Changing to Modal Coordinates

Now the system is in state space form, suppose we change to diagonal coordinates:

$$\begin{eqnarray*} {\bf z_{n}} &=& {\bf D}{\bf z_{n-1}} \\ y_{out} &=& {\bf C^\prime z_{n}} \end{eqnarray*}$$

where

$$\begin{eqnarray*} {\bf D} = {\left [\begin{matrix} \left [\begin{matrix} \lambda_{1} & & & & \cr & \lambda_{1}^{*} & & & \cr & & \ddots \cr & & \cr & & & \lambda_{N} & \cr & & & & \lambda_{N}^{*} \cr \end{matrix}\right ] & \cr & \left [\begin{matrix} \epsilon_{1} & & \cr & \ddots & \cr & & \epsilon_{N} \cr \end{matrix}\right]\end{matrix}\right ]} \end{eqnarray*}$$

The $N$ complex-conjugate-pair eigenvalues $\lambda_{i}$ represent the modal frequencies and dampings

$$\lambda_i = R_i e^{j\omega_i T}, \quad 0 < R_i < 1, \quad -\pi < \omega_i T < \pi$$

• The main (desired) effect of the third order partial derivative with respect to time is to make the damping factors $R_i$ be smaller for higher-frequency modes.

• A secondary effect is that the modal frequencies $\omega_i$ are slightly shifted.

• Another secondary effect of the third-order time derivative is the introduction of $N$ real rapidly decaying modes, characterized by the $\epsilon_{i}$ . These eigenvalues are typically negligible.

We can reduce the size of the system to $2N$ , and further change coordinates so that the matrix is made up of $N$ 2 by 2 real ``modal'' blocks as follows:

$$\begin{eqnarray*} {\bf D^{''}} = {\left [\begin{matrix} \left [\begin{matrix} 0 & 1 \cr -\vert\lambda_{1}\vert^{2} & 2Re(\lambda_{1})\end{matrix}\right ] & & & \cr & \left [\begin{matrix}0 & 1 \cr -\vert\lambda_{2}\vert^{2} & 2Re(\lambda_{2})\end{matrix}\right ] & & \cr & & \ddots & \cr & & & \left [\begin{matrix}0 & 1 \cr -\vert\lambda_{N}\vert^{2} & 2Re(\lambda_{N})\end{matrix}\right ] \cr \end{matrix}\right ]} \end{eqnarray*}$$