Is it theoretically possible, that after discretization by using Talyor Series Expansion, a non-observable nonlinear system will became an observable?
It was proved, that used continuous model of PMSM is non-observable (see attached). I want to know, if resulting discrete system is observable or not. Any comment appreciated. Thanks.
Dear Karol Kyslan ,
as far as I understand both, controllability and observability, are mathematical properties that depend on the model rather than the system itself. For instance, for the same system, you could formulate both observable and nonobservable models. Now, regarding your specific question about the discretization. No, I don't see a way to obtain an observable system after discretization if the original model formulation was unobservable. Furthermore, the right application of the discretization technique should capture the same properties as the continuous one. For instance, stability, controllability and observability. Now, it is true that when you are dealing with nonlinear systems, both, controllability and observability, depend on the region you are exploring these two properties. Then, using the same model, observability or your system could deteriorate.
Some interesting papers dealing with that would be:
Article A set-theoretic approach to observability and its applicatio...
Article Reference Trajectory Design Using State Controllability for ...
Article Towards a controllability analysis of multiscale systems: Ap...
Dear Karol Kyslan,
Theoretically if the continuous system is non-observable after discretization the system stay non-observable, your attached file does not prove that there is a change in this property relating to PMSM.
However Taylor series expansion implies the existence of an infinite number of derivatives at the point.
For this raison we limit the number of harmonic components of the expansion so as not to have erroneous values during the reconstruction of the initial states from the outputs, often we can obtain false initial states which corresponds to an observability of system states.
Best regards
If you obtain an observable system, after discretization, for an originally non-observable system, this means that something is wrong in the discretization process. It is my understanding that the best scenario that you could ever hope for, after discretization, is to have a close approximation of the original continuous system, while preserving all its characterisitics.
Dear Karol Kyslan ;
Theoretically if the continuous system is non-observable after discretization the system stay non-observable.
Best regards
Some interesting links dealing with that would be:
https://en.wikipedia.org/wiki/Observability
https://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf
http://www.apmath.spbu.ru/ru/staff/gromov/files/ControlTheory_en_4.pdf
Best regards
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