To disturb or not to disturb?

So here is a recurrent question. You have some model that has a constant showing up on the differential equation and you’re not sure how to handle it. For example, your model looks like:


where $c$ is a constant. There are multiple ways you can handle this situation.

One way is to treat it as a disturbance signal. What do I mean by that is to compare the above equation with the one


in which $w(t)$ is a disturbance signal. Of course when $w(t)=c/b$, $t \geq 0$, then you recover your original equation. The advantage is that you can use the standard block-diagrams in the book, e.g. Fig 4.2, and their associated transfer-functions. You can also use linearity to calculate the combined response to $u$ and $w$, or $\bar{y}$ in case of a closed-loop. That’s one way among many others. Here’s another one.

How about a change-of-variables? If you let $z(t)=y(t)+c/a$ then 


and voila, the constant is gone! One limitation in this case is that it only works if $c$ is a constant, whereas the previous idea can be extended to handle any disturbance signal.

Variations of the above idea can be used to work with initial conditions in pretty much the same way.

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