Problem 1. (2 points) Consider the magnetic levitation system in Fig. 1. The equation of motion for the levitated ball is: m Light Source dw(t). dt Lens. where w(t) is the position of the ball measured from the bottom of the coil to the top of the ball, i(t) is the coil current, m is the ball's mass, g is the gravitational constant, and a is a constant determined via experiments. Let the state vector of the nonlinear system be given by z(t) = and the output be w(1). Coil i(t) mgaw (1) Ball = Photosensor Array Figure 1. Schematic of a magnetic levitation system. (t) = a) (0.5 point) Verify that the nonlinear state equation is Z (t) [00] [w] w (1) g output. Where are the poles and zeros located? 1 m , the input be v(t) = i(t), v (t) az(t), b) (1 point) Determine a linear state space representation of the nonlinear system that approximates the dynamics of the magnetic levitation system when the ball is 0.01 meter below the coil and it remains at that position. c) (0.5 point) Determine the transfer function of the linearized system. Clearly explain what are its input and