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Now we understand what least squares estimation
is and are comfortable with how to use it to determine an optimal
estimate of the parameters system.
You probably won't be surprized to learn that this point that there
are other complicating factors that could be considered, but that
I have left them out for this introduction. You will notice that
we have assumed that when we make a measurement, that our knowledge
of was exact and all the uncertainty/error was in .
We can reformulate the equations to handle the situation where the
error is in and not in , or even when there are uncertainties
in *both* *and* . We can also readily extend the
equations to handle the possibility that some data measurements
are more accurate than others. I have also not shown you how to
write any of this in matrix form. While the linear algebra
formulation is extremely powerful, I have discovered after years of
being a professor that the mere mention of the term ``linear algebra''
causes students to quake in fear. Its really not that difficult a
subject, but I now know better than to spring in on anyone without
some prior preparation beforehand.
Least squares is the analytic tool that
we need to create an adaptive PID controller. Our model will be
the PID equations and the parameters are the gains. What we will
do next time is to work out the optimal estimators for a PID
controller and discuss how the estimation equations can be written
in a form suitable for a real-time system.

Please don't hesitate to
contact me through Forth Dimensions or
via e-mail at `skip@taygeta.com` if you have any comments
or suggestion about this or any other ForthWare column.

Skip Carter is a scientific and software consultant.
He is the
leader of the Forth Scientific Library project, and maintains the
system `taygeta` on the Internet. He is also the President of the
Forth Interest Group.

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Skip Carter
2008-08-20