Stable state and signal estimation in a network context

TL;DR

This paper analyzes algorithms for joint state and input estimation in layered networks, addressing issues with unstable transmission zeros and proposing modifications that ensure asymptotic equivalence and unbiased estimates.

Contribution

It establishes the connection between original and outer factor-based estimation, demonstrating asymptotic equivalence and unbiasedness of the modified algorithms.

Findings

Outer factor estimates asymptotically match original system states.

Input signal estimates have stationary second-order statistics.

Modified algorithms provide unbiased state estimates for control.

Abstract

Power grid, communications, computer and product reticulation networks are frequently layered or subdivided by design. The layering divides responsibilities and can be driven by operational, commercial, regulatory and privacy concerns. From a control context, a layer, or part of a layer, in a network isolates the authority to manage, i.e. control, a dynamic system with connections into unknown parts of the network. The topology of these connections is fully prescribed but the interconnecting signals, currents in the case of power grids and bandwidths in communications, are largely unavailable, through lack of sensing and even prohibition. Accordingly, one is driven to simultaneous input and state estimation methods. We study a class of algorithms for this joint task, which has the unfortunate issue of inverting a subsystem, which if it has unstable transmission zeros leads to an unstable and unimplementable estimator. Two modifications to the algorithm to ameliorate this problem were recently proposed involving replacing the troublesome subsystem with its outer factor from its inner-outer factorization or using a high-variance white signal model for the unknown inputs. Here, we establish the connections between the original estimation problem for state and input signal and the estimates from the algorithm applied solely to the outer factor. It is demonstrated that the state of the outer factor and that of the original system asymptotically coincide and that the estimate of the input signal to the outer factor has asymptotically stationary second-order statistics which are in one-to-one correspondence with those of the input signal to the original system, when this signal is itself stationary. Thus, the simultaneous input and state estimation algorithm applied just to the outer factor yields an unbiased state estimate for control and the statistics of the interface signals.