Dynamics of Implicit Time-Invariant Max-Min-Plus-Scaling Discrete-Event Systems
Sreeshma Markkassery, Ton van den Boom, Bart De Schutter
TL;DR
This work analyzes autonomous time-invariant implicit MMPS systems for discrete-event dynamics. It develops solvability criteria via a permutation-based lower-triangular condition and introduces a normalization procedure that maps solvable implicit MMPS systems to zero-growth linear forms with polyhedral feasible regions, enabling computation of growth rates and fixed points through footprint-based linear programs. The authors derive a local stability framework by linearizing around fixed points and examining multiplicative eigenvalues of the resulting map, linking stability to the rank of constraint matrices. A case study on an urban railway network demonstrates the method’s practicality, revealing multiple fixed points and confirming stability of the linearized system. The results provide a foundation for closed-loop control design in MMPS-based discrete-event systems, with future work focusing on control-Lyapunov-function approaches.
Abstract
Max-min-plus-scaling (MMPS) systems generalize max-plus, min-plus and max-min-plus models with more flexibility in modelling discrete-event dynamics. Especially, implicit MMPS models capture a wide range of real world discrete-event applications. This article analyzes the dynamics of an autonomous, time-invariant implicit MMPS system in a discrete-event framework. First, we provide sufficient conditions under which an implicit MMPS system admits at least one solution to its state-space representation. Then, we analyze its global behavior by determining the key parameters; the growth rates and fixed points. For a solvable MMPS system, we assess the local behavior of the system around its set of fixed points via a normalization procedure. Further, we present the notion of stability for the normalized system. A case study of the urban railway network substantiates the theoretical results.