Recall from ``Closing the loop'' (FD XVIII No. 5)
the equation for a *proportional-integral-derivative* (PID)
controller is:

(4) |

where is the proportional gain, is the integral gain and is the differential gain.

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:

(5) |

this is a pretty generic model, as a example this can be thought of as a damped mass-spring system where is the mass, is the friction and is the spring constant. would represent the imposed external forces on the system (the input), and the solution to the equation would give the plants response to it.