Reconstructibility of time-invariant and periodic behavioural systems

Abstract

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.

Keywords:

• behaviours
• state space systems
• reconstructibility
• lifted system

Acknowledgements

The research of José C. Aleixo and Paula Rocha was partially supported by FEDER funds through COMPETE-Operational Programme Factors of Competitiveness (‘Programa Operacional Factores de Competitividade’) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (‘FCT-Fundação para a Ciência e a Tecnologia’), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.

Notes

Notes

1. Given a ring ℛ, a square matrix M ∈ ℛ ^g×g is said to be unimodular over ℛ if it has an inverse in ℛ ^g×g .

2. Where

3. ⌈·⌉ represents the ceiling function, i.e. the integer round-up defined as ⌈x⌉ = min{m ∈ ℤ: m ⩾ x}.

4. By the δ-reconstructibility of w ₂ from w ₁.

5. By the -reconstructibility of Lw ₂ from Lw ₁.