integrated run-to-run and on-line model-based control of particle size distribution for a semi-batch...

Computers and Chemical Engineering 26 (2002) 1117–1131

Integrated run-to-run and on-line model-based control of particlesize distribution for a semi-batch precipitation reactor

Kangwook Lee a, Jay H. Lee a,*, Dae R. Yang b, Alan W. Mahoney c

a School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USAb Department of Chemical Engineering, Korea Uni�ersity, Seoul 136-701, South Koreac School of Chemical Engineering, Purdue Uni�ersity, West Lafayette, IN 47907, USA

Accepted 22 February 2002

Abstract

The dynamics of particle size distribution (PSD) in a precipitation process are represented by a population balance (PB) andrelated differential–algebraic equations. The control of PSD is studied by using a closed-form solution of the PB. Batch-to-batchcontrol and on-line single batch control strategies are investigated for controlling a semi-batch reactor. A systematic integrationof the two strategies is shown to have a complementary effect on the control performance. © 2002 Elsevier Science Ltd. All rightsreserved.

Keywords: Model predictive control; Semi-batch; Batch-to-batch; Precipitation; Extended Kalman filter; Least-squares

www.elsevier.com/locate/compchemeng

1. Introduction

Particle or slurry handling is ubiquitous in chemicalindustries. It is particularly important in commonlyfound processing steps like crystallization, precipita-tion, and polymerization. These processes are operatedin continuous, semi-batch, or batch mode. The particlesize distribution (PSD) is one of the most importantproperties to control in these processes. PSD is typicallycharacterized by the mean and variance, or sometimesupper and lower particle sizes, but the entire shape ofthe distribution can affect the product’s material andprocessing properties. Final product PSDs in most par-ticulate processes exhibit significant variability, owingto variations in the individual particle phenomena suchas nucleation, growth, aggregation, and breakage (Ran-dolph & Larson, 1988). Unfortunately, the control isseverely limited by the fact that PSD is very difficult tomeasure on-line and the available secondary measure-ments are insufficient for accurate inference of theentire PSD.

On the mathematical modeling side, the populationbalance (PB) relates the individual particle phenomenawith the behavior of the entire population (e.g. thePSD). The PB has to be coupled with the materialbalance (MB), because the particle growth consumesthe reactants and also the nucleation, growth, aggrega-tion, and breakage all depend on the bulk phase prop-erties. Usually, the PB is represented by a partialdifferential equation (PDE) of particle size and time.The MB, on the other hand, is expressed as ordinarydifferential equations (ODEs) in most cases. Thegrowth and nucleation kinetics of particles are oftenbased on empirical correlations.

The solution methods for population balance equa-tions (PBE) are reviewed in several papers (Ramkr-ishna, 1985; Rawlings, Miller & Witkowski, 1993;Braatz & Shinji, 2000). In choosing a solution methodfor on-line optimization and control, we need to con-sider both the calculation accuracy and computationalburden. Rigorous calculation does not necessarily meanan accurate simulation of the process, when the modelhas substantial error in it.

If we neglect the complicating phenomena like aggre-gation and breakage, the most important unknownsfrom the viewpoint of modeling are the growth andnucleation rates. So the estimation of growth and nu-

* Corresponding author. Tel.: +1-404-385-2148; fax: +1-404-894-2866

E-mail address: [email protected] (J.H. Lee).

0098-1354/02/$ - see front matter © 2002 Elsevier Science Ltd. All rights reserved.PII: S0098 -1354 (02 )00030 -3

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K. Lee et al. / Computers and Chemical Engineering 26 (2002) 1117–11311118

cleation kinetics is the key step for building a predictionmodel for PSD (Randolph, White & Low, 1981). Usu-ally, some predetermined form of rate law is assumedand parameters are fitted but a nonparametric functioncan also be estimated using the ‘inverse problem’ tech-niques (Ramkrishna, 2000). Data-fitted kinetic parame-ters are likely to have errors in them. The kineticparameters are typically determined by steady stateexperiments, but the steady state experiments cannot beperformed very easily (Rawlings et al., 1993). Theparameters can also be affected by disturbances, soon-line estimation of kinetic parameters may be needed.

It is the usual practice to control the PSD by con-trolling its characteristic variables (e.g. the mean) orother easily measured variables (e.g. temperature, pres-sure, and concentrations) of the bulk phase. Unfortu-nately, these approaches are often inadequate for finecontrol of PSD, because (1) the PSD is affected bymany factors such as temperature, concentrations, andmixing, which cannot all be measured and controlledvery precisely; and (2) control of the mean and varianceof the PSD is generally insufficient for the control ofthe entire distribution.

Recent development of on-line instrumentation forparticle sizing makes it possible to use on-line PSD dataand on-line feedback control of PSD is becoming arealistic possibility. Even with an accurate on-line PSDmeasurement, the use of a conventional feedback con-trol strategy (e.g. multi-loop PID control) may be un-suitable, because the PSDs for different size rangescannot all be controlled independently. In addition,characteristics of most particulate systems are nonlinearand time-varying; a fixed parameter controller based onan off-line-data-fitted model may perform poorly over acourse of large time period. The problem is betterhandled by using an optimization-based control tech-nique coupled with state estimation (Rawlings et al.,1993). However, it requires an on-line optimizationcalculation at each time step and hence the computa-tional burden is not always manageable. Significantsimplification of the model or the simulation/optimiza-tion algorithm may be needed for on-line control.

In addition, many chemical processes are operated inbatch or semi-batch mode. In these modes of operation,batch-to-batch variations can be significant and are ofprimary concern. In most industrial cases, the batch-to-batch variations are strongly auto-correlated - a factthat provides a rationale for the practice of usingprevious batch results to adjust the recipe of a subse-quent batch. The error that cannot be removed byon-line batch control can be eliminated or reduced bythe so called ‘batch-to-batch’ control. The idea ofcombining batch-to-batch control with on-line controlhas begun to appear in some recent papers (Lee, Chin,Lee & Lee, 1999).

In this work, we investigate the problem of con-trolling the final PSD in a semi-batch precipitationprocess. We use a closed-form solution of the PBE,which is derived using the ‘method of characteristics.’The closed-form solution enables us to perform simula-tions very quickly and accurately, and hence providesus with a convenient test-bed for the control study.However, it is limited to the case of no aggregation/breakage and a separable form of the growth rate.Despite the limited generality, the simulation method isadopted here as our main goal is to evaluate the prosand cons of various model-based control strategies andthe example chosen still reflects many of the complexi-ties that make the control of PSD a very difficultproblem in practice.

We develop and test three different PSD controlstrategies that are suited to semi-batch processes. Thefirst is a batch-to-batch control method that uses off-line PSD measurements only. The second is an on-linecontrol method, in which the nucleation and growthrate kinetic parameters are estimated by an extendedKalman filter (EKF) using on-line PSD measurementsand the predicted final PSD is controlled through sim-ple least-squares calculation based on an empiricalmodel. Finally, the benefit of combining the batch-to-batch control with the on-line single batch control isexamined. The simulation result on a calcium carbonatereactor shows that the batch-to-batch control and on-line single batch control methods can be combined tocomplement each other.

2. Modeling of precipitation process

2.1. Model of calcium carbonate precipitation reactor

The overall reaction, that takes place in the precipita-tion process we study, is as follows:

Ca(OH)2+Na2CO3�CaCO3+2NaOH (1)

The feeds are a solution of sodium carbonate and asolution of calcium hydroxide at certain concentrations,and the main product is calcium carbonate. The PSD ofprecipitated calcium carbonate is the main variable, wewant to control. The precipitation occurs, when thecalcium ions and carbonate ions are present at super-saturated concentration levels. Supersaturation impliesthat the ionized species are present in the solution suchthat the solubility of the species is exceeded. It isassumed that the ionization reactions are fast comparedwith the precipitation and the ionization reactionsreach equilibrium instantaneously. Also, the perfectmixing in the reactor is assumed. Then the MB of theprecipitation reactor are written as follows:

K. Lee et al. / Computers and Chemical Engineering 26 (2002) 1117–1131 1119

d(VCj)dt

=qjfCj

f−qCj−kaV��

0

G(L, t)n(L, t)L2dL

(2)

dVdt

=q f−q (3)

where j=1, 2 (for Ca(OH)2 and Na2CO3). The integralterm in Eq. (2) represents the consumption due to theparticle growth. The rest of the notation is defined inTable 1.

Note that the integral of Eq. (2) contains the particlesize (L), the PSD (n), and the growth rate of particle(G). The above MB should be coupled to the PBE of:

�Vn(L, t)�t

+V� [G(L, t)n(L, t)]

�L=Vh(L, n, t)−qn(L, t)

(4)

where the right-hand side of Eq. (4) is a term involvingaggregation and breakage of the particles. It dependson the number density, reactor volume, and outlet flow.The PBE is a PDE with corresponding initial conditionn(L, t0)=nt 0

(L) and boundary condition n(L0, t)=nL 0

(t) (i.e. the number of nucleated particles). L0, thenucleated particle size, is assumed to be 0 (i.e. negligiblysmall) in this study.

Typically, nucleation and growth rates of precipita-tion and crystallization processes are represented bysemi-empirical power laws. There exist two kinds ofnucleation mechanism: primary nucleation which is in-duced by supersaturation without particles, and sec-ondary nucleation which is related to the existingparticles in the reactor. A nucleation model has to takeinto account the both phenomena. Growth rate is afunction of supersaturation and particle size. In thiswork, we adopt the following particular nucleation ratemodel used in (Eek, Dijkstra & van Rosmalen, 1995).

nL 0(t)=

1G(0, t)

��

0

anC sbnnL2.5dL (5)

Also, the following power law is used for the growthrate.

G(L, t)=atC sbt

11+exp(−aL(L−bL))

(6)

where Cs is the supersaturation of the solute, which isdefined as �C1C2−1 (expressed in terms of the nor-malized concentration) and an, bn, at, bt, aL, bL areparameters. The specific values we used for the parame-ters and the constants appearing in Table 1 will begiven later.

The kinetic equations have strong nonlinearity due tothe power terms. The nonlinearity of the kinetic modeland the coupling with the MB make the problemdifficult. Despite the nonlinearity, linear control tech-niques can still be effective for feedback control, as wewill see later. The PBE is the computationally demand-ing part of the precipitation reactor model. For thepurpose of model-based control, the calculation burdenfor simulating this process model cannot be overlooked.Thus an analytical approach for solving the PBE isinvestigated in the next section.

2.2. Closed-form solution of PB equation

Generally, the PBE can be handled by converting itto a set of ODEs. Various forms of finite elementmethod (FEM) and finite difference method (FDM)have been used for this purpose. With these techniques,one is left with a large number of ODEs, which canrequire exorbitant computation time for simulation andoptimal control calculation. The details on solutiontechniques for various classes of the PBE can be foundin a recent book by Ramkrishna (2000).

If the right-hand side of Eq. (4) is a linear or anindependent function of number density, a closed-formof the solution, which involves an integral along thecharacteristic curve, can be obtained using the methodof characteristics (Varma and Morbidelli, 1997). Whenaggregation and breakage are negligible and the growthrate takes on a separable form of GtGL as in Eq. (6), inwhich Gt is the time-dependent part of the growth rateand GL is the size-dependent part, the solution can bewritten as:

n(L, t)=V(t0)V(t)

nt 0(Lb)

GL(Lb)GL(L)

for Lb(L, t)�0 (7)

n(L, t)=V(tb)V(t)

nL 0(tb)

GL(0)GL(L)

for tb(L, t)� t0 (8)

The t0 is the start time of growth reaction. Lb and tb

are the birth size and birth time of the particle of size Lat time t, respectively, and can be obtained by solvingthe following equations:� L

L b

1GL(l)

dl=� t

t 0

Gt(�)d� (9)

Table 1Variables and parameters of the MB

DescriptionVariable/parameter

Cjf Concentration of species j in the jth feed

streamCj Reactor concentration of species jq j

f Feed flow rate of stream jq f Total feed flow rate

Total outlet flow rateQVolume of contents in the reactorVArea factorka

Characteristic particle sizeLG(L, t) Growth rate of particle

PSD, number of particles per volume ofn(L, t)solvent per particle size

K. Lee et al. / Computers and Chemical Engineering 26 (2002) 1117–11311120� L

0

1GL(l)

dl=� t

t b

Gt(�)d� (10)

For the particles generated by nucleation, the birthsize, Lb, is always equal to 0, and for the initial seedparticle the birth time tb is equal to t0. Detailed deriva-tion of the above analytical solution is presented inAppendix A.

The assumption of negligible agglomeration andbreakage indeed limits the usefulness of the abovesolution approach. Generally speaking, aggregationand breakage of particles are observed when the parti-cle density is high (Kataki & Tsuge, 1990). If wemaintain the particle density low, we can avoid theaggregation and breakage phenomena in the reactor.For the calcium carbonate precipitation, breakage isthought to be a negligible phenomenon but the particledensity in the reactor could be such that agglomerationis significant (Collier & Hounslow, 1999). In the case ofsize dependent growth, there are no generally agreedtheoretical kinetics. The separable form is the exclu-sively used empirical form.

2.3. Combining PB and MB equations

In order to simulate the precipitation reactor, theMB and PBE should be solved together. The nucleationand growth rate expressions in Eqs. (5) and (6) can besubstituted into Eqs. (7) and (8). The integral equationsof Eqs. (9) and (10) must be solved for tb and Lb toevaluate Eqs. (7) and (8) for each L and t. Note that,while GL is known ahead of time and therefore, theleft-hand sides of Eqs. (9) and (10) can be integratedanalytically, Gt is coupled with the MB as it depends onthe concentrations and it must be integrated along withthe MB equations. To facilitate the calculation of tb

and Lb, we propose to add the following equation tothe ODEs.

dIGt

dt=Gt(t) (11)

Then, Eqs. (9) and (10) become:

IL(L, Lb)=IGt(t) (12)

IL(L, 0)=IGt(t)−IGt

(tb) (13)

where IL denotes the integral in the left side of Eqs. (9)and (10). Given IGt

(t), the first equation is just anonlinear algebraic equation for Lb. The second canalso be solved for tb conveniently if we store the timehistory of IGt

as we integrate the ODEs for MB.The integral term in Eqs. (2) and (5) can be approxi-

mated by appropriate discrete sums, e.g.:��

0

GnL2 dL� �nsize

i=1

n(Li, t)Gt(t)GL(Lt)Li2�L (14)

where nsize is a large number and is selected as 100 inthis study. {Li, i=1, …, nsize} represent the discretizedvalues of L that are densely and evenly spaced; here thespacing was chosen as �L=0.2. The number and spac-ing are chosen to span the viable range of particle size.For notational convenience, let us denote n(Li, t) asni(t) from here on. Similarly, n0(t) is used to denote thenucleated particle density of n(0, t).

Substituting the above approximation and the rateexpressions into the ODEs and the solutions of thePBE, we obtain the following set of equations, we canuse for simulation:

dC1

dt=

q1f (t)

V(t)C1

f (t)

−ka��j=1

nsize nj(t)at(�C1(t)C2(t)−1)bt

1+exp(−aL(Lj−bL))Lj

2�L�

−C1(t)V(t)

(q1f (t)+q2

f (t))

dC2

dt=

q2f (t)

V(t)C2

f (t)

−ka��j=1

nsize nj(t)at(�C1(t)C2(t)−1)bt

1+exp(−aL(Lj−bL))Lj

2�L�

−C2(t)V(t)

(q1f (t)+q2

f (t))

dVdt

=q1f (t)+q2

f (t)

dIGt

dt=at(�C1(t)C2(t)−1)bt and for i=1, ..., nsize

0=ni(t)−

��

V(t0)V(t)

n(Lb(Li, t), t0)1+exp(−aL(Li−bL))

1+exp(−aL(Lb(Li, t)−bL))for Lb(Li, t)�0

V(tb(Li, t))V(t)

�0(tb(Li, t))1+exp(−aL(Li−bL))

1+exp(aLbL)for tb(Li, t)� t0

0=Li−Lb(Li, t)

−exp(−aL(Li−bL))−exp(−aL(Lb(Li, t)−bL))

aL

−IGt(t)

0=Li−exp(−aL(Li−bL))−exp(aLbL)

aL

−IGt(t)

+IGt(tb(Li, t)) (15)


Fig. 1. Comparison of the closed-form PBE solution based simulationvs. the FEM based simulation.

composed of q1f , q2

f . However, upon careful inspection,one realizes that the above is really not a DAE systemsince the algebraic equations involve past �alues ofsome of the variables, e.g. V(t0), V(tb), IGt

(tb), n(0, tb)and n(L, t0). If we store the time history of V, IGt

andn(0, t) as we perform the integration, as well as theinitial PSD n(L, t0), the above can be simulated usingan implicit integration method developed for solvingDAEs. The stored variables will be discrete in time andparticle length, but we can use interpolation to find thepast values needed in each iteration step.

Note that the number of discretization points for theparticle size determines the size of the algebraic equa-tions that must be solved at each time instant inaddition to the resolution of PSD. If a very fine resolu-tion is required, the computational burden can be veryhigh even for the analytical solution based simulationand one may need to resort to a model reduction ofsome sort. In this work, the calculation speed andaccuracy were found to be acceptable with 100 dis-cretization points.

If the computational burden is a concern, one canalso try an explicit integration method, in which casethe algebraic equations are solved just once at thebeginning of each integration step and held constant.This should work well as long as the algebraic variablesthat enter the differential equations (the integral termsrepresenting the rate of consumption of the species dueto the growth in this case) change significantly moreslowly than the differential variables and/or their effecton the integration of the ODES is relatively small. Thiswas found to be the case for the particular calciumcarbonate system we studied.

Fig. 1 shows the final PSD result of a simulation weperformed with the analytical solution in Eq. (15). Thesimulation was done in MATLAB™. The simulation con-dition and parameter values, we used are summarizedin Table 2. The feed flow rates and concentrations wereheld constant over the simulation period. Also shown inFig. 1 is the result of a FEM simulation based on 100finite elements with linear basis functions. We can seethat the two simulations agree in terms of the finalPSD. In both simulations, the integration was doneusing (ODE15S in MATLAB). The calculation time forthe simulation for one batch of length 30 h on PentiumIII 800 MHz PC was about 2.0 min for the analyticalsolution based approach and about 56.7 min for theFEM approach. The advantage of the FEM approachover the analytical solution based approach is that itcan be used to simulate more complex cases involvingagglomeration and breakage.

2.4. Recursi�e formulation

An alternative way to simulate the process is to useEq. (15) in a recursive fashion. Here the initial PSD

Table 2Simulation condition and kinetic parameter values used for theprocess simulations

Variable/parameter Value

6.0, 6.0Feed concentrations of Ca(OH)2, Na2CO3

Feed flowrates of Ca(OH)2, Na2CO3 0.3, 0.3Initial reactor concentrations of Ca(OH)2, 5.0, 5.0

Na2CO3

Initial reactor volume 10.0(0.01/�(3))Initial PSD×L2e−L

20Maximum particle sizeNumber of discretized particle size 100Simulation time 30Number of discretized simulation time 600Nucleation rate parameters (an, bn) 1e−3, 1

3e−2, 1, −3,Growth rate parameters (at, bt, aL, bL)17

Area factor 1

All variables are given in normalized values and therefore, unit-less.

where:

n0(t)=an

at

(�C1(t)C2(t)−1)bn−bt [1+exp(aLbL)]

×��j=1

nsize nj(t)Lj2.5�L

�The above has close resemblance to a system of

DAEs in the form of:

x� = f(x, z, u)

0=g(x, z) (16)

where the differential variable x consists of C1, C2, V,IGt

, the algebraic variable z consists of ni(t), Lb(Li, t),tb(Li, t), i=1, …, nsize, and the input variable u is


(nt 0(L)) is re-initialized at each sampling time. In this

formulation, one has to store the past values of V, IGt

and n0(t) for one sample interval (denoted as �ts here-after) only. After integrating the equations for onesample interval, the final PSD becomes the initial PSDfor the next time interval. Along with the initial PSD,IGt

should be reset to 0 at the start of each sampleinterval. The main motivation for considering this wayof simulation is not to increase the accuracy or effi-ciency of simulation but to be able to represent theprocess as a discrete time state-space system to whichconventional state estimation and control techniquescan be applied in a natural manner. Let us denote thevector containing the number densities at all the dis-crete points of L, i.e. n= (n1, …, nnsize). Then, thesample-to-sample behavior can be represented by:

xk+1=Fx(xk, nk, uk)

nk+1=Fn(xk,nk,uk) (17)

where Fx(xk, nk, uk) and Fn(xk, nk, uk) denote theresulting x and n when the system of equations in Eq.(15) is integrated for one sample interval with thestarting state and PSD of xk and nk.

3. State and parameter estimation using EKF

Usually, the detailed kinetics of precipitation or crys-tallization are known imperfectly and there are othersources of uncertainties in the reaction (e.g. impuritiesof feed, stochastic behavior of nucleation and growth,etc.). Since the kinetics and other parameters have astrong effect on the behavior of PSD, it is helpful toestimate these uncertain parameters, preferably on-line.In addition, not all the state variables are directlyavailable through on-line measurements. Hence, it is ofinterest to develop a technique to estimate the state andparameters of the model based on the availablemeasurements.

Note that, even if accurate on-line PSD measure-ments were possible, the PSD at the final time still hasto be predicted based on the estimated states andparameters for the purpose of control. Of course, inreality, the direct on-line PSD measurements may notbe available at all or may be available with substantialbias and noises.

3.1. EKF formulation

Let us collect the unknown parameters into a vector,which is represented by symbol � hereafter. One prob-lem with using Eq. (15) directly for state estimation isthat the integration requires the past values of the stateand algebraic variables. In other words, it does not fitinto the standard system description of:

xk+1=F(xk, uk, �) (18)

yk=h(xk, �) (19)

because, the integration of the ODEs in Eq. (15) requireevaluation of ni(t), which in turn requires the pastvalues of some of the state variables as well as theinitial PSD.

As we mentioned earlier in the simulation section,one can reset the initial condition for the integration ofEq. (15) at the start of each sample step. This naturallyyields the standard system description in the form of:

xk+1=Fx(xk, nk, �k, uk)+�k

nk+1=Fn(xk, nk, �k, uk)

�k+1=�k+�k (20)

The above is basically the same equation as Eq. (17)but the argument � is added to the integration opera-tors to emphasize the dependence of system behavioron the uncertain parameters. It is assume that � is heldconstant during integration for each sample interval.However, we allow the uncertain parameters to varyfrom the kth sample time to the next by the amount of�k. �k added to the equation is the state noise. We mayalso express the measurements generally as:

yk=h(xk, nk, �k)+�k (21)

where �k is the measurement noise. In the lack of betterknowledge, �k, �k, and �k can be treated as indepen-dent, identically distributed (white noise) sequences ofzero mean and covariances of R�, R� and R�, respec-tively. Typically, the state estimation problem is solvedby using the EKF, which is based on the linearizationof the above equation with respect to the availableestimate at each time step (Lee & Ricker, 1994).

With the model arrangement of Eq. (20), the numberdensities will be updated indirectly through x and �.That is because the random noises enter into thesestates only. In other words, even if very accurate on-line PSD measurements were available, the numberdensity will not be updated from the measurement datadirectly. Even though it is physically true that the PSDvariables are affected through x and �, the estimatedvalues can still have large errors because of the fact thatthe model structure may not be perfect. Hence, it isbetter that the PSD variables are reset according to the(‘filtered’) measurements of the PSD variables. One canmake sure that this is achieved by adding the extra statenoise directly into the equation for n, i.e.

nk+1=Fn(xk, nk, �k, uk)+ek (22)

where ek is a white noise. Since the independent noisethat enter into n, these states will be directly resetaccording to its measurements (after the measurementnoises are filtered out, of course).


Table 3Diagonal elements of the covariance matrices used for the EKFdesign

C2 V anVariable atC1 n

1e3 1e−3 1e2�0 1e11e3 1e−21 11R�

1e−2R� 1e−3Re 1e−8R� 1

Fig. 3. Root mean squares of the innovation term with time-invariantnoise covariance matrix.

3.2. Simulation result

We tested the performance of EKF with the processcondition in Table 2. The uncertain parameter vector �

consists of an and at. The initial estimates of an and at

are chosen as 0.8e−3 and 2.4e−2, respectively. Thenumber densities of particles are assumed to be mea-sured, so the number of measured data is 101 at eachsampling time. We assume that the measurement has�1% noise. The Jacobian matrix is calculated numeri-cally (by perturbing each state variable and calculatingthe sensitivity by integrating the equations for onesample interval).

We tested the performance of the EKFs based on therecursive models Eqs. (20) and (22). The diagonal ele-ments of the covariance matrices used for the EKFdesign are summarized in Table 3. �0 represents theinitial covariance matrix, R� is the state noise covari-ance matrix, R� is the parameter noise covariance ma-trix, and R� is the measurement noise covariancematrix. Re is the covariance matrix of the extra noiseterm in Eq. (22) (which is added directly to the numberdensity equation).

Fig. 2(a) shows the behavior of estimated parameters(for an, at) during a course of batch. The estimates

showed large oscillations that increase with time. Tounderstand the reason for this, we plotted the timeprofile of root mean squares of the innovation, definedas Ek+1=yk+1− yk+1�k. Fig. 3 shows the exponentialincrease in the innovation, which is mainly due to theincreased particle size and total number of particles.The increased innovation makes the update based onthe local linearized model less accurate thereby causingthe oscillation.

As a remedy, we tuned the EKF with a time-varyingfactor ��, which multiplies the measurement noise co-variance. It is calculated according to:

��=�

nsize

i=1

ni(k−1)

�nsize

i=1

ni(1)(23)

Fig. 2. EKF estimation results with (a) time-invariant noise covariance matrix; (b) time-varying noise covariance matrix.


Notice that, if the number density did not change withtime, �� would remain constant. Since the numberdensity values grow with time, �� also grows with time.

Fig. 2(b) shows the estimation results with the tuningof R�=��

5I(101) (where I(101) is an identity matrix ofdimension of 101). This reduces the gain for the updatein the later part of batch and thereby suppresses theoscillation in the estimates. However, it also impairs theEKF’s ability to track parameters in the later part.

Fig. 4 shows the prediction error for the final PSDfor two cases, the case when the independent noise terme is added to the state n as in Eq. (22) and the casewhen it is not as in Eq. (20). We can see that theprediction for the former case is far better.

4. Control of batch/semi-batch precipitation reactor

4.1. Batch-to-batch control

On-line measurement of PSD is often unavailable,unreliable, or inaccurate. More reliable and accuratemeasurements of PSD can be obtained after the com-pletion of a batch through laboratory analysis. Thus inmost cases, feedback control during a batch may not bea practical option. Without on-line control, the qualityof the product will depend completely on the recipedecided off-line, i.e. before the batch begins. In the faceof disturbances that perpetuate over a number ofbatches, the final PSD can be controlled by using theresult from an off-line analysis of the final product ofthe previous batch to modify the recipe of a new batch.This is referred to as ‘batch-to-batch control’ or ‘run-to-run control.’ A simple method to update the recipebased on an off-line PSD measurements is as follows.

1. After the completion of the lth batch, calculate themodel prediction error by comparing the off-linefinal PSD measurement nl

f with the model predictionnl

f :

E� l=nlf − nl

f (Ul) (24)

Here l is the batch index. nf is a vector containingthe final PSD (volume or number density) for vari-ous discrete sizes. U is a vector that contains all theadjustable parameters for the recipe (e.g. deviationof the input profiles from the nominal ones forvarious time intervals). Argument Ul is added for nl

f

just to clarify that the prediction is made with therecipe parameters used for the lth batch.

2. Then, the corrected prediction for the l+1th batchis

nl+1f = nl+1

f (Ul+1)+E� l (25)

nl+1f is the measurement-corrected model prediction

of the final PSD for the l+1th batch. It is supposedhere that, without further modification to the recipe(i.e. with Ul+1=Ul), the same prediction error inthe final PSD will repeat in the subsequent batch.Other extrapolation models can also be used ifappropriate.The updated recipe can be obtained bysolving:

min�U l+1

{�nf− nl+1f �Q

2 +��Ul+1�R2 } (26)

where nf is the target final PSD and �Ul+1 is thechange in the recipe parameter vector between thel+1th and lth batch. Here, conventional quadraticminimization of the PSD error is shown where Qand R are the weighting matrices for the PSD errorand the size of recipe change.

3. Repeat with l=l+1.Even though we adopt here the quadratic minimiza-

tion, many different forms of objective function can beused. Also, various constraints can be added to theoptimization. This optimization problem can be solvedusing the nonlinear model nl+1

f = f(Ul+1) directly,which is defined through the MB and PBE of Eq. (15),or a linearized model:

nl+1f � f(U� )+

� �f�U

U=U�

(Ul+1−U� ) (27)

where U� is the nominal recipe parameter vector or:

nl+1f � f(Ul)+

� �f�U

U=U l

(Ul+1−Ul) (28)

With linear constraints on Ul+1, the optimization is aquadratic program. If we use the nonlinear model ofEq. (15) directly for the control calculation, we have tosolve a nonlinear optimization at each time step, whichrepresents a large computational burden. Even thoughthe linearized model may be only a crude approxima-tion, it can be effective for control with the feedback ofthe final PSD result.

Fig. 4. Error in the final PSD prediction at various time points by theEKF formulations.


Fig. 5. The detailed scheme, we used for integrating on-line controland batch-to-batch control.

have the so called ‘shrinking horizon’ control strategy.�U+ means the deviation from the nominal trajectory,which is fixed or transferred from the previous batch atthe batch start time.

One notable point is that, even with exact on-line PSDmeasurements, the final PSD cannot be predicted per-fectly in general. This is because, unlike a continuousprocess, a batch process is time-varying and the errordoes not remain at a constant value until the end. Hence,adding the current feedback error directly to the predic-tion of the final PSD does not give the correct result. Abetter strategy is to model the possible disturbances asstochastic states and employ state estimation usingtechniques like EKF (as we have done in Section 3).

Some variations are possible within the above basicformulation, in terms of how information gets trans-ferred from batch to batch. One can reset the nominalrecipe, initial parameter values, and covariance matricesall back to some fixed values at the start of each batch.This way, each batch will be completely independent.One can also transfer the updated recipe and/or theterminal parameter estimates from the previous batch tothe new batch as the nominal recipe and the initialparameter values. Then, successive batches will be linkedto each other.

Though on-line measurements can enhance the pre-diction of final PSD through state estimation, estimationerrors as well as unmodeled disturbances and dynamicswill still cause some errors in the final PSD prediction.So perfect control is not possible in general, no matterhow many same batches are repeated. This is the mainshortcoming of on-line control.

4.3. Integration of batch-to-batch control and on-linecontrol

Given the shortcomings of the batch-to-batch andon-line control strategies, it is natural for us to explorethe possibility of combining them. Because on-line con-trol can respond to disturbances immediately and batch-to-batch control can correct any bias left uncorrected bythe on-line controller, which may be due to unmodeleddisturbances, parameter errors, and dynamics, the com-bined scheme can potentially complement each other torender the benefits of the both. However, one must becareful in integrating the two as they can potentiallyconflict with each other.

The integration we propose in this paper is describedin Fig. 5. If the estimated error calculated from thebatch-to-batch controller is not added to the predictionby the EKF, the on-line control calculation ends up‘undoing’ the correction made by the batch-to-batchcontroller and the benefit of the integration gets lost. Byincorporating this error, the prediction for the on-linecontrol can be enhanced through the measurement ofthe final PSD of the previous batch. The same

If the off-line PSD measurement contains significantnoise and/or significant disturbances exist that last onlyone batch, an appropriate filter should be applied to themeasurements to extract only the long-term trend of theerror.

The main shortcoming of batch-to-batch control isthat the correction is not made until the next batch.Hence, at least one batch will yield significant error if alarge disturbance happens. In addition, it cannot handledisturbances that change from batch and batch in acompletely random fashion and may actually amplifytheir effect.

4.2. On-line control of final PSD

Recently, some new developments in the area ofon-line PSD measurement have begun appearing in theliterature (Braatz & Shinji, 2000). If on-line measure-ment of PSDs can be made on a reliable basis, or if thePSDs can be inferred accurately from other on-linemeasurements, one can explore the possibility of imple-menting on-line control that adjusts the batch recipewhile the batch is going on.

On-line batch control can be established in a mannersimilar to the batch-to-batch control formulation above.The following optimization can be solved at severalimportant time points during a batch.

min�U+(i )

{�nf− nf(i )�Q2 +��U+(i )�R

2 } (29)

U+(i ) contains the recipe parameters that remain ad-justable at the ith time point of control execution. nf(i )is the prediction of the final PSD for the on-going batchbased on the state estimate at the ith time. Hence we


Table 4Kinetic parameter values used for the process and model in thecontrol simulations

bn atParameters btan

Process 11e−3 3e−2 10.95 2.4e−20.8e−3 0.9Model

Fig. 6. Final PSD errors under batch-to-batch control with differentweighting factor �R.

Table 5Weighting factors for the optimization

ValueWeighting factor

PSD error (Q) 10×I(101)�R×diag(1, 2, 3, 4, 5, 6)5Size of recipe change (R)

I(n) is an identity matrix of size n.

applies to the update of the recipe parameters. Thispoint will be demonstrated in the simulation study.

4.4. Simulation results of control performance

We next tested the performance of the aforementionedcontrol schemes on the calcium carbonate precipitationprocess through simulation. The basic process conditionunder which we test the control strategies is the same asthat we used in the earlier simulations (Table 2) exceptfor the nominal feed flow rate. In the control study, westart with a constant feed flow rate profile of 0.25 for thetwo feed flows. We consider the scenario that the modelcontains some errors in the kinetic parameters of thegrowth and nucleation rates, causing errors in the finalPSD. The parameter values assumed for the model andthe real plant are shown in Table 4.

Control was done by manipulating the feed flow ratesof the reactants at six uniformly spaced time points(t1=0 h, t2=5 h, t3=10 h, t4=15 h, t5=20 h, t6=25h). Hence, we optimize the inputs as piecewise constantflow profiles-with each piece lasting 5 h. The recipeparameter vector U then would consist of 12 variablesthat represent the deviations of the two flow rates fromthe nominal values for the six time intervals. Consideringthe physics of the process, we assumed that the two feedflows would be varied together and hence reduced therecipe parameters by half in the subsequent simulationstudies.

4.4.1. Batch-to-batch controlFirst, we applied the batch-to-batch control method.

Here we used the linear model structure of Eq. (27) inthe optimization. The gain matrix (�f/�U)U=U� wascalculated by simulating the final PSD results afterperturbing the inlet flow rates by varying amounts andthen performing least-squares regression with the data.Also, since we adopted a linear control model, the recipeupdate at the start of each batch was calculated throughsimple least-squares. The final PSD (i.e. the numberdensity) measurements were assumed to contain negligi-ble noise. As the semi-batch reactor volume increases,the PSD control becomes more difficult because itbecomes more difficult to introduce concentrationchanges with limited flow rates. Therefore, we need tocontrol the process as soon as possible before the reactorvolume builds up or we need to constrain the inlet flowsuch that the reactor volume does not build up tooquickly. Based on this, we tuned the controller throughthe weighting factors and constraints. Tables 5 and 6show the weighting factors and constraints we adopted.

Fig. 6 shows the control result with various weightingfactors. As we expected, the control result shows a largeerror in the first run but the error decreases quickly asbatch runs are repeated. A larger weighting factor onthe recipe change (i.e. a larger �R) leads to moreconservative adjustments and therefore, slower conver-

Table 6Constraints for the optimization

�U(5)�U(4)�U(3)�U(2)�U(1)Constraints for control outputs �U(6)

0.5 0.05 0.030.4 0.3Upper bound 0.1Lower bound −0.03−0.3 −0.2 −0.1 −0.05 −0.05

1e−21e−21e−21e−2 1e−2Rate of change (�U (i+1)−�U(i ))


Fig. 7. The final PSD and the input profiles for the first, fourth, seventh and the final batch.

gence. However, it also gives increased robustness tomodel errors and measurement errors. Fig. 7 shows thefinal PSD (in volume density) of the first, fourth, sev-enth and final run against the reference PSD (when�R=100). It also displays how the feed flow profiles aretransformed from the initial run to the final run. It isnotable that the batch-to-batch control scheme basedon the very crude approximation of the model (i.e.linear gain matrix identified through least-squares) isworking very effectively.

4.4.2. On-line single batch controlNext we tested the control strategy based on on-line

PSD measurements. It was assumed that the PSD mea-surement can be made every 3 min. We assumed thePSD measurement has �1% measurement noise. Themodel parameters and the plant condition remain thesame as in the previous simulations for the run-to-runcontrol. For the EKF calculation, the model state wasaugmented with only two of the four parameters thatcontain errors. The other two parameter errors were left‘unmodeled.’ This is more realistic as it is generallyimpossible in practice to identify all error sources andinclude them in the model.

We then applied the EKF estimation technique. TheEKF was tuned in the same way as in the earliersimulation (in Section 3.2). Due to the ‘unmodeled’disturbances (the errors in the parameters bn and bt),the modeled parameters an and at converged to differentvalues from the earlier EKF simulation in Section (3).The control adjustments were made with 5 h intervals.At the start of each interval, we integrated the modeluntil the batch end time, starting with the state estimateprovided by the EKF and using the nominal inputprofiles, and then calculated optimal deviations fromthe nominal profiles (i.e. optimal values of �U+(i ))through the same least-squares calculation as before.

We compare two different implementations of theEKF-based on-line control scheme. In the first imple-mentation, we reset the nominal input flow profiles andparameter values to the original values at the start ofeach batch. In the second, we carry forward the ad-justed input flow rate profiles as the nominal profiles forthe subsequent batch and we use the previous batch’sfinal time estimates of the parameters as the startingestimates for the new batch. Fig. 8 compares the resultsfrom the two implementations. The transfer of previousbatch information results in a better rejection of thefirst batch error. The scheme shows gradual improve-ment down to a very small error, while the no trans-fer case leaves a significant, lasting error that re-mains uncorrected. However, if there is a disturbancethat occurs only for a single batch, the corrected

Fig. 8. Comparison of the final PSD errors achieved with the twooptions for transferring recipe and estimate from batch-to-batchwithin the EKF-based on-line control scheme.


Fig. 9. Comparison of the final PSD errors over ten batches under thethree control strategies.

fluctuation is mild enough that it is not a cause forconcern in this case.

4.4.3. Integrated controlFinally, we test the strategy that integrates batch-to-

batch control and on-line control. In this method, theinput profiles calculated by the batch-to-batch con-troller is used as the nominal input trajectories for theon-line controller. Also, the prediction error of the finalPSD for the previous batch, as calculated from theoff-line measurements, is added as a bias value to theon-line predictions made by the EKF. With this, weexpect to see the on-line control’s advantage of immedi-ate correction being combined with the benefit of grad-ual reduction to the minimum error afforded by thebatch-to-batch control. Fig. 9 indeed confirms this.

For achieving the complementary effect of the twoearlier strategies, some care is needed in the integration.Fig. 10 shows the result when the recipe is updatedthrough the batch-to-batch control calculation but theprediction error calculated from the off-line analysis isnot forwarded to the EKF calculations of the subse-quent batch. As we can see, with this implementation,the EKF tries to ‘undo’ the corrections made by thebatch-to-batch controller and therefore, most of thecomplementary effect is lost.

4.4.4. Disturbance lasting for a single batchNext, we test how disturbances that occur just for a

single batch affect the batch-to-batch behavior of finalPSD under the different control strategies. The scenariohere is that the kinetic parameter an of the actualprocess changes to 1.3e−2 for the seventh batch andswitches back to the original value at the eighth batch.Fig. 11 shows the final PSD results under the threecontrol strategies.

parameter estimates and the recipe would be biased forthe next batch; hence the transfer of the recipe andparameter estimates would serve to introduce furthererrors in the subsequent batches. Some filtering may benecessary in the transfer.

We see from Fig. 9 that, compared with the run-to-run control strategy, the on-line scheme immediatelyreduces the PSD error starting with the first batch butshows some fluctuation that continues on. It is due tothe fact that the EKF does not give perfect estimationbecause of the unmodeled disturbances and measure-ment noises. Of course, we can decrease the fluctuationby gradually reducing the error covariance to zero,which turns off the EKF, but the controller would thenlose its ability to respond to new disturbances. The

Fig. 10. Comparison of the final PSD errors achieved by transferring and not transferring the prediction error used for the batch-to-batch controlcalculation to the next batch.


Fig. 11. Final PSD errors under the three strategies when a newdisturbance enters the seventh batch and disappears afterward.

control. The difference with the on-line control result inthe seventh batch can be attributed to the difference inthe starting recipes and parameters. The effect of thedisturbance does carry over to the next batch due to thebatch-to-batch control component but the effect is di-minished more rapidly compared with the plain batch-to-batch control.

4.4.5. Effect of measurement noiseOn-line measurements of PSDs are expected to be

inaccurate due to the limitations in measurement reso-lution as well as unreliable measurement conditions(e.g. high slurry concentration, agglomerated particle,temperature fluctuation). We need to consider the effectof large measurement noises on the controllers’performance.

Fig. 12 shows the effect of 20% measurement noise.The on-line control strategies are tested and all simula-tion conditions are the same as before except for themeasurement noises. As we can see, the performance ofthe on-line control strategy degrades significantly.

Of course, one can detune the EKF by increasing themeasurement noise covariance matrices. However, avery large measurement noise covariance decreases theon-line control’s sensitivity and hence leaves largeparameter errors.

In the case of integrated control with 20% measure-ment noise, the control performance does not deterio-rate much compared with the case of 1% measurementnoise. It shows that the batch-to-batch control action(based on the off-line measurement data) included inthe integrated control allows the scheme to reduce theerror to a small level despite the degraded quality ofon-line data.

5. Conclusions

In this paper, we presented computationally efficientmodel-based techniques for controlling the final PSDsin a semi-batch precipitation process. We examinedthree different strategies: the ‘run-to-run’ control whichuses off-line PSD measurements; the on-line controlwhich uses on-line PSD and other process measure-ments through EKF; and finally the strategy that inte-grates the both. We used the closed-form solution ofthe PBE, which is limited to the case of no aggregation/breakage, in order to simplify the simulation and alsothe control computation for the various forms of con-trol. However, the strategies are general and can beapplied to semi-batch processes described by PBEs ofmore complex types. The case study on the calciumcarbonate precipitation reactor showed that the com-plementary effect can be achieved by integrating thetwo strategies.

Fig. 12. Comparison of the final PSD errors in the case of 20%measurement noise.

In the case of batch-to-batch control, the disturbanceof the seventh batch resulted in a large error in thatbatch. The error of seventh batch causes incorrectrecipes in subsequent batches. The errors of subsequentbatches decrease by the batch-to-batch control action.

The EKF-based on-line batch control method resultsin a significant (but less than in the case of batch-to-batch control) error in the seventh batch. Also, theerror has much less effect on the subsequent batches. Inthis case, the input profiles and estimates at the end ofeach batch are transferred to next batch, meaning thestarting estimates and recipes are biased in the newbatch. Despite this, the errors of initial profiles andestimates are nearly eliminated at the end of the batch.

In the case of integrated control, the disturbance inthe seventh batch was rejected through the on-line


Acknowledgements

J.H. Lee gratefully acknowledges the support fromNational Science Foundation (CTS-0096326).

Appendix A. Analytical solution of PBE

In the absence of aggregation and breakage, the PBequation for the PSD including general size- and time-dependence of growth and volume change is written

�(nV)�t

+�GnV

�L=0 (30)

with the initial and boundary conditions:

n(L, t0)=nt 0(L) at t= t0 (31)

n(0, t)=nL 0(t)=

b(t)G(0, t)

at L=0 (32)

The first of these corresponds to the initial PSD presentin the system, and the second to the nucleation rate ofnew particles of size zero.

By rearranging Eq. (30) as:

�n�t

+G�n�L

= −nV

dVdt

−n�G�L

(33)

it is seen to be a first order PBE of the form

P(n, L, t)�n�t

+Q(n, L, t)�n�L

=R(n, L, t) (34)

with P=1, Q=G(L, t), and R(n, L, t)= − (n/V)(dV/dt)−n(�G/�L). Note that n indicates n(L, t), andnot a general dependence on the entire number densityhere. The equation is readily solved using the methodof characteristics; the characteristic curves are given bythe solution to the ODE:

dLdt

=G(l, t) (35)

The notation L(t ; Lb, tb) is used for the solutioncorresponding to the initial condition L=Lb at t= tb.This can arise from the boundary condition of nucle-ation when Lb=0 and the initial condition when tb=t0.

Along the characteristic:

dndt

= −nV

dVdt

−n�G�L

(36)

� n(L(t; Lb, tb),t)

n 0(L b, t b)

dn �

n �=

−� t

t b

�d log V(t)dt

t= t�

+�G(L, t �)

�LL=L(t�; L b, t b)

dt � (37)

Integrating and rearranging:

n(L(t ; Lb, tb), t)=n0(Lb, tb)V(tb)V(t)

exp�

−� t

t b

�G(L, t �)�L

L=L(t�;L b,t b)

dt �n(38)

This is the form of the solution required for generalgrowth rates. If the growth rate is separable, that is:

G(L, t)=GL(L)Gt(t) (39)

the solution is considerably simplified to:


exp�

−� t

t b

Gt(t �)�GL(L)

�LL=L(t�;Lb,t b)

dt �n

(40)

A change of variables, dL �=GL(L �)Gt(t �)dt �, can beused to further simplify the solution. The change ofvariables cannot be applied in the nonseparable case, asan implicit relationship for t � in terms of L � is thenintroduced.


exp�

−� L

L b

1GL(L �)

�GL(L �)�L �

dL �n

(41)


GL(Lb)GL(L)

=n0(Lb, tb)V(tb)V(t)

Gt(t)GL(Lb)Gt(t)GL(L)

(42)


G(Lb, t)G(L, t)


G(Lb, tb)G(L, tb)

(43)

with:

n0(Lb, tb)=

� nt 0(Lb) if L(t ; Lb, tb)�L(t ; 0, t0) (initial condition)

nL 0(tb) if L(t ; Lb, tb)�L(t ; 0, t0) (boundary condition)

(44)

The implicit solution for the characteristics them-selves is also simplified and becomes:

� L(t;Lb,tb)

L b

dL �

GL(L �)=� t

t b

Gt(t �)dt � (45)


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integrated run-to-run and on-line model-based control of particle size distribution for a semi-batch...

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