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.

I

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)…

G - Itza; (closed loop forks is szI÷s+⇒⇒ (at * "

ie )p

vwit 4EwiaE=u! → WE -223ftFEWtar PM = -2k =WILE - 2b) RECALL :-

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:

G --9up! CBP?

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

left) - vczltjw th ft -- ntjy]If Int - octet ') & u

is analytic ( co poles ! ) forthe E >o

,then

④em .

- ti FELL[ a bit caution at G --al]

log G = M + g Srg G

>

My Gcjw ) - f-Jodha Windu wth)M - log ( Gl ; u -- btwYw) w

'- w - e

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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)

O

J W Calder = 072

* -507

-- wlu) fla -a)duxis

•

a Iz f- Ca) .

Gain-Phase relationship and bandwidth

Recall that bandwidth frequency is defined as having

-

the

Hagwait Iii -e- I '¥EEgTmHz✓

If KG = -j ( ie . Arg KG = - go; I KEHL)( open loop relation)

zthen HEI -- s --

'II : HIETT 's

try 1¥ - - 450 . ⇒ If I KG Goyal , Arg KG -too , wow-- weIn general be E w Bw 12 we -