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(ELEC271)[2005](s)final~1833^_91436.pdf
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ELEC271 Automatic Control Systems
Final Exam, Spring 2005
Attempt all questions. They have equal weights. Only the .ve questions in which you score the most will be counted.

Figure 1: Feedback system for stabilization.
1
1. In Figure 1, let P (s)= 1 and C(s) = . Compute bP,C .
ss+2
2. Compute .
2 (s . 1)(s . 2) (s . 10).

n
.
s2 +2ns + 2 (s + 1)(s + 2) (s + 10)
n 2
3. Let U(s) be an all-pass transfer function, i.e., |U(j)| = 1 for all R.
(a) Let G1(s) and G2(s) be arbitrary nonzero transfer functions so that U(s)G1(s) and U(s)G2(s) do not have pole/zero cancellation. Prove
(U(s)G1(s),U(s)G2(s)) = (G1(s),G2(s)).
(b) Show that
.s . 1 s . 2..s . 2.

, =. 1,.
s +1s +1 s . 1
(c) Use the result in (a) to compute
.(s . 2)10 s . 1(s . 2)10 s . 2.

, .
(s + 2)10 s +1(s + 2)10 s +1 1
4. Consider a plant P (s) = . Design a unity feedback controller C(s) so that position
s2 . 1error constant of the loop transfer function P (s)C(s) is in.nity and the closed loop poles are
.1, .2, .3, .4. What is the velocity error constant of the loop transfer function?
1
5. Consider again Figure 1. Assume that P (s)= 3 . Design a controller C(s) to minimize
s
1 .

J =(.y1(t).2 + 64.y2(t).22).
w1(t)=27(t) +(.y1(t).2 + 16.y2(t).22).
2 42 w1(t)=0
w2(t)=0 w2(t)=2(t)
4
6. Design the optimal robust controller for plant P (s)= 2 .
s
1