Poles of a State Space Filter

In this section, we show that the *poles of a state-space model* are given
by the *eigenvalues* of the state-transition matrix
.

Beginning again with the transfer function of the general state-space model,

we may first observe that the poles of are either the same as or some subset of the poles of

(They are the same when all modes are

where denotes the

Thus, *the eigenvalues of the state transition matrix
are the
poles of the corresponding linear time-invariant system*. In
particular, note that the poles of the system do not depend on the
matrices
, although these matrices, by placing system zeros,
can cause pole-zero cancellations (unobservable or uncontrollable
modes).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University