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Homework 4 due Wednesday, December 11 at 5 PM.
An important corollary to the Principle of the Argument is
Rouche's theorem ("Walking the Dog Around the Pole Theorem")
If we have functions f,g which are analytic on a simple closed
contour C, and meromorphic inside the contour C, and if on the
contour C, then both f and f+g have the same value of:
Proof:
By the principle of the argument:
(argument of a product is the sum of the arguments)
Rouche's Theorem and Complex Root FindingMonday, November 25,
20131:54 PM
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Any winding around the origin must certainly intersect the
negative real axis ,but this can't happen so:
Rouche's theorem can be used to show that an analytic function
near an nth order zero locally maps the neighborhood of the zero in
an n-to-1 way to a neighborhood of the origin. (Again no nasty
folding). See the readings.
Principle of the argument and Rouche's theorem are very useful
in finding roots (zeros) of complex functions.
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Principle of the argument and Rouche's theorem are very useful
in finding roots (zeros) of complex functions.
Simple example: Consider the function
How many zeroes does f(z) have inside the unit disc?
To address this question of how f(z) behaves inside the unit
circle, the above theorems tell us it can be very informative to
simply look at how f(z) behaves on the unit circle. Let This
function h clearly has 16 zeros inside the unit circle, and no
poles. (
Notice that where
On the contour C,
have the same value for the difference in the number of zeros
and
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have the same value for the difference in the number of zeros
and poles inside the contour:
That is, all 16 roots of f(z) must be inside the unit
circle.
A less trivial example:
Consider the following delay-differential equation:
You can think about this as modeling a simple control system,
with response coefficient k, a lag response time of 1, and forcing
g(t).
A Laplace transform technique applied to this equation gives the
following Laplace transform for the solution:
Laplace transforms are very useful for stability analysis
because, under fairly general conditions, the stability of a system
is determined by the location of the singularities of the Laplace
transform in the complex plane.
Singularities in the RHP imply unstable system.If all the
singularities are in the LHP, then this implies stable system.If
some singularities lie on the imaginary axis, but the rest are in
the LHP, then neutral stability (persistent oscillations).
Typical forcing functions g will give rise to a Laplace
transform that has no singularities in the RHP. Then stability
properties of the response will be determined entirely by where the
transfer function (which corresponds to the dynamics) vanishes.
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transfer function (which corresponds to the dynamics)
vanishes.
That is, for stability analysis, we are interested in whether
the function f(s) = has any zeroes in the right half plane (where s
is a complex variable).
One could try to approach this by just trying to find zeroes
numerically but this runs the risk of missing zeroes by just not
sampling densely enough. Let's try to get some theoretical handle
on the stability.
Triangle inequality is usually a good place to start to get some
simple results. Assume that k is a positive real constant.
(Could reverse the role of the terms in the triangle inequality
argument to rule out f(s) vanishing near the origin, but we won't
pause to do this.)
Any instability in our system will therefore correspond to
having f(s) having some zero inside the region shaded blue above.
We'll use the
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having some zero inside the region shaded blue above. We'll use
the principle of the argument to count zeros inside this
region.
We'll find it convenient actually to apply the principle of the
argument to the contour denoted in red above. This is fine because
we know that any zeroes we find inside this contour will correspond
to instability. (The idea behind modifying the contour is that
often it's easier to deal with Rouche's theorem and/or principle of
the argument when one of the terms in the function dominates the
other over at least part of the contour, and this can often be
achieved by making the contour larger.)
We will now calculate the winding number of the image of around
the origin under the mapping .
Note that for Rouche's theorem, it is not apparent how to apply
because neither term in f(s) dominates over the whole contour.
Start by observing that, because the mapping is just a
perturbation of the identity over
What about the image of the piece This one requires more
delicate attention. Let's look at it parametrically:
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The image in the w plane takes the form of a cycloid, with
shape
depending on the size of k.
Intersection on real axis?
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Despite the loops, still not enclosing the origin becauseit does
not cross the negative real axis.
Check: The real axis is crossed whenever v(t) = 0.
So long as
the above graphical analysis indicates that all solutions
to v(t*) = t* - k sin t*=0 will satisfy u(t*) > 0. That is,
as long as
all
crossings of the real axis are on the positive real axis.
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One can show in this latter case that the negative real axis
will be crossed once
That doesn't guarantee a
nontrivial winding number. But what does guarantee the
nontriviality of the winding number is that all crossings of the
negative real axis are downward.
Consider:
Therefore we have shown, through this analysis, that the
transfer
function f(s) does not have zeroes in the RHP so long as
, that it
has a zero on the imaginary axis when
and does have zeoes in
the RHP when
Therefore the delay-differential equation is:
stable when
neutrally stable when
unstable when
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