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Global and Local Error-Tolerant Decentralized State Estimation under Partially Ordered Observations

Dajiang Sun, Christoforos N. Hadjicostis, Zhiwu Li

TL;DR

The paper tackles current-state estimation in a discrete-event system (DES) operating under potential tampering of observations in a decentralized, DO-based framework. It models errors with error relation matrices $[R]^{c_u}$ for global and local tampering and proposes four synchronizers via two design philosophies: system modification and S-builder-based inference, all governed by an estimation-by-release paradigm. It introduces modified systems $G_g$ and $G_l$, GETO and LETO sequences, and corresponding $E_g$/$E_l$-TS-builders and synchronizers to compute error-tolerant state estimates under bounded costs, with explicit complexity discussions. The framework supports robust, scalable state estimation in distributed DES, with applications to security-aware distributed control, fault diagnosis, and privacy-preserving sensing in cyber-physical systems operating under partially ordered observations.

Abstract

We investigate decentralized state estimation for a discrete event system in a setting where the information received at a coordinator may be corrupted or tampered by a malicious attacker. Specifically, a system is observed by a set of (local) observation sites (OSs) which occasionally send their recorded sequences of observations to the coordinator that is in charge of estimating the system state. The malfunctions and attacks, referred to as errors in this paper, include symbol deletions, insertions and replacements, each of which bears a positive cost. Two types of errors, global errors and local errors, are proposed to describe the impact of errors on decentralized information processing. Global errors occur when all OSs record the same error, while local errors occur when different OSs record different errors. Distinguishing these types of errors is important for a proper design of decentralized information processing (so as to be more resilient and better equipped to handle attacks and failures). For each type of error, we propose two methods to efficiently perform state estimation: one based on appropriately modifying the original system and the other based on inferring the matching behavior of the original system. For each method, we adopt an estimation-by-release methodology to design an algorithm for constructing a corresponding synchronizer for state estimation.

Global and Local Error-Tolerant Decentralized State Estimation under Partially Ordered Observations

TL;DR

The paper tackles current-state estimation in a discrete-event system (DES) operating under potential tampering of observations in a decentralized, DO-based framework. It models errors with error relation matrices for global and local tampering and proposes four synchronizers via two design philosophies: system modification and S-builder-based inference, all governed by an estimation-by-release paradigm. It introduces modified systems and , GETO and LETO sequences, and corresponding /-TS-builders and synchronizers to compute error-tolerant state estimates under bounded costs, with explicit complexity discussions. The framework supports robust, scalable state estimation in distributed DES, with applications to security-aware distributed control, fault diagnosis, and privacy-preserving sensing in cyber-physical systems operating under partially ordered observations.

Abstract

We investigate decentralized state estimation for a discrete event system in a setting where the information received at a coordinator may be corrupted or tampered by a malicious attacker. Specifically, a system is observed by a set of (local) observation sites (OSs) which occasionally send their recorded sequences of observations to the coordinator that is in charge of estimating the system state. The malfunctions and attacks, referred to as errors in this paper, include symbol deletions, insertions and replacements, each of which bears a positive cost. Two types of errors, global errors and local errors, are proposed to describe the impact of errors on decentralized information processing. Global errors occur when all OSs record the same error, while local errors occur when different OSs record different errors. Distinguishing these types of errors is important for a proper design of decentralized information processing (so as to be more resilient and better equipped to handle attacks and failures). For each type of error, we propose two methods to efficiently perform state estimation: one based on appropriately modifying the original system and the other based on inferring the matching behavior of the original system. For each method, we adopt an estimation-by-release methodology to design an algorithm for constructing a corresponding synchronizer for state estimation.
Paper Structure (13 sections, 11 theorems, 24 equations, 6 figures, 1 table, 3 algorithms)

This paper contains 13 sections, 11 theorems, 24 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Model
  4. Decentralized Observation-Based Information Processing
  5. Error Model and its Impacts on Sequences
  6. State Estimation Under Global Errors
  7. DO-E$_g$SE from the Perspective of System Modification
  8. DO-E$_g$SE from the Perspective of S-builder Modification
  9. State Estimation Under Local Errors
  10. DO-E$_l$SE from the Perspective of System Modification
  11. DO-E$_l$SE from the Perspective of S-builder Modification
  12. Discussion and Comparison of two methods
  13. Conclusion

Key Result

Lemma 1

Consider a sequence of events $t$ occurring in system $G$ such that the E$_g$SI-state is $\tau_g$ w.r.t. an ERM $[R]^{c_u}$. Given its modified system $G_g$, the DO-E$_g$SE after the coordinator receives $\tau_g$ is given by

Figures (6)

  • Figure 1: (a) NFA model $G$ where $\Sigma_1=\{\alpha_{12},\beta_{13}\}$, $\Sigma_2=\{\alpha_{12},\sigma_2\}$, and $\Sigma_3=\{\beta_{13}, \gamma_3\}$ discussed in Example \ref{['E1-withERM']} and (b) modified system $G_g$ w.r.t. the ERM $[R]^2$ (dotted lines used to indicate that these transitions are the results of error actions).
  • Figure 2: The synchronizer of the modified system $G_g$ w.r.t. the given $\tau_{g}$ and $Q_0=\{q_0\}$, in which $c_g(\tau_g)=\operatorname{UR}(Q_0\times\{0\})=\{(q_0,0),(q_1,0)\}$. The corresponding set of state estimates is displayed next to each SI-state.
  • Figure 3: The E$_g$T-synchronizer of the system $G$ w.r.t. the given $\tau_{g}$ and $Q_0$. The state estimate is displayed next to each state, where $\widetilde{T}_{g,0}=\widetilde{\tau}_g$ is the initial state with $\widetilde{c}_g(\widetilde{\tau}_g)=\operatorname{UR}(Q_0)=\{q_0,q_1\}$ and $\widetilde{T}_{g,e}=\{\widetilde{\tau}_{18}, \widetilde{\tau}_{19}\}$ is the set of marked and ending states. The states in each layer are listed to the side.
  • Figure 4: (a) The observation automaton $G_o$ of $G$ in Fig. \ref{['E1-withERM']}(a) and (b) the modified system $G_l$ w.r.t. the ERM $\{[R_i]\}^2_{i\in\{1,2,3\}}$ in Example \ref{['EX-local-1']} (dotted lines are used to indicate that these transitions are the result of error actions; to keep the diagram concise, the transitions in the modified system are represented by labels A-E, shown on the right).
  • Figure 5: The E$_l$-synchronizer of the modified system $G_l$ w.r.t. the given $\tau_{l}$ and $\operatorname{UR}(Q_0\times\{0\})=\{(q_0,0),(q_1,0),(q_4,1)\}$. For the sake of brevity, in this diagram, we write the state with cost $(q,c)$ in the form of $q^c$. The corresponding set of state estimates is displayed next to each E$_l$SI-state and states in each layer are listed at the bottom of the figure.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Remark 1
  • Example 1
  • Lemma 1
  • Theorem 1
  • Example 2
  • Definition 3
  • Lemma 2
  • Definition 4
  • ...and 24 more