Conclusion

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 $x$ was exact and all the uncertainty/error was in $z$. We can reformulate the equations to handle the situation where the error is in $x$ and not in $z$, or even when there are uncertainties in both $x$ and $z$. We can also readily extend the equations to handle the possibility that some $x,z$ 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.