Nyquist
– Relate closed-loop stability to the shape of the frequency-response curves of P
and C, called the Nyquist Stability Criterion
– Today’s Proof: Assumes linear differential equations govern P and C
ME190L
• Result is true for more complex linear systems (linear partial differential equations,
convolution systems, Desoer-Callier algebra)
Nyquist Stability Criterion
– Natural notion of robustness margins emerges
– Since frequency-responses can be obtained experimentally, closed-loop stability
of interconnections can be ascertained from the individual blocks’ frequency
response functions without ever explicitly finding the governing ODE (simply must
assume that system is governed by an ODE)
UC Berkeley
Fall 2010
http://jagger.me.berkeley.edu/~pack/me190L
Harry
Nyquist
Hendrik
Bode
Copyright 2007-10, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike
License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to
Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
Phase change and encirclements
Bounded closed contour, Σ , not passing through
the origin 0, traversed by s.
The origin, 0, is either
Phase change and encirclements
Σ
Outside Σ (does not encircle 0)
– Inside Σ
• Σ encircles 0 clockwise (CW)
• Σ encircles 0 counterclockwise (CCW)
– Outside Σ
What is the net change in
Inside Σ,
CW
as s traverses Σ?
Inside Σ,
CCW
Outside Σ
Phase change and encirclements
Inside Σ (Σ encircles 0 one time, CW)
Phase change and encirclements
Inside Σ (Σ encircles 0 two times, CW)
1
Phase change and encirclements
Inside Σ (Σ encircles 0 three times, CW)
Phase change and encirclements
Inside Σ (Σ encircles 0 one time, CW)
Net Phase change versus encirclements of 0
Summary: The phase change as a complex number traverses
a bounded, closed path Σ is equal to 2πN , where N is the
number of counterclockwise encirclements of 0 by Σ
Phase change and encirclements
Inside Σ (Σ encircles 0 two times, CW)
Phase change and encirclements
Inside Σ (Σ encircles 0 one time, CW)
pre-Nyquist: Argument Principal
Basic principle, preliminary to Nyquist stability criterion
– Rational function G(s)
– Simple, closed contour Γ, in the complex plane, not passing through
any poles or zeros of G
Im
Γ
s
G
Im
G(s)
Re
Re
– Map points s on Γ (complex numbers) into their value G(s)
– The closed contour G(Γ ) is defined by mapping all of Γ by G
Γ
Remark: negative clockwise encirclements are the same as
positive counterclockwise encirclements
G
G(Γ)
– Convention: Γ is traversed clockwise
• by mapping, this then defines the direction that G(Γ ) is traversed
2
pre-Nyquist: Argument Principal
Argument Principal: simplest case
Basic principle, preliminary to Nyquist stability criterion
– Rational function G(s)
– Simple, closed contour Γ, not passing through any poles or zeros of G
– The closed contour G(Γ ) is defined by mapping Γ by G, Γ is traversed
clockwise, and this then defines the direction that G(Γ ) is traversed
Γ
G
G(Γ)
– Define
Simple closed contour, Γ , traversed
clockwise by s.
Fixed complex number r, two possibilities
– Inside Γ
– Outside Γ
What is the net phase change in (s-r) as s
traverses Γ?
r inside Γ
• The number of poles of G inside Γ is denoted nP
• The number of zeros of G inside Γ is denoted nZ
Γ
r outside Γ
Result
– The closed contour G(Γ ) encircles the origin nZ-nP times, clockwise
– called “Cauchy’s Argument Principle”
Equivalently
– closed contour G(Γ ) encircles the origin nP-nZ times, counterclockwise
Argument Principal: general case
Argument Principal
Map Γ by G, giving G(Γ)
Map Γ by G, giving G(Γ)
Γ
Γ
G(Γ)
Here G is the product/quotient of many phasors. At any value
off s, the
th angle
l off G(s)
G( ) iis
G(Γ)
Since Γ does not pass through any poles or zeros of G, the
curve G(Γ) iis nonzero, and
db
bounded.
d d D
Define
fi
Theorem (argument principal):
Observation for later: If Γ passes through a zero of G, then
the curve G(Γ) passes through 0
Application to Closed-loop Stability analysis
The open-loop gain is
-
Define
Nyquist contour, “enclosing the right-half plane”
Standard ΓR contour, 3 segments,
Segment 1 corresponds to s=jω, with ω
ranging from 0 to R.
1
3
G(s):= 1 + L(s)
Γ is a contour that “encloses the entire right-half-plane”
2
R
Clearly
Apply argument principal with:
– on this segment, G(Γ) is just G(jω), with ω
ranging from 0 to R, so all frequency-response
Segment 2 corresponds to s=Rejθ with θ
ranging from π/2 to –π/2
π/2
– on this segment, G(Γ) collapses to a single
point, since R is large
Segment 3 corresponds to s=jω, with ω
ranging from -R to 0.
– on this segment, G(Γ) is just G(-jω), with ω
ranging from 0 to R. But G has real coefficients,
so G(-jω) is the complex-conjugate of G(jω), so
that G(Γ3) is the reflection (across real axis) of
G(Γ1)
3
Nyquist Stability Theorem
-
1
Nyquist Stability Theorem: Proof
1
2
R
2
R
Nyquist Stability Theorem: Assume L has
no poles on imaginary axis (we’ll deal with
that next time). Then
3
Nyquist Stability Theorem: Assume L has no poles
on imaginary axis. Then
3
Observation: If L(ΓR) passes through -1, then the closed-loop system has a
pole on ΓR
Nyquist contour, “enclosing the right-half plane”
Nyquist analysis w/ Standard ΓR contour determines the presence (or lack
thereof) of closed-loop poles inside ΓR
But the goal is to determine existence of poles
in right-half-plane, which is larger than ΓR
R3
R2
R1
2
Solution: Do the analysis for arbitrarily large R
– If there are any closed-loop poles in RHP, then a
large enough R will enclose them, and the analysis
((which detects closed-loop
pp
poles in ΓR) will detect
them.
– If there are no closed-loop poles in RHP, then for
every R>0, the analysis will conclude that there are
no closed-loop poles inside ΓR
Analysis for arbitrarily large R is easy
2
2
– Segment 2 corresponds to s=Rejθ with θ ranging
from π/2 to –π/2
– on this segment, as R takes on larger and larger
values, G(Γ) collapses to a single point
4