A linear dynamical system can be described using a state-space model represented by the following equations [32]:

where \( x(t) \in \mathbb{R}^n\) is the state vector at time \( t\) representing the physical parameters involved in the dynamical system, \( u(t) \in \mathbb{R}^m\) is the input vector (control input) at time \( t\) , \( y(t) \in \mathbb{R}^N\) is the output vector at time \( t\) , \( A \in \mathbb{R}^{n\times n}\) is the state transition matrix, \( B \in \mathbb{R}^{n\times m}\) is the input matrix, and \( C \in \mathbb{R}^{N\times n}\) is the output matrix. The state transition matrix \( A\) governs the dynamic behavior of the system, while the matrices \( B\) and \( C\) define how the inputs affect the state and how the state contributes to the output, respectively.

In practical applications, system models and measurements are subject to noise. To capture this realistic condition:

• \( \boldsymbol{\nu}(t) \in \mathbb{R}^n\) is the process noise, typically assumed to be zero-mean Gaussian noise
• \( \boldsymbol{\mu}(t) \in \mathbb{R}^N\) is the measurement noise, also assumed to be zero-mean Gaussian noise

This formulation is the typical model for filtering techniques, such as the Kalman filter, and fault detection over CPS. Understanding these linear systems is foundational for designing and implementing distributed filtering and estimation techniques that can efficiently track the (physical) state of the system in noise-corrupted setups.

A representation of linear systems and the study of their properties can be effectively approached through structured systems theory [33, 34, 35, 36, 37]. In structured systems theory, the behavior and dynamics of a linear system can be visually represented using a directed graph, known as the system digraph. This representation highlights the interconnections among state variables and outputs, and models the structural information contained in the system matrix \( A\) and output matrix \( C\) through their zero/nonzero patterns [38].

A system digraph is a directed graph that represents the relationships between the state variables and the outputs of a system, as shown in Fig. 3 for an illustrative example. The construction process follows these principles:

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• Nodes: The nodes in the digraph are associated with the state variables, representing the individual components of the system state vector;
• Directed links: The directed links denote the influence or control relationships among these variables as defined by the entries of the system matrix; and
• Construction rules: The system digraph is constructed by creating nodes for each state variable and drawing directed links according to the nonzero entries in the matrix \( \mathcal{A}\) .

The entries of the system matrix \( A\) dictate how the state variables interact with each other, i.e.,

• If an entry \( a_{ij}\) is nonzero, there is a directed link from state node \( j\) to state node \( i\) in the digraph, indicating that the state \( x_j\) influences the state \( x_i\) ;
• Conversely, if \( a_{ij} = 0\) , there is no direct influence from state \( x_j\) to state \( x_i\) in the system.

Similar to the system matrix, the output matrix \( C\) can be represented by its zero/nonzero pattern as follows:

• A nonzero entry \( c_{ij}\) indicates a directed link from state node \( j\) to output node \( i\) in the digraph, representing that the state \( x_j\) directly influences the output \( y_i\) ; and
• This establishes how each state variable contributes to the observable outputs, which is crucial for tasks such as estimation and fault detection.

This graph-theoretic representation provides valuable insights into the structural properties of the system, enabling the analysis of observability, controllability, and other fundamental system characteristics that are essential for distributed algorithm design.

2.2.1.1 Graph-Based Representation

Single time-scale algorithms balance estimation accuracy with communication efficiency by performing a single consensus iteration between consecutive system measurements. This approach is particularly valuable in bandwidth-constrained environments while still maintaining robust state estimation capabilities.

Given a linear dynamical system in the form (1) with sensor measurements as described in (2), distributed estimation aims to locally infer the global state of the system using only partial state measurements available at each node. Each agent must combine its local observations with information received from neighboring nodes to construct an accurate global state estimate.

Below, we present a representative example of a single time-scale distributed estimation technique that illustrates the core principles of this approach. This distributed estimator operates through a structured two-step process that alternates between consensus on predictions and measurement updates:

1. Consensus on a-priori estimates: Agents (i.e., an entity comprised of a sensor and communication capabilities) share a-priori estimates (or predictions [114, 115]) over the network \( \mathcal{G}\) as follows:

   \[ \begin{eqnarray} \widehat{\mathbf{x}}_i(t|t-1) = \sum_{j\in\mathcal{N}_i} w_{ij}A\widehat{\mathbf{x}}_j(t-1|t-1), \end{eqnarray} \]

   (10)

   where \( \widehat{\mathbf{x}}_i(t|t-1)\) represents the priori estimate of state \( \mathbf{x}\) at time \( t\) , using all the mesurements of node \( i\) , and its neighbors \( \mathcal{N}_i\) at time \( t-1\) . The consensus matrix \( W=[w_{ij}]\) satisfies the stochasticity conditions described in Section 2.3. As it is clear from Eq. (10), only one step of consensus is performed between two steps of system dynamics \( t-1\) and \( t\) (two consecutive sampling times), which confirms the single time-scale setup as illustrated in Fig. 7.

2. Measurement update: Agents/sensors share their measurements over the network \( \mathcal{G}\) and update their priori estimates as follows:

   \[ \begin{eqnarray} \widehat{\mathbf{x}}_i(t|t) =\widehat{\mathbf{x}}_i(t|t-1) + K_i \sum_{j\in \mathcal{N}_i}C_j^\top \left(\mathbf{y}_j(t)-C_j\widehat{\mathbf{x}}_i(t|t-1)\right), \end{eqnarray} \]

   (11)

   with \( \mathbf{y}_j(t)\) as the measurement of node \( j\) at time-step \( t\) and \( K_i\) as the local gain matrix at node \( i\) . This step is also called innovation-update in some literature [116, 117, 118, 119, 120], just to mention a few.

The dynamics of the distributed estimation error evolves as

where the error vector \( \mathbf{e}(t)\) (at all nodes) is defined as,

\( {\zeta}(t)\) collects the noise terms as

\( D_C\) is defined as

and the block-diagonal gain matrix in the form

3.1.1.1 Distributed Gain Matrix Requirements:

3.1.1.2 Stability and Distributed Observability:

3.1.1.3 Varying Observability Requirements in Literature:

3.1.1.4 Network Design for Observability Recovery:

3.1.1.5 Bounded Error and Local Observability:

The existing CPS literature on single time-scale distributed estimation and filtering encompasses diverse approaches addressing specific challenges inherent in distributed systems. Below, we categorize and review the key contributions in this domain:

3.1.2.1 Performance-Focused Innovations:

3.1.2.2 Resource-Efficient Communication Strategies:

3.1.2.3 Resilience to Network Failures:

3.1.2.4 Heterogeneity and Optimization Considerations:

Double time-scale algorithms represent a fundamentally different approach to distributed estimation, characterized by their intensified communication pattern between system dynamics updates. In these scenarios, agents perform multiple iterations of consensus and communication between consecutive time steps of system dynamics, as illustrated in Fig. 7. This communication-intensive phase is commonly referred to as the "consensus loop" in distributed filtering literature.

This approach presents a clear trade-off in distributed estimation design: while it imposes significantly higher communication load and network traffic on the multi-agent system, it offers two substantial advantages. First, it effectively relaxes the strict observability requirements that constrain single time-scale methods. Second, it demonstrates superior error performance, achieving more accurate state estimates compared to single time-scale approaches under equivalent system conditions. The key is that agents run multiple consensus rounds–more than the network diameter–in the interval between two consecutive system measurements. This multi-step consensus has opposing effects: it imposes higher communication/computation burden on agents, but simultaneously ensures that information originating from any agent reaches all other agents, effectively making the entire network’s measurements globally observable. This global information sharing compensates for limited local observability.

Below, we present a canonical implementation of a double time-scale distributed filter that exemplifies the core principles of this approach, as detailed in seminal works by Olfati-Saber and He et al. [159, 93]. This distributed estimator operates through two distinct phases that separate local prediction from network-wide consensus:

1. Local priori estimate: This step includes no consensus iteration and is performed locally at every node \( i\) as follows:

   \[ \begin{align} \widehat{\mathbf{x}}_i(t)= A \widehat{\mathbf{x}}_i(t-1)+{C}_{i}^\top(\mathbf{y}_i(t)-C_{i}A\widehat{\mathbf{x}}_i(t-1)), \end{align} \]

   (16)

   where \( \widehat{\mathbf{x}}_i(t)\) represents the priori estimated state of node \( i\) and \( \mathbf{y}_i(t)\) is its measurement at time-step \( t\) .

2. Consensus loop update: \( \mathcal{L}\) iterations of consensus update are then performed to average the estimate values as follows:

   \[ \begin{align} \widehat{\mathbf{x}}_{i,l}(t)=\widehat{\mathbf{x}}_{i,l-1}(t)-\epsilon\sum_{j \in \mathcal{N}_i}w_{ij}\left(\widehat{\mathbf{x}}_{i,l-1}(t)-\widehat{\mathbf{x}}_{j,l-1}(t)\right), \end{align} \]

   (17)

   where \( \widehat{\mathbf{x}}_{i,l}(t)\) represents the state estimate of node \( i\) after \( 1\leq l\leq \mathcal{L}\) communication and consensus iterations at time-step \( t\) , while \( {\epsilon}\) is a small positive constant that controls the rate of convergence. This consensus loop is over the neighbouring set \( {\mathcal{N}_i}\) of node \( i\) , and the last consensus term \( \epsilon\sum_{j \in \mathcal{N}_i}w_{ij}\left(\widehat{\mathbf{x}}_{i,l-1}(t)-\widehat{\mathbf{x}}_{j,l-1}(t)\right)\) ensures that the estimate of node \( i\) moves toward the average of its own and neighbours’ local estimates after \( \mathcal{L}\) iterations. For this, the consensus matrix needs to satisfy the stochastic condition in Section 2.3. As it is clear from Eq. (17), \( \mathcal{L}\) steps of consensus are performed between two steps of system dynamics \( t-1\) and \( t\) (two consecutive sampling times) and then the algorithm moves to the next sampling time-step \( t+1\) . This represents the double time-scale setup as illustrated in Fig. 7.

A critical parameter in double time-scale algorithms is the number of consensus iterations \( L\) performed between consecutive system dynamics updates. In the literature, it is typically assumed that \( L \geq d_G\) , where \( d_G\) denotes the diameter of the sensor network \( G\) (defined as the longest shortest path between any two nodes in the network). This requirement ensures complete information propagation across the network – every node’s data eventually reaches every other node during a single system time step.

This comprehensive information sharing fundamentally addresses the observability challenges that plague single time-scale approaches. By executing multiple consensus iterations, the double time-scale approach effectively transforms a partially observable system into a fully observable one from each node’s perspective. The consensus loop essentially functions as an information diffusion mechanism, redistributing measurements throughout the network and ensuring that every agent has access to sufficient information to reconstruct the global state.

The observability benefits of this approach yield two significant advantages:

1. Relaxed Network Connectivity Requirements: Strong connectivity of the network \( G\) becomes sufficient to ensure both observability and error stability. Unlike single time-scale approaches where specific topological structures must be carefully designed to guarantee observability [133, 121], double time-scale methods are more tolerant of arbitrary network configurations.
2. Enhanced Estimation Stability: The \( L\) iterations of communication and consensus make the information of every sensor accessible to every other sensor between consecutive system dynamics updates. This comprehensive data sharing ensures the system becomes observable to every sensor, eliminating the observability concerns that persist in single time-scale estimation.

Table 1 provides a comparative analysis of single time-scale and double time-scale protocols, highlighting the fundamental differences in computation rate, network connectivity requirements, and communication overhead per sample. This comparison illustrates the explicit trade-off between communication efficiency and observability guarantees that system designers must consider when selecting an appropriate distributed estimation approach.

The literature on double time-scale estimation protocols encompasses diverse approaches addressing various challenges in distributed filtering. Below, we categorize and analyze key contributions across several technical domains:

3.1.5.1 Advanced Estimation Frameworks:

3.1.5.2 Communication-Efficient Adaptations:

3.1.5.3 Adaptability to Heterogeneous Networks:

3.1.5.4 Robustness Against Network Imperfections:

The observationally redundant design represents a sophisticated approach to enhancing reliability and fault tolerance in distributed estimation systems. This methodology systematically incorporates strategic sensor redundancy based on observational equivalence principles, creating inherently resilient monitoring networks capable of maintaining estimation performance despite sensor failures or attacks.

The fundamental concept behind observationally redundant design is the strategic deployment of sensors that observe structurally or functionally equivalent aspects of the system state. When multiple sensors can provide observationally equivalent information about critical system states, the network maintains complete observability even if individual sensors malfunction or are compromised. This architectural redundancy fundamentally differs from simple replication, as it is based on mathematical equivalence relationships that preserve the system’s structural properties.

The notion of equivalence relation, ‘\( \sim\) ’, in set theory and abstract algebra is defined as having three properties: reflexivity, symmetry, and transitivity [188]. Towards observational equivalence in state estimation, reflexivity implies that every state is equivalent to itself, i.e. \( x_i \sim x_i\) ; symmetry implies that if \( x_i \sim x_j\) then \( x_j \sim x_i\) ; and transitivity implies that if \( x_i \sim x_j\) and \( x_j \sim x_m\) , then \( x_i \sim x_m\) . With these notations, the observational equivalence of two state nodes and the associated measurements are defined as follows. Let \( C_i\) denote a row vector of size \( n\) with only non-zero at \( i\) th entry denoting measurement of state \( x_i\) . Observational equivalence among two states, \( x_i\sim x_j\) , is defined as

It can be easily verified that the above definition follows three properties of transitivity, reflexivity, and symmetry.

3.1.6.1 Graph-Theoretic Classification for Observational Equivalence

3.1.6.2 Algebraic Interpretation and Redundant Observability Design

From an algebraic graph-theoretic perspective, the system’s structural properties have precise mathematical interpretations that inform redundant design strategies. The contractions in \( G_A\) directly correspond to rank-deficiency regions in the system matrix \( A\) , while the SCC decomposition reflects the irreducibility properties of \( A\) . Fig. 10 provides a visual illustration of these correspondences, demonstrating how graph structures map to algebraic properties.

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A fundamental result in structural observability theory establishes that for a given system digraph \( G_A\) , outputs from a strategically minimal set of nodes are sufficient to ensure full system observability:

• One state node from every parent SCC \( S^p_l\) ;
• One state node from every contraction \( C_l\) .

This minimal output set guarantees \( (A,C)\) -observability. More significantly, states within the same contraction \( C_l\) or within the same parent SCC \( S^p_l\) exhibit observational equivalence – they provide structurally identical information about the system’s behavior.

3.1.6.3 Designing \( q\) -Redundant Observable Systems

3.1.6.4 Network Connectivity for Distributed Implementation

Classical Least-Mean-Square (LMS) formulations traditionally operate under centralized processing assumptions, where either

• All sensor observations are collected and processed at a central node, or
• Local LMS estimators operate independently with their outputs subsequently fused at a central location.

This centralized paradigm, while conceptually straightforward, faces significant scalability and reliability limitations in modern cyber-physical systems. Contemporary applications increasingly demand truly distributed processing architectures, where

• Data is inherently distributed across a network of agents or sensors;
• Each node has access only to its local measurements and information from immediate neighbors;
• No central fusion center exists or is desirable due to robustness concerns; and
• Global parameter estimation must emerge from purely local computations and limited inter-node communication.

Following the seminal diffusion adaptation framework developed in [195, 196, 197], we consider a connected network of \( N\) agents indexed by \( k \in \{1, 2, …, N\}\) . Each agent \( k\) maintains the following:

• A twice-differentiable local cost function \( J_k(\boldsymbol\omega) \in \mathbb{R}\) , often referred to as its utility function;
• Access to local measurements that inform this cost function; and
• Communication links with neighboring agents that enable collaborative estimation.

Fig. 11 illustrates a typical network topology for such systems, with bi-directional communication links between neighboring agents represented by single lines. This network structure fundamentally shapes how information diffuses through the system during the estimation process.

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In this distributed setting, the collective goal is to estimate a global parameter vector that optimizes the aggregate cost function across all agents given by

where the goal is to identify its unique minimizer, denoted by \( \boldsymbol{\omega}^\star \) .

To enable fully distributed estimation, we introduce a set of non-negative combination weights \( \{c_{k\ell} \geq 0\}\) that govern information exchange among neighboring nodes. These weights must satisfy

for each node \( k = 1,2,…,N\) , where \( \mathcal{N}_k\) denotes the neighborhood of node \( k\) (including \( k\) itself). This constraint ensures that the resulting combination matrix \( C\) is right-stochastic, a property essential for the convergence of diffusion strategies.

Using these weights, we define for each node \( \ell\) a localized cost function that aggregates weighted costs from its neighborhood, i.e.,

This formulation allows us to re-express the global cost function in terms of these localized costs as follows:

3.2.2.1 Gradient-Based Distributed Optimization

Diffusion Least Mean Squares (DLMS) algorithms represent a practical implementation of the distributed optimization framework introduced previously. These methods enable networks of agents to collaboratively estimate a global parameter vector through a combination of local adaptation and strategic information exchange with neighboring nodes. By extending the classical LMS algorithm to decentralized network settings, diffusion strategies achieve robust estimation performance while maintaining the computational simplicity that makes LMS attractive for real-time applications.

3.2.3.1 Algorithmic Variants and Information Flow Patterns

3.2.3.2 ATC Algorithm Formulation

While the diffusion strategies discussed previously provide a solid theoretical foundation, their practical implementation faces a significant challenge: the need for statistical moments \( \{R_{u,k}, \boldsymbol r_{du,k}\}\) to evaluate gradient vectors. In real-world applications, these moments are rarely available a priori and must be estimated from available measurements. This section examines how stochastic approximation techniques transform theoretical diffusion algorithms into practical adaptive implementations suitable for deployment in uncertain environments.

3.2.4.1 From Exact to Stochastic Gradient Approximations

3.2.4.2 Simplified Implementation Variants

The diffusion strategies discussed thus far assume that all network nodes collaborate to estimate a single global parameter vector. However, many practical applications involve scenarios where different nodes need to estimate distinct yet related parameter vectors – a paradigm known as multitask learning. This section examines how diffusion strategies can be extended to support collaborative estimation in multitask environments while exploiting inter-task relationships to enhance overall system performance.

3.2.5.1 Multitask Learning Framework

3.2.5.2 Mathematical Formulation

3.2.5.3 Taxonomy of Distributed Learning Architectures

The estimation objectives and resulting algorithms depend fundamentally on the network’s collaborative structure, as illustrated in Fig. 14. This figure presents three principal architectural paradigms as follows:

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• Single-task networks: All nodes collaborate to estimate a common global parameter vector, representing the conventional diffusion approach discussed earlier;
• Multi-task networks: Each node estimates its own distinct parameter vector, with information exchange leveraging inter-task relationships to improve estimation performance; and
• Clustered multitask networks: Nodes are organized into clusters, with members of each cluster pursuing a common parameter while different clusters maintain distinct objectives. This hybrid architecture accommodates varying degrees of parameter sharing across the network.

The clustered multitask framework represents a generalized approach that encompasses both single-task and multitask models as special cases, offering a flexible framework for modeling complex distributed estimation problems with varying degrees of parameter relatedness.

Building upon the multitask framework introduced previously, this section examines specific algorithms and techniques for implementing diffusion-based learning in multitask environments. We focus particularly on clustered multitask networks, which provide a flexible architecture for balancing localized specialization with collaborative learning across related tasks.

3.2.6.1 Clustered Optimization Formulation

3.2.6.2 Inter-Cluster Regularization

3.2.6.3 Global Network Optimization

3.2.6.4 Distributed Implementation

The performance characteristics of diffusion-based estimation algorithms are strongly influenced by two critical design factors: the combination matrix that governs information exchange among neighboring nodes and the underlying network topology that defines the communication infrastructure. This section examines how these factors impact estimation performance and presents strategies for optimizing them in practical implementations.

3.2.7.1 Performance-Topology Relationships

3.2.7.2 Combination Weight Design Strategies

The core diffusion adaptation strategies discussed previously have inspired numerous extensions and enhancements to address specific challenges in distributed estimation. This section surveys recent advances that extend diffusion approaches to handle various practical constraints, improve performance in challenging environments, and enhance resilience against both natural and adversarial disturbances.

3.2.8.1 Probabilistic Extensions

3.2.8.2 Communication-Efficient Implementations

3.2.8.3 Handling Missing and Censored Data

3.2.8.4 Other Relevant Literature

Moving horizon estimation (MHE) is an optimization-based state estimation framework that explicitly incorporates system constraints, nonlinear dynamics, and bounded disturbances by solving a finite-horizon estimation problem at each sampling time [249]. In distributed settings, MHE provides a flexible alternative to Kalman-filter-based approaches, particularly for constrained nonlinear networked systems. Consider the discrete-time nonlinear system as,

In centralized MHE, the state estimate at time \( t\) is obtained by solving an optimization problem over a sliding horizon of length \( L\) as given below [250]:

where \( P\) , \( Q\) , and \( R\) are positive definite weighting matrices, \( \widehat{x}_{k-L}\) is a-priori estimate, and \( \mathcal{X}\) and \( \mathcal{V}\) denote the admissible constraint sets. The estimate \( \widehat{x}(t)\) is given by the optimizer’s terminal state.
A common distributed MHE approach decomposes the centralized cost into local objective functions [251, 252, 253, 254]. At agent \( i\) , the local MHE problem over horizon \( L\) is given by:

In consensus-based distributed MHE, consensus constraints are imposed and the coupling constraints are relaxed by adding disagreement penalties to the local cost functions [255]. Alternatively, ADMM-based distributed MHE introduces local copies of the state and corresponding Lagrange multipliers to enforce agreement among agents [256, 257]. This results in iterative local MHE updates combined with neighbor communication of primal and dual variables until convergence (or until a predefined number of iterations) is reached.

Consensus + Innovation filtering extends the linear filtering framework to nonlinear systems by combining two complementary mechanisms:

• A consensus term that fuses estimates from neighboring nodes, promoting agreement across the network, and
• An innovation term derived from nonlinear measurement models and local observations.

In this approach, each agent updates its state estimate based on a weighted combination of its neighbors’ estimates (the consensus component) and the innovation from its local nonlinear measurements. The consensus step ensures that the global estimate of the nonlinear system state is maintained with reasonable consistency across all agents, while the innovation term incorporates new measurement information to refine accuracy.

This methodology has been successfully applied to various nonlinear estimation problems, with notable implementations and extensions described in [118, 117, 258]. The framework’s flexibility makes it particularly suitable for systems with moderately nonlinear dynamics or measurement models, offering a natural extension of linear consensus-based techniques.

The Extended Kalman Filter (EKF) represents one of the most widely used approaches for nonlinear state estimation in centralized settings. It operates by linearizing the nonlinear system around the current state estimate, applying standard Kalman filter equations to this linearized model, and then updating the state estimate accordingly.

Although traditionally implemented in centralized architectures, the EKF has been successfully adapted for distributed scenarios through several key innovations:

• Each node maintains its own local EKF to process nonlinear measurements and update its state estimate;
• Nodes periodically share their state estimates and associated uncertainty information with neighbors; and
• Various fusion rules combine local and neighboring estimates to refine the overall estimation accuracy.

This distributed EKF approach has been implemented with numerous variations, including consensus-based fusion [259], information-form implementations [260], covariance intersection methods [261], and adaptive architectures [262, 263]. These approaches balance estimation accuracy with communication efficiency, enabling effective nonlinear state estimation across networks with diverse topologies and resource constraints.

The Unscented Kalman Filter (UKF) represents an alternative approach to nonlinear filtering that avoids explicit linearization. Instead, the UKF employs a deterministic sampling technique to capture the statistical properties of the state distribution:

• A set of carefully selected sample points (sigma points) is chosen to represent the state distribution;
• These points are propagated through the exact nonlinear system dynamics; and
• The transformed points are used to reconstruct the posterior mean and covariance.

This approach typically achieves higher accuracy than the EKF for systems with significant nonlinearities, as it better captures the effect of nonlinear transformations on probability distributions.

Distributed implementations of the UKF [264, 265, 266, 267, 268] follow similar principles to distributed EKF approaches, with nodes maintaining local UKFs and exchanging information with neighbors. The primary difference lies in the local filtering algorithm, with UKF-based methods generally offering improved performance for highly nonlinear systems at the cost of somewhat increased computational complexity.

For systems with severe nonlinearities or non-Gaussian noise characteristics, particle filters provide a powerful estimation framework. These methods represent probability distributions using sets of weighted samples (particles) rather than parametric distributions [269], enabling them to capture multi-modal and heavily skewed distributions that arise in many complex nonlinear systems.

In distributed implementations [270, 271, 272, 273, 274, 275], particle filtering operates through several coordinated mechanisms:

• Each node maintains its own local set of particles representing possible system states;
• Nodes exchange particle information or statistical summaries with neighbors according to various communication protocols;
• Fusion algorithms combine local and neighboring particle representations to refine the state estimation; and
• Resampling strategies prevent particle degeneracy while maintaining estimation accuracy.

Distributed particle filtering approaches offer unparalleled flexibility for handling complex nonlinear dynamics and non-Gaussian uncertainties. However, this flexibility comes at the cost of increased computational and communication requirements compared to EKF and UKF-based methods. Recent research has focused on developing communication-efficient variants that preserve estimation quality while reducing resource demands, making distributed particle filtering increasingly practical for resource-constrained cyber-physical systems.

These four complementary approaches to distributed nonlinear filtering provide system designers with a rich toolkit for addressing nonlinear estimation challenges across diverse application domains. The choice among these methods typically depends on the specific characteristics of the nonlinear system, the available computational and communication resources, and the required estimation accuracy.

Observer-based fault detection has emerged as a powerful methodology for real-time monitoring of cyber-physical systems in the presence of faults. This approach leverages the distributed state estimation techniques discussed in previous sections, extending them to enable fault detection capabilities. The core principles of this approach include:

• Continuous state estimation: Distributed observers/estimators continuously track the system state based on available measurements and system models;
• Residual generation: By comparing actual measurements with those predicted by the observers, residual signals are generated that quantify the discrepancy between expected and observed system behavior;
• Threshold-based detection: When residuals exceed certain thresholds (determined either probabilistically or deterministically), fault conditions are flagged; and
• Distributed implementation: Each node maintains its own local observer/estimator, enabling localized fault detection while strategic information sharing enhances system-wide diagnostic capabilities.

The consensus-based distributed estimator presented in equations (10)-(11) provides a natural framework for implementing distributed fault detection. This estimator can be applied directly to fault-corrupted measurements, serving the dual purpose of state tracking and fault detection. When faults occur, they manifest as anomalies in the estimation process that can be detected through careful analysis of estimation errors.

The resulting error dynamics follow equation (12), with fault terms incorporated into the \( \boldsymbol{\zeta}(t)\) term. Under fault-free conditions, the steady-state estimation error has been shown to satisfy \( \lim_{t\rightarrow\infty} \mathbb{E}(\mathbf{e}_i(t)) = 0\) [114, 121], providing a baseline expectation for normal system operation.

The error covariance in this case is defined as \( \Sigma_e := \lim_{t\rightarrow\infty} \mathbb{E}(\mathbf{e}_i(t)^{\top}\mathbf{e}_i(t))\) and satisfies the relationship [121, 283, 284]:

where

• \( b := \|(W \otimes A)-K D_C(W \otimes A)\|_2 < 1\) ,
• \( a_1 := \|I_{Nn}-K D_C\|^2_2\) ,
• \( a_2 := \|K\|^2_2\) ,
• \( \Sigma_{\mu} := \text{diag}[\sum_{j\in N_i} \mathbf{C}^{\top}_j \Sigma^j_{\mu}\mathbf{C}_j]\) where \( \Sigma^j_{\mu}\) denotes the \( j\) th diagonal entry of \( \Sigma_{\mu}\) as the variance of the measurement noise term \( \boldsymbol{\mu}\) , and
• \( \Sigma_{\nu}\) represents the variance of the process noise term \( \boldsymbol{\nu}\) .

This analytical characterization of estimation error behavior provides the foundation for designing effective fault detection mechanisms that can discriminate between normal system variations due to noise and actual fault conditions.

In the context of residual-based fault detection, two main paradigms have emerged in the literature, each representing a different philosophy regarding how detection mechanisms process temporal information:

• Stateless fault detection: This approach makes detection decisions based solely on the current residual values, without incorporating historical information. Detection thresholds are typically designed based on statistical properties of the residuals under fault-free conditions; and
• Stateful fault detection: This approach maintains historical information about residual behavior over time, often using sliding windows or exponentially weighted averages. By incorporating temporal information, stateful methods can detect subtle faults that develop gradually or manifest intermittently.

These complementary approaches offer different trade-offs between detection speed, sensitivity, and false alarm rates, providing system designers with flexible options to match specific application requirements. The following sections examine each of these detection paradigms in greater detail, exploring their mathematical formulations, implementation considerations, and performance characteristics.

The core mechanism of stateless detection begins with residual generation. At each local node \( i\) , the residual is defined as the difference between the actual measurement and the estimated output:

Under fault-free conditions, this residual is primarily influenced by system and measurement noise. However, when faults occur, they introduce additional components that shift the residual statistics in detectable ways.

Given the error dynamics established previously, the residual at sensor \( i\) (which may include a possible fault \( f_i\) ) at time \( t\) can be expressed as

where \( \hat{A} := (W \otimes A - KD_C(W \otimes A))\) represents the effective system matrix for the error dynamics, and \( \zeta_i(t)\) encompasses various noise and interaction terms.

For effective detection and isolation of faults at specific nodes, the gain matrix \( K\) must be carefully designed to ensure that faults at one node have minimal impact on the residuals at other nodes. This is achieved by satisfying the constraint:

for sufficiently small \( \varepsilon < 1\) as a design parameter. This constraint ensures that the fault-related term \( C_i^{\top}K_iC_jf_j\) in the residual \( r_i(t)\) due to a fault \( f_j\) at node \( j\) is scaled down by a factor of \( \varepsilon\) compared to the fault-related term in the residual \( r_j(t)\) of sensor \( j\) itself. This isolation property is crucial for determining which specific node has experienced a fault. The gain design can be formulated as a Linear Matrix Inequality (LMI) problem [283, 284].

The threshold-based detection operates on statistical hypothesis testing principles. In the absence of faults, the residual follows a zero-mean Gaussian distribution whose variance is determined by system and measurement noise characteristics. When a fault occurs, the residual becomes biased, following a Gaussian distribution with non-zero mean.

As an example consider the observer in [285, 286, 284]. Given the noise variance \( \Sigma_\nu\) and \( \Sigma_\mu\) and the residuals \( r_i(t)\) from Eq. (61), the detection threshold for a detection-level \( {m \in \mathbb{R}_{>0}}\) is ,

where \( {\kappa = erf (\frac{m}{\sqrt{2}})}\) is detection probability (with \( erf (\cdot)\) as the Gauss error function) and \( \|\Sigma_e\|_2\) from (59).

By applying a statistical hypothesis test with these two distributions (fault-free and faulty), we can determine the probability that an observed residual indicates a fault. Specifically, if the magnitude of the residual \( |r_i(t)|\) exceeds the threshold \( m\Sigma_r^i\) , then a fault is detected with probability \( \kappa\) , while the probability of a false alarm is \( 1-\kappa\) . This is better illustrated in Fig. 18.

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The parameter \( m\) in the threshold definition directly controls the sensitivity of the detection system, creating an explicit trade-off between detection sensitivity and false alarm rates. Larger values of \( m\) result in higher thresholds, reducing false alarms but potentially delaying fault detection or missing smaller faults. Conversely, smaller values of \( m\) increase detection sensitivity but at the cost of more frequent false alarms.

For practical implementations, \( m\) is typically chosen to achieve a desired false alarm rate. Common threshold configurations use integer values of \( m\) , with each value corresponding to a specific detection probability \( \kappa\) and false alarm rate as shown in Table 4. For instance, \( m=2\) corresponds to a 95.4% detection probability with a 4.6% false alarm rate, while \( m=3\) provides a 99.7% detection probability with only a 0.3% false alarm rate.

These statistically designed thresholds provide a rigorous foundation for fault detection decisions, allowing system designers to explicitly balance detection sensitivity against false alarm resilience based on specific application requirements and operating contexts.

The foundation of stateful detection lies in quantifying deviations between expected and actual system behavior over an extended time period. This is accomplished through distance measures that aggregate residual information across multiple time steps. The basic distance measure for a sliding time window of length \( \mathcal{T}\) is defined as

This measure effectively normalizes each squared residual by its expected variance and sums these values over the time window. The normalization ensures that the measure appropriately accounts for the anticipated noise characteristics of each measurement, while the summation captures persistent deviations that may indicate fault conditions.

The statistical properties of this distance measure provide a rigorous foundation for fault detection decisions. Under fault-free conditions, the summation of squared normalized random variables from a normal distribution follows a Chi-squared distribution with \( \mathcal{T}\) degrees of freedom (denoted by \( \chi^2_\mathcal{T}\) ), with an expected value of \( \mathbb{E}(\iota_i^\mathcal{T}) = \mathcal{T}\) [278, 287].

This property enables the application of well-established statistical tests to evaluate how well the observed residuals conform to their expected distribution. When system behavior deviates from nominal conditions due to faults, the distance measure tends to increase beyond what would be expected from noise alone, providing a statistical basis for fault detection.

Similar to stateless detection, stateful approaches require carefully designed detection thresholds that balance sensitivity against false alarm rates. For a Chi-squared detector with a desired false alarm rate of \( 1-\kappa\) , the appropriate threshold is given by

where \( \Gamma^{-1}(\cdot,\cdot)\) denotes the inverse regularized lower incomplete gamma function. If the computed \( \chi^2_\mathcal{T}\) detector value exceeds this preset threshold \( \theta_\kappa^\mathcal{T}\) , a fault is detected with probability \( \kappa\) .

The relationship between the false alarm rate and threshold can be expressed as

where \( \gamma(\cdot,\cdot)\) denotes the lower incomplete gamma function.

A significant enhancement to the basic time-window approach is the incorporation of temporal weighting that places greater emphasis on recent measurements [288]. This weighted distance measure is defined as

where \( 0 < \varrho \leq 1\) serves as a weighting factor that determines how rapidly the influence of past measurements diminishes. This detector, known as the weighted sum of Chi-squared distributions [287], offers improved sensitivity to recent changes in system behavior while still considering historical context.

The corresponding detection threshold for this weighted measure is

with the false alarm rate relationship given by

Statefull detection approaches introduce additional design parameters beyond those in stateless methods, creating expanded opportunities for performance optimization:

• Window length \( \mathcal{T}\) : Longer windows provide greater noise suppression and sensitivity to subtle faults, but increase detection latency and computational requirements. The window length should be chosen based on the time scale of expected faults and the desired balance between sensitivity and responsiveness;
• Weighting factor \( \varrho\) : Values closer to \( 1\) result in more uniform weighting across the window, while smaller values emphasize recent measurements more strongly. This parameter allows the detector to be tuned for different fault development rates, from abrupt changes to gradual drift; and
• Detection probability \( \kappa\) : As with stateless detection, this parameter directly influences the trade-off between false alarms and missed detections, though stateful methods generally achieve better performance at the same false alarm rate compared to stateless approaches.

The integration of temporal information in stateful detection provides a powerful complement to instantaneous stateless methods. By leveraging both approaches in combination, distributed fault detection systems can achieve robust performance across a wide range of fault scenarios, from sudden, large-magnitude faults to subtle, slowly developing anomalies.

Communication and computational resources represent critical constraints in many CPS deployments. Several research directions have emerged to address these limitations:

• Event-triggered FDI mechanisms [289, 290, 291] optimize communication efficiency by transmitting information only when significant changes occur or when predefined conditions are met. These approaches substantially reduce bandwidth requirements while maintaining detection performance, making them particularly valuable for wireless sensor networks with energy constraints.
• Low-complexity detection algorithms [292] focus on minimizing computational demands through streamlined processing techniques, enabling implementation on resource-constrained embedded platforms common in distributed sensing applications.

Practical CPS deployments invariably encounter various forms of uncertainty that can compromise detection reliability. Several research streams have developed robust approaches to maintain detection performance despite these challenges:

• Model uncertainty resilience [293] enables detection algorithms to function effectively despite imperfect system models, parameter variations, or unmodeled dynamics. These approaches typically incorporate adaptive mechanisms or robust design techniques that account for bounded uncertainty; and
• Communication imperfection handling [294] addresses practical networking challenges such as packet drops, transmission delays, and intermittent connectivity. These methods maintain detection functionality despite unreliable information exchange among distributed nodes.

Beyond algorithmic innovations, several research directions have explored broader architectural approaches to distributed fault detection:

• Heterogeneous system designs [295, 296] accommodate networks composed of diverse sensor types, varying processing capabilities, and mixed measurement modalities. These approaches often employ unknown input observer design techniques to handle the complexity introduced by system heterogeneity; and
• Integrated control and detection frameworks [297] simultaneously address the challenges of distributed control and fault detection, creating unified approaches that maintain both control performance and fault detection capabilities in a coordinated manner.

Several innovative detection paradigms have emerged that extend distributed fault detection beyond traditional approaches:

• Probabilistic threshold-based methodologies have been developed for various system classes, including the following:
• Full-rank dynamical systems [286], where complete state information can be reconstructed from available measurements;
• Rank-deficient systems [298], where structural limitations prevent complete state reconstruction from any single measurement set; and
• Nonlinear dynamical systems [299, 300], which incorporate detection mechanisms that account for nonlinear system behaviors.
• Data-driven approaches leverage advances in machine learning and statistical analysis:
• Clustering-based methods [301] identify and group similar fault patterns, enabling more nuanced detection and classification;
• Machine-learning techniques [302] employ supervised and unsupervised learning to identify complex fault signatures without requiring explicit system models; and
• Data-driven frameworks [303, 304, 305] develop detection algorithms directly from system data, particularly valuable when accurate analytical models are unavailable or prohibitively complex.

These diverse research directions collectively advance the state-of-the-art in distributed fault detection, providing system designers with a rich toolkit of methodologies that can be selected and combined based on specific application requirements.

The integration of renewable energy sources and electric vehicles [308, 309, 310, 311] has fundamentally transformed traditional power networks, introducing several key challenges that distributed algorithms are uniquely positioned to address:

• Intermittent generation: Renewable sources like wind and solar exhibit inherent variability and limited predictability, requiring sophisticated estimation techniques to forecast generation patterns and balance loads;
• Geographic dispersion: Generation assets are increasingly distributed across wide geographic areas, making centralized monitoring impractical;
• Heterogeneous components: Modern grids incorporate diverse technologies with varying characteristics, communication capabilities, and control requirements; and
• Critical reliability requirements: Power infrastructure must maintain high availability while preventing cascading failures that could affect large service areas.

Fig. 19 illustrates a representative distributed monitoring architecture for renewable-energy grids incorporating wind farms, solar installations, and conventional generation. In this configuration, a geographically distributed sensor network enables localized processing and collaborative state estimation through distributed algorithms.

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This approach delivers several key advantages compared to traditional centralized architectures:

• Enhanced resilience: The distributed processing architecture eliminates single points of failure, preventing global system shutdown when individual components malfunction;
• Localized fault management: Distributed fault detection enables precise isolation of faulty assets [298], containing problems before they propagate through the network;
• Scalable deployment: The distributed architecture naturally scales with system expansion, accommodating new generation sources without requiring fundamental redesign; and
• Efficient communication: By processing data near its source, the system minimizes communication bandwidth requirements and reduces latency.

The power industry has adopted various distributed estimation approaches to address specific operational challenges:

• Renewable generation forecasting: Distributed filtering techniques help predict generation patterns from variable renewable sources by combining local measurements from wind turbines and solar panels with meteorological data [312, 313];
• Power quality monitoring: Distributed processing enhances the accuracy of voltage and frequency estimation across the grid, improving power quality assessment capabilities [87, 314];
• Network state estimation: Various distributed approaches have been developed for monitoring the operating condition of power networks, including the following:
• Distributed estimation methods for control centers [315];
• Distributed maximum a posteriori estimation [316];
• Distributed Kalman filtering in batch-mode regression form [317];
• Distributed unscented information filtering [318]; and
• Reliable estimation over unreliable communication networks [319].

Beyond physical state monitoring, distributed algorithms play a crucial role in optimizing power system economics through distributed energy management and economic dispatch. These applications ensure that electricity generation and consumption are balanced economically and optimally [117, 320].

The economic dispatch problem is mathematically formulated as an optimization of generation costs:

where state \( x_i\) represents the generated power at node \( i\) and \( P_{mis}\) is the power mismatch between the generated power and demand. The cost function typically takes a quadratic form:

with parameters \( \gamma_i\) , \( \beta_i\) , and \( \alpha_i\) defined based on the generator type (fueled by oil, gas, coal, just to mention a few) as documented in [321, 322].

Consensus-based distributed estimation protocols, such as the consensus + innovation approach [117], can iteratively solve this optimization problem in a fully distributed manner. This decentralized approach allows individual generators to optimize their output based on local costs while accounting for neighboring units’ output levels [323, 324]. Furthermore, integrating interval observers for fault diagnosis significantly enhances the robustness of distributed economic dispatch against potential system anomalies [325].

In microgrids – localized energy systems composed of multiple interconnected distributed generation units – distributed fault detection and estimation techniques are especially valuable. These approaches enable quick localization and isolation of faults, preventing cascading failures that could compromise the entire microgrid [326, 327, 328]. The distributed nature of these techniques aligns perfectly with the inherently decentralized architecture of microgrids, providing natural fault containment boundaries and facilitating autonomous operation during main grid disconnection.

Social networks manifest across an extraordinarily wide spectrum of contexts, spanning human interactions, animal communities, economic systems, market behaviors, online platforms, citation networks, and numerous other domains [329]. These networks serve as fundamental infrastructures for understanding how information, influence, and behaviors propagate through interconnected social entities.

The underlying social phenomena of interest exhibit similar diversity, including the following:

• Collective decision processes: Voting behaviors, consensus formation, and group judgment;
• Coordinated movements: Flocking, herding, and synchronized behaviors;
• Information diffusion: Rumor propagation, viral content spread, and information cascades;
• Economic patterns: Stock price movements, market trends, and consumer behaviors; and
• Community evolution: Formation, growth, and dissolution of social groups.

Each state within these social dynamics typically represents an opinion, belief, preference, or behavioral characteristic of an individual actor or community member. The interconnected nature of these states – with each potentially influencing and being influenced by others – creates complex dynamical systems that present unique monitoring and analysis challenges.

Fig. 20 illustrates an innovative cyber-social system architecture that leverages distributed estimation for monitoring social dynamics. In this framework, a network of computational agents – whether autonomous monitoring entities or specialized algorithms implemented within digital platforms [6] – observes the states of individuals across a social network and collaboratively processes this information.

The cyber layer (green nodes and connections) forms a processing infrastructure that monitors the social layer (blue and red nodes with blue connections), where

• Red nodes represent influential actors who primarily impact others’ states;
• Blue nodes represent more passive actors whose states are predominantly influenced by others;
• Green monitoring agents collect data from selected individuals; and
• Green connections enable information exchange among monitoring agents.

This architecture enables distributed analysis of social phenomena without requiring centralized data collection, preserving privacy while providing meaningful insights into collective behaviors and opinion dynamics [285, 330].

The evolution of states (opinions, beliefs, preferences) across social networks follows various models that capture how individuals influence each other over time. Several well-established linear models include the following:

• Friedkin-Johnson social influence networks [331], which balance individual stubbornness with social influence;
• The French model [332], which focuses on interpersonal pressure toward uniformity; and
• The DeGroot model [333], which implements weighted averaging of neighboring opinions.

These models are typically linear as in Section 2.1 and share structural similarities with the consensus dynamics described in Section 2.3, though they typically incorporate additional parameters to capture the complex nature of social influence. A representative state update equation in these models takes the form [6, 329]:

where \( x_i\) represents the state (opinion) of individual \( i\) at time step \( t\) , and \( 0 \leq a_{ij} \leq 1\) denotes the influence weight that neighboring individual \( j\) exerts on individual \( i\) ’s state.

The complexity and diversity of social systems have inspired numerous specialized adaptations of distributed estimation and detection techniques:

• Trust-weighted distributed filtering [334] adjusts the influence of different information sources based on their assessed reliability, similar to how individuals in social networks assign varying credibility to different peers;
• Discrete distribution estimation [335] enables monitoring of categorical beliefs or preferences that don’t follow continuous state models;
• Interaction graph-based learning [336, 329, 285] leverages the structure of social connections to improve estimation accuracy;
• Observability-based tracking approaches examine opinion dynamics through various interaction models:
• Ergodic interaction models [337] that capture time-varying influence patterns;
• Cooperative/antagonistic interaction frameworks [296] that differentiate between positive and negative social influences; and
• Structural analysis methods [330, 6] that identify critical monitoring points in complex social networks.
• Gossip-based learning algorithms [338, 339, 340] mimic the natural information diffusion processes in social networks, where information spreads through pairwise interactions; and
• Bias detection algorithms [341, 284] identify potentially manipulated or misleading information flows within social monitoring systems.

These approaches demonstrate how distributed estimation and detection techniques can be effectively adapted to the unique challenges of social system monitoring, enabling insights into collective behaviors while respecting the privacy and autonomy of individual actors. As social interactions increasingly occur across digital platforms that generate vast quantities of behavioral data, distributed approaches to social system monitoring will continue to gain importance for both research and practical applications.

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Localization and tracking encompass related but distinct objectives in cyber-physical systems:

• Localization determines the position of an object or agent within a defined environment, serving as the foundational capability for navigation, interaction, and coordination; and
• Target tracking extends localization by continuously monitoring an object’s state (position, velocity, and potentially acceleration) over time, enabling trajectory prediction and informed decision-making based on anticipated movements.

Both capabilities represent quintessential cyber-physical tasks that integrate physical sensing with computational algorithms to achieve robust performance under uncertainty. The distributed nature of modern sensing systems, where multiple sensors observe a target from different locations, creates both opportunities and challenges for estimation accuracy, reliability, and scalability.

In this section, we focus specifically on relative localization, which determines an entity’s position relative to reference points or other entities within the network. This approach leverages geometric principles such as triangulation [342] and multilateration [343] based on distance measurements or angle calculations [344] from known locations, as illustrated in Fig. 21.

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As a representative example of distributed localization, we examine time-difference-of-arrival (TDOA) measurements. This approach determines the target position by comparing the arrival times of signals at different sensors [345], eliminating the need for precise time synchronization between the target and sensing network.

Consider a network of \( n\) sensors with positions \( \mathbf{p}_i = (p_{x,i}; p_{y,i}; p_{z,i})\) performing localization of a mobile target. The measurement process follows these steps:

1. Each sensor receives a beacon signal (with known propagation speed \( c\) ) from the mobile target;
2. Each sensor records the time \( t_i = \frac{1}{c}\|\mathbf{p}(t) - \mathbf{p}_i(t)\|\) corresponding to the distance between itself and the target;
3. Sensors share their time measurements and positions with neighboring sensors in set \( \mathcal{N}_i\) ;
4. By subtracting measurements, each sensor computes TDOA values, which convert to range-difference information:

where the observation matrix \( C_i\) is defined as

with \( \mathbf{p}_{j,i} := \mathbf{p}_{j}-\mathbf{p}_{i} := (p_{x,j,i}; p_{y,j,i}; p_{z,j,i})\) defined as the relative position vector between sensors \( j\) and \( i\) .

For fixed sensor positions, the bias term \( \|\mathbf{p}_j\|^2 - \|\mathbf{p}_i\|^2\) remains constant and known. By incorporating this known bias, the TDOA measurement can be expressed in standard form:

where \( \boldsymbol{\mu}_i(t)\) represents the additive measurement noise.

In real-world tracking scenarios, target dynamics are generally unknown and must be approximated using mathematical models. The most widely used target motion models follow the general linear form:

where \( \mathbf{x}(t)\) represents the target state vector, matrices \( A\) and \( G\) describe the transition and input dynamics, and \( \boldsymbol{\nu}(t)\) represents process noise or random inputs.

Three primary models dominate the literature [346, 347, 348, 349, 350, 351, 343]:

• Nearly-Constant-Velocity (NCV) Model: This approach models position and velocity in 3D space with state vector

  \[ \begin{align} \mathbf{x} = \begin{pmatrix} p_x \\ p_y \\ p_z \\ \dot{p}_x \\ \dot{p}_y \\ \dot{p}_z \end{pmatrix}. \end{align} \]

  (76)

  The transition and input matrices are defined as

  \[ \begin{align} A = \begin{pmatrix} \mathbf{I}_3 & T\mathbf{I}_3 \\ \mathbf{0}_3 & \mathbf{I}_3 \end{pmatrix}, ~ G = \begin{pmatrix} \frac{T^2}{2}\mathbf{I}_3 \\ T\mathbf{I}_3 \end{pmatrix}, \end{align} \]

  (77)

  where \( \mathbf{I}_3\) and \( \mathbf{0}_3\) represent 3×3 identity and zero matrices, and \( T\) denotes the sampling interval.

• Nearly-Constant-Acceleration (NCA) Model: This model incorporates acceleration terms, expanding the state vector to

  \[ \begin{align} \mathbf{x} = \begin{pmatrix} p_x \\ p_y \\ p_z \\ \dot{p}_x \\ \dot{p}_y \\ \dot{p}_z \\ \ddot{p}_x \\ \ddot{p}_y \\ \ddot{p}_z \end{pmatrix}, \end{align} \]

  (78)

  with corresponding matrices

  \[ \begin{align} A = \begin{pmatrix} \mathbf{I}_3 & T\mathbf{I}_3 & \frac{T^2}{2}\mathbf{I}_3 \\ \mathbf{0}_3 & \mathbf{I}_3 & T\mathbf{I}_3 \\ \mathbf{0}_3 & \mathbf{0}_3 & \mathbf{I}_3 \end{pmatrix}, ~ \text{ and } ~ G = \begin{pmatrix} \frac{T^2}{2}\mathbf{I}_3 \\ T\mathbf{I}_3 \\ \mathbf{I}_3 \end{pmatrix}. \end{align} \]

  (79)

• Singer Model: This probabilistic approach enhances the NCA model by incorporating a maneuvering parameter \( \alpha = \frac{1}{\theta}\) , where \( \theta\) represents the maneuver time constant:

  \[ \begin{align} A = \begin{pmatrix} \mathbf{I}_3 & T\mathbf{I}_3 & \frac{\alpha T-1+e^{-\alpha T}}{\alpha^2}\mathbf{I}_3 \\ \mathbf{0}_3 & \mathbf{I}_3 & \frac{1-e^{-\alpha T}}{\alpha}\mathbf{I}_3 \\ \mathbf{0}_3 & \mathbf{0}_3 & e^{-\alpha T}\mathbf{I}_3 \end{pmatrix}, ~ G = \begin{pmatrix} \mathbf{0}_3 \\ \mathbf{0}_3 \\ \mathbf{I}_3 \end{pmatrix}. \\\end{align} \]

  (80)

These models are interrelated and can transform into one another under specific conditions. For instance, the Singer model approaches the NCV model as the maneuver time constant \( \theta\) decreases, and converges toward the NCA model as \( \theta\) increases [352]. Larger \( \theta\) values represent gradual, predictable maneuvers, while smaller values indicate abrupt, evasive movements.

The literature on distributed algorithms for localization and target tracking has expanded significantly in recent years, addressing various aspects of the tracking problem:

• Consensus-based localization and tracking [342, 353, 354, 355] leverages the consensus techniques discussed in Section 3.1 to enable collaborative position estimation across sensor networks;
• TDOA-based distributed tracking has been developed for both delay-free networks [348, 349, 356, 357] and time-delayed setups [358, 359, 343], addressing practical implementation challenges in real-world sensing systems;
• Advanced filtering approaches for distributed tracking include:
• Distributed maximum likelihood Kalman filters [360] that optimize estimation under probabilistic uncertainty models;
• Consensus Kalman filters [361] that combine consensus mechanisms with optimal filtering;
• Adaptive filters [362] that dynamically adjust to changing target dynamics; and
• Event-triggered filters [363, 364] that reduce communication overhead by transmitting only significant updates.
• Specialized tracking scenarios have been addressed through the following approaches:
• Nonholonomic target models [365] for targets with constrained motion capabilities;
• Asynchronous communication networks [366] that accommodate realistic timing variations;
• Tracking under cyber attacks [367] to maintain performance despite malicious interference;
• Fault-tolerant approaches for leader-follower structures [368]; and
• Belief propagation techniques [369] that leverage probabilistic graphical models for efficient information fusion.

These diverse approaches demonstrate how distributed estimation and detection techniques can be effectively applied to the fundamental challenge of target tracking, enabling robust performance across a wide range of sensing conditions, target behaviors, and network configurations.

Mixed traffic transportation systems consist of connected autonomous vehicles (CAVs) and human-driven vehicles (HDVs) sharing the same roadway, see Fig. 22. In these settings, accurate knowledge of other vehicles’ states (e.g., positions, velocities) is crucial for safe driving. Traditional centralized estimation methods collect all sensor data at a central unit to infer traffic states, but they face scalability issues and vulnerability to single-point failures. To address these challenges, distributed estimation has emerged as a promising alternative, where individual CAVs estimate the state of HDVs via local sensing and information exchange with neighboring CAVs, see [370, 371, 372, 373] for details.

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To formulate the most general scenario, consider the system-measurement model (1)-(2) as a general model for group of \( N\) HDVs. The dynamics of every HDV is modelled by the NCV and NCA dynamics described by Eqs. (76)-(79), where the HDV’s state is a variable in \( \mathbb{R}^m\) . The state of HDVs is then tracked by every CAV \( i\) denoted by  \( \widehat{\mathbf{x}}^i_{k|k}\in\mathbb{R}^{N m}\) as the estimate of \( \mathbf{x}_k\in\mathbb{R}^{N m}\) . Every CAV uses all the available measurements over the communication network. Then, the global estimate of the states of HDVs is defined as,

which represents the estimate of the global state of HDVs as a networked system defined as,

Then, the mixed traffic ITS dynamics associated with \( \underline{\mathbf{x}}_{k}\) is [370]:

which follows the sochasticity of \( W\) matrix defined by (8). Then, the distributed estimation of the mixed traffic ITS modelled by a network of \( n\) CAVs tracking the state of \( N\) HDVs follows the observability of the pair

with \( D_C\) representing the shared measurements as defined in (14). This is referred to as distributed observability in Section 3.1.1. Then, a distributed estimator, e.g., (10)-(11), can be adopted to address this tracking problem.

Cooperative ITS represents a transformative approach to transportation management that integrates advanced communication technologies with distributed computational capabilities. By enabling vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communications [374], these systems create rich information ecosystems that support real-time decision-making across multiple scales – from individual vehicles to entire transportation networks.

In this cooperative framework, distributed algorithms play a critical role in the following:

• Enhancing scalability to accommodate growing numbers of connected vehicles and infrastructure elements;
• Improving robustness against individual component failures or communication disruptions;
• Enabling localized decision-making while maintaining global coordination;
• Reducing latency for safety-critical applications through edge processing; and
• Preserving privacy by minimizing the centralized collection of sensitive movement data.

A fundamental challenge in managing transportation networks involves accurate estimation of critical traffic states, including vehicle speeds, traffic density, and travel times. Distributed algorithms address this challenge by aggregating data from heterogeneous sources – including roadside cameras, in-vehicle GPS units, infrastructure sensors, and mobile devices – to create comprehensive yet efficient traffic models [375, 376].

Various distributed fusion techniques have been developed for traffic state estimation, e.g.

• Model-based approaches that leverage dynamic traffic flow models:
• Distributed unscented Kalman filtering [377] for nonlinear traffic dynamics;
• Consensus filter-based distributed variational Bayesian algorithms [378] that incorporate uncertainty quantification; and
• Parallelized particle filtering [379] for handling complex, non-Gaussian traffic patterns.
• Constraint-based methods that incorporate physical limitations and boundary conditions:
• Probabilistic-constrained distributed fusion filters [380] that maintain estimation consistency; and
• Set-membership filtering with attack detection capabilities [381] to ensure reliable operation despite malicious interference.
• Resource-aware frameworks that balance computational loads:
• Distributed dynamic computation offloading [382] that optimizes processing across available resources.

Precise vehicle positioning represents a critical capability for advanced transportation systems, particularly for autonomous driving applications. Distributed approaches enable vehicles to share positioning data with nearby units, significantly improving localization accuracy beyond what individual vehicles could achieve in isolation [383, 384].

These collaborative systems employ sophisticated sensor fusion techniques that integrate data from multiple sources, including the following:

• Global Navigation Satellite Systems (GNSS);
• Inertial Measurement Units (IMUs);
• Light Detection and Ranging (LiDAR) sensors;
• Computer vision systems; and
• Infrastructure-based reference points.

By combining these diverse information sources through distributed algorithms, vehicles can achieve robust localization even in challenging urban environments with GPS signal obstruction, adverse weather conditions, or ambiguous visual features [385].

Beyond individual vehicle capabilities, distributed algorithms enable system-wide coordination that improves overall traffic efficiency:

• Adaptive signal control systems allow vehicles to communicate expected arrival times at intersections, enabling dynamic adjustments to signal timing that reduce unnecessary stops and congestion [386, 387]; and
• Real-time routing optimization leverages distributed fusion of traffic data to identify congestion patterns and suggest alternative pathways, reducing travel times across the network [388, 389].

These distributed approaches offer significant advantages over traditional centralized traffic management by

• Responding more rapidly to changing conditions;
• Scaling efficiently as network complexity grows;
• Continuing to function even when some components fail; and
• Adapting to localized traffic patterns while maintaining global coordination.

Vehicle platooning represents one of the most promising applications of distributed algorithms in transportation systems. This approach coordinates multiple automated vehicles travelling in close proximity, as illustrated in Fig. 23, offering benefits including reduced fuel consumption through improved aerodynamics, increased road capacity, and enhanced safety through coordinated maneuvers.

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Platooning is typically facilitated through vehicular ad-hoc networks [390] that enable real-time communication among vehicles and with surrounding infrastructure. Within these networks, distributed algorithms support several critical functions:

• Distributed observer-based tracking enables vehicles to maintain precise inter-vehicle spacing while accounting for dynamic environmental conditions [391, 392, 370];
• Security-enhanced platooning algorithms address the unique vulnerabilities introduced by vehicle connectivity:
• Resilient distributed event-triggered approaches that maintain functionality despite denial-of-service attacks [393, 394];
• Secure platooning protocols designed to withstand various cyber threats [395];
• Robust estimation techniques that maintain reliability despite compromised measurements [396];
• Specialized detection algorithms for identifying false-data injection attacks [397]; and
• Finite-time attack detection and estimation methods that provide rapid response to potential threats [398].

The integration of distributed estimation and fault detection techniques into vehicle platooning systems creates resilient, adaptive formations that can maintain safe operation even under challenging conditions, including communication disruptions, sensor failures, or malicious interference. As transportation infrastructure continues to evolve toward greater connectivity and automation, the role of these distributed techniques will become increasingly central to ensuring reliable, efficient mobility.

As distributed systems continue to scale, communication efficiency becomes increasingly critical. Future research could focus on the following:

• Event-triggered protocols that dynamically adjust communication rates based on system conditions, reducing unnecessary data transmission while preserving estimation performance;
• Compressed sensing techniques that enable accurate reconstruction of system states from sparse measurements, minimizing the volume of data exchanged between nodes; and
• Strategic information sharing that optimizes which data to exchange based on information content rather than predetermined schedules, enhancing efficiency in bandwidth-constrained environments.

The increasing availability of operational data creates opportunities to enhance distributed algorithms through machine learning:

• Hybrid model-based and data-driven techniques that combine the interpretability of first-principles models with the adaptability of learning-based approaches;
• Distributed reinforcement learning for adaptive estimation and filtering that improves performance through experience without requiring centralized training; and
• Transfer learning methods that enable knowledge sharing between related but distinct monitoring tasks, reducing the data requirements for new applications.

As distributed algorithms process increasingly sensitive information, privacy and security considerations become paramount. Specifically, we have the following:

• Privacy-preserving distributed estimation techniques that enable collaboration without exposing raw measurements, using approaches such as differential privacy, secure multi-party computation, or homomorphic encryption;
• Resilient distributed algorithms designed specifically to withstand sophisticated cyber attacks, including data manipulation, topology poisoning, and inference attacks; and
• Trust mechanisms that dynamically assess the reliability of information sources and adjust fusion weights accordingly.

Emerging computational paradigms offer new possibilities for implementing distributed algorithms:

• Edge-cloud collaborative architectures that strategically partition computation between local devices and cloud resources based on latency requirements, computational intensity, and available bandwidth;
• IoT integration frameworks that accommodate the heterogeneity, resource constraints, and intermittent connectivity characteristic of IoT deployments; and
• Specialized hardware accelerators for distributed estimation that enable more efficient implementation of key computational kernels.

As distributed algorithms support increasingly critical applications, performance guarantees become essential to ensure the following:

• QoS-aware distributed designs that explicitly consider application-specific performance metrics during algorithm selection and parameter tuning;
• Resource allocation frameworks that dynamically balance computational and communication resources across competing objectives based on current system priorities; and
• Formal verification methods for distributed algorithms that provide provable guarantees about estimation performance, fault detection reliability, and system stability.

These research directions collectively represent pathways toward more efficient, reliable, secure, and capable distributed algorithms for cyber-physical systems. As these systems continue to grow in scale, complexity, and importance, advances in distributed estimation, filtering, and fault detection will play an increasingly critical role in ensuring their safe and effective operation.