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Proper transfer function

Example[edit]

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}}}}

isproper, because

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

isbiproper, 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} .

Anot 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} .

Implications[edit]

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.

References[edit]

• 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