Nyquist Plots, Examples

Reading: FPE, ch. 6

The basic setup:

Nyquist plot tells that the number of unstable (RHP) closed loop poles Z is equal to

Z=N+P,

where N is the number of (clockwise) encircling of -1/K; P is the number of unstable poles of open loop transfer function.

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Example 1

Reconstructing Nyquist plot: - starts at |G|=1/15, Arg=-180; - phase goes up to about -30 degrees, magnitude grows then starts to fall;- phase decreases to -90, magnitude goes to 0.

We know that P=2, so stability means N=-2, i.e. -1/K is surrounded by the contour twice counterclockwise. Doesn’t happen!

We see that the only information we need to recover the encircling numbers is at what magnitudes phase crosses some multiples of 180 degrees.

G - szsi-L.gs+ is C2pYihRHIN= - I

°¥NtP ?#>SISI- 4S -115+148-1) -= s3t@tk)s2- 4S -115-K '

I -4Stk 15-K

Stk> o K> - 8- 3k-47 - 4187k)- 1. ( 15- K) - -3K -4770Stk KC -47/3=-1543

.

Example 2

Reconstructing Nyquist plot: - starts at s=0, Arg=-180, |G|=1/3; - phase goes up a bit, then down again, crossing -180 at |G|=1/4; - tends to -270, |G| decreases to 0

We see that this happens for -1/3<-1/K<-1/4, i.e. 3<K<4…

We know that P=1, so stability means N=-1, i.e. -1/K is surrounded once counterclockwise by the contour.

Routh test:

G =1-(S- c) (5425-13)

-

-

12=1

°=z=n+p

-¥¥¥53+54 St (K -3)l l

l K-3 4 -K >O4-K K

'

-3 > O

L K-3

Example 3

Reconstructing Nyquist plot: - starts at |G|~.066, Arg=180; - phase goes down, to about 160, returns to 180 at |G|~.19 (magnitude growing); - goes up to about 255, then back to 180 as |G| converges to 0.

It is easy to see that P=2, so stability means N=-2. This covers the interval -.19<-1/K<-.066, so the stability happens at ~5.4<K<~15.(Easily confirmed with Routh criterion).

- 7=2⇐5¥'s

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8+85 - 4S -115+126-1)+85+42-415+15-Kl K-y

8 IFK15 - K > 0ksisgk

-yay9k > 47

- 9- c- Yrs - Yes15-K K> 475 47

g

Couple more Nyquist plots, - showing what happens when there are poles on the imaginary axis. They result in loops at infinity (obtained by circumventing those poles on the right).

+yr × Maneuvering around

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-3 ?s(52+44543546576)

Poles atit Doles att2j±2j- 2 1

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2 Seekers at n

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Reminder: Nyquist plots carries information about Phase and Gain margins:

- Phase margin is given by the arguments of the intersections of the Nyquist plot and the unit circle;

- Gain margin measures the ratios of -1 to the intersections of the plot with the negative axis…

One can consider further measures of the margins, - say, the closest to -1 point of the plot (“vector margin”)

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