# Derivation of the transfer function form with the system type number included

A transfer function is derived from an ODE with the Laplace transform as following...

Some times though there is a different form of the transfer function, one that includes the system type number $l$.

How is that form derived? As I have listed the way we go from an ODE to a transfer function, how is the system type number came to be included in it? g

• does this help? math.oregonstate.edu/home/programs/undergrad/… – Tony Stewart Sunnyskyguy EE75 Jun 25 '17 at 0:38
• @TonyStewart.EEsince'75 This is an example of using Laplace transform, I don't see how it can be relevant here. Although thanks for trying to help :) – Adam Jun 25 '17 at 0:39
• ok then you read en.wikipedia.org/wiki/… too? – Tony Stewart Sunnyskyguy EE75 Jun 25 '17 at 1:35
• @TonyStewart.EEsince'75 Suffice to say that I have 3 Control Engineering books open in front of me right now and 10 tabs in the browser. I have searched everywhere. This is not the standard derivation of the transfer function. The standard derivation is very easy. The only thing I don't understand is the case where the system type number appears in a particular form. I don't ask regarding the Laplace transform or the math included. – Adam Jun 25 '17 at 1:47
• The second TF is the forward path of the system. This is how type number is defined. – Chu Jun 25 '17 at 7:15

The type number is the number of pure integrators in the forward path, $\small G(s)$, of a closed loop system. This is equivalent to saying the number of 'free' $s$ terms in the denominator of $\small G(s)$ . Alternatively, the number of poles of $\small G(s)$ at the origin.
Hence if the CLTF is $\frac{G(s)}{1+G(s)H(s)}$, then the type number is the number of denominator free $s$ terms in $\small G(s)$. Note, type number is not determined by the open loop TF, $\small G(s)H(s)$, unless the system is unity feedback.