California University of Pennsylvania Department of Applied Engineering & Technology Electrical Engineering Technology < Use as a guide Do not copy and paste> EET 410 Design of Feedback Control Systems Spring 2012 Electromagnetic Ball Suspension System Project Proposal Submission Date: 02/21/2012 In Partial Fulfillment of the Requirements for EET 410 <Note: Absolutely no images on the cover page this is a formal report, not a marketing document> Team Members: Signatures: 1-2- 3-
Contents 1- Abstract 1 2- Introduction 2 2.1 Overview 2 2.2 Project Objectives 3 3- Description of the magnetic Levitation Project 4- Mathematical Model Development 5- Simulation 6- Next Design Steps 7- Preliminary Circuit Design (if any) 8- Design Development Phases and Timeline 9- Bill of Material 10- References (at least 4 follow the guidelines of how to cite references.) <Note, you may have sub-sections that is more likely in the progress and final reports>
1. Abstract Control systems have assumed an increasingly import role in the development and advancement of modern society. Particularly, every aspect of our day-to-day activities is affected by control systems. Control systems are found in all sectors of industry such as quality control of manufactured products, automatic assembly lines, space technology, medical technology, automotive systems, various transportation systems, and weapons systems. This project aims to demonstrate some of the concepts of control systems via the design of implementation of an electromagnetic ball suspension system. The system is unstable by its nature and it will be shown that with proper system modeling and controller design, the system will result in a stable response. 2. Introduction 2.1 Overview Magnetic levitation technology has been receiving increasing attention since it helps eliminate friction losses arising from mechanical contact. It has wide engineering applications such as high-speed trains, magnetic bearings and high precision platforms. A magnetic levitation system is one that suspends a ferromagnetic material with the aid of electromagnetism. Such systems have non-linear and unstable dynamics, which suggests the need for stabilizing controllers.
2.2 Project Objective The objectives of this project are modeling and control of a laboratory-scale magnetic levitation system. The system, examined in this project, facilitates the suspension of a hollow steel sphere at a desired vertical position. Suspension disturbances are accounted for in the controller design. The desired and actual ball position are compared and an error signal is generated for proper control action. The actual ball position is detected using photo emitter/detector cells. The optical transducer produces a voltage corresponding to the ball position. This voltage is fed back to the error generation subsystem to complete the closed-loop and provide stability. Figure 1 shows a graphical representation of the magnetic levitation system used in this project. Figure 1. Electromagnetic Ball Suspension System 2. Project Description This section covers the detailed description of the project including functional description, block diagrams, computer simulations and specifications. 2.1 Functional Description The inputs and outputs of the magnetic levitation system are shown in Table 1 below.
Table 1. Functional Description of the System Input Desired ball position External disturbance- exerted by the user Internal disturbance system parameter variations Output Actual ball position 2.2 Block Diagram Figure 2 shows a high-level block diagram of the system. The user enters the desired position of a steel ball. The system is affected by internal fluctuation in power supply voltage and any external disturbance signals applied by the user. The desired and actual ball position that is fed back are used to generate an error signal for the controller which produces a control signal for the electromagnet. ` Figure 2. Block Diagram Representation of the Magnetic Levitation System The system is composed of several subsystems as follows: Voltage reference signal This reference voltage signal represents the voltage needed to maintain the steel ball at the desired position (nominal position) Position Detection Subsystem Infrared emitter detector pair is used to sense the position of the steel ball. The output voltage of the detector varies as a function of the position. This variation is linear in a small range.
Error Detection Subsystem A difference amplifier is employed to output a voltage proportional to the difference between the reference position signal (desired position) and the actual voltage signal representing the actual ball position. Controller Subsystem This represents an analog compensator. Its input is the error signal from the difference amplifier and its output is the corrective action (voltage to the power amplifier) Power Amplifier Subsystem This circuit outputs a current to drive the coil of the electromagnet. The current at its output is proportional to the applied voltage at its input. The voltage at the power amplifier s input is from the controller/compensator. Electromagnet A coil made up of magnet wire. Its goal is to produce a magnetic force to suspend the steel ball. The magnetic force is proportional to the current flowing through the coil. The overall system block diagram is shown in Figure 3 below. Figure 3. Block Diagram of the System
3. Mathematical Model Development Figure 1 shows the diagram of a magnetic ball suspension system. The objective of the system is to control the position of the ball by adjusting the current in the electromagnet by adjusting the input voltage v(t). The differential equations that describe the system are: (1) And (2) Where: v(t): input voltage y(t): ball position (h) L: winding inductance m: steel ball mass R: winding resistance g: gravitational constant k: magnetic force constant At nominal conditions we have the following for the system variable: The assumptions are: L, R, and m are constant. It should be noted that nominal conditions may be calculated for different masses within the system capabilities (voltage and current ratings)
Define the following state variables to simplify the system representation: Therefore, the state equations are: (3) (4) (5) Linearize the system about the equilibrium point (nominal point) Will use the subscript 0 to designate equilibrium point. Given the values of R, L and m, the nominal value of the current is calculated as follows: Using equation (1), (6) And the nominal applied voltage is calculated from equation (2) to be: (7) We are concerned with small changes in state variables, thus, will replace all the variables with their corresponding small change at nominal conditions: Thus, Applying this concept to equations (3) through (5) results in:
(8) [ ] (9) (10) Express in State Space: [ ] [ ] [ ] [ ] (11) And the output being the ball position, y is given by: [ ] [ ] (12) 4. Simulation The system of equations obtained earlier can be implemented in Matlab to investigate the system s response and to aid in the development of a proper control strategy to achieve the desired performance, The system data is provided in Table 2 below.
displacement about nominal (meters) Table 2. Magnetic Levitation System Parameters (This data is obtained from CMU s web page ) System Parameter Value Units m: Steel Ball mass 0.05 Kg L: Coil Inductance 0.01 H R: Coil Resistance 1 Ω g: gravitational acceleration 9.81 m/s 2 constant k: magnetic force constant 0.0001 H/m or N/A 2 y 0 =h : Desired position 0.01 m The nominal current in the system is calculated using equation (6) The pen loop system is simulated using Matlab, The system has a pole in the RHP, thus resulting in unstable response. System Poles are: { 31.32, -31.32, -100} The step response of the open loop system is given in Figure 4 below. 0 transient response of the Ball-Suspension System -1000-2000 -3000-4000 -5000-6000 -7000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (sec.) (sec) Figure 4. Unstable Response of the uncompensated open loop MagLev System
5. Next Design Steps The next steps in the design will be as follows: Decide on a control strategy Design and test a proper compensator Design the magnetic levitation circuit including the synthesized compensator Implement the design and calibrate the system. Test the system response and verify operation. 7. Design and Development Phases Table 3 lists the major milestones of the project design and development phases. Table 3. Project Development Phases Design Phase Comments Completion Date Preliminary investigation of the project This will help identify the project and the best approach for the design and development completed Develop a mathematical model of the system This will help investigate the behavior of the systemsimulation completed Complete material 02/25/2012 acquisition Design a controller -use simulation to obtain 02/27/2012 best control strategy Fabrication Build assembly for the 03/05/2012 stand, the IR sensors, and for placement of circuitry Design and implement the The complete compensated 03/28/2012 electrical/mechanical system system is built and tested Test and refine the system 04/12/2012
4/15/2011 4/8/2011 4/1/2011 3/25/2011 3/18/2011 3/11/2011 3/4/2011 2/25/2011 2/18/2011 2/11/2011 2/4/2011 8. Design and Development Timeline The project development is illustrated in the Gantt chart below. Timeline for the Completion of the Magnetic Levitation Project Task Preliminary Project Planning Project Approval Acquistion of material Development of mathematical model Test the mathematical maodel and develop a controller Progress report Construct the system Test and verify operation Final demonstration Final report Figure 5. Gnatt chart for the development phases of the project. The project is to be completed (completely functional) by April 10 th 2011. 8. Bill of Material The Bill of Material (BOM) is illustrated in Table 4 below. Table 4. Bill of Material Item Description Quantity Cost/u Vendor nit LM741 Operational Amplifiers 6 $0.24 Available in-house 3N3055 NPN Power Transistor 1 $1.45 DigiKey
8. References Make sure to use several references. Also, you may use this link for how to cite references (References Citation Machine ) Use the MLA format http://libguides.calu.edu/citation