signals
are taken from voltage and current of the magnetic flux of the stepper motor.
vi
ABSTRAK
Kaedah kawalan sistem motor stepper boleh dijalankan dalam keadaan
lingkaran terbuka iaitu bermaksud sistem kawalan tidak memerlukan apa-apa
suapbalik signal masukan semasa operasi dijalankan. Dengan mengaplikasikan
urutan denyutan yang betul, stepper motor mampu beroperasi seperti kawalan
gerakan yang lain. Walaubagaimanapun prestasi tersebut tidak boleh mencapai pada
tahap yang tinggi. Ini memerlukan suapbalik masukan untuk mengimbangi kesilapan
semasa sistem pemanduan beroperasi. Dengan itu, alat pengesan atau encoder
diletakkan dalam sistem motor untuk menghasilkan suapbalik
dan seterusnya
membentuk sistem lingkaran tertutup sebagai pengimbangan kesilapan. Akan tetapi,
harga alatan ini adalah mahal, bersaiz besar and juga boleh mengurangkan prestasi
sistem. Oleh itu, projek ini mengetengahkan satu sistem tanpa alat pengesan dalam
sistem pemanduan motor stepper sebagai alternatif membentuk sistem lingkaran
tertutup di mana signal masukan diambil dari voltan dan arus magnetik fluk yang
dihasilkan oleh stepper motor.
vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF ABBREVIATIONS
xiii
INTRODUCTION
1.1
Background
1
1.2
Problem Statement
2
1.3
Objective of The Project
3
1.4
Scope of The Project
3
1.5
Methodology
3
LITERATURE REVIEWS
2.1
Introduction
6
2.2
Previous Research In Sensorless Technology
7
General Structured Unknown Input
7
2.2.1
Observer
2.2.2
Robust Speed Control Of DC Servo Motor
8
Based On Lyapunov’s Direct Method
2.2.3
Sliding Mode Based Disturbance Observer
For Motion Control
9
viii
2.2.4
Observer-Based Inertial Identification For
9
Auto-Tuning Servo Motor Drives
2.3
2.4
2.5
2.6
3
Observer
10
2.3.1
Model Reference Adaptive System (MRAS)
11
2.3.2
Unknown Input Observer
13
2.3.3
Disturbance Torque Observer
13
2.3.4
Instantaneous Speed Observer
14
Stepper Motor System
14
2.4.1
Stepper Motor Operation
15
2.4.2
Type Of Stepper Motor
17
2.4.2.1 Variable Reluctance Stepper Motor
17
2.4.2.2 Permanent Magnet Stepper Motor
18
2.4.2.3 Hybrid Stepper Motor
19
Controller And Driver
21
2.5.1
Current Controller
21
2.5.2
Feedback Linearizing Controller
21
2.5.3
Voltage Level Controller
22
2.5.4
Neural Controller
23
2.5.5
PID Controller
23
2.5.6
L6208 Chip Driver
24
2.5.7
Microstepping Driver
24
Encoder
25
2.6.1
Incremental Encoder
26
2.6.2
Absolute Encoder
26
2.6.3
Linear Encoder
27
2.6.4
Rotary Encoder
27
PROJECT BACKGROUND
3.1
Introduction
28
3.2
Sensorless System Control
29
3.2.1
MRAS Observer
29
3.2.2
Mathematical modeling of stepper motor
30
3.2.3
Stepper motor selection
33
ix
4
3.2.4
Stepper motor controller
35
3.2.5
Stepper motor driver
37
SOFTWARE AND HARDWARE CONSTRUCTION
4.1
4.2
Characteristic Of Hybrid Stepper Motor
42
4.1.1
Circuit Representation
42
4.1.2
Speed Characteristic
43
Simulation Implementation
46
4.2.1
Open Loop Simulation
46
4.2.2
Close Loop Simulation
50
4.2.2.1 Transformation Of Four Phase Into
50
Two Phase
4.2.2.2 MRAS Derivation
4.3
5
6
Hardware Implementation
54
57
4.3.1
Open Loop System
57
4.3.2
Close Loop System
59
RESULTS AND DISCUSSIONS
5.1
Introduction
61
5.2
Open Loop System
61
5.3
Close Loop System (Sensorless System)
65
CONCLUSIONS AND FUTURE WORKS
5.1
Conclusion
69
5.2
Future Works
70
REFERENCES
71
Appendices A - B
75
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
3.1
Sequence and Direction of Rotation HSM
34
3.2
Technical Specification of HSM
34
3.3
Full Step Mode
39
3.4
Half Step Mode
39
4.1
Parameters for Driver Block and Hybrid Stepper
50
Motor
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
Flow Chart Of The Methodology
5
2.1
Block Diagram Of The Observer And Adaptive
10
Controller
2.2
MRAS General Structure
12
2.3
From Controller (Logic Sequencer) To Motor
15
2.4
Input Controller
15
2.5
Cross Sectional Of Stepper Motor
16
2.6
Cross Section Of VR Stepper Motor
18
2.7
Permanent Magnet Stepper Motor Cross Section
19
2.8
Cross Section Hybrid Stepper Motor
19
2.9
Special Features In HSM
20
2.10
L6208 Functional Block Diagram
24
3.1
Block Diagram Of MRAS Estimator
30
3.2
Stepper Motor Selected
34
3.3
Stepper Motor Configuration
34
3.4
The Pin Diagram Of PIC
36
3.5
Block Diagram For PIC16F8X
37
3.6
a)Unipolar Drive b) The Effect On Motor
38
Performance Higher Supply Voltage And Larger
Series Limiting Resistance
3.7
Typical Motor Winding Connections
38
4.1
Circuit Model For One Phase Of A Hybrid Stepper
42
4.2
Simulink Block For Open Loop
46
4.3
Signal Generator Output
47
4.4
Look Under Mask For Four Phase HSM
48
4.5
a)Look Under Mask Of HSM b) Look Under Mask
49
xii
Of Model Discrete 4 Phases HSM
4.6
Equivalent Constant Air Gap For 4 Phase HSM
50
4.7
Block Diagram Of The Whole System
54
4.8
Basic Configuration Of MRAS
54
4.9
Simulink Block For Close Loop System
58
4.10
Circuit Simulation For Stepper Motor Driver
59
4.11
Waveform At Each Coil
59
4.12
Circuit Construction On a)Protoboard b)Strip Board
59
4.13
a)Pair Of Optical And 8 Slot Disc Rotary Encoder
60
b)Motor Shaft Was Slotted With Rotary Encoder Disc
5.1
Voltage Phase, Current Phase And Torque Of Open
62
Loop HSM
5.2
The Speed (Top) And Position (Bottom) Of The Open
62
Loop HSM
5.3
The Speed (Top) And Position (Bottom) Of The
63
MRAS System Without Feedback
5.4
The Comparison Between Open Loop System And
63
Open Loop MRAS System Without Feedback For
Speed (Top) And Position (Bottom)
5.5
Output Signal Of Rotary Encoder
64
5.6
The Speed (Top) And Position (Bottom) Of The
66
Sensorless System With Feedback
5.7
The Comparison Between Open Loop System And
66
Open Loop MRAS System With Feedback For Speed
(Top) And Position (Bottom)
5.8
Current Phase Of Simulation
67
5.9
Voltage Phase Of Simulation
67
5.10
Voltage Phase Of Hardware Implementation
68
xiii
LIST OF ABBREVIATIONS
ADC
-
Analog to Digital Converter
BCD
-
Binary Coded Decimal
CCW
-
Counter Clock Wise
CW
-
Clock Wise
DC
-
Direct Current
DIR
-
Direction
DMOS
-
Double-diffused Metal Oxide Semiconductor
DQ
-
Direct Quadrature
DSP
-
Digital Signal Processing
FPGA
-
Field- Programmable Gate Array
GS
-
General Structure
HSM
-
Hybrid Stepper Motor
IPM
-
Interior Permanent Magnet
LTI
-
Linear Time-Invariant
MOSFET
-
Metal Oxide Semiconductor Field Effect Transistor
MRAS
-
Model Reference Adaptive System
MSU
-
Microcontroller
PC
-
Personal Computer
PID
-
Proportional Integral Derivative
PLC
-
Programmable Logic Control
PM
-
Permanent Magnet
PWM
-
Pulse Width Modulated
UIO
-
Unknown Input Observer
VHDL
-
Very High-level Design Language
VR
-
Variable Reluctance
xiv
LIST OF APPENDICES
APPENDIX
A
TITLE
Source Code For Embedded MATLAB Function In
PAGE
74
Simulink
B
Subsystem Of Simulation
75
CHAPTER 1
INTRODUCTION
1.1
Background
In recent years, a real robust of motion control in mechatronics technique is
required in a very precise positioning and broad speed range applications. It means
that drive systems are robust-controllable for precise positioning and broad speed
range including from an ultra small to large positioning and ultra low to ultra high
speed range.
Both speed and positioning controller is very important for the
performance improvement of drive systems. One of the important motion controls is
to design a self reconfigurable controller such as electric motor controller for a
hybrid electric vehicle application. This system detects the current sensors failure
and will estimate the current successfully such that the motor continues working
safely. The motor model is used for estimating the currents and the phase are
estimated using Luenberger observer. The hall sensors with 60 degrees resolution
have been used for positioning sensor. [1].
Nowadays, the emerging control applications become advance as a power
assisting tools. An example is a power-assisted wheelchair where a controller is
considering necessary conditions for power assisting tools. For advanced controls of
a power-assisted wheelchair, the control for speed of power assisting motors is
needed. One of the features of a wheelchair is operating at very slow speed and even
stops frequently. Thus, an instantaneous speed observer is necessary for the control
of a power-assisted wheelchair since the instantaneous speed observer has fast
convergence speed, and applies it to gravity compensation controller of a powerassisted wheelchair especially when it goes on a hill [2].
2
Observer also called a sensorless system is a popular application in motion
control where the physical sensor such as encoder will not be used to obtain the
system feedback. Besides be able to remove space allocation for rotation-sensor
hardware, it also is able to eliminate mechanical adjustment and maintenance. The
other applications of the sensorless system are in city-scooter application, which has
been design of a sensorless scheme suitable for general applications where low speed
and standstill such as high speed operations are required. The observer detects the
rotor magnet flux components in the two-phase stationary reference frame using the
motor electrical equations [3].
The observer also used in solving the speed
estimation problem in high-power railway traction applications, including the very
low speed range. The full-order Luenberger observer design, based on voltage and
current models, is used as the optimal alternative instead of the conventional method
based on observer pole placement. [4].
1.2
Problem statement
Recently, there has been an increase attention of drives system in positioning
applications. This increasing is attributed of the developments in power electronics
and computing technology.
Together with faster computing capabilities, the
complex calculations can be performed in a shorter period of time. These advances
have opened up new opportunities for advanced control methods such as in nonlinear
and optimal control. Moreover, a self-tuning and adaptive methods which used to
determine unknown or slowly varying parameters are capable of adjusting the control
system to maximize performance with minimal or no operator intervention
automatically [1]. In motion control, accurate speed and positioning information is
necessary to realize high performance and precision control. Many techniques were
developed to achieve speed and positioning including mechanical sensors such as
shaft encoder or a resolver.
Nevertheless, the prices of these instruments are
expensive, bulky and degrade the system. Therefore, a sensorless motion control is
developed to replace the hardware part [5].
3
1.2
Objectives of the project
The objectives of the project are determined as:
i)
To investigate the performance of position control of stepper motor
using PIC controller
ii)
To study the application of sensorless in position control system of
stepper motor
1.3
Scope of the project
The scope of the project includes:
i)
To construct the hardware for stepper motor drive
ii)
To apply the sensorless method of MRAS to the system
iii)
To do performance comparison/analysis on position control of the
system.
1.4
Methodology
Figure 1.1 shows the flow chart of project methodology. It started off with
determine the objectives and scope of the project, literature review and to identify the
whole system requirement such as device/component, the type of drive system,
driver, controller and also the observer. The construction of the open-loop hardware
stepper motor system was initiated after procurement of the device has been done.
The construction of the circuit is including the controller, driver, interfacing between
computer and hardware. And, followed by the developing of the sequences of pulses
for stepper motor running in the open-loop system. The signal input or initial
position was sent to the driver to rotate the motor shaft to desire position. The
feedback from the output signal was then used to compensate the error at the
reference input. For the first stage, the encoder will be used to obtain the feedback
signal and then will be replaced by observer (MRAS observer) to form a close-loop
4
system. In order to get the MRAS observer, the mathematical modelling and the
derivation of the MRAS will be determined. These two experimental will be held to
get both result data. The data will be analysed and the performance comparison was
carried out between those two conditions.
5
Objective
Literature review
Identify :-the whole system
-type of observer, controller, motor
Modelling & Simulation for open
loop motor drive system
No
Valid?
Yes
Develop open loop hardware motor
system
Modelling and derivation MRAS observer
Apply MRAS observer into the system to develop close
loop system
No
Complete?
Yes
Comparison and Analysis
End
Figure 1.1 Flow Chart Of The Methodology
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Usually sensorless control is defined as a control scheme where no
mechanical parameters like, speed and torque, are measured.
In recent years,
observer is used to estimate parameters and states of electrical machines. Model
based sensorless schemes; generally, rely on accurate system modelling and accurate
model parameters values.
As in hybrid electric vehicle application, the phase
currents are estimated using Luenberger type observer. The state space equations in
this application have been decoupled and linearized using the feedback linearization
technique. Hall sensors which have the resolution of 60 degrees have been used
instead of high resolution position sensors. The method has been simulated in
Matlab/Simulink under different speed commands and different load torques. The
developed observer can also stand for the current sensors during the whole operating
time. The system drive is implemented on a interior permanent magnet (IPM) motor
and the digital signal processor [1].
Meanwhile in designing the brake-by-wire systems, a new angle-tracking
observer in which a closed-loop linear time-invariant (LTI) observer is integrated
with a quadrature encoder is introduced. Finite-gain stability of the proposed design
and its robustness to three different kinds of parameter variations are proven based
on theorems of input–output stability in nonlinear control theory. The performance
of observer and two other methods (a well-known LTI observer and an Extended
Kalman Filter) was examined to estimate the position and speed of a brake-by-wire
actuator [7]. Meanwhile, a sensorless controller for IPM synchronous motors has
7
been used in city-scooter application. In this application, the final solution consists a
hybrid technique that employing an adaptive observer for medium/high speed
operation and a signal injection based technique for low speed and standstill
operation. The observer detects the rotor magnet flux components in the two-phase
stationary reference frame using the motor electrical equations. The motor speed is
identified by a model reference adaptive scheme using an additional equation
obtained by a Lyapunov function [3].
2.2
Previous research in sensorless technology
2.2.1 General-Structured Unknown Input Observers
The author [9] using the framework of the general structured observer
and present a straightforward procedure for designing an Unknown Input
Observer (UIO) for a linear system subject to unknown inputs or uncertain
disturbances.
Necessary and sufficient conditions in terms of the
transmission zeros are provided for the existence of a full order UIO capable
of producing an asymptotic estimate of the state. An extension of the UIO,
called the extended UIO, is developed to estimate both the system state and
the unknown input simultaneously. Experimental results for a DC servo
motor system demonstrate the applicability of the proposed methodologies.
Based on the Luenberger observer the writer focused on a design in fullorder UIO based on the configuration of the General Structured (GS)
observer proposed by Cheok et al.
Since the existence conditions are
provided in terms of transmission zeros the same as the existence conditions
of the inverse system problem, the writer also propose an extended design of
the UIO, the extended UIO, for estimating both system states and unknown
inputs.
The estimate of unknown inputs can be further applied to
accommodate process uncertainty so as to produce robust control systems.
The writers have implemented the proposed UIO separately on a DC servo
motor system and a servo table system with external loading. The dominant
disturbance of the two systems is friction and the external loading,
8
respectively. Experimental estimations of states and disturbance are in good
agreement with measurements, thus confirming the applicability of the
developed UIO methodologies.
2.2.2 Robust Speed Control of DC Servo Motor Based on Lyapunov’s
Direct Method
The author [10] proposed the robust speed control method of DC
servo motor with disturbance torque observer, in which the disturbance
observer is used for compensating parameter variations and disturbance also
the stability of speed control system and current control system are
discussed. The useful features and validity of proposed control method are
confirmed by computer simulations.
The disturbance observer is
constructed using Gopinath’s observer design algorithm. The influence of
estimation error for disturbance torque is restrained by feedback gain based
on Lyapunov’s direct method. Lyapunov function is selected as a test
function for stability criterion and the feedback gain is determined to be
negative the time derivative of Lyapunov function. The writer used the
voltage source PWM (Pulse Width Modulated) inverter for power supply of
DC servo motor. But the method produced undesirable current fluctuation
because of system parameter variations. Therefore, current control method
taking account of the variations of motor parameters is necessary and the
writer presented the robust current control method of voltage source PWM
inverter with disturbance voltage observer. Armature current is controlled
to be exactly equal to the reference current under the variations of motor
parameters and operating conditions. In this control method, load circuit of
inverter is R-L load with emf. This is electrical equivalent circuit of DC
servo motor. The disturbance voltage generated by parameter variations is
estimated by disturbance voltage observer. The estimation voltage is used
as control input voltage of inverter. The influence of estimation error is
restrained by feedback gain based on Lyapunov’s direct method.
The
feedback gains are adjusted in on-line calculations to ensure system’s
9
stability with various disturbances. Since the feedback gains are determined
instantaneously with respect to state variables, feedback gains are not
excessive high-gain and therefore the stability of the system in digital
control is improved.
2.2.3 Sliding Mode Based Disturbance Observer for Motion Control
The author [11] focused on the experimental system which is a Digital
Signal Processing (DSP) controlled single degree-of-freedom motion
control system and consists of a conventional DC servo motor with
harmonic gear and encoder feedback. The inertia load is coupled by a
relatively rigid shaft.
With the help of the proposed disturbance
compensation, the servo system is forced to track a reference model. The
relatively big parameter perturbation, external disturbance and friction are
estimated and compensated.
The experimental system consists of a
conventional DC servo motor with harmonic gear and encoder feedback.
The inertia load is coupled by a relatively rigid shaft. The controller is
implemented by using a DSP as the computation engine. The DC motor is
supplied by a DC chopper. In the control design, a reduced order model is
used; the armature inductance and the flexibility of the shaft are ignored.
The system has a big friction, if less than 18% of the maximum voltage
input is switched to the motor, it cannot move because of the friction.
2.2.4 Observer-based inertial identification for auto-tuning servo motor
drives.
An observer-based auto-tuning scheme for servo motor drives is
presented by the author in [12]. The scheme is consisted of a state estimator
to estimate motor disturbance and two adaptive controllers to separately
adjust drive inertia and friction to their correct value. The servo control
loop is tuned automatically with the inertia found. For disturbance torque
10
estimator, the authors use a full state estimator and use it to calculate the
motor position. By using relation between motor speed and position,
߱௥ ൌ ‫ߠݏ‬௥
the estimator can be arrange into block diagram.
For actual external
disturbance and friction, the author assumed that disturbance torque vary
much slowly than the servo motor’s cycle time, then, its average over a
servo cycle can be calculated. The inertia can be estimated when the motor
is in acceleration and deceleration regions. From the block diagram shows
in Figure 2.1, motor current and speed information are used to estimate the
mechanical parameter influenced external disturbance and two adaptive
controllers are used to determine disturbance estimator.
Figure 2.1 Block Diagram Of The Observer And Adaptive Controller
2.3
Observer
The idea of observer was introduced because not all state variable feedback
could assign the closed loop poles arbitrarily. It is basic mechanism to estimate the
unmeasured state from the available output measurement or could called virtual
sensor or sensorless. The state space model is;
11
‫ݔ‬ሶ ሺ‫ݐ‬ሻ ൌ ‫ܣ‬௢ ‫ݔ‬ሺ‫ݐ‬ሻ ൅ ‫ܤ‬௢ ‫ݑ‬ሺ‫ݐ‬ሻሺͳሻ
‫ݕ‬ሺ‫ݐ‬ሻ ൌ ‫ܥ‬௢ ‫ݔ‬ሺ‫ݐ‬ሻሺʹሻ
And a general linear observer takes the form of;
‫ݔ‬ሶ ሺ‫ݐ‬ሻ ൌ ‫ܣ‬௢ ‫ݔ‬ොሺ‫ݐ‬ሻ ൅ ‫ܤ‬௢ ‫ݑ‬ሺ‫ݐ‬ሻ ൅ ‫ݕܬ‬ሺ‫ݐ‬ሻ െ ‫ܥ‬௢ ‫ݔ‬ොሺ‫ݐ‬ሻሺ͵ሻ
Where ‫ ܬ‬is the observer gain and ‫ݔ‬ොሺ‫ݐ‬ሻ is the state estimate. If ‫ ܬ‬ൌ Ͳ, the observer
degenerates into open-loop model as in (1). The term of
‫ݒ‬ሺ‫ݐ‬ሻ ‫ݔ ؜‬ሺ‫ݐ‬ሻ െ ‫ݔ‬ොሺ‫ݐ‬ሻሺͶሻ
is known as the innovation process which represent the feedback error between the
observation and the predict model output for nonzero ‫ ܬ‬in (3). The following result
shows how the observer gain ‫ ܬ‬can be chosen such that the error ‫ݔ‬ොሺ‫ݐ‬ሻ, defined as
‫ݔ‬෤ሺ‫ݐ‬ሻ ‫ݔ ؜‬ሺ‫ݐ‬ሻ െ ‫ݔ‬ොሺ‫ݐ‬ሻሺͷሻ
can be made to decay at any desired rate. Consider the state space model (1) – (2)
and an associated observer of the form (3). Then the estimate error ‫ݔ‬෤ሺ‫ݐ‬ሻ defined by
(5) satisfies
‫ݔ‬෤ሶ ሺ‫ݐ‬ሻ ൌ ሺ‫ܣ‬௢ െ ‫ܥܬ‬௢ ሻ‫ݔ‬෤ሺ‫ݐ‬ሻሺ͸ሻ
Moreover, provided the model is completely observable, then the eigenvalues of
ሺ‫ܣ‬௢ െ ‫ܥܬ‬௢ ሻ can be arbitrary assigned by choice of ‫[ ܬ‬12]. There are many types of
observer, depend on what type to observe as a feedback to system. The type of
observer to be used in application is depending on kind of parameter to be observer
and the availability of the parameter in the system.
2.3.1 Model reference adaptive system (MRAS) observer
Model Reference Adaptive Control (MRAC) also called MRAS is one
of the main approaches to adaptive control. The basic structure of MRAS
observer is shown is Figure 2.2. The reference model is chosen to generate
12
the desired trajectory, ‫ݕ‬௠ , that the plant output ‫ݕ‬௣ has to follow.
The
tracking error
݁ ‫ݕ ؜‬௣ െ ‫ݕ‬௠
represents the deviation of the plant output from the desired trajectory. The
purpose of MRAS is to design the controller and parameter adjustment
mechanism so that all signals in the closed-loop plant are bounded and the
plant output ‫ݕ‬௣ tracks ‫ݕ‬௠ as close as possible.
‫ݕ‬௠
Reference
model
‫ݎ‬
‫ݑ‬௣ Controller
ߠ
Adjustment
mechanism
݁
Plant
‫ݕ‬௣ ‫ݕ‬௣ ݁
Figure 2.2 MRAS General Structure
There are two categories of MRAS as direct or indirect and with
normalized or unnormalized adaptive laws.
For direct MRAS, the
parameter vector , of the controller is updated directly by an adaptive law.
For indirect MRAS, is calculated at each time, t, by solving a certain
algebraic equation that relates with the online estimates of the plant
parameters.
But both categories with normalized adaptive law, the
controller is motivated from the known parameter case is kept unchanged.
This design can be used with adaptive include gradient, least-squares and
SPR-Lyapunov design.
Meanwhile, with unnormalized adaptive laws,
controller is modified to lead to an error equation whose form allows the use
of the SPR-Lyapunov design approach for generating the adaptive law. In
13
unnormalized, the controller design is more complicated in both direct and
indirect case, but simpler in analysis and follows a consideration of a single
Lyapunov-like function [14].
2.3.2 Unknown input observer
The unknown input observer (UIO) was developed to estimate the
state of an uncertain system despite of the existence of unknown inputs or
uncertain disturbances. The problem of reconstruction of the state of a
dynamic system whose input is not measurable, which is importance in
practice since there are many situations where plant disturbance occurs or
part of the input of the system is inaccessible. Under such circumstances, a
conventional observer that requires knowledge of all inputs cannot be used
directly [9].
2.3.3 Disturbance torque observer
Generally, the motor control methods using disturbance observer is
used to compensate parameter variations and disturbance torque to achieve
high performance control of motor drives. This can be seen in the ability of
speed control drive system especially DC servo motor is affected by
variations and disturbance torque. As sensorless system, a mathematical
modelling is used for controlling the machines.
However, an accurate
parameter measurement is very difficult to obtain and it’s vary with
operation conditions such as temperature or saturation effects. This method
also considers error estimation since the estimated disturbance by
disturbance observer produce estimation error. The current tracking control
is very important on motor control, since motor torque is generated by the
armature current. The disturbance torque observer is constructed using
Gopinath’s observer design algorithm [10].
14
2.3.4 Instantaneous speed observer
Normally, a speed sensor based on rotary encoder can be used to
obtain speed information by counting an increased or decreased number of
encoder pulses in a sampling period of mechatro servo system. However, in
an ultra low speed range, a speed resolution of speed sensor based on rotary
encoder is easily lost. Hence, the instantaneous speed observer is used.
These methods have some effects of mechanical parameter variation and
disturbance torque. Therefore, the new instantaneous speed observer in
motor drives was deliberated, which mounts the electrical parameter
identification algorithm to cancel out the estimation error caused by both the
voltage drop of power converter and the armature resistance variation.
From the state equation of DC servo motor, the motor speed is obtained.
The differential state equation is transformed into the discrete state equation
by the backward difference approximation.
The speed observer is
constructed by using both the discrete state equation and the nominal
parameters of DC servo motor.
The speed observer estimates the
instantaneous value of motor speed in the sampling period and its input
variables are an armature voltage and an armature current [15].
2.4
Stepper motor system
A stepper motor is an electromechanical actuators, converts electrical pulses
into mechanical actions. When the right sequence pulses signal are applied to the
system, the shaft of a stepper motor rotates in discrete step. The stepper motors
revolution has various direct relationships to applied input pulses. The speed of the
shafts rotation is connected to the frequency of the input pulses and the length of
rotation is associated to the amount of input pulses applied. Meanwhile the direction
of the shaft rotation is depends on sequences of the input pulses. Not like other type
of motor, the stepper motor has no contacts or brushes. It is a synchronous motor in
order to rotate armature magnet through the magnetic field switching. The essential
function of a stepper motor is to translate switching excitation changes into precisely
15
defined increments of rotor position. Generally the stepper motor capable works as
an electric motor when the drive running without commutator. The rotor of the
stepper motor can be permanent magnet or variable reluctance motor which has a
toothed block of some magnetically soft material. Typically, all the windings in the
stepper motor are part of the stator. A stepping motor control system consists of
three basic elements; controller, driver and motor [16].
2.4.1 Stepper motor operation
A simple drive system of a stepping motor is expressed by the figure
shown that the stepper motor system can be an open loop system [16-22].
The block diagram is represented detailed in Figure 2.3 and Figure 2.4. The
figure represents the portion from logic sequence to motor.
Step
Controller
Motor
Driver
Direction
command
Controlsignal
Figure 2.3 From Controller (Logic Sequencer) to Motor
Phase
Input
controller
Controller
Input
controller
1
2
3
4
Figure 2.4 Input Controller
When a step command pulse is applied to the logic sequencer, the
states of the output terminal are changed to control the motor driver so as to
rotate the motor a step angle in the commanded direction. The rotational
16
direction is determined by the logic state at direction input, e.g. the H level
indicate High for CLOCK WISE (CW) rotation and L indicate Low for the
COUNTER CLOCK WISE (CCW) direction input. In the same application,
the logic sequence is unidirectional, having no direction signal terminal. If
one increment of movement is performed by one step, the block diagram of
figure represents the whole system. But, when an increment is performed
by two or more steps, another stage to produce a proper train of pulses is
needed to put before the logic sequencer, and this is represented in next
Figure 2.4.
This logic circuit is denoted the “input controller”.
In
sophisticated applications the function of input controller is carried out
pulse train to speed up, slew, and slow down the motor in the most efficient
and reliable manner. The step command pulses are given from an external
source, and it is expected that the stepping motor is able to follow every
pulse [5].
Stepper motor uses electromagnetic field to create the motor shaft turn
into a precise distance when a sequence of pulses is applied to the input
pulses. The action is like the same poles of a magnet keep away and
different poles are attracted. Figure 2.5 shows a typical cross-sectional view
of the rotor and stator of a stepper motor. From the diagram, the stator has
four poles, and the rotor has six poles. So the rotor will require 12 pulses of
electricity to move the 12 steps to make one complete revolution. The rotor
will move precisely 30 degrees (360°/12 steps) for each pulse of electricity
the motor receives.
However for different stepper motor has different
degree rotation for each pulse.
Figure 2.5 Cross Sectional Of Stepper Motor
17
When no power is supplied to the motor, the residual magnetism in
the rotor magnets will cause the rotor to detent or align one set of its
magnetic pole. This magnetic pole will be aligned with the magnetic poles
of one of the stator magnets. For the figure type as shown in Figure 2.5, the
rotor will have 12 possible detent positions. When the rotor is in a detent
position, it will have enough magnetic force to keep the shaft from moving
to the next position. This makes the rotor feel like it is clicking from one
position to the next as the rotation rotate the rotor by hand with no power
supplied [16].
2.4.2 Type of stepper motor
There are three basic stepper motor types. They are:
• Variable-reluctance
• Permanent-magnet
• Hybrid
2.4.2.1 Variable-reluctance (VR) stepper motor
The variable reluctance motor has been around for a long time in
stepper world. Figure 2.6 shows a cross section of a typical V.R.
stepper motor. This VR motor consists of a soft iron multi-toothed
rotor and a wound stator. When the stator windings are energized
with DC current the poles become magnetized. Rotation occurs when
the rotor teeth are attracted to the energized stator poles [18]. The
variable-reluctance (VR) stepper motor at its core basically differs
from the PM stepper in that it has no permanent-magnet rotor and thus
no residual torque to hold the rotor at one position when turned off.
This means the field strength can be varied. The stator of a variablereluctance stepper motor has a magnetic core constructed with a stack
18
of steel laminations. The rotor is made of demagnetized soft steel
with teeth and slots, or any other such magnetically permeable
substance, unlike PM stepper motors.
When the stator coils are
energized, the rotor teeth will align with the energized stator poles. In
the non-energized condition there is no magnetic flux in the air gap so
there is no detent torque. This type of motor operates on the principle
of minimizing the reluctance along the path of the applied magnetic
field. By alternating the windings that are energized in the stator, the
stator field changes, and the rotor moves to a new position [6].
Figure 2.6 Cross Section of VR Stepper Motor
2.4.2.2 Permanent Magnet (PM) stepper motor
The permanent magnet step motor also known as a “tin can” or
“canstock” is a low cost and low resolution type motor with typical
step angles of 7.5° to 15° or 48 – 24 steps/revolution respectively.
The PM motors have permanent magnets which added to the motor
structure. Contrary to the variable reluctance (VR), the rotor of the
PM motor no longer has teeth as in the motor structure. Instead the
rotor is magnetized with alternating north and south poles situated in a
straight line parallel to the rotor shaft. These magnetized rotor poles
provide an increased magnetic flux intensity and because of this the
PM motor exhibits improved torque characteristics when compared
with the VR type [17]. A PM stepper motor operates on the reaction
between a permanent-magnet rotor and an electromagnetic field.
Figure 2.7 shows a cutaway diagram of a typical permanent magnet
19
stepper motor and shows that the permanent magnet motor can have
multiple rotor windings, which means that the shaft for this type of
stepper motor will turn fewer degrees as each pulse of current is
received at the stator [16].
Figure 2.7 Permanent Magnet Stepper Motor Cross Section
2.4.2.3 Hybrid stepper motor
The hybrid stepper motor unites the well characteristics of both
the PM and VR type stepper motors. The rotor of the hybrid motor is
a multi-toothed like the VR motor and includes an axially magnetized
concentric magnet around its shaft. The teeth on the rotor provide an
even better conduit which helps guide the magnetic flux to preferred
locations in the air gap. As a result, the detent, holding and dynamic
torque characteristics of the motor increase when compared with both
the VR and PM types. Normally, the hybrid stepper motor is more
expensive than the PM stepper motor but provides better performance
with respect to step resolution, torque and speed. A Typical step
angles for the hybrid stepper motor is range from 3.6° to 0.9° or 100 –
400 steps per revolution respectively [17].
Figure 2.8 Cross Section Hybrid Stepper Motor
20
A typical hybrid motor is shown in Figure 2.8.
The stator
construction is similar to the permanent magnet motor where the rotor
is a cylindrical type and magnetized like the PM motor with numerous
teeth like a VR motor. The teeth on the rotor provide a better path for
the flux to flow through the preferred locations in the air gap. This
increases the detent, holding, and dynamic torque characteristics of
the motor compared to the other two types of motors. Even the hybrid
motor are more expensive compare to the other two motors, one of the
advantages of hybrid motors is it offer a smaller step angle compared
to the permanent magnet motor. This feature gives a precise position
compare with PM and VR motors. For low cost applications, the step
angle of a permanent magnet motor is divided into smaller angles
using better control techniques [16].
The two most commonly used types of stepper motors in the
practical application are the permanent magnet and the hybrid types.
If a designer is not sure which type will best fit his applications
requirements he should first evaluate the PM type as it is normally
several times less expensive. If not then the hybrid motor may be the
right choice. There stepper motor designs also come with special
feature i.e. the disc magnet motor. This feature contain the rotor is
designed as a disc with rare earth magnets as shown in Figure 2.9.
This motor type has a very low inertia and a optimized magnetic flow
path with no coupling between the two stator windings.
qualities are essential in some applications [17].
Figure2.9 Special Features In HSM
These
21
2.5
Controller and driver
2.5.1 Current controller
The controlling process include chopper control perform inside
controller. Normally, the controller contains 2 PWM channels for chopper
control where to control the current of 2 phase stepper motor. The process
of selecting a correct PWM duty cycle is under the influence of different
parameters (the angle of the magnetic vector, the amount of current through
each phase and shaft resonance considerations). In synchronization, the
rotor teeth are aligned with the magnetic flux vector and with a small
amount of angular error which depends on the amount of load. The profile
of the torque can be adjusted by controlling the current flowing through
each phase using a sin table. By controlling the angle, the zero crossing of
torque profile can be adjusted about the stator. The duty cycle of the PWM
signal controls each phase voltage that include the flux linkage, the induced
voltage and the applied phase voltage. Since the phase current depends on
both phase voltages, rotor speed and motor specific parameter, constant
current controlling requires feedback of current [28].
2.5.2 Feedback linearizing controller
A feedback linearization algorithm is used to control a permanent
magnet (PM) stepper motor for point-to-point moves of a linear positioning.
Issues such as rotor speed estimation from an optical encoder, the effects of
high-frequency noise on current measurements due to pulse-width
modulated (PWM) amplifiers, and the constraints of source voltage limits
are addressed and resolved.
The feedback linearizing controller used
guarantees global trajectory tracking as long as the amplifier limits are not
exceeded. A setup consists of a PM stepper motor with a encoder attached
to a linear positioning location, two PWM amplifiers, and a digital signal
processor for implementing the algorithm.
The point of feedback
22
linearization control is to find a (nonlinear) state-space transformation such
that, in the new coordinates, the nonlinearities may be cancelled out by state
feedback. For the PM stepper motor, the appropriate nonlinear coordinate
transformation is known as the direct-quadrature (DQ) transformation and is
presented with phase voltage and current.
The PWM amplifiers are used for actuation because of their high
efficiency.
Due to the switching nature of the PWM, there is a high
frequency signal contained in the measurements of the phase currents. To
remove this noise, analog filters are added to filter the currents before
sampling. However, these filters add significant phase shifts to the current
measurements that shown that these phase shifts distort the DQ
transformation to the extent that the exact linearization controller will not
work. A solution to the current measurement noise is to apply the DQ
transformation on the noisy currents and then filter them on the digital
signal processor (DSP) [29].
2.5.3 Voltage level controller
The voltage-level controller is designed using robust backstepping by
first considering mechanical dynamics and then “stepping back” to electrical
dynamics. The dynamics of the three mechanical position states x, y, and z
are essentially similar and partially decoupled (coupling being only through
current dynamics and disturbance forces and torque). The dynamics of each
mechanical position state is second order with force (or torque) considered
as input. Since currents are unmeasured, backstepping is carried out not
through these unmeasured states, but instead through the filter state. The
filter states act as estimates of currents. Hence, the voltage control inputs
will be designed by robust backstepping applied to the subsystems [30].
23
2.5.4 Neural controller
A neural controller of the stepper motor has been created and
implemented in a FPGA unit using a VHDL model is based on voltagecurrent control strategy,. The neural stepper motor controller developed
uses a FPGA alongside with a programmable peripheral interface unit
82C55 (PPI) to control the stator voltage and current induced. As a means
of communication and control of the 80C188EB-based overall system
controller, an application control program in Intel 80x86 family assembly
language is provided. The software module that was implemented in the
controller generates, through the 82C55 programmable interface adapter, the
triggering signals the drive (a high-voltage and high current ULN2004
circuit) of the stepper motor that positions the manipulator's end-tool [31].
2.5.5 PID controller
Over 90% of the controllers in operation today are Proportional
Integral Derivative (PID) controllers (or at least some form of PID
controller like a P or PI controller). The PID controllers are based on the
common control algorithm, due to their simplicity and robustness. The PID
controllers are widely use. Moreover, they are often incorporated into the
programmable logic controllers (PLCs) that are used to control many
industrial processes. The basic form of the PID controller is
௧
‫ݑ‬ሺ‫ݐ‬ሻ ൌ ‫ܭ‬௣ ݁ሺ‫ݐ‬ሻ ൅ ‫ܭ‬௜ න ݁ሺ߬ሻ݀ ൅ ‫ܭ‬ௗ
଴
݀݁ሺ‫ݐ‬ሻ
݀‫ݐ‬
Where u is the input to the plant, y is plant output, r is the reference
input and the error input to the PID is e = r - y. Meanwhile, Kp is the
proportional gain, Ki is the integrals gain, and Kd is the derivative gain [14].
In order to validate the effectiveness of the system, mostly a stepper motor
speed control was implemented with PID and any controller. By assigning
feedback signal element, the system output voltage could be monitored for
each controller action [30].
24
2.5.6 L6208 Chip Driver
The L6208 is a fully integrated stepper motor driver manufactured
with multipower BCD technology, which combines isolated DMOS power
transistors with CMOS and bipolar circuits on the same chip. The L6208
is a highly integrated, mixed-signal power IC that allows the user to easily
design a complete motor control system for two-phase bipolar stepper
motors. Figure 2.10 shows the L6208 functional block diagram. The IC
integrates eight power DMOS devices, a centralized logic circuit which
implements the phase generation, a PWM current control technique (quasisynchronous mode) for each of the two phases of the motor, plus other
added features for safe operation and flexibility [16].
Figure 2.10 L6208 Functional Block Diagram
2.5.7 Microstepping driver
Microstepping driver is the classical approach to drive a PM stepper
motor at high resolution. It deals mechanical steps of the motor itself by
25
injecting current in several coils simultaneously: the rotor simply aligns
itself with the magnetic field generated by the stator coils. This highresolution electronic driver has the advantage of being easily controlled with
digital pulses, since each pulse corresponds to a single microstep (of about
40 nm with the 256 microstep driver IM483H and the NEMA 17 stepper
motor selected). So as to track unpredictable stochastic trajectories, it is
proposed to close the loop on the estimated position error of the coarse
stage. Since the coarse actuator perturbs the overall controlled output with
its actions, a hybrid like nonlinear control law is proposed to minimize its
action. The coarse controller input signal is evaluated and compared to an
absolute threshold. The action is taken only when the input signal is greater
than a predefined level. The stepper closed-loop stage reacts to the error
signal only when it is above a certain limit. With the dead-zone controller,
the piezoelectric stack actuator is driven slowly back to its mid-position
when getting too close to its stroke limits. Thus, the controller design
amounts to fixing the limits. Then, generating the trigger of the coarse
moves whenever the coarse stage estimated error is larger than the
predefined limits. The main advantage of such a controller is the ease of
implementation of the control algorithm. The controller deals only with
digital lines, that are really robust to electromechanical perturbations and it
does not require a complicated control algorithm [27].
2.6
Encoder
Encoder is a device used to change a signal or data into a code. The most
application is used in high precision machining tools. In motion control, the encoder
will convert the analog signal into digital signal to keep track the increasing or
decreasing number of revolutions of the motor shaft has rotates from its initial
position. In close-loop system, the encoder output will be used as a feedback signal
to the controller reference input to compensate the error. There are many types of
encoder. The simple and most application encoder now days are in mouse pc which
it uses optical encoder or quadrature encoder.
26
2.6.1 Incremental encoder
This type of encoders determines the current position by stating a
single data and counting measuring steps.
The output signals from
incremented rotary encoders are manipulated by an electronic counter in
where the measured value is generated by counting increments. These
encoders form the majority of all rotary encoders.
The electronics
interpolation can be manipulated to get the desire resolution which means
the precision rotary encoders capable to design for angle measurement for
precise positioning. The higher resolution, the more precise positioning of
the motion control.
Petar Crnoˇsija et al. [31] use incremental encoder to analyse of the
torque characteristics and the optimal control angle of hybrid stepper motor
drives with a chopper amplifier and current controller.
The proposed
optimal control algorithms have been realized and tested for feasibility in a
motor drive containing a hybrid stepper motor with incremental encoder by
Phytron, a power supply (chopper amplifier) and current controller by
Phytron, a microcontroller card by Intel, and a PC.
The two-phase
incremental encoder with 500 pulses/revolution was mounted on the motor
shaft. The dependence of the optimal control angle on angular speed was
calculated from the known parameter values of the presented stepper motor
drive using the analytical expressions. The number of pulses counted by the
timer within one step was measured.
2.6.3 Absolute encoder
Absolute rotary encoders are derived from the design of the coded
disc to provide an angular position since it’s require no previous transfer to
provide the current position.
The code signal is processed within a
computer or in a numerical control and position value produced once power
is available. There are many versions of absolute encoders available today,
27
such as single-turn or multi-stage versions and each must be evaluated based
on its intended application.
Absolute encoders provide a continuous,
accurate measurement of the angle, but they are expensive [32].
2.6.4 Linear encoder
A linear encoder is a sensor, transducer paired with a scale that
encodes position. The sensor reads the scale in for converting the encoded
position by a digital readout. Atsushi Kato et al. [33] used optical linear
encoder together with motor driver and PC to form a Linear DC Solenoid
Motor system. The optical linear encoder puts on the tip of moving part
directly. Its resolution is 0.1um.
2.6.5 Rotary encoder
A rotary encoder is an electro-mechanical device. It used to convert
the angular position of a shaft into digital code. It also called shaft encoder
making it a sort of a transducer.
Rotary encoders serve as measuring
sensors for rotary motion. In motion control, the rotary encoder can gives
speed information by counting increased or decreased number of encoder
pulses in sampling period but in ultra low speed range, the speed resolution
is easily lost. Kiyoshi Ohishi et al. [15] used rotary encoder to detect speed
of the motor to identify the parameter identification algorithm, the voltage
drop and the armature resistance variation. The rotary encoder has 20000
pulses/revolution resolution.
CHAPTER 3
PROJECT BACKGROUND
3.1
Introduction
Stepper motor is a nonlinear electromechanical incremental motion device
and cam be used in positioning applications. This motor was designed to provide
precise positioning control to within an integer number of steps without using
sensors. In fact, the stepper motor is open-loop stable to any step position and thus,
no feedback is needed to control them. This motor is electronically commutated and
has no windings on the rotor. The motor do not have two major disadvantages of DC
brush motors i.e. mechanical wear of the commutator and limitation to a 50% duty
cycle to allow the armature windings to cool. These features give better reliability
and better heat dissipations the winding are located on the stator and not on the rotor.
For these reasons, many positioning systems now use stepper motors with the
addition of encoder (position) feedback. Furthermore, it more reliable and, being
brushless machines, require less maintenance. The ability to provide accurate control
over speed and position combined with small size and relatively low cost make
stepper motor a popular choice in a range of applications. In particular, permanent
magnet stepper motor deliver higher peak torque per unit weight and has a higher
torque to inertia ratio than DC motor [21, 22].
However, the performance of a stepping motor is limited under the openloop mode. The drive may fail to follow a pulse command when the frequency of the
pulse train is too high or inertial load is too heavy. Their step response has quite a bit
of overshoot and a relatively long settling time. Moreover the motor motion tends to
be oscillatory in open–loop drives. The performance of stepping motor can be
improved to a great extent by employing position feedback and/or speed feedback to
29
determine the proper phase(s) to be switched at proper timings. This type of control
is termed ‘closed–loop’ drive. The closed–loop control is advantageous over the
open loop control not only in that step failure never occurs but also that the motion is
much quicker and smoother. In closed-loop control, a position sensor is needed for
detecting the rotor position. As a typical sensor, nowadays, many types of encoder is
used and it is usually coupled to the motor shaft. Position and speed sensors have
some disadvantages; they are usually expensive, the sensor and the corresponding
wires will take up space and inn defective and aggressive environments, the sensor
might be the weakest part of the system [1, 8]. As an alternative, an observer is
introduced to overcome the problem. The observer also called sensorless system.
The application of sensorless drives system is widely used in the world such as
hybrid electric vehicle, power-assisted wheelchair, city-scooter and high-power
railway traction.
3.2
Sensorless system control
3.2.1 MRAS observer
MRAS (Model Reference Adaptive System) observer is one of the
method to estimate speed the of a drive machines that is directed towards
the high performance speed control without a mechanical sensor for speed
feedback. The method MRAS observers based on rotor flux as the error
vector. The MRAS observer used to estimate the rotor speed and rotor flux
angle is shown in below. The rotor flux angle will be used to detect the
position of the motor shaft. The observer is based on a current-model is
derived from rotor flux equation and the voltage-model is derived from
stator flux equation.
Figure 3.1 shown the block diagram of MRAS
estimator for speed and angle estimator. The rotor frame equations for the
current model are given by;
ߖ௥௖ ൌ
‫ܮ‬଴
݅
߬௥ Ǥ ‫ ݏ‬൅ ͳ ௦
30
Where;
߬௥ ൌ
௅ೝ
ோೝ
is the rotor time constant. As can be seen in the block diagram, the rotor
flux angle is derived from the current model since this is independent of the
reference voltage of the machine [23].
Adjustment
parameter
Controller
Input
Current
model
Input
Voltage
model
Figure 3.1 Block Diagram of MRAS Estimator
3.2.2 Mathematical modelling of stepper motor
The dynamic equations, derived in are cast in state-space form as
follows [1]:
െܴ
‫ۍ‬
‫ܮ ێ‬
݅௔
‫ێ‬
݀ ݅௕
൦ ൪ൌ‫Ͳ ێ‬
݀‫߱ ݐ‬
‫݇ ێ‬௠
ߠ
െ
‫ܬ ێ‬
‫Ͳ ۏ‬
Ͳ
െܴ
‫ܮ‬
݇௠
ܿ‫ߠݏ݋‬
‫ܬ‬
Ͳ
݇௠
‫ߠ݊݅ݏ‬
‫ܮ‬
െ݇௠
ܿ‫ߠݏ݋‬
‫ܮ‬
െ‫ܤ‬
‫ܬ‬
ͳ
ͳ
Ͳ‫ې‬
‫ې ۍ‬
‫݅ ۑ‬௔
‫ۑ ܮێ‬
Ͳ‫ ۑ‬Ǥ ൦݅௕ ൪ ൅ ‫ ۑͳێ‬Ǥ ሾ‫ݒ‬
௔
‫߱ ۑ‬
‫ۑ ܮێ‬
‫ۑ‬
‫ۑͲێ‬
Ͳ ߠ
‫ۑ‬
‫ےͲۏ‬
Ͳ‫ے‬
And the output vector;
‫ݓ‬
Ͳ Ͳ ͳ
ቂ ቃൌቂ
ߠ
Ͳ Ͳ Ͳ
݅௔
Ͳ ݅௕
Ͳ
ቃ ൦ ൪ ൅ ቂ ቃ ሾ‫ݒ‬௔
ͳ ߱
Ͳ
ߠ
‫ݒ‬௕ ሿ
‫ݒ‬௕
Ͳ
Ͳሿ
31
Where;
߱
ቂ ቃ is the output vector
ߠ
R is the resistance of the coils
L is the inductance of the coils
݇௠ is the motor torque constant depending on the
design of
the rotor
J is the inertia of the rotor and the load
ߠ(t ) is the actual rotor position
ߠ is the location of the coil a in the stator
߱ is the rotational velocity of the rotor
݅௔ is the current in the coil as function of time
The model described above conforms to the general state space model given
by;
x = Ax + Bu
y = Cx + Du
x, u and y are called the state, input and output vectors, respectively.
Where;
ଵ
‫ۍ‬௅‫ې‬
‫ێ‬ଵ‫ۑ‬
‫ ܤ‬ൌ ‫ ێ‬௅ ‫ ۑ‬is called the input matrix,
‫ۑ Ͳێ‬
‫ے Ͳۏ‬
Ͳ Ͳ
‫ܥ‬ൌቂ
Ͳ Ͳ
ͳ Ͳ
ቃ is called output matrix and
Ͳ ͳ
Ͳ
‫ ܦ‬ൌ ቂ ቃ is the direct transmission matrix.
Ͳ
The equation above can be developed from follow derivation.
The derivative of winding current:
݅‫ ܮ‬ൌ ܸ െ ܴ‫ ܫ‬െ ݇߱
‫߱ܬ‬ሶ ൌ ߣ‫ ܫ‬െ ܶ௅
32
For phase A of stepper motor:
ܸ௔ ൌ ‫ܮ‬
ௗ௜ೌ
ௗ௧
െ ܴ݅௔ െ
ௗఒೌ
ௗ௧
ൌ Ͳሺ͹ሻ
For phase B of stepper motor:
ܸ௕ ൌ ‫ܮ‬
ௗ௜್
ௗ௧
െ ܴ݅௕ െ
ௗఒ್
ௗ௧
ൌ Ͳሺͺሻ
The flux due to permanent magnet of rotor for each phase:
ߣ௔ ൌ ߣ௠ …‘•ሺܰ௥ ߠሻ
ߣ௕ ൌ ߣ௠ •‹ሺܰ௥ ߠሻ
And the voltage induced in winding each phase (A and B):
െ݀ߣ௔
ൌ ݇௠ ɘ •‹ሺܰ௥ ߠሻሺͻሻ
݀‫ݐ‬
െ݀ߣ௕
݁௕ ൌ
ൌ െ݇௠ ɘ …‘•ሺܰ௥ ߠሻሺͳͲሻ
݀‫ݐ‬
݁௔ ൌ
Where;
݇௠ ‫ܰ ؜‬௥ ߣ௠ and ߱ ൌ
ௗఏ
ௗ௧
The power absorbed (negative power) by winding from voltage source;
݅௔ ݁௔ ൅ ݅௕ ݁௕ ൌ ݅௔ ݇௠ ɘ •‹ሺܰ௥ ߠሻ െ ݅௕ ݇௠ ɘ …‘•ሺܰ௥ ߠሻ
But
߬௠ ൌ െ݅௔ ݁௔ െ ݅௕ ݁௕
Thus,
߬௠ ൌ െ݅௔ ݇௠ •‹ሺܰ௥ ߠሻ ൅ ݅௕ ݇௠ …‘•ሺܰ௥ ߠሻ
At equilibrium position;
ߠ ൌ Ͳǡ ݅௔ ൌ ݅௢ ǡ ݅௕ ൌ Ͳ
Thus,
߬௠ ൌ െ݅௔ ݇௠ •‹ሺܰ௥ ߠሻ
Substitute (9) into (7) and substitute (10) into (8). These equations can be
formed into mathematical modelling of stepper motor as,
33
݀݅௔
െ ܴ݅௔ ൅ ݇௠ ɘ •‹ሺܰ௥ ߠሻ
݀‫ݐ‬
݀݅௕
ܸ௕ ൌ ‫ܮ‬
െ ܴ݅௕ െ ݇௠ ɘ …‘•ሺܰ௥ ߠሻ
݀‫ݐ‬
݀ߠ
ൌ߱
݀‫ݐ‬
݀߱
ൌ െ݅௔ ݇௠ •‹ሺܰ௥ ߠሻ ൅ ݅௕ ݇௠ …‘•ሺܰ௥ ߠሻ െ ‫ܤ‬
‫ܬ‬
݀‫ݐ‬
ܸ௔ ൌ ‫ܮ‬
3.2.3 Stepper motor selection
The hybrid stepper motor is manufactured by RS modeled RS440-442
was choosen for the proposed project. This four phase hybrid stepper motor
are capable of delivering much higher working torques and stepping rates
than permanent magnet types. The stepper motor can produce 1.8° per step
meanwhile permanent maganet stepper motor can achive 7.5° and 1.5° per
step. Whilst at the same time maintaining a high detent torque even when
not energised. This feature is important for position integrity. The choosen
stepper motor are supplied in 8-lead wire configuration which allows the
maximum felxibility when connecting to the drive boards. Rear extension
shafts are proviede to enable connection of the drive requirements and
feedback devices. The picture of stepper motor was depicted in Figure 3.2
below [24].
Figure 3.2 Stepper Motor Selected
34
The 8 lead wire configuration exiting sequence and direction of
rotation when facing mounting flange end are shown in Figure 3.3 and
Table 3.1. Technical specification of the selection stepper is depicted in
Table 3.2;
Figure 3.3 Stepper motor configuration
Table 3.1 Sequence And Direction Of Rotation HSM
Step
Red
1
On
Green Black Yellow
2
On
On
3
4
Com
+dcV
On
On
On
On
On
Table 3.2: Technical Specification of HSM
Specification
Value
Rated voltage
5V
Rated current
0.5I
Resistance
10ohm
Inductance
6mH
Detent torque
5mHm
Holding torque
70mNm
Step angle accuracy
5%
Step angle
1.8°
Insulation class
B
CW
35
3.2.4 Stepper Motor Controller
The existing of various microcontrollers now days give numerous
embedded and intelligent applications can be built. The project will use the
PIC microcontroller that is produced by Microchip. PIC is one of the
microcontroller (MCU) types and MCU is a computer implemented on a
single very large scale integrated (VLSI) circuit. MCU have been used in
almost every application that requires certain amount of intelligence. This
embedded system can be found in keyboard, printers, home appliance and
nowadays, they have being used in luxurious car. MCU contains some of
the following peripheral components [16, 25];
-
Memory
-
Timers
-
Pulse width modulation (PWM)
-
Analog to digital converter (ADC)
-
Digital to analog converter (DAC)
-
Parallel I/O interface
-
Asynchronous serial communication interface (UART)
-
Synchronous serial communication interface
-
Direct memory access (DMA controller)
-
Memory component interface controller
-
Software debug support software
In order to run the real-time control algorithm and create pulse width
modulation (PWM) signals, a 8-bit microcontroller from Microchip is used.
Microchip has introduced six different lines of 8-bit MCUs including
PIC12XXX, PIC14000, PIC16C5X, PIC16CXX, PIC 17 and PIC18 [25].
For this project PIC16f877 was used in order to create PWM. It consists of
fixed-point and floating-point capability. This microcontroller combines
this real-time processing capability with controller peripherals to create a
suitable solution for a vast majority of control system applications.
36
• Flexible instruction set
• Inherent operational flexibility
• Comparative high-speed performance
• Cost effectiveness
The pin and block diagram for the PIC16F877 is shown in Figure 3.4[25].
The other specification of PIC16F877 is attached in Appendix.
Figure 3.4 The Pin Diagram of PIC
37
Figure 3.5 Block Diagram For PIC16F8X
3.2.5 Motor Driver
The normal way of driving a four phase stepper motor is shown in
Figure 3.6. This is commonly known as the ‘Unipolar Linear Drive’ as
shown in Figure 3.6(a). When energizing, the current in each winding flows
in one direction only, n (n1) and ܴ݊ are the sum of the external resistance
plus the winding resistance, ܴ. By supplying a higher value for n (i.e. larger
external resistance) and dc voltages, the torque speed characteristics can be
improved.
This is because of the rated voltages and current is to be
maintained for each winding. As a result, a 6V, 6ohm motor (1A per phase)
can be driven from a 6V dc. Alternatively, it can be driven from a 24Vdc
supply using 18ohm series resistance in the L/4R mode with much improved
38
performance. The effect on motor performance of higher supply voltages
and larger series limiting resistance is shown in Figure 3.6(b).
(a)
(b)
Figure 3.6 a) Unipolar Drive b) The Effect On Motor Performance
On Higher Supply Voltage And Larger Series Limiting Resistance
One of the disadvantages of stepper motor is losses track on drive
input at certain frequencies. This situation occurs because of the stepper
motor facing resonant problem and at the same time an audible vibration
come together. These frequencies can be avoided by driving the motor on
half step mode which can reduce the effect of resonance. Alternatively,
extra load inertia and external damping may be added to shift resonance
regions away from the operating frequency. When the winding of the
stepper motor are assigned (‫ ͳ׎‬െ ‫׎‬Ͷ) as shown in Figure 3.7, they can be
connected to the driver board according to the unipolar drive in Figure
3.6a).
Figure 3.7 Typical Motor Winding Connections
39
A specific switching sequence for the drive transistors Q1-Q4 needs to
be followed in order to step a motor in particular direction. The sequence
examples are shown in Table 3.3 and Table 3.4 which for full step mode and
half step mode respectively. It results in the rotor advancing through one
complete step at a time [24].
Table 3.3 Full Step Mode
Step
no.
Start
position
Repeating
sequence
Q1
Q2
Q3
Q4
On
Off
Off
On
1
On
Off
On
Off
2
Off
On
On
Off
3
Off
On
Off
On
4
On
Off
Off
On
5
On
Off
On
Off
Anti
clockwise
Clockwise
Table 3.4 Half Step Mode
Step
no.
Start
Repeating
sequence
Q1
Q2
Q3
Q4
On
Off
Off
On
1
On
Off
On
Off
2
Off
On
On
Off
3
Off
On
Off
On
4
On
Off
Off
On
5
On
Off
On
Off
6
Off
On
On
Off
7
Off
Off
On
Off
8
On
Off
On
Off
Anti
clockwise
Clockwise
The implementation plant of the project was shown in the Table 3.5
which divided into two stages. The first stage is from January to April and
second stage from July to November.
7. Report
6. Seminar Presentation
5. Slide preparation
4.Synopsis preparation
- Circuit simulation
- Driver
- Controller (PIC)
- Stepper motor
- Previous research
2.Stepper Motor Control
system
- Motor
- Observer
1. Literature review
2
Week
1
January
Month
3
4
1
2
February
3
4
Table 3.5 Project Implementation Plan
1
2
March
3
4
1
April
2
6. Writing
- Performance analysis
-Experimental result
5. Result
- MRAS application
- Hardware and software
4. Integration of the system
- Derivation
- Modelling
3. MRAS Observer
- Signal generation and controlling
2.Software development
- Validation
- Circuit construction
- Device procurement
1. Hardware part
Month
Week
1
July
2 3
4
August
1 2 3 4
September
1 2 3 4
October
1 2 3 4
November
1 2 3 4
CHAPTER 4
SOFTWARE AND HARDWARE CONSTRUCTION
4.1
Characteristics of hybrid stepper motor
Since the hybrid stepper motor is a combination of variable reluctance and
permanent magnet, the voltages induced in VR stepper motor must also be
considered. The variation of the phase inductance with rotor position will produce
the voltages that were induced by the rotor motion. If the stator and rotor teeth of
one phase are misaligned, the flux path will have high reluctance. For a given phase
current, a small flux links the winding, so the phase inductance is at its minimum
value. The opposite values occur for a fully aligned condition.
4.2.1 Circuit representation
In the hybrid stepping motor, the inductance of the windings is almost
independent of the rotor position and the average inductance specified by the
manufacturer. The phase circuit model shown in Figure 4.1 includes the total
circuit resistance (R) and the average winding inductance (L). The hybrid
motor has four phase windings that are mounted on separate stator poles.
Consequently for each winding, the phase circuit model must include the
resistance and inductance.
Normally, the inductance is specified by the
manufacturer and the resistance is specified in the drive circuit design.
43
R
L
Figure 4.1 Circuit Model For One Phase Of A Hybrid Stepper Motor
4.2.2
Speed characteristic of hybrid stepper motor
For the complete model, the voltages that were induced in the phase
winding by the rotor motion must be considered. The induced voltage occur
because of the permanent magnet flux linkage at each windings are varies
sinusoidally with the rotor position. The flux linkage for four phase hybrid
stepper motor with p rotor teeth can be described as;
߰஺ ൌ ߰ெ •‹ ‫ߠ݌‬
߰஻ ൌ ߰ெ •‹ሺ‫ ߠ݌‬െ ߨȀʹሻ
߰஼ ൌ ߰ெ •‹ሺ‫ ߠ݌‬െ ߨሻ
߰஽ ൌ ߰ெ •‹ሺ‫ ߠ݌‬െ ͵ߨȀʹሻ
(11)
Where ߮ெ , is the maximum flux linkage each winding,‫݌‬is number of rotor
teeth.
‫ ݌‬ൌ ͻͲȀ‫ݐ݄݈݃݊݁ ݌݁ݐݏ‬
ൌ
ͻͲ
ͳǤͺ
ൌ ͷͲ
(12)
‫ ߠ݌‬is integration of the average rotor velocity over one supply cycle with
respect to time. This will give the variation of rotor position with time as
expressed below;
44
݀ߠ
ൌ ݀݅‫݁ܿ݊ܽݐݏ‬Ȁ‫݁݉݅ݐ‬
݀‫ݐ‬
߱
ʹߨ
ߠ ൌ ሺʹߨȀ‫݌‬ሻȀሺ ሻ ൌ
‫݌‬
߱
Thus,
‫ ߠ݌‬ൌ ߱‫ ݐ‬െ ߜሺͳ͵ሻ
Where;
ߜ ൌ ‫ି݊ܽݐ‬ଵ ሺ߱‫ܮ‬Ȁܴሻ
The voltages in the phase windings are equal to the rate of change of flux
linkages.
Thus,
݀߰௔
݀ߠ
ൌ ‫߰݌‬ெ …‘•ሺ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
݀߰௕
݀ߠ
݁஻ ൌ
ൌ ‫߰݌‬ெ …‘•ሺ‫ ߠ݌‬െ ߨȀʹሻ
݀‫ݐ‬
݀‫ݐ‬
݀߰௖
݀ߠ
݁஼ ൌ
ൌ ‫߰݌‬ெ …‘•ሺ‫ ߠ݌‬െ ߨሻ
݀‫ݐ‬
݀‫ݐ‬
݀߰ௗ
݀ߠ
݁஽ ൌ
ൌ ‫߰݌‬ெ …‘•ሺ‫ ߠ݌‬െ ߨȀʹሻ ሺͳͶሻ
݀‫ݐ‬
݀‫ݐ‬
݁஺ ൌ
This four phase hybrid stepper motor is excited by positive and negative
current. With four steps will produce one excitation cycle. If the motor
running at stepping rate f, then the excitation cycle repeats at frequency of
f/4. Thus, for one excitation cycle, the angular frequency is given by;
݂
߱ ൌ ʹߨ ൈ ൬ ൰ ൌ ߨ݂Ȁʹ
Ͷ
From Figure.1.1, the instantaneous voltages and currents in each phase can
be expressed as below;
݀݅஺
൅ ݁஺
݀‫ݐ‬
݀݅஻
‫ݒ‬஻ ൌ ܴ݅஻ ൅ ‫ܮ‬
൅ ݁஻
݀‫ݐ‬
‫ݒ‬஺ ൌ ܴ݅௔ ൅ ‫ܮ‬
45
‫ݒ‬஼ ൌ ܴ݅஼ ൅ ‫ܮ‬
‫ݒ‬஽ ൌ ܴ݅஽ ൅ ‫ܮ‬
݀݅஼
൅ ݁஼
݀‫ݐ‬
ௗ௜ವ
ௗ௧
൅ ݁஽ ሺͳͷሻ
The fundamental currant component in phase for each phase can be
expressed as;
݅஺ ൌ ‫ݏ݋ܿܫ‬ሺ߱‫ ݐ‬െ ߜ െ ܽሻ
݅஻ ൌ ‫ݏ݋ܿܫ‬ሺ߱‫ ݐ‬െ ߜ െ ܽ െ ߨȀʹሻ
݅஼ ൌ ‫ݏ݋ܿܫ‬ሺ߱‫ ݐ‬െ ߜ െ ܽ െ ߨሻ
݅஽ ൌ ‫ݏ݋ܿܫ‬ሺ߱‫ ݐ‬െ ߜ െ ܽ െ ͵ߨȀʹሻሺͳ͸ሻ
Where ܽ is a phase angle, ߜis the load angle.
By taking phase A, the flux linked with phase is the product of current and
inductance;
߰஺ ൌ ‫ܮ‬஺ ݅஺ (17)
Where ‫ܮ‬஺ is the phase inductance with rotor teeth which is given by
‫ܮ‬஺ ൌ ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሺͳͺሻ
And the rate of changes in flux linkage with time can divided into two
portions, the first part is the voltage induced in the phase windings by the
rotor motion and the second is the changing current in the phase inductance
[33]. Thus can be described as;
݀߰஺
݀‫ܮ‬஺
݀݅஺
ൌ ݅஺
൅ ‫ܮ‬஺
݀‫ݐ‬
݀‫ݐ‬
݀‫ݐ‬
݀‫ܮ‬஺ ݀ߠ
݀݅஺
ൌ ݅஺
ൈ
൅ ‫ܮ‬஺
݀ߠ ݀‫ݐ‬
݀‫ݐ‬
߱
݀݅஺
ൌ ݅஺ ሺ‫ܮ݌‬ଵ …‘• ‫ߠ݌‬ሻ ൈ ൅ ‫ܮ‬஺
‫݌‬
݀‫ݐ‬
ൌ ߱‫ܮ‬ଵ ݅஺ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ‫ܮ‬஺
݀݅஺
ሺͳͻሻ
݀‫ݐ‬
46
4.2
Simulation Implementation
The
model
of
the
stepper
motor
system
was
simulated
in
Simulink/MATALB. There are two parts simulation, open loop and close loop
which means the MRAS observer is included.
4.2.1 Open loop simulation
Figure 4.2 Simulink Block For Open Loop
The Simulink model for open loop is presented in Figure 4.2. The
motor phases are fed by three H-bridge MOSFET PWM converters as
shown in Fig. 4.3. For simulation part, the DC bus is representing by a 28V DC voltage source. The motor currents are independently regulated by
three hysteresis-based controllers that generate the MOSFETs drive signals
by comparing the measured currents with their references. The ripple in the
current waveforms is controlled by the hysteresis band of the comparators.
The switching frequency is variable and dependent on the motor parameters.
47
Figure 4.3 Signal Generator Output
For this simulation, double-phase-on excitation scheme is used.
Square-wave current references are generated using the current amplitude
and the step frequency parameters specified in the dialog window. The
movement of the stepper drive is controlled by the STEP and DIR signals
received from Signal Builder block. The current amplitude and the stepping
rate are selected in the dialog mask to be 2A and 238.7 step/s respectively.
The STEP signal from the Signal Builder block controls the movement of
the stepper drive. A positive value (1.0) will make the motor rotates and a
zero value will stop the rotation. The DIR signal controls the rotation
direction. A positive value (1.0) will impose the positive direction while a
zero value will impose the reverse direction [34]. This is shown in Figure
4.3.
The simulation is done using a variable-step solver with a sampling
time of 1 us, providing acceptable accuracy for the PWM. If a high PWM
accuracy is required, a smaller step time can be used but the simulation will
be slower.
block
Additional
Figure 4.4 Look Under Mask For Four Phase HSM
49
In the Simulink block, when changing the two phase into four phase
stepper motor, the additional connections are needed in order to connect two
extra points in HSM. This change is required in order to synchronize of the
hardware application where the stepper motor is four phase type. These
connections are connected to the Driver where some additional blocks have
been done in Look Under Mask of the Driver. The additional blocks were
depicted in Figure 4.4.Figure 4.5 shows the Look Under Mask of the HSM
and Model discrete four phase HSM.
The phases of FEM input was
changed from [ 0 –pi/2] into [0 –pi/2 –pi pi/2] (which is equivalent to [0 –
pi/2 –pi –pi3/2] ) for four phase condition. Table 4.1 shows the parameters
used in simulation.
(a)
(b)
Figure 4.5 a)Look Under Mask of HSM b) Look Under Mask of
Model Discrete 4 Phases HSM
50
Table 4.1 Parameters for Driver Block and Hybrid Stepper Motor
Parameters
Driver Block
HSM
2
2
238.7
238.7
Ts
Ts
Phase current (A)
Stepping rate (step/s)
Sampling time (s)
4.2.1 Close loop simulation
The close loop system required the transformation of four phase
system to two phase system using ݀ߠ െ ‫ ߠݍ‬rotational condition.
4.2.2.1 Transformation of four phase into two phase system
Figure 4.6 shows the star connection of the stator windings, an
equivalent four-phase synchronous machine with constant air gap and
one pole pair.
݀ఏ
d axis
ߠ
݃஺
݃஻
݃஽
q axis
‫ݍ‬ఏ
݃஼
Figure 4.6 Equivalent Constant Air Gap For
4 Phase HSM
51
Based on the space-phasor theory, the space phasor ݃Ԧ
corresponding to four phase system of magnitudes ݃஺ ǡ ݃஻ ǡ ݃஼ ǡ and ݃஽
where ݃Ԧ can be stator current ݅௦ , voltage ‫ݒ‬௦ or flux ߰௦ . Thus, the
space phasor for voltage and current are given by;
ʹ
ʹ
෍ ‫ ݒ‬ൌ ሺ‫ݒ‬஺ ൅ ‫ݒ‬஻ ൅ ‫ݒ‬஼ ൅ ‫ݒ‬஽ ሻ ൌ ‫ݒ‬ௗ ൅ ݆‫ݒ‬௤
݇
݇
ʹ
ʹ
ଓԦ ൌ ෍ ݅ ൌ ሺ݅஺ ൅ ݅஻ ൅ ݅஼ ൅ ݅஽ ሻ ൌ ݅ௗ ൅ ݆݅௤ ሺʹͲሻ
݇
݇
‫ݒ‬Ԧ ൌ
Where ‫ݒ‬ௗ ,݆‫ݒ‬௤ ,݅ௗ and ݆݅௤ are the components in the complex plane
and ݇ is the number of phase. The homopolar components ‫ݒ‬଴ା and
‫ݒ‬଴ି are needed for the transformation of four phase system into two
phase system as;
ͳ
ሺ‫ ݒ‬൅ ‫ݒ‬஻ ൅ ‫ݒ‬஼ ൅ ‫ݒ‬஽ ሻ
݇ ஺
ͳ
ൌ ሺ‫ݒ‬஺ െ ‫ݒ‬஻ ൅ ‫ݒ‬஼ െ ‫ݒ‬஽ ሻ
݇
‫ݒ‬଴ା ൌ
‫ݒ‬଴ି
ሺʹͳሻ
These homopolar components are linked to each other together with
space phasor in matrix form where ݇ ൌ Ͷ as follow;
ሾ‫ݒ‬Ԧ ሿ ൌ ሾܽԦସ ሿሾ‫ ݒ‬ሿସ ൌ ൣ‫ܬ‬Ԧସ ൧ሾ‫ݒ‬ሿଶ
Where,
‫ݒ‬஺
‫ݒ‬ௗ
‫ݒ‬Ԧ
‫ݒ‬஻
‫ݒ‬
‫ݒ‬Ԧ ‫כ‬
ሾ‫ݒ‬Ԧሿ ൌ ൦ ൪ ǡ ሾ‫ݒ‬ሿସ ൌ ൦ ‫ ݒ‬൪ ǡ ሾ‫ݒ‬ሿଶ ൌ ቎ ௤ ቏
‫ݒ‬
‫ݒ‬଴ା
஼
଴ା
‫ݒ‬଴ି
‫ݒ‬஽
‫ݒ‬଴ି
ͳ
ͳ ͳ
ሾܽԦସ ሿ ൌ ൦
ʹ ͲǤͷ
ͲǤͷ
݅
െ݅
ͲǤͷ
െͲǤͷ
െͳ െ݅
െͳ
݅
൪
ͲǤͷ ͲǤͷ
ͲǤͷ െͲǤͷ
52
ͳ
ͳ
ൣ‫ܬ‬Ԧସ ൧ ൌ ൦
Ͳ
Ͳ
݅
െ݅
Ͳ
Ͳ
Ͳ
Ͳ
ͳ
Ͳ
Ͳ
Ͳ
൪
Ͳ
ͳ
ሺʹʹሻ
Thus,
ͳ
ሾ‫ݒ‬Ԧ ሿ ൌ ͲǤͷ ൦ ͳ
ͲǤͷ
ͲǤͷ
ͳ ݅
ͳ െ݅
ൌ൦
Ͳ Ͳ
Ͳ Ͳ
݅
െ݅
ͲǤͷ
െͲǤͷ
Ͳ
Ͳ
ͳ
Ͳ
‫ݒ‬஺
െͳ െ݅
‫ݒ‬
െͳ
݅
஻
൪൦ ൪
ͲǤͷ ͲǤͷ ‫ݒ‬஼
ͲǤͷ െͲǤͷ ‫ݒ‬஽
Ͳ ‫ݒ‬ௗ
Ͳ ‫ݒ‬௤
቏
൪቎
Ͳ ‫ݒ‬଴ା
ͳ ‫ݒ‬଴ି
Therefore, the formulae of transformation from four phase system into
two phase system or vice verse are;
ሾ‫ݒ‬ሿଶ ൌ ሾ‫ܣ‬ସ ሿሾ‫ ݒ‬ሿସ
and
ሾ‫ ݒ‬ሿସ ൌ ሾ‫ܣ‬ସ ሿିଵ ሾ‫ݒ‬ሿଶ Where;
ሾ‫ܣ‬ସ ሿ ൌ ሾ‫ܬ‬ସ ሿିଵ ሾܽସ ሿ and
ሾ‫ܣ‬ସ ሿିଵ ൌ ሾܽସ ሿିଵ ሾ‫ܬ‬ସ ሿ
Thus for transformation of the four phase hybrid stepper motor can be
explained as;
ሾ‫ݒ‬ሿଶ ൌ ሾ‫ܣ‬ସ ሿሾ‫ݒ‬ሿସ
ൌ ሾ‫ܬ‬ସ ሿିଵ ሾܽସ ሿሾ‫ݒ‬ሿସ
ͳ ݅
ͳ െ݅
ൌ൦
Ͳ Ͳ
Ͳ Ͳ
Ͳ
Ͳ
ͳ
Ͳ
Ͳ ିଵ
ͳ
Ͳ
ͳ
൪ ͲǤͷ ൦
ͲǤͷ
Ͳ
ͲǤͷ
ͳ
݅
െ݅
ͲǤͷ
െͲǤͷ
‫ݒ‬஺
െͳ െ݅
‫ݒ‬
െͳ
݅
஻
൪ ൦ ൪
ͲǤͷ ͲǤͷ ‫ݒ‬஼
ͲǤͷ െͲǤͷ ‫ݒ‬஽
53
‫ݒ‬஺
‫ݒ‬ௗ
ͲǤͷ
Ͳ
െͲǤͷ
Ͳ
‫ݒ‬
Ͳ
ͲǤͷ
Ͳ
െͲǤͷ ‫ݒ‬஻
ൌ቎ ௤቏ൌ൦
൪ ൦ ൪ ሺʹ͵ሻ
‫ݒ‬଴ା
ͲǤʹͷ ͲǤʹͷ ͲǤʹͷ ͲǤʹͷ ‫ݒ‬஼
‫ݒ‬଴ି
ͲǤʹͷ െͲǤʹͷ ͲǤʹͷ െͲǤʹͷ ‫ݒ‬஽
The complex equations of the four phase system into two phases are;
‫ݒ‬ௗ ൌ ͲǤͷ‫ݒ‬஺ െ ͲǤͷ‫ݒ‬஼ ൌ ͲǤͷሺ‫ݒ‬஺ െ ‫ݒ‬஼ ሻ
‫ݒ‬௤ ൌ ͲǤͷ‫ݒ‬஻ െ ͲǤͷ‫ݒ‬஽ ൌ ͲǤͷሺ‫ݒ‬஻ െ ‫ݒ‬஽ ሻ
(24)
The above equations are in fixed stator ݀ െ ‫ ݍ‬system. In order to
construct the system in rotational ݀ߠ െ ‫ ߠݍ‬system, the rotational
operator of ሾ‫ܦ‬ସ ሺߠሻሿ is used where the two phases can be constructed
from;
ሾ‫ݒ‬ଶ ሿଶఏ ൌ ሾ‫ܦ‬ସ ሺߠሻሿሾ‫ݒ‬ሿଶ
Where
ܿ‫ߠݏ݋‬
ሾ‫ܦ‬ସ ሿ ൌ ൦െ‫ߠ݊݅ݏ‬
Ͳ
Ͳ
ሾ‫ݒ‬ଶ ሿଶఏ
‫ߠ݊݅ݏ‬
ܿ‫ߠݏ݋‬
Ͳ
Ͳ
Ͳ
Ͳ
ͳ
Ͳ
Ͳ
Ͳ
൪
Ͳ
ͳ
‫ݒ‬ௗఏ
‫ݒ‬
ൌ ቎ ௤ఏ ቏ሺʹͷሻ
‫ݒ‬଴ାఏ
‫ݒ‬଴ିఏ
Thus,
‫ݒ‬ௗఏ
ܿ‫ߠݏ݋‬
‫ݒ‬௤ఏ
െ‫ߠ݊݅ݏ‬
቏ൌ ൦
቎
‫ݒ‬଴ାఏ
Ͳ
‫ݒ‬଴ିఏ
Ͳ
‫ߠ݊݅ݏ‬
ܿ‫ߠݏ݋‬
Ͳ
Ͳ
Ͳ
Ͳ
ͳ
Ͳ
Ͳ ‫ݒ‬ௗ
Ͳ ‫ݒ‬௤
቏
൪቎
Ͳ ‫ݒ‬଴ା
ͳ ‫ݒ‬଴ି
As a result, the rotational stators ݀ߠ െ ‫ ߠݍ‬for the voltages are;
‫ݒ‬ௗఏ ൌ ‫ݒ‬ௗ ܿ‫ ߠݏ݋‬െ ‫ݒ‬௤ ‫ߠ݊݅ݏ‬
‫ݒ‬௤ఏ ൌ ‫ݒ‬௤ ܿ‫ ߠݏ݋‬െ ‫ݒ‬ௗ ‫ ߠ݊݅ݏ‬ሺʹ͸ሻ
54
Since the derivation of voltages and currents are identical, thus the
rotational stators ݀ߠ െ ‫ ߠݍ‬for the currents are [35];
݅ௗఏ ൌ ݅ௗ ܿ‫ ߠݏ݋‬െ ݅௤ ‫ߠ݊݅ݏ‬
݅௤ఏ ൌ ݅௤ ܿ‫ ߠݏ݋‬െ ݅ௗ ‫ߠ݊݅ݏ‬
(27)
4.2.2.2 Derivation of Model Reference Adaptive Control (MRAS)
The whole system of the hybrid stepper motor including MRAS
become a close loop or a sensorless system is shown in the Figure 4.7
and the basic configuration of the MRAS is depicted in Figure 4.8.
Power supply
Controller
Driver
MRAS
ADC
Figure 4.7 Block Diagram Of The Whole System
Reference
model
Adjustable
model
Adaptive
mechanism
Plant
Figure 4.8 Basic Configuration of MRAS
55
The reference models are represented from equation (14) and (15) and
can be expressed as;
݀݅஺
݀߰஺
ൌ ‫ݒ‬஺ െ ܴ݅஺ െ ‫ܮ‬
݀‫ݐ‬
݀‫ݐ‬
݀߰஻
݀݅஻
ൌ ‫ݒ‬஻ െ ܴ݅஻ െ ‫ܮ‬
݀‫ݐ‬
݀‫ݐ‬
݀݅஼
݀߰஼
ൌ ‫ݒ‬஼ െ ܴ݅஼ െ ‫ܮ‬
݀‫ݐ‬
݀‫ݐ‬
݀݅஽
݀߰஽
ൌ ‫ݒ‬஽ െ ܴ݅஽ െ ‫ܮ‬
ሺʹͺሻ
݀‫ݐ‬
݀‫ݐ‬
And adjustable models can be expressed from equation (18) and (19)
as;
݀߰஺
݀݅஺
ൌ ߱‫ܮ‬ଵ ݅஺ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ሺ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
݀݅஻
݀߰஻
ൌ ߱‫ܮ‬ଵ ݅஻ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ሺ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
݀݅஼
݀߰஼
ൌ ߱‫ܮ‬ଵ ݅஼ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ሺ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
݀݅஽
݀߰஽
ൌ ߱‫ܮ‬ଵ ݅஽ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ሺ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
Since the system is using
models are reduced.
݀ߠ െ ‫ ߠݍ‬rotational system, the above
By substitution equation (24) into (26) the
models are simplified as below.
For reference model in ݀ߠ െ ‫ ߠݍ‬system are;
‫ݒ‬ௗఏ ൌ ͲǤͷሺ‫ݒ‬஺ െ ‫ݒ‬஼ ሻܿ‫ ߠݏ݋‬൅ ͲǤͷሺ‫ݒ‬஻ െ ‫ݒ‬஽ ሻ‫ߠ݊݅ݏ‬
‫ݒ‬௤ఏ ൌ ͲǤͷሺ‫ݒ‬஻ െ ‫ݒ‬஽ ሻܿ‫ ߠݏ݋‬െ ͲǤͷሺ‫ݒ‬஺ െ ‫ݒ‬஼ ሻ‫ߠ݊݅ݏ‬
By substituting equation (28), the flux linkages for reference models
become;
56
݀݅ௗఏ
݀߰ௗఏ
ൌ ‫ݒ‬ௗఏ െ ܴ݅ௗఏ െ ‫ܮ‬
݀‫ݐ‬
݀‫ݐ‬
݀߰௤ఏ
݀݅௤ఏ
ൌ ‫ݒ‬௤ఏ െ ܴ݅௤ఏ െ ‫ܮ‬
݀‫ݐ‬
݀‫ݐ‬
Meanwhile for adaptation model in ݀ߠ െ ‫ ߠݍ‬system are;
݅ௗఏ ൌ ͲǤͷሺ݅஺ െ ݅஼ ሻܿ‫ ߠݏ݋‬൅ ͲǤͷሺ݅஻ െ ݅஽ ሻ‫ߠ݊݅ݏ‬
݅௤ఏ ൌ ͲǤͷሺ݅஻ െ ݅஽ ሻܿ‫ ߠݏ݋‬െ ͲǤͷሺ݅஺ െ ݅஼ ሻ‫ ߠ݊݅ݏ‬ሺʹͻሻ
When substituting equation (29), the flux linkage for adaptation
models can be formed as;
݀݅ௗఏ
݀߰ௗఏ
ൌ ߱‫ܮ‬ଵ ݅ௗఏ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ሺ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
݀߰௤ఏ
݀݅௤ఏ
ൌ ߱‫ܮ‬ଵ ݅௤ఏ …‘•ሺ߱‫ ݐ‬െ ߜሻ ൅ ሺ‫ܮ‬଴ ൅ ‫ܮ‬ଵ •‹ ‫ߠ݌‬ሻ
݀‫ݐ‬
݀‫ݐ‬
The adaptive mechanism is using Proportional Integral (PI) controller
to turn the error state of reference and adaptive model to zero by
adjusting the input of adaptive model, which variable is the rotor
speed. The error between two models is the angle between two flux
linkage vectors and can be expressed as;
݁ఠ ൌ ߰෠௩ఏ െ ߰෠௜ఏ
The angle produced between two flux linkages was then used in
adjustment parameter to get the speed. Then, the PI controller was
used as speed regulation in order to regulate the state speed to follow
the actual speed.
equation (13).
The parameter adjustment was derived from
And by using the PI controller, the regulated
estimation speed as;
57
௧
߱
ෝ ൌ ሺ‫ܭ‬௣ ൅ න ‫ܭ‬௜ ሻ݁ఠ
଴
The position of the sensorless system can be achieved by differential
the speed [35, 36]. The Simulink block diagram for sensorless system
for hybrid stepper motor is depicted in Figure 4.9. The subsystems are
attached in Appendix.
4.3
Hardware implementation
4.3.1 Open loop system
The open loop of the hybrid stepper motor system is using PIC to get
the desired speed and position. Before proceed to develop the circuit, the
simulation of the circuit had been executed. The circuit simulation for
stepper motor driver was carried out with MultiSim7. Figure 4.10 shows
the circuit simulation for stepper motor driver with MultiSim7. The driver
circuit was selected from manufacturer recommendation but need to include
inverter as buffer (7404N).
This buffer is included to overcome the
overshoot at the outputs. The LED diodes are acting as coil for each stepper
motor winding since the MultiSim 7 do not have stepper motor tool. The
stepping signals to each coil of the stepper motor were examined by
oscilloscope.
Figure 4.11 shows the result from the driver circuit
simulation. The generated waveform is same as expected in Table 3.3 Full
step mode.
As a conclusion, the stepper motor driver can be used and the next
step is to develop an open loop system of the stepper motor system
hardware followed by two stages close-loop system. The first stage is using
the encoder to get the feedback input and then will be replaced with
observer as a feedback to the drive system.
construction of the hybrid stepper motor.
Figure 4.12 shows the
Figure 4.9 Simulink Block For Close Loop System
59
Figure 4.10 Circuit Simulation For Stepper Motor Driver
Figure 4.11 Waveform At Each Coil
(a)
(b)
Figure 4.12 Circuit Construction On A)Protoboard B)Stripboard
4.3.1 Close loop system
In order to make the system close loop, a rotary encoder is applied to
the open loop system. Rotary encoder is a sensor or transducer used to
convert the data of rotary motion into a series of electrical pulses that can be
60
readable by controller. This type encoder comes with eight slotted disc and
sensor board. The slotted disc has 35mm outsider diameter with eight slots
that provides sixteen transitions. The optical sensor is used to sense the 16
transitions. The pulses produced will be sent to controller to recognize the
rotary angle of the disc.
Therefore, the current position of the hybrid
stepper motor can be acquired by employ the rotary encoder. The rotary
encoder is depicted in Figure 4.13.
Rotor
shaft
(a)
(b)
Figure 4.13 a)Pair Of Optical And 8 Slot Disc Rotary
Encoder. b) Motor Shaft Was Slotted With Rotary Encoder Disc
In second stage, the encoder was removed and replace with the MRAS
system.
The hardware implementation for sensorless system can be
achieved via converting the simulation of sensorless Simulink/MATLAB
file into C language by using Real Time Workshop. Then compile the C
language by MicroC compiler before loaded into microcontroller. Since the
voltage phase and current phase of the hardware is analogue input, thus the
analogue to digital converter (ADC) is required. Hence, a PCI 1711 card
from Advantech was utilized.
CHAPTER 5
RESULT AND DISCUSSION
5.1
Introduction
The sensorless technique using Model Reference Adaptive System (MRAS)
for hybrid stepper motor has been developed. The sensorless technique purpose is to
replace the physical sensor or encoder to measure observable parameters. By using
these parameters, it can estimate speed or position using the measured states in form
of current or voltage. The simulation of the system is using the proper system model
and input in MATLAB. The hardware implementation also has been done in the
open loop system to obtain the measured value of the hybrid stepper motor speed.
5.2 Open loop
Figure 5.1 depict the output from open loop simulation with zero load
torque where it shows the voltages phase, currents phase and torque while in Figure
5.2 shows the speed and position respectively. From this figure, it shows that the
stepper motor moved forward where the position increase from 0 rad to 0.82 rad with
respected speed of 140 rad/s in 0.105s. Then it stays at same position for 0.055s
before it reverse the direction at time 0.153s and stop at 0.205s at 0.44 rad
62
Figure 5.1 Voltage Phase, Current Phase And Torque of Open
Loop HSM
Figure 5.2 The Speed (Top) And Position (Bottom) of The
Open Loop HSM
63
Figure 5.3 The Speed (Top) And Position (Bottom) of The MRAS
System Without Feedback
Figure 5.4 The Comparison Between Open Loop System And Open Loop MRAS
System Without Feedback For Speed (Top) And Position (Bottom)
64
Figure 5.3 shows the speed and position of the MRAS system without
feedback respectively. Many time simulations were throughout in order to obtain the
PI value to produce the best performance. The value of Proportional is 1120000 and
Derivative is 1. Generally the figure illustrates the MRAS system without feedback
is able to achieve the target position with slowly raising the speed. On the other
hand, the position is slowly increases when forward condition (DIR ‘1) and stay at
position 0.8 rad for almost 0.06s but very fast decrease when backward condition
(DIR ‘0’) and stop at low position i.e. 0.5 rad. Figure 5.4 show the comparison
between the open loop system of HSM and MRAS system without feedback.
For hardware implementation result, the signal was tapped from rotary
encoder point and observes the signal via oscilloscope. The data taken was shown in
Figure 5.5. From the graph, the period is 50ms (2.5scale box) and the rotary encoder
gives eight pulses per revolution.
Figure 5.5 Output Signal of Rotary Encoder
As a result, the speed of the hardware implementation can be achieved. The speed of
the rotary encoder can be obtained from revolution per minute equation.
Revolution per minute,
ܴܲ‫ ܯ‬ൌ ͸ͲȀሺ‫ ݊݋݅ݐݑ݈݋ݒ݁ݎݎ݁݌݁ݏ݈ݑ݌‬ൈ ‫݀݋݅ݎ݁݌‬
ൌ
͸Ͳ
ͺ ൈ ͷͲ݉‫ݏ‬
ൌ ͳͷͲ ‫݀ܽݎ‬Ȁ‫ݏ‬
65
5.3
Close loop (Sensorless system)
These close loop simulations were carried out with model reference
adaptive system (MRAS) with feedback in many time in order to obtain and verify
the performance of the speed and position as accurate as possible. The feedback is
the output of the MRAS into the parameter adjustment in the system. The speed and
position for MRAS system with feedback were depicted in Figure 5.6.
In general, both figures illustrate that the speed and position were capable to
follow the pattern of the system. Compare with the system without MRAS feedback,
it shows that the speed is faster rise up and slower fall down. The simulation shows
the MRAS system is able to increase position (DIR ‘1’) at speed around 140 rad/s.
The motion stays at position 0.8 rad at time 0.106s before it reverse the direction
(DIR ‘0’) from 0.158s to 0.206s and stop at position 0.45rad, which is higher from
MRAS without feedback. The comparison between open loop system HSM and
open loop MRAS system with feedback for speed and position was shown in Figure
5.7. The Proportional value for this simulation is 26000 and Derivative value is 1
which is lower than open loop MRAS without feedback.
The uneven speed and position occurs since the inner simulation of the
Stepper Motor Drive cannot be modified easily. Therefore, the voltage phase and
current phase were used are unexpected signal. Figure 5.8 and Figure 5.9 show the
current phase and voltage phase respectively that were used in sensorless system.
66
Figure 5.6 The Speed (Top) And Position (Bottom) Of The
Sensorless System With Feedback
Figure 5.7 The Comparison Between Open Loop System And Open Loop
MRAS System With Feedback For Speed (Top) And Position (Bottom)
67
Figure 5.8 Current Phase Of Simulation
Figure 5.9 Voltage Phase Of Simulation
68
Nevertheless, the hardware implementation for the sensorless system could
not be done due to lack of current sensor. Consequently, only the voltage phase data
can be acquired from ADC card. The voltage phases are shown in Figure 5.12.
Figure 5.10 Voltage Phase Of Hardware Implementation
CHAPTER 6
CONCLUSION AND FUTURE RESERACH
6.1
Conclusion
Generally, both open loop and close loop or sensorless systems of the hybrid
stepper motor were investigated.
Both simulation and hardware for open loop
system were shown that the speed and position can be getting easily. However, by
applying the sensorless system can improve the performance of the motion control.
In this project the MRAS was implemented to form a sensorless system for a hybrid
stepper motor. The MRAS can be acting as a non physical sensor in motion control.
Via MRAS, the speed and position ware estimated before applying both parameters
to the HSM system.
The main inputs of the MRAS system are current phase and voltage phase
were induced into the system to produce two flux linkages. The error between these
two flux linkages is the angle which was used in adjustment parameter to get the
speed estimation. Then by regulating with PI controller, the speed estimation will
follow the actual value and by doing differential on these value, the position
estimation will be getting. Overall, the speed estimation and position estimation
from developed MRAS in simulation is able to follow the pattern of the actual speed.
The speed of the MRAS are obtain to track the speed at average 140rad/s and
position along the forward (DIR ‘1’) and reverse direction (DIR ‘0’) with error
convergence to smaller value. The result can be improved by improving the voltage
and current phase input of the MRAS system.
70
The hardware implementation of the close HSM system was implemented
by using rotary encoder where it measures the speed at 150rad/s. By the way, the
sensorless system can be implemented if the current sensor is available. The ADC
card only acquire the voltage input. Moreover, the constructions of the circuit for the
current sensor take a quite long time.
6.2
Future Work
The availability of current sensor is one of the key to validate the developed
observer i.e. model reference adaptive system observer. The convenient current
sensor with small modification is available in the market, but the price quite
expensive in order to measure four current phase. Other than that, the current sensor
circuit also can be assembled by using operational amplifier.
In MRAS, besides by measuring the flux linkage as the error vector, as well
as the back EMF and reactive power by using additional full order adaptive observer
may one of the alternatives. Beside MRAS itself, other observer also can be used as
sensorless system such as Luenberger observer, Extended Kalman Filter (EKF) or
Sliding Mode Base observer.
To overcome the low performance of the PIC
controller, the faster microcontroller or DSP may be used. The Motorola
microcontroller is one of the best choices because it is specifically design for motor
control application.
71
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75
APPENDICES
A. Source code for embedded MATLAB function in Simulink
Embedded MATLAB
function [t,CosPT,w_new] =
Function#1
fcn(t_nminus1,w_in)
t=1e-6 + t_nminus1;
w_new=w_in;
R=0.7;
L=0.0014;
delta=atan((w_new*L)/R);
CosPT=0.0014*Cos(w_new*t - delta);
Embedded MATLAB
function L = fcn(theta)
Function #10
L=0.0014 + 0.01*sin(50*theta);
Embedded MATLAB
function [t,omega] = fcn(t_nminus1,w,angle)
Function #11
t=1e-6 + t_nminus1;
R=0.7;
L=0.0014;
P=50;
delta=atan((w*L)/R);
omega=2.05*(P*angle + delta)/t;
76
B. Subsystem of simulation
Subsystem
Subsystem1
Subsystem3
Subsystem2
74