I take it back. I was plotting the higher order coefficients for the
first time and now I have become skeptical about the published
coefficients above 4th order. Up to forth I think they still seem ok.
I've seem these values published a lot of places though, and its
hard to believe no one ever checked them before.
dave y.
Perhaps you should try to determine what the coefficients should be.
Start with the 3rd and 4th order and see if your answer is closer to
Graham and Lathrop's or mine. Perhaps our names can be referenced for
the next 50 years until someone else calculates these coefficients to
100 decimal places and orders up to 20 with what ever super is
available in the future.
Jerry, have you have Mathcad. Have you tried the minimization
function? If you have been following this thread then have you seen
any errors in my worksheets?
Peter Nachtwey
I haven't played with any of this or looked at your worksheets. Based on
history, I believe you are likely to be right.
Many DSP texts show the sampling period as an element of a filter's
gain. That's wrong from the point of view of dimensional analysis and
has generated many specious arguments in justification. The attitude
still seems to be "That's the way I learned it and tha't the way I
intend to teach it.
An illustration in several physics texts that purports to demonstrate
the influence of head on pressure shows a vertical pipe pierced with
holes along its length, the water streaming in flatter arcs from lower
holes. The paths taken by the streams is the same in texts published 30
years apart and many in between, but both observation and simple
calculation show it to be a fiction.
Have you examples to add?
Jerry

"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
I don't know what you are getting at. The sampling period certainly
must be taken into account.
A DSP PID may look something like this:
K0=Ki*T*+Kp+Kd/T
K1= Kp2*Kd/T
K2= Kd/T
u(n) = u(n1) + K0*e(n) + K1*e(n1) + K2*e(n2)
You can see the sampling period does play a part in the gains in the
difference equation.
The sampling period also affects how short the closed loop time
constants can be. A common error I see on PID tuning websites is
saying the sample period must be 10 times shorter than the open loop
time constants when they should be saying the sample period should be
10 times shorter than the shorter of the open loop time constants and
close loop time constants.
I can think of many errors that persisted for a length of time. There
has always been the world is flat, the universe revolves around the
earth, and orbits are perfect circles, there are 9 planets etc. What
can we do but keep questioning what we think we know .
Peter Nachtwey
Ok. I decided it was time to actually go and plug your coefficients
into Matlab and see what I got..
I got your coefficients from your posting for a fourth order (below)
num=1;
den=[1.0000 1.9520 3.3460 2.6470 1.0000];
I also used published coefficients for ITAE for a fourth order (below)
num=1;
den=[ 1 2.1 3.4 2.7 1 ];
I then computed a step response with the following time vector
dt=0.01;
t=[0:dt:2000*dt]';
Both looked good but the Nachtwey coefficients were slightly better.
I then computed the ITAE value (see previous posting) and got the
following
ITAE_Error = 4.6235
Nach_Error = 4.5864
Again the Nachtwey coefficients were slightly better.
Therefore I am confirming that your coefficients do a slightly better
job than the standard published values for fourth order, and it would
not be suprising to see a larger improvement for higher orders.
Good job! So now what?
dave y.
I could calculate the coefficients to for orders up to 8 easily enough
since I already have the basic work sheet done. You could verify my 3rd
order coefficients. I think my calculation benefit from have more computer
power.
Peter Nachtwey
I'm not sure where your other coefficients are, so I decided to come
up with the set in Matlab using 'fminsearch' function to minimize
itae error. I get the following:
Order Coefficients (highest to lowest)
2: 1.0000 1.5053 1.0000
3: 1.0000 1.7832 2.1713 1.0000
4: 1.0000 1.9407 3.3398 2.6424 1.0000
5: 1.0000 2.0403 4.4715 4.6390 3.2468 1.0000
6: 1.0000 2.0479 5.5317 6.7263 6.6983 3.7020
1.0000
7: 1.0000 2.0372 6.5533 8.8306 11.2594 8.4366
4.2744 1.0000
8: 1.0000 2.0481 7.5235 10.9512 16.7603 15.2920
10.9993 4.7288 1.0000
dave y.
Dave y, I verified this second order solution. I caluclated 1.50534.
I don't know how many digits are really significant. I haven't got
around to the other orders. I am surprised that the Graham and
Lathrop second order solution was so far off. It would help if you
published the minimum ITAE values to.
Peter Nachtwey
Certainly, but the equations must also be dimensionally correct and T in
the right place. See http://tinyurl.com/ygfwo8 and other similar posts.
The usual formulations lead to absurdities.
Jerry

"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
Would be interested to see some more on this?
Does the dimensional analysis come from analog or digital domains?
If the situation is hydrostatic then piercing the pipe at different depths
would result in a faster flow than at a point of lower pressure above it.
Force at point without hole pierced is F=PA and P=rho*g*H
...
This is an outline of the problem written by Robert BristowJohnson in
http://tinyurl.com/ygfwo8 on comp.dsp. There was a lengthy
backandforth discussion there a few years before that thread:
_________________________________________________________________________
Here is the mathematical expression of the sampling theorem:
x(t)*q(t) = T*SUM{x(kT)d(t  kT)} ..
x(t) >(*)> H(f) > x(t)
^ ''

' q(t) = T*SUM{ d(t  kT) } (SUMming over all k)
where d(t) = 'Dirac' impulse function
and T = 1/fs = sampling period
fs = sampling frequency
q(t) = T*SUM{ d(t  kT) } is a periodic function with period, T
and can be expressed as a Fourier
series. It turns out that ALL the
Fourier coefficients are equal to 1.
q(t) = SUM{ exp(j2n(pi)(fs)t } (SUMming over all n)
_________________________________________________________________________
now, in most or all DSP texts, the leading 'T' factor in
+inf
q(t) = T*SUM{ d(t  kT) }
k=inf
is left off. This should not be the case. If you leave it off, it's the
same as multiplying by 1/T and then your Fourier series coefficients
are not 1 but are 1/T which will, in the reconstruction brickwall LPF
filter, lead to a passband gain that is not 0 dB (or 1) but is also
not even dimensionally correct. so in the DSP texts, they put the 'T'
factor in the reconstruction filter where they should put the 'T'
factor in the sampling function, q(t).
_________________________________________________________________________
Jerry

"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
I think Setanta and I are interested in the how water flows out of a
tube with holes at differenet level. Inquiring minds want to know.
I must have been imagining things when I notice packing leaks at test
depth.
Peter Nachtwey
The holes are horizontal, the supply pipe vertical. (You can do this
experiment with an awl or ice pick and a paper milk carton.) At greater
distance from the surface, the stream velocity at the orifice increases,
so the lower streams have a flatter trajectory. Do the streams cross? If
so, what is the pattern? Many highschool texts in the 40s through the
middle 60s illustrated this, all with the same wrong pattern.
Let's hope so!
Jerry

"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
Someone earlier on was asking
Why Model stuff
The problem of water flowing through an orifice was modelled by Toricelli
Using a Large Tank with a circular orifice cut into the side of it.
He reasoned that by applying the Bernoulli/Energy equation between a point
[P1] at the top of the tank  which was a vertical distance H[m] above the
centre line of the orifice [P2]
For ideal conditions
Ideal conditions being
No Losses
Steady Flow
Analysis operated on a streamline between P1 and P2
V1: The velocity of fluid at the top of the tank is assumed zero (negligible
by comparison to V2 at the orifice)
The Pressure at P1 is equal to P2 (Both atmospheric)
===========================================Energies in the system
DeltaPressure + Delta(V^2/2g)+DeltaZ = 0
i.e. The system does not create or destroy energy
PressureEnergy term = 0
KineticEnergy term = ([V2^2/2g][V1^2/2g]) = ([V2^2/2g][0])
PotentialEnergy term = H
so that ideally: H = V2^2/2g
and the relationship between velocity and head is nonlinear
This is a very good reason for modelling
The units of the equation
H[m]=[(m/s)^2]/[m/s^2] = [m]
The constants are omitted as they are dimensionless and have no effect upon
the system rather than scaling.
Is what you are saying with regard to conversion of an analog to a digital
system that when the T is omitted *completely* (i.e. it is assumed to equal
to 1) that the effect of its omission is NOT only a scalar one.
Regards
Setanta
Dnia Thu, 19 Oct 2006 19:37:07 +0200, Peter Nachtwey
(...)
They probably derive this coeff. just like here
http://faculty.capitolcollege.edu/~andresho/tutor/mapleV/TUToptimum.htm
by hand.
Personally, I doubt that minimizing time dependent quality index such ITAE
is used to control motion systems (fast but with big overshooting,
no limit on driving signal  after controller in classical systems).
That is why I think these coefficients should be recalculated using
modern computers. I wonder if Maple or Mathematica can find symbolic
solutions. I tried using symbollic math to find the answer to the
second order example using Mathcad. Mathcad doesn't like taking the
integral of an absolute error. That is OK. There are brute force
iterative methods now.
I know I don't use minimum ITAE for anything. That is why I didn't
think these coefficients and technique had any value.
Peter Nachtwey
I agree if it comes to motion systems, motors etc.
Maybe ITAE is used in some other domain, e.g. teaching domain :)?
Designed ITAE controller is permanent and can not be tuned unlike PID,
this could be a serious disadventage.
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