Question 1 (60%)
Consider a unity feedback control system with the time delayed plant
G(s) = 4/(S^2+2S+4) e^-0.1s
and the PI controller
C(s) = kp+ ki/s
.
(a) Find the boundary and region in the kp – ki parameter space where phase margin
PM > 50o
. (20 pts)
(b) Find the boundary and region in the kp – ki parameter space where gain margin
GM > 3. (20 pts)
(c) Obtain intersections of PM and GM bounds. Select a test point that satisfies both
requirements and show frequency domain properties (PM and GM) of the closed-loop
system for this selected point. (20 pts)
(Hint: Plot kp and ki values from 0 to 2 in order to easily visualize regions in the parameter
space).
Question 2 (20%)
Consider a sensitivity weighting function WS(s) with the following properties:
– The maximum steady state error is limited to 0.1.
– The high frequency disturbance amplification is limited to 1.6.
– The minimum transition frequency is 10 rad/sec.
Obtain frequency magnitude plot of W-1
S
(s). Show the required properties on the plot.
Question 3 (20%)
Consider a complementary sensitivity weighting function WT (s) with the following properties:
– The unstructured model error is assumed to be smaller than 5% for frequencies ? 20
rad/sec.
– It is assumed that the unstructured model error does not exceed 180% for frequencies
? 20 rad/sec.
Obtain frequency magnitude plot of WT (s) and show the required properties on the plot.
Also, plot frequency magnitude plot for WT (s)?m(s) and show the scaling effect of WT (s)
on unstructured uncertainty ?m
