Functional observability and target state estimation in large-scale networks

The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that a targeted subset of state variables can be reconstructed from the available measurements. Here, we develop a graph-based theory of functional observability, which leads to highly scalable algorithms to 1) determine the minimal set of required sensors and 2) design the corresponding state observer of minimum order. Compared with the full-state observer, the proposed functional observer achieves the same estimation quality with substantially less sensing and fewer computational resources, making it suitable for large-scale networks. We apply the proposed methods to the detection of cyberattacks in power grids from limited phase measurement data and the inference of the prevalence rate of infection during an epidemic under limited testing conditions. The applications demonstrate that the functional observer can significantly scale up our ability to explore otherwise inaccessible dynamical processes on complex networks.

Parametric design to reduced-order functional observer for linear time-varying systems

This article studies the parametric design of reduced-order functional observer (ROFO) for linear time-varying (LTV) systems. Firstly, existence conditions of the ROFO are deduced based on the differentiable nonsingular transformation. Then, depending on the solution of the generalized Sylvester equation (GSE), a series of fully parameterized expressions of observer coefficient matrices are established, and a parametric design flow is given. Using this method, the observer can be constructed under the expected convergence speed of the observation error. Finally, two numerical examples are given to verify the correctness and effectiveness of this method and also the aircraft control problem.

Functional observer-based feedback controller for ball balancing table

The two degree of freedom ball balancing table (BBT) is a well-known didactic tool used to evaluate the effectiveness and performances of many control algorithms for dynamic systems. The present paper proposes to control the ball position of the BBT system via a linear feedback controller based on a functional observer. The parameters of the linear functional observer are determined by applying the direct method which requires neither a Sylvester equation resolution nor canonical transformations. The use of a digital controller has motivated the elaboration of the equations in the discrete time case. In this work, the BBT is tested in real-time to evaluate the proposed controller performances when stabilizing a ball on a reference point. This paper is a continuity of the previous work [12], in which only simulation results have been carried out.