I was reading a script about state space representation and at one point it was mentioned that the eigenvalues of the state matrix, in that specific case the complex conjugate pair $\lambda_{1/2}= -0.68 \pm 1.63j $ has a damping factor of about $.4$ .
How can you get the damping factor from a (complex) eigenvalue?
Best Answer
I assume you are talking about continous systems.
To get the damping, draw a line from the eigenvalue to the origin. The slope of that line is the (absolute value of the) damping factor.
Or, as formula: given the eigenvalues $\lambda_i = a_i + j b_i$, the damping factors are
$$ D_i = \frac{-a_i}{\sqrt{a_i^2 + b_i^2}} \tag{1} $$
In your case: $D_1 = \frac{-(-0.68)}{\sqrt{(-0.68)^2 + 1.63^2}} = D_2 = \frac{-(-0.68)}{\sqrt{(-0.68)^2 + (-1.63)^2}} = \frac{0.68}{\sqrt{0.68^2 + 1.63^2}} = 0.385 \approx 0.4$.
If you have a discrete system with sample time $T$, you can first convert the discrete eigenvalues to a continous version from which you can get the damping factors.
The continous counterpart of the discrete eigenvalues $\tilde{\lambda}_i = \tilde{a}_i + j \tilde{b}_i$, are
$$ \lambda_i = a_i + j b_i = \frac{\log(\tilde{\lambda}_i)}{T} = \frac{\log(\tilde{a}_i + j \tilde{b}_i)}{T} \tag{2} $$
with $\log$ the natural logarithm. Then you can use the same formula $(1)$ as before, just with $(2)$.