The paper gives conditions under which stability and spectrum are preserved for approximately reachable and observable realizations.
Consequently, we can only consider a single realization to perform the observable estimation.
The concepts of observability, reachability, and minimal realization are defined as follows: The system (1) is said to be reachable if the matrix [ B A B ⋯ A n − 1 B ] is of full row rank, and it is said to be observable if the matrix [ C ′ A ′ C ′ ⋯ (A ′ ) n − 1 C ′ ] ′ is of full column rank, where ′ means transpose.
However, if the signal received from the neighbor is noisy (for example, because of an additive random term such as (u_{2 1} + varepsilon_{2 1})), then the producer should place more weight on his or her own realization, even after controlling for observable differences.
One anomaly is that there can be two realizations, both approximately reachable and observable, but one is exponentially stable and the other is not (unstable).
As explained in Section "Experimental constraints", only one realization can be considered to perform the observable estimation.
In our context, this type of estimation is not possible, for the two following reasons: As explained in Section "Experimental constraints", only one realization can be considered to perform the observable estimation.
This state/signal realization is related to the de Branges Rovnyak model of a linear observable and backward conservative scattering input/state/output system whose scattering matrix is a given Schur class function in the same way as H Z) is related to H(A).
For a smooth system, an observable state remains observable throughout the system evolution with the exception of few selected realizations of the state vector.
We then turn to the inverse problem, and construct two passive state/signal realizations of a given passive future behavior W+, one of which is observable and backward conservative, and the other controllable and forward conservative.
