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Tim Wescott
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 Posted:
Thu Dec 09, 2004 3:07 am Post subject:
Name of "Ordinary" Root-Locus Plot |
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Does anybody here know if there's a name for the "ordinary" root locus
plot that one gets dragged through in control engineering school? I
mean the one where you have the sum of a polynomial plus a polynomial
times a constant, i.e. it's linear in the parameter being varied.
Just "root locus" plot won't do, because you can have a polynomial that
isn't linear in the parameter and make a perfectly nice little plot, you
just can't flog the subject endlessly until your students can draw plots
by hand in their sleep for systems with 10 poles and 5 zeros or whatever
other perverse variations your sadistic mind can dream up.
And yes, almost nobody uses them in control design. I'm presenting them
because it's a handy way to understand certain compensation schemes and
because its a very powerful way of visualizing what happens to your
system stability when you vary some parameter.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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Jerry Avins
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| Posted:
Thu Dec 09, 2004 4:14 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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Tim Wescott wrote:
| Quote: | Does anybody here know if there's a name for the "ordinary" root locus
plot that one gets dragged through in control engineering school? I
mean the one where you have the sum of a polynomial plus a polynomial
times a constant, i.e. it's linear in the parameter being varied.
Just "root locus" plot won't do, because you can have a polynomial that
isn't linear in the parameter and make a perfectly nice little plot, you
just can't flog the subject endlessly until your students can draw plots
by hand in their sleep for systems with 10 poles and 5 zeros or whatever
other perverse variations your sadistic mind can dream up.
And yes, almost nobody uses them in control design. I'm presenting them
because it's a handy way to understand certain compensation schemes and
because its a very powerful way of visualizing what happens to your
system stability when you vary some parameter.
|
How about "Ordinary root-locus plot"?
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Tim Wescott
Guest
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| Posted:
Fri Dec 10, 2004 9:20 pm Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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Tim Wescott wrote:
| Quote: | Does anybody here know if there's a name for the "ordinary" root locus
plot that one gets dragged through in control engineering school? I
mean the one where you have the sum of a polynomial plus a polynomial
times a constant, i.e. it's linear in the parameter being varied.
Just "root locus" plot won't do, because you can have a polynomial that
isn't linear in the parameter and make a perfectly nice little plot, you
just can't flog the subject endlessly until your students can draw plots
by hand in their sleep for systems with 10 poles and 5 zeros or whatever
other perverse variations your sadistic mind can dream up.
And yes, almost nobody uses them in control design. I'm presenting them
because it's a handy way to understand certain compensation schemes and
because its a very powerful way of visualizing what happens to your
system stability when you vary some parameter.
Found it -- somebody pointed out by email that the whole story is in the |
December IEEE Control Systems magazine, which was filed in geological
order on my kitchen table.
So now I know it was done by W. D. Evans, and my kitchen table is much
cleaner!
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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David Kirkland
Guest
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| Posted:
Fri Dec 10, 2004 11:49 pm Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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Tim Wescott wrote:
| Quote: | Tim Wescott wrote:
Does anybody here know if there's a name for the "ordinary" root locus
plot that one gets dragged through in control engineering school? I
mean the one where you have the sum of a polynomial plus a polynomial
times a constant, i.e. it's linear in the parameter being varied.
Just "root locus" plot won't do, because you can have a polynomial
that isn't linear in the parameter and make a perfectly nice little
plot, you just can't flog the subject endlessly until your students
can draw plots by hand in their sleep for systems with 10 poles and 5
zeros or whatever other perverse variations your sadistic mind can
dream up.
And yes, almost nobody uses them in control design. I'm presenting
them because it's a handy way to understand certain compensation
schemes and because its a very powerful way of visualizing what
happens to your system stability when you vary some parameter.
Found it -- somebody pointed out by email that the whole story is in the
December IEEE Control Systems magazine, which was filed in geological
order on my kitchen table.
So now I know it was done by W. D. Evans, and my kitchen table is much
cleaner!
Tim, would you care to enlighten the rest of us! What is an "Ordinary" |
Root Locus (not to be confused with Locust's) plot.
Thanks.
Cheers,
David |
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Tim Wescott
Guest
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| Posted:
Sat Dec 11, 2004 6:59 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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David Kirkland wrote:
| Quote: | Tim Wescott wrote:
Tim Wescott wrote:
Does anybody here know if there's a name for the "ordinary" root
locus plot that one gets dragged through in control engineering
school? I mean the one where you have the sum of a polynomial plus a
polynomial times a constant, i.e. it's linear in the parameter being
varied.
Just "root locus" plot won't do, because you can have a polynomial
that isn't linear in the parameter and make a perfectly nice little
plot, you just can't flog the subject endlessly until your students
can draw plots by hand in their sleep for systems with 10 poles and 5
zeros or whatever other perverse variations your sadistic mind can
dream up.
And yes, almost nobody uses them in control design. I'm presenting
them because it's a handy way to understand certain compensation
schemes and because its a very powerful way of visualizing what
happens to your system stability when you vary some parameter.
Found it -- somebody pointed out by email that the whole story is in
the December IEEE Control Systems magazine, which was filed in
geological order on my kitchen table.
So now I know it was done by W. D. Evans, and my kitchen table is much
cleaner!
Tim, would you care to enlighten the rest of us! What is an "Ordinary"
Root Locus (not to be confused with Locust's) plot.
Thanks.
Cheers,
David
|
The Evans root-locus plot is a plot, on the complex plane, of all the
possible roots of the system characteristic polynomial (AKA the transfer
function denominator) as a single gain in the system is varied. So if
you have a system
.---------.
| |
| |
------>O-------->| k G(z) |---------------->
^ | | |
| | | |
| '---------' |
| |
'----------------------------'
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de
where G(z) is a ratio of polynomials and k is a gain you'll know all the
possible root locations, and you'll have an idea of what you can do to
tune the system. You can, if you're determined, even find the gain
corresponding to a particular root location. Any one who's survived a
3rd-year controls class since about 1955 will know what I'm talking about.
I called it "ordinary" because I've also found it handy at times to plot
root loci for a parameter that doesn't vary linearly -- e.g. instead of
the system polynomial having coefficients that are of the form a_n + k
b_n one may be a_n + k^2 b_n and another may be a_n + k b_n, etc.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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Randy Yates
Guest
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| Posted:
Sat Dec 11, 2004 7:39 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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David Kirkland <spam@netscape.net> writes:
| Quote: | [...]
Tim, would you care to enlighten the rest of us! What is an
"Ordinary" Root Locus
|
I believe it's a type of tree infestation, affecting mainly North American
pines and hardwoods, but any where there is damp ground available it can
strike.
--
% Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and
%%% 919-577-9882 % Verdi's always creepin' from her room."
%%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO
http://home.earthlink.net/~yatescr |
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David Corliss
Guest
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| Posted:
Sun Dec 12, 2004 12:11 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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I would point out that although the poles and zeros of G(Z) are fixed, a
result of the physical properties of the plant, the overall locus, or
path, can be modified by the addition of pole-zero pairs, called filters:
to enhance response time ... make things faster
to change the damping ratio ...
to bring the system into stability ...
The governing law here is analogous to that of static charges on a
plane. Zeros attract poles ... poles repel poles ... zeros repel zeros.
The physical construction of the modifying filters is that they come in
pairs. You can't just arbitrarily place a single pole or zero on the
root-locus diagram. However, you can usually choose the location of the
pole or zero to suit your need. Is your inverted broomstick always
falling over due to a couple of poles right at the origin ... that is,
too close to the unstable right hand plane? No problem ... just add
couple of lead-lag pole-zero filters to the left of the origin ... so
that the unstable locus will be attracted into the region of stability
.... put the new zeros near the origin where they will be effective, and
the new poles way off to the left, where they will be out of the way.
Could you do any of this without a root-locus diagram? ... It would be
quite a trick. |
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Tim Wescott
Guest
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| Posted:
Sun Dec 12, 2004 12:59 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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David Corliss wrote:
| Quote: | I would point out that although the poles and zeros of G(Z) are fixed, a
result of the physical properties of the plant, the overall locus, or
path, can be modified by the addition of pole-zero pairs, called filters:
to enhance response time ... make things faster
to change the damping ratio ...
to bring the system into stability ...
|
Actually in the full presentation G(z) turns out to be the plant (as
reflected into the sampled-time domain) plus the controller. Or it may
be some other construct entirely, if you're investigating the impact of
moving a pole or a zero around, for instance.
| Quote: |
The governing law here is analogous to that of static charges on a
plane. Zeros attract poles ... poles repel poles ... zeros repel zeros.
The physical construction of the modifying filters is that they come in
pairs. You can't just arbitrarily place a single pole or zero on the
root-locus diagram. However, you can usually choose the location of the
pole or zero to suit your need. Is your inverted broomstick always
falling over due to a couple of poles right at the origin ... that is,
too close to the unstable right hand plane? No problem ... just add
couple of lead-lag pole-zero filters to the left of the origin ... so
that the unstable locus will be attracted into the region of stability
... put the new zeros near the origin where they will be effective, and
the new poles way off to the left, where they will be out of the way.
|
While you can't arbitrarily place just a zero you _can_ arbitrarily
place just a pole, as in a low-pass filter -- although in the
sampled-time domain you're often better with a zero at z=0 for a little
less lag.
| Quote: |
Could you do any of this without a root-locus diagram? ... It would
be quite a trick.
|
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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David Corliss
Guest
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| Posted:
Sun Dec 12, 2004 2:35 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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"While you can't arbitrarily place just a zero you _can_ arbitrarily
place just a pole, as in a low-pass filter -- although in the
sampled-time domain you're often better with a zero at z=0 for a little
less lag."
....
A single pole, say, as a result of a low pass filter, is usually
something that you have to 'contend with', as opposed to a filter pair
construct which imparts a beneficial effect.
By adding a single pole, you are creating an additional locus branch.
There may be instances where it would be useful. However, it would seem
to be more of a problem than a solution.
The decided advantage of the pole-zero pair filter is that the modifying
filter locus branch includes both a starting point (the pole) and an
ending point (the zero).
.... Say for example you have a single pole as a result of your low pass
sampling filter. Does this usually, or ever, cause a problem? How do you
deal with it? |
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Tim Wescott
Guest
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| Posted:
Sun Dec 12, 2004 3:20 am Post subject:
Re: Name of "Ordinary" Root-Locus Plot |
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David Corliss wrote:
| Quote: | "While you can't arbitrarily place just a zero you _can_ arbitrarily
place just a pole, as in a low-pass filter -- although in the
sampled-time domain you're often better with a zero at z=0 for a little
less lag."
...
A single pole, say, as a result of a low pass filter, is usually
something that you have to 'contend with', as opposed to a filter pair
construct which imparts a beneficial effect.
By adding a single pole, you are creating an additional locus branch.
There may be instances where it would be useful. However, it would seem
to be more of a problem than a solution.
The decided advantage of the pole-zero pair filter is that the modifying
filter locus branch includes both a starting point (the pole) and an
ending point (the zero).
... Say for example you have a single pole as a result of your low pass
sampling filter. Does this usually, or ever, cause a problem? How do you
deal with it?
|
It often makes sense to use low-pass filtering when you have noisy
feedback and a relatively high sampling rate. But there is a distinct
tradeoff on the loop bandwidth -- the extra pole is there and must be
contended with, no question.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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