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# Proper

 Contents

## 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

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

because

$deg(\textbf{N}(s)) = 4 \leq deg(\textbf{D}(s)) = 4$.

The following transfer function however, is not proper

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

because

$deg(\textbf{N}(s)) = 4 \nleq deg(\textbf{D}(s)) = 3$.

### Implications

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

$|\textbf{G}(\infty)| < \infty$

### Related

03-10-2013 05:06:04