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Proper transfer function
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<h2 id="example">Example</h2> <p>The following transfer function: </p> G ( s ) = N ( s ) D ( s ) = 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 {\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}}}} <p>is proper, because </p> deg ( N ( s ) ) = 4 ≤ deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=4\leq \deg({\textbf {D}}(s))=4} . <p>is biproper, because </p> deg ( N ( s ) ) = 4 = deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=4=\deg({\textbf {D}}(s))=4} . <p>but is not strictly proper, because </p> deg ( N ( s ) ) = 4 ≮ deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=4\nless \deg({\textbf {D}}(s))=4} . <p>The following transfer function is not proper (or strictly proper) </p> G ( s ) = N ( s ) D ( s ) = 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 {\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}}}} <p>because </p> deg ( N ( s ) ) = 4 ≰ deg ( D ( s ) ) = 3 {\displaystyle \deg({\textbf {N}}(s))=4\nleq \deg({\textbf {D}}(s))=3} . <p>A not proper transfer function can be made proper by using the method of long division. </p><p>The following transfer function is strictly proper </p> G ( s ) = N ( s ) D ( s ) = 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 {\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}}}} <p>because </p> deg ( N ( s ) ) = 3 < deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=3<\deg({\textbf {D}}(s))=4} . <h2 id="implications">Implications</h2> <p>A proper transfer function will never grow unbounded as the frequency approaches infinity: </p> | G ( ± j ∞ ) | < ∞ {\displaystyle |{\textbf {G}}(\pm j\infty )|<\infty } <p>A strictly proper transfer function will approach zero as the frequency approaches infinity (which is true for all physical processes): </p> G ( ± j ∞ ) = 0 {\displaystyle {\textbf {G}}(\pm j\infty )=0} <p>Also, the integral of the real part of a strictly proper transfer function is zero. </p> <ul><li><a href="https://web.archive.org/web/20160304220240/https://courses.engr.illinois.edu/ece486/documents/set5.pdf">Transfer functions</a> - ECE 486: Control Systems Spring 2015, University of Illinois</li> <li><a href="http://www.ece.mcmaster.ca/~ibruce/courses/EE4CL4_lecture9.pdf">ELEC ENG 4CL4: Control System Design Notes for Lecture #9</a>, 2004, Dr. Ian C. Bruce, McMaster University</li></ul>