Passivity for a thermal subsystem
Evgenii Rudnyi
I am working with electrothermal modeling. An example can be found in
“Efficient Electrothermal Simulation of Power Electronics for Hybrid
Electric Vehicle” at
http://modelreduction.com/Applications/Thermal.html
The question is how should I define passivity for a thermal subsystem?
I see serious problems if I use the criterion from the electrical
circuits in this case. The starting point for the passivity is the
statement that the system cannot generate energy and the equation for
the energy as follows
E = i_1*o_1 + ... i_n*o_n
where i_1 is the first input and so on, and o_1 is the first outputs
and so on. Then starting from this they come to positive definiteness
of the transfer function.
The equation above implies two things that are correct for an
electrical circuit
1) There are conjugated inputs and outputs. This actually contradicts
a bit to the original assumption of the control theory that one can
define outputs arbitrarily. Well, for an electrical circuit to define
a port is quite natural (still this implies some constraints on a
dynamic systems). The outputs must be connected with inputs correctly,
presumably based on physical considerations. If the order of inputs
and outputs in the equation above is changed, the system may become
not passive.
2) The product i_1*o_1 is energy. This is the case for a circuit
indeed.
However, both things seem not to be applicable for a thermal system.
In the case of a thermal system the product of input (power) by an
output (temperature) does not lead to the energy (W multiplied by K is
definitely not energy). This means that the second assumption is just
wrong and this makes the whole proof actually meaningless for the
thermal system. I mean it does not make sense to talk about not
producing energy anymore as the original equation is not the energy.
Then in electrothermal simulation, the temperature is actually
considered in the sense of information. There is nothing like a wire
in a circuit that actually couples the input and output. A transistor
dissipates power and then the temperature distribution changes the
properties of the transistor.
So, how should one treat passivity in this case?
Best wishes,
Evgenii
Tim Wescott
What are you trying to do that it matters?
Certainly any thermal sinking system is passive, in that you dump heat
into it and the heat goes away. Only when you couple it with some sort
of controlled thermal generator may it become active -- is this your
concern?
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
Evgenii Rudnyi
> What are you trying to do that it matters?
>
> Certainly any thermal sinking system is passive, in that you dump heat
> into it and the heat goes away. Only when you couple it with some sort
> of controlled thermal generator may it become active -- is this your
> concern?
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
I guess, yes. My colleagues, electrical engineers, have a criterion
for a passive system based on that the transfer function must be
positive definite. Then they state that if the subsystem is not
passive, then when one connects it to other subsystems, the whole
system may become unstable.
I do model reduction in order to reduce the dimension of the thermal
system resulting from finite elements, see for example
http://modelreduction.com/Applications/Thermal.html
So, what I hear from my colleagues is that the reduced system must be
passive. You can look for example in
Lorenzo Codecasa, Dario D’Amore, and Paolo Maffezzoni
Compact Modeling of Electrical Devices for Electrothermal Analysis
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND
APPLICATIONS, VOL. 50, NO. 4, APRIL 2003, p. 465
see Section 3 SPURIOUS ELECTROTHERMAL OSCILLATIONS.
So my goal was to understand better the formal definition of passivity
and along this way I see inconsistency, as Eq (2) in this paper is not
energy.
So the question is more how to put your words in a formal way, in
order to apply them formally.
Evgenii
RRogers
Thanks for the link! I am doing something along these lines and
perhaps I can avoid reinventing the wheel!
As far as your definition of "passivity" goes; can you provide a
link? I don't remember every seeing the one you mentioned. Power
dissipated is always potential X flow on a per port basis, or more
correctly the gradient of the energy with respect to time. In
electronics E x I (where I is signed indicating the direction) for
every port. The same reasoning is applied in thermal systems. In
either case draw a boundary around any subsection and calculate the
inflows and outflows of energy/time; that would include resistive
losses through the environment.. Then the imbalance must be energy
sources and sinks. In order to eliminate the time dependency we do a
Fourier transform and then figure out how to include the storage
systems in the algebra; capacitors, thermal specific heat,
inductors.
You can have thermal "storage" and loss; heat up a bar at one place,
save it in an insulator, and then use it to heat something else
later. But it's still passive because the energy can be reclaimed;
but that leads into areas you probably don't want to go right now.
I guess my point should be; you have to go to the Fourier xform to
have an algebraic formulation of passivity. Eliminating time allows
you to count not only the energy coming in now, but also the previous
energy that came in and was saved (or not).
In any case examine your i*o ... they are basically strange you
should always be able to talk about things like E=V^2*C/2 and dE/
dt=V*C*dV/dT=V*I for particular parts or ports; and sum things up.
this is a storage case (for a capacitor) and is assigned to the
imaginary direction in the Fourier xform plane. Sometimes it gets a
little strange but in the end it works out.
Back to your case. Your original "true" model should conserve energy
and your subspace model should also do this; otherwise you will have
ridiculous results. It's simply constraints placed upon the
mathematics. As far as I can see you simply take a large section of
the meshing, define a boundary, replace the insides of the boundary
with an approximation/model. Take the model and boundary and
proceed. The interior model should conserve energy and flow. Energy
should flow across the boundary and be stored or dissipated; probably
never generated spontaniously except for sources. But it must all be
accounted for in the model.
Did this make any sense?
Rrogers
RRogers
I just read the linked paper. It's great!
Rrogers
Tim Wescott
For the purposes of stability analysis passivity doesn't have to be
strictly in units of energy. It's nice if you can keep it that way,
because it's hard to argue against thermodynamics, but ultimately you
may have to change to other units to make the math tractable.
This concept comes from Lyapunov's method of stability analysis, which
requires you to find an energy-like function, but it doesn't strictly
require it to _be_ energy.
--
Tim Wescott
Evgenii Rudnyi
> > and along this way I see inconsistency, as Eq (2) in this paper is not
> > energy.
>
> > So the question is more how to put your words in a formal way, in
> > order to apply them formally.
>
> > Evgenii
>
> For the purposes of stability analysis passivity doesn't have to be
> strictly in units of energy. It's nice if you can keep it that way,
> because it's hard to argue against thermodynamics, but ultimately you
> may have to change to other units to make the math tractable.
>
> This concept comes from Lyapunov's method of stability analysis, which
> requires you to find an energy-like function, but it doesn't strictly
> require it to _be_ energy.
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
Thanks a lot. But then have I understood it correctly that we should
not take literatlly the definition that the passive system should not
generate the energy. I mean that we should just ignore it in the case
of the thermal subsystem.
In other words, what is called energy in system theory has nothing to
do with the energy in physics?
Evgenii
Evgenii Rudnyi
...
Well, in the finite elements the energy is conserved. So one should do
nothing special, just use software that uses finite elements.
My starting point was the paper
Lorenzo Codecasa, Dario D’Amore, and Paolo Maffezzoni
Compact Modeling of Electrical Devices for Electrothermal Analysis
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND
APPLICATIONS, VOL. 50, NO. 4, APRIL 2003, p. 465
see eq (2). This paper as reference just cites the PRIMA paper that in
turn cites two books on electrical circuits.
At present I am reading two papers of Prof Willems on dissipative
systems that I have found recently
J.C. Willems, Dissipative dynamical systems. Part 1: general theory.
Archive for Rational Mechanics and Anylisis 45(5) (1972) 321-351
J.C. Willems, Dissipative dynamical systems. Part II: linear systems
with quadratic supply rates. Archive for Rational Mechanics and
Anylisis 45(5) (1972) 352-393
http://homes.esat.kuleuven.be/~jwillems/
It seems that a passive system is the special case of a dissipative
system.
Evgenii
http://ModelReduction.com
Tim Wescott
So measuring the amount of Joules in a system will always count as a
measure of energy from a systems theory perspective, but you may come up
with other, systems-theory-legal definitions of "energy" that would have
a Physicist gagging, or pointing and laughing.
--
Tim Wescott
RRogers
Thanks for the Willems link. Starting from the earlier papers leads
to a more generalized formulation than what I knew.
I still don't understand
E = i_1*o_1 + ... i_n*o_n
where i_1 is the first input and so on, and o_1 is the first outputs
I guess I am misinterpreting "outputs". I have to presume you mean
the flow result (output) as a result of potential (input)?
Ray
Evgenii Rudnyi
> to a more generalized formulation than what I knew.
>
> I still don't understand
> E = i_1*o_1 + ... i_n*o_n
> where i_1 is the first input and so on, and o_1 is the first outputs
>
> I guess I am misinterpreting "outputs". I have to presume you mean
> the flow result (output) as a result of potential (input)?
>
> Ray
A standard way to write down a dynamic system is
dx/dt = A x + B u
y = C x
where u is a vector of input functions (inputs) and y is a vector of
output functions (output). In the case of electrothermal simulation
usually inputs are power dissipated by devices and y are device
temperatures.
Evgenii
http://ModelReduction.com
Evgenii Rudnyi
...
> > Thanks a lot. But then have I understood it correctly that we should
> > not take literatlly the definition that the passive system should not
> > generate the energy. I mean that we should just ignore it in the case
> > of the thermal subsystem.
>
> > In other words, what is called energy in system theory has nothing to
> > do with the energy in physics?
>
> "Energy" in systems theory is a generalization of "energy" in physics.
> So measuring the amount of Joules in a system will always count as a
> measure of energy from a systems theory perspective, but you may come up
> with other, systems-theory-legal definitions of "energy" that would have
> a Physicist gagging, or pointing and laughing.
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
Thank you very much. This was the reason for confusion that I thought
in terms of energy as accepted in physics. The explanation that
stability is based on Lyapunov functions and that they do not have to
be energy sounds okay for me.
From curiosity. When you say about a generalization of "energy" - what
then could be a meaning of a product power*temperature (that is W*K)?
Evgenii
http://ModelReduction.com
Tim Wescott
would guess that it's the input dimension times the output dimension.
So in a circuit you'd use volts * amps (which really does equal energy)
where with heat you might use power * temperature.
I think there _is_ a measure that ends up being a real energy in
thermodynamics -- something like net heat flow * thermal resistance?
But I'm no thermodynamicist.
--
Tim Wescott
RRogers
I see. I reviewed "Network Theory and Analysis" Anderson,
Vongpanitleed. I have forgotten a lot!
Ray
Evgenii Rudnyi
> > then could be a meaning of a product power*temperature (that is W*K)?
>
> Good question. I'm not the worlds best at that sort of analysis, but I
> would guess that it's the input dimension times the output dimension.
> So in a circuit you'd use volts * amps (which really does equal energy)
> where with heat you might use power * temperature.
>
> I think there _is_ a measure that ends up being a real energy in
> thermodynamics -- something like net heat flow * thermal resistance?
> But I'm no thermodynamicist.
>
> --
>
> Tim Wescott
>
> Do you need to implement control loops in software?
> "Applied Control Theory for Embedded Systems" gives you just what it says.
> See details athttp://www.wescottdesign.com/actfes/actfes.html
I used to be a thermodynamicist and I do not see any physical meaning.
But probably I have to start now pure formally from the viewpoint of
Lyapunov functions and forget about thermodynamics. It may help to
understand the phenomenon better.
Thank you,
Evgenii
JCH
"Evgenii Rudnyi" <use...@rudnyi.ru> schrieb im Newsbeitrag
news:e7174b39-948b-4825...@8g2000hse.googlegroups.com...
*** JCH
Forget Lyapunov functions. See
<cited>
http://en.wikipedia.org/wiki/Lyapunov_function
One must be aware that the basic Lyapunov Theorems for autonomous systems
are a sufficient, but not necessary tool to prove the stability of an
equilibrium. Finding a Lyapunov Function for a certain equilibrium might be
a matter of luck. Trial and error is the method to apply, when testing
Lyapunov-candidate-functions on some equilibrium.
</cited>
Use process identification methods for finding the transfer functions
(blackbox method).
Example
Using data points (red points) you can find the process transfer function
(black line):
1,000228E-06 y'''''' + 6,001139E-05 y''''' + 0,001500228 y'''' + 0,02000228
y''' + 0,1500114 y'' + 0,6000228 y' + y = 1,000063 (decimal commas)
See also
http://home.arcor.de/janch/janch/_news/20080904-transfer-function/
--
Regards/Grüße http://home.arcor.de/janch/janch/menue.htm
Jan C. Hoffmann eMail aktuell: ja...@nospam.arcornews.de
Microsoft-kompatibel/optimiert für IE7+OE7
Evgenii Rudnyi
...
> I used to be a thermodynamicist and I do not see any physical meaning.
> But probably I have to start now pure formally from the viewpoint of
> Lyapunov functions and forget about thermodynamics. It may help to
> understand the phenomenon better.
>
> *** JCH
>
> Forget Lyapunov functions. See
>
Thank you for the link.
...
> Use process identification methods for finding the transfer functions
> (blackbox method).
Sorry, but this does not suit my needs at all. I start with finite
element modeling - my clients already have an already developed finite
element model. Then I use model reduction to reduce its dimension -
see
for details. A short introduction is
From ANSYS to System Level Simulation
http://modelreduction.com/doc/papers/rudnyi08infoplaner.pdf
Antonio Perez
>>
>> What are you trying to do that it matters?
>>
>> Certainly any thermal sinking system is passive, in that you dump heat
>> into it and the heat goes away. Only when you couple it with some sort
>> of controlled thermal generator may it become active -- is this your
>> concern?
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Serviceshttp://www.wescottdesign.com
> I guess, yes. My colleagues, electrical engineers, have a criterion
> for a passive system based on that the transfer function must be
> positive definite.
Is this the kind of "passive" definition that you need?
http://en.wikipedia.org/wiki/Passive_component
Note: There is no multiplication of inputs by outputs!!
> ...Then they state that if the subsystem is not
> passive, then when one connects it to other subsystems, the whole
> system may become unstable.
That's is the usual understanding about "stable" systems.
Perhaps your colleagues and you are talking from different assumptions.
...
> So, what I hear from my colleagues is that the reduced system must be
> passive. You can look for example in
>
> Lorenzo Codecasa, Dario D’Amore, and Paolo Maffezzoni
> Compact Modeling of Electrical Devices for Electrothermal Analysis
> IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND
> APPLICATIONS, VOL. 50, NO. 4, APRIL 2003, p. 465
>
> see Section 3 SPURIOUS ELECTROTHERMAL OSCILLATIONS.
>
> So my goal was to understand better the formal definition of passivity
> and along this way I see inconsistency, as Eq (2) in this paper is not
> energy.
Sorry, I can't access such paper, do you have any web accessible reference
for such equation where inputs and outputs are multiplied?
JCH
On Sep 4, 12:04 pm, "JCH" <ja...@nospam.arcornews.de> wrote:
...
> I used to be a thermodynamicist and I do not see any physical meaning.
> But probably I have to start now pure formally from the viewpoint of
> Lyapunov functions and forget about thermodynamics. It may help to
> understand the phenomenon better.
>
> *** JCH
>
> Forget Lyapunov functions. See
>
> http://en.wikipedia.org/wiki/Lyapunov_function
Thank you for the link.
...
> Use process identification methods for finding the transfer functions
> (blackbox method).
Sorry, but this does not suit my needs at all. I start with finite
element modeling - my clients already have an already developed finite
element model. Then I use model reduction to reduce its dimension -
see
for details. A short introduction is
From ANSYS to System Level Simulation
http://modelreduction.com/doc/papers/rudnyi08infoplaner.pdf
*** JCH
Ok. You deal with 'boundary value differential equations'. That's a set of
linear equations to be solved.
A very simple example just for explanation could be
http://home.arcor.de/janch/janch/_news/20080905-adaptive-integration/
In this case 4 equations h = 0.2 = const.
The set of equations could be reduced to 3 if h(mean) = 0.3333
Choosing e.g.
h1=0.1
h2=0.2
h3=0.6666666...
This can be done with grid adaptation if the solution won't loose accuracy.
See Literature:
H.R. Schwarz, Numerische Mathematik, B.G. Teubner Stuttgart
8.1.5 Adaptive Quadraturverfahren (adaptive integration)
Evgenii Rudnyi
>
>
> *** JCH
>
> Ok. You deal with 'boundary value differential equations'. That's a set of
> linear equations to be solved.
>
> A very simple example just for explanation could be
>
> http://home.arcor.de/janch/janch/_news/20080905-adaptive-integration/
>
> In this case 4 equations h = 0.2 = const.
>
> The set of equations could be reduced to 3 if h(mean) = 0.3333
>
> Choosing e.g.
>
> h1=0.1
> h2=0.2
> h3=0.6666666...
>
> This can be done with grid adaptation if the solution won't loose accuracy.
>
> See Literature:
> H.R. Schwarz, Numerische Mathematik, B.G. Teubner Stuttgart
> 8.1.5 Adaptive Quadraturverfahren (adaptive integration)
>
> --
> Regards/Grüße http://home.arcor.de/janch/janch/menue.htm
> Jan C. Hoffmann eMail aktuell: ja...@nospam.arcornews.de
> Microsoft-kompatibel/optimiert für IE7+OE7
Yes, it is possible to reduce the number of nodes by using adaptive
finite elements. However, model reduction is of completely different
nature. It is an area of mathematics that searches for a low
dimensional approximation for a large-scale dynamic system. The origin
of ODEs is not that important. However, from experience one can state
that for finite element models the low-dimensional approximations do
exist. One can start with 1 millions degrees of freedom (nowadays this
is not something unusual for finite elements) and then obtain a low-
dimensional approximation of dimension 30-100.
You can find many examples at
http://modelreduction.com/Applications/
Evgenii
Evgenii Rudnyi
...
> Is this the kind of "passive" definition that you need?http://en.wikipedia.org/wiki/Passive_component
>
> Note: There is no multiplication of inputs by outputs!!
Well, on this page there are no equations at all, so it is hard to say
what they mean. I guess, the word passive has many different
meanings.
> > ...Then they state that if the subsystem is not
> > passive, then when one connects it to other subsystems, the whole
> > system may become unstable.
>
> That's is the usual understanding about "stable" systems.
> Perhaps your colleagues and you are talking from different assumptions.
This is why it is good to express things formally with equations. Then
it is possible to check the meaning.
...
> > So my goal was to understand better the formal definition of passivity
> > and along this way I see inconsistency, as Eq (2) in this paper is not
> > energy.
>
> Sorry, I can't access such paper, do you have any web accessible reference
> for such equation where inputs and outputs are multiplied?
You can look at the two fundamental papers of Willems
J.C. Willems, Dissipative dynamical systems. Part 1: general theory.
Archive for Rational Mechanics and Anylisis 45(5) (1972) 321-351
J.C. Willems, Dissipative dynamical systems. Part II: linear systems
with quadratic supply rates. Archive for Rational Mechanics and
Anylisis 45(5) (1972) 352-393
Both are available from his homepages
http://homes.esat.kuleuven.be/~jwillems/Publications/Publications.html#1972
This is a more general treatment in terms of dissipativity. But in the
second paper you will find so called the quadratic supply function.
This is exactly the multiplication of inputs and outputs and this is
the formal definition for passivity.
Evgenii
http://modelreduction.com/
Antonio Perez
> On Sep 4, 11:21 pm, Antonio Perez <ap235...@gmx.com> wrote:
> ...
>> Is this the kind of "passive" definition that you
>> need?http://en.wikipedia.org/wiki/Passive_component
>>
>> Note: There is no multiplication of inputs by outputs!!
>
> Well, on this page there are no equations at all, so it is hard to say
> what they mean. I guess, the word passive has many different
> meanings.
Granted, is a very non-formal statement, but:
I see an eq. ~ Ea= int.[ <v(t),i(t)>]
Called the "Available Energy" and understood as sum of the internal product
of every V(·) and I(·) pair for any possible initial condition.
It further states that if Ea is bounded the system is Passive.
It then goes on explaining that any "gain" on the system (transistors,
tunnel diodes) is normally associated with "active" systems.
Which is the "normal" understanding from an Electrical Engineering point of
view.
It's a common practice for Electrical Engineering to express input as
Voltages and calculate the output as Currents. This is used extensively in
network theory. An one Energy Storage function could be <v,i>.
This Energy concept is (somewhat) related to the Storage function on the
papers of Willems: Defined as 1/2*<x,Kx>, and understanding that what is
being stored is energy.
But to generalize that <x,y> (input times output) is allways an Storage
function seems to be the core of the problem.
Yes, an input could be Watts and the output could be ºK for thermal
analysis. But the Storage Function here is not Power * Temperature.
That's why you can't find a physical meaning to "power*temperature (that is
W*K)" because it is not an "Energy" storage function.
In it's simple form Power*Time or h*T could be construed as Energy
functions. A form of those could be used as an storage function. My
thermodynamics background is not enough to be certain.
> Both are available from his homepages
>
http://homes.esat.kuleuven.be/~jwillems/Publications/Publications.html#1972
>
> This is a more general treatment in terms of dissipativity. But in the
> second paper you will find so called the quadratic supply function.
> This is exactly the multiplication of inputs and outputs and this is
> the formal definition for passivity.
I have to disagree that <x,K-x> or <x,Qx> means (input * output):
output is (y = C x +D)
Qx is an Storage Function.
Is the Q matrix identical to the C matrix?
JCH
http://modelreduction.com/Applications/
*** JCH
This may be a thermodynamic analogy to your problem:
See usage
http://home.arcor.de/janch/janch/_news/20080906-thermodynamics/
(just 10 equations)
- Power Station System divided into subsytems (Page 1)
- Sum m_point = 0 (mass fow in kg/s) (Page 2)
- Sum m_point*h = 0 (heat flow in kg/s*kJ/kg=kJ/s) (Page 2)
h may be an analogy to electrical voltage (h = enthalpy)
m_point may be an analogy to electrical current
So far all is steady state!
Dynamical considerations by analytical derivations can't be sucessful.
Therefore I developed a process identification tool.
See again
http://home.arcor.de/janch/janch/_news/20080904-transfer-function/
You may get e.g. a DE of degree 6. But this can be reduced (e.g. 3)
Evgenii Rudnyi
> Voltages and calculate the output as Currents. This is used extensively in
> network theory. An one Energy Storage function could be <v,i>.
>
> This Energy concept is (somewhat) related to the Storage function on the
> papers of Willems: Defined as 1/2*<x,Kx>, and understanding that what is
> being stored is energy.
>
> But to generalize that <x,y> (input times output) is allways an Storage
> function seems to be the core of the problem.
There are two functions - supply rate and storage function - and they
are different.
input*output is the supply rate in terms Willems paper, not the
storage function.
> Yes, an input could be Watts and the output could be ºK for thermal
> analysis. But the Storage Function here is not Power * Temperature.
>
> That's why you can't find a physical meaning to "power*temperature (that is
> W*K)" because it is not an "Energy" storage function.
>
> In it's simple form Power*Time or h*T could be construed as Energy
> functions. A form of those could be used as an storage function. My
> thermodynamics background is not enough to be certain.
Yes, one explanation could be that we have to use other functions for
the supply rate and storage. Some problem along this way that I see,
is that a system could be dissipative for one choice and not
dissipative for another choice. At the same time, stability should be
free of such ambiguity.
> > Both are available from his homepages
>
> http://homes.esat.kuleuven.be/~jwillems/Publications/Publications.htm...
>
>
>
> > This is a more general treatment in terms of dissipativity. But in the
> > second paper you will find so called the quadratic supply function.
> > This is exactly the multiplication of inputs and outputs and this is
> > the formal definition for passivity.
>
> I have to disagree that <x,K-x> or <x,Qx> means (input * output):
> output is (y = C x +D)
> Qx is an Storage Function.
>
> Is the Q matrix identical to the C matrix?
This is a storage function that is expressed in terms of the state
vector and this is clearly not input*output. I have meant the supply
rate <u, y>. Here one find the product input*output. Sorry for being
not clear.
Evgenii
Evgenii Rudnyi
...
> Yes, it is possible to reduce the number of nodes by using adaptive
> finite elements. However, model reduction is of completely different
> nature. It is an area of mathematics that searches for a low
> dimensional approximation for a large-scale dynamic system. The origin
> of ODEs is not that important. However, from experience one can state
> that for finite element models the low-dimensional approximations do
> exist. One can start with 1 millions degrees of freedom (nowadays this
> is not something unusual for finite elements) and then obtain a low-
> dimensional approximation of dimension 30-100.
>
> You can find many examples at
>
> http://modelreduction.com/Applications/
>
> *** JCH
>
> This may be a thermodynamic analogy to your problem:
>
> (just 10 equations)
>
> - Power Station System divided into subsytems (Page 1)
> - Sum m_point = 0 (mass fow in kg/s) (Page 2)
> - Sum m_point*h = 0 (heat flow in kg/s*kJ/kg=kJ/s) (Page 2)
>
> h may be an analogy to electrical voltage (h = enthalpy)
> m_point may be an analogy to electrical current
I am afraid that we consider different cases. My case is described in
“Efficient Electrothermal Simulation of Power Electronics for Hybrid
Electric Vehicle”. Please follow the link at
http://modelreduction.com/Applications/Thermal.html
Evgenii
JCH
http://modelreduction.com/Applications/Thermal.html
*** JCH
Sorry, what I was thinking of can be seen and is lectured in:
Two lectures on the model order reduction:
Background material
http://scripts.mit.edu/~mor/wiki/index.php?title=Background_material
Lecture 1
http://web.mit.edu/mor/papers/6336/sma-5211-lec-mit-1390-02dec2003-2000-4cifp.ram
My basic idea: Up to Page 22 (inclusive)
http://web.mit.edu/mor/papers/6336/lec24.ppt
You may find your solution on Page 23 and higher.
Normally, physics are applied for finding solutions. Then the input/output
behavior is calulated and proved by measuring.
I use real measured data and find the appropriate differential equation DE.
This DE can be reduced and applied for automatic control systems.
See example
http://home.arcor.de/janch/janch/_control/20071123-pd2(pid)z1z2/
Measured values represent a system 'for sure'. But physics approaches do not
for complicated systems.
Evgenii Rudnyi
...
> *** JCH
>
> Sorry, what I was thinking of can be seen and is lectured in:
>
> Two lectures on the model order reduction:
>
> Background materialhttp://scripts.mit.edu/~mor/wiki/index.php?title=Background_material
>
> Lecture 1http://web.mit.edu/mor/papers/6336/sma-5211-lec-mit-1390-02dec2003-20...
>
> My basic idea: Up to Page 22 (inclusive)http://web.mit.edu/mor/papers/6336/lec24.ppt
>
> You may find your solution on Page 23 and higher.
Yes, the group of Prof White also uses Krylov subspaces for model
reduction. But not mode superposition - this just does not work for
thermal problems. You can find my tutorial on model reduction at
http://modelreduction.com/doc/teaching/eurosime/
The mathematics is very similar - so called implicit moment matching.
By the way, there a recent course on model reduction by Tamara
Bechtold on model reduction at
http://www-num.math.uni-wuppertal.de/Lectures/specialtopics07.html
> Normally, physics are applied for finding solutions. Then the input/output
> behavior is calulated and proved by measuring.
>
> I use real measured data and find the appropriate differential equation DE.
> This DE can be reduced and applied for automatic control systems.
>
> See examplehttp://home.arcor.de/janch/janch/_control/20071123-pd2(pid)z1z2/
>
> Measured values represent a system 'for sure'. But physics approaches do not
> for complicated systems.
I have nothing against system identification. But this does not fit my
problem at all. By definition my clients start with the finite element
model. It has been already developed and this is given. The problem is
to determine a compact model statring not from the measurements but
from the finite element model.
It is possible to imagine that first one simulates the finite element
model and then uses system identification. However, this happens to be
much longer as using Krylov subspaces for linear models. So, in this
particular case model identification is not a good solution.
Evgenii
> --
> Regards/Grüße http://home.arcor.de/janch/janch/menue.htm
> Jan C. Hoffmann eMail aktuell: ja...@nospam.arcornews.de
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