The differential equation model for a certain position control system for a metal cutting tool is
where the actual tool position is x; the desired position is xd (t); and Kp, KI, and KD are constants called the control gains. Use the Laplace transform method to find the unit-step response [that is, xd(t) is a unit-step function].
Use zero initial conditions. Compare the response for three cases.
a. Kp = 30, KI = KD = 0
b. Kp = 27, KI = 17.18, KD = 0
c. Kp = 36, KI = 38.1, KD = 8.52
