How solve the equation $Ax^B+C = y$

Question

I am doing one sensor calibration. For that I need to calculate $A, B, C$. The equation is of the format

$$y = A x^B + C$$

for different values of input $x$, sensor will give me $y$. How can I calculate $y$? How many input/output values are required to get $A,B,C$?

Answer 1

You can eliminate $A$, and $C$ using three points, by

$$\frac{y_2-y_1}{x_2-x_1}=\frac{x_2^B-x_1^B}{x_1^B-x_0^B}.$$

This is a nonlinear equation in $B$, that will require a numerical solver.

If by chance you can choose the values of $x$, and adopt $x_1:=\sqrt{x_2x_0}$, you get

$$\frac{x_2^B-\sqrt{x_2x_0}^B}{\sqrt{x_2x_0}^B-x_0^B}=\sqrt{\frac{x_2}{x_0}}^B$$ and you can solve for $B$ by logarithms.