Transfer Function Classification

Transfer function

A transfer function $P(s)$ or $G(z)$ is:

• Rational if $P(s)=\frac{N(s)}{D(s)}$ or $G[z]=\frac{N[z]}{D}[z]$ where $N,D$ are polynomials
• Real if in addition the coefficients of $N,D$ are all in $\mathbb{R}$
• Proper if an addition the $\text{deg}(D)\ge \text{deg}(N)$.
• Strictly proper if it is proper and $\text{deg}(D)>\text{deg}(N)$.

Example:

$$P(s)=\frac{s^3+2s+1}{s^3+s+2}$$

• Is rational, real, proper, but not strictly proper.

Example:

$$\frac{z^2+1}{z^3+z-3}$$

• Is rational, real, proper, and strictly proper.