In
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, a proper transfer function is a
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
in which the
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
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, because
:
.
is biproper, because
:
.
but is not strictly proper, because
:
.
The following transfer function is not proper (or strictly proper)
:
because
:
.
A not proper transfer function can be made proper by using the method of long division.
The following transfer function is strictly proper
:
because
:
.
Implications
A proper transfer function will never grow unbounded as the frequency approaches infinity:
:
A strictly proper transfer function will approach zero as the frequency approaches infinity (which is true for all physical processes):
:
Also, the integral of the real part of a strictly proper transfer function is zero.
References
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
{{DEFAULTSORT:Proper Transfer Function
Control theory