ABSTRACT
The state space is applied in the design of Magnetic Suspension System, where a
controller is designed with the aim of making the system stable and providing the
performance specifications of settling time less than or equal to 0.5 secs, Maximum
overshot less than or equal to 5 percent and Steady State error less or equal to 1 percent.
A reference input was introduced which help to make steady state error equal to zero
from a value of about 100%. Also an observer is designed which estimate the state
variable that can not be measured. The designed controller had a gain of 9.6, which was
used to determine the gain of the power amplifier used in the system. The designed
controller caused the overshoot and settling time to reduce from undefined values to
4.1% and 0.22 seconds respectively which conformed with the performance
specifications. The observer also made it possible to observe the convergence of the
actual and estimated values of the state variables in less than 0.5 sec. Prototype of
magnetic suspension system was constructed and the system was able to suspend a ball
of mass of 28g at a distance of 1.2cm below the coil.
TABLE OF CONTENTS
Title page i
Declaration ii
Certification iii
Dedication iv
Acknowledgments v
Table of Contents vi
List of Figures ix
List of Tables xi
Symbols and Abbreviation xii
Abstract xiii
CHAPTER ONE: General Introduction
1.1 Introduction 1
1.2 Background Information 1
1.3 Motivation 2
1.4 Statement of Problem 3
1.5 Aim and Objectives 4
1.6 Significance of the Studies 4
1.7 Thesis Outline 4
CHAPTER TWO : Literature Review And Theoretical Background
2.1 Introduction 6
2.2 Review of Past Works in the Area 6
2.3 Theoretical Background 8
2.3.1 State 8
2.3.2 State Variables 8
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2.3.3 State Vector 8
2.3.4 State Space 9
2.3.5 State Space Equation 9
2.3.6 The Mathematical Relationship between Magnetic Flux Density (B)
and Temperature 9
2.3.7 The Magnetic Flux Density on the Axis of a Circular Coil 10
2.3.8 Determination of Step Response 12
2.3.8.1 Maximum Overshot 13
2.3.8.2 Settling Time 13
2.2.8.3 Steady State Error (SSE) 13
CHAPTER THREE: Methodology
3.1 Introduction 14
3.2 Development of Mathematical Model 18
3.3 State –Space Model 22
3.4 Controllability and Observability 26
3,4,1 Controllability 26
3.4.2 Observability 26
3.5 Controller Design using Pole Placement 27
3.5.1 Performance Specification and System Response 28
3.6 The Reference Input 35
3.7 Observer Design 37
3.8 System Block Diagram 39
3.9 Design of Infrared Emitter 39
3.10 Design of Signal Detector 40
3.11 Design of Comparator 41
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3.12 Design of Compensator 42
3.13 Design of Output Amplifier 43
3.14 Design of Coil driver and Electromagnet 44
3.15 Design of DC Power supply 45
3.16 Design of Adjustable D.C Power Supply 46
3.17 Implementation of Designed Controller 47
3.18 Temporary Construction of the Circuit on breadboard 48
3.19 Permanent Construction of the Circuit on Veroboard 49
3.20 Design of the Casing 49
CHAPTER FOUR: Performance Evaluation, Results And Analysis
4.1 Introduction 50
4.2 Performance Evaluation 50
4.3 Analysis of Designed Controller and an Observer 50
4.4 Complete Circuit Diagram 54
CHAPTER FIVE: Summary, Conclusion And Recommendation
5.1 Summary 58
5.2 Limitation 58
5.3 Conclusion 58
5.4 Suggestion for Further Work 59
References 60
Appendix 62
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Introduction
The chapter one gives an overview of the magnetic suspension system, motivation and
statement of the problems, aim and objectives and significance of study.
1.2 Background Information
Magnetic suspension is a means by -which metallic object is suspended with no support
other than magnetic fields. The electromagnetic force is used to counteract the effects of
the gravitational force (Dolga and Dolga, 2007). There are several practical applications
of the magnetic suspension system and this includes; the Magnetic Levitation (MagLev)
train, which is a high-speed train that runs using electromagnetic principle of levitation.
The train floats above the guide rail and the polarity of the magnet is used to move the
train. Because the MagLev train is not in contact with the track, there is no friction and
thus the MagLev train can travel faster than conventional trains. Another application is
frictionless bearing, in which the use of magnetic suspension reduces wear and tear on
the bearing since there is no contact with other metallic parts (Glavin, 2005). The
magnetic suspension system is an unstable non-linear system. Therefore, it is always a
challenging effort to design a feedback controller to control the position of the suspended
object. In recent years, many approaches have been reported. Hurley and Wolfe (1997)
described the linearization of the plant by examining perturbation around the operating
point. Compensation is achieved by implementing proportional plus derivative (PD)
control. Glavin (2005) carried out work on state space control of a magnetic suspension
system in which a controller was designed and system behaviour was simulated using
MATLAB/simulink and Pspice
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This research work deals with the design of magnetic suspension system using statespace
concept. In addition, a prototype model will be constructed. The dynamic
compensation of the system will be illustrated by working with state variables of the
system. A simplified magnetic suspension system is shown in Figure 1.1
Figure 1.1 Simplified magnetic suspension system.
Coil acts as electromagnetic actuator, while an optoelectronic sensor determine the
position of the ferromagnetic ball. By regulating the electric current in the circuit through
a controller, the electromagnetic force can be adjusted to be equal to the weight of the
steel ball, thus the ball will levitation in an equilibrium state.
1.3 Motivation
A control system provides the means by which any quantity of interest in a machine,
mechanism can be maintained or altered in accordance with a desired manner (Nagrath
and Gopal, 2005). The methods of achieving this objective usually involve the use of
certain control strategies such as discrete PID controller and state feedback control
techniques. More so, control systems play a vital role in the advancement of engineering
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technology such as in magnetic suspension system. However, research is still on going in
this area because it allows the study of various control strategies. Therefore, the design
of magnetic suspension system based on classical methods (Bode plot, polar plot, root
locus etc) do not permit the control engineer the ability to specify all poles in the control
system for the systems higher than second order and also it is applicable to only linear
system. Hence, the need to design and analyse the system using state-space technique
and MATLAB, which will allow the specifications of all the poles of the system, even
for higher order systems irrespective of whether the system is linear or non-linear.
Linear system; A system is called linear if the principle of superposition applies. The
principle of superposition state that the response produced by the simultaneous
application of the two different forcing functions is the sum of the two individual
responses while Non-linear system the response to two inputs cannot be calculated by
treating one input at a time and adding the results.
1.4 Statement of Problem
The magnetic suspension system is an unstable non-linear system. Therefore, it is
always a challenging effort to design a feedback controller to control the position of the
suspended object. This work involves the application of state space concepts in the
design and analysis of a magnetic suspension system. The mathematical model would be
developed, representation of the model in the state space form, verification of
controllability and observability of the system using Kalma’s method. Controller would
be designed by using a pole placement technique so as make the system stable; observer
would also be designed in order to estimate the state variables that can be measured. The
following components of feedback control would be designed. And these are reference
input, position sensor, comparator, and control mechanism.
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1.5 Aim and Objectives
This work aimed at construction of prototype non-linear dynamic model of
magnetic suspension system where a non-linear state-space transformation is used to
linearized the system. The prototype system suspended a mass of 28 gram at a distance
of 1.2cm below the coil. Critical issues in the construction of magnetic suspension
system considered includes weight of the ball, power requirement to energize the coils
in order to produce the required field to levitate the ball, the time and the temperature at
which the field will become weak such that the ball drops.
1.6 Significance of the Studies
The Magnetic Suspension System is a classical non-linear control problem just like other
non-linear systems such as Ball -and –Beam system and Inverted pendulum. These
systems are generally regarded as unstable. It is very important that different approaches
are adopted in modelling these systems and designing controller that will ensure system
stability. The State-Space approach is adopted in modelling the magnetic suspension
system and designing the controller and observer.
1.7 Thesis Outline
This work comprises of five chapters. The first chapter one gives an overview of the
Magnetic Suspension system, motivation and statement of the problems. Chapter two
covers related works that have been done and theoretical background. Chapter three
focuses on modelling and analysis of Magnetic Suspension System, development of
mathematical model, representation of the model in the state space form, performance
specification, verification of controllability and obervability of the system using Kalma’s
method whether a solution exist to the problem stated, achieving the set specifications
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with aid of a controller, the introduction of reference input to remove steady state error
and design of an observer that estimate the state variables that are not available for
control and measurement purpose. The design of position sensor, reference input,
comparator, control mechanism, and implementation of controller. Chapter four
highlights the result, analysis and experimental measurements. Chapter five center on the
significance of the study, limitation and recommendation for further work.
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