2)    You are to optimize the turntable for moves of 45° and 90°. For all the following parts provide plots of the angular displacement, the angular velocity, and the drive value. Show all calculations and provide your reasoning for any assumption.

a.    Define a proportional gain K_p based on an assumed linear range, then perform angular moves of 45° and 90°.

• Select your own linear range. A general guideline is to select a linear range close to the smallest move one expects to move.

b.    Perform angular moves of 45° and 90° with double the value K_p used in part a.

c.    Using the value K_p from part a, calculate a value of K_d assuming a theoretical damping factor ζ= 0.9, then perform angular moves of 45° and 90°.

d.    Repeat part c but with K_d equal to 0.5X (ζ= 0.45) and 2X (ζ= 1.8) of what was calculated in part c.

e.    Repeat part c with the addition of integral gain K_i to eliminate any offsets. Describe any limits you put on the integral gain. Compare results with and without integral gain.

f.     Comment on what you would do next to optimize your control response.