Equation for a smooth staircase function I am looking for a smooth staircase equation $f(h,w,x)$ that is a function of the step height $h$, step width $w$ in the range $x$. 

I cannot use the unit step or other similar functions since they are just one step. I have been experimenting with various sigmoid curves and while I can get a single smooth step I cannot get to realize the staircase shape. The closest staircase function I have found is given in this paper in equation (18) and depicted in Fig. 4 and it is a close example of what I want (i.e generate a staircase in the range $x$ for arbitrary step heights and widths) but it is not smooth at all.

Regarding smooth steps, a likely starting point I found is here but it gives a smooth function of just a single step. I have been unable to modify the equation to make it into a staircase. I would like to specify arbitrary step heights and widths and generate a smooth staircase in the range $x$ specified.  

Edit (Extra info):
The smooth function I mention above has the problem that the upper, horizontal line is not equal in length to the lower, horizontal line which is why I have been unable to adapt it into a staircase function 

Edit 2
Including some pictures 



Edit 2
Plot of $s$ with a steep slope showing a different width on the first horizontal line
Here is an example based on Math536's answer: Wolfram link

$$f(h,w,a,x) = h \left[\frac{\tanh \left( \frac{ax}{w}-a\left\lfloor \frac{x}{w} \right\rfloor-\frac{a}{2}\right)}{2\tanh\left(\frac{a}{2}\right) } + \frac{1}{2} + \left\lfloor \frac{x}{w} \right\rfloor\right]$$

Where h is the step height, w is the period, and a is the smoothness
Stair cases of Height=H and Width =W may be represented by:

$$S\left(x\right)=H\left(\frac{x}{W}+\frac{1}{2}-\frac{\arctan\left(\tan\left(\pi\left(\frac{x}{W}+\frac{1}{2}\right)\right)\right)}{\pi}\right)$$

Here you can observe https://www.desmos.com/calculator/nxmj1yqrph