On the structure of polyhedral positive invariant sets with respect to delay difference equations

Mohammed-Tahar Laraba 1 Sorin Olaru 2 Silviu-Iulian Niculescu 1
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France, SUPELEC, CNRS - Centre National de la Recherche Scientifique : UMR8506
Abstract : This chapter is dedicated to the study of the positive invariance of poly-hedral sets with respect to dynamical systems described by discrete-time delay difference equations DDE. Set invariance in the original state space, also referred to as D-invariance, leads to conservative definitions due to its delay independent property. This limitation makes the D-invariant sets only applicable to a limited class of systems. However, there exists a degree of freedom in the state-space transformations which can enable the positive invariant set-characterizations. In this work we revisit the set factorizations and extend their use in order to establish flexible set-theoretic analysis tools. With linear algebra structural results, it is shown that similarity transformations are a key element in the characterization of low complexity invariant sets within the class of convex polyhedral candidates. In short, it is shown that we can construct, in a low dimensional state-space, an invariant set for a dynamical system governed by a delay difference equation. The basic idea which enables the construction is a simple change of coordinates for the DDE. The obtained D-invariant set exists in the new coordinates even if its existence necessary conditions are not fulfilled in the original state space. This proves that the D-invariance notion is dependent on the state-space representation of the dynamics. It is worth to recall as a term of comparison that the positive invariance for delay-free dynamics is independent of the state-space realization.
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Mohammed-Tahar Laraba, Sorin Olaru, Silviu-Iulian Niculescu. On the structure of polyhedral positive invariant sets with respect to delay difference equations. Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016, 2017. ⟨hal-01817035⟩

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