134 Stable rational functions ${\cal S}$

Definition (Stable rational functions)

Let $LHP=\{s\in{\bf C}:\Re s<0\}$ be the open left half plane. A rational function $f(s)$ is said to be stable if

(i) $f(s)$ is proper, and

(ii) all the poles of $f(s)$ are the open left half plane.

The space of stable rational functions is denoted ${\cal S}$ .

Equivalently, $f(s)=g(s)/h(s)$ is stable if

(i) ${\hbox{degree}}(g(s))\leq{\hbox{degree}}(h(s))$ , and

(ii) all the zeros of $h(s)$ have $\Re s<0$ , so $h(s)$ is stable.

So a polynomial $h(s)$ is stable as in 134, if and only if $1/h(s)$ is a stable rational function as in 136.)