How to model or represent mathematically a periodic sigmoid like function

[acordes]

For an sales incentive system I'd like to model a sigmoid function that repeats periodically and increases thereby by a step function.
Any idea?
An approximation is okay for me as I'm aware of the fact that the sigmoid input ranges from minus infinity to plus infinity and that the function is convergent towards 0 and 1.

[Rick_SAS · Accepted solution]

I guess you mean that it is a sigmoid on each periodic domain and overall it is a monotonic nondecreasing function.

The main thing you need to do is identify a base function and the domain of periodicity.  I have chosen the base function to be 

f(x) = (1 - cos(pi*x)) / 2  for x in [0, 1]

which is a monotonic function that has the range [0, 1].

You can then extend the base function by adding a step function to it.

One step function is s(x) = floor(x). 

/* The base function has domain [0, 1] and also range [0, 1].
   The base function is
   f(x) = (1 - cos(pi*x)) /2
*/
data PeriodicSigmoid;
   pi = constant('pi');
   do x = 1 to 6 by 0.01;
      z = x - floor(x);     /* z is always in [0,1] */
      y = floor(x) + (1 - cos(pi*z)) /2;  /* add step function to base function */
      output;
   end;
run;

proc sgplot data=PeriodicSigmoid;
   series x=x y=y;
   xaxis grid;
run;