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modelicaeigenvaluelinearization

Modelica linear analysis

I am new into Modelica/Dymola-modeling. I started to model fluid models for my master-thesis (pipes, heaters, control). My professor suggested: Check the eigenvalues of the systems and the ratio of them (and if the systems is stiff). So did the following:

  • Linear analysis
  • Full linear analysis
  • SimulationSetup: linearizeAtInitial = false; t_linearize = 1000 (that the system is in operation mode (after initialization)

The simulation restuls are Ok.

But the linear analysis says: The system is not stable --> what does that mean?

And regarding eigenvalues i get the following result:

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I have to say that I am not really into numerics. So these are my questions:

  1. Does that mean that the system has 23 eigenvalues?
  2. If I want to specify how "stiff" the system is, I guess i have to check the biggest and smallest value?
  3. What are the eigenvalues 12-23 --> not stable, not stabilizable, not detectable?

Any recommendations for a (simple) introduction book/chapter that explains general things like :linearication of Models, eigenvalues, etc.?

Thank you very much for your help.

Solution

  • An eigenvalue with real part < 0 is stable; and the ratio between the smallest one and the largest one can be seen as the "stiffness".

    An eigenvalue with real part > 0 is clearly instable.

    Eigenvalues with real part == 0 are more complicated; they are written as "not stable" - but are on the border between stable and instable.

    Since the imaginary parts are also zero they are pure integrators, and a fluid tank that can be seen as an integrator; which explains why you have so many eigenvalues at zero. (If the imaginary part is non-zero they are un-damped oscillators.)

    Thus I would skip the zero eigenvalues in the analysis.

    The "detectable/controllable" refer to the case when your model have outputs and inputs (and stabilizable is a combination of them).