I am trying to optimize the integrated absolute error IAE and integrated
time absolute error functions. These are ways of determining best gains for
control transfer functions. In the IAE the absolute value of the error is
added to a sum. In the ITAE the absolute value of the error at each time
period is multiplied by the time and added to a sum. Normally I use the sum
of squared error SSE and it works well and optimizes easily. The SSE
squares the error at each time period and adds it to as sum. The SSE
results in a continuos and smooth function However, the IAE and ITAE do not.
It seems that the discontinuity cause by the abs function GREATY hinders the
efficiency of quasi Newton methods of finding minimums. Is this my
imagination or is it real?

Also, these coefficients that result in the minimum IAE or ITAE were calculated about 50 years ago. There were computers back then but I doubt they had the techniques or the processing power to compute these coefficients. They must have been calculated by hand. How does one integrate functions ,that use absolute functions, by hand?. It seems to me that one would need to do a separate integration each time the sign of the error changed. These guys must have been very dedicated. However, the coefficients that have been handed down over the years are not quite right. I can prove it for orders up to 5 but the time it takes to find the best coefficients by using optimizing functions is very long. My Mathcad doesn't seem to converge if for 6th order models and above.

Peter Nachtwey

Also, these coefficients that result in the minimum IAE or ITAE were calculated about 50 years ago. There were computers back then but I doubt they had the techniques or the processing power to compute these coefficients. They must have been calculated by hand. How does one integrate functions ,that use absolute functions, by hand?. It seems to me that one would need to do a separate integration each time the sign of the error changed. These guys must have been very dedicated. However, the coefficients that have been handed down over the years are not quite right. I can prove it for orders up to 5 but the time it takes to find the best coefficients by using optimizing functions is very long. My Mathcad doesn't seem to converge if for 6th order models and above.

Peter Nachtwey