Background 1
Balancing systems, such as the traditional cart-pole and ball and beam systems, present significant challenges for control engineers The ball and plate system, an advanced version of the ball and beam, features a four-degree-of-freedom (DOF) setup where a ball rolls freely on a rigid plate, making it more complex due to its multi-variable coupling This under-actuated system operates with only two actuators and requires two control inputs for stabilization The high-speed movement of the ball adds to the difficulty of designing an effective controller, which is why these systems are rarely utilized in laboratory settings The plate is pivoted at its center, allowing for manipulation of its slope in two perpendicular directions.
A servo system is comprised of a motor controller card and two servo motors designed to tilt a plate It utilizes an intelligent vision system for measuring the ball's position via a CCD camera The primary challenge in motion control is to accurately position the ball on the plate, accommodating both static positions and predefined paths The plate's slope can be adjusted in two perpendicular directions, allowing the tilting of the plate to facilitate the ball's movement (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
2 the fields of path planning, trajectory tracking and friction compensation (Oriolo & Vendittelli, 2005)
In recent years, various control methods for the ball and plate system have emerged Knuplež et al (2003) proposed a controller design based on classical and modern control theory for a two-dimensional electro-mechanical ball and plate system Bai et al (2006) introduced a supervisory fuzzy controller that addressed both the set-point and tracking problems through a two-layer approach Wang et al (2008) developed a nonlinear velocity observer for output regulation, estimating ball velocities using a state observer Hongrui et al (2008) implemented a double feedback loop system to regulate the ball's position, utilizing recursive back-stepping design for the external loop and a switching control scheme for the inner loop Additionally, Dong et al (2009) contributed a proportional-integral-differential neural network controller based on a genetic algorithm for the ball and plate system.
Previous studies have treated the ball and plate system as a single loop structure To enhance control of this system, a double feedback loop structure—comprising a loop within a loop—has been proposed (Liu & Liang, 2010).
Significance of Research 2
The ball and plate system is a crucial model in control education, widely used in both undergraduate and postgraduate studies for teaching and testing control algorithms It serves as a benchmark nonlinear plant, offering greater complexity than the traditional ball and beam system due to the coupling of variables In this system, the ball moves freely without the ability to perceive its environment, meaning it cannot independently control its behavior.
The system faces three key control challenges: ball position control, trajectory tracking, and obstacle avoidance Position control aims to ensure the ball reaches a specified point quickly and accurately, while trajectory tracking focuses on maintaining a defined path at high speeds Obstacle avoidance involves determining the optimal route for the ball in complex environments based on specific criteria These challenges serve as effective benchmarks for evaluating various control schemes, making the design of a suitable controller to address these issues a significant 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.
Problem Statement 3
The ball and plate system is a two-dimensional electromechanical device characterized as a nonlinear, multivariable, and unstable system with two inputs and two outputs As an under-actuated system, it has more degrees of freedom than available actuators To achieve effective control, a double feedback loop structure is implemented, consisting of an inner and outer loop Given the uncertainties related to friction, parameter variations, and measurement delays, nonlinear control methods are essential for both loops The inner loop functions as an angular position controller for the plate's inclination, while the outer loop is responsible for managing the linear position of the balls on the plate.
This study focuses on developing the inner loop of a ball and plate system using a linear algebraic approach A comprehensive transfer function is selected to minimize the integral of time multiplied by absolute error (ITAE), utilizing a two-parameter configuration.
The implementation of compensators derived from the solution of a Diophantine equation is essential for the ball and plate system To address model parameter uncertainties and external disturbances, including friction and variations in the ball's parameters, a robust controller technique based on H-infinity control is utilized in the outer control loop This design ensures effective management of the challenges posed by these factors, enhancing the system's overall performance.
Aim and Objectives 4
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 research aims to enhance the control of the ball and plate system by designing the inner loop with a linear algebraic method and the outer loop utilizing H-infinity sensitivity functions Additionally, it seeks to create a virtual reality model of the system using Virtual Reality Modeling Language (VRML) alongside a graphical user interface (GUI) simulation in MATLAB 2013a The performance of this developed model will be validated through comparisons with the studies of Ghiasi and Jafari (2012) and Negash and Singh (2015), focusing on metrics such as settling time, trajectory tracking error, and maximum overshoot.
Introduction 5
The literature review provides a comprehensive overview of essential concepts and examines related research It begins by analyzing key theories and significant works that underpin the success of this study, followed by a discussion of similar research efforts in the field.
Review of Fundamental Concepts 5
Ball and Plate System 5
The ball and plate system, an advanced two-dimensional variant of the ball and beam system, exhibits strong coupling and nonlinear dynamics, making it a highly unstable plant influenced by friction effects This system serves as an essential benchmark for evaluating the effectiveness of control algorithms, particularly in friction compensation The control target is a plate with two rotating axes, monitored by a digital camera positioned above to track the copper ball's relative position to the plate's center The primary control objectives include stabilizing the ball at a specific location on the plate and enabling it to follow a predetermined trajectory (Wang et al., 2014) Stabilization control focuses on maintaining the ball's position, while trajectory tracking control ensures the ball adheres to a given reference path (Wang et al., 2007).
The ball and plate system allows for simultaneous static and dynamic position tracking control The control process involves several key steps: first, a position sensor, typically a camera or touch-screen device, detects the ball's position on the plate This position data is then relayed to the control system, which calculates the reference angle for either position commands or trajectory tracking When a camera is used, image processing techniques are employed to identify the ball and ascertain its position from the captured image Finally, two actuators, usually DC motors or stepper motors, rotate the plate in two orthogonal directions to balance the ball at the desired position or along a specified trajectory, with a pneumatic device being used in a rare instance.
Figure 2.1 shows a typical laboratory version of the ball and plate system by HUMUSOFT (Andinet, 2011)
Figure 2.1: The Ball and Plate System by HUMUSOFT (Andinet, 2011)
The nonlinear model of the ball and plate system presented in equations (2.1) to (2.4) as described in (Andinet, 2011):
m x b 2 J b J Px x 2 m xx b x m xy b y m xy b xy y m gx b cos x x (2.3)
m y b 2 J b J Py y 2 m yy b y m xy b x m xy b xy x m gy b cos y y (2.4)
In equations (2.1) to (2.4) ,m b (kg) is the mass of the ball; J b kgm 2 is the rotational moment of inertia of the ball; ,
The rotational moment of inertia of the plate is denoted as J kgm, while R m b represents the radius of the ball The ball's position along the X-axis is indicated by x m, and its position along the Y-axis is represented by y m The velocity and acceleration of the ball along the X-axis are given by x m s and x m s respectively, while y m s and y m s denote the velocity and acceleration along the Y-axis The plate's deflection angle about the X-axis is described by θ x (in radians), with θ x indicating the angular velocity of this deflection Similarly, θ y represents the deflection angle about the Y-axis, and θ y signifies the angular velocity of the deflection in that direction.
Y-axis; x Nm is the torque exerted on the plate in X-axis direction and y Nm is the torque exerted on the plate in Y-axis direction
Equations (2.1) and (2.2) illustrate the relationship between the ball's acceleration and the plate's deflection angle and angular velocity Meanwhile, Equations (2.3) and (2.4) detail how the dynamics of plate deflection are influenced by external driving forces and the position of the ball (Duan et al., 2009).
To simplify the mathematical modelling of the ball and plate system, the following assumptions are made (HUMUSOFT Ltd, 2012): i) Ball-plate contact is not lost under any circumstances
8 ii) No sliding of the ball on the plate is allowed iii) All friction forces and torques are neglected iv) Plate angles and area limitations are not considered
Based on the above mentioned assumptions, Figure 2.2 shows the rigid body model for the ball and plate system
The ball and plate system can be mathematically analyzed as a simplified particle system comprising two rigid bodies The plate is constrained by three geometric limits for translation along the X, Y, and Z axes, and it has a rotational limit about the Z-axis It possesses two degrees of freedom (DOF) for rotation around the X and Y axes Meanwhile, the ball is restricted in its translation along the Z-axis and has two degrees of freedom in translation along the X and Y axes Overall, the model exhibits four degrees of freedom (DOF), with the generalized coordinates defined as q1 = x, q2 = y, q3 = θx, and q4 = θy.
Consider the state variable assignment (Fan et al., 2004)
From equations (2.3) and (2.4), since the mass and the moment of inertia of the ball are negligible compared to the moment of inertia of the plate, then,
(2.9) The state space equations of the ball and plate system can be expressed as (Fan et al., 2004):
To develop a linearized model of the ball and plate system, several assumptions are made, as noted by Andinet (2011) Firstly, it is assumed that the motor can precisely control each movement of the ball, ensuring that no steps are skipped and that the magnitude of the plant moment remains unaffected.
In the analysis of the ball and plate system, the rotor's position is primarily influenced by the inputs θx and θy rather than the torque moments τx and τy, as the load moments do not impact motor positioning Consequently, equations (2.3) and (2.4) are omitted from the study In a steady state, the plate must maintain a horizontal position, with both inclination angles set to zero Assuming the angle remains within ±5°, the sine function can be simplified to its argument Additionally, in equations (2.1) and (2.2), the small velocities θx and θy are negligible when squared or multiplied together.
Based on the above assumptions, the linearized, simplified and uncoupled ordinary differential equations are (Andinet, 2011):
By substituting the moment of inertia of the ball into equations (2.15), (2.13), and (2.14), we observe that the state space description is divided into two distinct parts This division arises from the simplified assumptions that allow for the independence of motion along the X-axis and Y-axis (Andinet, 2011).
The system functions as two distinct yet simultaneous systems, allowing for the use of similar but independent controllers to manage each coordinate of the ball's motion (Awtar et al., 2002).
Based on the linear model, a preliminary controller is designed with the scheme of a
„loop within a loop‟ The first step involves the design of an inner loop where the encoder feedback is sent to the dc motors to achieve a servo position control
The control system design involves integrating an inner loop within an outer loop that regulates the ball's position The subsequent step is to develop a controller for the outer loop, utilizing the transfer function that relates the ball's position to the corresponding plate angle (Awtar et al.).
The overall control scheme consists of an outer loop controller that calculates the necessary angle for the plate to balance the ball, while the inner loop controller executes the movement of the plate to that angle Although the inner loop aims to perform this action instantaneously, real-world limitations prevent this from happening (Awtar et al., 2002) This process is illustrated in Figure 2.3.
Figure 2.3 Control Scheme (Awtar et al., 2002)
To effectively balance the ball, it is essential to maintain a significantly higher speed in the inner loop compared to the outer loop, as demonstrated by Awtar et al (2002).
Most physical systems exhibit nonlinear characteristics, prompting engineers to linearize them around a nominal operating point for analysis This approach allows for the examination of the resulting linear model, which is defined within the framework of linear systems To qualify as a linear system, it must adhere to two key principles: superposition and homogeneity (Khalil & Grizzle, 1996).
The principle of superposition states that for two different inputs, x and y , in the domain of the function f , (Khalil & Grizzle, 1996)
f x y f x f y (2.18) The principle of homogeneity states that for a given input, x , in the domain of the function f , and for any real number k, (Khalil & Grizzle, 1996)
Any function that does not satisfy superposition and homogeneity principle is nonlinear
It is worth noting that there is no unifying characteristic of nonlinear systems, except for not satisfying the two above-mentioned principles (Khalil & Grizzle, 1996)
To simplify the analysis of the nonlinear ball and plate system, it is essential to linearize the system around its operating point This operating point is defined as the state where all input and state variables are initialized to zero (Khalil & Grizzle, 1996).
A system is said to be controllable if a control vector u t exist that will transfer the system from any initial state x t 0 to some final state x t in a finite time interval (Burns, 2001)
A system is said to be completely state controllable if the Kalman controllability matrix (Burns, 2001):
M c B AB A B (2.20) is of full rank, that is it contains n linearly independent column or row vectors
A system is said to be observable if at time t 0 , the system state x t 0 can be exactly determined from observation of the output y t over a finite time interval (Burns, 2001)
A system is said to be completely state observable if the kalman observability matrix (Burns, 2001):
M C A C A C (2.21) is of full rank, that is, it contains n linearly independent rows or columns vectors
The concepts of controllability and observability are crucial in control systems, as they help determine the existence of a viable control solution from the outset While most physical systems exhibit controllability and observability, their mathematical models often do not If a model is identified as uncontrollable and unobservable, it indicates that the model fails to accurately represent the physical system, potentially leading to suboptimal control solutions (Ogata, 2002).
The condition for stability is that all the poles must lie in the left hand of the s-plane (LHP), as shown in Figure 2.4
Figure 2.4: Root Locus Plot of System Indicating Region of Stability (Dingyu et al.,
Ball and Plate System Modelling 56
This section explains the decomposition of the nonlinear equation of the ball and plate system and linearization of the decomposed nonlinear equation of the ball and plate system.
Decomposition of the Ball and Plate System 56
As discussed in subsection 2.2.1, the mathematical model of the ball and plate system can be decomposed into X-axis and Y-axis as follows:
The ball and plate system can be divided into two independent subsystems: the X-axis and Y-axis This allows for a focused analysis on the X-axis direction.
From the state space model of the ball and plate system as given in equation (3.1) and (3.2), and substituting the value of B and J b as given in equation (2.15) as:
However, since the ball and plate system was decomposed into the X and Y-axis, the state space model along the X-axis is given as:
Linearization of the Ball and Plate System 57
The ball and plate system was linearized by using the approximate moment of inertia for a solid ball as defined in equation (2.15) This value was then substituted into equations (2.1) and (2.2) under the assumption that the plate's inclination angle remains within ±5 degrees, allowing for the simplification of the sine function, where sin(θ) can be approximated by its argument.
(3.6) ii) The velocities x and y are small and have negligible effect when squared and multiplied together
Then, equations (2.16) and (2.17) are written as:
The differential equations for the X-axis and Y-axis, represented by equations (3.8) and (3.9), were utilized to estimate the states of the ball and plate system, including x, y, ẋ, and ẏ By incorporating the system inputs θx and θy, the transfer function for the system can be formulated.
Subsequent analysis was made using the linearized model of the ball and plate system due to its symmetrical nature.
Controllability and Observability Test for the Ball and Plate System 58
To perform position and trajectory tracking analyses, controllability and observability tests were conducted on the system to determine its effectiveness Figure 3.1 illustrates the test code used for the ball and plate system, highlighting its controllable and observable characteristics.
Figure 3.1: Controllability and Observability Test Code