Factoring Higher Order Polynomials: A Practical Method for Control System Design

[Granger]

Homework Statement

Determine the parameters of the PI controller such that two of the closed-loop poles of the transfer function Gclr(s) correspond to the poles of a second order LTI system with the following specifications: i) overshoot S% = 25%; and ii) settling time ts(5%) = 120 s.

Homework Equations

3. The Attempt at a Solution [/B]

I have the following system:

$$G(s)=\frac{2K(s+z)}{1000s^3+110s^2+(1+2K)s+2Kz}$$

Now I need to have this system meeting certain specifications (not relevant to what I'm asking now) such as a certain value of overshoot and of settling time. And I have to determine z and K for that.

Anyway for that my guess is that I must decompose my transfer function into the sum of a first order system and a second order system (without zeros I suppose) and compare the second order parcel to the generic analysis of second order system. In theory I know that's what I have to do, but I'm a bit stuck. How do I decompose my denominator?

My first attempt consisted of making z=0. There I would have a transfer functios of

$$G(s)=\frac{2Ks}{1000s^3+110s^2+(1+2K)s}$$

And by cancelling the zero at the origin (it's valid to do that right?)

$$G(s)=\frac{2K}{1000s^2+110s+(1+2K)}$$

And since the static gain of this system is $$\frac{2K}{1+2K}$$ we must have, after some algebraic manipulation:

$$ G(s)=\frac{2K}{(1+2K)} \frac{0.001}{\frac{s^2}{1+2K}+ \frac{0.11s}{1+2K} + 0.001} $$

Now the second term of this product corresponds to the standard second order system without zeros. I should now apply know formulas and obtain the desired overshoot and settling time.

Is this correct?
But now can't z be different than zero? How do I deal with that in that case?

Then I made another try where I calculate the parameters of a second order system without zeros has to be to meet those specifications. I came up with a denominator of the system $$ s^2 + 0.04999 s + 0.00383$$. and two complex conjugate poles $$ s = - 0.025 \pm j0.05663$$. Now how can I relate this denominator to the transfer function denominator and therefore determine my parameters?

Thanks in advance!

 

[DaveE]

This may help, a practical method for factoring polynomials. If not for this problem, then for another one in your future.
https://www.coursera.org/lecture/co...l-factoring-of-higher-order-polynomials-6vTUc