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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...
...t[\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 [449, 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 [449, p. 136].

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