MM functions, formed by finite composition of the operators min, max and translation, represent discrete-event systems involving disjunction, conjunction and delay. The paper shows how they may be formulated as homogeneous rational algebraic functions of degree one, over (max, +) algebra, and reviews the properties of such homogeneous functions, illustrated by some orbit-stability problems.

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule ${𝒮}^n$ over a semiring $𝒮$ is rational if it has a generating family that is a rational subset of ${𝒮}^n$, ${𝒮}^n$ being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules...

This paper discusses the properties of reachability and observability for linear systems over the max-plus algebra. Working in the event-domain, the concept of asticity is used to develop conditions for weak reachability and weak observability. In the reachability problem, residuation is used to determine if a state is reachable and to generate the required control sequence to reach it. In the observability problem, residuation is used to estimate the state. Finally, as in the continuous-variable...

Repetitive processes constitute a distinct class of 2D systems, i.e., systems characterized by information propagation in two independent directions, which are interesting in both theory and applications. They cannot be controlled by a direct extension of the existing techniques from either standard (termed 1D here) or 2D systems theories. Here we give new results on the design of physically based control laws. These results are for a sub-class of discrete linear repetitive processes with switched...