What an interesting system you must be working on!
It'll have the form (At^(n-1)+Bt^(n-2)+etc)(e^etc + cos(etc + etc*t)). Your
handbook won't cover this because multiple same-frequency resonances in a
control system are quite rare, so you need to find the answer with one of
the same ways that you can find the transform of a single 2nd-order pole
- Factor the pole into it's complex conjugate pair, to give you (s + c) /
((s + q + jp)^n (s + q - jp)^n)
- Find the partial fraction expansion of the expression. Your math package
will probably choke.
- Find the inverse transform using the expression for multiple 1st-order
- Convert the various complex conjugates back to real numbers using Euler's
- Verify your arithmatic by seeing that all the imaginary numbers cancel
- you'll probably have to go back to the partial fraction expansion step
once or twice
- Once everything checks out you're done!
- Clean all the scattered paper from the floor,
- retrieve the documentation for your math package from the trash,
- and your computer if necessary,
- eat something nutritious,
- let your friends know you're alive and sane,
- have a beer.
Well, actually I'm not working on a system. I'm writing a paper ;-) and I'm
trying to study the transfer function as much general as possible.
As you can understand, I can't apply the method you suggest. I have to
invert the general transfer function I wrote, not a particular instance of
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