When we convert a transfer function of LTI system into state space model, is there any loss of information due to the cancellation of pole zero? Transfer function doesn't support unobservable poles.
No, you don't lose information when you convert from a transfer function to a state-space model (A,B,C,D) because you can always recover the transfer function from
C(SI-A)^{-1}B+D. You lose information when you transform from a non observable or non controllable state-space model to a transfer function model due to pole-zero cancellation.
@Nandan, I do not think that any loss of information appears. Someone should take care for concrete system for which the state-space model is to be adopted, that all state-space variables should be real physical quantities and measurable!
Possibly... if you transform your state-space model (A,B,C,D) into a transfer-function, you'll lose information about the states. So, if your states represent physical values (as opposed to e.g. modal states in the Jordan canonical form), this information is not available in the transfer-function.
@Marc, You are righ! But,in my answer I was speaking about the opposite transformation, from Transfer Function to State-Space-Model!
Please, could my downvoters gives some explanation for such doing?
from State-space model to transfer function.... loss of information is about controllable and observability of the system.
No, you don't lose information when you convert from a transfer function to a state-space model (A,B,C,D) because you can always recover the transfer function from
C(SI-A)^{-1}B+D. You lose information when you transform from a non observable or non controllable state-space model to a transfer function model due to pole-zero cancellation.
@Paulo and @Guillaume, you are absolutely right. I have sad the same in my previous answers, but they were downvoted!!!
I also agree with Lopes dos Santos answer. The transfer function has full information on a controllable and observable ( and minimal , i.e. of minimum order state vector) system state-space realization but not on those nonminimal realizations of the same transfer function . Of couse, unless yo know exactly (from supplementary information on the system) which are the zero-pole cancellations, if any. In this case, you could also reconstruct the whole state of the non--minimal realization which corresponds with such cancellations starting with the transfer function containing a polynomial cancellation.
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