Clustered model reduction of positive directed networks

Takayuki Ishizaki (1) , Kenji Kashima (2) , Antoine Girard (3) , Jun-Ichi Imura (1) , Luonan Chen (4) , Kazuyuki Aihara (5)

1 TITECH - Tokyo Institute of Technology [Tokyo]
2 Graduate School of Information Science and Technology [Osaka]
3 CASYS - Calculs Algébriques et Systèmes Dynamiques
4 LRI - Laboratoire de Recherche en Informatique
5 IIS - Institute of Industrial Science

This paper proposes a clustered model reduction method for semistable positive linear systems evolving over directed networks. In this method, we construct a set of clusters, i.e., disjoint sets of state variables, based on a notion of cluster reducibility, defined as the uncontrollability of local states. By aggregating the reducible clusters with aggregation coefficients associated with the Frobenius eigenvector, we obtain an approximate model that preserves not only a network structure among clusters, but also several fundamental properties, such as semistability, positivity, and steady state characteristics. Furthermore, it is found that the cluster reducibility can be characterized for semistable systems based on a projected controllability Gramian that leads to an a priori H2-error bound of the state discrepancy caused by aggregation. The efficiency of the proposed method is demonstrated through an illustrative example of enzyme-catalyzed reaction systems described by a chemical master equation. This captures the time evolution of chemical reaction systems in terms of a set of ordinary differential equations.

Mots clés: Positive linear systems; Model reduction; Network clustering; Chemical master equations

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

Takayuki Ishizaki, Kenji Kashima, Antoine Girard, Jun-Ichi Imura, Luonan Chen, et al.. Clustered model reduction of positive directed networks. Automatica, 2015, 59, pp.238-247. ⟨10.1016/j.automatica.2015.06.027⟩. ⟨hal-01276233⟩