Force-Driven-Mass Diagonalization Example

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

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

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].