Stability from nyquist plot
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Stability from Nyquist plot. The complete Nyquist plot: Plot G ( j ω ) for ω = 0 + to + ∞ Get complex conjugate of plot, that’s G ( j ω ) for ω = 0 – to – ∞ If G(s) has pole on j ω -axis, treat separately Mark direction of ω increasing Locate point: –1.

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Stability from Nyquist plot

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Stability from nyquist plot

Stability from Nyquist plot

The completeNyquist plot:

  • Plot G(jω) for ω = 0+ to +∞

  • Get complex conjugate of plot,that’s G(jω) for ω = 0– to –∞

  • If G(s) has pole on jω-axis, treat separately

  • Mark direction of ω increasing

  • Locate point: –1


Stability from nyquist plot

Encirclement of the -1 point

  • As you follow along the G(jω) curve for one complete cycle, you may “encircle” the –1 point

  • Going around in clock wise direction once is +1 encirclement

  • Counter clock wise direction once is –1 encirclement


Stability from nyquist plot

Nyquist Criterion Theorem

# (unstable poles of closed-loop)Z= # (unstable poles of open-loop)P+ # encirclementNor: Z = P + NTo have closed-loop stable:need Z = 0, i.e. N = –P


Stability from nyquist plot

  • That is: G(jω) needs to encircle the “–1” point counter clock wise P times.

  • If open loop is stable to begin with, G(jω) cannot encircle the “–1” point for closed-loop stability

  • In previous example:

    • No encirclement, N = 0.

    • Open-loop stable, P = 0

    • Z = P + N = 0, no unstable poles in closed-loop, stable


Stability from nyquist plot

Example:


Stability from nyquist plot

As you move around

from ω = –∞ to 0–,

to 0+, to +∞, you go

around “–1” c.c.w.

once.

# encirclement N = – 1.

# unstable pole P = 1


Stability from nyquist plot

  • i.e. # unstable poles of closed-loop = 0

  • closed-loop system is stable.

  • Check:

  • c.l. pole at s = –3, stable.


  • Stability from nyquist plot

    • Example:

    • Get G(jω) forω = 0+ to +∞

    • Use conjugate to get G(jω) forω = –∞ to 0–

    • How to go from ω = 0– to ω = 0+? At ω ≈ 0 :


    Stability from nyquist plot

    # encirclement N = _____

    # open-loop unstable poles P = _____

    Z = P + N = ________

    = # closed-loop unstable poles.

    closed-loop stability: _______


    Stability from nyquist plot

    • Example:

    • Given:

      • G(s) is stable

      • With K = 1, performed open-loop sinusoidal tests, and G(jω) is on next page

    • Q:1. Find stability margins

      • 2. Find Nyquist criterion to determine closed-loop stability


    Stability from nyquist plot

    • Solution:

      • Where does G(jω) cross the unit circle? ________Phase margin ≈ ________Where does G(jω) cross the negative real axis? ________Gain margin ≈ ________Is closed-loop system stable withK = 1? ________


    Stability from nyquist plot

    Note that the total loop T.F. is KG(s).

    If K is not = 1, Nyquist plot of KG(s) is a scaling of G(jω).

    e.g. If K = 2, scale G(jω) by a factor of 2 in all directions.

    Q: How much can K increase before GM becomes lost? ________

    How much can K decrease? ______


    Stability from nyquist plot

    Some people say the gain margin is 0 to 5 in this example

    Q: As K is increased from 1 to 5, GM is lost, what happens to PM?

    What’s the max PM as K is reduced to 0 and GM becomes ∞?


    Stability from nyquist plot

    • To use Nyquist criterion, need complete Nyquist plot.

      • Get complex conjugate

      • Connect ω = 0– to ω = 0+ through an infinite circle

      • Count # encirclement N

      • Apply: Z = P + N

    • o.l. stable,P = _______

    • Z = _______

    • c.l. stability: _______


    Stability from nyquist plot

    Correct

    Incorrect


    Stability from nyquist plot

    • Example:

    • G(s) stable, P = 0

    • G(jω) for ω > 0 as given.

    • Get G(jω) forω < 0 by conjugate

    • Connect ω = 0– to ω = 0+.But how?


    Stability from nyquist plot

    Choice a) :

    Where’s “–1” ?

    # encirclement N = _______

    Z = P + N = _______

    Make sense? _______

    Incorrect


    Stability from nyquist plot

    Choice b) :

    Where is“–1” ?

    # encir.N = _____

    Z = P + N= _______

    closed-loopstability _______

    Correct


    Stability from nyquist plot

    Note: If G(jω) is along –Re axis to ∞ as ω→0+, it means G(s) has in it.

    when s makes a half circle near ω = 0, G(s) makes a full circle near ∞.

    choice a) is impossible,but choice b) is possible.


    Stability from nyquist plot

    Incorrect


    Stability from nyquist plot

    • Example:G(s) stable,P = 0

    • Get conjugatefor ω < 0

    • Connect ω = 0–to ω = 0+.Needs to goone full circlewith radius ∞.Two choices.


    Stability from nyquist plot

    Choice a) :

    N = 0

    Z = P + N = 0

    closed-loopstable

    Incorrect!


    Stability from nyquist plot

    Choice b) :

    N = 2

    Z = P + N= 2

    Closedloop has two unstable poles

    Correct!


    Stability from nyquist plot

    Which way is correct?

    For stable & non-minimum phase systems,


    Stability from nyquist plot

    • Example: G(s) has one unstable pole

    • P = 1, no unstable zeros

    • Get conjugate

    • Connectω = 0–to ω = 0+.How?One unstablepole/zeroIf connect in c.c.w.


    Stability from nyquist plot

    # encirclement N = ?

    If “–1” is to the left of A

    i.e.A > –1

    thenN = 0

    Z = P + N = 1 + 0 = 1

    but if a gain is increased, “–1” could be inside,N = –2

    Z = P + N = –1

    c.c.w. is impossible


    Stability from nyquist plot

    If connect c.w.:

    For A > –1N = ______

    Z = P + N

    = ______

    For A < –1N = ______

    Z = ______

    No contradiction. This is the correct way.


    Stability from nyquist plot

    Example: G(s) stable, minimum phase

    P = 0

    G(jω) as given:

    get conjugate.

    Connect ω = 0–

    to ω = 0+,


    Stability from nyquist plot

    If A < –1 < 0 :N = ______Z = P + N = ______stability of c.l. : ______

    If B < –1 < A : A=-0.2, B=-4, C=-20N = ______Z = P + N = ______closed-loop stability: ______

    Gain margin: gain can be varied between (-1)/(-0.2) and (-1)/(-4),

    or can be less than (-1)/(-20)


    Stability from nyquist plot

    If C < –1 < B :N = ______Z = P + N = ______closed-loop stability: ______

    If –1 < C :N = ______Z = P + N = ______closed-loop stability: ______


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