Modal Analysis
Extracting a state-space model, as described under state-space extraction, produces a discrete state matrix. Its eigenvalues describe how the system behaves without simulating it: which oscillations exist, how fast each decays, and which states are involved in each one.
From discrete to continuous eigenvalues
The extracted model is discrete, so its eigenvalues $z$ live in the complex plane where stability means $|z| < 1$. That is awkward to read, because the quantities of interest are a frequency in hertz and a damping ratio, both of which are natural in the continuous plane.
Because the model was discretised with the trapezoidal rule, the mapping back is its inverse, the bilinear transform
This maps the interior of the unit disc onto the left half plane exactly, so a mode that is stable in one description is stable in the other, with no threshold effects at the boundary. From $\lambda = \sigma + j\omega$ the damped frequency is $\omega / 2\pi$ and the damping ratio is $-\sigma / |\lambda|$.
The mapping is exact for the discretisation used, not an approximation of it. What it cannot undo is the frequency warping the trapezoidal rule introduced in the first place: a continuous mode at a frequency approaching the Nyquist rate is represented at a shifted frequency in the discrete model, and mapping back returns the shifted value rather than the original. Modes well below Nyquist are unaffected; modes near it should not be read literally.
What the eigenvectors say
The eigenvalues say which modes exist but not which parts of the system take part in them. The right eigenvectors describe how each mode appears in the states, the left eigenvectors describe how strongly each state excites each mode, and the product of the two, element by element,
is the participation factor of state $k$ in mode $i$.
Participation factors are the useful output. A poorly damped oscillation is a number; knowing that two particular machine rotor states dominate it is actionable. They are also dimensionless and normalised in a way that makes them comparable across states with different units, which raw eigenvector entries are not.
The computation requires the eigenvector matrix to be invertible. It is not, when the state matrix is defective, meaning it has a repeated eigenvalue without a full set of independent eigenvectors. This is not a numerical failure but a property of the system, and it is the one case where participation factors are not defined at all.
Choice of frame
The states of the extracted model are in whatever frame each component works in, which for a network containing machines means several rotating frames turning at different speeds plus the network’s own. Eigenvalues of that system are still correct, but the modes mix frames, and a mode’s frequency is then relative to whichever frame its dominant states live in.
Transforming everything into one common frame before the analysis removes that ambiguity, at the price of choosing the frame and its initial angle. The two choices are therefore: analyse in the native frames and read each mode relative to its own states, or transform to a single frame and read every frequency against the same reference. The second is what makes modes from different machines directly comparable.
Limits
The analysis is linear and local. It describes the system as it is at the operating point where the model was extracted, and says nothing about behaviour after a large disturbance moves it elsewhere. A system can be comfortably damped at its nominal point and not at another, so a single modal analysis is evidence about one condition rather than about the system.