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Typical State-Space Diagonalization Procedure

As discussed in [452, p. 362] and exemplified in §C.17.6, to diagonalize a system, we must find the eigenvectors of $ A$ by solving

$\displaystyle A\underline{e}_i = \lambda_i \underline{e}_i

for $ \underline{e}_i$ , $ i=1,2$ , where $ \lambda_i$ is simply the $ i$ th pole (eigenvalue of $ A$ ). The $ N$ eigenvectors $ \underline{e}_i$ are collected into a similarity transformation matrix:

$\displaystyle E= \left[\begin{array}{cccc} \underline{e}_1 & \underline{e}_2 & \cdots & \underline{e}_N \end{array}\right]

If there are coupled repeated poles, the corresponding missing eigenvectors can be replaced by generalized eigenvectors.2.12 The $ E$ matrix is then used to diagonalize the system by means of a simple change of coordinates:

$\displaystyle \underline{x}(n) \isdef E\, \tilde{\underline{x}}(n)

The new diagonalized system is then
$\displaystyle \tilde{\underline{x}}(n+1)$ $\displaystyle =$ $\displaystyle \tilde{A}\, \tilde{\underline{x}}(n) + {\tilde B}\, \underline{u}(n)$  
$\displaystyle \underline{y}(n)$ $\displaystyle =$ $\displaystyle {\tilde C}\, \tilde{\underline{x}}(n) + {\tilde D}\,\underline{u}(n),$ (2.13)

$\displaystyle \tilde{A}$ $\displaystyle =$ $\displaystyle E^{-1}A E$  
$\displaystyle {\tilde B}$ $\displaystyle =$ $\displaystyle E^{-1}B$  
$\displaystyle {\tilde C}$ $\displaystyle =$ $\displaystyle C E$  
$\displaystyle {\tilde D}$ $\displaystyle =$ $\displaystyle D.
\protect$ (2.14)

The transformed system describes the same system as in Eq.(1.8) relative to new state-variable coordinates $ \tilde{\underline{x}}(n)$ . For example, it can be checked that the transfer-function matrix is unchanged.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University