Control with Big Polynomials

Bucek (textbook author) explained how to find system frequency and damping when the characteristic equation is quadratic. And when it's 3rd order,
as a second-order system with integral control, he factors out a first-order system, assuming the remainder is zero. Then he wants that pole on the real axis, and that becomes a design constraint.
But what should I do when my characteristic equation is fourth or fifth order? Just start factoring out first-order terms with no remainder, and put them all on the real axis?
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Gregory L. Hansen wrote:

I'm not sure what you mean by "wants that pole on the real axis" -- if you have a polynomial with real coefficients all of the roots will either be real or will come in complex pairs, so if you have an odd number of roots one of them must be on the real axis.
For _any_ system you want to take the roots of the characteristic polynomial and see where they land.
I _think_ you then want to know how to predict performance, or get an idea of how well the system will behave? If so, you want to look for two things: 1, a "dominant" pole or pole pair, and 2, that the poles all have adequate damping ratios.
In general if you have a pole pair with a low damping ratio then you just don't want to implement that system. There are exceptions but at your level just stay away.
The dominant pole will often be the pole closest to the imaginary axis (for a continuous time system) or the unit circle (for sampled time). If the system has a zero that lies close to the "slowest" pole, however, then that pole's contribution will be masked and it will be the closest "unmasked" pole.
When things get complex like this there often isn't a pole or pole pair that you can point to and say "the system acts like that". Once things get past 2nd order it's often best to just plot the response in the time and frequency domains and see what you have.
If you are interested in designing a compensator for a plant with a complex response your best bet is often to put the plant response on a Bode plot and design a compensator in the frequency domain. Alternately you can use the new robust control theory that's lying around to design your compensator.
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Well, what he really says is
"If the damping ratio of the second-order system is between zero and one, then the optimal location of the first-order pole is equivalent to the real part of the pole locations for the second-order system. This will guarantee the quickest response possible with this type of control scheme."
Can't say I'm sure what he's talking about. But this is from the system characteristic equation, which includes control that will be designed such that the above condition is true.

I'm thinking in terms of critical damping, rather than fastest possible correction. I want to zero out the error signal, and want to reduce rapid changes in the control voltage.

I think I understand poles pretty well. The zeroes still mystify me. Except that I know a zero can cancel a pole, and I can recite reasons not to try to design a system that way.

That seems like the brute-force way to do it. I was hoping there was some magic trick I could use, like setting the multiplier of some term to something to put a constraint on my parameters.

For some real fun, how would you control a bicycle or a unicycle? Because to turn right on a bicycle, you first have to turn left, to get the lean. To speed up on a unicycle, you first have to slow down, so you can lean into it and you don't fall on your ass when you accelerate.
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This is actually fairly simple to do with pole placement techniques. You should be covering this when you get to state space methods. You can put the poles where you want, or, if you don't know where they belong, optimal control theory will do that for you.
dave y.
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