ENME 585 - Quiz 2
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C(D)
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P(D)
1. In the feedback system shown above, proportional control 𝐶(𝐷) = 𝑘𝑃 is applied to the plant
𝐷+1
𝑃(𝐷) = 𝐷2 +1.
a) Plot the closed-loop poles as 𝑘𝑃 > 0 varies, showing correct departure and arrival angles.
b) For some 𝑘𝑃 > 0, the closed-loop system has a pair of repeated poles. These repeated
closed-loop poles occur at what point on the real axis?
c) What value of 𝑘𝑃 gives the repeated poles in part (b)?
d) For a general value of 𝑘𝑃 , find the steady-state error 𝑒𝑠𝑠 corresponding to the following
inputs (or indicate if it does not exist):
i. 𝑟 = [1], 𝑤 = 0.
ii. 𝑟 = 0, 𝑤 = [1].
iii. 𝑟 = [𝑡], 𝑤 = 0.
iv. 𝑟 = 0, 𝑤 = [𝑡].
2. Suppose instead that proportional-integral control 𝐶(𝐷) = 𝑘𝑃 +
Question 1, with 𝑘𝐼 = 𝑘𝑃 .
𝑘𝐼
𝐷
is applied to the plant of
a) Plot the closed-loop poles as 𝑘𝑃 > 0 varies, showing correct departure and arrival angles.
b) For a general value of 𝑘𝑃 , find the steady-state error 𝑒𝑠𝑠 corresponding to the following
inputs (or indicate if it does not exist):
i. 𝑟 = [1], 𝑤 = 0.
ii. 𝑟 = 0, 𝑤 = [1].
iii. 𝑟 = [𝑡], 𝑤 = 0.
iv. 𝑟 = 0, 𝑤 = [𝑡].
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3. Suppose proportional control 𝐶(𝐷) = 𝑘𝑃 is applied to the plant 𝑃(𝐷) =
𝐷−1
(𝐷+1)2 (𝐷+2)
in the
feedback loop shown on the previous page.
a) Plot the closed-loop poles as 𝑘𝑃 > 0 varies.
b) Find the smallest value of 𝑘𝑃 > 0 that gives an unstable closed-loop system.
c) For the value of 𝑘𝑃 found in (b), find the locations of all 3 closed-loop poles.
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