On Control Networks Over Finite Lattices

TL;DR

This paper develops an algebraic framework using semi-tensor products to model, analyze, and control networks over finite lattices, with algorithms for structure recovery and verification.

Contribution

It introduces a novel algebraic state space approach for control networks over finite lattices and algorithms for lattice structure identification and verification.

Findings

Algebraic state space representation for networks over finite lattices

Algorithms for recovering lattice structures from network data

Numerical examples demonstrating the effectiveness of the methods

Abstract

The modeling and control of networks over finite lattices are studied via the algebraic state space approach. Using the semi-tensor product of matrices, we obtain the algebraic state space representation of the dynamics of (control) networks over finite lattices. Basic properties concerning networks over sublattices and product lattices are investigated, which shows the application of the analysis of lattice structure in the model reduction and control design of networks. Then algorithms are developed to recover the lattice structure from the structural matrix of a network over a lattice, and to construct comparability graphs over a finite set to verify whether a multiple-valued logical network is defined over a lattice. Finally, numerical examples are presented to illustrate the results.