Development and Implementation of a High-Performance Sensor System for an Industrial Polymer Reactor

Development and Implementation of a High-Performance SensorSystem for an Industrial Polymer Reactor

Martin Pottmann

E.I. DuPont de Nemours and Company, Wilmington, Delaware 19880

Babatunde A. Ogunnaike*

Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

James S. Schwaber

Department of Pathology, Anatomy and Cell Biology, Thomas Jefferson University,Philadelphia, Pennsylvania 19107

Because of increasingly stringent demands on product quality, process measurements of highprecision, accuracy, and reliability are required in many processes where process variables aretraditionally difficult to measure. In many applications, several different measurements of certainkey process variables are available from various sources, including on-line analyzers, lab analyses,inferential measurements, and models of all types. Because these individual sensors often differsignificantly in terms of their static and dynamic characteristics, precision, accuracy, samplingrate, measurement time delay, etc., in practice, only one of them is utilized for process monitoringand control purposes; the other sensor measurements are ignored. However, significantlyimproved estimates of the process variables can be obtained by utilizing all of the availablesensor information simultaneously, i.e., by combining (fusing) these dissimilar measurementsinto a single estimate in a robust and fault-tolerant fashion. There is evidence that such sensorfusion operations are integral to the operation of biological control systems, and it is from oneof these systems (the blood pressure control system) that we have drawn inspiration fordeveloping the technique discussed in this article. A sensor fusion approach is developed usingstochastic systems theory (in particular, Kalman filtering). The advantages of using multiple,redundant sensors, as well as the potential improvements achievable by combining delayed ormultirate sensor information with that of a single nominal sensor, are then quantified.Robustness is achieved by augmenting the nominal technique with failure detection, classifica-tion, and compensation schemes, based on statistical hypothesis testing. The performance ofthe techniques is demonstrated on an actual plant application in which improved feed compositionestimates are obtained for an industrial ethylene copolymerization reactor.

1. Introduction

In many processes, the main barrier to obtaininghigh-quality measurements is the fact that many keyprocess variables are difficult to measure online. Reli-able measurements can only be obtained by augmentingonline sensors with laboratory analyses, or by inferen-tial deduction from auxiliary measurements. Therefore,techniques are needed that are capable of combininginformation about a process variable from a variety ofsources, including online instruments, offline analyzers,and models of all types (e.g., fundamental models,artificial neural networks, regression models). Thespecific sensor fusion problem addressed in this articlecan be stated as follows: “Given measurements of asingle process variable that are available from differentsources, at possibly different rates, and with differentmeasurement delays, form a composite measurement ina robust and failure tolerant fashion.”

Many practical examples exist in which measure-ments of a key process variable are available from

several different sources: For instance, compositionmeasurements may be provided online by several gaschromatographs, in addition to laboratory analysisperformed at irregular and infrequent intervals. Viscos-ity can be measured online via viscometers; however, afaster and more-robust indication may be availablethrough measuring the pressure drop across a pipe.Both online measurements are typically augmentedwith infrequent laboratory analyses.

Recently, there has been growing interest in the useof multiple sensors to increase the capabilities of intel-ligent systems,1 as well as to improve the robustness ofcomplex dynamical systems. Typical domains of ap-plication for sensor fusion techniques are robotics2 andinertial navigation,3 as well as target detection andtracking.4 Applications in the process industries typi-cally involve the related problems of inferential mea-surements,5,6 data reconciliation,7,8 and sensor valida-tion,9 but rarely with the sensor fusion problem statedpreviously.

Generally, sensor fusion may be considered as amethodology for integrating signals from multiplesources,10 with the following potential advantages:information from redundant or complementary sensors

* To whom correspondence is to be addressed. Tel.:(302)-831-4504. Fax: (302)-831-1048. E-mail: [email protected].

2606 Ind. Eng. Chem. Res. 2005, 44, 2606-2620

10.1021/ie049614t CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 12/09/2004

can be utilized, resulting in reduced uncertainty, andincreased precision and reliability in the event of asensor failure. Moreover, information may be providedwith a smaller delay or at a higher rate by multiplesensor arrangements (e.g., multiple online analyzers)than by a single sensor. The common principle utilizedby most sensor fusion techniques is that of measure-ment redundancy, which can be present in two forms:

(1) Direct Redundancy. Multiple instruments provideestimates of the same process variable. However, theseredundant measurements may be available at differentfrequencies or with different time delays. This type ofredundancy is the focus of this article.

(2) Analytical Redundancy. In addition to a directmeasurement, redundant information about a processvariable may be available from measurements of other“analytically” related variables.

Sensor fusion methods range from applications ofprobabilistic models (such as Bayesian reasoning), least-squares techniques (such as Kalman filtering), to ar-tifical intelligence approaches.10 In what follows, webriefly review several approaches that are currentlyavailable for sensor fusion and fault detection in directlyredundant sensor systems.

If more than two redundant measurements of aprocess variable are available, simple “voting logic” canbe used to eliminate faulty sensors. In this approach,sensors that do not agree with the majority of theremaining sensors, are considered faulty (majority vot-ing). However, voting sytems perform poorly in detecting“soft failures”, such as small sensor biases.11

As an extension to voting logic, fuzzy logic techniquescan be applied to data fusion problems. Fuzzy logicinvolves extensions of Boolean logic to a continuous-valued logic via the concept of membership functions.12

For instance, Hong and Wang13 proposed a techniquefor fusing both noisy and fuzzy data, which uses bothKalman filtering and fuzzy arithmetic.

Expert system approaches have been proposed in thecontext of sensor validation. The approach of Lee14 isbased on causal relations between sensors. In particular,the expected value of a sensor output can be determinedfrom measurements of all the remaining sensors whenthey are in a causal relation. Individual sensors areassigned validity levels corresponding to the number ofindependent sources of supportive evidence that aparticular sensor reading is valid.14

Applications of artificial neural networks (ANNs) insensor fusion and failure detection are also quite com-mon.6,15 For instance, ANNs have been used6 to combinemeasurements from several analyzers and standardreactor measurements to predict key process variablesfor online monitoring of a fermentation process. Naiduet al.15 used a back-propagation network for sensorfailure detection based on Fourier analysis of a diag-nostic signal.

Kalman filtering provides a convenient framework forperforming sensor fusion, and several applications havebeen reported in the literature.5,16,17 For instance, aKalman filter has been used in the context of inferentialestimation of a process variable from secondary mea-surements that are available more frequently than theprimary measurement.5 In another approach, a mainKalman filter that processes all measurements is usedin conjunction with a bank of auxiliary Kalman filtersthat process subsets of the measurements;16 failure

detection is performed by checking the consistencybetween the state estimates of the main Kalman filterand the corresponding estimates of the auxiliary filters.This technique is based on the premise that sensorfailures have only a small effect on the overall estimateprovided by the main Kalman filter, whereas the effectson the auxiliary filters that contain the faulty sensorswill be more significant. Similarly, decentralized Kal-man filters have been shown to have improved faulttolerance, when compared to a single (“centralized”)Kalman filter.17

Yet to be addressed in a coherent, systematic manneris the problem of combining measurements from differ-ent sources in an actual industrial setting, where sensorcharacteristics are not known perfectly, and robustnessto sensor biases and other partial sensor failures arecritical to process operation. This is the focus of thispaper. First, we address the nominal sensor fusionproblem and then explicitly address the robust problemby augmenting the nominal approach with sensorfailure detection, classification, and compensationschemes based on statistical hypothesis testing.

The remainder of this article is organized as follows.In section 2, we discuss an example of sensor fusion inbiological systems and identify one of the underlyingprinciples that may be utilized for process applications.Section 3 presents a dynamic sensor fusion approachbased on Kalman filtering, whereas section 4 examinesrobustness issues in the presence of sensor failures. Theapplication study presented in section 5 demonstratesthe performance of the proposed techniques on anindustrial copolymerization reactor.

2. Sensor Fusion in Biological Systems

Biological control systems face problems that aresimilar to those encountered in chemical process control.However, biological systems seem to solve these prob-lems much more efficiently than currently availableindustrial control approaches. Current knowledge ofbiological control systems indicates that the high levelof reliability and accuracy required of measurements ofcritical physiological variables is achieved by highlyintegrated measurement schemes with large arrays ofpartially redundant sensors. Moreover, these systemsare highly adaptive and remarkably tolerant to sensorand actuator failure. Thus, biological control systemsmay provide a viable source of inspiration for developingsensor fusion techniques. Referring to the fault-tolerantsignal processing capabilities of neural systems, it hasbeen recognized18 that “robust neural mechanisms em-ploy a combination of (i) a large number of inputscarrying overlapping but nonidentical information, and(ii) a limitation of the effect that each input can have onthe output by a suitable nonlinearity.”

In the following, we will discuss an example of sensorfusion in biological control systems and note several ofits characteristics that may be responsible for achievingproperties (i) and (ii) cited in the previous paragraph.

2.1. The Baroreceptor Reflex. The baroreceptorreflex (or baroreflex for short) is responsible for short-term blood pressure regulation. In the baroreflex, arraysof pressure sensitive neuronal stretch receptor endingslocated in the aortic arch and carotid sinus transmitblood pressure information to other neurons (so-called“second-order neurons”) in the brain. Information istransmitted via voltage signals that have the form of

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2607

trains of voltage spikes of very short duration, whichare referenced as “action potentials”. Whenever such avoltage spike is generated, a neuron is said to fire anaction potential. The frequency with which these actionpotentials are generated is directly related to thepressure at the location of the receptor: If the bloodpressure is low, the firing frequency of these barore-ceptive neurons will be low, whereas for higher pres-sures, the firing frequency will increase, up to amaximum frequency (or saturation frequency).

In a particular region in the brainsthe so-called NTS(nucleus tractus solitarius)sthese baroreceptor signalsare integrated with other sensory signals that indicateblood composition and other physiological variables.These integrated signals are then processed further bytwo parallel control systems that eventually send sig-nals to the heart, to maintain the blood pressure at adesired level. See refs 19 and 20 for a more-detaileddiscussion of the baroreceptor reflex and potentialapplications in chemical process control.

Several properties of the baroreceptor reflex seem tobe particularly relevant in the context of sensor fusion:

(1) The baroreceptive neurons are organized in theform of a large number of neurons with narrow indi-vidual operating ranges, which have the highest sensi-tivity around a nominal operating point (i.e., pressure)and decreasing sensitivity away from this nominaloperating point. The individual baroreceptors havepartially overlapping operating ranges such that thedesired measurement range is completely spanned bythe receptor ensemble rather than by single neurons.

(2) The frequency with which a baroreceptor cangenerate and transmit action potentials is boundedbetween a lower limit (the frequency at which thereceptor generates purely random action potentials thatare unrelated to the blood pressure) and an upper limitat which the receptor saturates. Beyond this saturationlimit, an increase in pressure does not further elevatethe firing frequency, resulting in a sigmoidal-shapedfiring characteristic as shown in ref 21.

(3) The baroreflex possesses a high degree of robust-ness, because the removal of a significant fraction ofbaroreceptive neurons has a relatively small effect onthe overall control system performance.22

(4) The baroreceptor reflex apparently performs out-lier detection; data that are inconsistent with othersensory inputs are “ignored” at the central signalprocessing stage.22

Note that property 1 provides the brain with a largenumber of inputs carrying overlapping but nonidenticalinformation about the blood pressure. The steady-statefiring characteristics of the individual receptors, whichessentially have a sigmoidal shape, as discussed in thepreviously mentioned property 2, can be thought of asrepresenting a nonlinearity that can limit the effect thateach input (a baroreceptor signal) can have on theoutput (the next signal processing stage).

Unlike the baroreceptors, whose pressure-responsefunctions have been well-defined, very little is knownabout the processing of the baroreceptor signals withinthe brain. The first stage of this signal processing occursin the so-called second-order neurons in the NTS.Electrophysiological recordings were undertaken byRobert F. Rogers at DuPont to define the properties ofthese neurons. The results of this work indicate thatsecond-order neurons receiving baroreceptor inputs arecapable of providing the direction and magnitude of

pressure changes but are unlikely to provide a reliableencoding of mean pressure.23,24 These findings result inthe hypothesis that the central nervous system extractsand separates different pieces of information providedwithin it and uses them for different functions.24

Despite these recent findings, the actual mechanismsfor signal processing within this part of the brain arestill poorly understood. Therefore, our objective is notto “copy” the neuronal signal processing within the NTS,but rather to mimic some of the strategies or generalprinciples (e.g., architecture) of the baroreceptor reflex.As an example of this approach, we discuss in thefollowing paragraphs how one of the general character-istics of the baroreceptor reflex (the utilization of largenumbers of sensors with narrow individual operatingranges) can be exploited in designing multisensorsystems.

2.2. Idealized Sensor Configurations. Here, weconsider a multisensor system, which can be thoughtof as being derived from the multisensor array seen inthe baroreflex by making the following crude ap-proximations: (i) the sigmoidal sensor characteristicsare replaced by piecewise linear ones, and (ii) there isno overlap between the individual sensor operatingranges (i.e., the entire operating range is divided intosegments of identical width and every sensor providesmeasurements only in its assigned local operatingrange). To quantify the advantages of such a multisen-sor system over a sensor system in which all sensorscover the entire measurement range, we consider thefollowing example. We compare two sensor systems thateach consist of two sensors (see Figure 1). Sensor system1 utilizes two sensors that have the same global operat-ing range; its sensors are called Type 1 sensors. Sensorsystem 2 consists of two sensors, one of which coversthe lower half of the overall operating range and theother sensor covers the upper half. Its sensors are calledType 2 sensors.

Figure 1 shows the raw sensor output (for example,a voltage), as a function of the process variable P, forboth the sensors of sensor system 1 (solid line) andsensor system 2 (dashed lines). If we assume that,independent of the sensor type, the same raw measure-ment error is present (characterized by the standarddeviation σV), the standard deviation of the estimatedprocess variable that results from a Type 2 sensor (i.e.,σP) is only half the standard deviation associated witha Type 1 sensor (2σP). The variance of the compositemeasurement of sensor system 1 (σ1

2) (obtained as the

Figure 1. Multisensor systems.

2608 Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005

average of the two Type 1 measurements) is given by

whereas the variance σ22 of the “combined” sensor

system 2 measurement (which is simply obtained fromone of the two Type 2 sensors) is

Thus, the advantage of limited operating ranges isbetter precision. However, note that, in this example,there is no overlap between the sensor operating ranges.Therefore, the reliability of sensor system 1 can beexpected to be superior to that of sensor system 2,because there are two sensors available at every pointof the input range. The advantage of overlapping sensoroperating ranges is that redundancy is introduced and,therefore, the robustness of the measurement systemis improved. This leads to the conclusion that, with acombination of partially overlapping sensors with lim-ited operating ranges, we can either increase robustnessat the expense of precision or increase precision at theexpense of robustness.

These concepts obviously have significant implicationsfor sensor design (choosing the number and configura-tion of sensors required to achieve specified precisionand robustness goals). However, they also result in thecomplementary “analysis problem”: how does one sys-tematically combine the multiple measurements froma given multisensor configuration? How do the variousdesign variables affect achievable precision and robust-ness? A detailed discussion of these and other issuesconcerning the analysis of multisensor configurationsis provided in the next section.

3. A Dynamic Sensor Fusion Approach

The sensor fusion approach developed in this sectionrequires two types of information: (i) a process model,i.e., a description of how the true process variable ηevolves over time, and (ii) sensor models, i.e., represen-tations of how the individual sensors respond to changesin the process variable η.

3.1. Process Model. The process model consideredhere is the general linear state-space model of the form

where xp is the state vector of the process model, u avector of process inputs, vp an uncorrelated, white (i.e.,independent, identically distributed) zero-mean randomdisturbance, and η the true value of the process variable.Output disturbances are not included, because η rep-resents the true (noise-free) value of the process vari-able. We will subsequently use the notion of a “preciseprocess model” if the effects of the stochastic modelcomponents on the process variable η are small, whereasa model will be referenced as being “imprecise” if thevariance of the stochastic components at the output islarge.

In cases where a complete process model that containsdeterministic and stochastic components as in eqs 3 and4 is not available, the process variable η can be describedas a purely stochastic process. A generally accepted andnonrestrictive assumption is that η(k) varies relatively

smoothly with k. A natural class of such smoothlyvarying random processes is the class of martingales,25

for which the following is true:

The simplest martingale is the random walk process,which is defined by

where vp(k) is an uncorrelated, white, zero-mean ran-dom sequence with a variance of E[vp(k)2] ) Pp(k). Notethat this process model has been introduced here as onethat, in the absence of a deterministic model, requiresthe least amount of extraneous detail to describe asmoothly varying random process. Moreover, becausea random walk process is nonstationary, the estimateof η that is based on this model and derived by theoptimal filter discussed below will be unbiased, if theactual process variable is asymptotically constant.

In practice, some knowledge about the process timeconstants and operating conditions can provide guidancein selecting adequate values for Pp(k). By choosing avery small value for Pp, the estimate will rely heavilyon the smooth variation model implied by eq 6, whereas,by choosing very large values for Pp (relative to theprecision of the sensors), the estimate of η will mainlyrely on measurements.

3.2. Sensor Models. The individual sensors areassumed to be linear and can be described by standardlinear state-space models. Examples of such sensormodels include the following:

(a) Unbiased Static Sensor. The measurement uncer-tainty is completely captured by additive measurementnoise, and the sensor model consists of a single outputequation:

(b) Static Sensor with Constant Bias. A bias state isintroduced as

and the output (as defined by eq 7) is modified to includethe bias state,

(c) Static Sensor with Time-Varying Bias. The sim-plest representation of a time-varying bias is again therandom walk process, which leads to the followingsensor model:

where E[vi(k)2] characterizes the “severity” of the ran-dom walk.

Next, we combine the individual sensor models intoa single model of the corresponding multisensor systemby defining the state vector,

σ12 ) 2σP

2 (1)

σ22 ) σP

2 (2)

xp(k + 1) ) Ap(k)xp(k) + Bp(k)u(k) + Fp(k)vp(k) (3)

η(k) ) Cp(k)xp(k) (4)

E[η(k) | η(k - 1), ..., η(1)] ) η(k - 1) (5)

η(k + 1) ) η(k) + vp(k) (6)

yi(k) ) η(k) + wi(k) (7)

âi(k + 1) ) âi(k) (8)

yi(k) ) η(k) + âi(k) + wi(k) (9)

âi(k + 1) ) âi(k) + vi(k) (10)

yi(k) ) η(k) + âi(k) + wi(k) (11)

xs(k) ) [x1(k)T ‚‚‚ xn(k)T ]T (12)


Explicitly indicating the potential time dependenceof the system matrixes, the model of the multisensorsystem can be written as

where vs and w are uncorrelated, white, zero-meanrandom disturbances.

3.3. Solution of the Dynamic Sensor FusionProblem. The process and sensor models can becombined by introducing the following matrices:

It is entirely reasonable to assume that the process andsensor disturbances are uncorrelated. Hence,

with the overall state and disturbance vectors definedas

the combined model can be written as

with

We also assume that the measurement noise (w(k)) andthe state disturbances (v(k)) are uncorrelated, i.e., cov-[v(k), w(k)] ) 0. Based on the description of the overallsystem, the Kalman filter26 provides the optimal stateestimate (i.e., the estimate with the minimum estima-tion error variance among all unbiased estimates). Itcan be written in the following form:

where K is the Kalman filter gain and ∑ (with ∑(0) )∑0) is the so-called a priori covariance of the state

estimation error; it is based on measurements up to y(k)but not including y(k + 1). Equivalent Kalman filterformulations that are based on an a posteriori covari-ance, which includes knowledge of the most-recentmeasurements (i.e., up to y(k + 1)), are also available.The a posteriori estimation error covariance T is relatedto the a priori covariance, via

Recall that the true process variable η(k) is a knownlinear combination of a subset of the state variables ineq 17, as stated in eq 4. Therefore, the optimal estimateof η is obtained as a linear combination of the stateestimates provided by the Kalman filter.

3.4. Practical Considerations. We now considerseveral specific practical situations that the sensorfusion approach will face. In particular, the followingcases will be addressed:

(a) Static Sensors. The sensor dynamics can beneglected, resulting in a sensor model as described byeq 7.

(b) Delayed Sensors. The sensors provide delayedmeasurements but are otherwise static.

(c) Multirate Sensors. In many applications, some ofthe available measurement devices (e.g., a chemicalanalyzer) do not provide measurements at every sam-pling instant but at multiples of the basic samplinginterval. Although such instruments also often producedelayed measurements, we will only consider undelayedmultirate sensors, for the sake of clarity.

(d) Reference Sensor. In addition to online sensors, alaboratory analysis of a process sample is performed atinfrequent and irregular time intervals, which providesan unbiased “reference” measurement.

3.4.1. Static Sensors. If we assume that the trueprocess variable η is generated by a random walk (seeeq 6), the combined system consisting of process andsensor models (eq 7) can be written as

with

Note that if there was a deterministic model component,it could be omitted entirely for investigations concerningestimation precision.27 Therefore, the results derivedhere are also valid if there are underlying deterministicmodels, whose uncertainties can be described by eq 6.

The Kalman filter equations (eqs 21-23) can now beused to determine the estimate of the process variable,η̂(k). Because the multisensor system in eqs 25-27 istime-invariant, the Kalman filter gain and covariancewill converge to their steady-state values, under reason-ably mild conditions (for details, see ref 28, pp 252 ff).In this situation, it is common practice to use the steady-state Kalman filter for state estimation, obtained byiterating eq 22, or directly by letting ∑(k + 1) ) ∑(k) )

xs(k + 1) ) As(k)xs(k) + Bs(k)η(k) + Fs(k)vs(k) (13)

y(k) ) Cs(k)xs(k) + Ds(k)η(k) + w(k) (14)

A(k) ) [As(k) Bs(k)Cp(k)0 Ap(k) ]

B(k) ) [ 0Bp(k) ]

C(k) ) [Cs(k) Ds(k)Cp(k) ] (15)

F(k) ) [Fs(k) 00 Fp(k) ] (16)

x(k) ) [xs(k)xp(k) ]

v(k) ) [vs(k)vp(k) ] (17)

x(k + 1) ) A(k)x(k) + B(k)u(k) + F(k)v(k) (18)

y(k) ) C(k)x(k) + w(k) (19)

cov[v(k), v(k)] ) P(k)

cov[w(k), w(k)] ) Q(k)

x(0) ) x0 (20)

x̂(k + 1|k + 1) ) A(k)x̂(k|k) + B(k)u(k) + K(k + 1) ×[y(k + 1) - C(k + 1)A(k)x̂(k|k)] (21)

Σ(k + 1) ) A(k){Σ(k) - Σ(k)C(k)T[C(k)Σ(k)C(k)T +Q(k)]-1C(k)Σ(k)}A(k)T+ F(k)P(k)F(k)T (22)

K(k + 1) ) Σ(k + 1)C(k + 1)T[C(k + 1)Σ(k + 1)C(k +1)T + Q(k + 1)]-1 (23)

T(k) ) [I - K(k)C(k)]Σ(k) (24)

η(k + 1) ) η(k) + v(k) (25)

y(k) ) Cη(k) + w(k) ) [1l1 ]η(k) + w(k) (26)

cov[v(k), v(k)] ) P

cov[w(k), w(k)] ) Q ) [Q1· · · Qn

] (27)


∑ in eq 22, which results in the algebraic equation (with∑ being scalar)

The solution of eq 28 immediately yields the steady-state Kalman filter gain K, as

The steady-state a posteriori estimation error varianceT is given by

These results are now used to determine the perfor-mance gain that can be achieved using redundantsensors with identical characteristics. For this purpose,we define an efficiency factor φ1:

where σi2 is the error variance associated with estimates

provided by an ensemble of i identical sensors. Figure2 shows the efficiency factor φ1, as a function of the totalnumber of sensors. Not surprisingly, the potentialimprovement is dependent on the ratio P/Q1, whichcharacterizes the precision of the sensors, relative to theprecision of the process model. In particular, if there isno process knowledge available (i.e., P/Q1 f ∞), theefficiency factor increases linearly with the total numberof sensors. Because, in this case, the optimal estimateis simply obtained by averaging the individual mea-surements, it follows that

As the process model becomes more precise, comparedto the sensors, the advantages of using multiple sensorsdiminish gradually. For the limiting case P/Q1 f 0, theefficiency factor φ1 reduces to

3.4.2. Delayed Sensors. Here, we consider delayedbut otherwise static sensors. Starting with a random

walk process model (eq 25) with E[v1(k)2] ) P1, and oneundelayed static sensor as in eq 7, with E[w1(k)2] ) Q1,we now add a second static sensor with a measurementdelay of θ sampling periods:

with E[w2(k)2] ) Q2. Introducing the following statevector,

yields the overall system description

where the noise terms are characterized by

The steady-state Kalman filter equations are again usedto assess the improvement in the estimation precisionthat can be achieved by adding a delayed sensor. Forthis purpose, an efficiency measure (φ2) is introduced:

where σ12 is the error variance in estimating η if a single

undelayed sensor is used and σθ2 is the corresponding

variance if a second sensor with a delay θ is added.The results for the case, Q1/Q2 ) 10 are shown in

Figure 3. The efficiency factor φ2 is shown as a functionof the time delay of the secondary sensor, for fivedifferent values of the P1/Q1 ratio, which characterizesthe relative precision of the primary sensor versus theprecision of the process model. The results show thatadding a delayed sensor is advantageous only if thereis some process knowledge available, because a delayedsensor can only contribute to the estimate of the processvariable if reliable prediction beyond the delay ispossible. Moreover, the improvements that can beobtained from adding a delayed sensor increase withincreasing precision of the process model (i.e., decreas-ing P1/Q1 ratio).

3.4.3. Multirate Sensors. In addition to a primarysensor, which provides measurements at every samplinginstant, one multirate sensor is available that providesmeasurements every M sampling instants, where M isthe period of the multirate sensor. For a discussion ofmultirate state estimation, see, e.g., ref 29.

Here, again, we assume a stochastic process modelof the form of eq 25. Because the sensors are assumed

Figure 2. Effect of adding redundant sensors.

ΣCT(CΣCT + Q)-1CΣ ) P (28)

K ) ΣCT[CΣCT + Q]-1 (29)

T ) (1 - KC)Σ (30)

φ1 )σ1

2

σn2

)T(1 sensor)T(n sensors)

(31)

φ1 ) n (32)

φ1 ) xn (33)

y2(k) ) η(k - θ) + w2(k) (34)

x(k) ) [η(k) η(k - 1) ‚‚‚ η(k - θ) ]T (35)

x(k + 1) ) [11 · · ·1 0 ]x(k) + [v1(k)

0···0

] (36)

y(k) ) [1 0 ‚‚‚ 00 0 ‚‚‚ 1 ]x(k) + w(k) (37)

cov[v(k), v(k)] ) P ) [P1

0· · ·

0]

cov[w(k), w(k)] ) Q ) [Q1Q2 ] (38)

φ2 )σ1

2

σθ2

(39)


to be static (and do not exhibit any time delays), η isthe only state variable of the combined process andsensor model. The output equation is dependent onwhether one or both measurements are available at aparticular sampling instant. Thus, we obtain the fol-lowing time-varying system description:

with

where y(k) ) [y1(k) y2(k)]T, w(k) ) [w1(k) w2(k)]T, andC(k) is a periodically time-varying matrix,

Because of the periodicity of C(k), the Kalman filterconverges to a stable periodic filter after the initialtransients have died out. It is this stable periodicsolution that can again be used to assess the perfor-mance of the multisensor system.

The periodic Σ(k) is obtained by iterating eq 22 untilit converges to a stable periodic solution; and theperiodic solutions for K(k), and T(k), then follow directlyfrom eqs 23 and 24. For periodic Kalman filter applica-tions, also see refs 30 and 31. Alternatively, the periodictime-varying system could be “lifted” into a time-invariant system by considering the output as consistingof all measurements made during one sensor period.27

This latter approach provides computational advan-tages, because the steady-state Kalman filter is deter-mined directly for the lifted time-invariant system,instead of iteratively from eq 22. However, because

computational aspects are of minor concern for this(offline) study, we adopt the approach of iterating eq22.

Again, an efficiency factor (φ3) is introduced, as

where σ12 is the estimation error variance if a single

sensor with sensor period M ) 1 is used, and σjM2 is the

average value of the variance if a multirate sensor withperiod M > 1 is added to the primary sensor. Theaverage σjM

2 has been introduced, because the multi-sensor system is now periodic and obtaining a meaning-ful performance measure requires an averaging of theestimation error variance over one period M of themultirate sensor after eq 22 has converged.

Figure 4 shows, for Q1/Q2 ) 10, the efficiency factorφ3, as a function of the period of the multirate sensor,for different values of P/Q1. It can be concluded that theaddition of a multirate sensor is always advantageous.Even if there is no process knowledge available (P/Q1f ∞), the additional measurements provided by themultirate sensor every M sampling instants will reducethe estimation error variance. Naturally, these improve-ments diminish with increasing sampling period of themultirate sensor. In analogy to the case of adding adelayed sensor, the improvements are dependent on theprecision of the process model, because of the need forpredicting the process variables during the intersampleperiod between multirate measurements.

3.4.4. Reference Sensors. The key characteristicsof a reference sensor are (i) it is static and unbiased,(ii) its time delay is large, compared to the samplinginterval of the online sensors, and (iii) the measurementtime delay is not necessarily constant, because the timeneeded to analyze a process sample may vary. Givenproperty (ii), one could consider adding as many delayedversions of η(k) as state variables as called for by themaximum possible time delay associated with thereference sensor. However, this would be impractical,because an excessive numbers of state variables maybe required in some cases. Instead, we propose toincorporate the reference sensor in the following man-ner:

Figure 3. Effect of adding a delayed sensor (Q1/Q2 ) 10).

η(k + 1) ) η(k) + v(k) (40)

y(k) ) C(k)η(k) + w(k) (41)

cov[v(k), v(k)] ) P

cov[w(k), w(k)] ) Q ) [Q1Q2 ] (42)

C(k) ) {[1 1]T (for k ) nM, n∈ N )

[1 0]T (otherwise)(43)

Figure 4. Effect of adding a multirate sensor (Q1/Q2 ) 10).

φ3 )σ1

2

σjM2

(44)


(1) The original system description (without thereference sensor) is augmented by the output equationcorresponding to the reference sensor,

and the measurement noise covariance matrix Q isaugmented by one additional diagonal element, Qref )E[wref(k)2]. A time-varying output matrix C(k) is usedto indicate when a reference measurement is available.

(2) At time instants at which a sample for a laboratorymeasurement is taken (for example, kref), the followingvariables are stored in memory:

In addition, all measurements that are obtained frominstant kref up until the reference measurement becomesavailable, have to be stored in memory.

(3) At sampling instants at which a reference mea-surement becomes available (for example, k), we “re-wind” the Kalman filter to sampling instant kref byretrieving the aforementioned variables from memory.A new output matrix C(kref) that takes into account thenow-available reference measurement is obtained, andthe filter gain K(kref) and state estimates x̂(kref|kref) arerecalculated.

(4) Based on the updated Kalman filter at time kref,and all measurements obtained between kref and k, thefilter is then iterated forward to the current samplinginstant k.

This procedure generates a new sequence of stateestimates, including estimates of the process variable,for the time period between taking the sample and thecurrent sampling instant. Because reference measure-ments are available rather infrequently, this approachcan be expected to be computationally more efficientthan the approach of introducing a potentially largenumber of additional state variables.

4. Robustness Issues

The Kalman filter is based on assumptions (e.g.,uncorrelated random noise) that, quite often, are vio-lated in practice. Therefore, careful attention must begiven to issues concerning robustness in the presenceof any deviations from nominal system behavior. Inparticular, the following situations will arise frequently:

(a) Measurement Outliers. Measurements with “un-usually large” errors constitute deviations from thenominal sensor description.

(b) Sensor Degradation. The performance of individualsensors may degrade over time, i.e., the measurementerror variances may increase and/or the bias charac-teristics may change considerably.

(c) Changes in the Process Behavior. The filter isbased on a particular process model. However, theprocess may change its dynamic behavior over time, asperiods of operation close to a particular operating pointmay be followed by periods of frequent process changesand transitions.

Robustness in the face of the aforementioned sce-narios is achieved via failure detection, classification,and compensation techniques, as discussed below.

4.1. Failure Detection. The innovations ω(k) aredefined as the difference between sensor readings at

time instant k and the predicted values based oninformation up to k - 1:

If there is no mismatch between model assumptions andthe true system, the innovation sequence is a whitenoise sequence with known covariance. However, if thetrue process does not match the process model, then theKalman filter loses its optimality and the innovationswill be correlated. Therefore, the innovations can beused for failure detection or for simply detecting changesin the process; for instance, tests for whiteness of theinnovations, or tests that detect shifts in the mean orincreases in the variance of the innovations, have beensuggested.32

Here, we consider the so-called standardized innova-tions,

which provide a normalization such that cov[ω̃(k),ω̃(k)] ) I, where I denotes the identity matrix.

The standardized innovations are more convenient foruse in fault detection, because the statistical propertiesof the standardized innovations remain constant, eventhough the system description may be time-varying. Anydeviations of the true system behavior from the modeldescription will be reflected in changes in the statisticalproperties of the (standardized) innovations. If suchchanges are detected, it can be concluded that a failurehas occurred somewhere in the system. However, themore specific task of isolating the failure origin fromthe innovations is a rather difficult problem in general.Yet, in the present application, because of the particularstructure of multisensor systems, which can be charac-terized by exhibiting very weak interactions betweenthe model components associated with the individualsensors, we will make the assumption that a change inthe statistical properties in one component of theinnovation sequence can be associated with a fault inthe corresponding sensor. Because any changes in theprocess behavior that are not captured by the processmodel would also affect the innovations, we mustmonitor the adequacy of the (stochastic) process model.In particular, we have to detect rapid process changes,because such changes would affect the statistical prop-erties of the innovations, and false indications of sensorfailures could occur.

4.1.1. Process Model Validity. If a purely stochasticprocess model is used, the problem of monitoring themodel validity involves estimating the covariance ma-trix that is associated with the process model Pp. For arandom walk process, which is a white noise process,the following approach can be used for estimating Pp:The model described by eq 6 is completely characterizedby a single parameter, Pp ) E[vp(k)2], which can beestimated online. For this model, we obtain

Assuming for the moment that the process model isadequate, the combined measurement obtained from theKalman filter is an unbiased estimate of the true process

yref ) η(k) + wref(k) (45)

kref, y(kref), C(kref - 1), Σ(kref), x̂(kref - 1| kref - 1)(46)

ω(k) ) y(k) - C(k)x̂(k|k - 1)

) y(k) - C(k)A(k)x̂(k - 1|k - 1) (47)

ω̃(k) ) [C(k)Σ(k)C(k)T + Q]-1/2ω(k) (48)

E[(η(k) - η(k - r))2] ) E[(vp(k) + vp(k - 1) + ‚‚‚ +

vp(k - r))2] ) rPp (49)


variable. Therefore, we replace η(k - i) with the corre-sponding estimates to obtain

where ση2 is the estimation error variance. Note that ση

can be time-varying (due to the possible time-varyingnature of the system description) and may not beperfectly known. Fortunately, its value is not requiredfor estimating Pp, if a sufficiently large value for r isused. For large values of r, rPp dominates over 2ση

2 andeq 50 can be used to derive an upper bound for Pp froma sequence of η estimates of length N, according to

4.1.2. Detection Algorithm. To detect sensor fail-ures, and, at the same time, to be able to track rapidprocess changes, the following tests are implemented.

(1) Outlier Test. Whenever the standardized in-novation associated with a particular sensor is “unusu-ally large”, the corresponding measurement is verylikely to be an outlier. This leads to the following test:Sensor i is considered faulty if

where Ri is an appropriately chosen threshold. Knowingthat, under the assumption of a perfect model, thevariances of the standardized innovation sequences areunity, the thresholds Ri can be easily chosen a priori.

(2) Bias and Variance Tests. Biases whose magni-tudes are smaller than the thresholds for outlier detec-tion can be detected by analyzing sequences of pastinnovations (with any outliers detected in test 1 re-moved from the sequence). The statistical test used todetect shifts in the mean of a random variable isdescribed in Appendix A.1. Note that this test assumesthat the variance of the sequence is knownswhich isthe case here, at least under the perfect model assump-tions. However, because there will always be someprocess/model mismatch, the assumed variances (de-noted as {γi} for future reference) should rather beviewed as tuning parameters for the sensor fusionapproach. They can be determined by observing theinnovations of the Kalman filter for actual process data,i.e., by processing a “test” data set.

The innovation sequence can also be used to detectsensor degradation, because an increase in the mea-surement error variance of one sensor will primarilyresult in an increase in the variance of the correspond-ing innovation sequence ω̃i. To detect this type of sensordegradation, we use the test described in Appendix A.2,which again requires the specification of the assumedvariances of the standardized innovations.

(3) Model Validity Test. The outcome of the afore-mentioned tests is dependent on the validity of themodel characterizing the dynamics of the true processvariable. For the case of the random walk process model,we monitor the model validity by estimating Pp online,

using eq 51. The process model is considered inadequateif

where Pp0 is the nominal value of Pp used by the Kalman

filter. Note that we only consider the model inadequateif the estimated variance is larger than that assumedby the model. Although the Kalman filter will also besub-optimal if the actual process variance Pp is smallerthan Pp

0, we consider Pp0 as a lower bound, to ensure

that the filter remains “alert” and can respond quicklyto process changes, even though the process variablemight have hardly changed for an extended period oftime.

If the process model is determined to be inadequate,the system description is updated by introducing theestimate P̂p(k), instead of Pp

0, in matrix P in eq 20.Furthermore, the detection mechanisms for soft sensorfailures (test 2 previously described) are disabled tem-porarily, because inadequate model assumptions arevery likely to affect the innovations. Note that, althoughthe process description is updated whenever a rapidprocess change is detected, the process model may stillbe inadequate, because sudden changes in the processdynamics cannot be captured immediately by P̂p. How-ever, to be robust in the face of gross measurementoutliers, the outlier test (test 1) is still performed.

(4) Test on Bias Estimates. As an additionalrobustness measure, we also monitor the bias estimatesthat are provided by the Kalman filter. A very simplefailure detection scheme is to consider a sensor to befaulty if its bias exceeds a specified threshold, i.e., if

Because the ranges of the bias magnitudes for whichthe sensors can function properly are dependent on thesensor type, different thresholds Fi may be chosen forthe individual sensors. The thresholds {Fi} are againdetermined by monitoring past sensor performance.

4.2. Failure Classification and Compensation.Because all online sensors are assumed to exhibit time-varying biases, even under nominal conditions, theclassification task is reduced to discriminating betweenfaulty and normal operation. The failure classificationproblem is thus completely addressed by the previouslydiscussed failure detection scheme.

Whenever one of tests 1, 2, or 4 detects a sensorfailure, the faulty sensor is excluded from contributingto the estimation process by setting the appropriateelements of the output matrix C(k) to zero. Thus, themeasurements of faulty sensors do not enter the esti-mation process.

5. Application Study: Sensor Fusion for anEthylene Copolymerization Process

The production of ethylene copolymers of consistentcomposition requires tight control of the reactor feedcomposition. In this particular example, an ethylenecopolymer grade was produced with vinyl acetate as theco-monomer. The process variable to be monitored is thefeed composition, as characterized by the mole percent-age of vinyl acetate. Several indicators of this variableare available:

E[(η̂(k) - η̂(k - r))2] ) rPp + 2ση2 (50)

P̂p(k) ≈∑

k)r+1

N

[η̂(k) - η̂(k - r)]2

(N - r)r(51)

||ω̃i(k)|| g Ri (52)

P̂p(k) > Pp0 (53)

||â̂i(k|k)|| g Fi (54)


(1) Two gas chromatographs with sampling intervalsof 60 s and measurement time delays of 210 and 270 s,respectively;

(2) A Fourier Transform Near InfraRed (FTNIR)measurement, which is available every 30 s and has nomeasurement time delay associated with it;

(3) A process model that provides composition esti-mates every 60 s; this estimate is also undelayed;

(4) In addition to these frequent indicators, samplesof the polymer product are taken approximately everyhour and lab analyses are performed, the results ofwhich become available ∼30-90 min after taking thesamples. This laboratory measurement is consideredunbiased. Therefore, it can be used as a referencemeasurement for sensor fusion. A summary of thesensor characteristics is given in Table 1. The measure-ment error variances σi

2 in this table were estimatedfrom the given data record.

The first step in applying the proposed sensor fusiontechnique is to obtain an approximate model descriptionof the process and the multisensor system.

Note that one of the available indicators representsa process model that could be used as a deterministicprocess model and augmented with a stochastic uncer-tainty description. Here, however, we choose to treatthe model-based indicator (or “soft sensor”) as a mea-surement device, and we use the random walk model(eq 6) to describe the process dynamics. In the currentapplication, it is known a priori that all sensors, exceptfor the laboratory measurement, exhibit time-varyingbiases. Therefore, the following sensor model is sug-gested, in which the sensor outputs (yi, i ) 1, ..., 4) aredescribed by

where θi is a measurement time delay, wi is randommeasurement noise, and âi is a time-varying sensor biaswhich is again assumed to be generated by a randomwalk process (eq 6). The unbiased reference sensor isdescribed by eq 7. The system, which consists of the trueprocess variable and the measurements obtained fromsensors 1-5 can be described by a single dynamic modelby introducing the following state vector x(k):

where θmax is the largest of the individual sensor timedelays (except for the reference sensor). Using this statevector, the combined process and sensor model can bewritten in standard state-space form:

where v(k) ) [vp(k) 0 ‚‚‚ 0 v1(k) ‚‚‚ v4(k)]T and w(k) )[w1(k) ‚‚‚ w4(k)]T. The covariance matrixes associated

with v(k) and w(k) are assumed to be diagonal, i.e., cov-[v(k), v(k)] ) diag(Pp 0 ‚‚‚ 0 P1 ‚‚‚ P4) and cov[w(k), w(k)]) diag(σ1

2 ‚‚‚ σ52). The output matrix C(k) is time-

varying, because the sampling rates are not identicalfor all sensors. Therefore, C(k) is dependent on whichmeasurements are available at a particular samplinginstant. Using the system description given as eqs 57and 58, the Kalman filter (see eqs 21-23) providesestimates of the current state of the system, based onmeasurements up to time k. Because the time delayassociated with the reference sensor is extremely large(>100 sampling instants), it would be impractical toincorporate the reference sensor by adding delayedversions of η(k) as state variables. Here, we use theapproach of rewinding the Kalman filter when a refer-ence measurement becomes available, as discussedpreviously.

5.1. Model Parameters. Although the system ma-trices A and C directly follow from the dynamic char-acteristics of the multisensor system, there are severaladditional “tuning” parameters that must be specified.Although the approaches suggested here are by nomeans intended to be generally valid, they seemedappropriate for the case study considered, and theyresulted in satisfactory performance of the sensor fusiontechnique.

One important guideline in determining “tuning”parameters for robust sensor fusion is to avoid relyingon the results obtained from the filter in estimatingthese parameters. The tuning parameters consideredare given as follows:

(a) The process model parameter Pp. Before imple-menting the Kalman filter, the only estimates of theprocess variable available are the measurements. There-fore, eq 51 is used to estimate Pp by replacing η̂(k) withyi(k). This is done for all sensors i, and the results areaveraged.

(b) The variances Pi associated with the randomsequences generating the biases. To estimate Pi, refer-ence measurements are needed. At sampling instantsat which reference measurements were taken, thesensor biases are approximated by

and, for consecutive reference measurements obtainedat samples k1 and k2, respectively, estimates of Pi areobtained from

which are, again, averaged over several referencemeasurements.

(c) The measurement error variances Qi. These pa-rameters might be given in the sensors' specificationsthat are provided by the sensor manufacturer, or,alternatively, the sensor variances can be estimatedfrom measurements during time periods in which therewere hardly any changes in the process variable andthe bias.

This procedure is not applicable to the referencesensor. There, it is necessary to process several samplestaken at the same time to estimate the sensor variance.

5.2. Results. We now discuss some results that arerepresentative of the sensor fusion technique’s perfor-mance when it was applied to actual process sensor data

Table 1. Summary of Sensor Characteristics

sensornumber sensor type

samplingrate

timedelay σi

2 biased

1 GC 1 60 s 210 s 4 × 10-3 yes2 GC 2 60 s 270 s 2 × 10-3 yes3 FTNIR 30 s 0 s 1 × 10-2 yes4 model 60 s 0 s 5 × 10-3 yes5 lab measurement ∼60 min 30-90 min 2 × 10-2 no

âi(k) ≈ yref(k) - yi(k) (59)

P̂i ≈ [âi(k2) - âi(k1)]2

k2 - k1(60)

yi(k) ) η(k - θi) + âi(k) + wi(k) (55)

x(k) )[η(k) η(k - 1) ‚‚‚ η(k - θmax) â1(k) ‚‚‚ â4(k) ]T (56)

x(k + 1) ) Ax(k) + v(k) (57)

y(k) ) C(k)x(k) + w(k) (58)


covering a time period of eight consecutive days. Thefollowing parameters were chosen:

(a) The sampling period for the algorithm is chosenas ∆t ) 30 s. Hence, the time delays are θ1 ) 7, θ2 ) 9,θ3 ) θ4 ) 0.

(b) Additional model parameters (besides matrices Aand C) required for the sensor fusion approach weredetermined by following the guidelines given in theprevious section: The variance of the random sequence,which is assumed to generate the true process variable(see eq 6) was determined as Pp ) 0.003. The variancesof the random sequences that are assumed to generatethe sensor biases were determined as P1 ) P2 ) P3 ) 5× 10-4 and P4 ) 1 × 10-3.

(c) The window size for performing the bias andvariance tests on the innovations is NW ) 25. The samewindow size is used to estimate Pp online, according toeq 51.

(d) The thresholds {Ri} for test 1 on the magnitude ofthe innovations are chosen as Ri ) 5, i ) 1, ..., 4 (i.e., 5times the theoretical standard deviation).

(e) The assumed variances of the standardized in-novations required by the bias and variance tests arechosen as γi ) 1.25 (i.e., 25% above their theoreticalvalues of 1).

(f) The thresholds {Fi} on the magnitude of biasestimates (test 4) are chosen as Fi ) 3.

In the following results, we assume that the labora-tory measurements are undelayed, because the actualdelays of these measurements were unknown. Althoughthe sensor fusion was performed for the entire set ofsensor data, only a few representative data sequencesare chosen here, to demonstrate the following: (i) theperformance of the sensor fusion approach under “nor-mal” condition; (ii) the detection and removal of mea-surement outliers; (iii) the detection of gradual sensorfailures; and (iv) the detection of “abnormal” processbehavior.

5.2.1. Data Set 1. Here, we examine the performanceof the sensor fusion scheme during transitions betweentwo different polymer products. Figure 5 shows that theprocess transition seems to be followed quite accuratelyby the fused measurement. Again, the model parameterPp is estimated online (see Figure 6). During the initialphase of the transition, P̂p(k) exceeds its nominal value,and the estimates are used to update the process

description. The online estimation of Pp seems to becapable of tracking the changes in the process behavior,as indicated by the fact that the failure detection schemedid not trigger a sensor failure in any of the sensors.

5.2.2. Data Set 2. Figure 7 illustrates the detectionof a sudden shift in one sensor and a gradual failure ofanother sensor. The combined measurement does notseem to be affected by the severe degradation of sensor2, nor is it affected by the sudden shift in sensor 3 at t≈ 154.5, which was detected by the outlier test (test 1).Test 2 was capable of detecting an imminent failure ofsensor 2 very promptly, because the measurement errorvariance of sensor 2 exhibited a significant increasestarting at a value of t ≈ 156.

5.2.3. Data Set 3. This data set contains multiplesensor failures. The performance of the sensor fusionand fault detection schemes is shown in Figure 8. Thelower trace of the figure shows the decisions of the fault

Figure 5. Sensor fusion during a process transition (data set 1).

Figure 6. On-line estimation of the model parameter Pp (dataset 1).

Figure 7. Sensor fusion for data set 2.


detection scheme. Note that the plot only shows theoverall decisions. However, it is interesting to indicatewhich tests triggered at a particular time. At thebeginning of this data set, sensor 2 is detected as beingfaulty, as indicated by test 4 (test on the bias magni-tude). Shortly before t ) 182, sensor 4 fails, which isdetected initially by test 1, and then by test 4. Becausethe magnitude of the bias estimate remains above thethreshold of test 4 (F4) for some time, test 4 continuesto be triggered until t ≈ 182.2. After that, a sensorfailure of sensor 4 continues to be indicated by test 2,because the bias is changing rapidly, resulting in aninflation of the variance of the corresponding innovationsequence.

6. Summary and Conclusions

We have presented a sensor fusion approach that hasbeen developed using stochastic system theory butinspired by experimental evidence of integrated fusionof all available disparate sensory inputs in the controlfunction performed by baroreflexes, which are thebiological systems responsible for short-term blood-pressure control in mammals. More specifically, weaddressed the problem of how to combine measurementsof a process variable that are available from differentsources into one single, robust, and more-precise esti-mate; moreover, we have done so within the industrialcontext where the unavoidable imperfections and strin-gent requirements on robustness to sensor bias andother partial sensor failures must be addressed explic-itly.

We derived algorithms for obtaining optimum esti-mates of the true process variable that are more precise,more accurate, and more failure tolerant than thatobtainable from each individual sensor. The advantages

of utilizing multiple redundant sensors, as well asdelayed sensors and multirate sensors, have beenquantified. Robustness was achieved by augmenting thenominal technique with failure detection, classification,and compensation techniques, based on statistical hy-pothesis testing. The performance of the proposedtechniques was demonstrated via an actual applicationon an industrial ethylene copolymerization process.

The primary limitation of the technique is related toits direct connection with the Kalman filter. Eventhough we explicitly discussed how to make the algo-rithm robust to the key idealized assumptions underly-ing the Kalman filter, it is not clear how the techniquewill perform on a severely nonlinear process. However,the application on the industrial ethylene copolymerprocess provides confirmation that the technique will,in fact, function well on actual industrial processes, andit has been tested during grade transitions when processnonlinearities can be expected to be present.

Although the entire development was inspired by thebaroreceptor reflex, not much biological detail of theblood-pressure control system is contained in the algo-rithm. It would be interesting, as more-detailed knowl-edge of the biological system’s sensor fusion mechanismbecomes available, to revisit the problem and develop atruly biomimetic algorithm that faithfully mimics “insilico” the biological sensor fusion mechanism withsignificant biological fidelity.

Nevertheless, even in its current stochastic systemstheory manifestation, the technique explicitly addressesonly the analysis problem: i.e., given the multisensorconfiguration, it produces the optimal fused estimate.A possible next step is to investigate how this frame-work can be used for the design problem: i.e., given aspecified objective of system precision and robustness,determine the number, characteristics, and configura-tion of the sensor system required to meet the objective.

Appendix A. Development of the StatisticalTests

A.1. Bias Test. Several failure detection mechanismsdescribed in this article are used to perform statisticaltests on the recent history of random variables. First,we develop a statistical test that determines whether asample (which, in this case, corresponds to a sequenceof consecutive observations) has a zero or nonzero mean.This test will be performed for data sequences ofincreasing length, starting with data corresponding tothe current sampling instant and successively includingmore past data until a decision can be made with acertain confidence or until the entire data set (of lengthNW) has been analyzed. Such a sequential procedureensures that biases are detected as soon as possible.

Before describing the sequential test implementation,we discuss the basic tests for data sequences of constantsize N. Let us assume for the moment that the elementsof the set X (for example, xi) are normally distributedrandom variables with an unknown mean µ and unityvariance N(µ, 1). As will be seen later, this assumptionof normality is not required to apply the bias test.However, we will derive this test using the normalityassumption. The following two tests are performed inparallel.

A.1.1. Test 1. This test is designed to detect a nonzeromean of the data sequence X. The hypothesis that themean of the sequence is identical to zero (H0: µ ) 0) istested versus the alternative hypothesis that the mean

Figure 8. Sensor fusion for data set 3.


is unequal zero (H1: µ * 0). The likelihood ratio for adata sequence of length N is33

where p0 and p1 are Gaussian probability densities thatcorrespond to hypotheses H0 and H1, and µ̂ is themaximum likelihood estimate of the new mean, µ̂ )∑i)1

N xi/N. After simplifying eq 61 and noting that,under hypothesis H0, the random variable

the likelihood ratio test can be stated as follows:

Here, c1 is chosen to result in a specified probability ofa Type I error (accepting H1 when H0 is true). Note thatthis test is capable of detecting changes in the mean ofthe sequence X with a specified confidence. However,the test does not provide a measure of confidence inhypothesis H0 if H1 has not been accepted. In this case,the test simply concludes that there is not sufficientevidence to accept H1; however, H0 may be true or not.Therefore, we apply a second test as follows.

A.1.2. Test 2. This test is designed to detect zero-mean sequences. The hypothesis that the absolute valueof the mean is larger than a specified minimum value(H2: |µ| > µmin) is tested versus the hypothesis that thesequence has a zero mean (H0: µ ) 0). A minimumnonzero bias (µmin) must be specified in this case, toenable the rejection of H0, given a finite data set. Notingthat, under hypothesis H2, the random variable

the following test can be stated:

Again, c2 is chosen to yield a desired significance levelof the test. The parameters of tests 1 and 2 must beselected appropriately, such that contradictory decisions(hypotheses H0 and H1 are accepted simultaneously) are

avoided. Hence, the parameters c1, c2, µmin, and N mustsatisfy the relation

Because the threshold for test 1 is constant and thethreshold for test 2 is monotonically increasing with N,it is sufficient that eq 66 holds for the maximum windowsize NW. If eq 66 holds as an equality at the maximumwindow size NW, a conclusive decision (i.e., either H0 orH1 is accepted) is guaranteed.

By inspecting eqs 62 and 64, we notice that thestatistical tests described here should be applicable toany type of random sequences, as long as ∑i)1

N xi/xN isclose to the normal distribution indicated in eq 62 (oreq 64). Now, it follows from the central limit theorem(see, e.g., ref 25, p 175) that, for any sequence ofindependent, identically distributed random variableswith finite means and unity variance, but irrespectiveof the actual distributions, the statistics given in eqs62 and 64 approach the corresponding normal distribu-tions for N f ∞. Therefore, from a practical point ofview, the bias test can be applied, even if the randomvariables involved follow distributions that are quitedifferent from the normal distribution, provided thatlarge window sizes N are used.

The tests are applied in a sequential fashion to thesequence X and the parameters of the statistical testsare chosen such that eq 66 is satisfied as an equalityfor N ) NW. The sequential test starts with a windowsize of N ) 1 and performs tests 1 and 2. If neither testaccepts its alternative hypothesis (H1 and H0, respec-tively), the number of values used in the data sequenceis increased by one (which corresponds to including morepast data) and the tests are performed again. Thewindow size N is increased until one of the tests acceptsits alternative hypothesis. By design, this must occurat the maximum window size NW, at the latest. Theimplementation of the statistical tests is graphicallyillustrated in Figure 9.

Note that, in the sequential implementation of tests1 and 2, several hypothesis tests are typically performedbefore a decision is made. Therefore, the probability ofa Type I error in the sequential implementation will belarger than the value specified for a single hypothesistest. Using c1 ) c2, these limits have been adjusted for

L(0; x1, x2, ..., xN)

L(µ̂; x1, x2, ..., xN)) ∏

i)1

N p1(xi)

p0(xi)

) exp[-∑i)1

N

xi2

2+

∑i)1

N

(xi -µ̂)2

2] (61)

∑i)1

N

xi

xN∼ N(0, 1) (62)

accept H1 if

|∑i)1

N

xi|

xN> c1; do not accept H1 otherwise

(63)

∑i)1

N

xi

xN∼ N(µ, 1) (64)

accept H0 if

|∑i)1

N

xi|

xN< µminxN - c2;

do not accept H0 otherwise (65)

Figure 9. Illustration of sequential hypothesis testing (bias test).

c1 g µminxN - c2 (∀N) (66)


window sizes of N > 1, such that the probability of aType I error (rejecting H0, although it is true) in thesequential test implementation remains constant at adesired level. These thresholds for various Type I errorprobabilities (5%, 2.5%, 1%) are summarized in Table2.

Monte Carlo simulations are performed to evaluatethe performance of the proposed sequential test. Figure10 shows the results of such simulations for a Type Ierror probability of 1%. The data were drawn randomlyfrom N(1, 0), and biases of increasing magnitudes havebeen added to the data sequences. The percentage ofdecisions where the tests detected bias in the given datasequence is plotted versus the actual bias in the data.As can be seen, the performance of the tests is depend-ent on the maximum window size NW.

A.2. Variance Test. Here, we intend to derive a testthat will detect an increase in the variance of a randomvariable. The basic test used is the following.

A.2.1. Test 3. Under the hypothesis H0, that thesequence of observations of a random variable X followsa normal distribution with known variance σ2 butunknown mean, the statistic

where

follows a central ø2 distribution, with N - 1 degrees offreedom (ø2(N - 1, 0)).33 Under the alternative hypoth-esis H1sthat the variance is, in fact, larger than the

assumed valuesthe statistic Γ(N) would be inflated.This suggests the following test:

where the threshold c(N) is typically chosen by specify-ing a desired Type I error probability. This test isstrongly dependent on the normality assumption and,therefore, is somewhat “risky” to apply in practice,where deviations from normality are very common. Inparticular, both strong asymmetries and outliers arelikely to cause difficulties for such a test. Although wedo not provide comprehensive measures that wouldavoid problems with this test that result from anydeviations from normality, it needs to be noted that,within the sensor fusion techniques described in thisarticle, this variance test is not applied to “raw processdata”, but only to data that have been cleaned ofoutliers. Therefore, the test can be expected to performwell, even in the presence of certain types of non-normality that are observed in practice.

To detect changes in the variance quickly, the test isperformed in a sequential fashion for data sequencesof increasing length (i.e., data sets that include moreand more past information). Similar to the sequentialtest for bias detection, the test is implemented in thefollowing fashion. Starting with the statistic at the twomost recent sampling instants (i.e., N ) 2), the windowsize N is increased until hypothesis H0 is rejected. IfH0 is not rejected at the maximum window size NW, thenH0 is accepted.

The thresholds c(N) are determined as follows: First,a sequence c0(N), 2 e N e NW is determined from thepercentage points of the appropriate ø2 distribution, byspecifying a desired Type I error probability for test 3(rejecting H0 when H0 is true). However, as has beennoted previously, the Type I error probabilities in thesequential test implementation do not coincide withthose for a data set of fixed size. Hence, the thresholdsc0(N) must be adjusted, such that the Type I errorprobability in the sequential test implementation equalsa certain desired value. This adjustment is performedby increasing the entire sequence of thresholds c(N) bya constant factor, such that the overall test achieves thedesired Type I error probability:

The constant f2 values have been determined numeri-cally via Monte Carlo simulations; they are summarizedin Table 3.

Nomenclature

A, B, C, D, F ) system matricesf1, f2 ) multipication factorsH ) hypothesisk ) sampling instant

Figure 10. Performance of bias test (Type I error probability )1%).

Table 2. Thresholds for Sequential Bias Test with Type IError Probabilities of 5%, 2.5%, and 1%

threshold c1 ) c2

window size, NW 5% 2.5% 1%

1 1.96 2.24 2.583 2.25 2.54 2.82

10 2.51 2.78 3.0925 2.68 2.93 3.2450 2.76 3.02 3.32

100 2.84 3.10 3.40

Γ(N) ) ∑i)1

N (xi - xj)2

σ2(67)

xj )1

N∑i)1

N

xi

Table 3. Correction Factors f2 for the Thresholds of theSequential Variance Test for Type I Error Probabilitiesof 5%, 2.5%, and 1%

f2

window size, NW 5% 2.5% 1%

2 1.0 1.0 1.05 1.26 1.23 1.17

10 1.33 1.29 1.2425 1.37 1.32 1.2750 1.38 1.325 1.275

100 1.39 1.33 1.28

reject H0 if Γ(N) > c(N) (68)

c(N) ) f2c0(N) (for N ) 1, ..., NW) (69)


K ) Kalman filter gainM ) period of multirate sensorn ) number of sensorsN ) length of a data sequenceNW ) window size (for statistical tests)P ) covariance matrix associated with vQ ) covariance matrix associated with wT ) a posteriori estimation error covariancev ) random disturbancew ) random measurement noisex ) state vectorxi ) state vector associated with sensor ix̂ ) state estimateψ ) measurement vectorRi ) threshold for outlier testâi ) sensor biasγi ) assumed variance of innovation sequence∆t ) sampling periodε ) estimation errorη ) true process variableη̂ ) estimated process variable (using all available sensors)η̂i ) estimated process variable (using sensor i only)θ ) time delay (in multiples of ∆t)µ ) meanν ) threshold for sensor consistencyFi ) threshold for bias magnitudeσ ) standard deviationΣ ) a priori estimation error covarianceτ ) time constantφ ) efficiency factorω ) innovationω̃ ) standardized innovation

Subscripts

c ) combinedi ) associated with sensor ip ) processref ) reference sensors ) sensor system

Acknowledgment

We thank Cliff Phares (DuPont Packaging & Indus-trial Polymers) for providing the process data for section5, for many discussions on how to make the approachwork in practice, and for his assistance in makingpossible the first online implementation of the tech-nique.

Literature Cited

(1) Luo, R. C.; Kay, M. G. Multisensor Integration and Fusionin Intelligent Systems. IEEE Trans. Syst., Man Cybernet. 1989,19, 901-931.

(2) Crowley, J. L.; Demazeau, Y. Principles and Techniques ofSensor Data Fusion. Signal Process. 1993, 32, 5-27.

(3) Kerr, T. Decentralized Filtering and Redundancy Manage-ment for Multisensor Navigation. IEEE Trans. Aerosp. Electron.Syst. 1987, 23, 83-119.

(4) Dasarathy, B. V. Decision Fusion Strategies in MultisensorEnvironments. IEEE Trans. Syst., Man Cybernet. 1991, 21, 1140-1154.

(5) Mitchell, A.; Willis, M. J.; Tham, M. T.; Johnson, N.; Skorge,L. M. Inferential Estimation of Viscosity Index on a LubricantProduction Plant. In Proceedings of the American Control Confer-ence, Seattle, WA, 1995; IEEE: Piscataway, NJ, 1995; pp 24-28.

(6) Cimander, C.; Carlsson, M.; Mandenius, C.-F. Sensor Fusionfor On-Line Monitoring of Yoghurt Fermentation. J. Biotechnol.2002, 99, 237-248.

(7) Ramamurthy, Y.; Sistu, P. B.; Bequette, B. W. Control-Relevant Dynamic Data Reconciliation and Parameter Estimation.Comput. Chem. Eng. 1993, 17, 41-59.

(8) Tjoa, I. B.; Biegler, L. T. Simultaneous Strategies for DataReconciliation and Gross Error Detection in Nonlinear Systems.Comput. Chem. Eng. 1991, 15, 679-690.

(9) Negiz, A.; Cinar, A. Monitoring of Multivariable DynamicProcesses and Sensor Auditing. J. Process Control 1998, 8, 375-380.

(10) Sasiadek, J. Z. Sensor Fusion. Ann. Rev. Control 2002, 26,203-228.

(11) Willsky, A. S. A Survey of Design Methods for FailureDetection in Dynamic Systems. Automatica 1976, 12, 601-611.

(12) Takahashi, K.; Nozaki, S. From Intelligent Sensors toFuzzy Sensors. Sens. Actuators A 1994, 40, 89-91.

(13) Hong, L.; Wang, G.-J. Centralised Integration of Multi-sensor Noisy and Fuzzy Data. IEE Proc.sD: Control Theory Appl.1995, 142, 459-465.

(14) Lee, S. C. Sensor Value Validation Based on SystematicExploration of the Sensor Redundancy for Fault Diagnosis KBS.IEEE Trans. Syst., Man Cybernet. 1994, 24, 594-605.

(15) Naidu, S. R.; Zafiriou, E.; McAvoy, T. J. Use of NeuralNetworks for Sensor Failure Detection in a Control System. IEEEControl Syst. Mag. 1990, 10 (4), 49-55.

(16) Da, R.; Lin, C.-F. Sensor Failure Detection with a Bank ofKalman Filters. In Proceedings of the American Control Confer-ence, Seattle, WA, 1995; IEEE: Piscataway, NJ, 1995; pp 1122-1126.

(17) Sun S.-L.; Deng, Z.-L. Multi-Sensor Optimal InformationFusion Kalman Filter. Automatica 2004, 40, 1017-1023.

(18) Faggin, F.; Mead, C. VLSI Implementation of NeuralNetworks. In An Introduction to Neural and Electronic Networks;Academic Press: San Diego, CA, 1990; pp 275-292.

(19) Henson, M. A.; Ogunnaike, B. A.; Schwaber, J. S.; Doyle,F. J. The Baroreceptor Reflex: A Biological Control System withApplications in Chemical Process Control. Ind. Eng. Chem. Res.1994, 33, 2453-2466.

(20) Pottmann, M.; Henson, M. A.; Ogunnaike, B. A.; Schwaber,J. S. A Parallel Control Strategy Abstracted from the BaroreceptorReflex. Chem. Eng. Sci. 1996, 51, 931-945.

(21) Seagard, J. L.; van Brederode, J. F. M.; Hopp, F. A.; Dean,C.; Gallenburg, L. A.; Kampine, J. P. Firing Characteristics ofSingle-Fiber Carotid Sinus Baroreceptors. Circulat. Res. 1992, 66,1499-1509.

(22) Karemaker, J. M. Neurophysiology of the BaroreceptorReflex. In The Beat-by-Beat Investigation of Cardiovascular Func-tion; Clarendon Press: Oxford, U.K., 1987; pp 27-49.

(23) Rogers, R. F. Arterial Baroreceptor Signal Processing inthe Nucleus Tractus Solitarius, Ph.D. Dissertation, University ofPennsylvania, Philadelphia, PA, 1995.

(24) Rogers, R. F.; Rose, W. C.; Schwaber, J. S. Encoding ofCarotid Sinus Pressure and dP/dt by Myelinated BaroreceptorTarget Neurons in the Nucleus Tractus Solitarius. J. Neurophysiol.1996, 76, 2644-2660.

(25) Grimmett, G. R.; Stirzaker, D. R. Probability and RandomProcesses, 2nd ed.; Clarendon Press: Oxford, U.K., 1992.

(26) Gelb, A. Applied Optimal Estimation, MIT Press: Cam-bridge, MA, 1980.

(27) Tyler, M. L.; Morari, M. Estimation of Cross-DirectionalProperties: Scanning vs Stationary Sensors. AIChE J. 1995, 41,846-854.

(28) Goodwin, G. C.; Sin, K. S. Adaptive Filtering, Prediction,and Control; Prentice Hall: Englewood Cliffs, NJ, 1984.

(29) Zambare, N.; Soroush, M.; Ogunnaike, B. A. A Method ofRobust Multi-rate State Estimation. J. Process Control 2003, 13,337-355.

(30) Rawlings, J. B.; Chien, I.-L. Gage Control of Film andSheet Forming Processes. AIChE J. 1996, 42, 753-766.

(31) Campbell, J. C.; Rawlings, J. B. Predictive Control ofSheet- and Film-Forming Processes. AIChE J. 1998, 44, 1713-1723.

(32) Mehra, R. K.; Peschon, J. An Innovations Approach toFault Detection and Diagnosis in Dynamic Systems. Automatica1971, 7, 637-640.

(33) Hogg, R. V.; Craig, A. T. Introduction to MathematicalStatistics, 4th ed.; Macmillan: New York, 1978.

Received for review May 10, 2004Revised manuscript received August 23, 2004

Accepted August 30, 2004

IE049614T


Development and Implementation of a High-Performance Sensor System for an Industrial Polymer Reactor

Documents

Transcript of Development and Implementation of a High-Performance Sensor System for an Industrial Polymer Reactor