I would like to fit data to this model:
$$y = c_0 \cdot \tanh (t - c_1) + c_2$$
When I have determined $c_0$, $c_1$ and $c_2$ how can I estimate a time-axis scale which relates to the change in $y$ from one asymptote to the the other?
To provide some context, the function represents temperature change by PID control; I'd like to estimate how long the temperature takes to change from one setpoint (asymptote) to the next.
By analogy to an exponential relationship such as:
$$ y = c_2 + \left(c_1 - c_2 \right) \cdot \exp^{-\left( t/c3 \right)}$$
The asymptote in $y$ is reached within an acceptable margin after $T = kc_3$ where $k$ is between 5 and 7.