Proper transfer function

The following transfer function:

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

is proper, because

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

is biproper, because

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

but is not strictly proper, because

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

The following transfer function is not proper (or strictly proper)

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

because

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

A not proper transfer function can be made proper by using the method of long division.

The following transfer function is strictly proper

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

because

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

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

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

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

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

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

• Transfer functions - ECE 486: Control Systems Spring 2015, University of Illinois
• ELEC ENG 4CL4: Control System Design Notes for Lecture #9, 2004, Dr. Ian C. Bruce, McMaster University