Recall from ``Closing the loop'' (FD XVIII No. 5)
the equation for a proportional-integral-derivative (PID)
The quantity is an error signal that is the difference between the commanded input, and the output of the controlled system . In our earlier investigations we were not much concerned with the internals of the controlled system (often called the plant), all we needed was the output signal. Figure 1 shows a generic schematic of our controller and plant.
For an adaptive system, we need to have an additional equation that describes how the plant behaves so that we can properly adjust the parameters of the controller. So now we have two equations to consider, the controller and the plant. The design of the adaptiveness depends upon the form of both equations. For our example plant, we will use the second order differential equation: