Distributed Error-Identification and Correction using Block-Sparse Optimization

TL;DR

This paper tackles distributed fault-detection, identification, and reconstruction (FDIR) in networks of multi-agent Cyber-Physical Systems with nonlinear inter-agent measurements. It reformulates FDIR as recovering a block-sparse error vector $\mathbf x$ from nonlinear measurements via a hybrid of sequential convex programming (SCP) and the alternating direction method of multipliers (ADMM), enabling distributed processing without anchors. The authors connect the problem to rigidity theory, characterize the search-space as a lower-dimensional submanifold, and show that sparsity regularization can yield accurate fault localization and error recovery through a scalable, network-local algorithm. They develop a two-loop algorithm (outer SCP and inner ADMM) with a thresholding interpretation, and validate it on a 20-UAV scenario with distance measurements, demonstrating correct fault identification and robust reconstruction. The work provides a practical, scalable framework for anchor-free FDIR in nonlinear measurement networks, with potential extensions to dynamic and Bayesian settings.

Abstract

The conventional solutions for fault-detection, identification, and reconstruction (FDIR) require centralized decision-making mechanisms which are typically combinatorial in their nature, necessitating the design of an efficient distributed FDIR mechanism that is suitable for multi-agent applications. To this end, we develop a general framework for efficiently reconstructing a sparse vector being observed over a sensor network via nonlinear measurements. The proposed framework is used to design a distributed multi-agent FDIR algorithm based on a combination of the sequential convex programming (SCP) and the alternating direction method of multipliers (ADMM) optimization approaches. The proposed distributed FDIR algorithm can process a variety of inter-agent measurements (including distances, bearings, relative velocities, and subtended angles between agents) to identify the faulty agents and recover their true states. The effectiveness of the proposed distributed multi-agent FDIR approach is demonstrated by considering a numerical example in which the inter-agent distances are used to identify the faulty agents in a multi-agent configuration, as well as reconstruct their error vectors.

Figures (6)

• Figure 1: Two configurations of a rigid graph that produce the same measurement vector $\mathbf y$, but are not related to each other via rigid translations or rotations, are said to be flip ambiguous.
• Figure 2: The intersection of a generic two-dimensional manifold (yellow surface) with the set of $2$-sparse vectors is a one-dimensional manifold (red curves). Its intersection with the set of $1$-sparse vectors is a zero-dimensional manifold (red points).
• Figure 3: The black dots correspond to the true states of the multi-agent system, whereas the yellow discs represent the agents' estimated states. The red arrows depict the non-zero blocks of the reconstructed error vector after $60$ iterations (in total) of the ADMM loop.
• Figure 4: The accuracy of the reconstructed error vector at each agent. The red lines correspond to the agents in $\mathcal{D}$ (i.e., the faulty agents), and the vertical grey lines indicate the iterations where the constraint was re-linearized.
• Figure 5: The fault-detection threshold $1/\rho$ (blue dashed line) and the residual energy $\|\mathbf A_i^\top \mathbf b_i\|$ are visualized for each agent; the definitions of $\mathbf A_i$ and $\mathbf b_i$ are given in (\ref{['eq:A_i-b_i']}). The red lines correspond to the agents in $\mathcal{D}$ (i.e., the faulty agents), and the vertical grey lines indicate the iterations where the constraint was re-linearized.

Theorems & Definitions (18)

• Remark 1
• Remark 2
• Lemma 1: Regular Level Set Theorem lee2012
• Theorem 1: Characterization of the Search-Space
• proof
• Remark 3: Regular Values are Almost-Everywhere
• Remark 4
• Remark 5
• Proposition 1
• proof