What is a "numerically fragile" model, and how to avoid them in the MSL #4577

I'm not sure whether we can always avoid numerically fragile models. Sometimes this depends on parameters - as we have seen in #4559 where changing parameters of the diodes reduces chattering.
We have usages of components that are stable and others that are really challenging, see discussion in #4485. It deals with IdealizedOpAmpLimited - which is the only OpAmp model in MSL with saturation and finite amplification (this is essential for some applications). This component has been "enhanced" (and detoriated!) after the first implementation - we have to redesign it.
But the question is: Can we avoid numerically fragile models by different implementation of the components or is it also a question of the system (i.e. the example to be simulated)?
See also #4477 which triggerd my implementation of chaotic systems which shows that for certain parameter values we get periodic results and for other parameter values we get chaotic behaviour, i.e. for smallest variations of initial conditions (or a different solver algorithm!) we get drastically different trajectories.
I would appreciate to discuss with other Modelicans - how can we reach them?

Note that the problem we had in #4559 was not that a small variation of inputs/parameters/initial conditions caused a large deviation of the solution. The problem was that the formulation of the model around the diode switching was ill-conditioned due to the resistance and the conductance being too close to zero. If you set Ron = 0, Goff = 0, the model is theoretically singular. If those parameters are set to non-zero but too small values, the model is practically nearly singular due to very large condition numbers of some implicit equations.

The symptom in this case was chattering, but it was not chattering caused by a system with a discontinuous RHS ODE system, as in models that have sliding modes; as I understand, it was an ill-conditioned situation caused by too small Ron and Goff. Making them larger (but still negligible from a practical point of view) solved the problem. I believe we should check all condition numbers during the numerical solution process, and reject models that lead to condition numbers larger than, say, 1e9 or something like that.

Chaotic systems are a different story. Their problem is that their trajectories are inherently unstable. I would not classify them as "fragile" or "ill-conditioned".

It's a pity that experiments and components both are named "model" although they are very different:
A component can't be simulated. You have to add the ambient of an application to get an experiment and simulate,
Are we talking about components or experiments?
If we're talking about experiments, I agree with @casella : We should dig into condition numbers (sigh, I'm not very experienced in numerical mathematics). Could Tolerance have an influence on condition numbers?
To classify the different problems:
#4559 shows that changing parametrization (Ron and Goff) of the experiment could improve numerical condition.
#4485 shows that a component (OpAmp) that works fine in one application causes problems in another experiment. It would be a pity to reject an experiment that is no big deal in reality like the multi-vibrator or the signalGenerator.
#4477 is about a chaotic model. As I wrote in #4637 I think for most such experiments we might find a parameterization that allows compariability and leave it to the user to change parameters to obtain chaotic behaviour.

Looking more closely at SPMP_Braking #4559 seems to suggest why it is ill-conditioned and how it relates to the chattering.

SMPM_Braking

Notice how the current doesn't seem to increase linearly after the event, but it looks more like a quadratic function.

• That will make it harder to compute, and since it is close to a zero-crossing it will increase the risk of chattering.
• It also indicates why it is temporary, since quadratic functions are only problematic close to zero.

That also suggests that a quick-fix would be to introduce a new iteration variable in the IdealDiode:

 Real i2;
equation
 i- Goff*Vknee=noEvent(abs(i2))*i2;

That crazy idea seems to work as far I could tell, indicating that it is an ill-conditioned system of equations (it's well-known that quadratic systems are ill-conditioned near zero) leading to chattering.

Obviously changing the ideal diode in that way in general is problematic; but it may serve as a starting point for a good solution - and also dismiss bad ideas.

I don't know why the solution has a current that increases like that.

@HansOlsson do I understand correctly: i2 is just an additional variable (unit sqrt(A)) making the iteration easier?
I tried it and I see:
With the original diode I see Sizes of linear systems of equations: {2}.
With the "new" diode I see Sizes of nonlinear systems of equations: {1, 1, 1, 1, 1, 1} and the simulation time gets higher.
Astonishing: You zoomed in at a point in time when mechanical speed is nearly zero. Therefore induced voltage is nearly zero and nearly nothing should happen. Ok maybe challenging in the rectifier with current and voltage near zero.
With the "new" diode the chattering still happens:
SMPM_Braking

BTW: If you want to have in iteration variable i2 shouldn't we write:
Real i2 = noEvent(abs(i - Goff*Vknee))*(i - Goff*Vknee);
Sizes of nonlinear systems of equations: { }

... and @casella is right: Higher (but technically speaking neglectible) Ron=1e-4 and Goff=1e-4 solves the problem completely,
even without additional iteration variable:
SMPM_Braking
What's astonishing: Sizes of nonlinear systems of equations: { } - why that? Due to change of Ron and Goff?

Here's a post by Christopher Rackauckas that proves that for chaotic models (resp. models with parameterization that leads to chaotic behaviour) comparability totally depends on accuracy. He calls the point in time where chaos starts "Lyapunov time". And this Lyapunov time depends on the algorithm, tolerance and accuracy (computer system and OS). So for me the safe solution to achieve comparability is to compare with parameterization that leads to periodic results and leave it to the user to vary parameters and investigate the path to chaos.
https://www.juliabloggers.com/how-chaotic-is-chaos-how-some-ai-for-science-sciml-papers-are-overstating-accuracy-claims/

Maybe the only way to check whether a model (with a certain parameterization!) is numerically fragile or not is to simulate with different algorithms, different tools and carry out a sensivtivity analysis: How large are the deviations of trajectories for small changes of parameters and / or initial conditions?
Bear in mind all our models are deterministic, and talking about chaos means "deterministic chaos", i.e. the trajectories lie in a certain region but which of the possible trajectory is hit is impossible to predict.

I don't think chaotic models are "numerically fragile". That's the way they behave. In real life. I mean, their behaviour is truly unpredictable, that's what Lorentz's butterfly was all about.

To me, "numerically fragile" means that the real system has well-defined behaviour, but the model is prone to crash or become singular or produce bogus results because some parts of them are ill-posed.

Then I see 2 types of "numerically fragile" models;

1. The one @casella did describe "The real system has well-defined behaviour, but the model is prone to crash or become singular or produce bogus results because some parts of them are ill-posed." BTW what is the definition of "well-defined"? FInite results? Predictible results? Repeated experiments show the same results?
2. "Chaotic models" show only for a certain parameter range deterministic chaos, for other parameters they show periodic results. Yes to produce chaotic results some prerequisites have to be implemented in the model, for some models by intention, for some models by chance. Deterministic chaos means that though the model algorithm is determinstic the results are unpredictable, since small changes in parameters, initial values, algorithm and tolerance lead to finite but unpredictable different results.
   Isn't the second case also "numerically fragile"? Interesting to read: Christopher Rackauckas' post mentioned above.