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Distributed Algorithms for Filtering, Estimation, and Fault Detection over Cyber-Physical-Systems: A Tutorial and Survey

Mohammadreza Doostmohammadian, Mahdi Shamsi, Hadi Zayyani, Nader Meskin, Hamid R. Rabiee, Sergio Pequito, Usman A. Khan

arXiv:2609.29640v1eess.SYcs.SYeess.SPmath.OC

TL;DR

Existing surveys cover distributed optimization but not the distinct observability, faulty-data, and robustness challenges of estimation and fault detection. This survey synthesizes foundational theory, distributed algorithms, and applications, concluding with a unified perspective on monitoring and control in large-scale systems.

  • Problem

    Existing surveys cover distributed optimization but not the distinct modelling, observability, and robustness challenges of decentralized state and parameter estimation and fault or attack detection.

    Source: §1 · p. 7

  • Method

    The survey integrates theoretical foundations with analyses of estimation, filtering, and fault-detection methods and their practical applications.

    Source: §1 · p. 7, §6 · p. 91

  • Results

    The survey provides a unified perspective on how distributed algorithms support robust monitoring and control of complex, large-scale systems.

    Source: §6 · p. 91

  • Takeaways & Limitations

    The review identifies observability analysis, consensus under faulty data, and robustness guarantees as distinct challenges for decentralized estimation and detection.

    Source: §1 · p. 7

    Distributed particle filtering requires more computation and communication than EKF- and UKF-based methods.

    Source: Distributed Particle Filtering · p. 59

Abstract

from arXiv · show

This survey provides overview of distributed estimation, filtering, and fault detection techniques in the context of cyber-physical systems (CPS). To establish a strong foundation, we first define essential aspects of linear dynamical systems and related graph theoretic concepts, emphasizing observability conditions that are key for local state estimation. After discussing consensus algorithms, this survey highlights single-time and double-time-scale consensus-based estimation and filtering approaches. We provide a detailed comparative analysis of these methodologies, examining their computational requirements, communication overhead, and performance characteristics in resource-constrained environments. We further explore different diffusion-based estimation techniques and observationally redundant designs to enhance resilience and robustness against failures and adversarial attacks. In addition, we investigate distributed fault detection methods that enable local isolation of faults over large-scale CPS. We present both stateless and stateful detection mechanisms, along with threshold-based techniques that balance detection accuracy and false alarm rates. These approaches are essential for maintaining system integrity and preventing cascading failures in critical infrastructure. This survey concludes with an exploration of diverse real-world applications. We examine implementation challenges and algorithm adaptations in smart grid and power networks, social systems, target tracking and localization, and intelligent transportation systems. By bridging theoretical insights with practical applications, this survey and tutorial provides valuable understanding and research directions in the field of distributed algorithm design for CPS, offering both newcomers and experienced researchers a comprehensive resource for addressing current challenges and future opportunities.

Highlights

The section establishes graph-based foundations for distributed CPS algorithms, including observability, consensus, and Laplacian properties. It then surveys consensus- and diffusion-based methods, fault detection, and applications.

Distributed Algorithms for Filtering, Estimation, and Fault Detection over Cyber-Physical-Systems: A Tutorial and Survey

  • The survey classifies consensus-based filtering into single- and double-time-scale approaches and compares their computational and communication requirements.It also discusses their applicability in resource-constrained environments.Source: Distributed Algorithms for Filtering,… · p. 2, §1 · p. 7
  • The survey analyzes diffusion-based filtering convergence and robustness, including performance in dynamic network topologies with varying connectivity.It also reviews observationally redundant designs for resilience against failures and disturbances.Source: Distributed Algorithms for Filtering,… · p. 2, §1 · p. 7
  • The survey examines distributed fault-detection methods, including threshold-based techniques for real-time fault identification and isolation.It highlights observer-based detection and isolation of faulty nodes to help prevent cascading failures.Source: Distributed Algorithms for Filtering,… · p. 2, §1 · pp. 7–8
  • The survey examines applications in smart grids and power networks, social systems, target tracking and localization, and intelligent transportation systems.It presents these applications as a bridge between theoretical approaches and practice.Source: Distributed Algorithms for Filtering,… · p. 2, §1 · p. 8

1. Introduction

  • The survey addresses estimation and detection challenges distinct from optimization, including reconstructing unobservable states and distinguishing system dynamics from malicious attacks.It identifies observability analysis, consensus under faulty data, and robustness guarantees as specific challenges for decentralized sensor networks.Source: §1 · p. 7

2.1. Linear Dynamical Systems

  • Linear dynamical systems provide the physical-process model, while system digraphs encode state and output relationships through the matrices’ zero/nonzero patterns.These graph representations support analysis of structural properties relevant to distributed algorithm design.Source: §2.1 · pp. 9–10

2.2. Observability Conditions

  • A linear system is observable when its observability matrix has full rank; otherwise, some states cannot be inferred from output measurements.The full-rank condition is evaluated relative to the dimension of the state vector.Source: §2.2 · p. 11

2.2.1. Structural Observability

  • Structural observability assesses state inference from matrix patterns and graph connectivity, making it useful when system parameters are uncertain.A structural observability matrix must have full structural rank; each state must also have a path to an output node.Source: §2.2.1 · p. 12
  • Each state must belong to a cycle or lie on a path to an output for structural observability; DFS/BFS checks paths, while structural-rank algorithms check cycles.Results for structural controllability can be adapted by reversing path or link directions.Source: §2.2.1 · p. 13

2.3. Consensus Algorithms

  • Consensus has agents iteratively update states from neighbors’ information to reach agreement, with delays, asynchronous updates, packet drops, and link failures posing communication challenges.Consensus may use stochastic weights, including Metropolis-Hastings or Wasserstein average consensus; algorithms also vary in convergence time, synchrony, and linearity.Source: §2.3 · pp. 13–15

2.4. Algebraic Graph Theory

  • The Laplacian encodes network topology, and its second-smallest eigenvalue λ2 indicates connectivity and consensus speed: λ2 > 0 means connected, while λ2 = 0 indicates disconnection.For connected graphs, the Laplacian has one zero eigenvalue; larger λ2 is associated with faster convergence.Source: §2.4 · pp. 16–17
  • Graph connectivity is the minimum number of links whose deletion disconnects a graph; higher algebraic connectivity suggests greater resilience to failures and more robust consensus dynamics.A graph is k-connected when at least k link deletions are required to disconnect it.Source: §2.4 · p. 17

3. Distributed Filtering and Estimation

This section surveys distributed filtering and estimation methods, relating consensus, diffusion, redundant sensing, and nonlinear extensions to trade-offs in communication, observability, and accuracy. It contrasts single- and double-time-scale consensus and details the observability and gain-design constraints of a representative single-time-scale estimator.

3. Distributed Filtering and Estimation

  • The survey covers consensus- and diffusion-based estimation, redundant sensing, nonlinear filtering, and trade-offs among communication efficiency, observability, and accuracy.It also discusses distributed observers alongside estimation-based methods as a coherent framework.Source: §3 · p. 18

3.1. Consensus-based Techniques

  • Single time-scale methods use one consensus update between system samples, minimizing communication but potentially requiring stronger observability; double time-scale methods use repeated inner-loop updates, typically improving accuracy at higher communication cost.Figure 4 depicts the one-update and repeated-update schedules.Source: §3.1 · p. 19, Figure 4 · p. 19

3.1.1. Single Time-Scale Algorithms

  • A representative single time-scale estimator alternates consensus on a-priori estimates with a measurement update using shared measurements and local gains.The single consensus step occurs between consecutive system-dynamics updates; the measurement step is also called an innovation update in some literature.Source: §3.1.1 · p. 22
  • A block-diagonal gain matrix preserves distributed design but may require specialized cone-complementarity optimization with linear matrix inequalities; stability is equivalent to observability of (W ⊗ A, DC).The stability condition is stated for equation (12).Source: §3.1.1 · p. 23
  • Single-time-scale agents may lack enough local measurements for observability, making communication-network design important for restoring distributed observability.Proposed approaches include sharing measurements and predictions or using structural rank to assess communication needs; structurally rank-deficient systems require more extensive sharing.Source: §3.1.1 · pp. 23–24

Literature Review on Single Time-Scale Approaches

The survey reviews single-time-scale estimation innovations and then contrasts double-time-scale methods, which exchange heavier communication for relaxed observability requirements and more accurate estimates.

Literature Review on Single Time-Scale Approaches

  • Finite-time fusion removes asymptotic convergence limits, while resilient H∞ filtering and attack-mitigation extensions address disturbances, uncertainty, and attacks.These filtering approaches minimize worst-case estimation error.Source: Literature Review on Single Time-Scale… · pp. 24–25
  • Event-triggered transmission reduces communication overhead, while delay-tolerant methods compensate for variable latency using timestamps, buffers, and prediction.The approaches target infrequent data changes, limited bandwidth, and wireless sensor networks with variable latency.Source: Literature Review on Single Time-Scale… · p. 25
  • Filters adapting to link or node failures and adaptive consensus [153, 154] support estimation as network topology changes.These methods address unreliable communications and partial network degradation.Source: Literature Review on Single Time-Scale… · p. 25
  • Heterogeneity-aware filtering and sensor selection or placement [156, 157, 158] address differing sensor capabilities while balancing estimation accuracy against communication and energy costs.Sensor selection and placement use observability to identify informative sensors.Source: Literature Review on Single Time-Scale… · p. 25

3.1.2. Double Time-Scale Algorithms

  • More than the network diameter’s worth of consensus rounds between measurements relaxes single-time-scale observability constraints and improves estimation accuracy, at higher communication and computation cost.The consensus rounds let information from any agent reach all others, compensating for limited local observability.Source: §3.1.2 · p. 26
  • Double-time-scale filtering separates local prediction from a consensus loop in which agents repeatedly exchange estimates between system updates.The canonical implementation is attributed to Olfati-Saber and He et al..Source: §3.1.2 · pp. 26–27

Observability Advantages of Double Time-Scale Approaches

Double-time-scale consensus can spread measurements across a network to address local observability limits, while its higher communication demands motivate efficiency adaptations. The section also surveys structurally grounded sensor redundancy and diffusion-based estimation.

Observability Advantages of Double Time-Scale Approaches

  • Assuming L ≥ dG consensus iterations between system updates, information can propagate across the network, addressing observability challenges faced by single-time-scale methods.dG is the network diameter; this iteration condition ensures each node’s data reaches every other node during one system time step.Source: Observability Advantages of Double… · p. 27
  • Strong network connectivity is sufficient for observability and error stability in double-time-scale methods, which are more tolerant of network configurations than single-time-scale approaches.Repeated consensus makes each sensor’s information accessible to every other sensor between system updates.Source: Observability Advantages of Double… · pp. 27–28
  • Table 1 compares single- and double-time-scale filters by computation complexity and network-connectivity × communication-rate per sample; the comparison frames communication efficiency against observability guarantees.The passage identifies the comparison dimensions but supplies no cell values.Source: Table 1 · p. 28, Observability Advantages of Double… · p. 28

Literature Survey on Double Time-Scale Techniques

  • Double-time-scale literature includes moving-horizon estimation, event-triggered communication, and filtering adapted to heterogeneous sensor reliability.Event-triggered strategies transmit when changes are significant or estimation errors exceed thresholds, aiming to save communication while preserving performance.Source: Literature Survey on Double Time-Scale… · pp. 28–29
  • Distributed filters maintain functionality despite link failures or packet losses; other approaches address DoS attacks [175, 176, 177], adversarial inputs, and communication delays.Source: Literature Survey on Double Time-Scale… · pp. 29–30

3.1.3. Observationally Redundant Design

  • Observationally redundant design uses structurally or functionally equivalent observations to preserve system observability if individual sensors malfunction or are compromised.The design is based on mathematical equivalence relationships rather than simple sensor replication.Source: §3.1.3 · pp. 30, 31, 33
  • A q-redundant observable design places q+1 outputs in every contraction and parent SCC across q+1 sensors, preserving observability after any q sensor or output failures.Distributed implementation also calls for q-node-connectivity and q-link-connectivity, retaining connectivity after removal of any q−1 nodes or links [191].Source: §3.1.3 · pp. 33–34

3.2. Diffusion-based Techniques

  • Diffusion-based estimation is presented as an adaptive-filtering alternative to consensus, particularly suited to parameter estimation in dynamic environments.The survey examines its theoretical foundations, implementation variants, and performance characteristics.Source: §3.2 · p. 34

From Centralized to Distributed Estimation

This section surveys the progression from local, neighbor-based estimation to diffusion-based distributed estimation and filtering for shared, clustered, and task-specific objectives. It covers design adaptations for communication limits, imperfect data, noise, and attacks, including nonlinear filtering approaches that use approximations for tractable collaboration.

From Centralized to Distributed Estimation

  • Distributed estimation replaces centralized fusion with local processing: nodes use their measurements and immediate-neighbor information, while global estimates emerge through local computation and limited communication.Centralized processing is described as facing scalability and reliability limitations in modern cyber-physical systems.Source: From Centralized to Distributed… · pp. 34–35

Distributed Optimization Framework

  • In the distributed optimization framework, agents use neighborhood-weighted local costs and gradient updates; replacing the unknown optimum with neighbor intermediate estimates makes diffusion implementable and improves stability.Combination weights are constrained so the resulting matrix is right-stochastic, a property described as essential for diffusion convergence.Source: Distributed Optimization Framework · pp. 36–38

3.2.1. Diffusion LMS

  • Both ATC and CTA yield unbiased estimates under standard assumptions, but ATC consistently achieves lower mean-square error through adaptation before spatial mixing.CTA remains useful when architectural simplicity, reduced communication, or privacy is prioritized over optimal MSE performance.Source: §3.2.1 · p. 40

3.2.2. Adaptive Diffusion Strategies

  • Adaptive diffusion replaces unavailable statistical moments with instantaneous local gradient estimates, enabling practical ATC and CTA implementations from measurements.Smaller step sizes generally improve steady-state accuracy but slow convergence; simplified variants reduce inter-node communication for resource-constrained applications.Source: §3.2.2 · pp. 42–44

3.2.3. Multitask Diffusion Problem

  • Multitask diffusion extends collaboration to nodes estimating distinct but related parameter vectors, using regularization and information exchange to balance task accuracy and knowledge transfer.The framework accommodates correlated phenomena, differing objectives, and partially correlated parameters across the network.Source: §3.2.3 · pp. 44–45
  • Clustered multitask networks generalize single-task and multitask models by letting clusters share parameters while different clusters retain distinct objectives.This architecture accommodates varying degrees of parameter relatedness across the network.Source: §3.2.3 · pp. 46–47

3.2.4. Multitask Diffusion Learning

  • Clustered multitask learning combines local data adaptation and within-cluster estimate combination with regularized knowledge transfer across cluster boundaries.A quadratic penalty encourages neighboring clusters to maintain similar parameters when supported by the data, while allowing differentiation; the distributed updates implement these three operations.Source: §3.2.4 · pp. 47–49

3.2.5. Effect of Combination Weights and Network Topology

  • Network topology trades estimation performance against resource costs: dense connections can improve convergence and resilience but raise implementation costs, while sparse designs can degrade if essential information paths are lost.Sparse topologies can reduce communication overhead and power consumption, but require careful design to preserve information flow.Source: §3.2.5 · p. 50
  • Common combination-weight rules include uniform, Laplacian, and Metropolis schemes; separate strategies use similarity between neighboring nodes’ estimates or a minimum-distance criterion to prevent impulsive-noise propagation through the network.The neighborhood-based approach is computationally efficient and requires no global network information, but may be suboptimal when nodes vary in reliability or relevance.Source: §3.2.5 · pp. 50–52

3.2.6. Extensions of Diffusion-based Techniques

  • Partial diffusion strategies transmit only subsets of parameter-vector entries during each communication round, significantly reducing bandwidth requirements; notable implementations include PDLMS and PDRLS.Source: §3.2.6 · p. 53
  • Practical sensing environments face censoring and missing data; specialized diffusion extensions address them with censored estimation, bias-compensated algorithms [217], and multiple imputation for MAR scenarios.Source: §3.2.6 · pp. 53–54
  • Adaptive censoring approaches selectively activate sensors based on information content.This is an adaptive participation strategy for managing node involvement.Source: §3.2.6 · p. 54
  • Diffusion extensions adapt participation rates to estimation error, trading energy consumption against accuracy.Low-cost sampling dynamically adjusts node participation.Source: §3.2.6 · p. 54
  • For impulsive noise and censored observations, extensions use robust updates and losses, MCC filters, and partial-visibility frameworks.Signal-flow analysis and thresholding-based support identification also feature in partial-visibility diffusion.Source: §3.2.6 · p. 54
  • To counter sensor, link, and channel attacks, diffusion methods use ALRT-based detection or adaptive credibility weights to reject unreliable data.These methods target resilience with low complexity or improved robustness and accuracy; malicious agents can disrupt estimation through bias or delay.Source: §3.2.6 · p. 54

3.3. Nonlinear Models

  • Distributed nonlinear filtering addresses CPS whose nonlinearities cannot be adequately captured by linear models, enabling collaborative state estimation in complex nonlinear environments.Unlike linear filtering, nonlinear filtering typically needs approximation techniques for tractable distributed estimation; the section examines four principal approaches with distinct advantages for different scenarios.Source: §3.3 · p. 55

Distributed Moving Horizon Estimation for Nonlinear Systems

Moving horizon estimation (MHE) offers a flexible optimization-based alternative to Kalman filtering for constrained nonlinear networked systems. Distributed formulations decompose local estimation problems and coordinate agents through consensus penalties or ADMM.

Distributed Moving Horizon Estimation for Nonlinear Systems

  • MHE incorporates system constraints, nonlinear dynamics, and bounded disturbances by solving a finite-horizon estimation problem at each sampling time.The survey primarily develops its foundations and algorithms for linear systems; nonlinear estimation is discussed briefly as context and an extension.Source: Distributed Moving Horizon Estimation… · p. 55
  • Centralized MHE estimates the state by optimizing over a sliding horizon of length L, subject to system dynamics and admissible state and residual constraints.The optimizer’s terminal state is the estimate; P, Q, and R are positive-definite weighting matrices.Source: Distributed Moving Horizon Estimation… · p. 56
  • A common distributed MHE approach decomposes the centralized cost into local objective functions, with each agent solving a local problem over horizon L [251, 252, 253, 254].The local formulation uses agent-specific states, measurements, and measurement models.Source: Distributed Moving Horizon Estimation… · p. 56
  • Consensus-based distributed MHE adds disagreement penalties to local costs, whereas ADMM-based MHE uses local state copies and Lagrange multipliers to enforce agreement.ADMM alternates local MHE updates with neighbor exchanges of primal and dual variables until convergence or a preset iteration limit.Source: Distributed Moving Horizon Estimation… · p. 56

Consensus + Innovation Filtering for Nonlinear Systems

Consensus + Innovation filtering extends linear filtering to nonlinear systems by combining neighbor-estimate fusion with locally derived measurement innovations. The two terms support network-wide consistency while incorporating new local measurement information.

Consensus + Innovation Filtering for Nonlinear Systems

  • Consensus + Innovation filtering combines a consensus term that fuses neighboring estimates with an innovation term derived from nonlinear measurement models and local observations.The approach extends the linear filtering framework to nonlinear systems.Source: Consensus + Innovation Filtering for… · pp. 56–57
  • Each agent updates its estimate using a weighted combination of neighbors’ estimates and the innovation from its local nonlinear measurements.The consensus step maintains reasonable network consistency, while local innovation refines accuracy.Source: Consensus + Innovation Filtering for… · p. 57
  • The framework has been applied to nonlinear estimation problems and is described as suitable for systems with moderately nonlinear dynamics or measurement models.The survey presents it as a natural extension of linear consensus-based techniques.Source: Consensus + Innovation Filtering for… · p. 57

Distributed Extended Kalman Filtering

Distributed EKF implementations let nodes process nonlinear measurements locally, exchange estimates and uncertainty information, and fuse information across neighbors. Variants balance estimation accuracy with communication efficiency across network topologies and resource constraints.

Distributed Extended Kalman Filtering

  • The EKF estimates nonlinear states by linearizing the system around the current estimate, applying Kalman filter equations to the linearized model, and updating the estimate.The survey identifies EKF as a widely used approach in centralized settings that has been adapted to distributed scenarios.Source: Distributed Extended Kalman Filtering · p. 57
  • Each node maintains a local EKF, processes nonlinear measurements, and updates its state estimate; nodes periodically share estimates and uncertainty information with neighbors.Fusion rules combine local and neighboring estimates to refine overall estimation accuracy.Source: Distributed Extended Kalman Filtering · p. 57
  • Distributed EKF variants include consensus-based fusion, information-form implementations, covariance intersection, and adaptive architectures.The survey describes these approaches as balancing estimation accuracy with communication efficiency across diverse topologies and resource constraints.Source: Distributed Extended Kalman Filtering · pp. 57–58

Distributed Unscented Kalman Filtering

Distributed UKF methods use propagated sigma points to reconstruct state statistics and generally offer improved performance for highly nonlinear systems at increased computational cost. Distributed particle filters represent local state distributions with weighted samples and trade flexibility for greater computation and communication demands.

Distributed Unscented Kalman Filtering

  • UKF sigma points represent the state distribution, propagate through exact nonlinear dynamics, and reconstruct the posterior mean and covariance.Distributed nodes maintain local UKFs and exchange information with neighbors; compared with EKF, UKF generally improves performance for highly nonlinear systems at somewhat greater computational complexity.Source: Distributed Unscented Kalman Filtering · p. 58
  • The UKF typically achieves higher accuracy than the EKF for systems with significant nonlinearities by better capturing nonlinear transformations of probability distributions.Source: Distributed Unscented Kalman Filtering · p. 58

Distributed Particle Filtering

  • Particle filters represent state distributions with weighted samples, enabling them to capture multimodal and heavily skewed distributions in systems with severe nonlinearities or non-Gaussian noise.Distributed variants maintain local particles, exchange particle information or summaries, fuse representations, and resample to prevent particle degeneracy.Source: Distributed Particle Filtering · pp. 58–59
  • Distributed particle filtering offers flexibility for complex nonlinear dynamics and non-Gaussian uncertainties but requires more computation and communication than EKF- and UKF-based methods.Communication-efficient variants aim to preserve estimation quality while reducing resource demands; method choice depends on system characteristics, available resources, and required accuracy.Source: Distributed Particle Filtering · p. 59

4. Distributed Fault Detection

Distributed fault detection and isolation supports monitoring across networked CPS without centralized processing. The section motivates this approach through diverse fault sources and the risks and limitations of large-scale centralized monitoring.

Motivation and Challenges

  • Smart grids, robotic networks, and industrial automation face faults from environmental disturbances, component degradation, sensor malfunctions, communication failures, and subsystem interactions.Source: Motivation and Challenges · pp. 59–60
  • Undetected localized faults can cascade through interconnected systems, degrading performance, causing downtime or safety hazards, and potentially triggering catastrophic failure.Source: Motivation and Challenges · p. 60
  • Traditional centralized fault detection approaches struggle to process the volume and diversity of measurements in large-scale systems.Source: Motivation and Challenges · p. 60
  • Centralized monitoring creates a single-point vulnerability, can congest bandwidth and add unacceptable latency, and typically lacks graceful degradation when system components become inaccessible or compromised.Source: Motivation and Challenges · p. 60

Distributed Fault Detection Paradigm

Distributed fault detection extends distributed estimation and filtering across networked nodes, primarily through observer-based techniques. Nodes process local measurements, share diagnostic information, and fuse assessments to support system integrity.

Distributed Fault Detection Paradigm

  • Local processing distributes computational burden, avoids a single point of failure, and limits communication to relevant diagnostic information; the stated objectives include reliability, availability, maintainability, and safety.Source: Distributed Fault Detection Paradigm · pp. 60–61
  • The survey emphasizes distributed observer-based fault detection for its theoretical foundations and practical effectiveness, while also covering alternatives for specific application scenarios.Source: Distributed Fault Detection Paradigm · p. 61
  • Nodes maintain local observers or estimators using system models and available measurements, then generate residuals by comparing predicted and actual behavior.Source: Distributed Fault Detection Paradigm · p. 61
  • Detection logic uses residual patterns, neighboring nodes exchange relevant diagnostics, and nodes fuse local and neighbor assessments to enhance detection reliability.Source: Distributed Fault Detection Paradigm · p. 61

Mathematical Formulation of Distributed Fault Detection

This section surveys distributed fault-detection methods that use local state estimates, residuals, and threshold rules to identify sensor-specific faults while accounting for noise. It covers stateless and stateful detectors, model-based approaches for different system structures, and data-driven methods that learn fault patterns or detection algorithms from data.

Mathematical Formulation of Distributed Fault Detection

  • The fault model allows independently corrupted sensor measurements, treating faults as random-valued and potentially nonzero-mean, larger, and more persistent than zero-mean Gaussian measurement noise.Faults may appear as sudden jumps or slow drifts, whereas noise is described as smaller, bounded or of known variance, and high-frequency.Source: Mathematical Formulation of… · p. 62
  • Distributed observers estimate system state, generate residuals by comparing observed and predicted measurements, and flag faults when residuals cross local thresholds, with strategic information sharing supporting network-wide diagnosis.Under fault-free conditions, the steady-state expected estimation error is stated to be zero, providing a baseline for distinguishing noise-driven variation from faults.Source: Mathematical Formulation of… · p. 63

4.1. Stateless Detection

  • In stateless detection, each decision uses only the current residual, and m = 2 corresponds to a 95.4% detection probability with a 4.6% false alarm rate, while m = 3 gives 99.7% and 0.3%, respectively.Larger m raises the threshold and reduces false alarms but can delay or miss smaller faults; smaller m increases sensitivity and false alarms.Source: §4.1 · pp. 65, 67, Table 4 · p. 67

4.2. Statefull Detection

  • Stateful detection aggregates variance-normalized squared residuals over a window, using their fault-free chi-squared distribution to detect persistent deviations; temporal weighting can emphasize recent measurements.Longer windows improve noise suppression and sensitivity to subtle faults but increase latency and computation, while the highlighted discussion presents stateful and stateless methods as complementary across abrupt and gradual faults.Source: §4.2 · pp. 68–70

4.3. Relevant Literature

  • Event-triggered FDI transmits only after significant changes or preset conditions to reduce bandwidth, while low-complexity algorithms [292] target embedded platforms with limited computation.The survey identifies wireless sensor networks with energy constraints as a setting where event-triggered methods are particularly valuable.Source: §4.3 · p. 70
  • Robust distributed FDI work addresses imperfect models and unmodeled dynamics, as well as packet drops, delays, and intermittent connectivity, to preserve detection under uncertainty and communication failures.Source: §4.3 · pp. 70–71
  • Architectural approaches accommodate heterogeneous sensors and measurements using techniques such as unknown input observers, or integrate distributed control and fault detection in coordinated frameworks.Source: §4.3 · p. 71
  • Full-rank-system methods assume complete state information can be reconstructed from available measurements.This distinguishes them from systems where structural limitations prevent complete reconstruction from a single measurement set.Source: §4.3 · p. 71
  • Rank-deficient-system methods address structural limitations that prevent complete state reconstruction from any single measurement set.Source: §4.3 · p. 71
  • Nonlinear-system approaches incorporate detection mechanisms that account for nonlinear system behaviors.Source: §4.3 · p. 71
  • Clustering-based methods group similar fault patterns to support more nuanced detection and classification.Source: §4.3 · p. 71
  • Machine-learning techniques use supervised or unsupervised learning to identify complex fault signatures without explicit system models.Source: §4.3 · p. 72
  • Data-driven frameworks develop detection algorithms directly from data, especially when analytical models are unavailable or prohibitively complex.Source: §4.3 · p. 72
  • These research directions collectively offer a toolkit of distributed fault-detection methods that designers can select and combine for application requirements.Source: §4.3 · p. 72

5. Applications

This section surveys applications of distributed estimation, filtering, and fault detection for practical monitoring and control across power networks, social systems, target tracking and localization, sensor networks, and transportation systems. It covers vehicle-state estimation, traffic management, localization, and vehicle-platooning coordination and safety under challenging sensing and network conditions.

5.1. Smart Grid and Power Network

  • Distributed power-grid monitoring addresses renewable variability, geographic dispersion, heterogeneous components, and reliability needs while enabling resilience, scalable deployment, efficient communication, and localized fault management.Distributed processing avoids single points of failure, scales with new generation sources, reduces communication bandwidth and latency, and helps isolate faults before they propagate.Source: §5.1 · pp. 73–74
  • Consensus+innovation lets generators optimize economic dispatch using local costs and neighbors’ outputs, while interval observers aid fault diagnosis and distributed microgrid methods isolate faults and support operation during main-grid disconnection.Economic dispatch balances generation and consumption economically; distributed fault management provides fault-containment boundaries in microgrids.Source: §5.1 · pp. 75–76

5.2. Social Systems

  • Social-network agents collaboratively monitor individual states without centralized data collection, supporting analysis of collective behavior and opinion dynamics while preserving privacy.Social states can represent opinions, beliefs, preferences, or behaviors, with individuals influencing one another.Source: §5.2 · pp. 77–78
  • Social-system adaptations include trust-weighted filtering, categorical-belief estimation, interaction-graph learning, gossip-based learning, and detection of potentially manipulated information flows [334, 335, 336, 329, 285, 338, 339, 340, 341, 284].These approaches adapt distributed estimation and detection to social monitoring while accounting for source reliability, non-continuous states, network structure, pairwise information exchange, and misleading flows.Source: §5.2 · pp. 78–79

5.3. Target Tracking and Localization

  • Relative localization uses distance or angle measurements with triangulation or multilateration, while distributed sensors collaboratively estimate targets under uncertainty to support tracking and trajectory-informed decisions.Localization determines position; tracking extends it by monitoring position, velocity, and potentially acceleration over time.Source: §5.3 · p. 80
  • TDOA localization compares signal arrival times across sensors without requiring precise target-network synchronization, while target tracking commonly uses NCV, NCA, and Singer motion models.Singer dynamics approach NCV as the maneuver time constant decreases and NCA as it increases; larger values indicate gradual maneuvers, smaller values abrupt ones.Source: §5.3 · pp. 81, 82, 84

5.4. Intelligent Transportation Systems

  • CAVs estimate HDV states through local sensing and neighbor information exchange, using distributed observability to support network-wide tracking.This distributed approach avoids centralized estimation’s scalability issues and single-point failure vulnerability.Source: §5.4 · pp. 85, 87
  • Distributed traffic-state estimation fuses heterogeneous sources, including cameras, GPS, infrastructure sensors, and mobile devices, using model-based, constraint-based, and resource-aware methods.Examples include distributed unscented Kalman filtering, consensus variational Bayesian algorithms, particle filtering, and probabilistic-constrained fusion.Source: §5.4 · pp. 87–88
  • Distributed sensor fusion improves vehicle localization in obstructed-GPS, adverse-weather, or visually ambiguous urban environments.Systems combine GNSS, IMUs, LiDAR, computer vision, and infrastructure reference points.Source: §5.4 · p. 89
  • Distributed traffic management uses adaptive signal timing and real-time rerouting to reduce stops, congestion, and network travel times while responding rapidly and scaling with complexity.It draws on vehicle-reported arrival times and distributed fusion of traffic data.Source: §5.4 · p. 89
  • Transportation algorithms should continue functioning when components fail and adapt to localized traffic patterns while maintaining global coordination.Source: §5.4 · p. 89
  • Distributed observer-based tracking maintains inter-vehicle spacing under dynamic conditions, while related methods address denial-of-service attacks, compromised measurements, and rapid finite-time attack detection.These techniques support resilient vehicle platooning.Source: §5.4 · p. 90
  • Integrating distributed estimation and fault detection enables adaptive platoon formations to maintain safe operation under communication disruptions, sensor failures, or malicious interference.This application supports connected and automated mobility under challenging conditions.Source: §5.4 · p. 90

6. Conclusion and Future Directions

The survey brings together theoretical foundations, algorithmic developments, and practical applications of distributed estimation, filtering, and fault detection for CPS. It presents a unified perspective on using distributed algorithms to monitor and control complex, large-scale systems.

6. Conclusion and Future Directions

  • The survey examines theoretical foundations, algorithmic developments, and practical applications of distributed estimation, filtering, and fault detection within CPS.These domains are synthesized as interconnected parts of the survey.Source: §6 · p. 91
  • The survey synthesizes insights across these domains into a unified perspective on distributed algorithms for CPS.Source: §6 · p. 91
  • Distributed algorithms are presented as a means of robust monitoring and control for complex, large-scale systems.Source: §6 · p. 91

Key Contributions and Insights

The survey’s contributions span mathematical foundations, algorithm comparisons, resilience and fault-management mechanisms, resource analysis, and applications across multiple domains. Together, these insights connect distributed algorithm theory with design and implementation considerations for CPS.

Theoretical Foundations

  • Theoretical foundations integrate linear dynamical systems, graph theory, and observability analysis to explain when and how distributed estimation can succeed in networked environments.Source: Key Contributions and Insights · p. 91

Algorithm Taxonomy

  • The taxonomy distinguishes single-time-scale from double-time-scale consensus approaches and clarifies trade-offs between communication efficiency and estimation performance.It is intended to help designers select algorithms for application constraints.Source: Key Contributions and Insights · p. 91

Resilience Mechanisms

  • Diffusion-based filtering and observationally redundant designs are analyzed as mechanisms for resilience against failures, disturbances, and malicious attacks.The survey notes their value for critical infrastructure reliability.Source: Key Contributions and Insights · p. 91

Fault Management

  • Localized fault detection and isolation can prevent cascading failures by detecting and isolating faults at their source.These approaches are described as maintaining system integrity when individual components malfunction.Source: Key Contributions and Insights · p. 92

Resource Optimization

Real-World Applications

  • Case studies across smart grids, social systems, target tracking, and intelligent transportation illustrate practical applications of theoretical advances.Source: Key Contributions and Insights · p. 92

Future Research Directions

The survey identifies remaining challenges and future research directions for distributed algorithms in CPS. Proposed directions address communication, data-driven methods, privacy and security, computing architectures, and performance guarantees.

Communication Efficiency

  • Communication-efficiency research could use event-triggered protocols, compressed sensing, and information-based sharing to reduce transmission or exchanged data.Source: Future Research Directions · p. 92

Data-Driven Approaches

  • Data-driven directions include hybrid model-based and learning methods, distributed reinforcement learning, and transfer learning for related monitoring tasks.Source: Future Research Directions · p. 93

Privacy and Security

  • Privacy and security proposals include privacy-preserving estimation, resilience to cyber attacks, and trust mechanisms that adjust information-fusion weights.Source: Future Research Directions · p. 93

Computational Architecture

  • Computational architecture directions include edge-cloud partitioning, IoT frameworks for heterogeneous and intermittently connected deployments, and specialized hardware accelerators.Source: Future Research Directions · pp. 93–94

Quality of Service Guarantees

  • Quality-of-service work could guide algorithm selection and tuning with application-specific metrics, alongside resource allocation across competing priorities and formal verification.Source: Future Research Directions · p. 94
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