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

Question

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?

Answer 1

When dealing with block diagrams be aware of the fact that you can not only derive the general transfer function (input-outuput) but also the disturbance transfer function, the error transfer function, the noise transfer function and some others. The one which you should derive depends on what you would like to study for the particular given system. Each one describes the impact of its component on the output of the system. For example, the noise transfer function describes the impact of the noise on the system’s output.

For the system shown, you should follow the classical procedure and come up with an expression like this one:

$$ Y(s) = (...)R(s) + (...)T_{out}(s) $$

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 ($R(s)=0$).