and $T$ is the sequence of observed altitudes from a trajectory of the dataset. We suppose that $(j-i) \leq |\mathcal{H}(\theta,s_0)| \leq |T|$ and that the timestamp associated to the point $\mathcal{H}(\theta,s_0)_n$ is the same than $T_n$. When fitting the whole climbing phase, some difficulties with eq.\[eq:opt\] may arise. First, we need to give a termination criterion when simulating the hybrid system that is to stop when the TP reaches the level flight. Depending on the parameters, the cardinal of the resulting sequence of points will be different. So, it might be necessary to add points to the real trajectory T since the TP can reach the top of climb after T, e.g. a high value for the mass parameter. To compute this error, we simply add points at the level flight until the TP reaches it. Inversely, we add these points to the TP if it reaches the top of climb before the real trajectory. Besides, $i < j$ are bounds to restrain the optimization on any subset of contiguous points of the trajectory. Thereafter, this notation will be useful for the online predictor.

   Time after takeoff     Nominal (FL)       Tuned (FL)
  -------------------- ------------------ -----------------
         2min.          4.9195 (3.1422)    3.0929 (2.3133)
         5min.          