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Monotone linear dynamical systems over dioids. (English) Zbl 1059.93093
Benvenuti, Luca (ed.) et al., Positive systems. Proceedings of the first multidisciplinary international symposium on positive systems: Theory and applications (POSTA 2003), Rome, Italy, August 28–30, 2003. Berlin: Springer (ISBN 3-540-40342-6/pbk). Lecture Notes in Control and Information Sciences 294, 39-45 (2003).
Summary: Linear systems over naturally ordered dioids are other kinds of positive systems than the usual ones over the semiring $$(\mathbb{R}_+, +,\cdot)$$. In this short paper we study some monotonicity concepts of linear systems over dioids inspired by results on monotonicity of Markov chains which are also particular cases of positive systems. We derive a necessary and sufficient condition for monotonicity in a simple case which requires strong assumptions on the dioids (lattice distributivity and invertibility of the multiplication law). The result suggests links between monotonicity and positive invariance, which plays an important role in control theory, and also with aggregation problems of linear systems (i.e., conditions for the existence of aggregated variables and their linear dynamics from which the complete behavior can be retraced).
For the entire collection see [Zbl 1031.93004].

MSC:
 93C65 Discrete event control/observation systems