Why matlab returns a damp ratio as 1 for a over damped system?

Question

I have a second order system with the transfer function as: "1/(s^2 + 5.45s + 1)", I thought it should be an over-damped system, but when I use the damp function in MatLab, it returns the damp ratio zeta as 1 with 2 different natural frequency, what does that mean?

">> $display(G6)$

$G6 = \dfrac{1}{s^2 + 5.45s +1}$

Continuous-time transfer function.

$damp(G6)$

Eigenvalue Damping Frequency

-1.90e-01 1.00e+00 1.90e-01
-5.26e+00 1.00e+00 5.26e+00

(Frequencies expressed in rad/seconds)"

Answer 1

According to Matlab doumentation, the damping is calculated per pole. and it is given by the formula

$$ \zeta = -\cos(\angle s) $$ where s is the pole location. Since both poles of an over damped system are on the (negative) real axis, the damping is 1 for both poles.

For over damped systems, the poles are purely real and one cannot naturally pair the poles as in the case of complex conjugates. This may be why Matlab decides to report the damping per pole.