Consider the disk drive depicted in Figure 3.11. Disk where Read/write head V Figure 3.11: Disk...

 

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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 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