Granting the beginner's mistake of confusing
E (not really the culprit here), let's look at your integrand:
Simplify[Abs[(b*(1 - e^((-b)*s)))/b - h[s, t]]]
(* ==> Abs[1 - e^(-b s) - HeavisideTheta[s] HeavisideTheta[-s + t]] *)
it becomes clear that the integral over $s$ to $\infty$ is not convergent for $b\neq 0$. So you should go back and check your math. When you encounter such integration problems, it's a good idea to test the expression for some reasonable numerical choice of the parameters (
t in particular)!
In this case you perhaps confused multiplication by the
HeavysideTheta terms with subtraction.
Because I'm guessing that this is a math error on the user end, I'm proposing to close this question as too localized.