Linear interpolation for PID controller?

Question

Mathematics

Good afternoon! There is such an interesting task, there is a calculated schedule of some process P , there is a schedule of the real process Pr . The calculation chart has some allowable range of delta value change (+-1). It is necessary to find a piecewise linear function for programming a linear PID controller so that this piecewise linear function (some approximation between the real and the calculated graph) does not go beyond the delta (+ -1) range. Someday I will learn how to insert a picture like this: