Scale-Free Estimation of the Average State in Large-Scale Systems

Abstract

This paper provides a computationally tractable necessary and sufficient condition for the existence of an average state observer for large-scale linear time-invariant (LTI) systems. Two design procedures, each with its own significance, are proposed. When the necessary and sufficient condition is not satisfied, a methodology is devised to obtain an optimal asymptotic estimate of the average state. In particular, the estimation problem is addressed by aggregating the unmeasured states of the original system and obtaining a projected system of reduced dimension. This approach reduces the complexity of the estimation task and yields an observer of dimension one. Moreover, it turns out that the dimension of the system also does not affect the upper bound on the estimation error.

Subgraph Detection for Average Detectability of LTI Systems

“…Based on the recently proposed notion of average detectability [9], [10], we proposed three methods to choose which nodes to Fig. 16.…”

“…Proof. Even though this result can be derived as a consequence of the theory in [9], [10], we prefer to include a self-contained proof, which is instructive and will inspire our results in Section 3.…”

“…However in numerous cases, there is no need to reconstruct the state of each node, but only an aggregation of these states. Recently, Niazi et al [9], [10] took advantage of this observation and proposed the notions of average observability and average detectability. These notions refer to the reconstruction of the average state of the unmeasured nodes respectively in closed-loop and in open-loop.…”

Average State Estimation in Large-Scale Clustered Network Systems

“…Moreover, the dimension scales with v, which may yield an observer of very high dimension. Therefore, it is computationally feasible to design Ω of order r − 1 by choosing V 2 ∈ R (r−1)×(r−1) in (11) and (15) such that lim sup t→∞ ẽ ζ (t) is minimum, [34], under the assumption that σ(t) is bounded for all t ≥ 0.…”

Structure-based Clustering Algorithm for Model Reduction of Large-scale Network Systems

“…Therefore, the clustering problem is formulated as a minimization of the distance from lumpability, which is the difference between the states of the projected system and the reduced system. Such an approach is quite reasonable, for instance, in the estimation of the average states of multiple clusters in a network system, [18]- [20].…”