Model Decomposition of Timed Event Graphs under Partial Synchronization in Dioids

Abstract

Timed Event Graphs (TEGs) are a graphical model for decision free and time-invariant Discrete Event Systems (DESs). To express systems with time-variant behaviors, a new form of synchronization, called partial synchronization (PS), has been introduced for TEGs. Unlike exact synchronization, where two transitions t 1 , t 2 can only fire if both transitions are simultaneously enabled, PS of transition t 1 by transition t 2 means that t 1 can fire only when transition t 2 fires, but t 1 does not influence the firing of t 2 . Under some assumptions, we can show that the dynamic behavior of a TEG under PS can be decomposed into a time-variant and a time-invariant part. The time-invariant part can be interpreted as a standard TEG. Moreover, it is shown that the tools introduced for standard TEGs can be used to analyze the overall system.

Model decomposition of timed event graphs under periodic partial synchronization: application to output reference control

“…Proposition 3 (Trunk et al 2018) An ω-periodic T-operator v ∈ T ω has an ω-periodic canonical form given by a finite sum I i=1 δ τ i Δ ω|ω δ τ i , where…”

“…Proposition 5 (Trunk et al 2018) A periodic partial synchronization of a transition by the signal S in Definition 1 has an operator representation given by…”

“…Therefore, the transfer function matrix H of a TEGPS can be obtained by y = H u = CA * Bu. In Trunk et al (2018) it was shown that the entries of the transfer function matrix are ultimately cyclic series in T per [[γ ]]. In order to compute this transfer function matrix, to compute system compositions, and to obtain control, we have to perform addition, multiplication and the Kleene star operation of series in T per [[γ ]].…”

“…Proposition 11 (Trunk et al 2018) A series s = m ω Qb ω ∈ T per [[γ ]] can be expressed with a multiple period nω by extending the core matrix Q, i.e., s = m ω Qb ω = m nω Q b nω , where Q ∈ M ax in [[γ, δ]] nω×nω is given by…”

“…A preliminary version of this work has been reported in Trunk et al (2018), where the modeling process of a TEGPS in the dioid T per [[γ ]] was established and a decomposition into an invertible time-variant and a time-invariant part was discussed. The purpose of this paper is to introduce optimal output reference control for TEGsPS based on the model in the dioid T per [[γ ]].…”