160 CHAPTER 4. FUZZY CONTROL
4.18. Magnetic levitation
The mathematical model of this system can be obtained as follows. For
the steel ball,
Mz®=Mg−F
For the electrical circuit, using Kirchoffís voltage law, we can write
L
di
dt
+Ri=V
whereLis the coil inductance andRis the coil resistance. The coupling
equation where the forceF is related to the currentiis given by
F=km
i^2
z^2
wherekm is the magnetic constant. Assume that the ball position is
measured using an optical sensor. Use the following set of parameters in
your model setup.
M Mass of steel ball 20 milligrams (mg)
km Magnetic constant 2. 058 x 10 −^4 N(m/A)^2
R Coil resistance 0. 92 Ohms (Ω)
L Coil inductance 0. 01 millihenry (mH)
i Coil current [0,3]Amps (A)
V Coil voltage [0,5]Volts (V)
g Gravitational constant 9 .80665 m/s^2
z Ball position [min, max] [3,7] cm
(a)Linearizethesystemattheminimumandmaximumballpositions.
For each linearization, design a PID controller using a suitable set
of criteria, and simulate the performance of the system for various
ball positions around the point of linearization. How well do the two
separate linear models compare? Is it sufficient to obtain a single
linear model that will perform equally well?
