Consider a feedback-based causal LTI system with input x(t) shown in Figure 6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is to use the error e(t) to drive these, so as to make y(t) track x(t). Y(s) X(s) E(s) (a) Find the transfer functions H(s) = X (8) and He(s) = X(3) (b) Now set G(s) = 10. This is termed “proportional feedback,” where the feedback is propor- tional to the error. (i) Find y(t) and e(t) for x(t) = sin 10t. = (ii) How do your answers change (provide a qualitative discussion) for x(t) = sint and x(t) = sin 100t? (iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values as t∞. (iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as t→ ∞. Hint: You can simply use the final value theorem in (iii). 2 (c) Redo (b)(iv) for G(s) = 10+ ("proportional plus integral" feedback). S
Consider a feedback-based causal LTI system with input x(t) shown in Figure 6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is to use the error e(t) to drive these, so as to make y(t) track x(t). Y(s) X(s) E(s) (a) Find the transfer functions H(s) = X (8) and He(s) = X(3) (b) Now set G(s) = 10. This is termed “proportional feedback,” where the feedback is propor- tional to the error. (i) Find y(t) and e(t) for x(t) = sin 10t. = (ii) How do your answers change (provide a qualitative discussion) for x(t) = sint and x(t) = sin 100t? (iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values as t∞. (iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as t→ ∞. Hint: You can simply use the final value theorem in (iii). 2 (c) Redo (b)(iv) for G(s) = 10+ ("proportional plus integral" feedback). S
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i only want part b and c please. i want a step by step solution and explanation
Transcribed Image Text:Consider a feedback-based causal LTI system with input x(t) shown in Figure
6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is
to use the error e(t) to drive these, so as to make y(t) track x(t).
(a) Find the transfer functions H(s) =
Y(s)
X(s)
and He(s) =
=
E(s)
X(s)*
(b) Now set G(s) = 10. This is termed “proportional feedback," where the feedback is propor-
tional to the error.
(i) Find y(t) and e(t) for x(t)
sin 10t.
(ii) How do your answers change (provide a qualitative discussion) for x(t)
sin 100t?
=
sint and x(t)
(iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values
as t∞.
(iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as
t → ∞.
Hint: You can simply use the final value theorem in (iii).
(c) Redo (b)(iv) for G(s) = 10+ 2/3 ("proportional plus integral" feedback).
Transcribed Image Text:x(t) —
e(t)
G(s)
y(t)
1/s
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Step 1: Summarize the given information.
VIEWStep 2: Determine transfer functions.
VIEWStep 3: Determine outputs for x(t) = sin(10t).
VIEWStep 4: Explain system outputs for different sinusoidal signals.
VIEWStep 5: Determine output for step input, x(t) = u(t).
VIEWStep 6: Find asymptotic error for ramp input, x(t) = tu(t).
VIEWStep 7: Find asymptotic error for proportional plus integral feedback.
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