Consider the disk drive depicted in Figure 3.11. Disk where Read/write head V Figure 3.11: Disk Drive Problem The horizontal position of the head shown in Figure 3.11 is driven by a motor which may be modelled as a double integrator. Let y denote the head position, and u denote the motor voltage. Then A = * = Ax + Bu y = Cx 4-88---8- Drive Motor. B = C = [20]. 0 - [J Sensor The track-following problem is that of holding the head above a given track on the disk while reading from or writing to the disk. Let the rotational velocity of the disk be wo (according to [30], between 2500 and 4000RPM.) Due to the fact that the disk is not centered perfectly, the tracks defined as circles on the disk actually trace out ellipses with respect to a stationary head position. Hence, the track following problem requires that the head position track a periodic signal satisfying 2 = F2, F = r = Gz For purposes of illustration, assume wo 1. (a) Form the augmented state equations for this system. (b) Derive general state equations for the closed loop response to the refer- ence command. , z(0) = zo (c) Design a stabilizing state feedback so that the control input in response to the reference r(t) = sint + 2 cost satisfies the bound |u(t)| < 5, and so that the output converges to its steady state value within 5 seconds (cf. Figure 3.12). Hand in your values of state feedback gain and a plot showing that the specifications have been satisfied. (d) Explain why, as shown in Figure 3.12, the asymptotic values of u(t) and y(t) are 180 out of phase. response of disk drive system to reference time (e) Using a Bode plot of the transfer function from r to the tracking error e, show that the use of the augmented oscillator introduces a zero at the oscillator frequency. -reference r ...input u --output y DOX Figure 3.12: Reference Tracking for Disk Drive