Re: PID autotuning - not working for heating application

I might have missed something significant here.

It is my assumption that a system with a pure time delay is inherently
non-linear.

Let's take my shower example with a pure delay in the pipes ...

With no delay, you can just say that

   Temperature(t) = Valve_Position

or perhaps with a little thermal mass thrown in you can say that:

   d Temperature / dt = K * (Valve_Position - Temperature)

where of course I'm assuming that valve position and water temperature are
the same thing.

The first is I think a 0'th order linear differential equation and the
second is a 1st-order LDE.

But how would you linearize a system with a pure time delay, exactly?

The shower example with a pure pipe delay between the shower valve and my
skin is fine.

Thanks, Datesfat

Datesfat Chicks wrote:

Superposition is sufficient proof of linearity. What comes out of a pipe
(assuming that there is no mixing in transit) is almost a delayed linear
superopsition of what is pushed into it, but it is not linear because it
is not a pure delay. When the input velocity increases because both hot
and cold water are flowing, the delay time decreases. Superposition
doesn't strictly apply because the time to look isn't well defined.

Any delay pushes a servo system toward unstable. That's not a linearity
problem.


There's also the time it takes the valve to move.


It's already linear. Just nasty.


But, as I wrote above, a pipe is onlt a pure delay as long as the flow
is constant.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Tue, 10 Nov 2009 13:24:00 -0500, Datesfat Chicks wrote:


Well, if it's already linear you don't linearize it.

Take the system y = h(x, t) ==> y(t) = x(t - td).  Testing this with
superposition we get

y1(t) = x1(t - td),
y2(t) = x2(t - td),

y1(t) + y2(t) = x1(t - td) + x2(t - td)

which is both h(x1, t) + h(x2, t) and h(x1 + x2, t) -- therefore the
system is linear.

Note that as Jerry points out a shower isn't necessarily a linear system,
unless your shower valve insures a constant flow and the pipes don't have
any turbulence.  Let a vastly simplified version be

y(t) = x(t - kd * x(t)),

(this doesn't capture the delay behavior in even a perfect pipe)

Then we try superposition:

y1(t) = x1(t - kd * x1(t)),
y2(t) = x2(t - kd * x2(t)).

This does _not_ equal the system output to the sum:

ys(t) = x1(t - kd * (x1(t) + x2(t))) + x2(t - kd * (x1(t) + x2(t))).

so this system isn't linear -- but not for the reason that you thought.

--
www.wescottdesign.com

I have recently done a thermal MIMO  PID controller that ended up
preforming adequately despite using very simple controls.
Some comments:
Even the simplest differential description ends up with an infinite
number of state/poles.
Most real thermal systems have little tabs and things that foul up
theoretical analysis.
Therefore: you can start with simple mathematical models to estimate
requirements but you always end up with approximations.
Pole zero analysis in this case is almost worthless except to roughly
get started.
Bode and/or Nichol's chart analysis (I used both) works very well;
but ..
You have to get and use the experimental data.  You can use that
directly or find a sufficiently good model for the system.
You should establish a "process" for the tuning and experiments; the
system you take the data on will undoubtedly not be the one that ends
up being manufactured.
Gotcha's:  Scilab's system identification processes are unstable
dealing with this type of system.  They can be used to attempt
modelling but tread carefully and double check.
When taking the data, the room/environmental temperature will do
everything it can to confound the experiment.
Don't worry about the lower frequencies, go to where the phase starts
to shift significantly.
For the Bode/Nichols derived compensation just redo the experiment
(which you probably will) to clarify the standard compensation region
round the Bode criterion; 180 degrees +- one or two decades.
Try to give at least hints to how the tuning was done for the
"outsourced" maintenance people who have to maintain the tuning after
the mechanical assembly is altered; unless you want to come back and
start over yourself in a year.

Really, really examine the code to make sure you don't "windup".   I
was forced to rely on programmers in another group and I had study the
experimental results for a while to realize that the anti-windup code
just clipped the output not the integrator.

Ray

I agree with the last paragraph.
However, I have had a lot of success with identifying systems poles
and zero.   I can then place both where I want with the controller
gains.

I didn't know Scilab has a system identification function, but I have
used the lsqrsolve and optim successfully.

Peter Nachtwey

Interesting, I have thought about going that route but opted for a
more conventional process;  System Identification routines.  But that
wasn't very satisfactory.  I have a problem in that I like to continue
along routes until I really understand why they don't work.  Sometimes
I think that half my brain is autistic.
Once I get my system identification code reorganized (with or without
a gui) I plan to test it against my data and some available test cases
from NICONET.   Although they don't seem to be MIMO.  In biological
testing equipment you are forced into MIMO situations in order get the
required temperature accuracy over large testing areas and
environmental conditions.  In addition mammalian reactions are tuned
to constant temperature within a narrow band; 37degC in our case
(presuming no aliens in the group).   I was actually looking forward
to doing that; I had never had use MIMO before.  Wasn't so enthused
after a while; the design process is a lot more complicated and the
tools were not robust.
Once I resolve (or at least identify) the problems perhaps I will
compare the results with lsqrsolve.  If your interested I will post a
link here; but don't expect anything soon.  I am just settling into
Mexico, and am not as fast as I used to be.

Ray

When you have MIMO test data why don't you share it with us.  I would
like to have a crack at too.
It would be helpful to know what I am fitting data too though so I can
get the general form the equations right.  I don't know anything about
your field of study.

The trick is how you use optim() and lsqrsolve(). The best system
identification uses Runge-Kutta to integrate the model's system of
differential equations.

For MIMO systems you will need to use optim().  optim() can optimize a
cost function.  lsqrsolve() requires two arrays of data, the actual
data and the estimated data.  I don't know how you would do this if
you have two sets of actual data and two sets of estimated data.

Peter Nachtwey

I don't quite understand your approach; it seems different from what I
had in mind.  I have multiple sets of experimental data consisting of
three stimulus/drive columns and three columns of resulting temerature
data; together with a multitude of other columns of other temperature
readings for thermal design of the overall assembly.
     My hypothetical approach  to raw curve fitting type of modeling:
Write out the ABCD equations with unknown coefficients and try to find
the coefficients; which are linear (superficially) coefficients
applied to the data.  Having an adequate model in hand, then I thought
I  would use optim() to find the control gains in the closed loop.
This is not what you are describing.  My formulation was just a
passing thought and certainly has a lot of problems I haven't
resolved.
Your comments don't fall in line with this, so why not tell me yours.

Brief technical details follow (of interest only to those who enjoy
these things):
     The system consists of three heaters and three sensors; actually
far more sensors for the data, but the others were temporary and
informational for the rest of the machine and not used in control.
The system consists of a disk holding something like 20 test strips
and rotating the strips under a dispenser and then under an optical
head; so each of the test strips rotated to have a drop of sample
deposited and then put under the optical head to monitor the reaction
development.   One of the heating systems was a buffer to isolate the
test disk from the room.  The other two are more precise and localized
controls that control the sample tray fairly precisely to 37 degC.
The reason for two heaters: one  controls most of the circular sample
disk consisting of 20 or so test strips that have been entered; the
other heater brings the incoming test strips up to temperature from
the room temperature when they are inserted.  The original specs were
that the samples had to be at 37degC +- .1degC when the reaction was
occuring, warmup in 5 minutes, ambient/room temperature 18degC to
30degC.   I designed the control system to be .02 degC accurate at the
tray thermistor, control loop closure at power up inside of two
minutes, PID controls around the principal MIMO directions (the
thermistors were placed reasonably close to the individual heaters).
The last part was to make the programming (done by another group) and
maintenance easier; requiring less skilled people and the end
performance was adequate.  The problems involved were:
1) I couldn't put the thermistors where I actually wanted them,
2) I couldn't be hyper conservative and truly insulate the assembly
( the mechanical people had more than enough problems) so I had to
rely on chunks of metal smoothing out the spatial frequencies.  Of
course the assembly had variations across it anyway.
3) I never had the final machine available during testing because the
mechanical people needed to know about thermal problems before the
design was finished, and I didn't want to be the person holding up
release after the machine was finished.  The was only a problem during
testing since one set of readings would be different from a set taken
later.
4) The sys-id routines were not robust and had to be watched very
carefully.  In fact I ended up using the DC gains of the models as the
first quality determiner.   Then I would look at various residuals to
determine the real quality.  Usually the test data was split in half
(or so) so the model wouldn't be just regurgitating data back to me.
The first half was used to determine a model and then the model was
used to predict the second half; the resulting residual time series
were then examined.  I wanted the residuals to be below .1 degC  (1
part out of 370) or so but never got there due to inadequacies in the
model, and I had to settle for 2 degC; the slack/error was taken up
when the loops were closed.  Apparently the sys-id routines want
random inputs; whereas people are more comfortable with large step
inputs.  I have both types of data.
5) All of the heater systems talk to each other and the environment
thermally; the reason for MIMO approach.
6) Severe organizational problems with people who had never done
instrument design before (:  That's a different story.

    What is driven home is the fact that you are just looking for an
adequate model of reality in thermal situations; not looking for
"truth".   The mechanical assembly can not reduced to anything less
than a FEA analysis; which I couldn't get the department to
institute.  It's not a trivial thing to incorporate in a design
process.  Having done a partial survey I think COMSOL is a pretty good
multiphysics tool and does have the ability to incorporate spice
models between objects like a thermistor (actually a point) and a
heater.

And so on, I have more information.  None of this relates to any
proprietary information; except if I come up with a better process I
can answer questions from the engineer who has to redo the system
after they make changes to the mechanical design.  The design changes
are inevitable and occasionally people get back to me with  questions.

If you really want some data I can post it on an FTP sight.   The
project is done and I am retired so there is no hurry.  The data is
not clean and has a lot of confounding disturbances; OTOH there is a
lot of it :)  I am still interested in determining a better process
for establishing good models; although I am inclined to fix up the sys-
id functions so that higher order approximations don't lead to
(wildly) worse and worse predictions.  That is just nonsense.
Be aware that my criteria are DC gain and residuals;  and any comment
on the modelling will probably be oriented around that.  If your
interested in my code; my SCILAB program does produce a lot of
outputs, BODE and Nichols charts; but is not finished code in the
sense that some parameters are done with I/O, and some parameters are
entries in the code.  There are shortcomings, I never did a good Bode
plot of the raw data, just of the models.  I kept meaning to but that
requires a lot of filtering to be meaningful.

Hope I haven't bored you to much Peter.

Ray

See simple example with differential equation of order 2:

* http://home.arcor.de/janch/janch/_control/20081123-real-system-model/

I try to find the best possible process transfer function (page 1) by using
approximation methods on the basis of some measured values (page 2).

Thereafter I have a benchmark test scheme (page 3) with a program (page 4)
that automatically finds the best PID parameters using the IAE criteria.

This could be done for process identifications up to differential equations
of degree 6.


--
Regards JCH





 

clip..........

Okay  I have uploaded the file that corresponds to step inputs.  This
one is fairly clean.
http://www.plaidheron.com/ray/temp
static-test.jpg
static-test.xls
Should get you there.  If there is a permission problem let me know; I
will resolve.

The .jpg is a graph to get the idea.  T-11 is included to verify the
environment didn't change much.
The .xls is: sheet 1 graphs, sheet static-test is the long
experimental run covering about 4 hours
Cols: T-1,2,3  are the three direct thermistors used later for control
Cols: M,N,O are the PWM drives, 0-100%, to the corresponding heaters;
the trailing columns can be ignored
The intermediate columns are various sensors distributed away from the
actively controled points.

Let me know and I (or you ) can cross-verify your model against other
experimental runs.

I have other experimental data sets that are less clear; some are
basically random inputs to try to satisfy the sys-id programs.

Ray

Basically refering to

* http://home.arcor.de/janch/janch/_control/20081123-real-system-model/

Can you approach the best possible ODE (process transfer function) in a
range of order <= 6?

C6 y'''''' + C5 y''''' + C4 y'''' + C3 y''' + C2 y'' + C1 y' + y = K

Decimal commas!

Example data points: 30

0 0
0,062 0
0,124 0,002
0,187 0,012
0,249 0,04
0,311 0,093
0,373 0,17
0,435 0,266
0,498 0,373
0,56  0,48
0,622 0,581
0,684 0,671
0,746 0,748
0,809 0,811
0,871 0,861
0,933 0,899
0,995 0,929
1,057 0,95
1,12  0,966
1,182 0,977
1,244 0,984
1,306 0,99
1,368 0,993
1,431 0,996
1,493 0,998
1,555 0,999
1,617 1
1,679 1
1,741 1
1,804 1,001


--
Regards JCH

My solution see down here:































































Decimal commas!
1,048734E-06 y'''''' + 6,2427E-05 y''''' + 0,001548347 y'''' + 0,02048154
y''' + 0,1523982 y'' + 0,6047773 y' + y = 1,000953

 

We seem to have a disconnect here.
The system is MIMO which means that a finite model would have a set of
simultaneous differential equations.  In my case three independent
variables drives and three dependent variables; leading to three
simultaneous differential equations whose order varies with the number
of state variables needed for an adequate description.  Including the
room temperature we actually have four drives.  Including the various
components inside the instrument (motors, solenoids, and doors) we
would have more; but for the sake of simplicity I took 3 drives and 3
sensors and treated the other drives as disturbances.  A design
assumption that could have been rendered wrong by results; but then I
would have had to add more sensors and possibly more heaters.
The reason for the 3 heaters and sensors is to establish control over
extended mechanical assemblies having basically an infinite numbers of
internal states.  Although the higher order states are rapidly
suppressed by the heat equation when the metal thermal time constant
is short.
As an illustration: The simple case of the sun warming a piece of
ground through the seasons.  The result is basically that a 20 degC
surface variation causes .5 degC variation 2 meters down with a six
month lag; with the transfer function having an infinite number of
poles and a continuously rolling phase shift going through 180 deg
over and over.  This imposes constraints when you are trying to hurry
it up via control systems.  These numbers are "representative" since I
am remembering; I do have the book Bell Labs book somewhere that
solves the equation.
Alternately: Writing the Green's function for the internal temperature
of a bar heated at the surfaces requires an infinite degree polynomial
resulting in an infinite number of poles in the Laplace xform.  But
the significance of higher poles drops down exponentially, so they
don't matter unless you try to wrap a control loop and close the loop
with time constants that are comprable.

And so on
Ray

We seem to have a disconnect here.
The system is MIMO which means that a finite model would have a set of
simultaneous differential equations.  In my case three independent
variables drives and three dependent variables; leading to three
simultaneous differential equations whose order varies with the number
of state variables needed for an adequate description.  Including the
room temperature we actually have four drives.  Including the various
components inside the instrument (motors, solenoids, and doors) we
would have more; but for the sake of simplicity I took 3 drives and 3
sensors and treated the other drives as disturbances.  A design
assumption that could have been rendered wrong by results; but then I
would have had to add more sensors and possibly more heaters.
The reason for the 3 heaters and sensors is to establish control over
extended mechanical assemblies having basically an infinite numbers of
internal states.  Although the higher order states are rapidly
suppressed by the heat equation when the metal thermal time constant
is short.
As an illustration: The simple case of the sun warming a piece of
ground through the seasons.  The result is basically that a 20 degC
surface variation causes .5 degC variation 2 meters down with a six
month lag; with the transfer function having an infinite number of
poles and a continuously rolling phase shift going through 180 deg
over and over.  This imposes constraints when you are trying to hurry
it up via control systems.  These numbers are "representative" since I
am remembering; I do have the book Bell Labs book somewhere that
solves the equation.
Alternately: Writing the Green's function for the internal temperature
of a bar heated at the surfaces requires an infinite degree polynomial
resulting in an infinite number of poles in the Laplace xform.  But
the significance of higher poles drops down exponentially, so they
don't matter unless you try to wrap a control loop and close the loop
with time constants that are comprable.

And so on
Ray

If you can't find one differential equation (process transfer function) as
part of a set you won't be able to solve anything.

See basics and decoupling of MIMO system:

* http://home.arcor.de/janch/janch/_control/20091117-mimo-system/


--
Regards JCH






 

clip......

JCH & Peter

This thread is getting long and unfocused.  Rather than talk about
doing something perhaps we could start a new thread or blog, and
actually have some contests and results doing system-id on some data
sets?  Nothing formal, just trials and analysis to improve our grasp
of the problems.  A reference site is NICONET but that is real data
and the "truth" is unknown, although I think they have some results
for comparison.  In any case dividing the data up into analysis and
prediction parts allows an objective criteria of tracking accuracy.
In addition we could construct various systems (say circuits or
something like what CLF showed), run simulations, present the data,
and see if the others can reconstruct the source of the data.  A
variation is that the subjects of the test can specify the type of
drives and we can see what problems/solutions various experimental
designs present.
Experiment design is a crucial part of system identification.

JCH
 *http://home.arcor.de/janch/janch/_control/20091117-mimo-system/
Perhaps my spam filters are blocking but all I see is a complicated
block diagram.
I hope that you don't mind if I disagree with your flat statement.
Among other things the heat equation is quite explicit and succinct;
but the Laplace transform (or any other form of solution) has an
infinite number of poles/states.  Having a good differential equation
form doesn't guarantee simplicity.  In other cases, i.e. distributed
systems, the situation can also lead to complications.
In any case actually doing some test cases would be more interesting
than abstract talking.

Ray

[...]

I haven't sent more. Have again a close look to:

* http://home.arcor.de/janch/janch/_control/20091117-mimo-system/

This block diagram shows you eliminating coupling superposition. The MIMO
system 'should act' as if there were just two separate control loops not
interconnecting. Each of them can be optimized separately if the process
transfer functions are known. Therefore you MUST have a good process
identification.

Controller y1 influences process value x2 and vice versa!

Be aware that this is just an EXAMPLE for 2 input and 2 output signals. We
have to find 4 differential equations for this EXAMPLE using process
identification methods.

Physical derivation is difficult. Therefore measured (true) values should be
used for finding the differential equations.


--
Regards JCH


 

When starting the identification process the system must be at steady
state.  The three temperature sensors are at different temperatures.
That could be steady state for a combination of heater outputs but it
is hard to know.  If all the heaters started at the same ambient
temperature then I know the system was at steady state.

Peter Nachtwey

      Okay, I will post that experiment but it's not as clean.  Since
I only had shared access to the prototype I couldn't let the machine
cool down long enough for a real restart, and (of course) the room
temperature changed.   These thermal systems have really long "tails";
some sections (plastic) absorb heat and let it out very slowly.

Ray

      Okay, I will post that experiment but it's not as clean.  Since
I only had shared access to the prototype I couldn't let the machine
cool down long enough for a real restart, and (of course) the room
temperature changed.   These thermal systems have really long "tails";
some sections (plastic) absorb heat and let it out very slowly.

Ray