Re: PID autotuning - not working for heating application
Tim Wescott
> "Frank W." <frankw...@mailinator.com> wrote in message
> news:7lggquF...@mid.dfncis.de...
>>
>> Since all PID temperature controllers have Autotune, there must be a
>> solution for this problem. Any ideas?
>
> As you probably know from control theory, the basic theory of a PID
> controller is that you have a system described by a set of linear
> differential equations that is inherently unstable or has some
> performance problems. As a result you strap a PID controller onto it
> (with said controller also described by its own linear differential
> equations), and the resulting system (now described by linear
> differential equations which are a mathematical mix of the underlying
> system and the PID controller) has better characteristics.
>
> Did you notice that there is a word that appears many times in my
> description above?
>
> Want to guess what the word is?
>
> That word is "linear".
>
> A system with a time delay is not described by linear differential
> equations. Strapping a PID controller onto it is bad math.
>
> One of the more classic examples is a shower or an industrial process
> that mixes fluids of varying temperature and the sensor is located
> substantially downstream from the mixing value. This is a pure time
> delay. My shower at home is like that. I turn the water a little
> hotter. Nothing happens. I turn it a little more hotter. Nothing
> happens. Then I turn it a little more hotter. Then the wave of hot
> liquid hits me and I scream in agony.
>
> Over time, I've adapted to my shower. I don't burn myself anymore.
>
> I think the control algorithms you want to use for a system like yours
> fall outside the range of PID. I'm sure there is a body of theory that
> covers it, but I don't know what that is.
>
> I would heat the system full bore for a fixed period of time, then stop
> and wait to see how the temperature catches up. And work from there.
>
> The best control strategy for that system isn't going to be PID. That
> is a non-linear system.
I've been resisting forking this over into the control newsgroup: now
it's compelling.
Systems with delay can be perfectly linear, as well as time invariant --
they just can't be described by ordinary differential equations with a
finite number of states.
To be linear, a system only needs to satisfy the superposition property.
A delay element satisfies superposition just fine.
And while a PID controller may not be the theoretically best controller
for a system with delay, in many cases it's not a bad choice at all. PID
controllers can and will give perfectly satisfactory service with plants
that have significant delay. The thousands, if not millions, of PID
controllers in mills and factories around the world that are controlling
plants whose responses are dominated by delay certainly belie any
declaration that the PID controller isn't a good choice to control a
plant with delay.
None of the above is intended to minimize the difficulty in analyzing and
designing a truly optimal controller for a plant with pure delay --
that's an exercise that can make your brain hurt, and fast. And nothing
of the above is intended to chase you away from taking plant delays more
directly into account if a discrete-state controller such as a PID won't
let you eke the performance that you need out of your plant.
But in the absence of significant nonlinearities or time varying behavior
you can use all the analysis tools that are suitable for linear time
invariant systems on a system with delays just fine. You can do good
design work, without ever having to explicitly write out the differential
equations, much less solving them.
So if you don't want to get lost in Mathemagic Land searching for
performance that isn't necessary for your product's success, a good ol'
PID controller may be exactly the optimal controller -- in terms of
adequate performance and reasonable engineering time -- even if it
doesn't satisfy any egghead academic measure of "optimal" for the
particular plant you're trying to control.
Datesfat Chicks
news:rPCdnSIjis5QNWTX...@web-ster.com...
>
> Systems with delay can be perfectly linear, as well as time invariant --
> they just can't be described by ordinary differential equations with a
> finite number of states.
Hi Tim,
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
Jerry Avins
> "Tim Wescott" <t...@seemywebsite.com> wrote in message
> news:rPCdnSIjis5QNWTX...@web-ster.com...
>>
>> Systems with delay can be perfectly linear, as well as time invariant
>> -- they just can't be described by ordinary differential equations with a
>> finite number of states.
>
> Hi Tim,
>
> I might have missed something significant here.
>
> It is my assumption that a system with a pure time delay is inherently
> non-linear.
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.
> 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.
There's also the time it takes the valve to move.
> 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?
It's already linear. Just nasty.
> The shower example with a pure pipe delay between the shower valve and
> my skin is fine.
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.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Tim Wescott
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.
RRogers
> On Tue, 10 Nov 2009 11:54:36 -0500, Datesfat Chicks wrote:
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
pnachtwey
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
RRogers
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
pnachtwey
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
RRogers
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
JCH
"RRogers" <rero...@plaidheron.com> schrieb im Newsbeitrag
news:d45695c8-7d48-4841...@f20g2000prn.googlegroups.com...
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
RRogers
> ...
>
> read more »
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
JCH
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
pnachtwey
> 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.
respect to each sensor
and then find the FOPDT or SOPDT coefficients that work
For the first temperature sensor you have a FOPDT formula that looks
like
t1'=A1*t1+B11*u1(t-dt11)+B12*u2(t-dt12)+B13*u3(t-dt13)+C
Where:
t1 is the temperature a sensor 1
A1 is the system time constant at temperature sensor 1. This is
basically exp(-t/tau1).
B11 is the input coupling of heater 1 to sensor 1.
B12 is the input coupling of heater 2 to sensor 1.
B13 is the input coupling of heater 3 to sensor 1.
u1(t-dt11) is the heater 1 signal for time t.
dt11 is the dead time from heater 1 to sensor 1.
C is the ambient temperature. It had better be the same for all
test unless the ambient temperature is really changing.
It is easy to ID B11 B12 and B13 if they are turned on 1 at
a time but the starting point should be ambient temperature or
some steady state. When done you would have this
t1'=A1*t1+B11*u1(t-dt11)+B12*u2(t-dt12)+B13*u3(t-dt13)+C
t2'=A2*t2+B21*u1(t-dt21)+B22*u2(t-dt22)+B23*u3(t-dt23)+C
t3'=A3*t3+B31*u1(t-dt31)+B32*u2(t-dt32)+B33*u3(t-dt33)+C
All the coefficient could probably be ID at once but then it would
be much harder to get exact values. It is best to do small sections
at a time and rely on superimposition.
The way I ID a system is like this
http://www.deltamotion.com/peter/PDF/Mathcad%20-%20Sysid%20SOPDT.pdf
1. On page 1/10 I define the ideal SOPDT system. I chose different
value to to see how the well the system identification works under
different conditions Notice that there is dead time and I don't assume
all the poles are at the same location like others on this newsgroup.
2. At the bottom of page 2/10 I generate the test data that is later
to be used for system identification. I add noise the to ideal data
just to simulate reality a bit. The CO(t) function is a few steps.
The function can be arbitrary but I have found that the excitation is
critical to the identification. Dead times and time constants are
determined more accurately if the are step or rapid changes. The gain
and ambient coefficients are determined more accurate if the are steps
at different levels.
3. One page 3/10 I plot and save the generated test data. I can post
it on my FTP site for you to practice with. Notice that this data has
dead time and two poles that aren't at the same location. I could
have added more noise but the quasi-Newton method seems to filter it
out well.
4. One page 4/10 the system identification is done. Mathcad's Minerr
function can be like either Scilab's optim() function or lsqrsolve
function depending on the option chosen. I chose the quasi-Newton
optimization which is similar to the optim() function. Runge-Kutta is
used to integrate the differential equation. The differential equation
doesn't need to be linear. I could easily put a none linear term in
there like one that changes the gain as a function of temperature.
This happens with heat exchangers because of the LMTD. Fluid systems
are often of the form
v'=g/m-K*v^2. It is easy to ID non linear system IF you know the
general form of the equation and just need to ID the constants.
Notice that the ID'd poles are closer together than the real poles. I
have notice that system identification tends to ID the poles closer
together than what they really are. Notice that I all ID a dead time
and an ambient temperature. This is something that JCH does not do.
At the bottom of the MSE(), mean squared error function, is where I
calculate the mean squared error between the estimated temperature and
the actual or test data temperature. The Minerr function adjusts
Kp,t1,t2,thetap, and C till the MSE is minimized. You can see the
results are not perfect but that is reality.
5. On page 5/10 the actual or perfect response is compare to the
estimated response. The response looks close, almost identical, even
though the system identification puts the estimated poles closer
together. Also notice that a good system identification routine can
ID systems that are excited by more than just a step change. In fact
they must must be able to do system identification with arbitrary
excitation. Above I said the excitation is the key to doing system
identification. One key is the make multiple steps at different
levels. This is very important in computing the gain and computing
the gain when it isn't linear. Heat exchanger's gain changes because
of LMTD. ( log mean temperature difference ).
6. On page 6/10 PID gains are calculated using the estimated plant
parameters found by system identification. My formula is a little
more complex that the IMC formulas but the response is faster/better
for the same closed loop time constant. I doubt the extra complexity
in the formula is worth the effort for most applications.
7 Page 7/10 simulates the PID control of the original system using the
gains calculated from the system identification. Notice that feed
back noise is simulated as well as the dead time.
8. Page 8/10. The simulation show the response. The response isn't
perfect because there was noise in the original data used to do the
system identification. The system identification is not perfect
because the poles are closer together than they should be and I
simulated noise on the feedback but this is closer to reality.
9 Page 9/10 uses the internal model gain formulas that I got from the
www.controguru.com site. They work well too and are much simpler they
don't work quite as mine. I should have provided a IAE value for my
gains and the IMC gains for comparison.
10 page 10/10 shows the IMC response is a little slower but most would
be please with it.
I would use the above technique one at time with each heater and
temperature sensor. Actually one can excite each of the heaters one
at a time but the data for the 3 temperature sensors at the same time.
I posted a link to a scilab program that does the same thing many
years ago but no one seemed interested.
JCH, you should copy this so your program can handle dead times,
arbitrary inputs, and poles that are not all at the same place. What
you appear to be missing the quasi-Newton code( BFGS) or Levenberg-
Marquardt code that allows you to do proper optimization. I bet you
use a grid search.
Peter Nachtwey
pnachtwey
> clip..........
>
> > ...
>
> > read more »
>
> Okay I have uploaded the file that corresponds to step inputs. This
> 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
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
RRogers
> "RRogers" <rerog...@plaidheron.com> schrieb im Newsbeitragnews:3d4e61d7-69d7-4431...@x5g2000prf.googlegroups.com...
>
>
>
> > clip..........
> >> ...
>
> >> read more »
>
> > 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.
>
> Basically refering to
>
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
RRogers
>
>
>
> > clip..........
> >> ...
>
> >> read more »
>
> > 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.
>
> Basically refering to
>
We seem to have a disconnect here.
RRogers
> don't work quite as mine. I should have provided a IAE value for my
> gains and the IMC gains for comparison.
> 10 page 10/10 shows the IMC response is a little slower but most would
> be please with it.
>
> I would use the above technique one at time with each heater and
> temperature sensor. Actually one can excite each of the heaters one
> at a time but the data for the 3 temperature sensors at the same time.
>
> I posted a link to a scilab program that does the same thing many
> years ago but no one seemed interested.
>
> JCH, you should copy this so your program can handle dead times,
> arbitrary inputs, and poles that are not all at the same place. What
> you appear to be missing the quasi-Newton code( BFGS) or Levenberg-
> Marquardt code that allows you to do proper optimization. I bet you
> use a grid search.
>
> Peter Nachtwey
Peter,
Sorry I didn't answer earlier; I was answering JCH and putting
the data up. I considered the route you illustrated, and perhaps I
should have tried harder. But in your example in the above text you
implicitly assumed (by writing the equation that way) that a good
description was accessible through a single state variable per heater/
sensor, and I ran into intellectual problems trying to have the
flexibility for extension.
I did read your earlier posting and will reread it.
The problem I had was this:
Suppose the correct set of terms for sensor one was:
x'=Ax+Bu
y=Cx+Du
Where u is heater power, y is the sensor readings, and x is the
internal state vector larger than u or y . Now a set of individual
SISO readings and using supposition would result in individual state
vectors xi and Ai,Bi,Ci,Di . Some the state vectors might be shared
between the individual responses and some not. How do you determine
which ones are shared or not? I am sure I can make up a circuit with
discrete components that would illustrate this. This interpretation
problem could be resolvable, but I didn't see how to do it. If I am
missing your point please post an example with more state variables.
It does seem as though some generalization of Thevin's equivalence
circuit theorem might be possible and applicable. Maybe using
Telegin's theorem? But these thoughts are just meanderings of my
mind.
Ray
RRogers
Peter,
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
RRogers
RRogers
RRogers
> > 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
>
> Peter,
> 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
Well I looked around, while I do have SIMO heater by heater data the
subject heater input is random trying to obtain information the sys-id
routines like.
Incidentally: In case I forget; some of the data was taken has a
problem which I found out after much work and threatening to sue the
programmers; the PWM percentages were rounded down to units not tenths
and such. That's the reason for the second set of PWM data.
Maybe I should have quit when they separated the programming from
engineering (: Endeavour to write and check your own control and
monitoring algorithms; you will have a happier life.
Ray
RRogers
> > 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
>
> Peter,
> 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
Well I looked around, while I do have SIMO heater by heater data the
JCH
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
RRogers
>
> 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
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
pnachtwey
initial state must be known and usually that is to ensure the system
is at a steady state where the derivatives are 0. It is too bad the
data got truncated too. I work with motion control systems. It is
easy to make sure the system is stopped before getting into excitation
procedure. If the ambient temperature or C changes during the test
then that must be recorded too. Then it isn't a parameter to be
determined.
Are the time periods seconds? If so then time constants are long.
The long time constants make it hard to find the difference between
small changes in estimated time constants. The optimizing routines
calculate a sum of squared error. I like to think of the SSE as the
elevation in a multiple dimension terrain and the optimizer is seeking
the valley floor or pit. The slope of the SSE is checked in all
dimensions, one for each parameter being optimized. If the slopes
are very flat it is hard for the optimizing routine to get to a final
best position. The truncating data will make this almost impossible
because small difference make a big difference on a flat 'desert'
We have a motor that we put a relatively heavy disk on. The weight
increases the time constant to about 1 minutes which is forever in a
motion control system. This system doesn't ID with a consistent set
of numbers but all seem to work. It appear that there are some steep
hills but once the SSE gets off the steep hills the valley is more
like a flat 'desert'. The long time constants do make finding the
lowest part in the 'desert' difficult. However, any solution on the
desert floor seems to work equally well as the elevations are all the
same but the question is will they work equally well under all motion
conditions. This is odd but true. We have put a lot of effort in to
exciting the motor so that the SSE 'desert floor' has more of a slope.
Motors with short time constants are ID with more consistent
parameters.
Ray, you had the cards stacked against you, but your main problem was
with the data, not optim() or lsqrsolve(). The superimposed method
would work. The MIMO problem was not the biggest problem you had.
Hopefully this explains why auto tuning sometimes doesn't always
work.
Peter Nachtwey
JCH
"RRogers" <rero...@plaidheron.com> schrieb im Newsbeitrag
[...]
>
> *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 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
RRogers
I am sure you don't want to get into this area but....
For a multi-state SISO system with poles on the negative real axis
there is a mathematical theory that is guaranteed to produce a
convergent series of models; I think it was extended to complex poles
as well. It is based up Muntz polynomials being converted to an
orthogonal polynomial series; really posynomials.
Good news and bad news from this theory. Bad news: almost any
sequence of poles will work. Good news: almost any sequence of poles
will work.
That means that you can guide the process and make mistakes; the
mistakes will be washed out later. But the result, while accurate,
will not necessarily be physically meaningful. It also implicitly
means that there are an infinite number of perfectly accurate models;
which might throw optimization procedures off a little. To summarize
you can estimate the two dominant poles, then by subtracting them out
of the data in a precise manner find another pole, use that in a
precise manner, and so on; the "precise manner" is the generation of
the orthogonal polynomial sequence. The most obvious application
would be to use the impulse response; but I think I see how to extend
it to arbitrary inputs.
I have been thinking about applying it to the standard sys-id process
to make successive approximations converge; get better and better as
the order increases. I haven't actually managed to get insight on
how to do this for MIMO approximations.
Enough said
Ray
pnachtwey
They are all independent but the results at the temperature sensors
will be from the sum of the 3 heaters. This should hold true unless
there is something that I don't know where superimposition doesn't
apply. The states for a system of SOPDT equations would simply have
the temperature and the temperature rate at each of the 3 temperature
sensors. I don't see how the temperature at one sensor will affect
another temperature since the temperature sensors are not heat
sources.
Peter Nachtwey
RRogers
> They are all independent but the results at the temperature sensors
> will be from the sum of the 3 heaters. This should hold true unless
> there is something that I don't know where superimposition doesn't
> apply. The states for a system of SOPDT equations would simply have
> the temperature and the temperature rate at each of the 3 temperature
> sensors. I don't see how the temperature at one sensor will affect
> another temperature since the temperature sensors are not heat
> sources.
>
> Peter Nachtwey
Your right; and I see how the dependent/independent has to be handled
for state-space representation. A state space reduction to a non-
singular matrix is required to make A nonsingular in the ABCD
representation.
> > t1'=A1*t1+B11*u1(t-dt11)+B12*u2(t-dt12)+B13*u3(t-dt13)+C
> > t2'=A2*t2+B21*u1(t-dt21)+B22*u2(t-dt22)+B23*u3(t-dt23)+C
> > t3'=A3*t3+B31*u1(t-dt31)+B32*u2(t-dt32)+B33*u3(t-dt33)+C
As a note:
In more complicated systems you need additional terms on the left.
m1*t1''' + n1*t1'' + p1*t1'
m2*t2''' + n2*t2'' + p2*t2'
Ray