Proper transfer function

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In control theory, a proper transfer function is a transfer function in which the degree of the numerator does not exceed the degree of the denominator.

[edit] 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 .

[edit] Implications

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

|\textbf{G}(\pm j\infty)| < \infty

[edit] See also

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