```Modeling & Control
of
Magnetic Levitation System
By
Date: March 11th 2003
Outline
• Overview
• Project Description
-
Objective
Functional Description
System Block diagram
Mathematical Model
Software Flowchart
• Project Status
• Updated timeline
• References
Overview

Magnetic Levitation Systems
• Definition
• Nature: non-linear and open-loop unstable

Objective of the project:
• Modeling: Derive mathematical equations that describe
the system dynamics and validate the model experimentally
• Controller: Design stabilizing controller and implement the
controller equations on an 8051 micro-controller to suspend
the ball at a desired position
Functional Description

Inputs:
 Desired vertical position of the ball
 Disturbances could be
• Externally: Varying the disturbance input to the plant
• Internally: From the internal system such as power
supply fluctuation

Output:
• Actual ball position

Feedback:
• Analog signal corresponding to the ball position
System Layout
Plant
HW + SW
Controller
Ball position: 22.5 mm
‘C’= Restart
8051UC
Mathematical Model

There are two sets of
equations that govern the
dynamics of the system:
• Electrical equation- relates the
ball’s suspension X with the
input voltage e.
Obtaining KVL around circuit
e = R. i + L.di/dt - (Lo. x0.i/x^2)dx/dt (1)
Mathematical Model
• Mechanical equation- relates the
resultant force applied to the ball
with the distance from the
Electromagnetic
electromagnet
force
EF= C (i/x)2
From the force diagram
F= GF- EF = m.g - C (i/x)
2
(2)
Gravitation force
GF = m*g
Mathematical Model
• Linearization of the mathematical model
To develop a linear model of the plant about a specified
operating point Xo.
• Method of linearization
Use of Taylor series expansion to linearize the model
Fmag= C(I/x)^2
Delta F = 2C{ io/Xo2 delta I - io^2/Xo^3 delta X)
Where Xo, io are the parameters at the linearized operating
point
Mathematical Model
(Experimental determination of Variables)
• Magnetic force constant
- Plant is highly sensitive to temperature.
- Several attempts were taken to calculate the plant’s
constant.
Method:
- An analog controller was connected to stabilize the
plant.
- The gain of the controller was adjusted till the ball
slightly suspended upwards. The ball was assumed to
be at mechanical equilibrium(F=0)
- The position of the ball and current flowing through
coil were recorded and substituted in the previous
equation.
-2
- C was found to be 1.477x10^-4 N.m2.A
Mathematical Model
(Experimental determination of Variables)
• Coil inductance vs. Ball distance from
electromagnet
• Exhibits a non-linear relationship
• The coil inductance was measured at different ball
positions.
• Found to be fairly constant about the operating
range of 18 to 27mm. (Full-range inductance
variation=210uH)
• After linearization, the inductance was set to be
constant( = 296.74mH)
Mathematical Model
(Experimental determination of Variables)
• Coil inductance vs. Ball distance from the electromagnet
C o il In d u c ta n c e v s B a ll D is ta n c e
y = 0 .0 0 1 x
2
R
2
- 0 .0 7 6 1 x + 2 9 8 .1 2
= 0 .9 9 5 8
2 9 8 .5
298
S e rie s 2
297
P o ly. (S e rie s 2 )
2 9 6 .5
296
B a ll d is ta n c e fro m th e c o il (m m )
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
2 9 5 .5
1
C oil Inductance (m H )
2 9 7 .5
Mathematical Model
(Experimental determination of Variables)
• Calibrating the sensor
• Highly sensitive to opaque objects
Method of calibration:
• A transparent tube-calibrator was used to place
the ball at different positions.
• Assumption taken that the calibrator’s interference
is small. This assumption had to be verified
experimentally by removing the ball and placing
the tube-calibrator between the sensor plates.
• The relationship between the position of the ball
and the sensor’s voltage was found to be linear as
expected.
Mathematical Model
(Experimental determination of Variables)
Sensor voltage vs. Ball distance about the operating point
Mathematical Model
(Experimental determination of Variables)
• Combining all the equations and substituting the
measured constants, the overall plant’s transfer
function was obtained analytically.
X(s)
-------=
E(S)
0.4838
------------------------------------------------------------------------------
0.006232 s^3 + 0.4386 s^2 - 5.29 s - 372.3
Mathematical Model
(Experimental determination of Variables)
• Root locus sketch of the plants’ poles (Notice the
plant’s zeroes are located @ infinity)
Mathematical Model
(Experimental determination of Variables)
• Step response of the open-loop transfer function.
Mathematical Model
(Experimental determination of Variables)
• Next step in the mathematical modeling is verifying
the model experimentally, which is the current task
being performed.
Closed-loop frequency response data were measured
and recorded for graphical analysis. More data needs
to be measured to finalize the modeling, and obtain
the actual transfer function of the plant.
Software Flowchart
Desired
position
Xi
Actual
Position, Xo
Error Signal
E= Xi-Xo
Multiplexed 8-bit A/D
Converter
LCD
displaying desired and
actual ball position
Sampled
error Signal
e(k)
Control
Algorithm
u(k)=f [e(k)]
u(k)
8-bit D/A
Converter
u(t)
Software

8051 micro-controller
• Developed assembly language modules for sampling an input
signal via the A/D converter and producing it at the D/A
converter. User-interface module is fully completed.
• Managed to code a software skeleton for implementing a
hypothetical digital filter (code modification will be done
later for the actual filter).
• Verified the digital filter code by running it on the simulator,
but was unsuccessful with real-time signals.
Software

Signal sampled through the A/D and produced via the D/A
converter.(Freq=7Khz)
Updated
Project timeline

March
• Finalize the modeling phase and obtain the
validated transfer function of the plant.
•
Design a stabilizing controller and verify its
response using Matlab
• Adjust the controller to account for stability, noise
rejection, phase margin and overshoot
specifications.
Updated
Project timeline

April
• Design hardware components required by the
controller.
• Finish coding and debugging the digital filter.
• Test the overall software and verify its
operation.
• Interface the hardware and software and
verify the overall system performance.
Updated
Project timeline

May
• Continue debugging the overall system.
References


Barie and Chiason, International Journal of System
Sciences, 1996, vol 27
D’Azzo and Houpis, Linear control system analysis and
design: Conventional and Modern.

Dempsey, Bradley Univ. EE431-EE432 Lecture notes,

Grabbe, Ramo and Wooldridge, Handbook of automation,

Wong, IEEE Trans. on Education, 1986, vol. E29#4.
2003.
computation and control, vol.1
The End
Questions
Thank you

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