Problem 3 (15 points) Given the Nyquist plots and loop transfer functions given
ID: 2078908 • Letter: P
Question
Problem 3 (15 points) Given the Nyquist plots and loop transfer functions given below, use the Nyquist stability criterion to determine whether the corresponding closed-loop system is BIBO stable Assume the unity negative feedback (i.e the standard feedback we studied in class). Ex plain your answers, and if the closed-loop system is not stable, state the number of unstable closed-loop poles. (8 1)( 2) (a) (5 points) L(s) (s 3)( 4) Nyquist Diagram 0.8 0.6 0.4 0.2 E 0.2 0.4 0.6 0.8 0.8 0.6 -0.4 -0.2 1.8 1.6 1.4 1.2 Real AxisExplanation / Answer
Ans)
a) For the system given open loop poles are s=-3,-4 all are on left half of s-plane so there are no unstable open loop poles Hence P=0
No of clock wise encirclements of point -1+j0 in nyquist plot is N=1
No of closed loop unstable poles is Z=N+P=1+0=1 unstable pole
As there are unstable poles as per nyquist criteria the system is not BIBO stable i.e unstable
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b)
For the system given open loop poles are s=3,4 all are on right half of s-plane so these are unstable open loop poles Hence P=2
No of clock wise encirclements of point -1+j0 in nyquist plot is N=-2 (negative because anti clock wise)
No of closed loop unstable poles is Z=N+P=-2+2=0 unstable poles
As there are no unstable poles as per nyquist criteria the system is BIBO stable
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c)
For the system given open loop poles are s=-3,-4 all are on left half of s-plane so there are no unstable open loop poles Hence P=0
No of clock wise encirclements of point -1+j0 in nyquist plot is N=2
No of closed loop unstable poles is Z=N+P=0+2=2 unstable poles
As there are unstable poles as per nyquist criteria the system in not BIBO stable i.e unstable
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