Standard Form for second order systems.

[zoom1]

Suppose there's a system with given uncertain parameters. And I would like to obtain certain Rise time, max. over shoot, settling time by adjusting those parameters.

Let's say this is the second order system;

T(s) = (ks + c) / (s² + as + b)

First of all; for a second order system there is a standard form which is;

Wn² / s² + 2ζωns + ωn²

As we have to have the Wn² in the numerator, it's not that way always. Just like in the example. So, what am I suppose to do at that point ?

If the transfer function T(s) looks like exactly the standard form, I could get the desired values by changing parameters. I think.

 

[NascentOxygen]

First of all; for a second order system there is a standard form which is;

Wn² / s² + 2ζωns + ωn²

Parentheses, please!

ωn² / ( s² + 2ζωns + ωn² )

The constant in the numerator doesn't affect ζ, nor ωn, nor parameters such as % overshoot, rate of gain fall-off, etc., since these are calculated as ratios. The numerator is just a scaling factor for the plots.

The general expression is: A.ωn² / ( s² + 2ζωns + ωn² )

where A can be seen to be the low frequency gain of this system.