ABSTRACT
This research work presents the development of a position and trajectory tracking control of ball and plate system. The ball and plate control system was considered as a double feedback loop structure (a loop within a loop), for effective control of the system. The inner loop was designed using linear algebraic method by solving a set of Diophantine equations. The outer loop was designed using H-infinity sensitivity approach. A virtual reality model of the ball and plate system using the virtual reality modelling language (VRML) and graphical user interface (GUI) based simulation model of the system were developed in MATLAB 2013a. The results of the simulation of the system showed that the plate was stabilized at 0.3546 seconds and the ball was able to settle at 1.7087 seconds. The trajectory tracking error of the system using the H-infinity controller was 0.0095 m. The improvements in terms of trajectory tracking error and settling time of the system when compared with the single loop H-infinity (SLH) controller are 71.8% and 60.5% respectively. The improvements when compared with the double loop structure using fuzzy sliding mode controller are 52.5% and 51.2% in terms of the trajectory tracking error and settling time respectively.
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TABLE OF CONTENTS
TITLE PAGE
DECLARATION i
CERTIFICATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT vi
TABLE OF CONTENTS vii
LIST OF FIGURES xi
LIST OF TABLES xiii
LIST OF APPENDICES xiv
LIST OF ABBREVIATION xv
CHAPTER ONE: INTRODUCTION
1.1 Background 1
1.2 Significance of Research 2
1.3 Problem Statement 3
1.4 Aim and Objectives 4
CHAPTER TWO: LITERATURE REVIEW
2.1 Introduction 5
2.2 Review of Fundamental Concepts 5
2.2.1 Ball and Plate System 5
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2.2.1.1 Control system design 11
2.2.2 Nonlinear Systems 12
2.2.3 Controllability and Observability 13
2.2.3.1 Stability 14
2.2.3.2 Trajectory and motion tracking 16
2.2.3.3 Path following 17
2.2.4 Types of Controllers 18
2.2.4.1 H controller 19
2.2.4.2 H Mixed sensitivity problem 22
2.2.5 Linear Algebraic Method 23
2.2.5.1 Transient and steady-state requirements 25
2.2.5.2 Implementation by two-parameter configuration 27
2.2.5.3 Actuator parameters 31
2.2.5.4 Inner loop design 33
2.2.6 Virtual Reality Modelling Language (VRML) as a 3-D Modelling Tool 35
2.2.7 Simulink® 3D Animation 37
2.2.8 Graphical User Interface (GUI) 38
2.3 Review of Similar Works 39
CHAPTER THREE: MATERIALS AND METHODS
3.1 Introduction 55
3.2 Methodology 55
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3.2.1 Ball and Plate System Modelling 56
3.2.2 Decomposition of the Ball and Plate System 56
3.2.3 Linearization of the Ball and Plate System 57
3.2.4 Controllability and Observability Test for the Ball and Plate System 58
3.3 Selection of the Actuator Parameters 59
3.3.1 Two-Port Parameter Configuration 63
3.4 Determination of the H Controller 64
3.5 Development of the Virtual Reality (VR) Model of the Ball and Plate System
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3.6 Development of the Simulation Environment in MATLAB Simulink 66
3.6.1 Development of the Inner Loop Controller 66
3.6.2 Development of the Outer Loop Controller 67
3.6.3 Development of the Ball Dynamics of the Ball and Plate System 68
3.6.4 Development of the Reference Signal for the Trajectory Tracking 70
3.7 Graphical User Interface (GUI) of the Ball and Plate System 70
3.8 Performance Evaluation 71
3.8.1 Trajectory Tracking Error 71
3.8.2 Transient Response 71
3.9 Comparison of Results 71
CHAPTER FOUR: RESULTS AND DISCUSSION
4.1 Introduction 72
4.2 Result of the Controllability and Observability Test on the System 72
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4.3 Result of the Actuator Parameter 72
4.4 Result of Two-Port Parameter Configuration 73
4.5 Result of the H Controller 75
4.6 Result of the Virtual Reality (VR) Model 76
4.7 Result of the Trajectory Tracking of the Ball and Plate System 78
4.8 Result of the Graphical User Interface (GUI) for the Circular Trajectory
Tracking 79
4.9 Result of the Circular Trajectory Tracking Using H-infinity Controller
Considering the Ball and Plate System as a Single Loop System 79
4.10 Comparison of the Results 80
4.10.1 Comparison of the controllers based on the Step Response Performance
Index 81
4.10.2 Comparison of the Developed Controller with that of Negash and Singh
(2015) 82
CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion 83
5.2 Limitation 83
5.3 Significant Contributions 83
5.4 Recommendations for Further work 84
REFERENCES 85
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CHAPTER ONE
INTRODUCTION
1.1 Background Balancing systems are one of the most popular and challenging test platforms for control engineers. Such systems are like the traditional cart-pole system (inverted pendulum), the ball and beam system, double and multiple inverted pendulums (Mohajerin et al., 2010). The ball and plate system is a generalization of the famous ball and beam benchmark system. The latter is a two degree of freedom (DOF) system consisting of a ball that can roll on a rigid beam, while the former is a four DOF system consisting of a ball that can roll freely on a rigid plate (Moarref et al., 2008). However, it is more complicated than the ball and beam system due to its coupling of multi-variables. This under-actuated system has only two actuators and is stabilized by just two control inputs (Ghiasi & Jafari, 2012). Since the movement of the ball over the plate can reach high speeds, the design of a suitable controller for this system is a major challenge; therefore, these systems are not commonly used in laboratories (Galvan-Colmenares et al., 2014). The system consists of a plate pivoted at its centre such that the slope of the plate can be manipulated in two perpendicular directions (Dong et al., 2011). A servo system consists of motor controller card and two servo motors to tilt the plate. Intelligent vision system is used for measurement of a ball position from a CCD camera. The problem of the motion control of this system is to control the position of a ball on a plate for both static positions and desired paths. The slope of the plate can be manipulated in two perpendicular directions, so that the tilting of the plate will make the ball move on the plate (Dong et al., 2011).
The ball and plate system finds application in areas like humanoid robot, satellite control, rocket system and unmanned aerial vehicle (UAV) (Mukherjee et al., 2002) in
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the fields of path planning, trajectory tracking and friction compensation (Oriolo & Vendittelli, 2005). Various control methods have been introduced in the recent years for the ball and plate system. A controller design for two dimensional electro-mechanical ball and plate system based on the classical and modern control theory was proposed by (Knuplež et al., 2003). A supervisory fuzzy controller for studying motion control of the system included the set-point problem and the tracking problem along desired trajectory which was composed of two layers as described by (Bai et al., 2006). A nonlinear velocity observer for output regulation of the ball and plate system where the ball velocities were estimated by state observer was proposed by (Wang et al., 2008). In the work of (Hongrui et al., 2008), the position of the ball was regulated with a double feedback loop system, in which recursive back-stepping design was employed for the external loop, while switching control scheme was employed in the inner feedback loop. Also, proportional-integral-differential neural network controller based on genetic algorithm is another piece of work done on ball and plate system by (Dong et al., 2009). However, previous research works considered the ball and plate as a single loop structure. For an effective control of the ball and plate system, a double feedback loop structure, that is, a loop within a loop is considered (Liu & Liang, 2010).
1.2 Significance of Research
The ball and plate system is one of the most popular and important models in control education, which is used at undergraduate and post graduate studies in teaching and testing of control algorithms and also it is a benchmark nonlinear plant, this is because it is more complex than the traditional ball and beam system due to the coupling between the variables. The ball is able to move freely and has no ability to recognise the environment, as a result, the ball cannot control its behaviour by itself. However, the
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control problems of the system consist of the ball‟s position control, trajectory tracking and obstacle avoidance. The ball‟s position control problem is to make the ball arrive at a point accurately and as soon as possible, the trajectory tracking problem is to make the ball follow some defined path accurately and at high speed, and the obstacle avoidance control problem is to find an optimum path for the ball in a complicated environment based on some certain criteria. All these problems are good benchmarks to proof-test the capability of the different control schemes. As a result, designing a suitable controller that will solve these problems is a major challenge. In this research, the study of the first two problems, which is position and trajectory tracking has been carried out, considering the system as a double loop structure.
1.3 Problem Statement The ball and plate system apparatus is a two dimensional electromechanical device which can be considered as a nonlinear, multivariable (with two inputs and two outputs) and unstable system. The system is under-actuated as it possesses more degrees of freedom than the number of available actuators. For an effective control of the ball and plate system, a double feedback loop structure, that is, a loop within a loop is considered. However, due to the existence of uncertainties due to friction, parameter uncertainties, measurement time delays, practical applications requires nonlinear control methods that will be adopted in the design of the inner and outer loops. In order to fulfil these requirements, the inner loop is designed as an actuator (angular) position controller for the plate inclination, while the outer loop is designed so as to control the balls (linear) position on the plate.
This research is aimed at designing the inner loop of the ball and plate system based on linear algebraic method. An overall transfer function is chosen that minimizes the integral of time multiplied by absolute error (ITAE), and a two parameter configuration
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is used for the implementation of the compensators that will be obtained from the solution of a Diophantine equation. A robust controller technique, based on H-infinity control technique is implemented in the outer control loop of the ball and plate system, due to the problem of model parameter uncertainties and external disturbances, such as friction between the ball and the plate and parameter variations on the ball, it is necessary to design the outer loop with a controller that will take care of the above problem.
1.4 Aim and Objectives The aim of this research is the development of a position and trajectory tracking control scheme for the ball and plate system using a double feedback loop structure. The objectives of the research are as follows:
i) To design the inner loop using linear algebraic method and the outer loop using H-infinity sensitivity function for a more effective control of the ball and plate system.
ii) To develop a virtual reality model of the ball and plate system using the virtual reality modelling language (VRML) and its graphical user interface (GUI) based simulation model using MATLAB 2013a.
iii) Validation of the performance of the developed model by comparison with the work of Ghiasi and Jafari (2012) and Negash and Singh (2015) based settling time, trajectory tracking error and maximum overshoot as performance metrics.
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