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(ELEC271)[2008](s)final~1833^_71952.pdf
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ELEC271 Automatic Control Systems
Final Exam, Spring 2008
1
1. Consider the unity feedback system shown in Figure 1. Here P (s) = and C(s)= K, where K is a pure
s(s + 4)
gain. Denote the transfer function from r(t) to y(t) by G(s).
(a) Design K such that the system is critically damped.
.
(b)
Design K such that the overshoot of y(t) when r(t) is a step is e.

(c)
Design K to maximize the velocity error constant Kv of the loop transfer function L(s)= P (s)C(s) subject to .G(s). = 1.

(d)
Design K to minimize
J = .e(t).22 + .u(t).2



2
when r(t)= (t). (For (d), the sketch of the design is su.cient. Finding the exact K is probably too hard without a computer.)
r
-

Figure 1: A unity feedback system.
2. Let G1(s) and G2(s) be two stable systems. Assume that the step response of G1(s) does not have overshoot and that the impulse response of G2(s) is always nonnegative. Assume for the sake of simplicity that G1(0)=1 and G2(0) = 1. Show that the step response of G1(s)G2(s) does not have overshoot.
s . 1
3. Consider the plant P (s)= .
s(s . 2)
(a)
Is it possible to design a unity feedback controller so that the closed loop step response is free of overshoot?

(b)
Is it possible to design a 2DOF controller as in Figure 2 so that the closed loop step response is free of undershoot.

(c)
Design a .rst order 2DOF controller as in Figure 2 so that the closed-loop characteristic polynomial is (s + 1)(s + 2)(s + 3), the output y(t) tracks a step in the reference r(t), and the transfer function from r(t) to y(t) has poles at .1 and .2.

(d)
With this design, does the closed-loop step response have overshoot? Give a brief justi.cation of your answer.



Figure 2: 2DOF control system. 11
4. In Figure 3, let P (s) = . We have two stabilizing controllers C1(s)=1 and C2(s) = . Compute bP,C1 and
ss +2
bP,C2 . Which controller is better in terms of the robustness of the closed-loop system?

w1

u1
-
-
P (s)
6.
y1
y2


C(s) ?
u2
w2
Figure 3: Feedback system for stabilization. s . 2
5. Consider the feedback system in Figure 3. For plant P (s)= , design C(s) to minimize
s(s . 3)
J =
. .
.
P (s)C(s) C(s) 1+ P (s)C(s) 1+ P (s)C(s)
P (s) .P (s)C(s)
1+ P (s)C(s) 1+ P (s)C(s)

. .
.
.
2
2
.