As an example of state-space analysis, we will use it to determine the frequency of oscillation of the system of Fig.G.3 [90].
Note the assignments of unit-delay outputs to state variables
and
.
From the diagram, we see that
and
In matrix form, the state time-update can be written
or, in vector notation,
We have two natural choices of output,
A basic fact from linear algebra is that the determinant of a
matrix is equal to the product of its eigenvalues. As a quick
check, we find that the determinant of
is
Since an undriven sinusoidal oscillator must not lose energy, and since every lossless state-space system has unit-modulus eigenvalues (consider the modal representation), we expect
Note that
. If we diagonalize this system to
obtain
, where
diag
,
and
is the matrix of eigenvectors of
,
then we have
where
If this system is to generate a real sampled sinusoid at radian frequency
, the eigenvalues
and
must be of the form
(in either order) where
is real, and
denotes the sampling
interval in seconds.
Thus, we can determine the frequency of oscillation
(and
verify that the system actually oscillates) by determining the
eigenvalues
of
. Note that, as a prerequisite, it will
also be necessary to find two linearly independent eigenvectors of
(columns of
).