Development of a Position and Trajectory Tracking Control of Ball and Plate System Using a Double Feedback Loop Structure
Development of a Position and Trajectory Tracking Control of Ball and Plate System Using a Double Feedback Loop Structure.
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.
TABLE OF CONTENTS
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
188.8.131.52 Control system design 11
2.2.2 Nonlinear Systems 12
2.2.3 Controllability and Observability 13
184.108.40.206 Stability 14
220.127.116.11 Trajectory and motion tracking 16
18.104.22.168 Path following 17
2.2.4 Types of Controllers 18
22.214.171.124 H controller 19
126.96.36.199 H Mixed sensitivity problem 22
2.2.5 Linear Algebraic Method 23
188.8.131.52 Transient and steady-state requirements 25
184.108.40.206 Implementation by two-parameter configuration 27
220.127.116.11 Actuator parameters 31
18.104.22.168 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
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
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
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
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
4.10.2 Comparison of the Developed Controller with that of Negash and Singh
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
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).
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Borah, M., Majhi, L., Roy, P., & Roy, B. (2014). Design of a Fractional Order PD Controller Tuned by Firefly Algorithm for Stability Control of the Nonlinear Ball and Plate System. Paper presented at the IEEE International Conference on Advanced Communication Control and Computing Technologies, Ramanathapuram, India.
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Cheng, C.-C., & Chou, C.-C. (2016). Fuzzy-Based Visual Servo with Path Planning for a Ball-Plate System. International Symposium on Intelligent Computing Systems (ISICS), 97-107. doi: 10.1007/978-3-319-30447-2_8
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