How to tune PID parameters to track a family of reference signals?
As already mentioned, a linear system that tracks a step reference, will track it for arbitrary values and time of application. Due to the internal model principle, in a feedback control system, if the denominator of the product
$$
C(s) P(s)
$$
contains a pole at $s = 0$, then it will robustly track a step function.
The control law
$$
C(s) = \frac{k_{d} s^{2} + k_{p} s + k_{i}}{s}
$$
has a pole at $s = 0$. If the plant  $P(s)$  does not have a zero at $s = 0$, then your system will track an arbitrary step function $R(s) = \frac{A}{s}$, if it is closed-loop stable.

Since the complementary transfer-function, from reference to output, is $Y(s)/R(s) = T(s)$, and
$$
T(s) = \frac{C(s) P(s)}{1 + C(s) P(s)} = \frac{\frac{1}{s} \bar{C}(s) P(s)}{1 + \frac{1}{s} \bar{C}(s) P(s)} = \frac{\bar{C}(s) P(s)}{s + \bar{C}(s) P(s)} ,
$$
this can be seen by using the final value theorem:
$$
\lim_{t \rightarrow \infty} y(t) = \lim_{s \rightarrow 0} s \frac{A}{s}\frac{\bar{C}(s) P(s)}{s + \bar{C}(s) P(s)} = A
$$