Stable State Estimation in a Network Context

Power grid, communications, computer and product reticulation networks are frequently layered or subdivided by design. The OSI seven-layer computer network model and the electrical grid division into generation, transmission, distribution and associated markets are cases in point. 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 are largely unavailable, through lack of sensing and even prohibition. Accordingly, one is driven to simultaneous input and state estimation methods. This is the province of this paper, guided by the structure of these network problems. We study a class of algorithms for this joint task, which if the system has transmission zeros outside the unit circle leads to an unstable and unimplementable estimator. Two modifications to the algorithm to ameliorate this problem were recently proposed involving (a) replacing the troublesome subsystem with its outer factor from its inner-outer factorization or (b) using the Kalman filter for a high-variance white signal model for the unknown inputs. The outer factor has only stable transmission zeros and so is stably invertible. The Kalman filter is stable by design. Here, we establish the connections between the original estimation problem for state and input signal and the outputs/estimates from the algorithm applied solely to the outer factor. We further show that both fixes coincide. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)