Input and State Observability of Network Systems With Time-Varying Topology

Abstract

We aim to study the problem of reconstructing the initial state as well as the sequence of unknown inputs [input and state observability (ISO)] for linear network systems having time-varying topology. Evolution of such systems can be represented by a collection of graphs $\lbrace \mathcal {G}_{k}\rbrace$ . We find conditions under which the system with a pattern of fixed zeros imposed by $\lbrace \mathcal {G}_{k}\rbrace$ is ISO: 1) for almost all choices of edge weights in $\lbrace \mathcal {G}_{k}\rbrace$ (structural ISO) and 2) for all nonzero choices of edge weights in $\lbrace \mathcal {G}_{k}\rbrace$ (strongly structural ISO). We introduce two suitable descriptions of the whole collection of graphs $\lbrace \mathcal {G}_{k}\rbrace$ called the dynamic graph and dynamic bipartite graph. Two equivalent characterizations of structural ISO are then stated in terms of the existence of linking and matching of a suitable size in the dynamic graph and in the dynamic bipartite graph, respectively. For strongly structural ISO, we provide a sufficient condition and a necessary condition, both concerning the existence of a uniquely restricted matching of a suitable size in the dynamic bipartite graph and in a subgraph of it. When there is no direct feedthrough of the input on the measurements, the two conditions can be merged to give rise to a necessary and sufficient condition.