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

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

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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

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  • \$\begingroup\$ does this help? math.oregonstate.edu/home/programs/undergrad/… \$\endgroup\$ – Sunnyskyguy EE75 Jun 25 '17 at 0:38
  • \$\begingroup\$ @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 :) \$\endgroup\$ – Adam Jun 25 '17 at 0:39
  • \$\begingroup\$ ok then you read en.wikipedia.org/wiki/… too? \$\endgroup\$ – Sunnyskyguy EE75 Jun 25 '17 at 1:35
  • \$\begingroup\$ @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. \$\endgroup\$ – Adam Jun 25 '17 at 1:47
  • \$\begingroup\$ The second TF is the forward path of the system. This is how type number is defined. \$\endgroup\$ – Chu Jun 25 '17 at 7:15
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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.

It can easily be shown that the type number defines the steady state response of the closed loop system to various deterministic input signals. Thus a type 1 system will have zero SS error to a step input, a type 2 will have zero SS error to a ramp input; a type 3 to a parabola etc.

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.

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