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Source for my new Modelica Book
https://mbe.modelica.university
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Invalid state space representation of Lotka-Volterrra equations ? #135

Closed tbeu closed 10 years ago

tbeu commented 10 years ago

In http://beta.book.xogeny.com/behavior/equations/population/ you say that "the Lotka-Volterra system of equations is actually non-linear". This is true since bilinear is non-linear in some case.

Later on in http://beta.book.xogeny.com/behavior/arrays/state_space/ you introduce the linear ABCD form of DOEs where system matrix A is either A(t) or time-invariant. However in ArrayEquations.StateSpace.Examples.LotkaVolterra you have an A = A(x), i.e. an A that depends on the state variables. This does not match the state space representation.

tbeu commented 10 years ago

I was reading the state space chapter again. The LotkaVolterra Modelica model is not referenced there. Instead it is only in the library. You could simply delete this model and you would be on the save side. However I suggest to not go the easy way. You could say that it is neither possible to represent the bilinear Lotka-Volterra equations as LTI nor as linear time-variant system and continue with the introduction of the concept of linearization.

tbeu commented 10 years ago

And what about the model ModelicaByExample.ArrayEquations.StateSpace.Examples.LotkaVolterra? It extends from ABCD model, i.e. is represented in state space form. This implies that A is constant or time-variant, doesn't it?

tbeu commented 8 years ago

While you say "In the section above we do not include a representation of the Lotka-Volterra equations in LTI form" the model ModelicaByExample.ArrayEquations.StateSpace.Examples.LotkaVolterra does exactly this (though not being mentioned).

xogeny commented 8 years ago

@tbeu I think what happened was that I included it and then thought it was confusing. But, I left the model there just as an example. Since it isn't in the book, I assume only an advanced reader is going to bother looking at models not mentioned in the book.

What might be useful is some kind of comment in the model explaining "This model is in ABCD form but is the A matrix included a nonlinear term. As such, it can be cast into a general ABCD form that allows the matrices to be both non-linear and time variant. This form is still useful in some cases because it can be used to compute properties that can be applied locally to some region of state space (e.g., natural frequencies)."

Does that address your concern?

tbeu commented 8 years ago

Indeed, such a commen to the model would be helpful.