Proper transfer function: Difference between revisions

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In [[control theory]], a '''proper [[transfer function]]''' is a [[transfer function]] in which the [[degreeDegree (mathematics)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.

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 is '''proper'''

== 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>

because

is '''proper''', because

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

The following transfer function however, is '''not proper'''

is '''biproper''', because

:<math> |\deg(\textbf{GN}(\inftys))| <= 4 = \inftydeg(\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 however, 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 approacesapproaches 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):

== Implications ==

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

A proper transfer function will never grow unbounded as the frequency approaces infinity.

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

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

== See also ==

[[Strictly proper]]

==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]]