On Networks Over Finite Rings

TL;DR

This paper introduces a framework for analyzing control networks over finite rings using semi-tensor product techniques, including a decomposition theorem that simplifies network analysis by breaking networks into sub-networks over factor rings.

Contribution

It develops a novel algebraic approach for control networks over finite rings, including a decomposition principle and representation theorem, extending existing methods from logical networks.

Findings

Decomposition theorem (DP) enables analysis of networks over product rings.

Control properties can be derived from sub-networks over factor rings.

Linear network control over product rings is thoroughly examined.

Abstract

A (control) network over a finite ring is proposed. Using semi-tensor product (STP) of matrices, a set of algebraic equations are provided to verify whether a finite set with two binary operators is a ring. It is then shown that the STP-based technique developed for logical (control) networks are applicable to (control) networks over finite rings. The sub-(control) network over an ideal of the bearing ring is revealed. Then the product ring is proposed and the (control) network over product ring is investigated. As a key result, the decomposition theorem, called the Decomposition Principle (DP), is proposed, which shows a (control) network over a product ring is decomposable into sub-(control)networks over each factor rings, which makes the (control) properties of a network can be revealed by its factor sub-networks over factor rings. Using DP, the control problems of a control network are investigated via its sub control networks. Particularly, using DP, the control of linear networks over product rings is discussed in detail. Finally, the representation theorem is presented, which shows that each (control) network over a finite set can be expressed as a (control) network over a product ring.