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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 $ A$ .

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

$\displaystyle H(z) = D + C \left(zI - A\right)^{-1}B,

we may first observe that the poles of $ H(z)$ are either the same as or some subset of the poles of

$\displaystyle H_p(z) \isdef \left(zI - A\right)^{-1}.

(They are the same when all modes are controllable and observable [37].) By Cramer's rule for matrix inversion, the denominator polynomial for $ \left(zI - A\right)^{-1}$ is given by the determinant

$\displaystyle D(z) \isdef \det(zI - A)

where $ \det(Q)$ denotes the determinant of the square matrix $ Q$ . (The determinant of $ Q$ is also often written $ \left\vert Q\right\vert$ .) In linear algebra, the polynomial $ D(z) = \left\vert zI-A\right\vert$ is called the characteristic polynomial for the matrix $ A$ . The roots of the characteristic polynomial are called the eigenvalues of $ A$ .

Thus, the eigenvalues of the state transition matrix $ A$ 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 $ B,C,D$ , although these matrices, by placing system zeros, can cause pole-zero cancellations (unobservable or uncontrollable modes).

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2015-03-04 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University