Reduced-Order Distributed Fault Diagnosis for Large-Scale Nonlinear Stochastic Systems

The majority of contemporary industrial and commercial control systems are composed of a large number of spatially distributed feedback modules with heterogeneous sensors, actuators, and controllers that exchange information over a band-limited communication network that is embedded within the system. These large-scale systems are characterized by high-dimensional state-spaces and nonlinear dynamics. Typical applications are water distribution networks, power grids, automated highway systems, swarms of unmanned aerial vehicles, and environmental control systems, just to name a few. Large-scale systems are much more vulnerable to faults since the effects of a single malfunction to an individual part may rapidly diffuse to the entire system due to the interconnection of the various subcomponents.

Availability, dependability, and resiliency are becoming major design goals for large-scale technological systems due to stringent economic, ecological, and safety demands. These attributes are of major importance, primarily for safety-critical systems, e.g., airplanes, automobiles, and nuclear reactors, since they ensure public safety. Therefore, there is a growing need for reliable real-time monitoring and supervision especially in the case of safety-critical systems. Fault diagnosis (FD) describes the dual objective of detecting the occurrence of a fault (detection) and identifying it (identification or isolation). A timely diagnosis of a fault mode may improve the system's availability and maintainability by avoiding down-times, breakdowns, and catastrophic failures rates.

Research in the field of FD has attracted significant attention since the beginning of the 1970s. The significant majority of existing FD methods [1–6] have a centralized architecture in the sense that the sensor measurements and the diagnostic algorithm are collected and executed by a singleton processing unit. Centralized FD is considered a matured field that has established reliable solutions to many engineering applications. However, the applicability of this traditional approach is limited to concentrated low-order systems.

Modern processes involve high-dimensional state-spaces as well as highly nonlinear dynamics. In the case of large-scale and spatially distributed systems, centralized FD becomes ill-suited. Every monitoring system has certain limitations in terms of computational power and communication bandwidth. When the dimensionality and complexity of the system increases, it is likely that these limitations will not be satisfied by a centralized configuration. The online monitoring of a high-dimensional system would require extensive computations from the central processing unit. Processes with geographically remote subcomponents necessitate long distance and energy demanding broadcasts or complex multihop routing protocols to transmit information to the central fusion center. In both cases, a centralized architecture exhibits poor scalability.

In the literature, the majority of distributed fault detection and identification methods are developed for discrete-event systems and for multiprocessor computing applications [7–9]. A growing interest in the development of distributed fault detection algorithms has also been reported by the wireless sensor networks community [10]. In this work our attention is spotlighted to model-based FD methods for dynamic systems that utilize a mathematical model of the process. A rudimentary classification of existing distributed FD model-based methodologies can take place based on the data type that is exchanged between the nodes of the diagnostic system. The diagnostic nodes (DNs) can exchange: raw measurements of the interconnected states [11–13], state estimates [14,15], or fault signatures [16]. The most prominent work on distributed, observed-based FD for nonlinear dynamic systems has been reported by Ferrari et al. [11] and Boem et al. [17]. The authors apply overlapping decomposition techniques to subdivide the monolithic process to a set of reduced order subsystems. Each subsystem is monitored by a local nonlinear observer. Seminal work in distributed estimation-based FD using Kalman filtering is reported in Ref. [18]. The algorithm is based on the distributed version of Kalman filter (KF) established by Olfati-Saber [19,20]. The KF is restricted to linear system, while linearization (extended KF) leads to high false alarms rates [21].

Foundational work on estimation-based FD for nonlinear systems that employ large number of correlated sensors is introduced in Refs. [22,23]. The author combines a distributed particle filtering algorithm for state estimation with fused hypothesis testing through likelihood tests, to determine the occurrence of fault modes. The proposed method is mainly geared toward relative low-order systems monitored by a high number of interconnected sensors. The layout of the algorithm does not accommodate subdivisions of the original process. The applied classical likelihood tests require a bank of estimators equal to the number of fault modes. Such replication of the state is unsuitable for large-scale systems. A full-order distributed failure-sensitive filter has also been introduced by Noursadeghi and Raptis [24,25]. In this scheme, a detection network is assigned to monitor the entire state of the monolithic system using only local measurements. Similarly to the work of Cheng, the authors did not considered any form of subdivision in the dynamics of the original process. Instead, the algorithm introduced in Refs. [24,25] provides an estimate of the entire state of the monolithic system.

The particle filter (PF) is an ideal estimator for fault diagnosis since it avoids linearity and Gaussian noise assumptions. A comprehensive analysis on distributed PF algorithms is given by Hlinka [26,27] and Mohammadi [28,29]. A distributed PF scheme for FD diagnosis that accounts for system decomposition is reported in Ref. [30]. The authors propose a hybrid modeling approach where every potential fault is treated as a system mode. This approach assumes that the transition probability between the fault modes is known a priori. This probabilistic information may not be available in most real-life applications.

In this work, we present a distributed, model-based and sequential fault diagnosis methodology for large-scale, stochastic nonlinear systems that are subject to multiple fault modes. This approach targets systems where the state dimension is significantly large (10² states and higher). A distributed version of the particle filtering method will serve as the foundation of the derived diagnostic algorithms. We introduce a reduced-order fault diagnostic algorithm that allows the subdivision of the original process dynamics to low-order interconnected subsystems with state overlap. A DN is assigned to monitor every partition of the monolithic system and triggers alarm indicators based on its local observations and information exchange between neighboring units. Each local failure sensitive filter outputs an estimate of the subsystem's state vector and the probabilities of failure of the local fault modes.

Our reduced-order FD technique achieves a dramatic decrease to the computational complexity of the original problem and provides significant design flexibility to the layout of the algorithm. The PF is an ideal estimator since it eliminates complex Lyapunov arguments that are required by the observed-based methods to guarantee convergence. A binary update rule is used to repopulate the particles and estimate the system modes without the need for transition probabilities. The failure sensitive filter can simultaneously detect and identify faults without the need for a bank of estimators. The proposed algorithm takes advantage of the decentralized architecture and computational strength of modern embedded systems such as wireless sensor networks and multicore processors.

This paper is organized as follows: A brief description of the PF algorithm is presented in Sec. 2. The synthesis of a centralized PF fault diagnosis algorithm is outlined in Sec. 3. The centralized algorithm serves as a benchmark framework for its distributed counterpart. The reduced-order distributed version of the failure sensitive filter is presented in Sec. 4. The performance of the proposed methodology is evaluated in Sec. 5 via numerical simulations. Finally, concluding remarks are given in Sec. 6.

Different variations of the PF algorithm exist depending on the choice of the importance density function and the resampling step. The most standard form of the PF algorithm is the sequential importance resampling filter (SIR). The SIR filter forms the foundation for some well-known PFs including the bootstrap filter [32], the auxiliary PF [33], and the regularized PF [33]. These PFs are derived using a suboptimal choice of the proposal pdf $q(x(k)|z(k−1),z(k))$⁠.

where $N(ei(k),0,Σω)$ is the normal distribution with zero mean and covariance matrix $Σω$ evaluated at the points $ei(k)=z(k)−h(xi(k))$⁠, where $ei(k)$ is the prediction error of the ith particle. For a detailed description of various PF algorithms and resampling techniques, the reader is referred to Ref. [33]. The pseudocode of the bootstrap filter and resampling are provided in Tables 1 and 2, respectively. The block diagram of the bootstrap algorithm is shown in Fig. 1.

This work extends the methodology introduced in Ref. [34] from one-dimensional fault-growth models to dynamic state-space systems of nonlinear processes introducing the centralized particle filtering fault diagnosis (CPFFD) algorithm. The CPFFD algorithms generate two outputs. The first is the system's state estimate from a sequence of noise infested measurements. The second output is a statistical characterization for the occurrence of each fault mode that can trigger fault alarms.

where the terms $x(k)∈ℝnx$⁠, $u(k)∈ℝnu$⁠, and $z(k)∈ℝnz$ refer to the state, input, and measurement vector, respectively; $f(k):ℝnx→ℝnx$⁠, and $h(k): ℝnx→ℝnz$ denote the known nonlinear functions of the system's healthy dynamics and measurement model, while $v(k)∈ℝnx$ and $ω(k)∈ℝnz$ stand for the process and measurement noise sequences, respectively.

It is assumed that the system is initiated from the healthy mode (⁠$β(·)=0$ at k = 0). Due to the random occurrence of the possible faults, the monolithic system may be viewed as a hidden Markov model, where the transition probabilities between the different system modes are unknown.

The proposed failure sensitive filter embeds the dynamics of the monolithic system S given in Eq. (8), as well as a binary variable (for every potential fault), that identifies the changes in the process dynamics expressed by the terms $β(k−k0j)$⁠. Hence, the binary state vector $bj(k)=[b1j(k)b2j(k)]T$⁠, with $b1j,b2j∈{0,1}$ and $j=1,…,M$⁠, is introduced to estimate the occurrence of each fault mode. More specifically, $b1j(k)=1$ indicates that the absence of failure mode j, while $b2j(k)=1$ denotes that the fault mode j is detected to the system. The continuous-valued states are coupled with the discrete-valued binary fault occurrence estimates resulting in a hybrid model.

where $ṽ(k)∈ℝnx$ and $ω̃(k)∈ℝnz$ are approximations of the failure sensitive filter's process and measurement noise, respectively. These noise sequences should be as close as possible to the actual ones (⁠$v(k)$ and $ω(k)$⁠). The nonlinear function $Φ:ℝ2→{[0 1]T,[1 0]T}$⁠, represents the evolution of the binary states driven by the identically independent distributed (i.i.d) uniform white noise $nj(k)$⁠. The function $Φ(·)$ is defined such that the previous state $bj(k−1)$ is randomly excited at each time step by $nj(k−1)$⁠. This random vector of $ℝ2$ is assigned to one of the binary states (normal/faulty operating condition) based on the distance metric of the perturbed vector $bj(k−1)+nj(k−1)$ to the coordinates $[0 1]T$ and $[1 0]T$⁠.

where $Z(k)=z(k), V(k)=[ṽ(k) n1(k)…nM(k)]$⁠, and $F(·), H(·)$ are nonlinear functions of appropriate dimensions and structure. The aforementioned definition will be used to ease the notation in subsequent parts of the analysis. The outputs of the CPFFD module are the estimation of the systems's state vector and the probabilities of failure of each fault mode. These probabilities are the expectations of the binary states $b̂2j(k)=E[b2j(k)|Z(k)]$⁠. This measure is used to trigger alarm indicators if its value exceeds a certain threshold $α∈(0;1)$ that marks the probability of detection (i.e., $b̂2j(k)<α$ indicates that the system is in healthy operating condition). With this layout, two or more different co-existing fault modes can be simultaneously detected. The pseudocode of the CPFFD algorithm is given in Table 3.

The probability of failure is a much more computationally attractive measure compared to classical change detection methods such as hypothesis testing. In the context of fault isolation, detection algorithms using hypothesis testing through logarithm likelihood ratio test requires the execution of a bank of estimators that is equal to the fault modes. For large-scale systems, this computational load is prohibited. The proposed CPFFD algorithm is significantly more efficient, since it increases the dynamics of the detector by only M binary state vectors.

The CPFFD algorithm described in Sec. 3 is not scalable or robust to complex large-scale dynamical systems that employ scattered measurement sensors over large geographical regions. For high-dimensional large-scale systems, this methodology becomes impractical due to limitations in the observation range of sensors, communication bandwidth, and computation power of the centralized computing node.

In this section, we present a reduced-order distributed particle filtering fault diagnosis (DPFFD) algorithm for large-scale nonlinear systems. The original diagnostic problem is subdivided to a number of lower-order interconnected fault sensitive filters. With this technique, each low-order filter can balance its computation power requirements and volume of data transfers. Similar to Ref. [37], we take into account subdivisions with state overlap. The states that are common between two or more subsystems are referred to as shared states. Shared states between subsystems appear when state variables are mutually monitored by sensors that correspond to different subsystems.

Here, we briefly illustrate the three most characteristic types of decomposition based on a similar description given in Ref. [37]. The most communication intensive decomposition involves nonoverlapping subsystems of order one (Fig. 2(a)). This fragmenting is the most computationally effective, however, most likely the communication limitations will be reached. On the contrary, the decomposition depicted in Fig. 2(b) provides a balanced compromise between computational labor and communication broadcasts. It is important to note that there exists a trade-off between computation power and communication capacity for the nodes of the network. The third case (Fig. 2(c)) is similar to the previous scenario with the difference that there is overlap between the subsystems. In principal, overlapping dynamics increase both the complexity and the communication requirements of the overall design. This additional complexity overhead is due to the fusion of the common measurements and shared state estimates between the nodes. The overlap can increase the fidelity of states and measurements that are exposed to higher uncertainty, since they are monitored by more than one sensor. This fact can justify the additional effort in terms of complexity and communications that stems from overlapping states. Decomposition techniques are out of the scope of this paper, and the interested readers are referred to Refs. [38,39].

Graph theoretical tools are deployed to represent the dynamical interdependence of the system's states [37]. A directed graph or digraph provides a pictorial representation of the system's structure [40]. The digraph of system S is defined as the pair $Gs=(Vs,Es)$⁠, where $Vs=χ∪V∪B$ is the set of vertices consisted of the system states $χ={Xj}j=1,…,nx$⁠, the noise inputs $V={vj}j=1,…,nx$⁠, and the scalar terms $B={βj}j=1,…,M$⁠. The set $Es$ represents the oriented edges defined by the ordered pairs ${νl,νm}$⁠, where $νl∈Vs$ and $νm∈χ$⁠. An oriented edge exists between the state X_l (or v_l and β_l) and state X_m, if the former appears at the dynamic equation of the latter. If an edge exists between vertices ν_l and ν_m, we call them adjacent and denote this relationship by $νl∼νm$⁠. We define the neighborhood $Vm⊆Vs$ of the vertex $Xm∈χ$ as the set $Vm={Xl∈χ|{Xl,Xm}∈Es}$ of all adjacent states to X_m. The digraph $Gs$ is also referred to as structural graph of the system S.

From a graph theoretical perspective, each subsystem S_I of the monolithic process is represented by a cut-point set of vertices $Vs(I)$⁠, where $Vs(I)⊆χ$⁠. Each cut-point set includes states that are observed locally by sensors of its corresponding subsystem. The components of χ that belong to the cut-point set $Vs(I)$ comprise the local states of subsystem S_I. States from subsystems with departing directed edges that enter the vertices of a cut-point set determine the interconnection variables or forcing terms.

where $uI(k)∈ℝnuI$ and $zI(k)∈ℝnzI$ refer to the control input and measurement vector of subsystem I, respectively. The nonlinear functions $fI(·,·,·) : ℝnxI×ℝndI×ℝnuI→ℝnxI$ and $hI(·) : ℝnxI→ℝnzI$ denote to the local subsystem and measurement dynamics; while $vI(k)∈ℝnxI$ and $ωI(k)∈ℝnzI$ stand for the subsystem and measurement noise, respectively.

Likewise to the centralized approach, the formulation of the reduced order local PF for fault diagnosis will include a vector of binary states to represent the absence or presence of each fault mode. The binary vector of failure mode j at subsystem I is represented by $bIj(k)=[b1,Ij(k) b2,Ij(k)]T$ with the local fault function $gIj(·)$⁠.

The aforementioned definitions are illustrated with a simple example. Consider a three-dimensional system with the global state vector $x(k)=[X1(k),X2(k),X3(k)]T$⁠, the noise vector $v(k)=[v1(k),v2(k),v3(k)]T$⁠, and the set of change step functions $B={β1(k),β2(k)}$⁠. The digraph of this example is shown in Fig. 3. Each sensor set monitors one subsystem. The monolithic system of the this example is decomposed into two subsystems represented by the cut-point sets $Vs(1)={X1,X2}$ and $Vs(2)={X2,X3}$⁠. The local dynamic models of these two overlapping subsystems are

where $XI=[(xI)T(bI1)T…(bI2)T…(bIM)T]T∈ℝnxI+2·M$⁠, $UI=[(uI)T (dI)T]T$⁠, $ZI=zI, VI=[ṽIT nIT]T$⁠, and $FI(·), HI(·)$ are nonlinear functions with appropriate dimensions and structure. The diagnostic algorithm includes the design of one DN for each subsystem S_I. Each DN consists of a processing unit that executes the local PF algorithm. The nodes can measure their own local states and communicate with their neighbors to obtain a processed estimate of their forcing term vector. The layout of the proposed DPFFD algorithm is depicted in Fig. 4. The algorithm can be separated into three main parts.

Particles update: In the first part, each DN executes a local bootstrap PF. For every subsystem, N_s particles are drawn according to the state transition propagation given in Eq. (15). This action requires estimates of the forcing terms $d̂I$ obtained by the neighbors of DN I. At time k − 1 all subsystems have already generated an estimate of their own states.

The local PF is concluded after the weight normalization and resampling steps of the bootstrap filter (Sec. 2). The outputs of each DN are an estimation of the subsystems' state vector $x̂I(k)$⁠, and the probabilities of failure $b̂2,Ij(k)=E[b2,Ij(k)|ZI(k)];j=1,…,M$⁠.

Shared states fusion: The last part of the reduced-order DPFFD algorithm involves the fusion of state variables that belong to overlapping cut-point sets. At each time step, the estimates of all DN are collected by a central fusion center that assembles the final global output of the diagnostic network. The data transmitted to the central unit contain only postprocessed information. This is the only centralized processing action that takes place on the fusion center and does not add significant computational overhead to the algorithm. A running average filter is executed between the shared states to calculate a common estimate of their value.

The block diagram, with the breakdown of the reduced-order DPFFD algorithm's steps, is depicted in Fig. 5. The corresponding pseudocode of the proposed algorithm is outlined in Table 4.