A remark on conditioned invariance in the behavioral approach

Conditioned invariance and detectability subspaces in the behavioral approach

“…Therefore V is conditioned invariant if and only if it is E-invariant, for some achievable error behavior E. By Proposition 21 this immediately leads to the following result (Pereira and Rocha, 2013).…”

“…A first attempt to define behavioral conditioned invariance was made in (Pereira and Rocha, 2013); however it turned out that this first definition was not strong enough since the simple existence of an observer guaranteed conditioned invariance. This weakness was due to the proposed definition of behavioral invariance that served as basis for the notion of conditioned invariance.…”

A Behavioral Approach to Estimation in the Presence of Disturbances

“…A full characterization of conditioned invariance in terms of the polynomial matrices associated to the relevant behaviors is given in the next proposition, [12].…”

“…Here, given our focus on estimation, we assume that the system variables are divided into measured variables, to-be-estimated variables and, possibly, disturbances (which play the role of unknown inputs in the classical approach). To investigate the behavioral estimation problem, we rely on the behavioral theory of observers introduced in [11] and [10], as well as on our previous results on conditioned invariant and behavioral detectability subspaces presented in [12]. Roughly speaking, an observer for a given behavior B is a second behavior B that shares the measured variables with B and produces a suitable estimate of the to-beestimated variables, in the sense that the error behavior (i.e., the set of all estimation error signals) has certain desired properties, see Figure 1: On the other hand, conditioned invariant and detectability subspaces are behaviors V e contained in the error behavior B e of an observer, such that the quotient behavior B e /V e has suitable properties: specifically, autonomy and stability.…”

“…Indeed, our first definition was based on an inadequate notion of behavioral invariance, which implied that the sole existence of an observer was sufficient to guarantee conditioned invariance. This problem was later solved in [12] by adjusting the definition of behavioral invariance.…”