stability from frequency response: bode and nyquist...
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Stability from Frequency Response: Bode and Nyquist Plots
Reading: FPE 6.4
Understanding Stability Margins
Design is not just building stable systems; it involves keeping them stable.Stability margins address this concern.Let’s start with a simple example
Stability is lost when the Nyquist plot crosses -1. Stability margins show, how far away from that dangerous value the plot is.
Phase margin (PM) is the amount by which the phase of KG(s) exceeds −180◦ when |KG(s)|=1
Gain margin (GM) is the factor by which one can scale the current transfer function, before instability.
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Nyquist plot^
fur KG =
stability is lost for k , when roots of 16It kf=o cross imaginary
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axis. ⇐ Esteykfljw) - -I . •y • w=o
Remarkably, the Gain and Phase margins can be read directly off the Bode plots for the open loop systems.
Magnitude plot tells us where the Nyquist plot will be crossing the unit circle.
Checking the phase plot at the corresponding frequencies tells us the Phase Margin.
Similarly, for the magnitude plot, the drop below 0 at the frequency where the phase hits -180 degrees yields GM.
Frown is good
is.
up from 1800
- is good
The GM yields, inter alia, the value of the critical gain, at which stability is lost; something we can also deduce from Root Locus (and Routh criterion).
The frequency at which the log of magnitude is equal to 0 is referred to as crossover frequency .
As the gain K increases, the argument of the transfer function at the crossover frequency approaches -180.
GM = ¥⇒ , for w* : Arg GCjw*)= - Iso
- we
IG Cjw e) I =L ⇒ if trg (Gcjwe)) → Isoinstability !
Example: standard transfer function
Standard considerations show that the RL never crosses imaginary axis; stability for all gains.
Bode plots:
Still, one can compute crossover frequency:It is given by
Recall the formulae for the overshot and the resonance peak: can be related to PM (only for the standard system though)…
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pm gRE-282
Mp -expf"¥⇒ overshot
aloof✓ for Esto Mr =zgt-yzresofaaftfuueh.usof k
> z
Important to remember, that the stability margins are heuristics, failing (in general) for more complicating transfer functions.
Here’s an example with
Is it stable? Unstable?
Nyquist plot shows what is going on:
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① ①a up ! Good
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Bode’s Gain-Phase theorem
An elegant theorem, a direct consequence of a standard fact from complex analysis (Kramer-Kronig theorem): for a function analytic in the upper half-plane, its imaginary part along the real axis recover its real part along the real axis.
In our situation, the analytic function is
Then Bode’s theorem implies that if the transfer function has all poles in the LHP
Here
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is analytic ( co poles ! ) forthe E >o
,then
④em .
- ti FELL[ a bit caution at G --al]
log G = M + g Srg G
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din -
- slope on Bode plot = dam-widdud-daaf-i.co'
uw cul - he (coth 'E) = be her e
witw
Bode’s Gain-Phase theorem
In particular, if the slope of the magnitude is approximately constant over a long enough interval,
the phase = 90 x slope
Very useful approximation…
Want PM at 90, - i.e. slope -1
(Because the weighting function is an approximation of the delta function)
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a Iz f- Ca) .
Gain-Phase relationship and bandwidth
Recall that bandwidth frequency is defined as having
-
the
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If KG = -j ( ie . Arg KG = - go; I KEHL)( open loop relation)
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