On the Design of Resilient Distributed Single Time-Scale Estimators: A Graph-Theoretic Approach

TL;DR

This work addresses resilient distributed state estimation for interconnected systems by leveraging graph-theoretic concepts. It introduces a single time-scale distributed estimator that avoids inner consensus loops and can maintain Schur stability under up to $q$ sensor/node and $q$ link failures, reducing communication relative to double time-scale methods. The framework relies on $q$-node and $q$-link connectivity, observational equivalence within parent SCCs, and the Kronecker product graph observability to ensure distributed observability with a block-diagonal gain $K$ obtained via LMIs. The approach enables scalable, robust estimation without requiring local observability at every sensor and is applicable to large-scale settings such as sensor networks, distributed target tracking, and intelligent transportation systems.

Abstract

Distributed estimation in interconnected systems has gained increasing attention due to its relevance in diverse applications such as sensor networks, autonomous vehicles, and cloud computing. In real practice, the sensor network may suffer from communication and/or sensor failures. This might be due to cyber-attacks, faults, or environmental conditions. Distributed estimation resilient to such conditions is the topic of this paper. By representing the sensor network as a graph and exploiting its inherent structural properties, we introduce novel techniques that enhance the robustness of distributed estimators. As compared to the literature, the proposed estimator (i) relaxes the network connectivity of most existing single time-scale estimators and (ii) reduces the communication load of the existing double time-scale estimators by avoiding the inner consensus loop. On the other hand, the sensors might be subject to faults or attacks, resulting in biased measurements. Removing these sensor data may result in observability loss. Therefore, we propose resilient design on the definitions of $q$-node-connectivity and $q$-link-connectivity, which capture robust strong-connectivity under link or sensor node failure. By proper design of the sensor network, we prove Schur stability of the proposed distributed estimation protocol under failure of up to $q$ sensors or $q$ communication links.

Figures (8)

• Figure 1: This figure illustrates two main strategies for distributed estimator/observer design in the literature. (Left) Double time-scale method with many steps of averaging (consensus) and communications between every two consecutive steps of system dynamics. The number of consensus steps is generally more than the diameter of the network. (Right) Single time-scale method (adopted in this work) with only one step of averaging (consensus) and communication, which imposes less communication and processing load on sensors/agents.
• Figure 2: This figure shows an example $3$-node-connected and $3$-link-connected graph. in this example, by removing any set of (up to) $3$ links, the remaining graph is still strongly-connected; for example, by removing the red links the left graph holds its strong-connectivity. Similarly, by removing any set of (up to) $3$ nodes, the remaining reduced-order graph is still strongly-connected; for example, by removing the red nodes (and their links) the right graph holds its strong-connectivity. Although this example shows an undirected network, the connectivity notions also hold for directed networks.
• Figure 3: (Left) Example system graph $\mathcal{A}$ with one set of parent SCC and observationally equivalent outputs $y_1$ and $y_2$. (Right) The associated system matrix $A$ and output matrix $C$.
• Figure 4: (Left) This figure shows an example system graph $\mathcal{A}$ with three sets of parent SCCs of size two and a pair of sensor measurements from each parent SCC; recall from Definition \ref{['defn_equiv']} that outputs of the same colour are observationally equivalent. (Right) The example sensor network to monitor this system graph is $2$-node-connected and $2$-link-connected. Thus, the distributed estimation network is resilient to the failure of $1$ sensor node (due to the size of parent SCCs and equivalent set of sensor outputs) or the failure of $2$ communication links.
• Figure 5: The evolution of MSE at six sensors tracking the state of system graph in Fig. \ref{['fig_graph_net']}.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof