Minimal Inputs/Outputs for a Networked System

TL;DR

This paper determines the minimal number of inputs and outputs needed for system controllability and observability based on the maximum geometric multiplicity of the state transition matrix, contrasting with NP-hard sparse matrix problems.

Contribution

It proves the minimal input/output count equals the maximum geometric multiplicity and provides a complete parametrization for such matrices.

Findings

Minimal inputs/outputs equal maximum geometric multiplicity

Contrasts NP-hard sparse matrix problems

Provides complete parametrization for minimal input/output matrices

Abstract

This paper investigates the minimal number of inputs/outputs required to guarantees the controllability/observability of a system, under the condition that its state transition matrix (STM) is prescribed. It has been proved that this minimal number is equal to the maximum geometric multiplicity of the system STM. The obtained conclusions are in sharp contrast to those established for the problems of finding the sparest input/output matrix under the restriction of system controllability/observabilty, which have been proved to be NP-hard, and even impossible to be approximated within a multiplicative factor. Moreover, a complete parametrization is also provided for the input/output matrix of a system with this minimal input/output number.