Question: The plant of Problem has a controller formed of an integrative unit (I) combined serially to a proportional plus integrative unit (P + I)-both control units have the same integrative constant KI. Considering unity feedback (Ks = 1), evaluate the steady-state error sensitivity to the derivative constant KD and the integrative constant KI.

Problem: A micro cantilever, such as the one sketched in Figure 11.68, receives an input displacement u from a rigid frame and, as a result, undergoes bending vibrations; the cantilever free end deflection is z. This deflection is sensed op to electronically by a photodiode whose output voltage is proportional (constant Ks) to the input cantilever free-end deflection. When these two units are connected in a standard non-unity-feedback control system with a summing point,

(a) Express the steady-state error in terms of the transfer functions involved and a generic reference signal. Use a lumped-parameter model for the cantilever with mass m, viscous damping coefficient c, and stiffness k- all expressed at the free end of the cantilever.

(b) For integrative (I) control, calculate the steady-state error when individually applying a unit step, a unit ramp, and a unit parabola input to this control system.