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Force-Driven-Mass Diagonalization Example

To diagonalize our force-driven mass example, we may begin with its state-space model Eq.$ \,$ (1.9):

$\displaystyle \left[\begin{array}{c} x_{n+1} \\ [2pt] v_{n+1} \end{array}\right]
\eqsp \left[\begin{array}{cc} 1 & T \\ [2pt] 0 & 1 \end{array}\right]\left[\begin{array}{c} x_n \\ [2pt] v_n \end{array}\right]
+ \left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right] f_n, \quad n=0,1,2,\ldots
$

which is in the general state-space form $ \underline{x}(n+1) = A\, \underline{x}(n) + B\,
\underline{u}(n)$ as needed (Eq.$ \,$ (1.8)). We can see that $ A$ is already a Jordan block of order 2 [452, p. 368]. (We can change the $ T$ to 1 by scaling the physical units of $ x_2(n)$ .) Thus, the system is already as diagonal as it's going to get. We have a repeated pole at $ z=1$ , and they are effectively in series (instead of parallel), thus giving a ``defective'' $ A$ matrix [452, p. 136].


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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