ramp function

The use of the ramp function is quite handy when dealing with many engineering systems, particularly when dealing with vibrations and control systems found in mechanical engineering. There are many other uses for ramp functions in other disciplines, especially electrical engineering, that I'm not very familiar with. I'm sure that mathematicians have also found many uses for them as well, because I'm pretty sure my differential equations and linear algebra professors blathered on about them while I was napping in their lectures. An explanation follows:

If we say that the Heaviside Function (aka step function) is:

     f(t) = u(t-a) = { 1 for t > a   (a >= 0) }
                     { 0 for t < a            }

then the graph of f(t) looks like:

            f
            ^
            |
           1+     +---------
            |     |
            |     |
     -------0-----+-------->t
            |     a
            |      

which has a Laplace transform given by:

      Y(s) = L(f(t)) = e^(-a*s) / s

Now let us say we have a graph, the ramp function, that looks like:

            f
            ^                  * 
            |                *
            |              * 
           1+            * 
            |          *
     -------0----+---+-------->t
            |    1   a
            |

then we can say the ramp function R(t-a) is given by:

       f(t) = R(t-a) = (t-a) * u(t-a) 

which has its Laplace Transform given by:

       Y(s) = L(f(t)) = e^(-a*s)/(s^2)

N.B.: The derivative of the ramp function is the Heaviside function: R'(t-a) = u(t-a).

The ramp function and the unit step function can be combined to greatly simplify complicated discontinuous piecewise functions. This is useful if we are trying to define a function such as:

            f
            ^                  
            |               
           1+     *                 +-------------
            |   *   *               |
            | *       *             |
     -------0-----+-----*-----+-----+------------>t
            |     1     2 *   3     4
            |               *       |
          -1+                 *******
            |

In the above example we can see that first we have a ramp function that begins at t=0 and ends at t=1, so our function for 0 < t < 1 is just R(t). In order to have our ramp function level off (go horizontal with f(t)=1 at t>1 ) we would need to subtract a single ramp function at that point which would be R(t-1) but since we are ramping down starting at t=1 instead of going horizontal, we need to subtract an additional R(t-1) from our function. So now we have our function f(t) defined between 0 < t < 3. If you are still paying attention that means f(t) = R(t) - 2 * R(t-1).

Now after t=3 the declining ramp levels off to horizontal so we need to add a single ramp function for when t>3, giving us R(t-3). Now f(t) = R(t) - 2 * R(t-1) + R(t-3), for 0 < t < 4. Starting at t=4 the function steps up from -1 to +1, so we need to add two step functions at this point, which would be 2 * u(t-4). Now the piecewise function in the graph can be simply defined on a single line:

      f(t) = R(t) - (2 * R(t-1)) + R(t-3) + (2 * u(t-4))

Now the Laplace transform of the function can be quickly and easily written down:

       L(f(t)) = Y(s) = (1/(s^2)) - 2*(e^(-s)/(s^2)) + (e^(-3*s)/(s^2)) + 2*(e^(-4*s)/s