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Unformatted text preview: Final Exam, EE3530, Fall 2004 Problem 1 (10 pts) Let G be a linear time-invariant stable system with frequency response shown by Figure 1. Let the input be
r(t) = [10 sin(t) + 5 cos(10t)]1(t). Determine the steady state response Bode Diagrams Fmrn: Uh) Phase (deg); Magnitude (dB) To: Y(1) Frequency (rad/sec) Figure 1: Problem 1 — Frequency Response Problem 2 (15 pts) Consider a system with transfer function 1 G = ——————————.
(s) s3 + ms2 + azs + a3 Show that the systemis stable if and only if a1 > O, alaz > (13 > 0. Problem 3 (25 pts) Consider a feedback system shown in Figure 2. The transfer function of
the plant is G (s) = with uncertain parameters a and ,6 such that 0 < a; S a g a2 and
0 < b1 3 ,6 5 b2 where a1, a2, b1, b2 are known constants. The transfer function of the PI
controller is C(s) = K (1 + where K > 0, T; > 0. Determine the conditions for K 1 and T1 so that the feedback system is stable for ANY value of the uncertain parameters in the bounded range. Ms) W +
t Figure 2: Problem 3 and Problem 4 Problem 4 Consider a feedback system shown in Figure 2. The transfer function of the plant is G (s) = 5
si+s+I ' (a) (10 pts) Suppose that the reference input signal is r(t) = 1(t) and that C(s) = K is a proportional controller with gain K. Can the steady state error be arbitrarily decreased by increasing the controller gain K? Justify your answer. (15 pts) Suppose that the reference input signal is r(t) = t 1(t) and that C(s) = g is
an integral controller with gain K. What is the smallest error can be achieved by increasing the controller gain K? Justify your answer. Problem 5 (25 pts) COnsider an electronic circuit system shown in Figure3. Assume that all resistors are of equal resistance R and that the capacitor constant C = %. Assume that NO. input voltage (i.e., r(t) = O) is applied to the circuit and that the switch is initially at OFF
state. Predict the consequence when you close up the switch. Justify your prediction. Figure 3: Problem 5 — Circuit System ...
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- Fall '07