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System Poles

Above, we found the transfer function to be

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

The poles of $ \mathbf{H}(z)$ are the same as those of

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

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 \left\vert zI - A\right\vert
$

where $ \left\vert Q\right\vert$ denotes the determinant of the square matrix $ Q$ (also written as $ \det(Q)$ .)


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Download StateSpace.pdf
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``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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