Nonlinear Control Sy.. - Free

1.9. EXAMPLES OF NONLINEAR SYSTEMS 23
1.9.1 Magnetic Suspension System
Figure 1.15: Magnetic suspension system.
Magnetic suspension systems are a familiar setup that is receiving increasing attention in
applications where it is essential to reduce friction force due to mechanical contact. Magnetic
suspension systems are commonly encountered in high-speed trains and magnetic bearings,
as well as in gyroscopes and accelerometers. The basic configuration is shown in Figure
1.15.
According to Newton's second law of forces, the equation of the motion of the ball is
V (t)
my = -fk + mg + F (1.27)
where m is the mass of the ball, fk is the friction force, g the acceleration due to gravity,
and F is the electromagnetic force due to the current i. To complete the model, we need to
find a proper model for the magnetic force F. To this end we notice that the energy stored
in the electromagnet is given by
E = 1Li2
where L is the inductance of the electromagnet. This parameter is not constant since it
depends on the position of the ball. We can approximate L as follows:
L = L(y) = 1+µy.
(1.28)