..and that is the million dollar question "the constants have to be determined" HOW !!

Your function indicates that you retain all the system states in some kind of array, but you don't, you only need the previous state with this control algorithm.

One method of tuning is to run a psudo random period square wave of control point, not sufficient to cause the system to saturate the controller, through the system and determine the impulse response, and therefore the transfer function from that, another is to use something like Ziegler-Nichols methods.

I have my own method: I give the job to my technician and tell him to do it ! We use a method which was originally in Astrom and Wittenmark, where we give the system a fixed period square wave and wait until we get a steady response out of the other end of the system. Which is fine if your system isn't whipping round in circles, but scary if it is.

Steve