Proper transfer function: Difference between revisions

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In [[control theory]], a '''proper transfer function''' is a [[transfer function]] in which the [[Degree of a polynomial|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.

#REDIRECT [[PROPER]]

The difference between the degree of the denominator (number of poles) and degree of the numerator (number of zeros) is the ''relative degree'' of the transfer function.

==Example==

The following transfer function:

:<math> \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}}</math>

is '''proper''', because

:<math> \deg(\textbf{N}(s)) = 4 \leq \deg(\textbf{D}(s)) = 4 </math>.

is '''biproper''', because

:<math> \deg(\textbf{N}(s)) = 4 = \deg(\textbf{D}(s)) = 4 </math>.

but is '''not strictly proper''', because

:<math> \deg(\textbf{N}(s)) = 4 \nless \deg(\textbf{D}(s)) = 4 </math>.

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

:<math> \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}}</math>

because

:<math> \deg(\textbf{N}(s)) = 4 \nleq \deg(\textbf{D}(s)) = 3 </math>.

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

The following transfer function is '''strictly proper'''

:<math> \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}}</math>

because

:<math> \deg(\textbf{N}(s)) = 3 < \deg(\textbf{D}(s)) = 4 </math>.

==Implications==

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

:<math> |\textbf{G}(\pm j\infty)| < \infty </math>

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

:<math> \textbf{G}(\pm j\infty) = 0 </math>

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

==References==

* [https://web.archive.org/web/20160304220240/https://courses.engr.illinois.edu/ece486/documents/set5.pdf Transfer functions] - ECE 486: Control Systems Spring 2015, University of Illinois

* [http://www.ece.mcmaster.ca/~ibruce/courses/EE4CL4_lecture9.pdf ELEC ENG 4CL4: Control System Design Notes for Lecture #9], 2004, Dr. Ian C. Bruce, McMaster University

{{DEFAULTSORT:Proper Transfer Function}}

[[Category:Control theory]]