Invariant sets for discrete time-delay systems: Set factorization and state representation

This paper deals with the study of the invariance of polyhedral 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. Hence an alternative solution based on the set factorization is established in order to obtain more flexible set characterization. With linear algebra manipulations and as a main contribution, it is shown that similarity transformations are a key element in the characterization of low complexity invariant sets. 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 artifact which enables the construction is a simple change of coordinates for the DDE. Interestingly, this D-invariant set will be shown to exist in the new coordinates even if in its original state space it does not fulfill the necessary conditions for the existence of D-invariant sets. This proves the importance of the choice of the state representation.