Can I always put a system in modal form?

Question

Given the transfer function of a system, can I always put the system in modal form? Are there exceptions?

Answer 1

You can always do it if the transfer function has non-repeated roots. For example, if you can put your transfer function into this form:

$$G(s) = \frac{Y(s)}{U(s)} = \frac{c_1}{s-p_1} + \dots + \frac{c_n}{s-p_n}$$

then, it is equivalent to:

$$\begin{align} \begin{bmatrix}\dot{x}_1 \\ \dot{x}_2 \\ \vdots \\ \dot{x}_n\end{bmatrix} &= \begin{bmatrix}p_1 & 0 & \dots & 0 \\ 0 & p_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & p_n\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} + \begin{bmatrix}1 \\ 1 \\ \vdots \\ 1\end{bmatrix}u \\ y &= \begin{bmatrix}c_1 & c_2 & \dots & c_n\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} \end{align}$$