What is the correct derivation of the magnetic levitation system dynamics?

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The equations in the slides form the lecture did not match the physical system. The equations should have been



 m \ddot z &= mg - k_m (k_A u)^2 / z^2 \\
 v_{ir} &= k_T z + v_0



where is the distance of the ball from the magnet (so positive z is down). The linearization of this equation around an equilibrium point gives a transfer function of


P(s) = \frac{-k}{s^2-r^2} \qquad k,r > 0


which corresponds to the plots that were shown. The computations requried for the plots can be found in L9_2_maglev.m.

--Richard Murray 08:01, 23 November 2006 (PST)