Nonlinear Control Sy.. - Free

24 CHAPTER 1. INTRODUCTION
This model considers the fact that as the ball approaches the magnetic core of the coil, the
flux in the magnetic circuit is affected, resulting in an increase of the value of the inductance.
The energy in the magnetic circuit is thus E = E(i, y) = 2L(y)i2, and the force F = F(i, y)
is given by
F(i, y) =
8E _ i2 8L(y) - -1 µz2
8y 2 ay 2(1+µy)2
Assuming that the friction force has the form
(1.29)
fk = ky (1.30)
where k > 0 is the viscous friction coefficient and substituting (1.30) and (1.29) into (1.27)
we obtain the following equation of motion of the ball:
1 .\µi2
my=-ky+mg-2(1+µy)2 (1.31)
To complete the model, we recognize that the external circuit obeys the Kirchhoff's voltage
law, and thus we can write
where
d _ d Mi
dt (LZ)
dt 1 + µy
Substituting (1.33) into (1.32), we obtain
v=Ri+at-(Li)
y(1+µy) dt+o9 (1+µy) dt
_ .µi dy .\ di
(1+µy)2 dt + 1+µy dt.
v = Ri - (1 + Uy)2 dt + l + µy dt
(1.32)
(1.33)
(1.34)
Defining state variables xl = y, x2 = y, x3 = i, and substituting into (1.31) and (1.34), we
obtain the following state space model:
it = x2
2 9
k
m x2
\µx3
2m(1+µx1)2
x3 =
1 + p xl
l1
[_RX3+ (l + xl)2 x2x3 + vJ .