Strong structural input and state observability of linear time-invariant systems: Graphical conditions and algorithms

Abstract

The paper studies input and state observability (ISO) of discrete-time linear time-invariant network systems whose dynamics are a↵ected by unknown inputs. More precisely, we aim at reconstructing the initial state and the sequence of unknown inputs from the system outputs, and we will use the term ISO when the input reconstruction is possible with delay one, namely the inputs up to time k 1 and the states up to time k can be obtained from the outputs up to time k, while the term unconstrained ISO will refer to the case where there is some arbitrary delay in the input reconstruction. We focus on the problem of s-structural ISO (resp. s-structural unconstrained ISO) wherein the objective is to find conditions such that for all system matrices that carry the same network structure, the resulting system is ISO (resp. unconstrained ISO). We provide first a graphical characterization for s-structural unconstrained ISO, and subsequently, su cient conditions and necessary conditions for s-structural ISO. For the latter, under the assumption of zero feedthrough, these conditions coincide and characterise ISO. The conditions presented are in terms of existence of suitable uniquely restricted matchings in bipartite graphs associated with the structured system. In order to test these conditions, we present polynomial-time algorithms. Finally, we discuss an equivalent reformulation of the main conditions in terms of coloring algorithms as in the literature of zero forcing sets.