Low Complexity Invariant Sets for Time-Delay Systems: A Set Factorization Approach

Abstract

International audienceThis chapter deals with the study of invariant sets for discrete time linear systems affected by delay. It establishes a new perspective on their structural properties via set factorization. This novel perspective describes, in a unified framework, different existing notions of invariant sets. Additionally, it is shown that the (possible non-minimal) state space representation is a key element in the description of low complexity invariant sets

Analysis of PWA control of discrete-time linear dynamics in the presence of variable input delay

“…Proof: See Appendix C. 3) Relationship between delay margins: The link between the two representations and their invariant sets has received a unifying characterization via set factorization in [17]. This relationship is formally stated in the next theorem and for the sake of brevity, the proof is omitted.…”

“…This relationship is formally stated in the next theorem and for the sake of brevity, the proof is omitted. For more details the reader is referred to [17] and the references therein.…”

“…The procedure describes, by means of set projections, all possible delay values for which the positive invariance (or alternatively D-invariance) of the state trajecto-ries is guaranteed. The relationship between delay margins in different state space representations is established and linked to set factorization [17]. Furthermore, we extend the delay margins procedure to the multisample case i.e.…”

Polyhedral Invariance for Convolution Systems over the Callier-Desoer Class

“…[2] and the references therein. The question is generalized in [18] to that of the determination of invariant sets, for a class of discrete systems with delays.…”

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

“…Recently, it has been recognized that D-invariance can be seen from the geometrical point of view as a factorization of invariant set in the extended state space [12]. It has been established that the extended state space invariance corresponds to a minimal factorization while D-invariance, under the constraints imposed by the dimension of the DDE, represents the maximal regular ordered factorization.…”