Unformatted text preview: ME 575 Handouts Nyquist Stability
Controller R(s) +
− Plant C G let L(s) = CG = Y(s) NC (s)NG (s) NL (s)
=
;
DC (s)DG (s) DL (s) characteristic eq: 1 + L(s) = 0 let F(s) = 1 +
ME575 Session 9 – Nyquist Stability NL (s) DL (s) + NL (s)
=
=0
DL (s)
DL (s) School of Mechanical Engineering
Purdue University Slide 1 Stability in Freq. Domain
• We want to know where the zeros of F(s) are.
• Use a MAPPING between the splane (where the
roots are) and the F(s)plane.
• Since we want stability, isolate the RHP with a
geometrically simple directed contour Γs. ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 2 1 ME 575 Handouts Effect of Zeros of F(s)
1) Zero outside contour: F(s) = s + a, a > 0
Im s Im F Re s ME575 Session 9 – Nyquist Stability ⇒ Re F School of Mechanical Engineering
Purdue University Slide 3 Effect of Zeros of F(s)
2) Zero inside contour: F(s) = s − a, a > 0
Im s Im F Re s ME575 Session 9 – Nyquist Stability ⇒ School of Mechanical Engineering
Purdue University Re F Slide 4 2 ME 575 Handouts Principle of the Argument (Cauchy)
Let F(s) be a singlevalued function that has a
finite number of poles in the splane. Choose a
closed path Γs in the splane such that it avoids
any poles or zeros of F(s).
Then the
corresponding contour ΓF mapped in the F(s)plane will encircle the origin NCW times in a
clockwise direction. ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 5 Application of P of A to Stability
Recall: R(s) + L(s) = CG
N (s)
=L
DL (s) Controller − let F(s) = 1 + L(s) = 1 + C Plant G Y(s) NL (s) DL (s) + NL (s)
=
=0
DL (s)
DL (s) zeros of F(s)
poles of F(s)
ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 6 3 ME 575 Handouts Relate to Loop Transfer Function
F(s) = 1 + L(s) ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 7 Nyquist Stability Criterion
NCW = NZ − NP
NCW = net # of CW encirclements of −1 by ΓL
NZ = # of closedloop poles encircled by ΓS
NP = # of openloop poles encircled by ΓS
A feedback system having NP openloop poles
in the RHP is stable if and only if the Nyquist
plot of L(s) encircles −1 NP times in a counterclockwise direction.
ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 8 4 ME 575 Handouts Steps in Sketching a Nyquist Plot
• Plot poles of L(s) in the splane.
• Draw the Nyquist contour Γs, indenting to the
right of any poles of L(s) on the imaginary
axis.
• Map contour Γs to L(s)plane.
• Apply encirclement condition. School of Mechanical Engineering
Purdue University ME575 Session 9 – Nyquist Stability Slide 9 Nyquist Example
L(s) = 2K
(2s + 1)(s + 1)(s / 2 + 1) Im s Im L Re s ME575 Session 9 – Nyquist Stability ⇒ School of Mechanical Engineering
Purdue University Re L Slide 10 5 ME 575 Handouts Nyquist Plot
① s = jω:
L( jω) = L( jω) = 2
(1 + j2ω)(1 + jω)(1 + jω / 2)
2 1 + (2ω)2 1 + ω2 1 + (ω / 2)2 ∠L( jω) = − tan−1 2ω − tan−1 ω − tan−1 ω / 2 ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 11 School of Mechanical Engineering
Purdue University Slide 12 Nyquist Plot
② s = Rejφ: ③ s = −jω:
ME575 Session 9 – Nyquist Stability 6 ME 575 Handouts Nyquist Stability Criterion School of Mechanical Engineering
Purdue University ME575 Session 9 – Nyquist Stability Slide 13 Nyquist Plot for Arbitrary K
Im s Im L Re s ME575 Session 9 – Nyquist Stability ⇒ School of Mechanical Engineering
Purdue University Re L Slide 14 7 ME 575 Handouts Relative Stability
Proximity to encirclement of −1 is a measure of
closeness to instability for the nominal system,
i.e., relative stability.
Relative stability is quantified via:
Gain Margin – the factor by which the openloop gain
opencan be increased at a phase of −180° before the
180°
system goes unstable.
Phase Margin – the amount by which openloop phase
opencan be decreased at unity magnitude before
system goes unstable.
Sensitivity Peak
ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 15 Margins and the Nyquist Plot
Lo ( s ) = ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University 3
(s + 1)3 Slide 16 8 ME 575 Handouts Margins and Bode Plots
Lo ( s ) = 3
(s + 1)3 Lo(jωgc) = 1 ∠Lo(jωpc) = −180° ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 17 Summary of Margins
• For stability, we want no encirclement of −1
(for minimumphase systems):
GM > 1
or GMdB > 0
PM > 0°
• As measures of relative stability, more positive
GM & PM imply farther away from instability:
GM indicates allowable extra gain
PM indicates allowable extra phase lag (time
delay)
ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University Slide 18 9 ME 575 Handouts Sensitivity Peak
Lo ( s ) = 3
(s + 1)3 1 + Lo ( jω) > η So ( jω) =
= ME575 Session 9 – Nyquist Stability School of Mechanical Engineering
Purdue University 1
1
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1 + Lo ( jω) η Slide 19 10 ...
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 Fall '10
 Meckl
 Trigraph, Purdue University, Nyquist plot, Nyquist stability criterion, Electrical parameters, School of Mechanical Engineering

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