Linear approximation for ACS?

Question

Mathematics

Good afternoon! There is such an interesting task, there is a calculated graph of some process (say, a semi-hyperbola), there is a graph of a real process (it turns out, approximately, the same semi-hyperbola, but with slight deviations in the sections). The calculation schedule has some (admissible range of value change + -1). For a linear PID controller, it is necessary to build a linear approximation, but there is a problem, the approximating straight line can go beyond the allowable range of the calculated graph. Is there some kind of mat apparatus for calculating the approximation passing through the allowable range? Thanks