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It is very rare to see a discussion of PID controllers
without also learning about how important it is to properly tune
the controller. Much of the literature talks about empirical
tuning of the controller.
The PID equation is based upon an integro-differential
equation and the ``gain'' factors are determined by a combination of
the desired gain plus the effect due to numerically approximating the
different parts of the equation.
Part of the preoccupation with empirical tuning PID controllers is
cultural and part of it is practical.
I have noticed that it is a rare engineer (as opposed to scientists)
that actually spends the time to analytically solve differential
equations, and not many scientists or engineers have the skills to solve
integro-differential equations in their bag of tricks.
Fortunately the equations are linear, which means that they are
solvable with such techniques as Laplace transforms.
A second reason
why PID controllers need to be empirically tuned is that often for
real-world systems the exact equations aren't really known, especially
given the uncertainties inherent in a system containing mechanical
devices and noise prone sensors.
It is also possible to design self-tuning controllers, by adding an
adaptive tuning section to the control software. We will leave the
topic of adaptive systems for another time.

** Next:** Experimenting with the sample
** Up:** number7
** Previous:** The PID controller
Skip Carter
2008-08-20