# How to calculate the open-loop transfer function with disturbance at the input of the plant

I have the system as follows:

I am trying to find the transfer function:

$$T(s) = \frac{Y(s)}{U(s)}$$

where $$Y(s)$$ is the output and $$U(s)$$ is the input of the plant.

The plant is $$H(s)$$.

I have got up until

$$\frac{Y(s)}{H(s)} = T_{\text{out}}(s) - U(s)$$

Moving $$U(s)$$ to the left by division and $$H(s)$$ to the right by multiplication gets me the transfer function, but there is a residual term $$\frac{T_{\text{out}}(s)}{U(s)}$$.

What can I do here?

$$Y(s) = (...)R(s) + (...)T_{out}(s)$$
I assume that R(s) is the reference input since the image doesn’t show the whole block diagram. Anyway, from this point on in order to derive the input-output transfer function you should set the disturbance term equal to zero ($$T_{out}(s)=0$$). In contrast, if you want the disturbance transfer function you should set the input term equal to zero ($$R(s)=0$$).