Question: Help adjusting an exponential function with horizontal asymptote

Hello, I'm just beginning to appreciate and enjoy the use of math to solve practical problems I encounter while working on sound synthesis and music composition, my main occupation. I've had no formal training on math but take it as a hobby and try to tutor myself with the occasional aid of books, videos and a couple of really patient friends. As they're currently not around, I bring here the issue I'm now struggling with:

I'm trying to model a function which would output the time taken for a voltage controlled amplifier to raise or decrease its volume from one level point to another at a given scaled rate (which you can set in the device, but is not expressed in db/sec, only with an integer from from 1 to 99). When the beginning and end level points are the min. and max. that the device allows, the time (y) to rate scale (x) function plots a nice exponential curve (of the form y=b*e^-cx). However, when the min and max levels are respectivly augmented or reduced and the volume range the amplifier has to cross gets smaller, the time taken at a given rate changes not as steadily. So for this situation, for each of the scaled rates I want to consider I draw a curve: on the x axis I plot increments of the amount summed or substrated to the min or max levels, and continue marking time on y. And then I get an exponential of the same form as before but only for some values of x: the curve suddenly flattens and approaches a horizontal asymptote when approaching low values. Rather by chance I figured that something with the form y=1+e^-e^x might to the trick. My question is how to adjust that function to my plotted curve, given that I know the actual function that leads the simple exponential portion: 16.7*e^-0.01x for x=22 onwards, y=14.7 from x=0 to x=22.

Any help would be greatly appreciated