On max-plus linear dynamical system theory: the observation problem

Vinicius-Mariano Gonçalves (1, 2) , Carlos-Andrei Maia (2) , Laurent Hardouin (1)

1 LARIS - Laboratoire Angevin de Recherche en Ingénierie des Systèmes
2 DEE - UFMG - Departamento de Engenharia Elétrica [Minas Gerais]

In this paper, we are interested in the general problem of estimating a linear function of the states for a given Max-Plus linear dynamical system. More precisely, using only the current and past inputs/outputs of the system, we want to construct a sequence that converges in a finite number of steps to the value given by a linear function of the states, for all initial conditions of the system. We provide necessary and sufficient conditions to solve this general problem. We also define and study a Max-Plus version of the well-known Luenberger observer, which is a subclass of the general problem that we are interested in, and we also provide necessary and sufficient conditions to solve this particular problem of observer synthesis. Finally, we show that there are important connections between results in the Max-Plus domain and associated results in the standard linear systems theory.

Mots clés: tropical algebra; Observer; Max-plus algebra; control

Domaines: Systèmes et contrôle [cs.SY]; Optimisation et contrôle [math.OC]

Vinicius-Mariano Gonçalves, Carlos-Andrei Maia, Laurent Hardouin. On max-plus linear dynamical system theory: the observation problem. Automatica, 2019, 107 (1), pp.103-111. ⟨10.1016/j.automatica.2019.05.026⟩. ⟨hal-02298411⟩