Minimal Inputs/Outputs for a Networked System

This letter investigates the minimal number of inputs/outputs required to guarantee system controllability/observability, under the condition that its state transition matrix (STM) is prescribed. A closed-form solution is derived. It has been proven that this minimal number is equal to the maximum geometric multiplicity of the STM, that is, the maximum of the dimensions of its eigenspaces. The obtained conclusions are in sharp contrast to those established for the problem of finding the sparest input/output matrix with a fixed number of inputs/outputs under the same restrictions. The latter has been proven to be NP-hard, and can only be approximated within a multiplicative factor. Moreover, a complete parametrization is also provided for the input/output matrix of a system with its number of inputs/outputs not smaller than this minimal value.