Re: ITAE poly/roots reference?



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.
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dave y. wrote:
...

I can cite many examples of errors that were copied from one text to another, sometimes spanning a generation.
Jerry
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dave y. wrote:

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
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Peter Nachtwey wrote:

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
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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= -Kp-2*Kd/T K2= Kd/T
u(n) = u(n-1) + K0*e(n) + K1*e(n-1) + K2*e(n-2)
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
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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.-
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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
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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.
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dave y. wrote:

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
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Peter Nachtwey wrote:

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
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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

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Setanta wrote:

...

...
This is an outline of the problem written by Robert Bristow-Johnson in http://tinyurl.com/ygfwo8 on comp.dsp. There was a lengthy back-and-forth 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 brick-wall 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
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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
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Peter Nachtwey wrote:

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 high-school texts in the 40s through the middle 60s illustrated this, all with the same wrong pattern.

Let's hope so!
Jerry
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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 non-linear
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
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Dnia Thu, 19 Oct 2006 19:37:07 +0200, Peter Nachtwey
(...)

They probably derive this coeff. just like here http://faculty.capitol-college.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).
--
Mikolaj

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Mikolaj wrote:

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
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napisa(a):

They will be probably very similar.

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.
--
Mikolaj

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