Background
I have 4 sets of data for a brushed DC motor all collected from the same experiment:
One alternative would be to calculate piecewise linear interpolation functions for each of the data sets, as you say. I would include quadratic and cubic splines within this same broad class. You might instead be able to fit reasonable functions to each of data sets using curvefit. Then, you could choose max(T-0) as the starting point for all of the fitted functions, and likewise choose min(T-N) as the ending point for all of the fitted functions.