On the Structure of Polyhedral Positive Invariant Sets with Respect to Delay Difference Equations

Abstract

This chapter is dedicated to the study of the positive 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. 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.