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.

A strictly proper transfer function is a transfer function where the degree of the numerator is less than the degree of the denominator.

Example

The following transfer function:

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

is proper, because

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

is biproper, because

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

but is not strictly proper, because

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

The following transfer function is not proper (or strictly 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

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

The following transfer function is strictly proper

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

because

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

Implications

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

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

A strictly proper transfer function will approach zero as the frequency approaches infinity (which is true for all physical processes):

${\textbf {G}}(\pm j\infty )=0$

Also, the integral of the real part of a strictly proper transfer function is zero.