Digital versatile disc (DVD) players incorporate several control systems for their operations. The control tasks include (1) keeping the laser beam focused on the disc surface, (2) fast track selection, (3) disc rotation speed control, and (4) following a track accurately. In order to follow a track, the pickup-head radial position is controlled via a voltage that operates a voice coil embedded in a magnet configuration. For a specific DVD player, the transfer function is given by

where x(t) = radial pickup position and v(t) = the coil input voltage (Bittanti, 2002).

a. Assume that the system will be controlled in a closed-loop configuration, such as the one shown in Figure P9.1. Assuming that the plant, P(s), is cascaded with a proportional compensator,
G_c(s) = K, plot the root locus of the system.

b. Repeat Part a using MATLAB if your root locus plot was
created by any other tool.

c. Find the range of K for closed-loop stability, the resulting damping factor range, and the smallest settling time.

d. Design a notch filter compensator so that the system’s dominant poles have a damping factor of ζ = 0.7 with a closed-loop settling time of 0.1 second.

e. Simulate the system’s step response for Part c using MATLAB.

f. Add a PI compensator to the system to achieve zero steady-state error for a step input without appreciably affecting the transient response achieved in Part b.

g. Simulate the system’s step response for Part e using MATLAB.

X(s) P(s) = V(s) 0.63 0.36 1+ 305.4 0.04 (1+ 305.42 248.2 248.22