To introduce the state$-$space control design method, we will use the

magnetically suspended ball as an example.

The current through the coils induces a magnetic force which can

balance the force of gravity and cause the ball $($which is made of a

magnetic material$)$ to be suspended in mid$-$air.

The equations for the system are given by:

m$($d$^(2)$h$)/($dt$^(2))=$mg$-($Ki$^(2))/($h$)$

V$=$L$($di$)/($dt$)+$iR

where h is the vertical position of the ball, i is the current through the

electromagnet, V is the applied voltage, m is the mass of the ball, g is

the acceleration due to gravity, L is the inductance, R is the resistance,

and K is a coefficient that determines the magnetic force exerted on

the ball.