ABSTRACT
This research is aimed at the development of a modified fruit fly optimization algorithm (mFFOA) for the determination of optimized weighting matrices (Q which is a positive definite matrix that penalizes the states and R which is also a positive definite matrix that penalizes the control inputs) of the linear quadratic regulator (LQR) to be used for the aircraft pitch control system (PCS).The standard fruit fly optimization algorithm (FFOA) is an optimization algorithm inspired by intelligent smell and vision behaviour of flies towards fruit. The FFOA suffers from the problem of lack of balance between exploration and exploitation as it has a higher rate of exploitation than exploration leading to a high probability of it being trapped in some local optimal. The mFFOA was developed by modifying the iteration factor of the search radius to a decreasing function to improve the exploration capability of the algorithm and then by introducing a linearly decreasing inertial weight in order to provide an efficient balance between exploration and exploitation of the FFOA. The mFFOA was benchmarked against the FFOA using ten optimization test functions (Ackley, Alpine, Eggcrate, Griewank, Pathologic, Rastrigrin, Rosenbrock, Schaffer, Sphere, and Whitley) and showed a 20% improvement in its convergence to global optima. The optimized values of the Q and R weighting matrices are obtained for ten test runs using mFFOA within a time of 126.9538s when compared with the 140.7819s taken using the FFOA approach. The proposed method reduced the time taken by the FFOA by 13.8281s.These matrices used to determine the LQR controller for the PCS showed a settling time of4.4456s when compared to 4.4764s obtained using FFOA. This showed a convergence of the solution search space using the LQR (mFFOA) having a more optimal time-to-solution.
TABLE OF CONTENTS
DECLARATION III
DEDICATION IV
CERTIFICATION V
ACKNOWLEDGMENT VI
ABSTRACT VII
TABLE OF CONTENT VIII
LIST OF FIGURES XII
LIST OF TABLES XIV
LIST OF ABREVIATIONS XV
CHAPTER ONE: INTRODUCTION
1.1 BACKGROUND OF RESEARCH 1
1.2 PROBLEM STATEMENT 3
1.3 AIM AND OBJECTIVES 4
1.4 METHODOLOGY 4
1.5 DISSERTATION ORGANIZATION 6
CHAPTER TWO: LITERATURE REVIEW
2.1 INTRODUCTION 7
2.2 REVIEW OF FUNDAMENTAL CONCEPTS 7
2.2.1 The fruit fly optimization algorithm (FFOA) 7
2.2.2 Adaptive search radius 11
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2.2.3 Linearly decreasing inertial weight 12
2.2.4 Standard optimization test functions 12
2.2.5 Linear quadratic regulator (LQR) controller 16
2.2.5.1 Determination of optimal values of weighting matrices in LQR problems 17
2.2.5.2 Methods of determining weighting matrices (Q and R) of LQR 18
2.2.6 Pitch control system of an aircraft 24
2.2.7 Performance metrics 27
2.3 REVIEW OF SIMILAR WORKS 30
2.3.1 Review of Research Works Based on Modification of FFOA 31
2.3.2 Review of Research Works Based on the use of Metaheuristics Search Algorithm for Determination of LQR Weighting Matrices (Q and R) 37
CHAPTER THREE: MATERIALS AND METHODS
3.1 INTRODUCTION 41
3.2 COMPUTER SYSTEM SPECIFICATION 41
3.3 INITIALIZATION OF FFOA, LQR AND PCS PARAMETERS 41
3.4 STANDARD FRUIT FLY OPTIMIZATION ALGORITHM 43
3.5 DEVELOPMENT OF THE MODIFIED FRUIT FLY OPTIMIZATION ALGORITHM 44
3.5.1 Decreasing iteration function for an adaptive search radius 44
3.5.2 LINEARLY DECREASING INERTIAL WEIGHT 45
3.6 PERFORMANCE EVALUATION 47
3.7 COMPARISON OF THE RESULTS OBTAINED FROM THE DEVELOPED MODIFIED FFOA WITH THE RESULTS OF THE STANDARD FFOA 47
3.8 DETERMINATION OF WEIGHTING MATRICES USING MFFOA 47
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3.9 PITCH CONTROL SYSTEM STABILIZATION 50
CHAPTER FOUR: RESULTS AND DISCUSSIONS
4.1 INTRODUCTION 56
4.2 PERFORMANCE EVALUATION OF MFFOA ON THE OPTIMIZATION TEST FUNCTION 56
4.2.1 Ackley function 57
4.2.2 Alpine function 58
4.2.3 Eggcrate function 59
4.2.4 Griewank function 60
4.2.5 Pathologic function 61
4.2.6 Rastrigin function 62
4.2.7 Rosenbrock function 63
4.2.8 Schaffer function 64
4.2.9 Sphere function 65
4.2.10 Whitely function 66
4.3 APPLICATION OF THE DEVELOPED MFFOA BASED LQR CONTROLLER FOR OPTIMAL DETERMINATION OF CONTROLLER PARAMETERS 67
CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS
5.1 SUMMARY 73
5.2 CONCLUSION 73
5.3 SIGNIFICANT CONTRIBUTIONS 74
5.4 RECOMMENDATION FOR FURTHER WORK 75
REFERENCES 76
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APPENDICES 80
APPENDIX A 80
APPENDIX B 83
APPENDIX C 86
CHAPTER ONE
INTRODUCTION
1.1 Background of Research
The linear quadratic regulator (LQR) is a linear optimal controller which ensures that a feedback control system returns to its state of equilibrium in the presence of disturbances or perturbations. The weighting matrices (Q which is a positive definite matrix that penalizes the states and R which is also a positive definite matrix that penalizes the control inputs) are the critical design parameters for the LQR and are usually selected by the designer (Hamidi, 2012) which may be time consuming and not necessarily lead to optimal solution. Recently, researches have been carried out for the determination of the optimized weighting matrices of the LQR controller using different computational intelligence (CI) methods which are useful in addressing these issues, such as: genetic algorithm (GA) ( Nagarkar & Patil, 2016 ;Gupta & Tripathi, 2014; Abdulla et al., 2013; Ghoreishi et al., 2011); particle swarm optimization (PSO) (Ghoreishi & Nekoui, 2012; artificial bee colony (ABC) (Ata & Coban, 2015); weighted artificial fish swarm algorithm (wAFSA) (Mu’azu et al., 2015; Salawudeen & Mu’azu, 2015), etc. For the purpose of this research work mFFOA is used to determine the weighing matrices of the LQR.
Pan (2011) developed the fruit fly optimization algorithm (FFOA), which is a metaheuristic search method that works based on the food finding behaviour of the fruit fly (Drosophila). The main inspiration of FFOA is that the fruit fly is superior to other species in sensing and perception, especially in osphresis and vision it can smell food source 40km away and it has the following advantages: simple structure, immediately accessible for practical applications, ease of implementation and speed to acquire solutions (Pan et al.,
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2014). Studies have shown that it can be used to solve real world and engineering optimization problems (Pan et al., 2014).
However, the FFOA suffers from the problem of lack of balance between exploration and exploitation in that it has a higher rate of exploitation than exploration leading to a higher probability of it being trapped into some local optimal (Rizk-Allah, 2016). It also has the drawback of having a fixed value of search radius (Pan et al., 2014). In order to address these issues, an iteration factor was introduced so that the search radius can be adaptively changed for different evolution phases (Pan et al., 2014). However, the iteration factor introduced by Pan et al., (2014) resulted in a larger step size which limits the exploration capability of the algorithm. This work intends to modify the adaptive behaviour of the FFOA by modifying the iteration factor of search radius introduced by (Pan et al., 2014), to a decreasing function in order to improve the exploration capability and also by introducing a linearly decreasing inertial weight in order to provide an efficient balance between exploration and exploitation of the FFOA.This research aims to apply the modified FFOA (due to its simplicity and its ability to improve the convergence time) to determine the optimized parameters of LQR (Q and R) for the pitch control system (PCS) of an aircraft in order to enhance its time-to-solution.
The PCS helps a pilot to control an aircraft by keeping the pitch attitude constant, that is, make the aircraft return to desired attitude in a reasonable length of time after a disturbance of the pitch angle, or make the pitch follow a given command as fast as possible (Ju & Mohamed, 2007). Many research works have been reported on controlling the pitch or longitudinal dynamic of an aircraft for the purpose of flight stability using LQR (Jisha &
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Aswin, 2015; Stefanescu et al., 2013; Wahid & Hassan, 2012; Wahid & Rahmat, 2010). The approach developed in this work improved the settling time of the PCS.
1.2 Problem Statement
The FFOA lacks balance between exploration and exploitation as such it has higher rate of exploitation than exploration leading to a higher probability of it being trapped into some local optimal. It also has the drawback of having a fixed value of search radius. In order to address this issue, an iteration factor was introduced (Pan et al., 2014) so that the search radius can be adaptively changed for different evolution phases. However, the iteration factor introduced results in a large step size which limits the exploration capability of the algorithm. Several CI methods have been used to determine the weighting matrices of the LQR controller for complex engineering benchmark system PCS in literature, but these methods require high computational time and are complex.
Thus this research developed an algorithm to determine the weighting matrices of the LQR controller for PCS using decreasing function iteration factor for the search radius to improve the exploration capability of the FFOA and linearly decreasing inertial weight to provide an efficient balance between exploration and exploitation for the FFOA. This improved the time-to-solution in the determination of the mFFOA-based LQR weighting matrices. Also, the effectiveness of the mFFOA-based LQR on the complex PCS system was investigated.
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1.3 Aim and Objectives
The aim of this study is to develop a modified fruit fly optimization algorithm (mFFOA) for the determination of the optimized weighting matrices (Q and R) of the LQR controller using the aircraft pitch control system (PCS) as a case study.
In order to achieve this aim, the following objectives were set:
1. Replication of the standard FFOA and development of the modified FFOA by the introduction of a linearly decreasing inertial weight and a decreasing iteration function.
2. Comparison of the performances of the standard FFOA and the modified FFOA using ten (10) benchmark optimization test functions (Ackley, Alpine, Eggcrate, Griewank, Pathologic, Rastrigrin, Rosenbrock, Schaffer, Sphere, and Whitley) using the convergence to the global optima as the performance metric.
3. Application of the mFFOA for the determination of the optimized weighting matrices (Q and R) of LQR controller for aircraft PCS model adopted from Jisha & Aswin (2015) and evaluating its performance using time-to-solution and settling time.
1.4 Methodology
The methodology adopted for this research work is as follows:
1. Replicate the standard FFOA, using the following steps:
a. Initialize parameters such as: maximum generation; maximum population; location; step and random reference
b. Initialize the fruit fly group i.e. random direction and distance of searching for food using the sense of smell of an individual fruit fly.
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c. Generate several fruit flies randomly around the fruit using osphresis for
foraging.
d. Evaluate the population to obtain the smell concentration values (fitness
value).
e. Determine the fruit fly with the maximum smell concentration among the
fruit fly swarm.
f. Retain the best smell concentration value and x, y coordinates, the fruit fly
swarm flies toward the position by vision.
2. Develop the modified FFOA:
a. Repeat step 1 (a) with random reference =
max
max
max
min
max
_
exp log
Iter
Iter Iter
where, is the search radius in each iteration, max is the maximum
radius, min is the minimum radius, Iter is the iteration number and
max Iter is the maximum iteration number.
b. Repeat step 1(b)
c. Introduction of linearly decreasing inertial weight
k
iter
w w
w w k
max
max min
max to step 1(c)
where, k w is the current inertial weigh, max w is the maximum generated
inertial weight, min w is the minimum generated inertial weight and k is
the kth position of inertial weight
d. Repeat Step 1(d-f)
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3. Compare the performance of the proposed mFFOA with that of the standard FFOA using ten benchmark optimization test functions (Ackley, Alpine, Eggcrate, Griewank, Pathologic, Rastrigrin, Rosenbrock, Schaffer, Sphere, and Whitley)
4. Determine the optimized LQR weighting matrices (Q and R) using the modified FFOA
5. Implement the LQR controller obtained in step (4) for the PCS model and evaluate the performance of the PCS performance using settling time and time-to-solution as performance metric when the system is subjected to step signal. All simulations will be carried out in MATLAB R2015a.
1.5 Dissertation Organization
The general introduction of this research has been presented in Chapter One. The remaining chapters are organized as follows: Firstly, a comprehensive review of related literature and important fundamental concepts about Fruit Fly Optimization Algorithm, LQR, PCS, Standard Optimization Test Functions, are carried out in Chapter Two. Secondly, a detailed approach and important mathematical models describing the development of the modified fruit fly optimization algorithm are presented in Chapter Three. Thirdly, the analysis, performance and discussion of the results are shown in Chapter Four. Finally, conclusion and recommendations of further work makes up the Chapter Five. The list of references pertinent to the research and MATLAB codes in the appendices are presented at the end of this dissertation.
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