Download Sample project proposal - Ghassan Salim

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 2013
Electromagnetic Ball Suspension System
Project Proposal
Submission Date: 02/11/2013
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>
Submitted by
<Names (no signatures)>
Contents
1- Abstract
2- Introduction
2.1
Overview
2.2
Project Objectives
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 final report>
1
2
2
3
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 and
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:
𝑚
𝑑 2 𝑦(𝑡)
𝑑𝑡 2
= 𝑚𝑔 − 𝑘
𝑖 2 (𝑡)
(1)
𝑦(𝑡)
And
𝑣(𝑡) = 𝑅𝑖(𝑡) + 𝐿
𝑑𝑖
𝑑𝑡
(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:
𝑣 (𝑡) = 𝑣0 .
𝑦(𝑡) = 𝑦0 = ℎ
𝑖(𝑡) = 𝑖0
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:
𝑥1 (𝑡) = 𝑦(𝑡), 𝑥2 (𝑡) =
𝑑𝑦(𝑡)
𝑑𝑡
, 𝑥3 (𝑡) = 𝑖(𝑡)
Therefore, the state equations are:
𝑑𝑥1 (𝑡)
𝑑𝑡
𝑑𝑥2 (𝑡)
𝑑𝑡
𝑑𝑥3 (𝑡)
𝑑𝑡
= 𝑥2 (𝑡)
=𝑔−
(3)
𝑘 𝑥32 (𝑡)
𝑚 𝑥1 (𝑡)
(4)
𝑅
1
𝐿
𝐿
= − 𝑥3 (𝑡) + 𝑣(𝑡)
(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),
𝑖0 = √
𝑚𝑔𝑥01
𝑘
= 𝑥30
(6)
And the nominal applied voltage is calculated from equation (2) to be:
𝑣 = 𝑅𝑖0
(7)
We are concerned with small changes in state variables, thus, will replace all the variables with
their corresponding small change at nominal conditions:
𝑑𝑥
𝑑(Δ𝑥)
Thus,
→
= (Δ𝑥́ )
𝑑𝑡
𝑑𝑡
Applying this concept to equations (3) through (5) results in:
Δ𝑥1̇ = Δ𝑥2
Δ𝑥2̇ = 0 −
Δ𝑥3̇ = −
𝑅
𝐿
(8)
2 Δ𝑥
𝑘 𝑥01 (2𝑥03 Δ𝑥3 )−𝑥03
1
𝑚
[
2
𝑥01
] Δ𝑥2
1
Δ𝑥3 + Δ𝑣
𝐿
(9)
(10)
Express in State Space:
Δ𝑥1̇
[Δ𝑥2̇ ] =
Δ𝑥3̇
𝑘
𝑚
0
1
2
𝑥03
2
𝑥01
0
[ 0
0
0
−2𝑘 𝑥03
𝑚 𝑥10
−𝑅
𝐿
]
0
Δ𝑥1
[Δ𝑥2 ] + [ 0 ] Δv
1⁄
Δ𝑥3
𝐿
(11)
And the output being the ball position, y is given by:
Δ𝑦 = [1
4.
0
Δ𝑥1
0] [Δ𝑥2 ]
Δ𝑥3
(12)
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.
Table 2. Magnetic Levitation System Parameters (This data is obtained from CMU’s web page )
System Parameter
m: Steel Ball mass
L: Coil Inductance
R: Coil Resistance
g: gravitational acceleration
constant
k: magnetic force constant
y0 =h : Desired position
Value
0.05
0.01
1
9.81
Units
Kg
H
Ω
m/s2
0.0001
0.01
H/m or N/A2
m
The nominal current in the system is calculated using equation (6)
𝑖0 = √
𝑚𝑔𝑥01
= 7.0036 𝐴
𝑘
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.
transient response of the Ball-Suspension System
0
displacement about nominal (meters)
-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:





7.
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.
Design and Development Phases
Table 3 lists the major milestones of the project design and development phases.
Table 3. Project Development Phases
Design Phase
Preliminary investigation of
the project
Develop a mathematical
model of the system
Complete material
acquisition
Design a controller
Fabrication
Design and implement the
electrical/mechanical system
Test and refine the system
Comments
This will help identify the
project and the best
approach for the design and
development
This will help investigate the
behavior of the systemsimulation
Completion Date
completed
completed
02/25/2012
-use simulation to obtain
best control strategy
Build assembly for the
stand, the IR sensors, and for
placement of circuitry
The complete compensated
system is built and tested
02/27/2012
03/05/2012
03/28/2012
04/12/2012
8.
Design and Development Timeline
The project development is illustrated in the Gantt chart below.
Timeline for the Completion of the Magnetic Levitation Project
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
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 10th 2011.
8.
Bill of Material
The Bill of Material (BOM) is illustrated in Table 4 below.
Table 4.
Item
LM741
3N3055
Bill of Material
Description
Operational Amplifiers
NPN Power Transistor
Quantity
6
1
Cost/u
nit
$0.24
$1.45
Vendor
Available in-house
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
Team Members:
Signatures and Dates:
1-
/
2-
/
3-
/