Name of "Ordinary" Root-Locus Plot

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

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

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.

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.

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.

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.

"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.