Proper transfer function

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.

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 approaches infinity.

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

See also

• Strictly proper