Re: Optimizing/minimizing functions that use absolute value.

JCH, would you stop posting links to your non-sense.  Start you own thread
if you want to do that.
I don't see what your link has to do with finding minimum IAE or ITAE
coefficients and it is a very bad example of how the IAE should be used.

What improvements?   All you have done is to make PID control look bad again
which really just shows you don't know how to use the PIDs or how to tune
them.


I don't see what is so difficult after one has the minimum IAE or ITAE
coefficients?  It is
calculating coefficients accurately that is difficult.  If one has the
minimum IAE coefficients for each order then all that is required is to plug
the plant parameters into the formulas and out pop the PID gains.  See page
5 of the link below.

It would be nice if the IAE and ITAE coefficients are calculated accurately
using modern software tools and computers once so everyone else can use
them.  Not every one has access to Maple or Mathematica and multicore 64 bit
computers.  One of my complaints is that ITAE coefficients appear to be
wrong and  that people keep using them without question.  I haven't looked
into the IAE coefficients yet.

BTW, I prefer critically damped systems. They are simple to calculate
because there is a binomial pattern to the coefficients.  If I make A1 and
A2 eqaul to 3 then I would have a critically damped response with no
ringing.


Those gains don't look right to me.  See the formulas on the bottom of page
5.  One can dispute the accuracy of my minimum IAE coefficients but they
must be close.  The ratio of your Ki of 1500 to a Kc ( Kp? ) of 1 is wrong.
You don't show the coefficients you use or the minimum IAE that  resulted.
In any case  your closed loop response looks a lot worse than mine and I bet
my closed loop IAE is much lower than your feed forward IAE..

The graphs are STILL mislabeled.  The main graph should be velocity and the
two graphs below it should be acceleration and jerk. I don't know why you
insist that position is proportional to the controller output instead of the
velocity.  When the power goes off the velocity goes to 0 and the position
coasts to a stop and stays where it is at.  When the power goes off, the
position doesn't go to 0 as in your example.

Notice also that at time 0.25 the 'position' moves +0.1 instantly due to a
disturbance.  This
requires an infinite amount of power.  The disturbance act on the plant or
actuator the same way the controller does.  You need to add another term to
your forcing function.  Notice that I added a disturbance to my similar.  I
admit it is rather arbitrary because you provide no information on the
distrubance.  I just designed mine so the actual velocity is displaced by a
litle more than 0.1 so it is similar to your disturbance.  The difference is
that my disturbance doesn't change the velocity instantly like your does.
The response of the system is taken into account instead of just changing
the position like you did.  Notice that my PID recovers MUCH faster than
yours.


How is it that the disturbances don't affect the feed forward control?
This is not realistic at all.  Also, the open loop system is already
critically damped.  So what is the point?  The controller just outputs a
open loop value proportional to the
desired velocity.  So what?  The one reason to have a controller is that it
can achieve a faster response than the open loop response.

What is the IAE for your feed forwards graph?  You don't show any of your
calculations so we can compae your feed forward response with a closed loop
response.  Your response is limited by the open loop response of the system.
The feed forward graphs shows how the open loop control out is limited to
0.9. I can keep increasing omega, see page 5, to make the response as
aggressive as is practical.  My control output can drive the plant or
actuator harder and reduce the my closed loop IAE below your feed
forward IAE.   This can give the closed loop control a big advantage when
making small changes.


Yes feed forwards are cool.  They seem to be your one trick.

To see how the coefficients are properly used see
ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20IAE3.pdf
Omega is necessary to make the response faster and the corner frequency or
bandwidth will roughly match omega. See tge Bode plot at the bottom.  The
equations on page 5 can be used to calculate the PID gains for ITAE, IAE or
critically damped response.  All one needs is the new set of coefficients.
Hopefully someone will find the .pdf example useful.  JCH should study my
.pdf but I bet he will continue to deny and defy reality as he has done for
the last 9 months..

Peter Nachtwey