a new multirate-measurement-based estimator: emulsion copolymerization batch reactor case study

A New Multirate-Measurement-Based Estimator: EmulsionCopolymerization Batch Reactor Case Study

R. K. Mutha and W. R. Cluett*

Department of Chemical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3E5

A. Penlidis

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

This paper presents the development of a state estimator for systems with multirate measure-ments. A fixed-lag smoothing based extended Kalman filter algorithm is proposed. Theperformance of the algorithm has been evaluated using a case study on a complex emulsioncopolymerization batch process. The proposed estimator uses measurements multiple times toachieve good convergence properties and robustness to state and measurement noise. Thesimulations also demonstrate that the performance of the proposed algorithm is superior to thebasic extended Kalman filter.

1. Introduction

One of the major obstacles in the control of polymerreactors is the lack of adequate and frequent measure-ments. In the literature, several estimation schemesbased on the extended Kalman filter (EKF) have beenproposed for process state estimation and subsequentinclusion in process control schemes in order to attemptto compensate for this lack of measurements.MacGregor et al. (1984) and Elicabe and Meira (1988)provided surveys on the estimation and control ofpolymerization reactors. Chien and Penlidis (1990)reviewed the impact of slow measurements on poly-merization reactor control and the need for using on-line state estimation. Schuler and Schmidt (1993) haverecently summarized the basics of state estimation andits application to polymerization reactors, bioreactors,and other chemical reactors. Dimitratos et al. (1994)have also recently reviewed various estimation andcontrol applications to emulsion polymerization reactors.Kozub and MacGregor (1992) developed a fixed-pointreiterated EKF for faster convergence of the states ofsemibatch polymerization reactors. They also demon-strated the importance of including nonstationary statesin the estimator to handle bias and model-processmismatch. Dimitratos et al. (1989) simulated an adap-tive hybrid EKF for emulsion copolymerization stateestimation. Subsequently, Dimitratos et al. (1991)applied this EKF to an experimental emulsion co-polymerization system. Adebekun and Schork (1989)also investigated the performance of an EKF appliedto a solution polymerization process and implementedthe estimator during feedback control of the reactor atthe simulation level.Systems with multirate measurements can be divided

into two categories: one with primary and secondarymeasurements and the second with fast (frequent) andslow (infrequent) measurements. In the first type,secondary inferential measurements are available morefrequently than the primary measurements, and the

estimator states can be made observable by usingsecondary inferential measurements even in the absenceof primary measurements (Gudi et al., 1995). In thesecond type of systems, the structure of the processmodel and, hence, the flow (direction of propagation) ofinformation is such that complete state observabilitycannot be met without the slow measurements. Poly-merization processes usually belong to this secondcategory of multirate systems. For instance, Schulerand Suzhen (1985) have shown that no information flowexists from the different moments of the chain lengthdistribution (quality part of the model) to conversion(productivity part of the model). Therefore, observabil-ity for the moments of the chain length distributioncannot be met by using density (conversion) measure-ments alone. One other point of clarification worthmentioning here is that fast measurements are assumedto be instantaneous in the sense that the measurementis taken and is available at the same sampling instant.On the other hand, slow measurements are taken at onesampling instant but are not available until some latersampling instant due to time delay associated withsample analysis, i.e., gel permeation chromatography(GPC).Very few of the estimation algorithms in the literature

are designed to handle systems with multirate meas-urements. One exception is the work by Ellis et al.(1988) where an EKF was designed for and applied to abatch polymerization system with fast and slow meas-urements. In their algorithm, the calculated stateestimates from the faster measurements are discardedwhen the slower measurements become available andthe estimator is applied from the time the slowermeasurements are taken. This discarding of the esti-mated states and restarting of the estimator from thetime the slower measurements are taken is inefficientand can lead to slower convergence, since useful infor-mation is unnecessarily being ignored. Also, this ap-proach does not address systems having several slowmeasurements each with a different delay.In this paper, a new extended Kalman filter algorithm

with fixed-lag smoothing (as opposed to fixed-pointsmoothing) for estimating the states of nonlinear sys-

* Telephone: 416-978-6697. Fax: 416-978-8605. E-mail:[email protected].

1036 Ind. Eng. Chem. Res. 1997, 36, 1036-1047

S0888-5885(96)00100-5 CCC: $14.00 © 1997 American Chemical Society

tems with multirate measurements is presented. Theproposed algorithm, which does not discard any usefulinformation and can handle several measurements withdifferent amounts of delay, will be shown to result inbetter convergence than the basic EKF algorithm withall measurements available without delay. The smooth-ing of the states can be implemented at all times(including startup), without any increase in computa-tional load over time, which would be the case with afixed-point smoothing algorithm. The algorithm is alsoapplicable to systems with variable measurement de-lays.To illustrate the use of the proposed estimator, a case

study on an emulsion acrylonitrile/butadiene copoly-merization batch system is presented. The organizationof the paper is as follows: section 2 briefly summarizesthe Kalman filter with fixed-lag smoothing, section 3presents the new extended Kalman filter algorithm withfixed-lag smoothing for multirate systems, section 4describes the model of the acrylonitrile/butadiene emul-sion copolymerization batch process, and finally, section5 evaluates the proposed algorithm through numeroussimulations on this batch process.

2. Kalman Filtering and Smoothing

Smoothing is defined as the estimation of past statesbased on past but more recent measurements (X(k -i|k) ∀ i g 1, where k is the current sampling instant).Filtering is defined as the estimation of current statesbased on past and current measurements (X(k|k)), andprediction is defined as the estimation of future statesbased on past and current measurements (X(k + i|k) ∀i g 1). The main motivation for using smoothing is thatsmoothed states actually use more data (by usingmeasurements beyond the time of the states beingestimated) and, therefore, have better convergenceproperties than filtered states. However, the smoothingalgorithms are related to the filtering problem and are,in fact, based on modifications to the standard Kalmanfiltering algorithm.An early survey paper on smoothing by Meditch

(1973) describes its application to linear and nonlineardynamic systems. State smoothing can be implementedin three different ways (Anderson and Moore, 1979;Elbert, 1984), each with a different objective. Thesethree smoothing algorithms are fixed point, fixed lag,and fixed interval. All three types of smoothing use theentire sequence of available measurements. Fixed-pointsmoothing estimates the state vector at a single fixedpoint in time, i.e., X(k0|k), where k0 is a sampling instantfixed in time (k0 < k). Fixed-lag smoothing estimatesthe state vector of the system at a specified time lagfrom the current sampling instant, i.e., X(k - SI|k),where SI denotes the smoothing interval. Fixed-interval smoothing estimates the state vector at allsampling instants including the current sampling in-stant, i.e., X(i|k) ∀ 0 e i e k.The algorithm proposed in this paper is based on

fixed-lag smoothing. Use of fixed-point and fixed-interval smoothing would result in an excessive com-putational load. In addition, it is well-known thatsmoothing algorithms in general have a negligible effecton the smoothed states beyond two to three times thedominant time constant of the Kalman filter (Andersonand Moore, 1979). The fixed-lag smoothing algorithmis well documented in the literature, e.g., Anderson andMoore (1979) and Elbert (1984). Therefore, only the key

features of this algorithm are summarized in the nextsubsection.2.1. Fixed-Lag Smoothing Algorithm. Consider

a time-varying, discrete-time linear dynamic systemwith n states,mmeasurements, and p inputs describedby the equations

where W(k) and V(k) are zero mean, white noisesequences with covariances equal to Q(k) and R(k),respectively. The dimensions of the above terms are asfollows: A (n × n), B (n × p), C (n × n), H (m × n), Q (n× n), R (m × m), X0 (n × 1), U (p × 1), W (n × 1), Y (m× 1), and V (m × 1). The notation A(k,k - 1), B(k,k -1), and C(k,k - 1) indicates the values of these matricesbetween sampling instants k - 1 and k. The determin-isticU(k) can be ignored without loss of generality. Fornonlinear systems, the above set of equations is obtainedby linearizing the system at the operating point.Smoothing of the states is achieved by using a state

vector augmentation technique, with the number ofaugmented state vectors being equal to the smoothinginterval, SI. The augmented system can be written as

The augmented state vectors in eq 2 are related accord-ing to

Therefore, the augmented system contains the SI previ-ous values of the state vector X0(k). Application of theKalman filter to this augmented system results insmoothed estimates of the states at a fixed lag of SI

X0(k|k - 1) ) A(k,k - 1)X0(k - 1|k - 1) +B(k,k - 1)U(k - 1) + C(k,k - 1)W(k - 1)

Y(k) ) H(k)X0(k) + V(k) (1)

[X0(k|k - 1)X1(k|k - 1)X2(k|k - 1)

···

XSI(k|k - 1)] )

[A(k,k - 1) 0 · · · 0 0I 0 · · · 0 00 I · · · 0 0···

···

···

···0 0 · · · I 0 ][X0(k - 1|k - 1)

X1(k - 1|k - 1)X2(k - 1|k - 1)

···XSI(k - 1|k - 1) ] +

[C(k,k - 1)00···0 ]W(k - 1)

Y(k) ) [H(k) 0 0 · · · 0][X0(k)X1(k)X2(k)

···

XSI(k)] + V(k) (2)

XSI(k) ) XSI-1(k - 1) ) ... ) X1(k - SI + 1) )X0(k - SI)

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1037

sampling intervals; i.e., XSI(k|k) ) X0(k - SI|k). Bydoing this, the order of the augmented system hasincreased SI times. However, the matrices of theaugmented system are sparse and result in relativelysimple equations. More details of the augmented Kal-man filter can be found in Elbert (1984).The two sets of equations used recursively to form the

covariance matrices for this smoothing algorithm aregiven as

and

where Pi,0 are the covariances of the Xith state(s) withthe X0 state(s). The smoothing (Kalman) gain matricesare given by the following set of equations:

Note that the first equation in each of the above sets ofequations (3), (4), and (5) corresponds to the standardKalman filtering equations without any smoothing.Equation set (5) indicates that there is only one matrixto be inverted, and it is the samematrix that would haveto be inverted even without augmentation. Therefore,the computational load due to matrix inversions has notincreased for the augmented system. The smoothed

estimates with the above Kalman gains are given bythe following equations:

where X0(k|k - 1) is computed using the process modelequation (1). The above implementation results in theestimation of the states at all lags less than or equal toSI. There are other approaches to the fixed-lag smooth-ing problem that yield only state estimates at lag SI.

3. EKF with Fixed-Lag Smoothing for MultirateSystems

This section develops the proposed extended Kalmanfilter algorithm with fixed-lag smoothing for multiratesystems, where the term “extended” refers to the factthat the algorithm is applied to a linearized version ofthe nonlinear system. The proposed algorithm is basedon a “boot-strap” application of filtering and smoothing.The value of the lag used for smoothing is a keyparameter of this algorithm and is referred to as thesmoothing interval (SI). Two EKF’s are used in thisscheme: one for filtering the states (EKF1) based onthe process model and the second for smoothing thestates (EKF2). EKF1 is used for estimating the statesat time k with the available measurements at time k,i.e., for calculating X(k|k). The standard extendedKalman filter is used for EKF1, and the states associ-ated with EKF1 are denoted by X1. The first equationfrom each set of equations (3), (4), (5), and (6) corre-sponds to the standard EKF algorithm and is used byEKF1. EKF2 is used for smoothing the states, i.e., forcalculating X(k - SI|k). The complete sets (3), (4), (5),and (6) compose the fixed-lag smoothing algorithm andare used by EKF2. The states associated with EKF2are denoted by X2. In general, the states of EKF2 area subset of the states of EKF1; i.e., it is often unneces-sary to smooth all of the states of X1.For multirate systems, a different number of meas-

urements is available at consecutive sampling instants.The EKF can easily accommodate this change in sizeof the measurement vector by using only the appropriaterows of H(k). Corresponding to these rows of H(k), asubmatrix of the noise covariance matrix R(k) is usedby the estimator. Then, the Kalman gain matrix iscomputed with the varying (in size) H and R matrices,where the number of columns in the Kalman gainmatrix is equal to the number of measurements.The proposed algorithm assumes that all states in

both EKF1 and EKF2, with X1 and X2 consideredseparately, are observable. Since the algorithm is basedon the extended Kalman filter equations, it is anapplication of a linear algorithm assuming piecewiselinearity of the nonlinear system. In general, linear

P00(k|k) ) [I - K0(k)H(k)]P00(k|k - 1)

P10(k|k) ) P10(k|k - 1)[I - HT(k)K0T(k)]

P20(k|k) ) P20(k|k - 1)[I - HT(k)K0T(k)]

l

PSI,0(k|k) ) PSI,0(k|k - 1)[I - HT(k)K0T(k)] (3)

P00(k|k - 1) )

A(k,k - 1)P00(k - 1|k - 1)AT(k,k - 1) +

C(k,k - 1)Q(k)CT(k,k - 1)

P10(k|k - 1) ) P00(k - 1|k - 1)AT(k,k - 1)

P20(k|k - 1) ) P10(k - 1|k - 1)AT(k,k - 1)

l

PSI,0(k|k - 1) ) PSI-1,0(k - 1|k - 1)AT(k,k - 1) (4)

K0(k) ) P00(k|k - 1)HT(k)[H(k)P00(k|k - 1)HT(k) +

R(k)]-1


R(k)]-1


R(k)]-1

l

KSI(k) ) PSI,0(k|k - 1)HT(k)[H(k)P00(k|k - 1)HT(k) +

R(k)]-1 (5)

X0(k|k) ) X0(k|k - 1) + K0(k)[Y(k) -H(k)X0(k|k - 1)]

X0(k - 1|k) ) X0(k - 1|k - 1) + K1(k)[Y(k) -H(k)X0(k|k - 1)]

X0(k - 2|k) ) X0(k - 2|k - 1) + K2(k)[Y(k) -H(k)X0(k|k - 1)]

l

X0(k - SI|k) ) X0(k - SI|k - 1) + KSI(k)[Y(k) -H(k)X0(k|k - 1)] (6)

1038 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

filters implemented on nonlinear systems can diverge,even though observability conditions are met.3.1. Proposed Algorithm. The convergence of the

estimated states, which are observable only with theslowmeasurements, to their true values depends on howfrequently slow measurements are taken and on theirrespective measurement delay. In polymerization proc-esses, measurements are often associated with longmeasurement delays and are taken infrequently. Theconvergence properties of these states can be enhancedby using the available measurements several times. Inother words, a slow measurement should be used ateach and every sampling interval until the next slowmeasurement becomes available, along with the newfast measurements, to estimate the states.The proposed algorithm tries to compensate for

infrequent slow measurements by repeated use of theavailable slow measurements. For the sake of clarityin presenting the algorithm, a two-rate measurementsystem will be considered where the first set of meas-urements is available instantaneously and the secondset is taken at a slower rate and is available with adelay. It is also assumed for simplicity that all of theslower measurements become available with the sameamount of delay equal to d sampling intervals.Consider that at sampling instant k, the following

variables are available from the previous samplinginstants: X1(k - SI|k - 1), K1(k), X1(k|k - 1), X2(k - i|k- 1) ∀ 0 e i e SI, P1(k|k), and Pi,0

2 (k|k) ∀ 0 e i e SI.The implementation of the proposed algorithm is asfollows:1. At sampling instant k, the fast (instantaneous)

measurements (yf(k)) are used for filtering the predictedstates of EKF1; i.e.,

2. This step is executed only when the slow measure-ments are taken. Calculate the entire Kalman gainmatrix (Ki

2(k) ∀ 0 e i e SI) of EKF2. For the slowmeasurements, the Kalman gain columns are stored ina buffer, Ki

2′(k). This buffer is referred as Ki2′(k - d)

after d sampling instants when these slow measure-ments become available.3. This step is executed only when slow measure-

ments (ys(k - d)) become available. Update the backend of the smoothed states (EKF2) as follows:

4. Using yf(k), EKF2 smooths the states to calculateX2(k - SI|k). This involves the execution of the sets ofequations (4), (5), (6), and (3) for X2.5. The smoothed states X2(k - SI|k) are used to

update the EKF1 states from X1(k - SI|k - 1) to X1′(k- SI|k).6. The states X1′(k - SI|k) are updated using all

measurements available at sampling instant k - SI togenerate X1(k - SI|k) as follows:

7. The states X1(k - SI|k) are now filtered by EKF1one sampling instant at a time until the currentsampling instant k is reached (X1(k|k)). As a result,EKF1 is implemented SI times during this step. Notethat EKF1 uses all measurements that are available ateach successive sampling instant. The state X1(k - SI+ 1|k) is stored in a buffer for use at the next samplinginstant in step 5.8. Predict X1(k + 1|k) using the process model and

calculate P1(k + 1|k), K1(k + 1), and P1(k + 1|k + 1)using the first equation from the sets of equations (4),(5), and (3), respectively. This step also involves thecalculation of A(k + 1, k) and H(k + 1).9. At the next sampling instant, i.e., k ) k + 1, repeat

from step 1.In the above algorithm, SI should at least be equal to

the largest delay (in this case d) associated with the slowmeasurements. By doing this, the slow measurementswill be used in the reintegration of EKF1 from k - SIto k. If the SI is greater than the measurement delayby an integer n, then the slow measurements will beused n + 1 times by the algorithm. Also, all of the fastmeasurements are used SI times by the algorithm. Theproposed algorithm is illustrated diagrammatically inFigure 1 for SI ) 4 and d ) 2.3.2. Implementation Issues. 3.2.1. Choice of A

and H during Reintegration of Smoothed States.There are three different ways of selecting the matricesA and H in step 7 of the algorithm: (1) use of matricesA(k,k - 1) and H(k) for all sampling intervals betweenk - SI and k; (2) use of matrices A(k - i, k - i - 1) andH(k - i) ∀ 0 e i e SI, which were calculated at eachsampling instant k - i in step 8; (3) the nonlinear model

Figure 1. Fixed-lag smoothing-based EKF for systems with multirate measurements. yf are the fast measurements, ys are the slowmeasurements, EKF1 is the filter, and EKF2 is the smoother.

X1(k|k) ) X1(k|k - 1) + K1(k)(yf(k) - yf(k))

X2(k - i|k - 1) ) X2(k - i|k - 1) +Ki-d2′ (k - d)(ys(k - d) - ys(k - d)) ∀ d e i e SI

X1(k - SI|k) ) X1′(k - SI|k) +K1(k - SI)(y(k - SI) - y(k - SI))


is relinearized at each sampling instant k - i ∀ 0 e i eSI, based on the updated value of X1, to compute the Aand H matrices.The first choice assumes that the process behaves

linearly over the smoothing interval period and may notgive good results if this assumption is invalid. Thesecond choice is a reasonable compromise betweenaccuracy and computational load. The third choiceshould result in the best performance, as it relinearizesthe process model at each and every sampling interval,based on the most recent state estimates. However, thisrequires the largest amount of computation. The thirdoption was used in the simulations described in section5.3.2.2. Justification and Computation of Ki

2′. Instep 3, the algorithm adjusts the states of EKF2 byincorporating the slow delayed measurements whenthey become available. This adjustment directly affectsthe state vector X1(k - SI|k). Any errors in thiscorrection of the states are minimized during reintegra-tion of the state vector up to X1(k|k) in step 7 of thealgorithm.The computation of Ki

2′(k - d) in step 2 is imple-mented as follows. Whenever a slow measurement istaken, EKF2 calculates the Ki

2 matrix assuming thatall measurements were available at that samplinginstant. The columns corresponding to the actuallyavailable measurements in Ki

2 are used in step 4, andthe remaining columns are stored in the buffer (Ki

2′).The model-based output predictions for these delayedmeasurements (ys) are also stored in a buffer. When-ever these delayed measurements become available, theKi2′ matrix and ys are used in step 3.3.2.3. Real-Time Implementation. Steps 1-7 of

the proposed algorithm would require computationaltime on the order of seconds (486DX2/66) between thetime the measurements are taken and the calculationof the new state estimates. At this point, any processcontrol algorithm would be executed followed by step 8during the intersample time. This computational loadcould be alleviated by modifying the point at which thecontrol algorithm is executed. Once the fast measure-ments yf(k) are available, step 1 is executed for calculat-ing X1(k|k) using the Kalman gain calculated in (theprevious) step 8 of the algorithm. At this point, theprocess control calculations can be performed and thenext control move can be implemented immediatelybased on these estimates. Then, the smoothing andreintegration of the process model (all the remainingsteps) are carried out during the intersample time. Ifthe estimator converges to the true state values, thedifference between the “ideal” implementation and thesuggested modification for real-time applications will benegligible.3.2.4. State Selection for EKF2. The selection of

states for EKF2 is another important issue for thisalgorithm. Not all states in the original system (eq 1)have to be smoothed. Only a subset of the statesassociated with EKF1 need to be selected for smoothing,in order to decrease computational load and memoryrequirements. The set of states in X2 must satisfyobservability requirements with the measurementsbeing used by EKF2. The main criterion is to includestates in EKF2 which are not affected by the fastmeasurements in order to improve convergence of theirestimates.3.2.5. Multirate Systems. For clarity only, the

algorithm was presented for a two-rate measurement

system. This can easily be generalized to multiratesystems (measurements taken at different rates andavailable with different delays). In the proposed algo-rithm, the delayed measurements are used by EKF1 asthey become available during reintegration of the statesfrom sampling interval k - SI to k in step 7. Thealgorithm does not even require the availability of adelayed measurement at a fixed amount of delay.Therefore, the algorithm lends itself naturally to ap-plications with variable measurement delays.3.3. Algorithm Simplification. The incorporation

of slow measurements in steps 2 and 3 can easily besimplified for certain cases. For strongly observablesystems, with large smoothing interval SI, use of asteady-state Kalman filter for EKF2 will not adverselyaffect the performance of the system. This is due to thefact that EKF2 eventually only affects the states at timek - SI. In fact, if SI is large, Ki

2 will tend to convergeto their steady-state values. The smoothed states willthen be used by the dynamic EKF1 to reintegrate theprocess from time k - SI to time k, using all of theavailable fast and slow measurements. Therefore, forsome systems, use of a steady-state smoother EKF2maynot significantly affect the performance of the algorithmbut will significantly decrease the complexity andcomputational load of the algorithm. In the simulationsdescribed in section 5, the full algorithm without anysimplifications was used.

4. Acrylonitrile/Butadiene Batch EmulsionCopolymerization Process Model

The proposed algorithm will now be evaluated on anacrylonitrile/butadiene emulsion copolymerization batchprocess. The process model characteristics are describedbelow.4.1. Model. A detailed model for acrylonitrile (AN)/

butadiene (Bd) emulsion copolymerization (NBR ornitrile rubber production) has been developed by Dubeet al. (1996). This comprehensive model allows for boththermal and redox initiation (“hot” and “cold” recipes).It accounts for both micellar and homogeneous nucle-ation, as well as for both water-phase polymerizationand reactions inside the polymer particles. In addition,reactions with both water- and monomer-soluble im-purities are included. The molecular weight part of themodel accounts for transfer to chain-transfer agent,transfer to polymer reactions, and terminal and internaldouble-bond polymerization in order to explain NBRbranching characteristics. The method of moments isused for the calculation of molecular weight averagesand average number of branch points.The model consists of 20 nonlinear ordinary dif-

ferential equations solved simultaneously using a stiff“ode” equation solver. Details about the model develop-ment and results from a sensitivity analysis (based onindustrial information) can be found in Dube et al.(1996) and Mutha (1996).4.2. Operating Conditions. The recipe used for the

isothermal, AN/Bd emulsion copolymerization batchprocess is shown in Table 1. The initial values used bythe estimator are presented in Table 2. The last entryin Table 1 (monomer-soluble impurities (MSI) flow rateinto reactor) represents an impurity disturbance con-tinuously entering the system over the course of thebatch which could be caused by an impurity leak, forexample.4.3. Measurements. The simulation results of

section 5 have been generated assuming that readily


available sensors (see Chien and Penlidis (1990)) areused. The density (F) of the polymer and, hence,conversion can be measured instantaneously and aretherefore used as a fast measurement in the simula-tions. Percentage-bound acrylonitrile (PA) in the poly-mer (a measure of copolymer composition), averageparticle size of the particles (Dp), and a measure of thepolymer viscosity (ν) are all slow delayed measurements.The polymer viscosity is related to the number-averagemolecular weight by a nonlinear correlation. In thesimulations, the base sampling interval was chosen as5 min, with density available every sampling intervalwithout delay. The sampling interval for the slowmeasurements and the measurement delay associatedwith these measurements ranged from 6 to 30 times thebase sampling interval.4.4. Estimator States and Observability. The

process states selected for estimation were based onobservability conditions. The states selected for thestate estimator EKF1 are the moles of CTA, MSI, [MA]p(or PA), [MB]p (or PB),Np, Vp, andQ0. Two nonstationarystates are also considered for inclusion in the estimatorto achieve integral action in the state estimates (Steph-anopoulos and San, 1984; Kozub and MacGregor, 1992).One of these nonstationary states (X1

s) was multiplica-tive with the total number of particles (Np), and thesecond of these states (X2

s) was multiplicative with thevolume of polymer particles (Vp).The states affecting conversion are observable by

using the fast measurement (density). However, esti-

mator states likeNp andQ0 (zeroth moment of the chainlength distribution) do not have any effect on the fastmeasurement (density) and, therefore, will not beobservable using the fast measurement alone. Duringthe filtering part of the algorithm when only fastmeasurements are used, these unobservable states willnaturally have a very low Kalman gain and, therefore,are almost unaffected. Since the state Q0, whichdirectly affects the number-average molecular weight,andNp are not observable from the density, these stateswill be included in EKF2. In addition, percentage-bound butadiene (PB) has also been included in EKF2.In this polymerization system, the weight-average

molecular weight and trifunctional and tetrafunctionalbranching frequencies are not observable. Including anappropriate sensor (in the future) could make some ofthese states observable. However, in these simulations,any states not included in the estimator will be pre-dicted solely using the model equations.Measurement signals are often noisy. Also, un-

accountable deviations from ideal conditions occur inalmost every application. These effects can be simu-lated by the addition of noise in the measurements andprocess states. Independent white noise sequences wereadded to all the measurements and to the state equa-tions for I, S, MSI, PA, and PB. These states were chosenas they affect all of the important product properties.The noise sequences were designed by multiplying awhite noise sequence of unit variance by a percentage(σ) of the corresponding state or measurement.

5. Simulation Results

The simulation examples illustrate the features of theproposed algorithm. They also show the algorithm’sability to effectively work with the slow measurementscurrently available in the polymer industry (Chien andPenlidis, 1990). The nonstationary state X1

s was al-ways included, but unless otherwise specified, thenonstationary state X2

s was not included. The simula-tion plots that follow present the most importantvariables of the system. MSI in the system is notobservable with the considered measurement vectorand, therefore, never converges to its true value. How-ever, the main effect of MSI is on the number of particlesgenerated, which is observable. The estimator accountsfor the effect of MSI by manipulating the number ofparticles and other related parameters of the system.The tuning matrices and the initialization of estimatorstates are listed in Table 2. Tuning of the estimatorwas carried out in a time-consuming, trial-and-errormanner.5.1. Base Case. Simulation runs were performed

with all three slow measurements being taken everysixth sampling interval. The measurement delay as-sociated with these measurements was also set as sixsampling intervals (30 min). State and measurementnoise at 6% ()3σ) of the respective states and measure-ments was used. A smoothing period of 12 samplingintervals was selected. The base case simulation wasperformed with a flow rate of 6.0× 10-5 mol/min of MSIinto the reactor. The performance of this base case iscontrasted with the case where there is no monomer-soluble impurity being added to the process, i.e., an MSIflow rate of 0 mol/min. The presence of MSI has anoften ignored and unappreciated effect on particlenucleation and, consequently, on other polymerizationvariables (Huo et al., 1988; Penlidis et al., 1988). Theperformance of these runs is contrasted in Figure 2.

Table 1. Recipe Used for Batch Acrylonitrile/ButadieneEmulsion Polymerization

ingredient initial true value

acrylonitrile 54700.0 molbutadiene 114 000.0 molwater 906 200.0 molemulsifier 675.0 molinitiator 27.0 molactivator 50.0 moltemp 285.0 Kchain-transfer agent 100.0 molMSI 0.0 molno. of particles/L of solvent 0.0vol of polymer particles 0.0 Lacrylonitrile in particle phase 0.0 molbutadiene in particle phase 0.0 molMSI flow rate into reactor 6.0 × 10-5 mol/min

Table 2. Covariance Matrices for Fixed-LagSmoothing-Based EKF

stateinitial

variance (P)state

variance (Q)state

initialization

Covariance State Matrices for EKF1NCTA 1.0 × 10-1 1.0 × 10-2 100 molNMSI 1.0 × 10-7 1.0 × 10-9 0 molPA 1.0 × 101 1.0 × 10-2 0 molPB 5.0 × 102 1.0 × 10-1 0 mol10-16Np 1.0 × 10-2 1.0 × 10-7 0Vp 1.0 × 10-1 1.0 × 10-2 0 LQ0 1.0 × 102 1.0 × 10-1 0 molX1s 1.0 × 10-5 1.0 × 10-8 1

X2s 5.0 × 10-7 1.0 × 10-8 1

Covariance State Matrices for EKF2PB 1.0 × 102 1.0 × 10-1

10-16Np 1.0 × 10-2 5.0 × 10-4

Q0 1.0 × 102 5.0 × 10-4

measurement variance (R)

density (F) 1.0 × 10-7

particle diameter (Dp) 1.0 × 10-10

PA 1.0 × 10-4

ln viscosity 5.0 × 10-5


The number of particles generated in these cases issubstantially different. In both cases, the estimatorstates converge to their true values, except for theparticle diameter. The convergence of the estimator forthe particle diameter will be discussed in more detailin section 5.4. The results in Figure 2 illustrate the factthat the estimator effectively handles the presence ofMSI, and in general, the results agree with experimen-tal results found in the literature (Huo et al., 1988;Penlidis et al., 1988).5.2. Effect of Smoothing Interval. Here, the base

case conditions were used except with state and meas-urement noise at 15% ()3σ) of the respective states andmeasurements. The smoothing interval was varied inthe simulations. Figure 3 shows the estimator perfor-mance with smoothing intervals of 6, 7, and 12. Theeffect of using a larger SI is most visible on theestimated number-average molecular weight whereconvergence to the true state is faster with smalleroscillations at a larger SI. Beyond a particular level ofsmoothing (7 in this case), the performance improve-ment is negligible.5.3. Effect of Measurement and State Noise. The

estimator performance was evaluated in this case at aneven higher noise level than used previously. The runpresented in section 5.2 was with 15% (3σ) noise in both

Figure 2. Base case and effect of impurity on convergence of the estimator. MSI ) 6.0 × 10-5 (estimator (- ‚ -), true (s)) (base case)and MSI ) 0.0 mol/min (estimator (•), true (- -)).

Figure 3. Effect of smoothing interval on convergence of theestimator. Smoothing interval of 6 (O), 7 (+), and 12 (×) againsttrue (s) values.


measurements and estimator states. These runs havebeen repeated here with 30% (3σ) noise in both meas-urements and estimator states. The estimator divergesand exhibits numerical difficulties for a smoothinginterval of 6, after 50 sampling intervals (Figure 4).With smoothing intervals of 12 and 18 (Figure 5), it isclear that the estimator’s performance improves withincreasing smoothing interval. At this noise level, itwas found that the performance of the estimator wasmarginal with a smoothing interval of 7. This set ofsimulations shows that the use of larger smoothingintervals results in better rejection of state and meas-urement noise by the estimator.5.4. Effect of Nonstationary States. In all the

simulation results presented in sections 5.2 and 5.3, anoffset was observable between the true and estimatedvalues of the particle diameter. In this case, it isdemonstrated that this offset can be reduced throughthe addition of X2

s. The effect of nonstationary stateshas been discussed by Stephanopoulos and San (1984)and Kozub and MacGregor (1992) and experimentallyverified by Dimitratos et al. (1991). Here, the base caseconditions were used with measurement and state noiseat 30% ()3σ) of the respective states and measurements.The performance of this case is compared with the case

where X2s was not included in Figure 6. The average

particle diameter shows better convergence to its truevalue with the additional nonstationary state.5.5. Effect of Model-Process Mismatch. In

practice, model-process structural mismatch and errorsin the values of some of the parameters of the modelare to be expected. These mismatches should be com-pensated for by the estimation algorithm. In this case,measurement and state noise of 15% ()3σ) was used,and both nonstationary states were included. Simula-tion runs were performed with all three slow measure-ments being taken every 16th sampling interval. Themeasurement delay associated with these measure-ments was also set as 16 sampling intervals (80 min).Smoothing intervals of 16, 24, and 32 sampling intervalswere examined. All runs were performed with a MSIlevel corresponding to 6.0 × 10-5 mol/min (same as basecase). Changes in the propagation rate constants areknown to significantly affect the reaction kinetics. Inthe true process model, the lumped propagation rateconstant was increased by a factor of 30% compared tothe base case. The performance of these runs is shownin Figure 7.The estimator performance with respect to the number-

average molecular weight is a function of the smoothinginterval. The estimator does not converge for a smooth-ing interval of 16 and has rather poor convergence witha smoothing interval of 24. However, it does convergewith a smoothing interval of 32. As in all previoussimulations, good convergence is observed for smoothingintervals greater than the largest sum of the samplinginterval plus measurement delay for each slow meas-urement considered separately. This is because whenthis choice for SI is made, at least one set of slowmeasurements is used at each and every samplinginterval by the estimator. For the simulations pre-sented in this case, the largest sum is equal to 16 + 16) 32.Whenever there is model-process mismatch, the

estimator will try to minimize the effect of the mismatchby biasing some of the states. The states which become(artificially) biased depend on the set of measurementsand the set of states being used in the filter. A largepart of this bias is normally absorbed by the non-stationary states. In these simulations, the number ofparticles in the reactor has been multiplied by a

Figure 4. Robustness to noise level. Measurement and statenoise of 30% (3σ). Smoothing interval of 6 (O) against true (s)values.

Figure 5. Robustness to noise level. Measurement and statenoise of 30% (3σ). Smoothing intervals of 12 (O) and 18 (+) againsttrue (s) values.

Figure 6. Effect of including an additional nonstationary state.Smoothing interval of 12, with (+) and without X2

s (O) againsttrue (s) values.


nonstationary state (X1s) and has, therefore, converged

to a biased value.The performance of the estimator for a system with

model-process mismatch can be contrasted with thebase case. For both the base case (Figure 2) and thesimulation with model-process mismatch (Figure 7),the estimator effectively tracks the number-averagemolecular weight (Mh n). However, this is achieved invery different ways. Mh n is the ratio of Q1 to Q0,multiplied with the effective molecular weight, Mweff.The Mweff is calculated as the sum of the individualmonomer molecular weights weighted by the copolymercomposition and can be directly related to the molefraction reacted acrylonitrile (FA) as follows:

The Q0 and Q1 for both cases are shown in Figure 8. Inthe base case, both Q0 and Q1 converge to their truevalues. However, when there is a mismatch in thepropagation rate constant, they do not converge to theirtrue values. Q0 is a state in the estimator, whereas Q1is not. When the propagation rate constant is increased,Mh n increases. Since Q1 is not a state in the estimator,the estimate of Q1 based solely on the model equationsremains unchanged, even though the kinetics of the trueprocess have changed. The estimator responds bybiasing Q0 and FA such that convergence of Mh n isachieved.5.6. Effect of Very Slow Measurements. This

case evaluates the effect of having very slow measure-ments on the convergence of the estimator states. Runswere performed with slow measurements being takenafter 30 sampling intervals. The measurement delaywas also set equal to 30 sampling intervals. State andmeasurement noise at 15% ()3σ) of the respective statesand measurements was added, and both nonstationarystates were included. Smoothing intervals of 30, 45, and

60 were used by the estimator. The same 30% mis-match in the lumped propagation rate constant waspresent in these simulations.The performance of the estimator for smoothing

intervals of 30, 45, and 60 is shown in Figure 9. Theestimator is not able to achieve good convergence withsmoothing intervals of 30 and 45 for the number-average molecular weight. The estimator diverges forthe smoothing interval of 60 in the number-averagemolecular weight. It is worth noting here that withoutthe propagation rate constant mismatch, the estimatorhad much better convergence properties for all SIvalues.By taking more frequent measurements but without

requiring any decrease in the measurement delay itself,convergence can be greatly improved even in thisdifficult case. Slow measurements taken every 30sampling intervals (every 150 min) having a measure-ment delay of 30 sampling intervals (150 min) implythat only one set of measuring devices is being used toanalyze all of the samples in series. If two sets ofmeasuring devices were available, the sampling fre-quency could be doubled. For this case, it would meanthat a measurement could be taken every 15 sampling

Figure 7. Effect of model-process mismatch in propagation rateconstant by 30%. Smoothing interval of 16 (O), 24 (+), and 32(×) against true (s) values.

Mweff ) MWmAFA + MWmB(1 - FA)

Figure 8. Performance of the estimator in trackingQ0 (estimated(O), true (s)) and Q1 (estimated (+), true (- -)). (a) Base case and(b) mismatch in propagation rate constant.


intervals (75 min), although the measurement will notbe available for 30 sampling intervals (150 min). Theperformance of the estimator with slow measurementstaken every 15 sampling intervals and a measurementdelay of 30 sampling intervals using a smoothinginterval of 45 is also shown in Figure 9. Now, theestimate of Mh n converges.These simulations demonstrate that using multiple

sets of measuring devices for measurements with largemeasurement delays can greatly improve on-line stateestimation. This situation is often seen in industrywhere some key measurements only become availableafter a long period of laboratory analysis and are thenconsidered obsolete (Ogunnaike and Gopalratnam, 1991).However, the above results suggest an alternativewhere, in spite of the long analysis time, the measure-ments can effectively be used for achieving good resultswith the proposed estimator.5.7. Comparison with Standard EKF. The per-

formance of the proposed estimator was finally com-pared with the standard EKF. The standard EKF hererefers to the filter implementation without any smooth-ing. Since this algorithm cannot cope with the delayedmeasurements, it will operate with the multirate meas-urements assuming all measurements are availablewithout measurement delay. This means that in thecomparisons to follow, the standard EKF has a signifi-cant advantage over the proposed algorithm in termsof availability of measurements.The simulations with the standard EKF were per-

formed with the same conditions as the base casedescribed in section 5.1. The standard EKF was tunedseparately in order to achieve its “best” performance.The tuning matrices are listed in Table 3. The perfor-mance of the standard EKF has been compared withthat of the proposed algorithm for the base case condi-tions in Figure 10. Although the standard EKF has the

advantage of working with undelayed measurements,its performance is poorer. Much better performance isachieved using the fixed-lag smoothing-based EKF sinceone of its main advantages is the ability to reuse ameasurement multiple times.The algorithm by Ellis et al. (1988) would at best

approach the performance of the standard EKF with allthe measurements available without measurement de-lay. Therefore, the simulation results of Figure 10illustrate the advantage of using the proposed algorithmover that of Ellis et al. (1988).5.8. Features of the Proposed Algorithm. Sev-

eral important features of the proposed fixed-lag reiter-ated EKF have been illustrated through an extensivecase study, and the results have important implicationsfor industrial practice. First, the estimator exhibitsgenerally good convergence to the true process statesand robustness to measurement and process noise.Second, the estimator is capable of handling systemswith multirate measurements. For systems wheremeasurement delays are very long and may be expectedto cause estimator divergence, the simulations demon-strate that estimator convergence can be achieved bytaking more frequent measurements and using multiplesets of measurement devices for their analysis. Third,the proposed algorithm can accept measurements witha variable measurement delay. This is because slowmeasurements are used only during the reintegrationof the smoothed states. At a given sampling instant, ifa measurement is available, the algorithm will use itfor filtering. If it is not available, the filtering will stillbe performed with the other measurements available.Of course, there is a cost associated with obtaining

these improvements in that the proposed algorithmrequires a rather large computational effort and thiscomputational load increases linearly with the smooth-ing interval. The memory requirements for the algo-rithm are higher than the standard EKF as all of themeasurements, selected states, and associated Kalmangain matrices are stored over a moving time window.However, these costs are not significant even at thistime with currently available computers. The otherdesign cost with using the proposed algorithm is theneed to select the states for the smoothing part of thealgorithm and to tune the smoother.Through usually not mentioned, the implementation

of such complex algorithms through programming inhigh-level languages takes considerable time and effort.Selection and use of proper tools, techniques, andprogramming languages are all important factors. Theproposed algorithm was predominantly written in C inorder to use its flexible dynamic allocation features. Theprograms were designed to be very flexible in terms ofchanging smoothing intervals, adding/deleting processstates, changing the interval at which slow measure-ments are taken and their associated measurement

Figure 9. Effect of very slow measurements. True values (s).(a) Smoothing intervals of 30 (O) and 45 (+). (b) Smoothinginterval of 60 (O). (c) Using two sets of measuring devices with asmoothing interval of 45 (O).

Table 3. Covariance Matrices for the Basic EKFa

state initial variance (P) state variance (Q)

NCTA 1.0 × 10-1 1.0 × 10-4

NMSI 1.0 × 10-7 1.0 × 10-9

PA 1.0 × 101 1.0 × 101PB 5.0 × 102 5.0 × 10-1

10-16Np 1.0 × 10-2 1.0 × 10-7

Vp 1.0 1.0 × 10-2

Q0 1.0 × 102 1.0X1s 1.0 × 10-4 1.0 × 10-11

a Same measurement variances as fixed-lag smoothing-basedEKF.


delays, etc. Use of an object-oriented programminglanguage like C++ would be a good choice for imple-menting the proposed algorithm.One final note should be made about the fact that the

algorithm performance has been illustrated throughsimulations on a specific process. Nonetheless, theemulsion copolymerization multirate process representsa very challenging and realistic example, and similarperformance can be expected with other systems.

6. Concluding Remarks

A fixed-lag smoothing-based extended Kalman filteralgorithm was proposed for systems with multiratemeasurements. The performance of the filter has beenevaluated through a case study on a complex emulsioncopolymerization batch process. The proposed estimatoruses measurements multiple times to achieve betterconvergence and is robust to state and measurementnoise. The estimator can also handle systems with largemeasurement delays. For cases where the estimatordiverges due to large measurement delays, the simula-tions demonstrate that convergence can be achieved byusing multiple measurement devices. The simulations

also demonstrate that the performance of the proposedalgorithm using delayed measurements is superior tothe standard extended Kalman filter with all measure-ments being available instantaneously.

Acknowledgment

The authors acknowledge financial support from theNatural Sciences and Engineering Research Council ofCanada and also thank Uniroyal Chemical, USA, andICI/Glidden, Worldwide, for useful discussions.

Nomenclature

(‚) ) estimate of (‚)(‚)1 ) (‚) parameter related with EKF1(‚)2 ) (‚) parameter related with EKF2A ) acrylonitrileB ) butadieneCTA ) chain-transfer agentDp ) diameter of particles (dm)FA ) mole fraction reacted acrylonitrileKi ) Kalman gain matrix for ith state(s)m ) monomer phase (as a subscript)Mh n ) number-average molecular weight

Figure 10. Comparison of the performance of basic EKF (- -) algorithm with the base case (•) of the proposed algorithm against true (s)values.


MSI ) monomer-soluble impuritiesMweff ) effective molecular weightNi ) number of moles of species inj ) average number of radicals per particleNp ) total number of particles in the reactor (per liter ofwater)

Pi ) moles of species i in particlePij ) covariance of ith states(s) with jth state(s)Q0 ) zeroth moment of chain length distribution (mol)Q1 ) first moment of chain length distribution (mol)Vp ) total volume of polymer particles (L)X(k) ) state estimates at time kXs ) nonstationary stateX(k|p) ) estimate of state x at sampling interval k basedon measurements until sampling interval p

yf ) fast (frequent) measurementsys ) slow (infrequent) measurementsF ) density (g/L)ν ) polymer viscosity (P)σ ) standard deviation

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Received for review February 20, 1996Revised manuscript received July 2, 1996

Accepted July 2, 1996X

IE9601007

X Abstract published in Advance ACS Abstracts, February15, 1997.


a new multirate-measurement-based estimator: emulsion copolymerization batch reactor case study

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