Graph-Theoretic Methods for Measurement-Based Input Localization in Large Networked Dynamic Systems

Abstract

In this paper we consider the problem of localizing disturbance inputs in first-order linear time-invariant (LTI) consensus networks using measurement-based graph-theoretic methods. We consider every node and edge of the network graph to be characterized with physical weights, and show that the resulting system dynamics can be represented in terms of an asymmetric Laplacian matrix Lm. Assuming the network graph to be divided into p coherent clusters, we next propose an input localization algorithm based on the properties of the weak nodal domains corresponding to the first p − 1 slow eigenvalues of Lm. The algorithm takes in sensor measurements of the states from selected nodes, runs a system identification routine to construct the input-output transfer matrix, and compares the signs of the residues of the component transfer functions to a nominal localization key to determine in which cluster(s) the disturbance input may have entered. We prove that for systems defined over a specific class of graphs, referred to as p-area complete graphs, the localization is unique. We also state the extension of this result for second-order synchronization networks. We illustrate the algorithms by applying them to large-scale power system networks.

Ensuring localizability of node attacks in consensus networks via feedback graph design

“…We find the optimal slow state feedback controller with the nodal domain constraints by solving (14). We choose ψ 2 = [+, +, −] T and use the constraints (15). Since the problem is still convex, we can use convex optimization packages such as cvx to solve the problem.…”

“…This is important because it means that we can efficiently guarantee the localizability of disturbances in three-area path networks by designing the aggregate edge weights according to (15). The next section illustrates these results with a wide-area control design for a three-area 30-node network.…”

“…In our previous work [1], [15] we developed graphtheoretic methods to localize the cluster in which a disturbance inputũ(t) enters whenū s (t) = 0. In this case, assuming the disturbance to be impulsive, the transfer function (TF) from inputũ(t) in area i to output y j (t) = e j x(t), j ∈ V j , i.e., any node in the j th area, can be written as…”

Location methods of oscillation sources in power systems: a survey

“…A graph-theoretic and measurement-based method is proposed in [69] for locating the input disturbance in large networked dynamic systems. This method first applies a system identification routine to construct the input-output transfer matrix and calculate the array of nominal localization keys.…”

Feedback through graph motifs relates structure and function in complex networks

“…To develop a concrete theory, we focus on the input-output properties of networks containing linear time-invariant (LTI) nodes. Such LTI networks are rooted in an extensive literature in control theory [22], with a broad range of applications including: consensus and cooperation in networked multiagent systems [23,24], fault detection and isolation [25], input localization [26], optimal control [27], neuronal and regional circuits in the brain [28][29][30]. Further, such linear models have been used to describe nonlinear systems around a steady state or periodic orbit [14,26,27,31].…”