Théorême de Cayley-Hamilton dans les dioïdes et application à l’étude des sytèmes à évènements discrets

Abstract

Discrete-event systems, when studied from a control-theorist’s point of view, can be represented by a linear dynamic system in the so-called max-algebra, or dioïd.

Some methods used in the usual linear-system theory still work in this algebra: z-transform, duality,... Problems arise when trying to reduce the state-dimension, or to define a canonical state-representation. This is due to the lack of an adequate theory of the rank, for matrixes, families of vectors, or linear operators in this algebra.

The Cayley-Hamilton theorem, which is a consequence only of combinatorial properties of matrix-calculus, is quite easy to prove in the max-algebra. Thus, a recurrent equation can be defined, which is satisfied by the transfer function of the system.

A conjecture is proposed:

In the max-algebra, a necessary condition for the state-representation of a SISO linear system to be minimal, is:

All non-decreasing solutions of the recurrent equation, deduced from the state-representation by applying the Cayley-hamilton theorem, have the same asymptotic growth-rate.