160 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

Dynamic Multiple Fault Diagnosis: MathematicalFormulations and Solution Techniques

Satnam Singh, Member, IEEE, Anuradha Kodali, Kihoon Choi, Krishna R. Pattipati, Fellow, IEEE,Setu Madhavi Namburu, Shunsuke Chigusa Sean, Danil V. Prokhorov, and Liu Qiao

Abstract—Imperfect test outcomes, due to factors such as un-reliable sensors, electromagnetic interference, and environmentalconditions, manifest themselves as missed detections and falsealarms. This paper develops near-optimal algorithms for dynamicmultiple fault diagnosis (DMFD) problems in the presence ofimperfect test outcomes. The DMFD problem is to determinethe most likely evolution of component states, the one that bestexplains the observed test outcomes. Here, we discuss four for-mulations of the DMFD problem. These include the deterministicsituation corresponding to perfectly observed coupled Markovdecision processes to several partially observed factorial hiddenMarkov models ranging from the case where the imperfect testoutcomes are functions of tests only to the case where the testoutcomes are functions of faults and tests, as well as the casewhere the false alarms are associated with the nominal (faultfree) case only. All these formulations are intractable NP-hardcombinatorial optimization problems. Our solution scheme canbe viewed as a two-level coordinated solution framework forthe DMFD problem. At the top (coordination) level, we updatethe Lagrange multipliers (coordination variables, dual variables)using the subgradient method. At the bottom level, we use adynamic programming technique (specifically, the Viterbi decod-ing or Max-sum algorithm) to solve each of the subproblems,one for each component state sequence. The key advantage ofour approach is that it provides an approximate duality gap,which is a measure of the suboptimality of the DMFD solution.Computational results on real-world problems are presented. Adetailed performance analysis of the proposed algorithm is alsodiscussed.

Index Terms—Dynamic faults, hidden Markov models, imper-fect tests, intermittent faults, multiple fault diagnosis.

Manuscript received August 29, 2007; revised February 24, 2008. Currentversion published December 17, 2008. This work was supported in part by theToyota Technical Center and in part by the Office of Naval Research underContract 00014-00-1-0101. Earlier versions of this paper were published atthe IEEE Aerospace Conference, Big Sky, MT, March 2007 and the DX-07Workshop, Nashville, TN, May 2007. This paper was recommended by Asso-ciate Editor G. Biswas.

S. Singh is with General Motors India Science Laboratory, Bangalore560066, India (e-mail: satnam.singh@gm.com).

A. Kodali, K. Choi, and K. R. Pattipati are with the Electrical and ComputerEngineering Department, University of Connecticut, Storrs, CT 06269 USA(e-mail: krishna@engr.uconn.edu).

S. M. Namburu, S. C. Sean, D. V. Prokhorov, and L. Qiao are with theTechnical Research Department, Toyota Motor Engineering and ManufacturingNorth America, Inc., Erlanger, KY 41018 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSMCA.2008.2007986

I. INTRODUCTION

ONLINE vehicle health monitoring and fault diagnosis isessential to improve the vehicle availability via condition-

based and opportunistic maintenance, and to reduce main-tenance and operational costs by seamlessly integrating theonboard and offline diagnosis, thereby reducing troubleshoot-ing time. During online (dynamic) fault diagnosis, the test out-comes are obtained over time as compared to static faultdiagnosis where the observed test outcomes are available as ablock. Online vehicle health monitoring heavily relies on the ex-tensive processing of data in real time, which is made possibleby smart onboard sensors. Using these intelligent sensors, thesystem parameters that are essential to vehicle fault diagnosiscan be transmitted to an onboard diagnostic inference engine.

A significant technical challenge in onboard vehicle healthmonitoring is the quality of tests. Generally, the tests are im-perfect due to unreliable sensors, electromagnetic interference,environmental conditions, or aliasing inherent in the signatureanalysis of onboard tests. The onboard tests can be symptoms,manifestations, alarms, or residuals generated from sensor datausing trending, range checking, dynamic thresholding, etc. Theimperfect tests introduce additional elements of uncertainty intothe diagnostic process: The pass outcome of a test does notguarantee the integrity of components under test because thetest may have missed a fault; on the other hand, a fail outcomeof a test does not mean that one or more of the implicatedcomponents are faulty because the test outcome may have beena false alarm. Hence, it is desired that an onboard diagnosticalgorithm should be able to accommodate missed detectionsand false alarms in test outcomes. The performance of onboarddiagnosis can be improved by incorporating the knowledge ofreliabilities of tests and by incorporating temporal correlationsof test outcomes.

The hidden Markov model (HMM) is a natural choice hereto represent the individual component states of the system. TheHMM is a doubly embedded stochastic process with an un-derlying unobservable (hidden) stochastic process (individualcomponent state evolution), which can be observed throughanother set of stochastic processes (i.e., uncertain test outcomesequences). The individual component state HMMs are coupledthrough the observation process. Consequently, the fault diag-nosis problem corresponds to a factorial HMM (FHMM), whereeach HMM characterizes the individual component states of thesystem. The sequence of uncertain test outcomes are probabilis-tic functions of the underlying Markov chains characterizingthe evolution of system states. In this paper, we investigate the

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SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 161

problem of determining the most likely states of components,given a set of partial and unreliable test outcomes over time.

A. Organization of This Paper

This paper is organized as follows. In Section II, we dis-cuss the previous research on multiple fault diagnosis (MFD).We formulate the NP-hard dynamic MFD (DMFD) problemwith imperfect test outcomes in Section III. Four formulationsof the DMFD problem are also discussed in Section III. InSection IV, we decompose the DMFD problem formulation 1using Lagrangian relaxation algorithm. The DMFD problemis decoupled into a set of parallel subproblems (involving dy-namic single HMM state estimation problems) using Lagrangemultipliers. A dynamic programming technique (the Viterbialgorithm) is used to solve each of the subproblems, and theirsolutions are used to update the Lagrange multipliers via thesubgradient method. Feasible (primal) solutions are constructedusing the dual solutions. In Sections V–VII, we discuss thedetails of DMFD problems 2, 3, and 4, respectively. OnlineDMFD problem is solved using a sliding window method,which is presented in Section VIII. The simulation results ofDMFD problem 1 is performed on several real-world data setsto validate our approach. Section IX discusses the simulationresults of both block and online DMFD problems. Finally, thispaper concludes with a summary and future research directionsin Section X.

II. PREVIOUS WORK

The MFD problem originates in several fields, such as med-ical diagnosis [1], [2], error correcting codes (ECCs), speechrecognition, distributed computer systems, and networks [3].The MFD problem in large-scale systems with unreliable testswas first considered by Shakeri et al. in [4]. They proposednear-optimal algorithms using Lagrangian relaxation and sub-gradient optimization methods for the static MFD problem. Inthe area of distributed system management, the MFD prob-lem is studied by Odintsova et al. in [3]. They utilized anadaptive diagnostic technique, termed active probing, for faultdiagnosis and isolation. A probe can be viewed as a test inour terminology; the purpose of a probe is to check the set ofsystem components on the probed path. The probe outcomesdetermine if one or more of the components on the probed pathare faulty or normal. Given the probe outcomes, a diagnosticmatrix (dependence matrix (D-matrix), diagnostic dictionary,reachability matrix) defining the relationship among the probesand component faults, as well as the initial system state, theydeveloped a sequential multifault algorithm to diagnose thesystem state. They considered the probe outcomes as beingdeterministic, which is analogous to the assumptions made inour Problem 4 and in the work described in [5]–[7]. In [8], Leand Hadjicostis applied graphical model-based decoding algo-rithms to the MFD problem in the presence of unreliable tests.They proposed a suboptimal belief propagation algorithm usedto decode low-density parity check codes. They considered afault model, where tests are asymmetric, i.e., the D-matrix isnot binary and the test outcomes are also unreliable, and theytermed it the Y model. Their implementation is parallel to

our Problem formulation 1; however, they considered only thestatic case.

The dynamic single fault diagnosis problem using HMMformalism was first proposed by Ying et al. [9], where it isassumed that, at any time, the system has at most one com-ponent state present. This modeling is somewhat unrealistic formost real-world systems. Another version of the dynamic faultdiagnosis problem was studied in [10]: Unknown probabilitiesof sensor error, incompletely populated sensor observations,and multiple faults were allowed, but the faults could only occuror clear once per sampling interval.

In the dynamic single fault framework [9], a hidden Markovmodeling framework was adopted, and a moving windowViterbi algorithm was used to infer the evolution of componentstates. In the multiple fault case, the state space of HMMincreases exponentially from (m + 1) to 2m, where m is thenumber of possible component states. Consequently, the HMM-based method would be viable only for small-sized systems.The solution method proposed in [10] is a multiple hypothesistracking approach, where at each observation epoch, k bestcomponent state configurations are stored. In that paper, themissed-detection/false-alarm process was a property of thesensor rather than the fault, with the effect that the underlyinginference process could not be decoupled into an FHMM. In[10], at each epoch, all candidate fault sets derived from the pre-viously identified faults are listed, based on at most one changeper epoch assumption. Then, of all k(m + 1) possible candidatesets, each has its score calculated; the candidate set whichobtains the highest score is selected as the inference result at theepoch, and the candidates with the k-best scores are updated.The method is equivalent to enumeration in a limited searchspace; consequently, it is either computationally expensive orfar from optimal. A major contribution of this paper is thatthe missed-detection/false-alarm process is modeled as beinga property of the component state: The model is perhaps lessrealistic, but the computational benefit of an FHMM is large.

Another approach, developed by Ruan et al. [11], decom-poses the original DMFD problem into a series of decoupledsubproblems, one for each epoch. For a single-epoch MFD,they developed a deterministic simulated annealing (DSA)method, which is inspired by its sibling stochastic simulated an-nealing and the approximate belief revision heuristic algorithm[1], [2]. The single-epoch MFD was extended to incorporatecomponent states of multiple consecutive epochs. In addition,they applied a local search and update scheme to further smooththe “noisy” diagnoses stemming from imperfect test results and,consequently, increase the accuracy of fault diagnosis.

The DMFD problem can be viewed as an FHMM, whichis discussed in the machine learning literature [12]. Here, theHMM state is factored into multiple state variables and repre-sented in a distributed manner. The authors discussed an exactalgorithm for inference computations in the FHMM framework.In this framework, inference and learning involves computingthe posterior probabilities of multiple hidden layers (or states),given the test outcomes. However, due to the combinatorialnature of the hidden state representation, the exact algorithm isintractable. They presented approximate inference algorithmsbased on Gibbs sampling and variational methods. The latter

162 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

Fig. 1. DMFD problem viewed as an FHMM.

methods are similar to Lagrangian relaxation, although moti-vated from a Fenchel duality perspective [1], [2], [4], [5], [13].

In this paper, we extend the work of Ruan et al. [11],Shakeri et al. [4], and Tu et al. [5] on MFD to solve the DMFDproblem by combining the Viterbi algorithm and Lagrangianrelaxation in an iterative way. Depending on the probabilisticassumptions on fault–test relationships and test outcomes, oneobtains various DMFD formulations. In summary, the con-tributions of this paper are the following: 1) a primal–dualoptimization framework to solve the DMFD problem; 2) fourformulations of the DMFD problem along with their solutions;3) simulation results on several real-world systems for thefirst and most general formulation of the DMFD problem;4) a comparison of the results between the subgradient andthe DSA methods [11]; and 5) simulation results, along withperformance analysis, of the online DMFD problem using asliding window method.

III. DMFD PROBLEM FORMULATIONS

The DMFD problem consists of a set of possible componentstates in a system and a set of binary test outcomes thatare observed at each sample (observation, decision) epoch.Formally, S = {s1, . . . , sm} is a finite set of m components(failure sources) associated with the system. The state of com-ponent si is denoted by xi(k) at epoch k, where xi(k) = 1if failure source si is present; otherwise, xi(k) = 0. Here,κ = {0, 1, . . . , k, . . . ,K} is the set of discretized observationepochs. The status of all component states at epoch k is de-noted by x(k) = {x1(k), x2(k), . . . , xm(k)}. We assume thatthe initial state x(0) is known (or its probability distributionis known). Component states are assumed to be independent.Each test outcome provides information on a subset of thecomponent states. At each sample epoch, a subset of testoutcomes is available. Tests are imperfect in the sense that theoutcomes of some of the tests could be missing, and tests havemissed-detection/false-alarm processes associated with them.The observations consist of imperfect binary test outcomes and

Fig. 2. Tripartite digraph for DMFD problem.

are characterized by sets of passed test outcomes Op and failedtest outcomes Of .

Our problem is to determine the time evolution of componentstates based on imperfect test outcomes observed over time.Fig. 1 shows the DMFD problem viewed as an FHMM. Thehidden component state of the ith HMM at time epoch k is de-noted by xi(k). Each component state xi(k) can be xi(k) = 1if failure source si is present; otherwise, xi(k) = 0. The obser-vations at each epoch are subsets of binary outcomes1 of testsO = {o1, o2, . . . , on}, i.e., oj ∈ {pass, fail} = {0, 1}. Fig. 2shows the DMFD problem as a tripartite digraph at epoch k.Component states, tests, and test outcomes represent the nodesof the digraph. Here, the true states of the component states andtests are hidden. P = {Pd, Pf} represents a set of probabilitiesof detection and false alarm, which is defined differently foreach of the DMFD problem formulations. Here, we assumedthat the probabilities P = {Pd, Pf} are available; otherwise,they can be learned during the training and validation. We also

1Extension to multivalued component states and test outcomes is straightfor-ward. In the general case, each component i will be in one of li states, and eachtest j has multiple outcomes oj . The detection capabilities of each test for acomponent is defined by an li × (oj − 1) matrix. Equation (12) needs to bemodified to account for multistate components and multioutcome tests.

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 163

define the matrix D = [dij ] as the D-matrix, which representsthe full-order dependence among failure sources and tests.

Each component state is modeled as a two-state nonhomoge-nous Markov chain. For each component state, e.g., for com-ponent si at epoch k, Π = (Pai(k), Pvi(k)) denotes the setof fault appearance probability Pai(k) and fault disappearanceprobability Pvi(k) defined as Pai(k) = Pr(xi(k) = 1|xi(k −1) = 0) and Pvi(k) = Pr(xi(k) = 0|xi(k − 1) = 1), respec-tively. These probabilities are required to model the intermittentfaults. Here, T = {t1, t2, . . . , tn} is a finite set of n avail-able binary tests, where the integrity of the system can beascertained. We denote the set of passed tests Tp and failedtests Tf . At each observation epoch, k ∈ κ, test outcomesup to and including epoch k are available, i.e., we let Ok ={O(b) = (Op(b), Of (b))}k

b=1, where Ok is the set of observedtest outcomes at epoch k, with Op(b)(⊆ O) and Of (b)(⊆ O)as the corresponding outcomes of sets of passed and failedtests at epoch b, respectively. The tests are partially observedin the sense that outcomes of some tests may not be available,i.e., (Op(b) ∪ Of (b)) ⊆ O. In addition, tests exhibit misseddetections and false alarms. Here, we also make the noisy-OR(“causal independence”) assumption [14].

Formally, we represent the DMFD problem as DM ={S, κ, T,O,D, P,Π}, where S = {s1, . . . , sm} is a finite setof m components (failure sources) associated with the system,κ = {0, 1, . . . , k, . . . ,K} is the set of discretized observationepochs, T = {t1, t2, . . . , tn} is a finite set of n available binarytests, O is a finite set of test outcomes up to and including epochK, D = [dij ] is the D-matrix, P = {Pd, Pf} is a set of proba-bilities of detection and false alarm, and Π = (Pai(k), Pvi(k))denotes the set of fault appearance probability Pai(k) and faultdisappearance probability Pvi(k). The DMFD problem can beformulated in the following ways, arranged from the general tosimplified.

Problem 1: The probability of detection (Pdij) and falsealarm probability (Pfij) are associated with each failed testand each failure source, i.e., Pdij = Pr(oj(k) = 1|xi(k) = 1)and Pfij = Pr(oj(k) = 1|xi(k) = 0) of a failure sourcesi and test tj . For notational convenience, when si does notaffect the outcome of test tj , we let the corresponding Pdij =Pfij = 0. This problem scenario frequently arises in medicalfault diagnosis. For example, the Quick Medical Reference,Decision-Theoretic database used in the domain of internalmedicine contains approximately 600 disease nodes (faultsor failure sources) and 4000 symptoms (tests) [1], [2]. Eachof the symptoms could have a probability pair (Pdij , Pfij)associated with the symptom and the disease node. Fig. 3 showsthe bipartite graph, where the edges represent the probabilitypair (Pdij , Pfij). These probabilities can be obtained from thetripartite digraph (Fig. 2) using the total probability theorem asfollows:

Pr (oj(k)|xi(k))

=∑

tj∈{0,1}Pr (oj(k), tj(k)|xi(k))

=∑

tj∈{0,1}Pr (oj(k)|tj(k)) Pr (tj(k)|xi(k)) . (1)

Fig. 3. Bipartite graph for the DMFD problem.

Problem 2: In situations where the probability of detection(Pdij) is associated with each failure source–test pair, but thefalse alarm probability is specified only for the normal systemstate, i.e., Pfj = P (oj(k) = 1|x1(k) = 0, . . . , xm(k) = 0),we obtain a slightly complicated variation of Problem for-mulation 1 (in terms of computational complexity, but not interms of parameterization). This type of scenario arises whenwe design class-specific classifiers that distinguish betweennormal system operation and failure source, si only, or whenthe false alarms are defined on an overall system basis. Here, theprobability pair (Pdij , Pfj) is associated with test outcomes tomodel imperfect test outcomes [4]. This model is also calledthe Z model in [8]. Similar to problem 1, the probability pair(Pdij , Pfj) is shown as edges between the hidden componentstates and test outcomes in Fig. 3, and they can be obtained fromthe tripartite digraph (Fig. 2) using the total probability theoremon the nodes of the test layer.

Problem 3: The detection probability (Pdj) and false alarmprobability (Pfj) are associated with each test tj only. Theprobability pair (Pdj , Pfj) is shown as the edges between thetests and test outcomes in the tripartite digraph (Fig. 3). Thisformulation is quite useful in classifier fusion using ECCs. Inthe ECC matrix, each column corresponds to a binary classifierwith the associated (Pdj , Pfj) pair. In this case, the fault–testrelationships are deterministic, but the test outcomes are unre-liable and depend on the concomitant test only. This type offormulation is also considered in [10].

This formulation provides a nice vehicle for the dynamicfusion of classifiers, where each column of the ECC matrix is aclassifier, and their associated probability pairs (Pdj , Pfj)areuncertainties associated with classifier outcomes. When thelearned parameters and the ECC matrix are fed as an input tothe DMFD algorithm, it performs a dynamic fusion of classi-fier outputs over time. Note that the sampling interval of thedynamic fusion algorithm can be different from the samplinginterval of the raw sensor data.

Problem 4: This is the deterministic case when tests areperfect, i.e., Pdij = 1 and Pfij = 0 [5]. This formulationreduces the tripartite digraph in Fig. 2 to a bipartite graphbetween the components and tests. This scenario is useful insituations where the tests are highly reliable (e.g., automatedtesting of electronic cards), and this leads to a novel dynamicset covering problem.

Next, we discuss the DMFD formulations in detail.

IV. DMFD PROBLEM 1

In this problem, we assume that the detection andfalse alarm probabilities (Pdij , Pfij) are associated with

164 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

Fig. 4. Detection and false alarm probabilities for problem 1.

each failure source and each test. Fig. 4 shows theseprobabilities.

The DMFD problem is one of finding, at each decision epochk, the most likely fault state candidates x(k) ∈ {0, 1}m, i.e.,the fault state evolution over time, XK = {x(1), . . . , x(K)},that best explains the observed test outcome sequence OK .We formulate this as one of finding the maximum a posterioriconfiguration

X̂K = arg maxXK

Pr(XK |OK , x(0)

). (2)

Applying the Bayes rule in (2), the objective function isequivalent to

X̂K = arg maxXK

Pr(OK |XK , x(0)

)Pr(XK |x(0)

).

With passed and failed test outcomes being conditionallyindependent given the status of component states (“the noisy-OR assumption”) and the Markov property of component stateevolution, the problem is equivalent to

X̂K = arg maxXK

K∏k=1

{Pr (Op(k)|x(k)) Pr (Of (k)|x(k))

×Pr (x(k)|x(k − 1))} (3)

where Op(k) ⊆ O and Of (k) ⊆ O denote the sets of passedand failed test outcomes at epoch k, respectively. To simplifythe problem, we maximize the ln (a monotonic function) of (3).We define a new function fk(x(k), x(k − 1)) as

fk (x(k), x(k − 1)) = ln {Pr (Op(k)|x(k)) Pr (Of (k)|x(k))

×Pr (x(k)|x(k − 1))} . (4)

Given the component state status x(k), the test outcomes areindependent. Consequently

Pr (Op(k)|x(k)) =∏

oj(k)∈Op(k)

Pr (oj(k) = 0|x(k)) (5)

Pr (Of (k)|x(k)) =∏

oj(k)∈Of (k)

Pr (oj(k) = 1|x(k)) . (6)

For test tj to pass at epoch k, it shall pass on all its associatedcomponent states so that

Pr (oj(k) = 0|x(k)) =m∏

i=1

Pr (oj(k) = 0|xi(k)) (7)

where

Pr (oj(k) = 0|xi(k)) ={

1 − Pfij , xi(k) = 01 − Pdij , xi(k) = 1

= (1 − Pdij)xi(k)(1 − Pfij)1−xi(k)

xi(k) ∈ {0, 1}. (8)

Evidently

Pr (oj(k) = 1|x(k)) = 1 − Pr (oj(k) = 0|x(k)) . (9)

In the same vein, the assumption of the independent evolu-tion of component states leads to

Pr (x(k)|x(k − 1)) =m∏

i=1

Pr (xi(k)|xi(k − 1)) (10)

where

Pr (xi(k)|xi(k−1))=

⎧⎪⎨⎪⎩1−Pai(k) xi(k−1)=0; xi(k)=0Pai(k) xi(k−1)=0; xi(k)=1Pvi(k) xi(k−1)=1; xi(k)=01−Pvi(k) xi(k−1)=1; xi(k)=1.

Equivalently

Pr (xi(k)|xi(k − 1))

= (1 − Pai(k))(1−xi(k−1))(1−xi(k))

×Pai(k)(1−xi(k−1))xi(k) · Pvi(k)xi(k−1)(1−xi(k))

×(1−Pvi(k))xi(k−1)xi(k), xi(k−1), xi(k)∈{0, 1}.(11)

Therefore, the problem that is equivalent to (3) is as follows:

X̂K = arg maxXK

K∑k=1

fk (x(k), x(k − 1)) (12)

where

fk (x(k), x(k−1))

=∑

oj∈Op(k)

m∑i=1

[xi(k) ln(1−Pdij)+(1−xi(k)) ln(1−Pfij)]

+∑

oj∈Of (k)

ln

[1−

m∏i=1

(1−Pdij)xi(k)(1−Pfij)(1−xi(k))

]

+m∑

i=1

{(1−xi(k−1)) (1−xi(k)) ln (1−Pai(k))

+ (1−xi(k−1)) xi(k) ln (Pai(k))

+ xi(k−1) (1−xi(k)) ln (Pvi(k))

+ xi(k−1)xi(k) ln (1−Pvi(k))} ,

x(k), x(k−1) ∈ {0, 1}m. (13)

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 165

The primal DMFD problem posed in (12) and (13) is NP-hard. Indeed, even the single-epoch problem, i.e., x̂(k) =arg maxx(k) fk(x(k), x̂(k − 1)), is NP-hard [4], which for allpractical purposes, means that it cannot be solved to optimalitywithin a polynomially bounded computation time.

A. Primal–Dual Optimization Framework

The NP-hard nature of the primal DMFD problem moti-vates us to decompose it into a primal–dual problem using aLagrangian relaxation approach [13]. By defining new variablesand constraints, the DMFD problem reduces to a combinatorialoptimization problem with a set of equality constraints. Theconstraints are relaxed via Lagrange multipliers. The relaxationprocedure generates an upper bound for the objective function.The procedure of minimizing the upper bound via a subgradientoptimization produces a sequence of dual and concomitantprimal feasible solutions to the DMFD problem. If the ob-jective function value for the best feasible solution and theupper bound are the same, the feasible solution is the optimalsolution. Otherwise, the difference between the upper boundand the feasible solution, termed the approximate duality gap,provides a measure of suboptimality of the DMFD solution;this is a key advantage of our approach. Another advantageof the primal–dual method is that, although the primal DMFDproblem is not concave, the dual DMFD problem is a piecewiseconvex function, which can be optimized via the subgradi-ent method. In order to write the primal DMFD problem,we define new variables Y K = {y(1), y(2), . . . , y(K)} andy(k) = {yj(k),∀oj ∈ Of (k)} such that

ln yj(k) =m∑

i=1

cijxi(k) + ηj ∀ oj ∈ Of (k)

where

cij = ln(

1 − Pdij

1 − Pfij

)ηj =

m∑i=1

ln(1 − Pfij). (14)

After simple algebraic manipulations of (13) and using (12)and (14), the primal problem can be written as

maxXK ,Y K

J(X,Y ) = maxXK ,Y K

K∑k=1

fk

(x(k), x(k − 1), y(k)

)(15)

where the component state sequence is XK ={x(1), x(2), . . . , x(K)}. Here, the primal objective function foran individual component state, i.e., fk(x(k), x(k − 1), y(k)),is defined as

fk

(x(k), x(k − 1), y(k)

)=

∑oj∈Op(k)

m∑i=1

cijxi(k) +m∑

i=1

μi(k)xi(k)

+∑

oj∈Of (k)

ln (1 − yj(k)) +m∑

i=1

σi(k)xi(k − 1)

+ γ(k) + g(k) +m∑

i=1

hi(k)xi(k)xi(k − 1) (16)

where

γ(k) =∑

oj∈Op(k)

ηj

μi(k) = ln(

Pai(k)1 − Pai(k)

)σi(k) = ln

(Pvi(k)

1 − Pai(k)

)hi(k) = ln

((1 − Pai(k)) (1 − Pvi(k))

Pai(k)Pvi(k)

)

g(k) =m∑

i=1

ln (1 − Pai(k)) . (17)

Note that the multiple HMMs are coupled here because theirstates are observed only via a set of test outcomes. In (16), theterms involving yj(k) and hi(k) show the coupling effects.

By appending constraints (14)–(16) via Lagrange multipliers{λj(k)}oj∈Of (k), the Lagrangian function L(X,Y,Λ) can bewritten as

L(X,Y,Λ) =K∑

k=1

fk

(x(k), x(k − 1), y(k)

)+

∑∀oj∈Of (k)

λj(k)

(ln yj(k) −

m∑i=1

cijxi(k) − ηj

)(18)

where Λ = {λj(k) ≥ 0, k ∈ (1,K), oj ∈ Of (k)} is the set ofLagrange multipliers. In (18), Lagrange multipliers {λj(k)}are nonnegative despite equality constraints (14), because theyj(k) needs to be nonnegative. Using the Lagrange multipliertheorem, we optimize the Lagrangian function in (17) w.r.t.yj(k) to obtain optimal y∗

j(k) as

y∗j(k) =

λj(k)1 + λj(k)

. (19)

The dual of the primal DMFD problem as posed in (15)–(17)can be written as

minΛ

Q(Λ)

subject to Λ = {λj(k) ≥ 0, k ∈ (1,K), oj ∈ Of (k)} (20)

where the dual function Q(Λ) is defined by

Q(Λ) = maxXK ,Y K

L(X,Y,Λ). (21)

The optimization problem described in (20) and (21) is thedual of the primal problem in (15)–(17) because it dualizes themaximization problem into a minimization problem.

Substituting (19) into (20) and simplifying further by re-arranging and combining the terms, we obtain the dualfunction as

Q(X,Λ) = maxXK

m∑i=1

Qi(Λ). (22)

166 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

Fig. 5. Decomposition of the original DMFD problem.

Here

Qi(Λ) =K∑

k=1

ξi (xi(k), xi(k − 1), λj(k)) +1m

wk(Λ) (23)

ξi (xi(k), xi(k − 1), λj(k))

=

⎛⎝ ∑oj∈Op(k)

cij + μi(k) −∑

oj∈Of (k)

cijλj(k)

⎞⎠xi(k)

+ σi(k)xi(k − 1) + hi(k)xi(k)xi(k − 1) (24)

wk(Λ) = γ(k) + g(k) +∑

∀oj∈Of (k)

λj(k) ln λj(k) − λj(k)ηj

−∑

∀oj∈Of (k)

(1 + λj(k)) ln (1 + λj(k)) (25)

represents the dual function for the ith component. The mainbenefit of (22) is that, now, the original problem is separable.As shown in Fig. 5, we employed the Lagrangian relaxationmethod to decompose the original DMFD problem into mseparable subproblems, one for each component state sequencexi, where xi = (xi(1), xi(2)), . . . , xi(K)), xi(k) ∈ {0, 1} andi ∈ {1,m}. This scheme can be viewed as a two-level coor-dinated solution framework for the DMFD problem. At thetop (coordination) level, we update Lagrange multipliers Λ ={λj(k), k ∈ (1,K), oj ∈ Of (k)} using the subgradient methodbased on the decoupled solutions of the individual subprob-lems. This level facilitates coordination among each of thesubproblems and can thus reside in a diagnostic control unit. Atthe bottom level, we use the dynamic programming technique(the Viterbi algorithm) to solve each of the subproblems witha computational complexity O(K), i.e., we optimize the ξi

function in (24) to obtain the optimal state sequence x∗i for

each component state, given a fixed set of Lagrange multipliersΛ = {λj(k), k ∈ (1,K), oj ∈ Of (k)}. The Viterbi algorithmis a dynamic programming technique to find the most likelyfault sequence [15]. It finds a recursive optimal solution tothe problem of estimating the state sequence of a finite stateMarkov chain observed in memoryless noise. The key featureof the Viterbi algorithm is that the objective function can bewritten as a sum of merit functions depending on one state andits preceding one. We obtain the optimal state sequence for eachcomponent state, i.e., X∗ = {x∗

1, x∗2, . . . , x

∗m}, using a binary

Viterbi algorithm. The key steps of the Viterbi algorithm aredescribed in Appendix I.

B. Approximate and Exact Duality Gap

After evaluating the optimum state sequence X∗ for fixedΛ, the problem reduces to one of minimizing the dual functionvalue Ql(Λ) = Q(X,Y,Λ) at iteration l, which is computedusing (22)–(25). Q∗ denotes the optimal dual function value,i.e., Q∗ = Q(Λ∗) = minΛ Q(Λ), where the dual problem isgiven by (22)–(25). The optimal primal solution is denotedby J∗ = J(X∗, Y ∗) = maxXK ,Y K J(X,Y ), where the primalproblem is given by (15)–(17).

The difference between the optimal dual and the primalfunction values, i.e., (Q∗ − J∗), is termed the exact duality gap.Since the DMFD problem is NP-hard, it is difficult to obtainthe global optimal solution J∗. However, we can obtain severalfeasible solutions from the dual solution and select the bestfeasible solution from the set. If Jf = J(Λ∗,Xf , Y f ) is thebest feasible value, then we have

Jf ≤ J∗ ≤ Q∗ ≤ Ql. (26)

Using this method, we can obtain an approximate dualitygap Q∗ − Jf = (Q∗ − J∗) + (J∗ − Jf ) ≥ 0, which providesan overestimate of the error between the global optimal solutionand the best feasible solution. To summarize, we update feasiblesolutions, i.e., Xf , Y f , and the lower bound Qlb as follows:If J(X∗(Λl), Y (X∗(Λl)) ≥ Qlb, then Xf = X∗(Λl), Y f =Y (X∗(Λl)) and

Qlb = Jf = J(X∗(Λl), Y (X∗(Λl)

). (27)

The upper bound Qub is obtained using the current dual valueQl as follows:

Qub = Qmin = min(Qmin, Ql). (28)

Since the dual function Ql(Λ) is a piecewise differentiablefunction of Lagrange multipliers Λ, this problem cannot besolved using differentiable optimization algorithms. We use asubgradient algorithm to compute a sequence of upper boundsfor Ql(Λ) [13]. The details of a subgradient method are de-scribed in Appendix II.

Fig. 6 shows the flowchart of our algorithm. There are fivemajor steps. In step 1, we initialize the Lagrange multipliersand the input fault universe, i.e., fault and test informationalong with the associated probabilities (Pdij , Pfij , Pai, andPvi). We also input test outcomes for the K epochs. In step 2,we run m binary Viterbi algorithms to obtain the optimal statesequences corresponding to the m faults. In step 3, we updatethe feasible solutions, i.e., Xf , Y f , and the lower and upperbounds, i.e., Qlb and Qub using (24)–(26). Next, the Lagrangemultipliers are updated using the subgradient method, whichis described in Appendix II. If stopping criteria, defined inAppendix II, are met, then the algorithm outputs the most likelycomponent state sequence for the m components.

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 167

Fig. 6. Flowchart of the algorithm.

Fig. 7. Detection and false alarm probabilities for problem 2.

V. DMFD PROBLEM 2

In this formulation, we define Pdij as Pdij = Pr(oj(k) =1|xi(k) = 1) and Pfj = Pr(oj(k) = 1|x1(k) = 0, x2(k) =0, . . . , xm(k) = 0). This scenario is shown in Fig. 7.

Pr (oj(k) = 0|x(k))

= (1 − Pfj)(1−x1(k)),...,(1−xm(k))m∏

i=1

(1 − Pdij)xi(k) (29)

Using a new variable z(k) =∏m

i=1(1 − xi(k)) and

Pr (oj(k) = 0|x(k)) = (1 − Pfj)z(k)m∏

i=1

(1 − Pdij)xi(k)

taking log

ln z(k) =m∑

i=1

ln (1 − xi(k)) (30)

ln (Pr (oj(k) = 0|x(k))) = z(k) ln(1 − Pfj)

+m∑

i=1

xi(k) ln(1 − Pdij). (31)

Following steps similar to those in problem 1, we have

ln (Pr (Op(k)|x(k)))

=∑

oj(k)∈Op(k)

ln (Pr (oj(k) = 0|x(k)))

=∑

oj(k)∈Op(k)

z(k) ln(1 − Pfj)

+∑

oj(k)∈Op(k)

m∑i=1

xi(k) ln(1 − Pdij)

= z(k)ηj(k) +m∑

i=1

∑oj(k)∈Op(k)

xi(k) ln(1 − Pdij) (32)

where ηj(k) =∑

oj(k)∈Op(k) ln(1 − Pfj) and z(k) is definedin (30). For failed tests

ln (Pr (Of (k)|x(k))) =∑

oj(k)∈Of (k)

ln (Pr (oj(k) = 1|x(k)))

=∑

oj(k)∈Of (k)

ln (1 − yj(k))

where yj(k) = Pr(oj(k) = 0|x(k)), and using (31)

ln (yj(k)) = z(k) ln(1 − Pfj) +m∑

i=1

xi(k) ln(1 − Pdij).

(33)

Here, the DMFD problem is equivalent to

X̂K = arg maxXK

K∑k=1

fk

(x(k), x(k − 1), y(k), z(k)

)(34)

where the primal objective function for an individual compo-nent state, i.e., fk(x(k), x(k − 1), y(k), z(k)) is defined as

fk

(x(k), x(k − 1), y(k), z(k)

)= z(k)ηj(k) +

m∑i=1

∑oj(k)∈Op(k)

xi(k) ln(1 − Pdij)

+∑

oj(k)∈Of (k)

ln (1 − yj(k)) +m∑

i=1

τi(k)xi(k)

+m∑

i=1

σi(k)xi(k − 1) +m∑

i=1

hi(k)xi(k)xi(k − 1) + g(k)

(35)

168 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

where

ln z(k) =m∑

i=1

ln (1 − xi(k))

ηj(k) =∑

oj(k)∈Op(k)

ln(1 − Pfj)

ln (yj(k)) = z(k) ln(1 − Pfj) +m∑

i=1

xi(k) ln(1 − Pdij)

τi(k) = ln(

Pai(k)1 − Pai(k)

)σi(k) = ln

(Pvi(k)

1 − Pai(k)

)hi(k) = ln

((1 − Pai(k)) (1 − Pvi(k))

Pai(k)Pvi(k)

)

g(k) =m∑

i=1

ln (1 − Pai(k)) . (36)

Appending constraints (30) and (33) via Lagrangemultipliers μ(k), {λj(k)}j∈Of (k), the Lagrangian functionL(X,Y, z,Λ) can be written as

L(X,Y, z,Λ)

=K∑

k=1

fk

(x(k), x(k − 1), y(k), z(k)

)+ μ(k)

(ln z(k) −

m∑i=1

ln (1 − xi(k))

)

+∑

∀oj∈Of (k)

λj(k) (ln yj(k) − z(k) ln(1 − Pfj))

−∑

∀oj∈Of (k)

m∑i=1

xi(k)λj(k) ln(1 − Pdij) (37)

where Λ = {μ(k), λj(k) ≥ 0, k ∈ (1,K), oj ∈ Of (k)} is theset of Lagrange multipliers. Using the Lagrange multipliertheorem, we optimize the Lagrangian function in (34) w.r.t.yj(k) to obtain optimal y∗

j(k) as

y∗j(k) =

λj(k)1 + λj(k)

(38)

and optimizing w.r.t. z(k), we obtain optimal z∗(k) as

z∗(k) =μ(k)

−ηj(k) +∑

∀oj∈Of (k)

λj(k) ln(1 − Pfj). (39)

The dual function Q(Λ) of problem 4 is defined by

Q(Λ) = maxXK ,Y K ,z

L(X,Y, z,Λ). (40)

Substituting (38) and (39) into (37) and simplifying furtherby rearranging and combining the terms, we obtain the dual

function as

Q(Λ) = maxXK

m∑i=1

Qi(Λ) (41)

where

Qi(Λ) =K∑

k=1

ξi (xi(k), xi(k − 1), λj(k), μ(k))

+1m

wk (λj(k), μ(k)) (42)

ξi (xi(k), xi(k − 1), λj(k), μ(k))

=∑

oj(k)∈Op(k)

xi(k) ln(1 − Pdij) + xi(k)τi(k)

−∑

oj(k)∈Of (k)

λj(k)xi(k)

× ln(1 − Pdij) + σi(k)xi(k − 1)+ hi(k)xi(k)xi (k−1)−μ(k) ln(1−xi(k)) (43)

wk (λj(k), μ(k))

= μ(k)

⎛⎜⎝ ηj(k) + ln (μ(k))−ηj(k)+

∑∀oj∈Of (k)

λj(k) ln(1−Pfj)

⎞⎟⎠+g(k)

+∑

∀oj∈Of (k)

[λj(k) ln λj(k) − (1 + λj(k))

× ln (1 + λj(k))] − μ(k)

×

⎛⎜⎝ ∑∀oj∈Of (k)

λj(k) ln(1 − Pfj)−ηj(k)+

∑∀oj∈Of (k)

λj(k) ln(1 − Pfj)

⎞⎟⎠.

(44)

The dual problem posed in (40)–(44) is separable, and it can besolved by following a procedure similar to that used for solvingproblem 1. The only difference is that we also need to updatethe Lagrange multiplier μ(k) using a subgradient method.

VI. DMFD PROBLEM 3

In this formulation, we consider the case where the proba-bilities of detection and false alarm (Pdj , Pfj) are associatedonly with each test tj (see Fig. 8). Formally, Pdj = Pr(oj(k) =1|tj(k) = 1) and Pfj = Pr(oj(k) = 1|tj(k) = 0). We canconvert these probabilities into a special case of problem 1 bycomputing (Pdij , Pfij) using the total probability theorem

Pr(oj(k)|xi(k))=∑

tj(k)∈{0,1}Pr(oj(k), tj(k)|xi(k))

=∑

tj(k)∈{0,1}Pr(oj(k)|tj(k))Pr(tj(k)|xi(k)) ,

Pdij = (dij)Pdj + (1 − dij)Pfj (45)

Pfij = (dij)Pfj + (1 − dij)Pdj . (46)

Here, D = [dij ] is the D-matrix. The solution of Problem 3 canbe obtained by substituting Pdij and Pfij in (45) and (46) inthe solution of problem 1.

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 169

Fig. 8. Detection and false alarm probabilities for problem 3.

VII. DMFD PROBLEM 4

Next, we consider the case when the system consists ofreliable tests and the fault–test relationships are deterministic,i.e., Pdij = 1 and Pfij = 0 for i = 1, . . . , m and j = 1, . . . , n,or equivalently, the D-matrix completely characterizes thefault–test relationships [5]. This formulation can be representedas a bipartite graph between the components and tests. In thiscase, if some tests have passed, then we can infer that all thefailure sources covered by these tests are good components.

Thus, we need to infer failed components from those coveredby the failed tests only, i.e., by excluding those componentscovered by the passed tests. Consequently, the size of theDMFD problem can be reduced by removing all failure sources{si|Pfij = 0, Pdij = 1, and tj(k) ∈ Tp(k)}. For each failedtest tj(k) ∈ Tf (k), the optimal solution contains at least onecomponent state xi(k) = 1 that satisfies dij = 1. Thus, theremust be one or more failure sources that cover the failed tests.Let us consider a matrix A, which has each row representing thelist of failure sources covered by a failed test. After excludingthe failure sources covered by the passed tests, the resulting ma-trix A is a binary matrix such that aij = dji. After substitutingPdij = 1 and Pfij = 0 in (13), the reliable test scenario witha binary D-matrix simplifies to a dynamic set covering problemwith the following objective function term at epoch k:

fk (x(k), x(k − 1)) =m∑

i=1

{μi(k)xi(k) + σi(k)xik − 1)

+ hi(k)xi(k)xi(k − 1)} + g(k) (47)

subject to the following constraints.A(k)x(k) ≥ e for tj(k) ∈ Tf (k) where e is a vector of one’s.

By appending constraints to (47) via Lagrange multipliers Λ ={λj(k) ≤ 0, k ∈ (1,K), tj ∈ Tf (k)}, the Lagrangian functionL(X,Λ) can be written as

L(X,Λ) =K∑

k=1

fk (x(k), x(k − 1))

+∑

∀tj∈Tf (k)

λj(k)

(1 −

m∑i=1

aji(k)xi(k)

). (48)

After rearranging the terms, the Lagrangian function of theoriginal problem is shown as a sum of the Lagrangian functionsof each subproblem as follows:

L(X,Λ) =m∑

i=1

Li(xi(k),Λ) (49)

where

Li (xi(k),Λ)=K∑

k=1

⎡⎣μi(k)xi(k)−∑

tj∈Tf (k)

aji(k)λj(k)xi(k)

+σi(k)xi(k−1)+ hi(k)xi(k)xi(k−1)

+1m

⎛⎝g(k)+∑

∀tj∈Tf (k)

λj(k)

⎞⎠⎤⎦. (50)

The dual function Q(Λ) is defined by

Q(Λ) = maxXK

L(X,Λ). (51)

Simplifying further by rearranging and combining the terms,we obtain the dual function as

Q(Λ) = maxXK

m∑i=1

Qi(Λ) (52)

where

Qi(Λ) =K∑

k=1

ξi (xi(k), xi(k − 1), λj(k)) +1m

wk(Λ) (53)

ξi (xi(k), xi(k − 1), λj(k))

= μi(k)xi(k)+σi(k)xi(k−1)+ hi(k)xi(k)xi(k−1)

−∑

tj∈Tf (k)

λj(k)aji(k)xi(k) (54)

wk(Λ) = g(k) +∑

∀tj∈Tf (k)

λj(k). (55)

The dual problem defined in (52)–(55) is separable. TheViterbi algorithm is used to solve each subproblem correspond-ing to each component state sequence {xi(k)}K

k=1. This algo-rithm can be viewed as a dynamic set covering problem, whichis NP-hard. Thus, the dynamic set covering problem is solvedby combining the Viterbi algorithm and Lagrangian relaxation.This generalizes Beasley’s Lagrangian relaxation algorithm forthe static set covering problem [5], [16] to dynamic settings. Wewill explore the applications of this algorithm in [18].

VIII. SLIDING WINDOW DMFD METHOD

During the online monitoring of a system, the observationsand potential fault sequences are usually very long. Hence, inorder to reduce the amount of computation and storage, theDMFD problem is solved using a sliding window method. Thewindow size W is selected based on the performance criteria,such as low classification error and low false isolation rate (FI).One of the key advantages of the sliding window method isthat Lagrange multipliers are available W − 1 samples ahead,which improves the speed of dual optimization. The slidingwindow method involves the following steps.

Step 1) Solve the DMFD problem for the window sizeW (W ≤ K). Make a decision at epoch k = 1.

170 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

TABLE ISMALL-SCALE SCENARIO FOR SIMULATIONS

Step 2) Move the window by one time epoch, i.e., k = 2 tok = W + 1.1) Initialize (W − 1) Lagrange multipliers using

previous window.2) Initialize component states at k = 2 using the

results of previous window.3) Solve the online DMFD problem using the data

from k = 2 to k = W + 1.4) Make a decision at epoch k = 2.

Step 3 Continue sliding the window until k = K − W + 1.The selection of window size is a key issue, andit depends on the system and fault behavior, i.e.,permanent or intermittent faults.

IX. SIMULATIONS AND RESULTS

We implemented and applied the solution of problem 1, themost general version of the DMFD problem formulation, to asmall-scale system and few real-world models.

A. Small-Scale System

We randomly generated a small-scale system to illustrate theinputs and outputs of our algorithm. The model was constructedfor a system with 20 components, 20 tests, and 20 observationepochs. Each component can have binary states, i.e., normaland faulty. The detection probabilities were set between 0.7 and0.9, and the false alarm probabilities were set between 0 and0.02, and the tests uniformly cover the component states. Thefault appearance and disappearance probabilities were variedbetween 0.0049 and 0.0051, and 0.00025 and 0.00033, respec-

tively. These probabilities were chosen such that the averagenumber of faults was two over a span of 20 epochs. The truefault state set and, accordingly, the test outcomes of each epochare generated using the aforementioned model parameters. Thestopping criteria as defined in Appendix II were used for thesubgradient method. We used the following metrics to evaluatethe performance of our algorithm.

CI: Correct isolation rate (CI) is the percentage of true faultstates which are detected by the algorithm at epoch k. Let x̂(k)be the fault state set at epoch k detected by the algorithm, andr(k) is the true fault state set at epoch k. Then, CI and theaverage

___CI over all epochs are obtained as follows:

CI(k) =|x̂(k) ∩ r(k)|

|r(k)| (56)

___CI =

∑Kk=1 CI(k)

K. (57)

FI: FI is the percentage of fault states which are falselydetected by the algorithm as fault state at epoch k. FI andaverage

___FI are computed as

FI(k) =|x̂(k) ∩ ¬r(k)||S| − |r(k)| (58)

___FI =

K∑k=1

FI(k)

K. (59)

Table I shows only partial data (five rows) of a small-scaleexample. Any component state can be detected by severaltests. For example, the state of component 1 can be detected by

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 171

TABLE IIRESULTS FOR SMALL-SCALE SCENARIO

Fig. 9. Approximate duality gap.

t3, t13, and t19 with (Pd1,3, Pf1,3) = (0.80, 0.01), (Pd1,13,Pf1,13) = (0.75, 0), and (Pd1,19, Pf1,19) = (0.74, 0), respec-tively. This implies that, when component state s1 occurs, testt3 detects it with probability 0.80, and if component states1 does not occur, test t3 has a 0.01 probability of falselyimplicating it. Similarly, test t13 detects s1 with probability of0.75 and has no false alarms. If s1 is not present at epoch k, thenat epoch k + 1, the probability of having s1 present is 0.005,while if s1 is present at epoch k, s1 has a 0.00026 probability ofdisappearing at epoch k + 1. The test outcomes at epochs k =1, 2, 3, 19, and 20 are shown in the table; test outcomes at otherepochs are not shown here for simplicity and for saving space.For example, at k = 1, the outcome of tests t3 is observed ashaving failed, while the outcome of all other tests except t3is observed as having passed. The DMFD problem here is toidentify the evolution of component states over the 20 epochs.The results for this model are shown in Table II. Here, J , Q, D,CI , FI , and t denote the primal function value, dual functionvalue, approximate duality gap, CI , FI , and computation timeper epoch. The primal and dual function values are computedusing (15)–(17) and (22)–(25), respectively. The approximateduality gap (D) is computed as a ratio of the difference betweenQ and J divided by the absolute value of the primal feasiblevalue J . The algorithms were implemented in MATLAB. Weused a standard PC having a Pentium 4 processor with 3.0-GHzclock speed and 512-MB RAM. The approximate duality gapis also shown in Fig. 9 is 13.2%. The duality gap reduces as

TABLE IIIREAL-WORLD MODELS

the number of iterations increases, and the subgradient methodconverges to the minimum dual function value.

B. Real-World Data Sets

Table III illustrates the model parameters of an automotivesystem, a document matching system (Docmatch), a powerdistribution system (Powerdist), a UH-60 helicopter transmis-sion system (Helitrans), and an engine simulator (EngineSim).Details of these models are provided in [5]. Here, m, n, andc denote the number of components (failure sources), numberof tests, and average number of intermittent faults that canoccur over a span of 100 epochs. The fault appearance prob-abilities (Pai) were computed based on the average number ofintermittent faults (c). These real-world systems are not idealbecause they have fewer tests as compared to failure sources;hence, some failure sources are not covered by any tests. Thefault disappearance probabilities (Pvi) were varied between0.0025 and 0.0049 to allow c intermittent faults, on average.The probabilities of detection and false alarm were varied asshown in Table III. Here,

__J ,

__Q ,

__D ,

___CI ,

___FI , and

__t denote

the average primal function value, average dual function value,average approximate duality gap, average CI, average FI, andaverage computation time per epoch. The maximum number ofsubgradient iterations was set at 80 and 100 Monte Carlo runswere used to generate the test outcomes.

Table IV shows the results obtained using the subgradi-ent (S) and DSA [11] methods. The subgradient method (S)achieves higher CIs as compared to DSA for all systems exceptHelitrans. However, the DSA method achieves better primalfunction value and is also effective in reducing the computationtime (t). Also, note that we can obtain a hybrid duality gapby taking the maximum primal solution from the subgradient(S) and DSA methods, and the dual function value from thesubgradient (S) method. The hybrid DSA–subgradient (HS)duality gaps are also shown in Table V. The average compu-tation time (

__t ) is measured in seconds. These numbers are

attractive practically, and they can be further reduced signifi-cantly by a careful implementation in the C language.

We also showed an application of the DMFD Problem 3formulation in our recent paper [17] where we performed thedynamic fusion of classifiers over time for automotive enginefault diagnosis. The temporal correlations considered by dy-namic fusion improve classification accuracy over a variety ofstatic fusion techniques (based on batch data).

172 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

TABLE IVRESULTS ON REAL-WORLD MODELS

TABLE VTYPE OF FAULTS

C. Sliding Window DMFD Results

Figs. 10–12 show the boxplots of CI and FI for variousreal-world models. These plots were obtained using the sliding-window DMFD method.

The boxplot shows the dispersion of the data with lines atlower quartile (25%), median, and upper quartile (75%). Thewhiskers are shown by extending the lines from each end of thebox, and the maximum length of the line is kept as a functionof the interquartile range. The outliers are shown as “o” in thefigures. The window size is selected such that it gives a high CIwith a small boxsize and a low FI with a low boxsize. The mostsuitable window size using the aforementioned criterion forAutomotive, Docmatch, Powerdist, Helitrans, and EngineSimsystems are 15, 10, 20, 15, and 15, respectively. For Helitransand EngineSim systems, window sizes of 15 and 25 achievesnearly the same performance; however, a smaller window sizeis preferred because it will reduce the time delay in making thediagnostic decisions.

Next, we perform simulations to study the effect of intermit-tent faults on the performance of the sliding-window DMFDmethod. The automotive system is used for simulations. Thefault appearance probability was kept so that, on average, threefaults occur over a span of 100 epochs.

Fig. 13 shows the CI for various fault behaviors. The resultsshow the mean value, and the vertical lines on the data pointsindicate the standard deviation. These results were obtainedusing 1000 Monte Carlo runs. The results demonstrate that

the algorithm achieves low variance when the fault behavioris highly intermittent.

The FI plot (Fig. 14) also illustrates the same behavior as theCI plot, i.e., the algorithm achieves the least variance for thehighly intermittent fault types. This illustrates that the DMFDalgorithm is highly suitable for intermittent faults.

D. Complexity

The algorithm presented here reduces the overall complexityfrom O(K(2m)) to O(K(m + Of )) where m is the numberof component states, K is the number of epochs, and Of isthe set of failed tests. More specifically, the complexities ofa binary Viterbi algorithm over all component states and thesubgradient method are O(Km) and O(KOf ), respectively,per iteration; this is a substantial improvement over extensiveapproaches based on exact inference.

X. CONCLUSION

In this paper, we discussed the problem of DMFD with im-perfect tests. The original DMFD problem is an intractable NP-hard combinatorial optimization problem. Using a Lagrangianrelaxation-based coordination framework, we decomposed theoriginal DMFD problem into parallel decoupled subproblemscoordinated via Lagrange multipliers. Each subproblem corre-sponds to finding the optimal state sequence of a fault withfixed Lagrange multipliers. The subproblems were solved usinga binary Viterbi decoding algorithm. The coordination amongthe subproblems was facilitated by Lagrange multipliers, whichwere updated using a subgradient method.

We discussed four formulations of the DMFD problem.Analogous forms of these formulations have been studiedwidely in fault diagnosis community in a static context andapplied in various fields. Here, we provided a unified formu-lation of all the MFD formulations in a dynamic context. The

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 173

Fig. 10. Boxplots of CI and FI for automotive and document matching system.

Fig. 11. Boxplots of CI and FI for power distribution and UH-60 helicopter transmission system.

first formulation refers to a generalized version of the DMFDproblem when the detection and false alarm probabilities areassociated with each test and fault. In the second formulation,the false alarm probability is associated with a fault-free caseonly. The solution to the second formulation was shown tobe quite similar to that of problem formulation 1, except forthe need to update an additional Lagrange multiplier. Thethird formulation considers the case where the uncertainties are

associated with only test outcomes. This models the dynamicfusion of classifier outputs. In the fourth formulation, we con-sidered the deterministic case, which led to a novel dynamic setcovering problem. We implemented the algorithm on severalreal-world data sets, and the results validated the theory. Thekey advantage of our approach is that the method provides anapproximate duality gap, which is a worst case indicator ofthe difference between the feasible and optimal solutions. Our

174 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

Fig. 12. Boxplots of CI and FI for engine simulator system.

Fig. 13. CI for various fault behaviors.

Fig. 14. FI for various fault behaviors.

results demonstrate that our algorithm achieves a high isolationrate as compared to the DSA method. The latter provides betterprimal function value as compared to the subgradient method.In this paper, we assumed that the DMFD model parameters

are known and that faults evolve independently and are coupledthrough the test outcomes via the diagnostic matrix (D-matrix).In our future work, we will implement techniques to learnthe DMFD model parameters from the observed test outcomesequences and relax the independence assumption to solve theDMFD problem when faults are dependent. Coupled HMMsoffer a promising platform for the solution of the dependentfault problem [19]. We will also focus on improving the primalsolution using a soft Viterbi algorithm.

APPENDIX ISOLVING SUBPROBLEMS USING VITERBI ALGORITHM

In this Appendix, we discuss the key steps of the Viterbi al-gorithm which is used to solve each subproblem correspondingto each component state sequence xi.

Initialization: In this step, the objective function is com-puted at k = 1 for each node (component state). It is assumedthat the initial state x(0) is known for all the component states.The maximum function value of ξi in (24) at time k is denotedby δk(xi(k)), and the value of xi where the function valueis maximum is denoted by ψk(xi(k)). For the binary case,we have used the notation δk(0) = δk(xi(k) = 0) and δk(1) =δk(xi(k) = 1). At time k = 1

δ1 (xi(1)) = ξi (xi(1), xi(0), {λj(1)})

=

⎧⎨⎩μi(1)−∑

oj∈Of (1)

cijλj(1)+∑

oj∈Op(1)

cij

⎫⎬⎭xi(1)

+ σi(1)xi(0) + hi(1)xi(1)xi(0)ψ1 (xi(1)) = φ, where xi(0) ∈ {0, 1}. (60)

Recursion: The recursion step involves maximizing theobjective function at each epoch k.

δk (xi(k)) =

⎧⎨⎩ ∑oj∈Op(k)

cij + μi(k)

⎫⎬⎭xi(k)

−∑

oj∈Of (k)

cijλj(k)xi(k)

+ maxxi(k−1)∈{0,1}

[δk−1 (xi(k−1))+σi(k)xi(k−1)

+ hi(k)xi(k)xi(k − 1)] (61)

SINGH et al.: DMFD: MATHEMATICAL FORMULATIONS AND SOLUTION TECHNIQUES 175

for 2 ≤ k ≤ K; xi(k) ∈ {0, 1}

ψk (xi(k)) = arg maxxi(k−1)∈{0,1}

[δk−1(xi(k − 1)) + σi(k)xi(k − 1)

+ hi(k)xi(k)xi(k − 1)] . (62)

Termination: This step computes the objective function attime epoch k = K.

F ∗ = maxxi(K)∈{0,1}

{δK (xi(K))]

x∗i (K) = arg max

xi(K)∈{0,1}[δK (xi(K))] . (63)

Optimal State Sequence Backtracking: The backtrackingstep computes the optimal state sequence by tracing the pathbackwards. The optimal state x∗

i(k) of ith fault at time epoch kis given by

x∗i(k) = ψk+1(xi(k + 1)∗), k = K − 1, . . . , 1. (64)

Similar to the recursion step, we can further simplify termi-nation and backtracking for the binary case.

APPENDIX IIUPDATING LAGRANGE MULTIPLIERS VIA

SUBGRADIENT METHOD

Lagrange multipliers are updated via

λl+1j (k) = max

(0, λl

j(k) + βl(k)dlj(k))

(65)

for j ∈ Of (k) and k ∈ (1,K) where the subgradients dlj(k) at

iteration l and epoch k are

dlj(k) = ln

(y∗

j(k))−

m∑i=1

cijx∗i(k) − ηj (66)

and step size βl(k) is

βl(k) = −υ(Ql − Q∗)Tf∑j=1

(dl

j(k))2 . (67)

Lagrange multipliers were initialized at iteration l = 0 withvalues equal to 0.5. Since the optimal dual function value isnot available, it is estimated using the primal feasible solutionJf and best current dual value Qmin using (27) and (28),respectively. We estimate the optimal dual function as

Q̂∗ =ω(Jf + Qmin)

2(68)

and initial value υ = 0.01 is used. If the best current dual valueQmin does not decrease in the previous 20 iterations of thesubgradient procedure with the current value of υ, then υ isreduced by a factor. To improve the subgradient convergence,

we also vary ω, which is increased or decreased based onwhether the dual function value is decreasing or not [13].We used the following stopping criteria for the subgradientmethod.

1) Stop if∑Of

j=1(dlj(k))2 = 0 since we cannot define a

suitable step size in this case.2) Stop if υ ≤ 10−4 because step sizes become too small.3) Stop if the number of iterations crossed the maximum

number of iterations, i.e., l ≥ 100.

ACKNOWLEDGMENT

Any opinions expressed in this paper are solely those of theauthors and do not represent those of the sponsor.

REFERENCES

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[2] F. Yu, F. Tu, H. Tu, and K. R. Pattipati, “Multiple disease (fault) diagnosiswith applications to the QMR-DT problem,” IEEE Trans. Syst., Man,Cybern. A, Syst., Humans, pp. 746–757, Sep. 2007.

[3] N. Odintsova, I. Rish, and S. Ma, “Multifault diagnosis in dynamic sys-tems,” in Proc. IM, 2005.

[4] M. Shakeri, K. R. Pattipati, V. Raghavan, and A. Patterson-Hine, “Optimaland near-optimal algorithms for multiple fault diagnosis with unreliabletests,” IEEE Trans. Syst., Man Cybern. C, Appl. Rev., vol. 28, no. 3,pp. 431–440, Aug. 1998.

[5] F. Tu, K. R. Pattipati, S. Deb, and V. N. Malepati, “Computationallyefficient algorithms for multiple fault diagnosis in large graph-basedsystems,” IEEE Trans. Syst., Man Cybern., vol. 33, no. 1, pp. 73–85,Jan. 2003.

[6] K. R. Pattipati and M. G. Alexandridis, “Application of heuristic searchand information theory to sequential fault diagnosis,” IEEE Trans. Syst.,Man Cybern., vol. 20, no. 4, pp. 872–887, Jul./Aug. 1990.

[7] V. Raghavan, M. Shakeri, and K. R. Pattipati, “Optimal and near-optimaltest sequencing algorithms with realistic test models,” IEEE Trans. Syst.,Man, Cybern. A, Syst., Humans, vol. 29, no. 1, pp. 11–27, Jan. 1999.

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[9] J. Ying, T. Kirubarajan, and K. R. Pattipati, “A hidden Markov modelbased algorithm for fault diagnosis with partial and imperfect tests,” IEEETrans. Syst., Man, Cybern. C, Appl. Rev., vol. 30, no. 4, pp. 463–473,Nov. 2000.

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[12] Z. Ghahramani and M. I. Jordan, “Factorial hidden Markov models,” inMachine Learning. Boston, MA: Kluwer, 1997.

[13] D. Bertsekas, Nonlinear Programming, 2nd ed. Belmont, MA: AthenaScientific, 2003.

[14] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plau-sible Inference. San Mateo, CA: Morgan Kaufmann, 1988.

[15] D. Forney, Jr., “The Viterbi algorithm,” Proc. IEEE, vol. 61, no. 3,pp. 268–278, Mar. 1973.

[16] J. E. Beasley, “An algorithm for set covering problems,” Eur. J. Oper. Res.,vol. 31, pp. 85–93, 1987.

[17] S. Singh, K. Choi, A. Kodali, K. Pattipati, S. M. Namburu, S. Chigusa,D. V. Prokhorov, and L. Qiao, “Dynamic fusion of classifiers for faultdiagnosis,” in Proc. IEEE SMC Conf., Montreal, QC, Canada, Oct. 2007,pp. 2467–2472.

[18] A. Kodali, S. Singh, K. Choi, K. Pattipati, S. M. Namburu, S. Chigusa,D. V. Prokhorov, and L. Qiao, “Diagnostic inference with nearly perfecttests,” in Proc. IEEE Aerosp. Conf., Big Sky, MT, Mar. 2008.

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176 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 39, NO. 1, JANUARY 2009

Satnam Singh (S’00–M’08) received the B.E.degree from the Indian Institute of Technology,Rookree, India, in 1996, the M.S. degree in elec-trical engineering from the University of Wyoming,Laramie, in 2003, and the Ph.D. degree in electri-cal engineering from the University of Connecticut,Storrs, in 2007.

He is currently a Senior Researcher with GeneralMotors India Science Laboratory, Bangalore, India.Prior to his graduate studies, he was a Design Engi-neer with Engineers India Ltd., New Delhi, India, for

four years.Dr. Singh is a reviewer for IEEE Potentials, IEEE TRANSACTIONS

ON SYSTEMS, MAN, AND CYBERNETICS, and IEEE TRANSACTIONS ON

INSTRUMENTATION AND MEASUREMENT. His research interests include datamining, graphical models, pattern recognition, fault detection and diagnosis,and optimization theory.

Anuradha Kodali received the B.E. degree inelectronics and communications engineering fromAndhra University, Visakhapatnam, India, in 2006.She is currently working toward the M.S. degree inelectrical and computer engineering in the Electricaland Computer Engineering Department, Universityof Connecticut, Storrs.

Her research interests include data mining, patternrecognition, fault detection and diagnosis, and opti-mization theory.

Kihoon Choi is currently working toward the Ph.D.degree in the electrical and computer engineeringin the Electrical and Computer Engineering Depart-ment, University of Connecticut, Storrs.

His research interests are in the areas of faultdiagnosis of complex systems, system real-time faultdiagnosis, and analysis and optimization of large-scale systems.

Mr. Choi is the recipient of the Best Techni-cal Paper Award at the 2004 IEEE AUTOTESTConference.

Krishna R. Pattipati (S’77–M’80–SM’91–F’95) re-ceived the B.Tech. degree in electrical engineeringfrom the Indian Institute of Technology, Kharagpur,India, in 1975 and the M.S. and Ph.D. degrees in con-trol and communication systems from the Universityof Connecticut (UConn), Storrs, in 1977 and 1980,respectively.

He is currently a Professor of electrical and com-puter engineering with the Electrical and ComputerEngineering Department, UConn. He has publishedover 370 articles, primarily in the application of

systems theory and optimization techniques to large-scale systems. His researchhas been primarily in the application of systems theory and optimizationtechniques to complex systems.

Dr. Pattipati was the Editor-in-Chief of the IEEE TRANSACTIONS ON

SYSTEMS, MAN, AND CYBERNETICS (SMC)—PART B: CYBERNETICS dur-ing 1998–2001. He is the recipient of the Centennial Key to the Future Award,in 1984, from the IEEE SMC Society, the Andrew P. Sage Award for theBest SMC Transactions Paper for 1999, the Barry Carlton Award for the BestAerospace and Electronic Systems Transactions Paper for 2000, the 2002 and2008 NASA Space Act Awards, the 2003 American Association of UniversityProfessors Research Excellence Award, and the 2005 School of EngineeringTeaching Excellence Award at UConn. He also the recipient of the BestTechnical Paper Awards at the 1985, 1990, 1994, 2002, 2004, and 2005 IEEEAUTOTEST Conferences, and at the 1997 and 2004 Command and ControlConferences.

Setu Madhavi Namburu received B.Tech. degreefrom Jawaharlal Nehru Technological University,Hyderabad, India, in 2002 and the M.S. degree fromthe University of Connecticut, Storrs, in 2006, bothin electrical engineering.

Currently, she is as a Research Engineer withToyota Motor Engineering and Manufacturing NorthAmerica, Inc., Erlanger, KY. Her research inter-ests include data mining, knowledge discovery, sys-tems analysis, simulation, and fault diagnostics andprognostics.

Shunsuke Chigusa Sean received the M.S. degree inelectrical engineering from Osaka University, Osaka,Japan.

He is a Senior Principal Research Scientist withthe Technical Research Department, Toyota Mo-tor Engineering and Manufacturing North America,Inc., Erlanger, KY. He started his automotive careeras an Autonomous Vehicle Control System Scientistand later as an Intelligent Diagnosis Scientist withToyota Motor Corporation, Japan.

Danil V. Prokhorov received the M.S. degree (withhonors) from the State Academy of Aerospace En-gineering (former LIAP), St. Petersburg, Russia, in1992 and the Ph.D. degree in electrical engineeringfrom Texas Tech University, Lubbock, in 1997.

He was with Ford Research Laboratory, Dearborn,MI, from 1997 until 2005. He has been engagedin application-driven studies of neural networksand training algorithms. He is currently a ResearchManager with the Technical Research Department,Toyota Motor Engineering and Manufacturing North

America, Inc., Erlanger, KY. His research interest is in machine learningalgorithms and their applications to decision making under uncertainty. He hasauthored 80 technical publications including several patents.

Dr. Prokhorov was the recipient of the International Neural Network Society(INNS) Young Investigator Award in 1999. He was the INNS-IEEE Interna-tional Joint Conference on Neural Networks (IJCNN) 2005 General Chair andthe IJCNN 2001 Program Chair. He has been a reviewer for numerous journalsand conferences, a program committee member for many conferences, and apanel expert for the National Science Foundation every year since 1995.

Liu Qiao received the B.S. degree in engineer-ing from Beijing University of Technology, Beijing,China, and the M.S. and Ph.D. degrees in electricalengineering from Tohoku University, Sendai, Japan.

He is a General Manager and Chief Technol-ogist with the Technical Research Department,Toyota Motor Engineering and Manufacturing NorthAmerica, Inc., Erlanger, KY. Switching from a uni-versity faculty position, he started his automotivecareer as an Advanced Automotive Control SystemExpert, Advanced Technology Manager, eBusiness

Manager, and Research Manager. He successfully led a Canadian hybrid vehicleproject and its market introduction. He is an active member/supporter of manyacademic associations.

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