(a) Explain how the Nyquist stability criterion can be used to determine the stability of a closed-loop feedback system. Formulate the criterion for a system with no right-half plane open-loop poles. (b) Consider the Nyquist plot of the frequency response of the system G(jw) = 500 shown in Figure Q4 below. (jw+1)(jw+10)2 i) Explain how the gain margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. What is the maximum value of an additional gain K for which the closed loop would still be stable?

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CALCULATE Q 4 PART A B i, ii

Q4
(a)
Explain how the Nyquist stability criterion can be used to determine the stability
of a closed-loop feedback system. Formulate the criterion for a system with no
right-half plane open-loop poles.
(b)
Consider the Nyquist plot of the frequency response of the system G(jw) =
500
shown in Figure Q4 below.
(jw+1)(jw+10)2
i)
Explain how the gain margin can be obtained from the Nyquist plot and
determine its approximate value from Figure Q4. What is the maximum
value of an additional gain K for which the closed loop would still be
stable?
0.5
-0.5
-1
-1
-0.5
0.5
1
Real Axis
Figure Q4
ii)
Explain how the phase margin can be obtained from the Nyquist plot
and determine its approximate value from Figure Q4. Assuming that the
corresponding point on the plot occurs at 4 rad/s, what is the maximum
delay which can be added to the system before the closed loop would
become unstable?
Sketch the approximate Bode diagram for the system in part (b) of the question,
clearly showing the asymptotes and corner frequencies. Indicate the gain and
phase margins in your diagram.
(d)
Explain how the bode plot changes for the system in part (b) when an additional
gain of K = 10 is added. Sketch the Bode diagram of the modified system and
comment on the stability of the closed loop in this case.
Imaginary Axis
Transcribed Image Text:Q4 (a) Explain how the Nyquist stability criterion can be used to determine the stability of a closed-loop feedback system. Formulate the criterion for a system with no right-half plane open-loop poles. (b) Consider the Nyquist plot of the frequency response of the system G(jw) = 500 shown in Figure Q4 below. (jw+1)(jw+10)2 i) Explain how the gain margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. What is the maximum value of an additional gain K for which the closed loop would still be stable? 0.5 -0.5 -1 -1 -0.5 0.5 1 Real Axis Figure Q4 ii) Explain how the phase margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. Assuming that the corresponding point on the plot occurs at 4 rad/s, what is the maximum delay which can be added to the system before the closed loop would become unstable? Sketch the approximate Bode diagram for the system in part (b) of the question, clearly showing the asymptotes and corner frequencies. Indicate the gain and phase margins in your diagram. (d) Explain how the bode plot changes for the system in part (b) when an additional gain of K = 10 is added. Sketch the Bode diagram of the modified system and comment on the stability of the closed loop in this case. Imaginary Axis
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