Let's use state-space analysis to determine the frequency of oscillation of the following system:
Note the assignments of unit-delay outputs to state variables and .
In matrix form, the state transition can be written as
or, in vector notation,
The poles of the system are given by the eigenvalues of , which are the roots of its characteristic polynomial. That is, we solve
for , , or, for our problem,
Using the quadratic formula, the two solutions are found to be
Defining , we obtain the simple formula
It is now clear that the system is a real sinusoidal oscillator for , oscillating at normalized radian frequency .
We determined the frequency of oscillation from the eigenvalues of . To study this system further, we can diagonalize . For that we need the eigenvectors as well as the eigenvalues.