(A,B,C,D)

57 Realization with a SISO

Proposition. The general strictly proper rational function

T(s)={{\sum_{j=0}^{n-1}\gamma_{j}s^{j}}\over{s^{n}+\sum_{j=0}^{n-1}\alpha_{j}s% ^{j}}}

is the transfer function of the SISO $(A,B,C,D)$ , where

A=\begin{bmatrix}0&1&0&\dots&0\\ 0&0&1&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&1\\ -\alpha_{0}&-\alpha_{1}&\dots&\dots&-\alpha_{n-1}\end{bmatrix},\quad B=\begin{% bmatrix}0\\ 0\\ \vdots\\ 0\\ 1\end{bmatrix},

C=\begin{bmatrix}\gamma_{0}&\gamma_{1}&\dots&\gamma_{n-1}\end{bmatrix},\quad D% =0.