transfer function and error gain function

simulate this circuit – Schematic created using CircuitLab

I know the difference between those two functions , but i want to see the relation between them , i think that the error gain function , is given by the transfer function , and the best transfer function which is

1. stable.
2. has an error gain function equal to 1.

if this is true , how can i derive the error gain function , from transfer function ?

• The stable output condition for an input of 1.0 is 0.5. Does that help at all? Regarding the rest of your question, I'm not entirely sure what you mean. – Andy aka Apr 10 '14 at 18:50
• @Andyaka since E(s) = Y(s) - X(s)G(S) , where X : is the desired output(input to system) , and G is the error gain function , then X(s)G(S) = X(s) . which is equal to Y(s) , then E(s) = 0. – hbak Apr 10 '14 at 19:02

Let me label your circuit nodes for better understanding of your question

simulate this circuit – Schematic created using CircuitLab

• Error gain function: a gain k that multiplies the error e = r-y
• Transfer function: the behavior of the closed-loop system y/r

Now,

$$y=k\cdot e$$ $$e=r-y$$ $$y=k\cdot (r-y)$$

So, the error gain function follows as

$$k=\frac{y}{r-y}$$

So, you don't even need the transfer function to establish the value of k. You just need y and r.

Anyway the transfer function is

$$TF=\frac{y}{r}=\frac{k}{k+1}$$

So, if you're given the transfer function (note that this is just a constant for this theoretical system) you can find k as

$$k=\frac{TF}{1-TF}$$

Note also that, with negative feedback, in stable system conditions, and with finite k:

$$0\leq TF<1$$

• the transfer function that i meant , that the function that you entered on it an input and give you an output , but the transfer function that you define , is the function that give it desired output r , and give you an output y , in another woed , you define the system above as transfer function , and i want the transfer function that described by this error gain function .... – hbak Apr 19 '14 at 9:33
• Can you be clearer? r is an input, not an output. Anyway the transfer function described by the error gain function (assuming you agree that this error gain function is just k, otherwise I don't understand your question) is what I defined as TF. – raggot Apr 28 '14 at 13:11