Proper transfer function

In control theory, a proper transfer function is a transfer function in which the 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.

If the degree of the numerator equals the degree of the denominator, the transfer function is biproper.

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

Example

The following transfer function:

$\textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}$

is proper, because

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

is biproper, because

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

but is not strictly proper, because

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

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

$\textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}$

because

$\deg(\textbf{N}(s)) = 4 \nleq \deg(\textbf{D}(s)) = 3$.

The following transfer function is strictly proper

$\textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}$

because

$\deg(\textbf{N}(s)) = 3 < \deg(\textbf{D}(s)) = 4$.

Implications

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

$|\textbf{G}(\pm j\infty)| < \infty$

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

$\textbf{G}(\pm j\infty) = 0$

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