Centralized and decentralized process and sensor fault ...

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Measurement xxx (2008) xxx–xxx

www.elsevier.com/locate/measurement

Centralized and decentralized process and sensorfault monitoring using data fusion based on adaptive

extended Kalman filter algorithm

Mohsen Mosallaei a,*, Karim Salahshoor a, Mohammadreza Bayat b

a Department of Automation and Instrumentation, Petroleum University of Technology, Tehran, Iranb Institute for Systems and Robotics (ISR), Instituto Superior Tecnico (IST), Universidade Tecnica de Lisboa (UTL), Lisbon, Portugal

Received 18 October 2007; received in revised form 23 January 2008; accepted 25 February 2008

Abstract

This paper presents an integrated design framework to utilize multi-sensor data fusion (MSDF) techniques for processmonitoring enhancement to detect and diagnose sensor and process faults. Two different distributed and centralized archi-tectures are presented to integrate the multi-sensor data based on extended Kalman filter (EKF) data fusion algorithm.The distributed integration architecture uses the state-vector fusion method, while the centralized integration architectureis based on the output augmented fusion (OAF) method. The usual approach in the classical EKF implementation is basedon the assumption of constant diagonal matrices for both the process and measurement covariances. This inflexible con-stant covariance set-up may cause degradation in the EKF performance. A new adaptive modified EKF (AMEKF) algo-rithm has been developed to prevent the filter divergence and hence leading to an improved EKF estimation. A set ofsimulation studies have been conducted to demonstrate the performances of the proposed adaptive and non-adaptive pro-cess monitoring approaches on a continuous stirred tank reactor (CSTR) benchmark problem. The sensor fault studiesinclude the sensor faults due to drift in calibration and drift in sensor degradation anomalies. Whereas, the process faultsconsist of four probable CSTR faults in cascaded single, double, triple and quadruple set-up.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Multi-sensor data fusion; Adaptive modified extended Kalman filter; Output augmented fusion; Continuous stirred tankreactor

1. Introduction

In modern chemical process industries, there hasbeen an ever-increasing push to optimize their pro-duction processes by satisfying the continuously

0263-2241/$ - see front matter � 2008 Elsevier Ltd. All rights reserved

doi:10.1016/j.measurement.2008.02.009

* Corresponding author. Tel.: +98 913 355 6310.E-mail address: [email protected] (M. Mosallaei).

Please cite this article in press as: M. Mosallaei et al., Centrament (2008), doi:10.1016/j.measurement.2008.02.009

tightening safety and environmental regulations.Industrial processes have become more highly inte-grated and complex and hence their proper moni-toring presents challenges that are not readilyaddressed using the conventional diagnosis by theoperators. To ensure that the process operationsare able to satisfy the increasingly stringent perfor-mance specifications, an accurate automated pro-cess monitoring is essential.

.

lized and decentralized process and sensor ..., Measure-

mailto:[email protected]

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The main goal of this automated process moni-toring is to ensure the success of plant operationsby recognizing anomalies of the behavior in real-time. It includes the tasks of fault detection to deter-mine the existence of faults due to sensor or processupsets and their fault diagnosis to find the root fac-tors causing such events.

This information keeps the plant operator andmaintenance personnel better informed of the statusof the process operation in order to make appropri-ate remedial actions to improve the abnormal behav-ior. As a result of this proper process monitoring,downtime is minimized, safety of plant operation isimproved and manufacturing costs are reduced.

The automated fault detection and diagnosis hasbeen an active area of research for many decadesand broad spectrums of methods have been devel-oped. Generally, depending on the rigorousness ofthe process knowledge employed, the existing meth-ods can be broadly classified into two maincategories of process history-based and processmodel-based methods. Each of these categories canfurther be divided into qualitative and quantitativeapproaches. The qualitative approaches involvefault trees [1], signed directed graph [2,3], fuzzy logic[4] and expert systems [5]. In many cases, however,these qualitative approaches simply give multipleinterpretations for a single event which is an inherentlimitation of the qualitative model-based methods [6].The quantitative model-based approaches, on theother hand, utilize the process model and on-linemeasurements to back-calculate crucial process vari-ables. These approaches include basically modeling,filtering and estimation techniques, where a widevari-ety of them have already been reviewed by [7–9].Among the existing quantitative model-based meth-ods, the Kalman filter variants have found widespreadapplications due to their simplicity and ability tohandle reasonable uncertainties and nonlinearities.

The purpose of this work is to present a newautomated fault detection and diagnosis systembased on an enhanced extended Kalman filter(EKF) estimator. The proposed methodology uti-lizes a multi-sensor data fusion (MSDF) techniqueto enhance the accuracy and reliability of state esti-mation in the process fault detection. The field ofmulti-sensor data fusion is fairly young and mostof its literature has dealt with military and civiliantarget tracking and autonomous robotics. This tech-nique seeks to combine data from multiple sensorsand related information to achieve improved accu-racies and more specific inferences. Thus, the main


problem in this paper is focused on the methodol-ogy by which the multi-sensor measurements canbe combined and processed. There are variousmulti-sensor data fusion approaches to resolve thisproblem. In this work, two different multi-sensorintegration architectures have been presented basedon the EKF data fusion algorithm. Using the EKF,two key architecture scenarios called as centralizedand decentralized or distributed methods have beendeveloped. In the centralized integration method,called as measurement fusion integration method,all the raw data from different sensors is sent to asingle location to be fused. While, in the distributedintegration method, called as state-vector fusionintegration method, the filtering process is dividedbetween some local EKF filters working in parallelto obtain individual sensor-based estimates andone master EKF filter to combine these local esti-mates to yield an improved global state estimate.However, the operation of the EKF algorithm asthe data fusion technique is critically influenced byuncertainty in covariance parameters of the processnoise (Q) and the observation error (R).

In this paper, an online adaptive scheme has beendeveloped to enhance the EKF estimation proce-dure based on the innovation and residualsequences of the EKF algorithm. As a consequence,three different process monitoring approaches,called as non-adaptive centralized output aug-mented fusion (NACOAF), adaptive centralizedoutput augmented fusion (ACOAF) and adaptivedistributed state-vector fusion (ADSVF) have beenproposed in this paper.

The paper is organized as follows. The model-based process fault monitoring problem is formu-lated in Section 2. In Section 3, first, a discrete-timeEKF algorithm is presented. Then, an adaptiveEKF is developed. Finally, two different centralizedand distributed multi-sensor integration architec-tures combined with the adaptive EKF data fusionwill be presented. Simulation studies illustratingthe performances of the three proposed processmonitoring approaches (i.e., NACOAF, ACOAFand ADSVF) are evaluated for sensor and processfault detection and diagnosis in Section 4. In Section5, the main concluding remarks are summarized.

2. Formulation of model-based process fault

monitoring problem

Instrumentation sensors are usually distributedthroughout the chemical process plants to meet


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both operational and safety requirements. However,this scheme introduces a number of complicationswhich makes the consolidation of the data fromthe located sensors a complicated task even for anexperienced engineer. Further complications includethe nature of information obtained from the sensorswhich is inherently incomplete, uncertain, andimprecise. Hence, it is imperative that a fusionmechanism be devised so as to combine data frommultiple sensors to minimize such imprecision anduncertainty, leading to a more comprehensive andunified view of the sensor data.

These conditions combined with the require-ments for a model-based approach to provide anyprocess failure detection and diagnosis informationmake the Kalman filter (KF) approach an idealsolution for the data fusion problem. However,the effectiveness of such approach depends to a largeextent on how redundant and complementary arethe information cues obtained from the installedsensors and the KF estimation. It is equally impor-tant to decide at what level of abstraction the fusionprocess is going to take place, e.g., at the measure-ment level, at the feature/state level, or at the deci-sion level.

The main issue in this model-based approachconcerns the ability to detect and diagnose the pro-cess faults using the dependencies between the dif-ferent process observed or estimated variables.These dependencies can be explored by consideringmathematical process and measurement models.

Assume that the process is monitored by N differ-ent sensors, described by the following general non-linear process and measurement models in discrete-time state-space framework:

xðkÞ ¼ f ðxðk � 1Þ; uðk � 1Þ; dðk � 1ÞÞ þ wðk � 1Þð1Þ

ziðkÞ ¼ hiðxðkÞÞ þ viðkÞ; i ¼ 1; . . . ;N ð2Þ

where f(�) and hi(�) are the known nonlinear func-tions, representing the state transition model andthe measurement model, respectively. xðkÞ 2 Rnx isthe process state-vector, uðkÞ 2 Rnu denotes themanipulated process variables, dðkÞ 2 Rnd representsthe process faults modeled by the process distur-bances, ziðkÞ 2 Rnzi are the measured variables ob-tained from the N installed sensors, w(k) and vi(k)indicate the stochastic process and measurementdisturbances modeled by zero-mean white Gaussiannoises with covariance matrices Q(k) and Ri(k),respectively.


Therefore, the process fault monitoring problemin this paper can be reduced to a design methodol-ogy to realize a data integration mechanism whichis able to fuse together the N noisy measured data(zi(k); i = 1, . . .,N), given in Eq. (2), to generatethe optimal detection and diagnostic estimationinformation (xðkÞÞ about the real-time status ofthe nonlinear process operation, described by Eq.(1).

The central challenges of this design problem,however, can specifically be expressed in terms ofthe data fusion algorithm by which the multi-sensormeasured data are fused together and the data inte-gration architecture approach to determine thefusion level and its implementation topology.

3. Multi-sensor data fusion technique based on

extended Kalman filter algorithm

Multi-sensor data fusion(MSDF) is a synergisticprocess, concerning the mechanism of fusing uncer-tain, incomplete, and sometimes conflicting datafrom a variety of disparate sensors in real time toextract a single compilation of the overall systemstatus for monitoring, control and decision-makingpurposes.

For a particular industrial process application,there might be plenty of associated sensor measure-ments located at different operational levels andhaving various accuracy and reliability specifica-tions. One of the key issues in developing a MSDFsystem is the question of how can the multi-sensormeasurements be fused or combined to overcomeuncertainty associated with individual data sourcesand obtain an accurate joint estimate of the systemstate vector. There exists various approaches toresolve this MSDF problem, of which the KF isone of the most significant and applicable candidatesolution.

3.1. Discrete-time extended Kalman filter

In most practical applications of interest, theprocess and/or measurement dynamic models aredescribed by nonlinear equations, represented inEqs. (1) and (2). This means that the non-linearbehavior can affect the process operation at leastthrough its own process dynamics or measurementequation. In such cases, the standard KF algorithmis often unsuitable to estimate the process statesusing its linearized time-invariant state-space modelat the desired process nominal operating point.



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Extended Kalman filter (EKF) gives a simple andeffective remedy to overcome such nonlinear estima-tion problem. Its basic idea is to locally linearize thenonlinear functions, described by Eqs. (1) and (2), ateach sampling time instant around the most recentprocess condition estimate. This allows the Kalmanfilter to be applied to the following linearized time-varying model:

xðkÞ ¼ AðkÞxðk � 1Þ þ BuðkÞuðk � 1Þþ BdðkÞdðk � 1Þ þ wðk � 1Þ ð3Þ

ziðkÞ ¼ H iðkÞxðkÞ þ viðkÞ; i ¼ 1; . . . ;N ð4Þ

where the state transition matrix A(k), the inputmatrices Bu(k) and Bd(k), and the observation ma-trix Hi(k) are the Jacobian matrices which are eval-uated at the most recent process operating conditionin real-time rather than the process fixed nominalvalues:

AðkÞ ¼ ofox

��xðkÞ

ð5Þ

BuðkÞ ¼ofou

��uðkÞ

ð6Þ

BdðkÞ ¼ofod

��dðkÞ

ð7Þ

HiðkÞ ¼ohi

ox

��xðkÞ; i ¼ 1; . . . ;N ð8Þ

In classical control, disturbance variables d(k) aretreated as known inputs with distinct entry in theprocess state-space model. This distinction betweenstate and disturbance as non-manipulated variables,however, is not justified from the monitoring per-spective using the EKF estimation procedure.Therefore, a new augmented state variable vectorx*(k) = [dT(k)xT(k)]T is developed by consideringthe process disturbances or faults as additional statevariables. To implement this view, the process faultsare assumed to be random state variables governedby the following stochastic auto-regressive (AR)model equation:

dðkÞ ¼ dðk � 1Þ þ wdðk � 1Þ ð9Þ

This assumption changes the linearized model for-mulations in Eqs. (3) and (4) to the following aug-mented state-space model:

x�ðkÞ ¼ A�ðkÞx�ðk � 1Þ þ B�ðkÞuðk � 1Þþ w�ðk � 1Þ ð10Þ

ziðkÞ ¼ H �i ðkÞx�ðkÞ þ viðkÞ; i ¼ 1; . . . ;N ð11Þ


Noting that:

A�ðkÞ ¼Ind�nd 0nd�nx

BdðkÞnx�nd AðkÞnx�nx

� �ð12Þ

B�ðkÞ ¼ 0nd�nu BuðkÞnx�nu� �T ð13Þ

H �i ðkÞ ¼ 01�nd H iðkÞ1�nx� �

ð14Þ

w�ðk � 1Þ ¼ wdðk � 1Þnd�1 wðk � 1Þnx�1� �T ð15Þ

where nx and nu denote the dimensions of the statevector (x) and the manipulated variables (u), respec-tively, and nd indicates the dimension of the distur-bance or non-manipulated variables (d).

In practice, the process dynamic model in Eq. (1)is of continuous-time nature. While, the measure-ments in Eq. (2) are available through the digitaldata-acquisition systems at discrete time instants.Furthermore, the EKF algorithm is implementeddigitally to provide a quick and accurate estimateof the process variables of interest. Therefore, anefficient formulation of the algorithm needs to bemade for a real-time practical implementation inorder to minimize the filter cycle time, while obtain-ing a reasonable state estimate accuracy. An appro-priate method can be used for numerical integrationof the continuous-time process model from onesample time to the next. In this paper, the simplefirst-order Euler integration algorithm has shownto be adequate. The time propagation equation forthe state covariance matrix P(k) is solved usingthe transition matrix technique [10]. This methodpreserves both the symmetry and the positive defi-niteness of P(k), and hence yields adequate estima-tion performance:

P�ðkÞ ¼ UðkÞPðk � 1ÞUTðkÞ þ QdðkÞ ð16Þwhere U(k) denotes the state transition matrix asso-ciated with A(k) for all the time interval s 2[(k � 1)Ts,kTs] which can be evaluated by:

UðkÞ ¼ I þ T sA�ðkÞ ð17Þ

where Ts is the sampling period and Qd(k) is calcu-lated as follows:

QdðkÞ ¼Z kT s

ðk�1ÞT s

UðkT s; sÞQðsÞUTðkT s; sÞds ð18Þ

As a result, Qd(k) can be obtained using the follow-ing trapezoidal integration scheme:

QdðkÞ ¼ ðUðkÞQðkÞUTðkÞ þ QðkÞÞ T s

2ð19Þ

The EKF is then implemented using the time updateequations which project the state and covariance



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estimates forward one time step, and the measure-ment update equations which correct the state andcovariance estimates using the latest measurementinformation, are summarized as follows:

EKF time update equations

(1) x��ðkÞ ¼ x�ðk � 1Þ þ T sf ðx�ðk � 1Þ;uðk � 1Þ; dðk � 1ÞÞ

(2) P�ðkÞ ¼ UðkÞP ðk � 1ÞUTðkÞ þ QdðkÞ

EKF measurement update equations

(1) KðkÞ ¼ P�ðkÞH �T

i ðkÞ½H �i ðkÞP�ðkÞH �T

i ðkÞþRðkÞ��1

(2) x�ðkÞ ¼ x�ðk � 1Þ þ KðkÞ½ziðkÞ � H �i ðkÞx��ðkÞ�(3) P ðkÞ ¼ P�ðkÞ � KðkÞH �i ðkÞP�ðkÞ

The covariance matrix can be initialized (P(0)) witha large value. This option, however, causes rapidfluctuations in the initial EKF state estimates andhence endangers the estimator convergence. Onthe other hand, choosing a small initial covariancematrix will make the estimator adaptation veryslow. Furthermore, when the process dynamicschange, the old estimated information will lose itssignificance as far as the new process dynamic isconcerned. Thus, there should be a means of drain-ing off old information at a controlled rate. Onesimple and useful way of rationalizing this desiredapproach is to modify the covariance matrix updaterelationship as follows:

P ðkÞ ¼ ½P�ðkÞ � KðkÞH �i ðkÞP�ðkÞ�=k ð20Þ

where 0 < k 6 1 behaves as the forgetting factorconcept in the weighted recursive least squares(WRLS) algorithm.

3.1.1. Adaptive extended Kalman filter (AEKF)

The operation of the EKF algorithm as the datafusion technique relies on the precise a priori knowl-edge of the process and measurement dynamic mod-els and their noise properties. The uncertainty in thecovariance parameters of the process noise (Q) andthe observation error (R) has a crucial impact on theEKF performance and may significantly degrade itsperformance. This is due to the fact that Q and R

influence the weight that the EKF applies betweenthe existing process information and the latest mea-surements. Hence, errors in any of them may causethe EKF to diverge.

The conventional way of determining Q and Rrequires good a priori knowledge of the process


noise and measurement error, which typically comesfrom intensive empirical analysis. In practice, how-ever, their values are generally assumed to be fixedduring the whole process of estimation time inter-val. The resulting EKF performance suffers due tothis inflexibility scheme, because, process and mea-surement noises are dependent on the applicationenvironment and process dynamics. Thus, their set-tings in different applications have to be done con-servative in order to stabilize the EKF for theworst case scenario, leading to performancedegradation.

In this paper, an online scheme is presented toprevent the EKF degradation and divergence. It iswell known that the innovation and residualsequences of the EKF are reliable indicators of itsfiltering performance. The innovation sequence isdefined as

giðkÞ ¼ ziðkÞ � H �i ðkÞx��ðkÞ ð21Þ

and the residual sequence as

eiðkÞ ¼ ziðkÞ � H �i ðkÞx�ðkÞ ð22ÞSubstituting the measurement model (Eq. (11)) intoEq. (21), gives:

giðkÞ ¼ ðH �i ðkÞx�ðkÞ þ viðkÞÞ � H �i ðkÞx��ðkÞ¼ H �i ðkÞ½x�ðkÞ � x��ðkÞ� þ viðkÞ

ð23Þ

On the other hand, we have:

e��k ffi x�ðkÞ � x��ðkÞ ð24Þ

e�k ffi x�ðkÞ � x�ðkÞ ð25Þ

P�ðkÞ ¼ E½e��k e��Tk � ð26Þ

P ðkÞ ¼ E½e�ke�Tk � ð27Þ

Ri ¼ E½viðkÞviðkÞT� ð28Þ

For an optimal EKF operation, the innovation andresidual sequences should be white Gaussian noisesequences with zero mean. On the basis of assumingthat w*(k) and vi(k) are uncorrelated white Gaussiannoise sequences and the orthogonality condition ex-ists between observation error and state estimationerror, the innovation covariance can be computedfrom Eq. (23):

E½giðkÞgTi ðkÞ� ¼ E½ðH �i ðkÞe��k ÞðH �i ðkÞe��k Þ

T�þ E½viðkÞvT

i ðkÞ�¼ H �i ðkÞE½e��k e��T

k �H �Ti ðkÞ þ RiðkÞ ð29Þ



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That is

RiðkÞ ¼ E½giðkÞgTi ðkÞ�

� H �i ðkÞP�ðkÞH �Ti ðkÞ; i ¼ 1; . . . ;N ð30Þ

Thus, when the innovation covariance E½giðkÞgTi ðkÞ�

is available, the covariance of the observation errorRi(k) can be estimated directly from Eq. (30). Calcu-lation of the residual covarianceE½giðkÞgT

i ðkÞ� nor-mally uses a limited number of samples of theinnovation sequence:

E½giðkÞgTi ðkÞ� ¼

1

M

XM�1

m¼0

giðk � mÞgTi ðk � mÞ ð31Þ

in which M represents the estimation window size.However, it is noted that Eq. (31) gives a valid resultwhen the innovation sequence is stationary andergodic over the M sample steps.

To improve the robustness of the adaptive filter-ing algorithm to innovation and residual covarianceestimations, a new process noise method is pro-posed here as follows:

giðkÞ � eiðkÞ ¼ H �i ðkÞ½x�ðkÞ � x��ðkÞ� ð32Þ

Noting that the process output variables in ourformulation are always the same as the process statevariables, implying that H �i ðkÞ ¼ I . Hence, Eq. (32)will be reduced to giðkÞ � eiðkÞ ¼ x�ðkÞ � x��ðkÞ.Thus, it can be written:

ðgiðkÞ � eiðkÞÞT ¼ ½x�ðkÞ � x��ðkÞ�T ð33Þ

Multiplying Eq. (32) into Eq. (33) and taking theexpectation operator from its both sides, gives:

ðgiðkÞ � eiðkÞÞðgiðkÞ � eiðkÞÞT

¼ ½x�ðkÞ � x��ðkÞ�½x�ðkÞ � x��ðkÞ�T ð34Þ

E½giðkÞgTi ðkÞ� þ E½eiðkÞeT

i ðkÞ�

¼ E½ðx�ðkÞ � x��ðkÞÞðx�ðkÞ � x��ðkÞÞT�¼ E½ðe��k � e�kÞðe��k � e�kÞ

T� ¼ P�ðkÞ þ PðkÞ ð35Þ

On the other hand, the residual covariance can beapproximated as follows:

E½eiðkÞeTi ðkÞ� ¼

1

M

XM�1

m¼0

eiðk � mÞeTi ðk � mÞ ð36Þ

When the residual covariance E½eiðkÞeTi ðkÞ�is avail-

able, the covariance of the process error Qd(k) canbe estimated from substituting equations (16),(31), (36) into Eq. (35):


QdðkÞ ¼ ½1

M

XM�1

m¼0

ðgiðk � mÞgTi ðk � mÞ

þ eiðk � mÞeTi ðk � mÞÞ�

� ½UðkÞP ðk � 1ÞUTðkÞ þ PðkÞ� ð37Þ

3.2. Multi-sensor integration architectures based on

Kalman filter data fusion algorithm

Multi-sensor data fusion can be done at a varietyof levels from the raw data or observation level tothe feature/state vector level and the decision level.This idea can lead to utilization of different possibleconfigurations or architectures to integrate the datafrom disparate sensors in an industrial plant toextract the desired monitoring information. UsingKalman filtering as the data fusion algorithm, mul-tiple sensors can be integrated in two key architec-ture scenarios called centralized method anddecentralized or distributed method. These methodshave been widely studied over the last decade[11,12]. In centralized integration method, all theraw data from different sensors is sent to single loca-tion to be fused, as shown in Fig. 1.

This architecture is sometimes called as measure-ment fusion integration method [11,12], in whichobservations or sensor measurements are directlyfuses to obtain a global or combined measurementdata matrix (H*). Then, it uses a single Kalman filterto estimate the global state vector based upon thefused measurement. Although this conventionalmethod provides high fusion accuracy to the estima-tion problem, the large number of states mayrequire high processing data rates that cannot bemaintained in practical real time applications.Another disadvantage of this method is the lack ofrobustness in case of failure in sensor or central fil-ter itself. For these reasons, parallel structures canoften provide improved failure detection and cor-rection, enhance redundancy management, anddecreased costs for multi-sensor system integration.As such, there has recently been considerable inter-est shown in distributed integration method inwhich the filtering process is divided between somelocal Kalman filters working in parallel to obtainindividual sensor-based state estimates and onemaster filter combining these local estimates to yieldan improved global state estimate, as shown inFig. 2. This architecture is sometimes called asstate-vector fusion integration method [11,12]. Theadvantages of this method are higher robustness


Fig. 1. Centralized integration architecture.

Fig. 2. Distributed integration architecture.


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due to parallel implementation of fusion nodes andlower computation load and communication cost ateach fusion node. It is also applicable in modularsystems where different process sensors can be pro-vided as separate units. On the other hand, distrib-uted fusion is conceptually a lot more complex andis likely to require higher bandwidth compared withcentralized fusion. However, a trade-off betweenbandwidth and performance is possible by lettingthe fusion nodes communicate at reduced rates [13].

3.2.1. Centralized integration method based on EKF

data fusion algorithm

In the presented centralized integration method,the data from all the local sensors is fed into a datafusion center where all the process measurementsare processed centrally by the EKF data fusionalgorithm to yield the global process state estimate.

Assume that the nonlinear dynamic process to bemonitored is described by the augmented state-space model, given in Eq. (10). Accordingly, theprocess measurement system, including N differentsensors, has the following dynamic model (Eq. (11)):

ziðkÞ ¼ H �i ðkÞx�ðkÞ þ viðkÞ; i ¼ 1; . . . ;N ð38Þ

where zi(k) represents the ith sensor measurementdata and H �i ðkÞ indicates its corresponding linear-ized measurement matrix.

The centralized fusion problem is concerned withdifferent formulation methods by which the obser-


vation data (zi, i = 1,. . ., N) can be combined totransform the N measurement equations describedby Eq. (38) into the following single measurementequation:

zðkÞ ¼ H �ðkÞx�ðkÞ þ vðkÞ ð39Þ

where H*(k) indicates a single observation matrix,called as global observation matrix and v(k) denotesthe overall measurement noise vector [v1(k) v2(k) . . .vN(k)]T based on the assumption that vi(k) areindependent.

3.2.1.1. Centralized integration method using outputs

augmented fusion (OAF) formulation. The observa-tion data (zi) can simply be combined by augment-ing the multi-sensor data, leading to the followingobservation model:

zðkÞ ¼ ½zT1 ðkÞ . . . zT

N ðkÞ�T ð40Þ

H �ðkÞ ¼ ½H �T1 ðkÞ . . . H �TN ðkÞ�T ð41Þ

RðkÞ ¼ diag½R1ðkÞ . . . RN ðkÞ� ð42Þ

It is then straightforward to apply the discrete-timeEKF algorithm as the global data fusion algorithmin centralized integration architecture to obtainfused measurement information.

This method makes use of all the raw measure-ment information in their original form withoutany dimension reduction in the observation model.



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Thus, it is applicable in all situations such as dissim-ilar sensors whose measurement matrices might beof different sizes. However, its computational loadincreases with adding more sensors.

3.2.2. Distributed state-vector integration method

based on the modified track-to-track fusion (MTTF)

algorithm

The standard track-to-track fusion (TTF) algo-rithm is an alternative approach to optimally com-bine the local state estimations. In this approach,the individual and local estimations (i.e., xiðkjkÞÞare fed back to the predictive stage of the Kalmanfiltering process corresponding to each individualsensor. This makes the state estimations xiðkjkÞ bemutually dependent. A better approach has beenintroduced by the MTTF algorithm in which theprediction procedure of the Kalman filteringwill be improved by the fused state estimationbX ðkjk � 1Þ at the last time step instead of the indi-vidual or local state estimation (i.e., xiðkjkÞÞ in thestandard TTF algorithm. In this paper, the MTTFalgorithm, shown in Fig. 3, has been employed forthe proposed distributed integration method. Thenew fused estimation bX ðkjkÞ and the covariancematrix of the fused estimation are derived by thefollowing equations:

bX ðkjkÞ ¼ x1ðkjkÞ þ1

N � 1

XN

i¼2

½P 1ðkjkÞ � P 1iðkjkÞ�

� ½P 1ðkjkÞ þ P iðkjkÞ � P 1iðkjkÞ � P i1ðkjkÞ��1

� ½xiðkjkÞ � x1ðkjkÞ� ð43Þ

PðkjkÞ ¼ P 1ðkjkÞ �1

N � 1

XN

i¼2

½P 1ðkjkÞ � P 1iðkjkÞ�

� ½P 1ðkjkÞ þ P iðkjkÞ � P 1iðkjkÞ � P i1ðkjkÞ��1

� ½P 1ðkjkÞ � P i1ðkjkÞ� ð44Þ

Fig. 3. The modified (T–T


4. Simulation studies

A major objective of this research is to investi-gate the usefulness of different multi-sensor integra-tion architectures based on the EKF data fusionalgorithm to detect and diagnose sensor and processfaults. The basic idea of the adapted monitoringapproach is to reconstruct the process states fromthe available multi-sensor data measurements inte-grated in two centralized and decentralized configu-rations using the EKF algorithm for both stateestimation and data fusion purposes. Two generaltypes of sensor and process faults are realized in asimulation continuous stirred tank reactor (CSTR)as a typical chemical benchmark problem.

4.1. CSTR plant description

The CSTR represented schematically in Fig. 4works under atmospheric pressure [14]. It is a coolingwater-jacketed reactor which involves an irreversibleand liquid phase exothermic reaction A ? B takingplace inside the reactor tank. The cooling jacket sur-rounding the reactor circulates the coolant water toabsorb the generated reaction heat.

Two proportional controllers are used to regulatethe reactor outlet temperature (T) and the liquidvolume (V) inside the reactor tank. The temperatureis controlled by manipulating the flow rate of thecoolant water (u1 = Fj) following through thejacket. While, the level in the reactor is controlledby manipulating the outlet flow rate (u2 = Fo) fromthe reactor.

The dynamic behavior of the CSTR is modeledby a system of differential equations translatingmolar and heat balances in the reactor. The derivedmodel is based on the following hypotheses:

� The reactor is perfectly stirred implying that thereaction mass temperature and concentrationsare homogeneous through the mass volume.

) fusion algorithm.


Fig. 4. Continuous stirred tank reactor.


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� The heat losses are negligible.� No phase change occurs in the reaction mass.� The density, the specific heat of the cooling and

the reaction mass heat are independent of thetemperature.� Constant holdup and perfect mixing are assumed

in the cooling jacket.

The resulting CSTR plant model equations canbe described by [14]:

dVdt¼ F i � F o ð45Þ

dðVCaÞdt

¼ F iCai � F oCa� V k0 expEa

RT

� �� Ca

ð46Þ

qcpdðVT Þ

dt¼ qcpðF iT i � F oT Þ � DHV

� ko expEa

RT

� �� Ca� Ua0ðT � T jÞ ð47Þ

qjV jcjdT j

dt¼ qjcjF jðT c � T jÞ þ Ua0ðT � T jÞ ð48Þ

F o ¼ 40� 10ð48� V ÞðProportional level controllerÞ ð49Þ

F j ¼ 49:9� 4ð600� T ÞðProportional temperature controllerÞ ð50Þ

The derived model equations are modified to be for-mulated in terms of the normalized and dimension-less states by the main process variables(Fi,Cai,Ti,Tc) to be monitored. The resulting stateequations include the explicit fault terms (DFi,DCai,DTi,DTc), as follows [15]:

dx1

dt¼ F i

V s

� F os

V s

ðu1 þ 1Þ

þ 1

1þ CaiCasþ T i

T s

DF i

1

V s

þ Cai

V sCas

þ T i

V sT s

� �ð51Þ


dx2

dt¼ F iCai

V sCas

� F osðu1 þ 1Þðx2 þ 1ÞV sðx1 þ 1Þ

� k0ðx2 þ 1Þ exp�Eaðx1 þ 1ÞRT sðx3 þ 1Þ

� �þ Cai

Cas þ Cai þ CasT iT s

DF i

1

V s

þ Cai

V sCas

þ T i

V sT s

� �þ DCai

F i

V sCas

ð52Þ

dx3

dt¼ F iT i

V sT s

� F osðu1 þ 1Þðx3 þ 1ÞV sðx1 þ 1Þ � DHCask0

qcpT s

�ðx2 þ 1Þ exp�Eaðx1 þ 1ÞRT sðx3 þ 1Þ

� �� Ua0ðx3 þ 1Þ

qcpV sðx1 þ 1Þ þUa0T jsðx4 þ 1Þ

qcpV sT s

þ DT iF i

V sT s

þ T i

T s þ CaiT s

Casþ T i

DF i

1

V s

þ Cai

V sCas

þ T i

V sT s

� �ð53Þ

dx4

dt¼ T cF js

V jT js

ðu2 þ 1Þ � F jsðu2 þ 1Þðx4 þ 1ÞV j

þ Ua0T sðx3 þ 1ÞqjcjV jT jsðx1 þ 1Þ �

Ua0ðx4 þ 1ÞqjcjV j

!

þ DT c

F jsðu2 þ 1ÞV jT js

ð54Þ

where

x1 ¼V � V s

V s

; x2 ¼VCa� V sCas

V sCas

; x3 ¼VT � V sT s

V sT s

;

x4 ¼T j � T js

T js

; u1 ¼F o � F os

F os

; u2 ¼F j � F js

F js

ð55Þ


Table 1Non-isothermal CSTR parameters

Notation Variable Steady state values

Fo Outlet flow rate 40 ft3/h (Fos)Cai Inlet reactant concentration 0.5 lb mol of A/ft3


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The relevant parameters for simulating the CSTRplant are tabulated in Table 1. The table includesthe steady-state condition of the reactor indicatedby the subscript ‘‘s” as Vs, Cas, Ts, Tjs, Fos, Fjs atnormal operating point.

(Cais)T Temperature of tank 600 �R (Ts)Fj Coolant flow rate 49.9 ft3/h (Fjs)V Volume of reactor 48 ft3 (Vs)Ca Outlet reactant

concentration0.245 lb mol of A/ft3(Cas)

Tj Temperature of the outletcoolant

594.6 �R (Tjs)

Ti Inlet feed temperature 530 �R (Tis)

Parameter values

Vj Volume of jacket 3.85 ft3

Ea Activation energy 30,000 Btu/lb molU Heat-transfer coefficient 150 Btu/h ft2 �RTc Temperature of the inlet

coolant530 �R

cp Heat capacity (process side) 0.75 Btu/lbm �Rq Density of process mixture 50 lbm/ft3

k0 Frequency factor 7.08 � 1010 h�1

R Universal gas constant 1.99 Btu/lb mol �Ra0 Heat-transfer area 250 ft2

DH Heat of reaction �30,000 Btu/lb molCj Heat capacity (coolant side) 1.0 Btu/lbm �Rqj Density of coolant 62.3 lbm/ft3

4.2. Formulation of plant state equations for faultmonitoring

The CSTR plant dynamic model in Eqs. (51)–(54) demonstrate its nonlinear dynamic naturewhich can be represented by the following generalnonlinear state equations (fi(�); i = 1, . . ., 4) in dis-crete-time domain:

x1ðk þ 1Þ ¼ f1ðxðkÞ; uðkÞ; dðkÞÞ þ w1ðk � 1Þ ð56Þx2ðk þ 1Þ ¼ f2ðxðkÞ; uðkÞ; dðkÞÞ þ w2ðk � 1Þ ð57Þx3ðk þ 1Þ ¼ f3ðxðkÞ; uðkÞ; dðkÞÞ þ w3ðk � 1Þ ð58Þx4ðk þ 1Þ ¼ f4ðxðkÞ; uðkÞ; dðkÞÞ þ w4ðk � 1Þ ð59Þ

where k denotes the sampling instants and

xðkÞ ¼ ½x1ðkÞ x2ðkÞ x3ðkÞ x4ðkÞ�T

ðState vectorÞ ð60ÞuðkÞ ¼ ½u1ðkÞ u2ðkÞ�T

ðInput vectorÞ ð61ÞdðkÞ ¼ ½DF iðkÞ DCaiðkÞ DT iðkÞ DT cðkÞ�T

ðProcess fault vectorÞ ð62Þ

wi(k � 1); i = 1, . . ., 4, describe the process noiseswhich have been added artificially to the CSTR pro-cess state model equations to include the real uncer-tainties faced in the practical situations. Theseprocess noises are assumed to behave as zero-meanwhite Gaussian noises with covariance matrix Q(k).Similarly, the output equation can be described bythe following general nonlinear model (h(�)), derivedfrom the CSTR dynamic model in Eqs. (51)–(54):

yðkÞ ¼ hðx�ðkÞÞ þ vðkÞ ð63Þ

where x*(k) = [dT(k) xT(k)]T represents the aug-mented state variable vector, v(k) = [v1(k), v2(k),v3(k),v4(k)]T has been added to represent the inevi-table measurement noises and y(k) denotes the out-put vector. v(k) is assumed to behave as zero-meanwhite Gaussian noises with covariance matrixR(k). Therefore, the nonlinear functions f(�) = [f1(�),f2(�), f3(�), f4(�)]Tand h(�) in state and output modelequations can be linearized at each sampling timearound the most recent process condition estimate,


leading to the augmented state-space model givenby Eqs. (10) and (11).

For computer simulation of the plant fault mon-itoring studies, the CSTR nonlinear model dynam-ics, described in Eqs. (56)–(59), are implementedusing s-function and SIMULINK facilities in MAT-LAB. The basic time unit is hours (h) and the sam-pling time is taken to be equal to 0.005 h.

4.3. Sensor fault detection and diagnosis

A sensor often comprises of different parts suchas a sensing device, transducer, signal processor,and communication interface. Any of these partsmay malfunction causing the sensor to generate sig-nals with unacceptable deviation from its normalcondition. A sensor working under this condition,i.e. faulty sensor, may cause process performancedegradation, process shut-down or even fatal acci-dents. A sensor is declared faulty when it displaysa non-permitted deviation from the characteristicproperties.

This study focuses on drift in sensor calibration(i.e., bias error) and drift in sensor degradation(i.e., excessive-variance noise) anomalies. A sensor


Table 2Sensor fault scenario in the key variable sensors

Sensor fault parameter The magnitude of fault

V T Ca Tj

Bias 5 0.03 �30 60Excessive-variance coefficient 0.5 0.01 2 2.5


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is biased if its reading differs by a constant valuefrom the actual value. The excessive-variance noisefault refers to the case where the sensor reading isaffected by an excessive-variance white noise. Thisfault might represent the gradual or incipient sensordegradation due to wear, aging, fouling or corrosionwhich is generally not detectable in their earlystages.

4.3.1. Drift in sensor calibration (bias error)

To carry out the sensor calibration fault detec-tion, a method is required to quantity the state der-ivation due to sensor fault occurrence. Thefollowing stochastic AR model is employed toexplain the time evolution of the sensor bias as extraprocess state variables

biðkÞ ¼ biðk � 1Þ þ N biðkÞ; i ¼ 1; . . . ; nb ð64Þwhere nb denotes the number of faulty sensors, andNbi(k) indicates a zero-mean white Gaussian noisewith covariance matrices Qbi(k) describing the biasmodel uncertainty. Therefore, the process state var-

Fig. 5. Sensor faults including bias and drift, (a) sens


iable vector in Eqs. (10) and (11) is augmented bythe sensor bias states, defined in Eq. (64).

4.3.2. Drift in sensor degradation (excessive-variance

noise)To simulate this type of sensor fault, the follow-

ing model equation is used to describe an added sen-sor white noise whose variance characteristic ischanged linearly with time evolution.

nðkÞ ¼ r0

k � 1

T max=1:5� 1N nðkÞ ð65Þ

where r0 shows the coefficient noise variance, Tmax

indicates the maximum number of measurement

or V, (b) sensor Ca, (c) sensor T, (d) sensor Tc.



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time samples, and Nn(k) is a zero-mean whiteGaussian noise with unit variance and uncorrelatedwith process and measurement noises.

4.3.3. Simulation and results

To illustrate the application of the fault detectionand identification procedures, a simulation studywill be conducted in this subsection. Consider theCSTR plant described in Section 4.1. A sensor faultscenario is simulated in which the four CSTR keymeasured variables (V,T,Ca,Tj) are corrupted withthe combined drift in their corresponding sensorcalibration (bias) and degradation (excessive-vari-ance) as summarized in Table 2.

The distributed state-vector fusion integrationmethod is used to detect and diagnosis the sensorcalibration and degradation faults. This is mainlydue to the distributed nature of this method inwhich the filtering process is done by parallel Kal-man filters, located locally at the individual sensorpositions. This makes it possible to detect and iso-

Fig. 6. Bias d


late the sensor faults in order to differentiate the realsensor data before integrating it at the fusion center.

In this approach, the sensor bias faults are esti-mated by the individual EKF estimation algorithmsusing the augmented state variable vector includingthe bias vector term. While, the sensor faults due todrift in sensors degradation are detected via theresidual errors (z� H �x) of each individual EKFestimates.

Fig. 5 illustrates the corrupted sensor measure-ments V,T,Ca,Tj due to the simultaneous occur-rence of both introduced sensor faults (describedin Table 2) including with the sensor data measure-ments due to actual fault-free for comparisonpurposes.

Figs. 6 and 7 demonstrate the resulting sensorfault detection and isolation in which both type ofsensor faults been estimated. As shown, the faultsdue to the sensor biases have accurately been esti-mated in Fig. 6. Fig. 7 depicts the estimated sensorfaults due to drift in the degradation.

etection.


Fig. 7. Drift detection.

Table 3Comparison of RMSE drift fault between proposed method andICA method

Measurementvariable

RMSE for the proposedmethod (%)

RMSE for theICA method (%)

V 3.5466 3.4415Ca 0.4462 0.6664T 3.7973 17.511Tj 3.6446 14.1558


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Table 3 summarizes the resulting drift detectionaccuracies calculated individually for each measuredvariable in terms of the root mean-squared error(RMSE) in the estimates (i.e., the difference betweenestimated and true values).

The obtained results indicate the superior effi-ciency of the proposed algorithm to detect and diag-nose both types of the sensor faults. To investigatethe capability of the presented algorithm to revealthe excessive-variance sensor fault, its performanceis evaluated and compared with that of the indepen-dent component analysis (ICA), as an emergingpowerful statistical signal processing technique,which is able to extract the hidden or underlyingsensor fault noise from the measured data usingthe FastICA algorithm [16]. The obtained resultsare depicted in Figs. 8–11 for the different key mea-sured variables.

A comparison of the RMSE measures for theboth approaches, indicate that the presented algo-rithm in this work is able to provide more accurateestimates with respect to the ICA method.


4.4. Process fault detection and diagnosis

4.4.1. CSTR process faults

Table 4 lists some of the probable CSTR processfaults.

4.4.2. Simulation and results

A series of simulation runs will be conducted onthe CSTR plant to evaluate and compare the effec-tiveness of the two multi-sensor distributed andcentralized integration approaches based on the



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Kalman filter data fusion algorithm. To carry outthe different simulation test runs under the sameprocess fault, a predefined multi-fault test scenariois programmed. First, the inlet reactant feed is chan-ged by a negative step of DFi = 10% at sampling

Fig. 8. Drift detection in the sensor (V). (a) Th

Fig. 9. Drift detection in the sensor (Ca). (a) Th


instant (t) of 5. Then, the inlet reactant concentra-tion undergoes a step change of DCai = 10% att = 10. Following that, the inlet temperature of thereactant undergoes a negative step change ofDTi = 10% at t = 15 and subsequently the inlet tem-

e proposed method. (b) The ICA method.




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perature of coolant goes up by a step change of DTc = 5% at t = 20. Thus, this fault test scenarioembodies a complete simulation picture in whichsingle, double, triple and finally quadruple processfaults occur sequentially after a 5 sampling timeinterval.

Fig. 10. Drift detection in the sensor (T). (a) Th

Fig. 11. Drift detection in the sensor (Tc). (a) Th


Three different simulation runs, corresponding tothree different process monitoring approaches, i.e.,non-adaptive centralized output augmented fusion(OAF) or (NACOAF) method, adaptive centralizedOAF (ACOAF) method and adaptive distributedstate-vector fusion (ADSVF) method, were carried




Table 4List of process faults

Fault number Fault name Notation

1 Low inlet feed of reactant dFi

2 High inlet concentration of reactant dCai

3 Low inlet temperature of reactant dTi

4 High inlet temperature of coolant dTc


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out. Fig. 12 demonstrates the resulting process faultestimation outcomes due to the foregoing threemonitoring approaches.

In order to clarify the assessment of the monitor-ing performances, the corresponding errors in theresulting fault estimation are depicted in Fig. 13.

Noting that all the fault monitoring approacheshave been evaluated on the same measured datafor the same realizations of the fault test scenario.The resulting observations demonstrate that theACOAF method generally present better fault esti-mation performance in terms of the resulting RMSEmeasures, summarized in Table 5.

However, ADSVF method has a lower computa-tion time and communication cost and benefits the

Fig. 12. Estimation of faults occurring at di


advantage of parallel distributed implementation,leading to a higher fault-tolerant capability.

5. Conclusions

This paper addresses the sensor and process faultmonitoring problem using multi-sensor data fusiontechnique based on EKF estimation algorithm. Adiscrete-time EKF approach has been adapted toenhance the robustness of its implementation. Theprocess and sensor bias faults have been treated asextra states from the monitoring perspective, lead-ing to a new augmented state-space model formula-tion. A new adaptive EKF algorithm has beendeveloped to include the time-varying nature of pro-cess and measurement noises in the estimation algo-rithm. To realize this uncertainty adaptation, theinnovation and residual errors information havebeen incorporated in the developed online relation-ships, to update the process and measurementcovariance matrices. This adaptive scheme increasesthe accuracy of the estimation and prevents the

fferent times with different magnitudes.


Fig. 13. The fault estimation errors and their RMSE.

Table 5Resulting RMSE measures and computation time

Method RMSE of process faults (%) Computation time (s)

dFi dCai dTi dTc

NACOAF 17.2971 88.5621 61.0648 17.8263 15.6941ACOAF 2.8565 2.8379 2.8471 1.2242 26.7952ADSVF 2.9819 3.9120 4.0590 1.2850 15.3771


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EKF algorithm to degrade its performance. Twodifferent centralized and distributed multi-sensorintegration architectures were presented based onthe non-adaptive and adaptive EKF data fusionalgorithms to monitor sensor and process faults.

A set of simulation studies were conducted on aCSTR benchmark problem to investigate the per-formances of the proposed monitoring methods.The sensor fault study focused on drift in sensor cal-ibration (i.e., bias error) and drift in sensor degrada-tion (i.e., excessive-variance noise) anomalies. Theprocess fault study included four probable CSTRprocess faults listed in Table 4.

The sensor bias faults were estimated accuratelyusing the individual EKF algorithms based on theaugmented state variable vector including the biasvector terms. While, sensor faults due to drifts wereestimated via the EKF residual estimation errors.Comparison of the resulting sensor drift fault mon-itoring to that of the ICA method, as a powerful


noise separation approach, revealed a better perfor-mance in terms of the obtained RMSE measure.

Three different proposed monitoring approaches,i.e. NACOAF, ACOAF and ADSVF, were testedon the CSTR plant to evaluate their performanceagainst process faults using a series of simulationstudies. The resulting performances, summarizedin Table 5, demonstrate that the ACOAF methodcan generally present a better fault estimation fromthe RMSE measure point of view. However, ADS-VF method has the advantages of parallel distrib-uted architecture, benefiting a lower computationtime and communication cost and higher fault-tol-erant characteristic.

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[2] M.A. Kramer, B.L. Palowitch Jr., A rule based approach tofault diagnosis using the signed directed graph, AIChE J. 33(7) (1987) 1067–1078.

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