Some time ago I was involved at work in making some mechanism to limit and control of an actuator based on a sensor reading $x$, a lower bound $a$, an upper bound $b$, and an "aggressiveness" constant $k$.
Following the given restraints: $$f(x) = \begin{cases} 1,& x \leq a \\ 0 \leq y \leq 1,& a \leq x \leq b \\ 0,& x \geq b \\ \end{cases}$$ And, of course, "We should be able to control how fast the effect settles in, how aggressive it is"
My solution was (and still is) this formula: $$f(x) = {\left(1 - \frac{x - a}{b - a}\right)}^{ k }$$
Example for $k=0.1, 0.5, 1, 2, 5$:
I have programmed it and named it "attenuation_curve", but I don't know if this is its correct name. What is the name of this family of functions or curves?
