Time domain response of this transfer function

Question

Would like to determine the time domain equation for this transfer function H(s) = a s / ( (a/b) s + 1) ) = Y(s) / U(s).

This answer on the Mathematics Stack exchange discusses the same equation, but doesn't work through finding the time domain equation. The question and answer describe that the transfer function corresponds to the derivative term of a PID controller and the denominator, (a/b) s + 1, implements a low pass filter to allow the derivative to be implemented for real systems.

Steps for computing the time domain response are given by this answer on the Mathematica Stack Exchange. This answer describes replacing s with dy/dt, s^2 with d^2/dt^2, etc. Using those steps gives the result:

y'(t) = b ( u'(t) - 1/a).

Having trouble understanding this result, because if u[t] is constant, then u'[t] is zero, and the above results gives y'[t] = -b/a, which suggest that y[t] is decreasing for all time. I expected y[t] to be zero. Is my expectation wrong? Or, perhaps, my calculation of the time domain expression is wrong.

To summarize my questions:

1. what is the time domain equivalent for the transfer function: H(s) = a s / ( (a/b) s + 1)
2. method(s) for computing this
3. for constant input, u(t) = constant, will y(t) be zero?

Answer 1

The following, taken from here, might be a satisfactory answer:

H(s) = Y(s) / U(s) = a s / ( (a/b) s + 1)

Y(s) ( (a/b) s + 1) = U(s) a s

changing to time domain and replacing Y(s) s by y'(t) and U(s) s by u'(t) gives:

(a/b) y'(t) + y(t) = a u'(t)

re-arranging to solve for y(t) give

y(t) = a (u'(t) - (1/b) y'(t) )

so the output of the transfer function in the time domain is a measure of the difference between the derivative of the input and the derivative of the output.

Answer 2

The transfer function is $$Y(s) = \frac{a s}{cs+1}U(s),$$ where $c=\frac{a}{b}$, and we want to find a way to represent it in the time domain without involving the time derivative $\dot{u}(t)$, especially since $u(t)$ is not necessary differentiable.

Well, the first option is to rewrite the system as a state-space model, then the answer becomes evident. Another option is to rewrite is as $$Y(s) = \frac{a}{c}\frac{cs+1-1}{cs+1}U(s) = \frac{a}{c}U(s) - \frac{a}{c} \frac{1}{cs+1}U(s),$$ and $y(t) = \frac{a}{c}u(t) - \frac{a}{c}z(t)$, where $$Z(s) = \frac{1}{cs+1}U(s).$$ The time-domain model for $z$ is straightforward.