Proper - Definition and Overview

Proper in control theory

Proper denotes a transfer function where the degree of the numerator does not exceed the degree of the denominator.

Example

The following transfer function is proper

<math> \textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}<math>

because

<math> deg(\textbf{N}(s)) = 4 \leq deg(\textbf{D}(s)) = 4 <math>.

The following transfer function however, is not proper

<math> \textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}<math>

because

<math> deg(\textbf{N}(s)) = 4 \nleq deg(\textbf{D}(s)) = 3 <math>.

Implications

A proper transfer function will never grow unbounded as the frequency approaces infinity.

<math> |\textbf{G}(\infty)| < \infty <math>

Related

Strictly proper.

Other possible meanings

PROPER relating to releases in the warez scene.