ABSTRACT
This research work is aimed at the development of an observer-based dynamic output feedback
controller for stabilization and tracking of nonlinear systems. The developed controller is
designed after the immersion-invariance and internal model principle (IMP) frameworks and targets
non-square systems such as rotational-translational actuator (RTAC), cart-driven inverted pendulum
(CIP) and quadrotor unmanned aerial vehicle (UAV). However non-square multiple-input
multiple-output (MIMO) systems such as the UAV represented the principal system of choice for
their structural properties. Non-square MIMO systems are systems that have more inputs than
outputs (over-actuated) or vice-versa (under-actuated) and reflect the structures of many real world
systems. The developed immersion invariance error feedback control law(IIEFCL) is used to solve
stabilization and robust tracking problems of non-square MIMO non-linear systems. The output
feedback internal model based observer is developed and tested with the RTAC, CIP and UAV
while the immersion invariance stabilizing controller is developed and tested on the RTAC system.
The output feedback controller showed good stability response on the selected models while the
immersion invariance method displayed a good transient phase stability and tracking results with
the addition of a robust state feedback feature to the underlying controller. The obtained settling
times for the output feedback stabilization results were 2.7s, 1.113s and 0.6435s respectively for the
three systems. The immersion-invariance control law acting as a robustifier to another controller
produced zero percent overshoot and tracking error. The results showed attainment of desired
stability and tracking and also quick convergence, disturbance rejection and handling of transient
oscillations such as finite time escape or transient instability phenomena, from which many nonlinear
systems do not recover after they occur. The IIEFCL was developed for the Quadrotor UAV
and the results obtained were compared with some other standard nonlinear controllers that have
been used in QUAV control. The metric for comparison was the integral of the squared control
input (ISCI) signal. Results obtained compared favourably with existing nonlinear control laws.
The IIEFCL showed the most improvement of 92.92% improvement over the backstepping conviii
trol law, it had a 72.92% improvement over the feedback linearization control law and the least
improvement was with respect to the sliding mode control law where only 66.225% improvement
was recorded. Simulations were made using Matlab/Simulink and embedded C++ tools.
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TABLE OF CONTENTS
Title page i
Declaration ii
Certification iii
Dedication iv
Acknowledgments v
Abstract viii
Table of Contents x
List of Figures xvii
List of Tables xx
List of Acronyms and Symbols xxi
Chapter 1: INTRODUCTION 1
1.1 Background to the Study 1
1.2 Motivation 8
1.3 Significance of Study 10
x
1.4 Statement of the Problem 10
1.5 Aim and Objectives 12
1.6 Methodology 12
1.7 Thesis Outline 13
Chapter 2: LITERATURE REVIEW 15
2.1 Introduction 15
2.2 Review of Fundamental Concepts 15
2.2.1 Output regulation 15
2.2.2 Output feedback-based regulation 27
2.2.3 Stabilization using immersion and invariance 31
2.2.4 Observers for control systems 33
2.2.5 MIMO dynamic model 41
2.3 Review of Similar Works 44
Chapter 3: METHODOLOGY 53
3.1 Introduction 53
3.1.1 Development of the output feedback controller 54
3.1.2 RTAC Parameters 61
3.1.3 Output regulation for the CIP system 62
3.1.4 Regulator equations for the CIP 65
3.1.5 CIP Parameters 67
3.1.6 Output regulator for the QUAV 68
3.1.7 Quadrotor UAV Parameters 71
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3.2 Development of immersion-invariance stabilizing controller 71
3.2.1 Immersion invariance control of the RTAC or TORA system 72
3.3 Development of the Immersion Invariance Output Error Feedback Regulator for
Non-Square Systems 77
3.3.1 Feedback stabilization control design 77
3.3.2 Stabilizing controller by Lyapunov approach 81
3.3.3 Revisiting output error feedback regulator design for the UAV 83
3.3.4 Computation of generalized immersion relations 86
3.3.5 Output error feedback control law 88
3.3.6 Immersion-invariance controller for the quadrotor 89
3.3.7 Actuator model formulation 90
3.3.8 Augmented system model 93
3.3.9 Selection of target dynamic for the quadrotor 94
3.3.10 UAV state feedback controller design for immersion invariance 95
3.3.11 Proof of boundedness and stability of immersion invariance control law 98
3.3.12 Synthesis of immersion invariance output feedback observer-based control 100
3.3.13 Design of the error feedback observer 101
3.4 Validation of the Developed Immersion Invariance Output Feedback Control Law 104
Chapter 4: RESULTS AND DISCUSSION 107
4.1 Introduction 107
4.2 Output Regulation Experiments 107
4.2.1 RTAC Regulation Experiments 108
4.2.2 CIP Output Regulation 111
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4.2.3 Quadrotor UAV Output Regulation 113
4.2.4 Summary of Output Regulation Experiments 116
4.3 Immersion Invariance Stabilization Experiments 117
4.3.1 RTAC Immersion Invariance 117
4.3.2 RTAC immersion invariance stabilization controller 117
4.3.3 Linear immersion invariance Tracking control with observation 121
4.3.4 Robust RTAC stabilization and tracking experiments with disturbance injection
121
4.4 Immersion Invariance Error Feedback Control Results 125
4.4.1 QUAV unforced stabilization results 126
4.4.2 Exosystem profile 127
4.4.3 Immersion-invariance error feedback regulator results 128
4.4.4 IIEFCL state output results 132
4.4.5 Characterization of the control laws 139
4.4.6 Comparison of results from various control laws 150
Chapter 5: SUMMARY AND CONCLUSIONS 152
5.1 Summary 152
5.2 Conclusions 153
5.3 Limitation 155
5.4 Contribution to Knowledge 156
5.5 Recommendation for Future Work 157
References 165
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Appendix A: Code listing for the RTAC Output Feedback Regulator Problem 167
A.0.1 RTAC parameters are defined here from Avis et al., 2013 167
A.0.2 Substitution of params 167
A.0.3 Definition of the exosystem matrix as either constant or variable 167
A.0.4 Computing controllability 167
A.0.5 Discussion 168
A.0.6 Computing observability 168
A.0.7 Computing observability 168
A.0.8 Construct a stabilizing gain Kx for A+BKx 168
A.0.9 Developed method for matching the poles 169
A.0.10 Computing the values in the gains L,L1,L2,G1,G2 169
A.0.11 Computing the gains L=L1 169
A.0.12 Computing the gains L1 and L2 for the matrix G2 170
A.0.13 Compute the L2 gain from developed method 170
A.0.14 Computing Kv 171
A.0.15 Compute the global K matrix Kavis=[Kxavis Kvavis] 171
A.0.16 Solve the sylvester equations from knowlwegde of Kx and Kv 172
A.0.17 Computing the other gain of the observer G1 172
A.0.18 Test for the controllability of Aobsv, Bobsv 172
A.0.19 Test for stability of G1 and G2 matrices 173
A.0.20 Check for Hurwitz stability of the closed loop system matrix below 173
A.0.21 Calling model here 173
xiv
CHAPTER ONE
INTRODUCTION
1.1 Background to the Study
The design of stabilizing controllers is a major task and requirement for the control engineer. While
notions of stability vary from linear to nonlinear systems, the main ideas and goals are the same;
which is to force the evolution of the trajectories of a system towards an equilibrium point or an
equilibrium set. This is the same goal in either linear bounded-input bounded-output (BIBO) stability
or more involved nonlinear counterparts like the Lyapunov stability variants and structural
stability. Output regulation, which is the core theme of this proposal, requires that for any system
to be successfully controlled, it must first be established to be stabilizable (Byrnes et al., 1990).
This stability being enforced in the presence of constraints such as known or unknown disturbances
and uncertainties. The system requirement is that all trajectories of interest or system states remain
asymptotically close to or converges exactly to the desired equilibrium point and remains there
for all time (Isidori & Byrnes, 1990). In the absence of stability, the ultimate desire for affecting
the behavior of any system through regulation using suitable control signals is not possible; such
regulation being demanded in the presence of the stated constraints which can interact with any
real system to cause instability and deviation from correct behavior. Examples of such uncertainties
and disturbances include; un-modeled dynamics, high frequency parasitic components, large
inputs and other complex coupling phenomena in multiple-input multiple-output (MIMO) systems
(Kokotovic, 1985).
The search for a globally stabilizing and asymptotically tracking feedback controller has seen various
control schemes put forward for consideration (Andrieu & Praly 2009; Astolfi & Praly 2015).
Another name for such a robust stabilization and tracking control scheme is output regulation. Pioneering
output regulation control designs include: linear single-input single-output (SISO) and
1
MIMO systems by Francis and Wonham (1976) and the equivalent nonlinear SISO and MIMO
treatment by Byrnes and Isidori (1990). This has also seen the achievement of various results
ranging from local, semi-global to global. However, while stabilization in the local sense has been
considered for non-linear systems since the early 1990’s, it is the stabilization in a large domain
such as semi-global stabilization by Teel and Praly (1993), Battilotti (2001) and global stabilization
by Chen and Huang (2004), Peixoto et al., (2009) that are good prospects in the search for a
global stabilizing feedback compensator. More specifically global robust regulation concerns have
dominated the research in many of the literature on stabilization of nonlinear systems (Astolfi &
Praly, 2017).
Continuing research in output regulation over the years has seen a stream of new results added by
Huang (2004), Chen and Huang (2004), Serrani (2005), Isidori (2011), Khalil and Praly (2014),
Astolfi and Praly (2015). These works contributed to the development of workable designs. Specifically
considering the analysis of the internal structure of a MIMO system and studies of how this
structure was affected by feedback and output injection. The obtained results were used to solve
such problems as stabilization, tracking, system decoupling and fault isolation. In the case of
non-linear systems, most of the methods for the analysis of the internal structure have been conveniently
extended beyond that originally developed by Francis and Wonham (1976). According
to Isidori (2011), recent works showed that the study of problems of output feedback design for
MIMO nonlinear systems has come to an almost complete stall. Some reasons given for this difficulty
in tackling the non-linear MIMO problem include simple and complex interconnections
between various states and inputs and difficulty in transferring observer control design from linear
to nonlinear MIMO systems (Isidori, 2011; Wang et al., 2014).
Therefore regulation of the output of any system must be preceded by satisfiable stabilization properties
in the implemented closed-loop control architecture if regulation of the system output must
be achieved. Figure 1.1 depicts such a control architecture that is well known in control systems
theory (Hoagg & Bernstein, 2006). The variables yc(t), e(t), u(t) and y(t) represent the command
2
or reference input, error signal, control input and output respectively. Such potential instability justifies
the case for the presence of a robust regulator that maintains the stability of the system state
and regulates the system output towards the asymptotic tracking of reference signals and rejection
of unwanted or undesired disturbances. In order to effectively handle a wide range of disturbances
Figure 1.1: Standard Regulator Structure (Hoagg and Bernstein, 2006)
and system uncertainties, a different architecture is needed as given by the internal model-based
output feedback controller. The design depicted in Figure 1.2 captures the internal model principle
Figure 1.2: Standard Output Feedback Regulator Structure (Valmorbida and Galeani, 2013)
(IMP) philosophy, which requires the resulting controller to have asymptotic tracking and disturbance
rejection properties (Kokotovic & Arcak, 2001). The internal model F, steady state control
and state variables uss and xss, plant and observer are P and ^ P respectively. Observer estimates
3
^x and ^!, unstable states ~x, exogenous signal ! and output error e. This is made possible by embedding
a copy of the actual system in the control architecture. This stabilizing control ensures the
tracking of a given reference trajectory so that the output error is regulated to zero. The parts of the
robust observer-based regulator consist of the plant, the observer which acts as the internal model,
the steady state stabilizer and steady state generator ( and respectively) and finally the exosystem
supplying the external inputs to the system. This framework also assumes knowledge of the reference
signals/disturbances.
In developing an observer-based output feedback control law, specific design methods depend on
the kind of information available for feedback (Astolfi & Praly, 2017). In general this technique
for robust regulation of a dynamic systems output has been categorized by Serrani (2005) into state
and output error feedback. The output feedback approach to regulation of a systems response is
more practicable in tackling the robust stabilization and tracking question. Considering that the
control system designer is assured of having the entire output present for measurement as against
the possibility of having individual state for measurement. Output feedback stabilizing regulator
also has other advantages which include; ease of practical implementation, multi-functionality
within the single design e.g. a single controller design being able to serve as both stabilizer and
steady state regulator for multiple manipulated variables and applicability for disturbance rejection
(Astolfi & Praly, 2017; Chang et al., 2015).
Therefore, the developed observer-based output feedback controller is used to achieve robustness,
adaptive behavior and importantly, to ensure global asymptotic stability (GAS) of system states.
Based on the principle of internal model, the output feedback controller ensures simultaneous robust
stabilization and tracking of reference inputs or rejection of reference disturbances (Francis &
Wonham, 1976). However, it often achieves this in a local domain of the work space (under small
signal perturbation). Another scheme for stabilization is the immersion and invariance method,
known for yielding globally stabilizing and adaptive designs for controllers in the presence of
parametric uncertainties (Astolfi & Ortega, 2003). Immersion and invariance method does not
4
however have the robustness characteristics of the internal model output feedback design.
The robust output regulator with structure shown in Figure 1.2 is proposed for modification. This
modification is necessary because it is not possible to always know every profile or function of the
disturbance or uncertainty to be encountered. Therefore while output measurement by an observer
is still necessary, the observer-based output feedback nomenclature remains open to increased robustness
and adaptive design features. The development of such stabilizing regulators has seen
the use of different implementation architectures using general principles of internal model control
(IMC) such as model predictive control (MPC) framework, one example of which is the model predictive
regulator of Aguilar and Krener (2014), state and error observer framework by Carnevale
et al., (2012), strictly adaptive framework, immersion-invariance framework by Astolfi and Ortega
(2003), disturbance observer (Shim et al., 2015).
Extending the treatment to systems with uncertainties, Wiese et al., (2015) and Shim et al., (2015)
have also addressed the problem of adaptive control designs. The uncertainties being sources of instability
in either the transient or steady operating phase is affected by the observer gains. Whereas
high gain designs which are ideal for local or semi-global stabilization are beneficial for quick
convergence, high gains have a detrimental effect in the transient regime where large error occurs.
High gains also have a detrimental effect on the measurement error in steady state regime where
the error (usually small), is derived from model uncertainties. Switched gain observer design has
also been studied by Ahrens and Khalil (2008).
Therefore, having established the existence of different approaches and the several attempts made
to obtain a globally stabilizing output feedback controller (Andrieu & Praly, 2009), this work
proposes another approach for non-square MIMO non-linear systems, based on the identified developments.
The approach will make use of a hybridization of techniques from both internal
model principle-based output-feedback and immersion-invariance stabilization theory. The developed
control scheme shall tradeoff the strengths and weaknesses of each method. While the
5
output regulation framework requires the unrealistic knowledge of every disturbance and uncertainty,
such requirements are absent for Immersion-Invariance stabilization. Therefore the strong
stability results from Immersion-Invariance are combined with the strong tracking results from
output regulation to produce an improved controller.
This work has given an alternative and detailed development of an immersion and invariance
observer-based output feedback regulator that has characteristics of global stability. The design
builds on the premise that global asymptotic stabilization implies global asymptotic tracking that
assures semi-global stabilization in output feedback (Teel & Praly, 1993). The current synthesis
of a stabilizing observer-controller framework has brought together established techniques which
are principally from internal model principle and immersion-invariance design. The design specifically
addresses stabilization and tracking a non-square MIMO nonlinear systems. Figure 1.3
summarizes the flow of information in an output-regulated feedback controller that is based on the
IMP. This flowchart starts with the definition of the plant, exosystem and controller, formation of
an autonomous closed loop system, analysis of the resulting structure of the resulting autonomous
system, solution of the regulator equations for the system. A, B and C are the system transmission
matrix, input matrix and output matrix respectively. Wr is the controllability or reachability
matrix, Kx, Kv are the stabilizing state and disturbance feedback gains. u is the regulator control
variable. L1, L2, G1, G2 are observer gains.
6
Figure 1.3: Output Error Feedback Regulator Flowchart
Similarly, the flowchart for setting up an Immersion and Invariance stabilization (Astolfi and
Ortega, 2003) scheme is shown in Figure 1.4. x is the system state in original coordinates, is system
state after immersion, K is stabilizing feedback matrix, f(:) and g(:) are state and input vector
fields. (:) is an immersion map and (:) is the zero manifold while (:; 🙂 is the off-manifold
control law. Following the establishment of the output regulation scheme using observer output
feedback and immersion invariance, a closed loop architecture was implemented. By extending
the standard architecture for observer-based output regulation, this work proposed an immersion
7
Figure 1.4: Immersion Invariance Flowchart
and invariance output feedback architecture for robust and adaptive stabilization of system state
and asymptotic convergence of system output error to zero.
1.2 Motivation
Investigations by Wang et al., (2014) in which the dynamic output feedback in MIMO system is
achieved in the context of a high gain observer (HGO) forms the principal motivation for this work.
Astolfi and Praly (2015) have also looked at the extension of the output feedback problem by in-
8
vestigating the effects of small un-modeled discrepancies which could also cover external sources
arising from reference signals and disturbance inputs. However, practice shows that not all disturbance
inputs are of small magnitude. In the case of MIMO non-linear systems that are controlled
via output feedback, the problem of asymptotic stabilization, despite of its obvious relevance in
dynamic systems remains unresolved. Literature evidence shows that inquiry in this direction has
received little attention (Wang et al., 2014). This is moreso for non-square MIMO systems where
works addressing observer-based output feedback for non-square uncertain systems are also limited
(Isidori, 2011; Shim et al., 2015).
While output feedback presupposes the existence of some form of estimator or observer of state,
certain constraints can be identified with the availability of such measurement (Nazrulla & Khalil,
2008). Firstly, not all states are usually available for measurement (Kazantis & Kravaris, 1998;
Kaldmae & Kotta, 2013), making the full state information problem of output regulation more ideal
than practical for the task. Aside from the full state information being rarely available, design methods
based on the availability of partial state information are important (Wang et al., 2014), as well
as designs that consider the output error information (Astolfi & Praly, 2017). Such designs have
proved to be more practically feasible and have necessitated output feedback techniques which
although applied successfully in SISO type systems, have not seen a similar level of progress in
nonlinear MIMO systems (Khalil & Praly, 2014; Isidori, 2011). More recent works have utilized
immersion (Astolfi and Ortega, 2003), embedding the partial or reduced state output observer as
an internal model of the system while respecting certain structural conditions that guarantees system
analysis. Other works have utilized non-linear coordinate transformation. These have become
more practical approaches to solving the dynamic Output Regulation problem but mainly in square
systems. Considering these facts, the problem of achieving robust stabilization-tracking via output
feedback is still largely open, especially for non-square MIMO non-linear systems (Wang et al.,
2014).
The question therefore becomes, how to achieve GAS in the presence of disturbances and model
9
uncertainties of varying magnitudes using a control law that is robust and adaptive under all operating
conditions? Specifically taking into account the inability to know accurately all possible
uncertainties that might enter into a system. One answer to this question was the control law provided
by the developed immersion-invariance output feedback controller. The appeal for such a
control law follows the established fact that stabilization and tracking without robustness and inherent
adaptation leads to instability, reduced control authority, loss of control, increased energy
usage and poor transient performance. Therefore the developed scheme did not need to have full
information of the possible uncertainties or disturbances to achieve the control goals.
1.3 Significance of Study
The significance of this work is the extension of the existing treatment of MIMO output regulation
or stabilization-tracking analysis to specifically address non-square MIMO nonlinear systems. The
utilized approach firstly, uses output-regulated output feedback control as a basis. Secondly, the
current treatment is extended in the particular direction immersion and invariance setting that allows
for design of globally stabilizing control laws. Lastly, identifying and making use of relevant
design parameters from the two previous techniques. It focuses on the development of a practical
framework to be applied towards achieving robust stabilization and tracking irrespective of
intervening uncertainties. This new framework is an Immersion-Invariance output feedback strategy.
Therefore, elements of the system which might prevent the attainment of global asymptotic
stability (GAS) such as finite time escape phenomena like peaking are addressed.
1.4 Statement of the Problem
Section 1.2 presented globally stabilizing control laws for nonlinear and linear systems as an open
problem in the context of nonlinear, non-square systems utilizing output feedback controllers. One
of the difficulties in achieving such stabilizing control laws using output feedback regulation concept
remains the existence and uniqueness of solutions to the regulator equations which is not
assured for all systems. Because, while it can be checked that such a regulator exists (neces-
10
sity), obtaining the regulator (sufficiency) remains a difficult task. Moreso, for non-square systems
where analysis sometimes requires squaring the plant before output feedback is applied. Seeing
the difficulty in achieving stability using the analytical route, certain literature developed a constructive
approach using eigenvalue pole placement and gain matrix designs which when properly
done, renders the system internally stable. The attraction for such a method rests with the fact
that if subsets of the system were designed to be stable then the global system when assembled
together retained such stability. The identified difficulty could be described in a sense as structural.
This description has meaning in non-square systems with respect to size of inputs and outputs.
Also more fundamental was the size reduction in systems with reduced rank leading to nominally
uncontrollable and unobservable systems. It has been observed that some of these characteristics
that are influenced by the system structure were readily identifiable using differential geometric
techniques for analyzing such nonlinear systems.
System stability in the global context requires control laws that are robust for all possible and acceptable
inputs. Therefore, the continued demand for controlled systems to have good stability and
tracking performance cannot be overemphasized. Also, overcoming the potentially destabilizing
effect of disturbances and uncertainties is key to driving the error term in any controlled system to
zero. The problem can therefore be summarized as the regulation of any given system’s state to an
invariant zeroing manifold from which it is impossible to escape and at the lowest possible control
cost. The implemented solutions used both state and output feedback principles to ensure the continued
internal stability and output regulation of the selected systems’ states. This was achieved
in the presence of structural imbalance inherent to non-square systems and also in the presence of
potential destabilizing factors such as external disturbances and inputs that were captured in the
utilized exosystem structure.
11
1.5 Aim and Objectives
The aim of this work is: development and synthesis of a dynamic output feedback observer for
stabilization and tracking in non-square MIMO nonlinear systems.
The objectives of the study are therefore as follows:
1. Development of an internal model based output regulation framework for stabilization of
non-square MIMO nonlinear systems.
2. Development of an immersion and invariance stabilization framework for non-square MIMO
nonlinear systems.
3. Development of an immersion and invariance output feedback control law for stabilization
and tracking of non-square MIMO nonlinear systems.
4. Validation of the immersion-invariance output feedback theory and control law using standard
benchmark non-square MIMO systems (cart-driven inverted pendulum (CIP) , rotational/
translational proof-mass actuator (RTAC) and a fixed rotary wing quadrotor unmanned
aerial vehicle (QUAV) .
1.6 Methodology
The methodology for implementation of the stated objective items are:
1. Internalmodel principle(IMP)-based output regulation framework for stabilization and tracking
(a) Adoption of relevant mathematical background for output regulation
(b) Analysis of necessary and sufficient conditions for existence of solutions for output
regulated control
12
(c) Implementation of the IMP regulation framework with internal model-based observer
on the following non-square dynamic: Rotational Translational Actuator (RTAC), Cart
Driven Inverted Pendulum (CIP), Quadrotor Unmanned Aerial Vehicle (QUAV).
(d) Simulation experiments of the output regulated control law in Matlab/Simulink
2. Immersion and invariance (I&I) stabilization framework
(a) Adoption of relevant mathematical preliminaries for immersion and invariance stabilization
(b) Analysis of the necessary and sufficient conditions for existence of solutions for an
immersion and invariance stabilizing control law
(c) Implementation of the immersion and invariance framework with standard observer on
selected benchmark non-square dynamic
(d) Simulation experiments of the control law in Matlab/Simulink
3. Development of an immersion and invariance output feedback framework
(a) Parameter identification for use in developed output regulated output feedback scheme
(b) Implementation of the immersion and invariance observer output feedback framework
on the non-square MIMO QUAV model.
(c) Simulation in Matlab/Simulink of the developed control law.
4. Validation on selected non-square dynamic system and fine-tuning(using neural tools) where
needed of the selected parameters in the developed controller of methodology item (3) to
ensure the goal of GAS and perfect tracking
1.7 Thesis Outline
The organisation of the thesis is as follows; Chapter One has laid-out in brief the overall goals to
be achieved with the introduction, motive, significance of study, statement of the problem, thesis
13
aim and objectives and ends with a discussion of the method used in achieving the set goals.
Chapter Two has given a detailed review of pertinent concepts and literature. Chapter Three has
addressed the materials and methods applied in-so-far as this thesis report is concerned, Chapter
Four presented the results for discussion and Chapter Five concludes the body of the report. All
cited references have been included in the references section while relevant additional material
such as codes and certain parameter derivations have been placed in the appendix section.
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