How to determine damping ratio of overdamped systems

Hey everyone,

I'm working with a second order system with two real negative poles -2.08051 and -14.41948. I was thinking of an over damped system, but my damping ratio is equal to 1. The pzmap does also indicates two real negeative poles.

This is the script I use: G1 = tf([0 0 30],[1 16.5 30]); [Wn,Z,P] = damp(G1);

Anyone suggestions?

Thank you in advance!

Answer 1

[Birdman]

Hi Sebastian,

Firstly I read your question incorrectly and answered it incorrectly, therefore I deleted my old answer.

An second overdamped system actually means that the poles are located on real axis and the damping ratio of the second order system is greater than 1. It has nothing to do with the places of the poles on the real axis. If two poles of second order system are located on the left hand side of the real axis, this means that the damping ratio is greater than 1. If the poles are the same, then it means the damping ratio is 1. You need the following to decide the damping ratio. Consider is as zeta. Since the system fits to the ideal second order system, you can use the following code:

syms zeta Wn
eq1=Wn^2==30;
eq2=2*zeta*Wn==16.5
sol=solve([eq1 eq2]);
%the positive ratio
vpa(sol.zeta(1),4)

You will see that it is greater than 1. Hope this helps and sorry for misunderstanding of mine again.

[Sebastiaan Verbesselt]

Hi Birdman, thank you very for your answer. It is much appreciated!

Answer 2

[Francesco Paparella]

OK, you can calulate daping ratio as you suggest, but question remains, why does Matlab have the convention that zta, as returned by [Wn,zta,p] = damp(G) cannot be greater than one? i.e. for any underdamped (or critically damped or undamped) system, the damping ratio is the geometric mean of the elemnets of Matlab's zta, but this is not the case for overdamped ... why is this?