On the Genericity Properties in Distributed Estimation: Topology Design and Sensor Placement

In this paper, we consider distributed estimation of linear, discrete-time dynamical systems monitored by a network of agents. We require the agents to exchange information with their neighbors only once per dynamical system time-scale and study the network topology sufficient for distributed observability. To this aim, we provide a novel measurement-based agent classification: Type- α,β, and γ, which leads to the construction of specific graph topologies:
G
_α
and
G
_β
. In particular, in
G
_α
, every Type-α agent has a direct connection to every other agent, whereas, in
G
_β
, every agent has a directed path to every Type-β agent. With the help of these constructs, we formulate an estimator where measurement and predictor-fusion are implemented over
G
_α
and
G
_β
, respectively, and show that the proposed scheme leads to distributed observability, i.e., observability of the distributed estimator. In order to characterize the estimator further, we show that Type-α agents only exist in systems with
S
-rank (maximal rank of zero/non-zero pattern) deficient system matrices. In other words, systems with full
S
-rank matrices only have Type-β agents, and thus, a strongly-connected (agent) network is sufficient for full
S
-rank systems-by the definition of
G
_β
above; however strong-connectivity is not necessary, i.e., there exist weakly-connected networks that result in distributed observability. Furthermore, we show that for
S
-rank deficient systems, measurement-fusion over
G
_α
is required, and predictor-fusion alone is insufficient. The approach taken in this paper is structural, i.e., we use the concept of structured systems theory and generic observability to derive the results. Finally, we provide an iterative method to compute the local estimator gain at each agent once the observability is ensured using the aforementioned construction.