If you have been following the thread of this column, you will recognize that we now have all the background we need to create an adaptive PID controller. With this installment we will proceed with its design. Writing this installment presented me with a difficult problem. Some of you are aware that I have been very gentle with the mathematics behind the topics that I have covered in the past. From the feedback that I have received, it would seem that in spite of my efforts to go easy, this column has a reputation for being a bit challenging. No matter how much you try to avoid or hide it, understanding adaptive controllers involves a lot of math. As I looked at how to present adaptive controllers I came to the conclusion that if you consider where these controllers get used (typically machinery, frequently dangerous machinery), one shouldn't be doing this as if it was from a cookbook. You really should know the math if you are doing this stuff. So be forewarned, there is a bit of math in this installment. There is enough here that we will be presenting the adaptive controller in two parts, this time we will do the math and next time we will look at an implmentation.