Reconstructibility of time-invariant and periodic behavioural systems

In this article, the properties of behavioural reconstructibility and forward-observability for systems over the whole time axis Z are introduced. These properties are characterised in terms of appropriate rank conditions, for the time-invariant case. A comparison is made with the existing results in the behavioural setting as well as in the classical state space framework. In the particular case of a periodic system, it is shown that there exists an equivalence between the reconstructibility of the periodic system and its associated lifted system, which is time-invariant. Furthermore, we prove that, for a classical state space system, state reconstructibility is equivalent to behavioural reconstructibility, regardless of the time varying or time-invariant nature of the system. This allows deriving rank tests for the cases of time-invariant and of periodic systems, rediscovering the already known results for state reconstructibility from an alternative perspective. The obtained results contribute to establishing links between two different settings, thus providing a better insight into the considered systems properties.