Control of Product Quality in Polymerization Processes

Control of Product Quality in Polymerization Processes

Francis J. Doyle IIIDepartment of Chemical Engineering

University of DelawareNewark, DE 19716

Masoud SoroushDepartment of Chemical Engineering

Drexel UniversityPhiladelphia, PA 19104

Cajetan CordeiroAir Products and Chemicals, Inc.

Allentown, PA 18195-1501

AbstractThe increasingly aggressive global competition for the production of higher quality polymer products at lower costs,along with a general trend away from new capital investments in the U.S., has placed considerable pressure on theprocess engineers in the U.S. to operate the existing polymer plants more efficiently and to use the same plant for theproduction of many different polymer products. The more efficient operation has been realized by better process controland monitoring while the available polymer product-quality sensors have been inadequate. Although many product qualityindices cannot be measured readily, they can be estimated/inferred in real time from the readily available measurements,allowing for inferential control of the polymer product quality. This paper presents a survey of the issues in controlling andmonitoring plant-product quality indices such as molecular weight, copolymer composition, and particle size distributionsin polymerization reactors. Examples will be given to illustrate some of the methods surveyed.

KeywordsParticle size distribution, Process control, Inferential control, Batch control, Population balance models, Polymerizationreactor control, Multi-rate measurements, Polymer product quality, Polymerization reactor monitoring

Introduction

A polymer product is composed of macromolecules withdifferent molecular weights, and the processability andsubsequent utility of a polymer product depends stronglyon the macromolecule distributions, such as molecularweight distribution (MWD), copolymer composition dis-tribution (CCD) [in copolymerization], and particle sizedistribution (PSD) [in emulsion polymerization]. For in-stance, in coatings, film formation, film strength, andgloss depend on the MWD, CCD, and PSD of the poly-mer. Since the distributions are influenced greatly bythe polymerization reactor operating conditions, the pro-duction of a high quality polymer requires effective mon-itoring and control of the operating conditions (Conga-lidis and Richards, 1998; Ogunnaike, 1995). The effectivemonitoring and control can be realized only when suffi-cient frequent information on the distributions is avail-able.

Polymerization reactors are a class of processes inwhich many essential process variables related to productquality cannot be measured or can be measured at lowsampling rates and with significant time delays. The lackof readily-available, frequent measurements from whichpolymer properties can be inferred, has motivated a con-siderable research effort in the following research direc-tions:

• The development of new on-line sensors [lists ofmany of the currently-available on-line sensors areprovided in (Ray, 1986; Chien and Penlidis, 1990)].

• The development of qualitative and quantitativerelations between easier-to-measure quality indicessuch as density, viscosity and refractive index, and

more-difficult-to-measure quality indices such asconversion and average molecular weights (Kiparis-sides et al., 1980; Schork and Ray, 1983; Canegalloet al., 1993; Soroush and Kravaris, 1994; Ohshimaet al., 1995; Ohshima and Tomita, 1995).

• The development of state estimators that are capa-ble of estimating unmeasurable polymer propertiesfrom readily available measurements. The availabil-ity of sufficiently-accurate, first-principles, mathe-matical models of many polymerization reactors hasmade possible the development of reliable state esti-mators for the reactors (Jo and Bankoff, 1996; Elliset al., 1988; Kim and Choi, 1991; Kozub and Mac-Gregor, 1992; Ogunnaike, 1994).

Product Quality in Polymerization Processes

Product quality is a much more complex issue in poly-merization than in more conventional short chain reac-tions (Ray, 1986). Because the molecular structure ofthe polymer is so sensitive to reactor operating condi-tions, upsets in feed conditions, mixing, reactor temper-ature and so on can change significantly critical molec-ular properties such as molecular weight distribution,copolymer composition distribution, copolymer chain se-quence distribution, stereoregularity, and degree of chainbranching.

The properties of a polymer product, such as the me-chanical properties and the characteristics in molding,having strong correlation with the molecular weight dis-tribution (MWD) of the polymer. Nunes et al. (1982)found that thermal properties, stress-strain properties,impact resistance, strength and hardness of films of poly-methyl methacrylate and polystyrene were all improved

290

Control of Product Quality in Polymerization Processes 291

by narrowing MWD. It is also generally said that thepolymer of long chain length gives superior mechanicalproperties to polymer products but has insufficient mold-ing characteristics. Then the molding characteristics canbe improved by blending short chain length polymer intothis long chain length polymer, while the good mechan-ical characteristics are kept. That is, the broader MWDcan be obtained by this blending. Therefore, the devel-opment of the methodology for adjusting MWD duringthe reaction to suitable one according to its use is de-sired, especially in producing high quality polymers.

Schoonbrood et al. (1995) studied the influence ofcopolymer composition and microstructure on the me-chanical bulk properties of styrene-methyl acrylatecopolymers. They found that copolymer compositiondrift has an influence on polymer mechanical propertiessuch as Young’s modulus, maximum stress, and elon-gation at break. In the case of copolymers that arehomogeneous with respect to chemical composition, (a)maximum stress and elongation at break depend on themolecular weight distribution, (b) Young’s modulus isindependent of copolymer composition and molecularweight distribution in the ranges studied, and (c) max-imum stress and elongation at break weakly depend onthe copolymer composition. In the case of copolymersthat are heterogeneous with respect to chemical composi-tion, copolymer microstructure affects strongly Young’smodulus, maximum stress, and elongation at break.

In paints and coatings, molecular weight, composition,and functional group distributions all play a key rolein polymer performance. For solution viscosity reasons,narrow molecular weight distribution is useful, but notevery paint or coating benefits from it. It depends onthe application. For example, air-dry paints benefit fromvery broad molecular weight distribution (Grady, 2000).

For processing and end-use performance of latex coat-ings, it is often advantageous to produce a latex withhigh solids content while maintaining viscosity within ac-ceptable limits. Latex particle size and particle size dis-tribution directly affect the relationship between solidsvolume fraction and rheological properties. The influ-ence of monodisperse latex particles on latex viscosityis described by the Dougherty-Krieger equation (Kriegerand Dougherty, 1959),

ηr =(

1− φ

φm

)−2.5φm

where ηr is the ratio of emulsion viscosity to that of thepure fluid (water for instance), φ is the volume fractionof solids, and φm is the maximum volume fraction of la-tex particles. For polydisperse systems, it has long beenestablished that blends of different size particles yieldviscosities which are lower than the viscosities of any ofthe monodisperse particles used to make nthe blend (forequivalent solids concentrations). Eveson and cowork-

ers (1951) suggested that a particle suspension with a bi-modal distribution can be regarded as a system in whichthe larger particles are suspended in a continuous phaseformed by suspension of the smaller particles in the fluidmedium. In other words, a suspension of smaller parti-cles behaves essentially as a fluid toward the larger par-ticles. Farris (1968) extended this line of reasoning to amultimodal blend of particle sizes with any number ofmodes and Parkinson et al. (1970) to a continuous parti-cle size distribution. In both cases, successive applicationof a monodisperse expression for relative viscosity to par-ticles of increasing size in a blend yielded an expressionfor relative viscosity of the form

ηr =∏

i

(1− φi

φm,i

)−kφm,i

where the φi are volume fractions of particles of a givensize in the particle size distribution. Although this ex-pressions is not directly applicable to the prediction ofviscosity for continuous latex distributions, the reason-ing behind its derivation suggests that control of theparticle size distribution would be an appropriate ap-proach to targeting desired latex rheological properties.In contrast, the more common approach of controllingmoments of the distribution, is only indirectly related totarget properties.

Classification of Variables in a PolymerizationPlant

A customer evaluates the quality of a polymer producton the basis of indices, end-product quality indices, thatare usually different from the product quality indices,plant-product quality indices, known to the plant processengineer. The end-product quality indices are related tothe final use of the polymer product and usually cannotbe measured in real time because of the complicated andslow measurement techniques needed or simply the in-ability to measure the quality indices until the final poly-mer product is formulated and used. On the other hand,polymer plants are operated at desired conditions by set-ting and regulating the plant variables (such as pressures,temperatures, and flow rates) that are measured readilyon-line. We will refer to these readily measurable vari-ables as basic plant variables to distinguish them fromthe plant-product quality indices and the end-productquality indices (end-use properties). These differenceslead us to categorize variables in a polymer plant intothe following three classes, intersections of which maynot be null:

• Basic plant variables,• Plant-product quality indices,• End-product quality indices.

Basic plant variables that can be measured readily on-line and whose values are set by the process engineer to

292 Francis J. Doyle III, Masoud Soroush and Cajetan Cordeiro

operate the plant at desirable operating conditions. Ex-amples of the basic process variables are temperatures,pressures, liquid levels, flow rates, and feed compositions.

Plant-product quality indices are usually monitoredby the process engineer to ennnsure proper operationof the plant. Measurements of these indices are rarelyavailable on-line and are usually obtained by laboratorysample analyses. Examples of these indices are viscosity,melt viscosity, density, copolymer composition distribu-tion, molecular weight distribution, melt index, copoly-mer chain sequence distribution, stereoregularity, parti-cle size distribution, porosity, surface area, and degree ofchain branching.

End-product quality indices, often referred to as cus-tomer specifications or end-use properties, quantify thequality of the final product. These indices are usually“abstract” (to the plant process engineer), and theirrelations to the plant-product quality indices are com-plex and not well-understood. In many cases, the re-lations are known qualitatively on the basis of experi-ence. It is important also to note that in cases wherethe relationships between these end-use properties andthe plant-product quality indices are known, they arenot “one-to-one”. The end-product quality indices arerarely measured off-line in the plant because the mea-surements usually cannot be made until the final poly-mer product is formulated and used. Furthermore, manyof these end-use properties (such as “softness”, “block-iness”, and “color”) are “categorical” but not quantifi-able in numerical form at the present time. Examples ofthe end-product quality indices (customer specifications)are adhesive strength, impact strength, hardness, elasticmodulus, flow properties (film blowing, molding, etc.)strength, stress crack resistance, color, clarity, corrosionresistance, abrasion resistance, density, temperature sta-bility, plasticity uptake, spray drying characteristics, andcoating and adhesion properties. More examples of theend-product quality indices can be found in (Nunes et al.,1982; Ray, 1986; Dimitratos et al., 1994).

One of the greatest difficulties in achieving quality con-trol of polymer end-products is our poor understand-ing of the quantitative relationship between (a) the end-product quality indices and (b) the plant-product qual-ity indices and the basic plant variables. The actualcustomer specifications are in terms of the end-productquality indices. Since the quantitative relationship is theleast understood area in polymerization reaction engi-neering, it is very hard to calculate the values of plant-product quality indices that corresponds to the actualcustomer specifications.

Mathematical Modeling

A major objective of polymerization reaction engineer-ing has been to understand how reaction mechanism, thephysical transport phenomena (e.g. mass and heat trans-

fer, mixing), reactor type and operating conditions affectthe plant-product quality indices. As discussed in (Ray,1991), various chemical and physical phenomena occur-ring in a polymer reactor can be classified into the fol-lowing three levels of modeling:

1. Microscale chemical kinetic modeling: Polymer re-actions occur at the microscale. If the elementaryreaction steps of a polymerization mechanism areknown, the distributions can be calculated in termsof the kinetic rate constants and the concentrationof the reactants. The available mathematical mod-els are statistical or are based on detailed speciesconservation methods. The most powerful approachto modeling polymerization kinetics is the detailedspecies balance method. Using the conservationlaws of mass, one can derive an infinite set of equa-tions for the species present in the reaction mixture.

2. Mesoscale physical/transport modeling: At thisscale, interphase heat and mass transfer, intraphaseheat and mass transfer, interphase equilibrium, mi-cromixing, polymer particle size distribution, andparticle morphology play important roles and fur-ther influence the polymer properties. For example,diffusion-controlled free-radical polymerizations aremanifestations of mesoscale mass transfer phenom-ena. For a comprehensive list of results available inthis area, the reader can refer to the excellent reviewpaper by (Kiparissides, 1996).

3. Macroscale dynamic reactor modeling: At themacroscale, one has to deal with the developmentof models describing the macromixing phenomenain the reactor, the overall mass and energy bal-ances, particle population balances, the heat andmass transfer from the reactor as well as the reactordynamics and control.

Population Balance Model

Population balance model descriptions have found a widerange of application in distributed process systems in-cluding crystallization, precipitation, and polymeriza-tion. An excellent treatment of the theoretical aspects ofthe subject is given in (Ramkrishna, 2000). In this paper,we focus on the application of population balance mod-els to a specific sub-class of polymerization systems—particle size distributions in an emulsion system. Withinthis class, there are two general categories of behaviors:zero-one and pseudo-bulk systems. When conditions aresuch that the rate of radical-radical bimolecular termi-nation within a latex particle is extremely fast relativeto the rate of radical entry into particles, evolution ofthe latex particle size distribution can be modeled as azero-one system (Gilbert, 1995). This model considerslatex particles containing either zero or one radical at agiven instant. The reasoning behind this model is thata particle will flip between two states, the zero and one


radical states, each time a radical enters the particle (orexits).

Latex particle size, monomer type and concentration,are among several key factors which strongly influencewhether a system approaches zero-one kinetics. For ex-ample, termination of radicals within small particles israpid because diffusion distance of the radical reactioncenters is small. Moreover, radical entry rates, accord-ing to the Smoluchowski relation, ke = 4πrsNADw (rs

is swollen particle radius) decrease with decreasing par-ticle size. In fact, many systems (styrene for example)approach zero-one kinetics during early stages of particlenucleation and growth when the size of particles is small.

To model a zero-one system, the particle size popula-tion is divided into a population containing zero radicals,n0(r) and a population containing one radical, n1(r).The one radical population is further divided into a pop-ulation containing a polymer radical, n1(r), which wouldnot readily diffuse out of the particle due to its size, anda population containing a monomer radical formed fromchain transfer reactions, nm

1 (r), which presumably canreadily exit particles.

The population, n1(r)dr, represents the moles(ornumber) of polymer particles per liter of water withunswollen particle radii between r and r + dr at timet. Population balance equations for a batch reactor aregiven by:

∂n0(r, t)

∂t= ρ(r) [n1(r) + nm

1 (r)− n0(r)] + ko(r) · nm1 (r)

+

∫ r/21/3

rnuc

r2B(r − r′, r′)

(r3 − r′3)2/3

[n0(r

′)n0(r − r′)+

n1(r′)n1(r

′ − r)]dr′

− n0(r)

∫ r∞

rnuc

B(r, r′)[n0(r

′) + n1(r′)

]dr′

∂n1(r, t)

∂t= ρinit(r) · n0(r)

− ρ(r) · n1(r)− ktrCpn1(r) + kpCpnm1

+

∫ (r3−r3nuc)1/3

rnuc

r2B(r − r′, r′)

(r3 − r′3)2/3n0(r

′)n1(r − r′)dr′

− n1(r)

∫ r∞

rnuc

B(r, r′)[n0(r

′) + n1(r′)

]dr′

− ∂

∂r[G(r) · n1(r)] +

[kw

p,jcrit−1CW [IMjcrit−1]

+

jcrit−1∑i=z

kem,iCmicelle [IMi]

]δ(r − rnuc)

Taking the first equation as an example, the terms pre-multiplied by ρ(r) represent radical entry into particles,the next term represents radical desorption from parti-cles, and the integral terms represent coagulation. The

second equation has an additional term that representsnew particle formation by micellular and homogeneousnucleation mechanisms.

Pseudo-bulk systems are characterized by slow radicaltermination within the particles relative to rapid entryof polymer radicals and re-entry of exited monomer rad-icals. In such systems, particles can contain more thanone radical at a given instant. Moreover, particles withzero, 1, 2, . . . radicals, switch identities(number of rad-icals) so rapidly that the evolution of the particle sizedistribution can be described by a single type of par-ticle with an average number of radicals, n(r). Again,latex particle size, monomer type and concentration arefactors which strongly influence whether a system ap-proaches pseudo-bulk kinetics. Specifically, larger parti-cle sizes increase radical entry rates and decrease radi-cal termination and desorption rates, all of which favorpseudo-bulk kinetics.

The gel effect also decreases radical termination rates.This phenomenon is operative in many polymer systemswhen monomer concentration in particles is low rela-tive to polymer concentration due to monomer depletion.Monomer depleted conditions often occur at the end ofa batch when particle sizes are large. Therefore, thegel effect often coincides with large particle size and canbe an additional factor which pushes a system towardspseudo-bulk kinetics.

A particle size distribution model for a pseudo-bulksystem is given by:

∂n(r, t)∂t

=12

∫ r−rnuc

rnuc

B(r − r′, r′)n(r′)n(r − r′)dr′

− n(r)∫ r∞

rnuc

B(r, r′)n(r′)dr′ − ∂

∂r[G(r) · n(r)]

+[kw

p,jcrit−1CW [IMjcrit−1]

+jcrit−1∑

i=z

kem,iCmicelle [IMi]]δ(r − rnuc)

where particle growth rate is given by

G(r) =kpCpn(r)ρpwm

4πr2Na

kp is the propagation rate constant, Cp the monomerconcentration in polymer particles, and n(r) the averageradical concentration.

In summary, at least for a batch polymerization, zero-one kinetics is expected to be operative at early stagesof polymerization when particle size is small and parti-cles are rich in monomer whereas pseudo-bulk kinetics isfavored in the latter stages of polymerization when par-ticle size is large and the gel effect is strong. Of courseneither of these models treats the more complicated in-termediate case wherein particles can contain 0, 1, 2, and


5 10 15 20 25 300

5

10

15

20

25

30Coagulation Effect, 5.0×10−3 persulfate, 3.0×10−2 SDS, 30% styrene

Particle radius / nm

Par

ticle

Num

ber

Den

sity

No CoagulationCoagulation

Figure 1: Simulation of zero-one model with andwithout coagulation 10 minutes after inception ofpolymerization.

3 radicals, for example, but for model based control pur-poses, we anticipate that the zero-one and pseudo-bulkmodels may be combined in a manner that adequatelyhandles the intermediate case as well. Details of boththese models can be found in (Gilbert, 1995).

Coagulation Coefficients. Coagulation is an ex-tremely important phenomena in emulsion polymeriza-tion, having a large impact on latex particle numbersand particle size distributions. Actually, colloidalparticles are thermodynamically unstable. Surfacetension between particle and bulk phase dictates thatfree energy decreases upon particle coagulation due todecreased surface area (Atkins, 1978):

dG = γdA

dG < 0 for dA < 0

Colloidal stability is a consequence of kinetics. The elec-trostatic charge on the surface of a surfactant stabilizedcolloidal particle represents a significant energy barrierto coagulation between two approaching particles. If thisbarrier is large, very few particles will have sufficient ki-netic energy to exceed the barrier and coagulate. Quan-titatively, this potential energy barrier can be calculatedusing DLVO theory. This is based on calculating the to-tal potential energy of interaction between two particlesas the sum of a van der Waals attractive potential andan electrostatic repulsion potential (Ottewill, 1982). De-tails can be found in (Coen et al., 1998b) and the final

result is

B(ri, rj) =2kBT

3ηWij

(2 +

rsi

rsj+

rsj

rsi

)Wij =

rsi + rsj

4κrsirsjexp

(Φmax

kBT

)where Wij is the Fuchs stability ratio and Φmax the maxi-mum potential energy with respect to particle separationdistance. One important characteristic of coagulation co-efficients is that values increase exponentially as particlesize is decreased. Also, for the smallest particles, thecoagulation coefficient is nearly independent of the ra-dius of the other particles with which the small particlescoagulate. This feature was exploited for determiningcoagulation coefficients in our simulations by approxi-mating the coagulation coefficients as depending only onthe radius of the smaller of any two given particles co-agulating. Figure 1 is a comparison of two simulatedparticle size distributions under identical operation con-ditions but with and without coagulation respectively.This figure highlights the influence that coagulation hason particle numbers and distributions during particle nu-cleation.

Calculation of coagulation coefficients in an on-linecontrol application is problematic because calculationof Φmax is an extremum problem over particle separa-tion distance and therefore, would be an embedded op-timization problem. Another limitation of the DLVOtheory is it does not account for shear effects which canmarkedly alter charge distributions surrounding particlesand cause greatly accelerated coagulation rates. Theseproblems are still open issues and feasible solutions willlikely involve significant empiricism.

Numerical Solution

Approaches to solving population balance equationsfound in the literature can be generally classified intoone of three distinct methods. Orthogonal collocation onfixed and moving finite elements has been used by severalauthors (Dafniotis, 1996; Rawlings and Ray, 1988a,b) tosolve population balance models. This moving finite ele-ment method overcomes some of the numerical instabil-ities and inaccuracies associated with more conventionaltechniques such as finite differences.

Dafniotis (1996) describes the numerical problems as-sociated with the solution of hyperbolic PDE’s in termsof wave theory. If the desired solution is expressed interms of a Fourier series, the behavior of a solution canbe examined by examining individual components of theseries, i.e., sine-cosine waves that propagate with specificphase and amplitude. Numerical operators such as finitedifferences, do not preserve the correct phase and am-plitude. Particularly, errors associated with the phaseare sometimes observed as spurious oscillations; referredto as dispersion. Also, when the amplitude of the nu-merical wave is damped relative to the exact solution


wave, discontinuities (for example boundary conditions)are smeared; referred to as numerical diffusion. Finally,hyperbolic partial differential equations often describethe propagation of near-shocks, or sharp wave frontswhich require adequate resolution in the region of theshock.

In Dafniotis (1996), a moving finite elementmethod(MFEM) is presented for solving the popu-lation balance equations for emulsion polymerizationand to address some of the inherent numerical problemsmentioned above. This method is based on the MFEMdeveloped by Sereno and coworkers (1991).

A less sophisticated but much easier method to set upis the finite difference method. Here, partial derivativesin the population balance equations are approximated byfinite differences. Gilbert (1997) has applied this methodto modeling particle size distributions for emulsion poly-merization systems.

A third method for solving population balance equa-tions is sometimes referred to as methods of classes.Here, the distribution is discretized into classes of par-ticles defined by finite particle size intervals. Mathe-matically, this involves transforming the partial differ-ential equations from a differential to integrated formover small intervals. The presumed advantages of thisapproach include transformation of integral terms intomore easily evaluated summation terms, elimination ofpartial differential terms with respect to particle sizeby forcing the discretization grid to move with parti-cle growth rate, and the ability to coarsen the grid andstill preserve key properties of the distribution such asmoments (Kumar and Ramkrishna, 1996, 1997).

State Estimation

The inadequacy of frequent measurements related to theplant-product quality indices in polymer processes, hasmotivated the use of state estimators in controlling andmonitoring the indices. The availability of sufficiently-accurate, first-principles, mathematical models for manypolymerization reactors has made possible the develop-ment of the state estimators/observers. An estimator,which is designed on the basis of the process model, es-timates unmeasured process variables from current andpast process measurements. State estimators have alsobeen used for sensor/plant fault detection and data rec-onciliation.

A major characteristic of polymerization reactors istheir complex nonlinear behavior. Phenomena such asmultiple steady states in continuouns stirred tank reac-tors, parametric sensitivity, and limit cycles are man-ifestations of the complex nonlinearity. Thus, reliablestate estimation in polymerization reactors requires non-linear models that can capture the complex nonlinearbehavior. Motivated by the need for nonlinear state es-timation, since the 1970s nonlinear state estimators have

been used for polymerization reactors (Jo and Bankoff,1996; Schuler and Suzhen, 1985; Ellis et al., 1988; Ade-bekun and Schork, 1989; Kim and Choi, 1991; Kozuband MacGregor, 1992; van Dootingh et al., 1992; Ogun-naike, 1994; Robertson et al., 1993; Liotta et al., 1997;Tatiraju et al., 1998a, 1999). In most of these studies, ex-tended Kalman filters (EKF’s) have been used for stateestimation.

Multi-Rate State Estimation

In polymerization reactors, most of essential measure-ments related to plant-product quality indices, such asthe leading moments of a MWD obtained by a gel perme-ation chromatograph (GPC), are available at low sam-pling rates and with considerable time-delays. On theother hand, measurements of basic plant variables suchas temperatures, pressures, and densities are usuallyavailable at high sampling rates and with almost no de-lays. Because the plant product quality indices are usu-ally not observable from the frequent, delay-free mea-surements alone, one has to design a multi-rate estima-tor/observer (i.e. one that uses both the frequent andinfrequent measurements), to provide reliable estimatesof the states, especially in the presence of model-plantmismatch and measurement noise. Multi-rate state es-timation in polymerization processes has received con-siderable attention (Elicabe and Meira, 1988; Ellis et al.,1988; Dimitratos et al., 1989; Kim and Choi, 1991; Ogun-naike, 1994; Liotta et al., 1997; Mutha et al., 1997; Tati-raju et al., 1998a, 1999). Multi-rate EKF’s have beenused in most of these studies. For example, Ellis et al.(1988) used a multi-rate EKF to estimate the unmea-surable process states continuously from the frequentlyavailable measurements of temperature and density andthe infrequent and delayed measurements of the averagemolecular weights (obtained by a gel permeation chro-matograph). Mutha et al. (1997) proposed the use ofa fixed-lag smoothing algorithm for multi-rate state es-timation in a polymerization reactor. Tatiraju et al.(1998a, 1999) developed a method of multi-rate nonlin-ear state estimation and applied it to a solution polymer-ization reactor with fast measurements of reactor tem-perature, jacket temperature and density, and slow mea-surements of the zeroth, first and second moments of thepolymer molecular weight distribution.

Inferential Control of Polymerization Re-actors

An inferential control system has been defined conven-tionally as one that requires an estimated or inferredvalue of a controlled output to calculate the value of amanipulated input (Joseph and Brosilow, 1978a,b; Se-borg et al., 1989; Marlin, 1995). Inferential control hasapplication in processes in which (a) measurement of acontrolled variable is not available frequently enough or


quickly enough to be used for feedback control, and (b)there are readily-available process measurements fromwhich the value of the controlled variable can be in-ferred or estimated. The design of an inferential con-trol system consists of two steps: (i) the synthesis of acontroller assuming that all the controlled outputs aremeasured readily, and (ii) the design of an estimator toestimate the controlled outputs that are not measuredreadily. An inferential control system should ensure zerooffset for all the controlled outputs. One of the indus-tries that has benefitted greatly from inferential controlis the polymer industry, where frequent measurementsof even plant-product quality indices are rare. Extensivereviews of recent advances in inferential control can befound in (Doyle III, 1998; Soroush, 1998b).

Multi-Rate Control

In the polymer industry, there are many processeswherein the choice of sampling rate is limited by theavailability of the output measurement. For example,composition analyzers such as gas chromatographs havea cycle time of say 5 to 10 minutes compared to a desiredcontrol interval of say 0.1 to 1 minute. If the controlinterval is increased to match the availability of mea-surements then control performance deteriorates signif-icantly. In addition to the slow measurements (whichare available at different low sampling rates and are de-layed), there are usually process variables such as tem-perature and pressure that can be measured at highsampling rates and with almost no time delays, lead-ing to multi-rate control problems. Successful recentimplementations of multi-rate control on polymerizationprocesses include (Ellis et al., 1994; Ogunnaike, 1994;Ohshima et al., 1994; Sriniwas et al., 1995; Crowley andChoi, 1996; Niemiec and Kravaris, 1997; Tatiraju et al.,1998b).

In the polymer industry, the problem of multi-ratecontrol has been addressed by the following control meth-ods:

• Cascade control

• Decentralized control

• State-estimator-based control

Cascade control systems have been used successfully tocontrol essential variables whose measurements are notavailable frequently. The slave controller regulates a setof basic plant variables and adjust the manipulated in-puts of process, while the master controller regulates theessential variables (usually plant-product quality indices)whose measurements are infrequent and delayed and cal-culates the set-points of the slave controller. While theinner loop is executed at the high rate at which the basicplant variables are measured, the outer loop is executedat the low rate at which the essential variables are mea-sured (whenever these measurements are available). An

advantage of this multi-rate control structure is its con-trol system integrity in the face of any unforeseeable fur-ther delay in the essential slow measurements; whetherthe slow measurements arrive or not, the inner loop isalways in place (Lee and Morari, 1990; Lee et al., 1992).

Decentralized control systems have been used in thepolymer industry to control process variables whose mea-surements are available at different rates. As in everydecentralized control system design, first the manipu-lated inputs and the controlled outputs should be paired.A single-input single-output controller is then designedfor each pair. Each of the SISO controllers is executed(takes action) at the rate at which the measurements ofthe corresponding controlled output are available. Anadvantage of these multi-rate control structures is alsoits control system integrity in the face of any unforesee-able further delay in the slow measurements; whetherthe slow measurements arrive or not, the “fast” loopsare always in place.

State-estimator-based multi-rate control systems in-clude a state estimator which estimates frequently allstate variables of the process from the available, fast andslow measurements. The frequent measurements and es-timates are then fed to a single-rate controller as if theprocess has only single-rate measurements. In contrastto the first two classes of the multi-rate control systemsthat can be non-model-based, the last class of the multi-rate systems have to be model-based (since they includeestimators).

Batch Control Issues

Model-based Optimization Approaches to BatchPolymerization Control

In the case of batch systems, one can formulate a clas-sical optimal control problem in an effort to control theendpoint properties of the batch. In a number of stud-ies, this is implemented in a receding horizon framework,yielding a so-called Model Predictive Controller (MPC).MPC utilizes a process model to compute a future open-loop control sequence which optimizes an objective func-tion, given past and current information of the system.The first control move is implemented and the optimiza-tion problem is re-solved at the next sampling time asupdated information becomes available.

Applications of MPC to semi-batch polymerizationsystems include (Russell et al., 1997), where linear MPCwas applied to a Nylon system using empirical modelsfor quality control. The primary modification to theMPC algorithm was the use of a shrinking horizon, orig-inally proposed in (Joseph and Hanratty, 1993). A sim-ilar formulation of MPC was adopted by Georgakis andco-workers (Liotta et al., 1997). In their work, a nonlin-ear formulation was proposed; however, they employeda “least-squares”-like analytical solution to the uncon-strained problem. Another notable citation is the work


in (Ettedgui et al., 1997), where a fed-batch reactor isstudied (in simulation) for the application of nonlinearmodel-based estimation and predictive control. In thatcase, the sequential solution and optimization techniqueof (Wright and Edgar, 1994) was employed.

In general, MPC is posed as an on-line optimizationproblem, typically requiring the solution of a constrainedlinear, quadratic, or nonlinear programming problem.The generalized optimization problem considered can beexpressed as:

minu

[maxΦi(x, u)] i = 1, nobj

subject to

x = f(x, u)gi(u) ≤ 0 i = 1, nineq

hi(u) = 0 i = 1, neq

Here, the vector u contains the values for the sequenceof manipulated variable moves over the batch cycle (e.g.,surfactant feed), nobj denotes the number of terms con-sidered in the objective function, and the constraintsdescribe the process model and corresponding operat-ing constraints. Several forms of the objective functioncan be considered. The following 1-norm type objectivecould be considered:

Φ =NE∑i=1

NJ∑j=1

∣∣nij − ntargetij

∣∣nscale

Here, nscale is a factor used to scale the objective func-tion values. A 2-norm objective can also be formulated;

Φ =NE∑i=1

NJ∑j=1

[(nij − ntarget

ij )nscale

]2

Many interesting polymer products have correspondingdistributions that are multi-modal in nature. These canbe produced, for example, by multiple surfactant addi-tions, sufficiently separated in time. Therefore, a poten-tially effective objective definition is to actually definemultiple objective functions, each tied to a particulardistribution mode, and perform a min-max optimization.For bimodal distributions, the objective can be expressedas follows:

Φ1 =N1∑i=1

NJ∑j=1

[(nij − ntarget

ij )nscale

]2

Φ2 =NE∑

i=N1+1

NJ∑j=1

[(nij − ntarget

ij )nscale

]2

minui,i=1,11

Φ = max (Φ1,Φ2)

Here, N1 represents the number of finite elements span-ning the lower particle size mode of the distribution.

Case Study I: Control of an EmulsionPolymerization Reactor

The control of particle size distribution as an end-objective in emulsion polymerization control is well mo-tivated in industrial practice, and has been well docu-mented in the literature (see, for example, the recentreview by Congalidis and Richards, 1998). The au-thors pointed out that “on-line control not only of aver-age polymer properties but also of polymer distributionssuch as the particle size. . . will become important”. Theycontinue: “The instrumentation and control methodolo-gies that will need be deployed to meet these needs isa challenging and vibrant area of investigation for aca-demic researchers and industrial practitioners alike.”

In this section, we present a case study for the opti-mal control of particle size distribution in a semi-batchstyrene polymerization reactor. The isothermal model ofCoen et al. (1998a) incorporates current theory on par-ticle nucleation, growth and coalescence mechanisms forstyrene at 50◦C and serves as the modeling basis for thisstudy. This model is a zero-one model, referring to theassumption of instantaneous bimolecular radical termi-nation in polymer particles, which gives rise to a mixedpopulation of particles containing either one or zero rad-icals. Parts of the model that deviate from (Coen et al.,1998a) are described in detail in (Crowley et al., 2000).

The basic control problem is defined as the achieve-ment of a target PSD at the end of the batch. For thisstudy, we consider the manipulation of both surfactantfeed rate (and/or concentration), and initiator feed rate.On-line measurement of the full PSD, for example bylight scattering, is assumed for this study.

Open-loop Optimization of PSD

Two distinct variables related to surfactant concentra-tions were optimized to match a target PSD. The firstvariable considered is surfactant feed rate. Specifically,the optimization routine calculates a sequence of 10 sur-factant feed flow rates(zero-order hold), each with a sam-ple hold time of 3 minutes, up to a final time of 30 min-utes. The target PSD was generated by simulating themodel up to 30 minutes, with values for the 10 surfactantfeed flow rates that yield a bimodal distribution. Con-trol trajectories were calculated by defining an objectivefunction in terms of simulated PSD deviations from a tar-get distribution, and minimizing the objective functionusing the sequential quadratic programming algorithm,FSQP.

In a different approach, we considered use of free sur-factant concentration, rather than surfactant feed rate,as the control variable. The reason behind this choiceis that free surfactant concentration above the cmc isthe essential driving force for particle nucleation. Thefree surfactant profiles consist of a sequence of first-orderholds (i.e. piecewise linear), with each hold spanning a 3


0 10 20 300

2

4

6

8x 10

−3

batch time

surf

acta

nt fe

ed r

ate target

suboptimal

0 10 20 300

0.005

0.01

0.015

0.02

0.025

batch timefr

ee s

urfa

ctan

t Con

c. m

ol l−

1

CMC

0 10 20 300

5

10

15

x 1017

batch time

part

icle

num

ber

0 10 20 300

0.5

1

1.5

2

2.5

3x 10

−7

particle radius, nm

part

icle

den

sity

, mol

l−1 n

m−

1

targetsuboptimal

Figure 2: Optimization of surfactant feed rate se-quence for target PSD; cmc= 3×10−3 mol/L, [I]0 =0.01 mol/L, [M ]0 = 2.59 mol/L.

minute interval. As in the previous case, 10 holds wereused to span a 30 minute control horizon. For this formu-lation of the optimization problem, the decision variablesare 11 free surfactant concentration nodes, spaced at 3minute intervals. Free surfactant concentration valuesbetween any two neighboring nodes are calculated sim-ply by linear interpolation between the two nodal values.Linked in this way, the nodes form a continuous, thoughnon-smooth, free surfactant profile.

The first optimization case involves computation of asequence of surfactant feed flow rates which drive the sys-tem to a target PSD. The target distribution was gen-erated by fixing the initial surfactant concentration attime= 0 and simulating the system out to 30 minuteswith one large surfactant addition at 21 minutes. Thisaddition is shown in Figure 2. Surfactant addition at 21minutes produces a shoulder in the distribution at theend of the 30 minute simulation, with a peak height lo-cated at a particle size of about 20 nm. The solid staircase profile for surfactant feed rate in this figure repre-sents the optimization solution. The 2-norm objectivewas used in this case. Although the optimizer appearsto drive the simulated PSD to the target, as seen in thisfigure, the final free surfactant concentration for the op-timized case is much larger than that of the target case.With disparity in surfactant levels at 30 minutes, thetwo distributions would diverge beyond this time. Thesedifferences can be improved through the use of inputblocking (Crowley et al., 2000).

The optimal solution is quite sensitive to the choice ofinitial conditions, as well as the particular blocking for-mulation. Consequently, a second control strategy wasconsidered. Because particle nucleation is dependent onthe free surfactant level relative to the critical micelle

concentration, a free surfactant trajectory is more closelyrelated to the physical phenomena than surfactant feedflow rates. Of course, ultimately the feed flow rates aremanipulated. A free surfactant trajectory represents ahigher control level in a cascade configuration, with feedrates being the lowest level control variable. A usefulproperty arises from this form. An intuitive initializationfor optimization computations is to set all free surfactantconcentrations in the control vector to the critical micelleconcentration. Sensitivity of particle nucleation rate tosurfactant perturbations is greatest at the CMC.

Local minima are often a cause of poor performance ingradient based optimization. For a bimodal distribution,it is possible to obtain suboptimal PSD solutions that aretrapped in a local minimum because surfactant pertur-bations which decrease target offset for one of the modesmay increase offset for the other mode. As an attempt topartially decouple this kind of interaction, we formulateda multi-function objective wherein each function is tiedto a particular mode of the distribution. As describedabove, the resulting optimization is a min-max problemwith two objective functions for a bimodal distribution.Figure 3 depicts the result of this case. When comparedto the PSD offsets seen using the 2-norm (Crowley et al.,2000), the target offset for the min-max optimization islower. Another advantage of this approach is that theoptimization is “well-behaved” in the sense that offset ofeach mode from the target is balanced due to the struc-ture of the objective function.

Batch-to-Batch Studies

In this section, a model refinement optimization tech-nique is described using information from historicalbatch data of emulsion polymerization in conjunctionwith a fundamental first-principles model to determinethe operating conditions necessary for a desired productquality.

The desired product quality must be a grade of prod-uct produced in the same range of operating conditionsused in the historical batch data. As a result of this con-dition, the desired PSD must have a similar character toknown PSDs produced by the process. A fundamentalfirst-principles model of emulsion polymerization whichaccounts for polymer particle nucleation and polymerpropagation exists. While these phenomena are well un-derstood, emulsion polymerization presents a challengingprocess to control due to difficulty in modeling complexbehavior such as particle aggregation and the significantnonlinear behavior involved in particle formation. Themethod described in this section seeks to combine thisfirst-principles model of emulsion polymerization with anMPLS model to find the optimal control input sequenceneeded to achieve a desired product quality in a semi-batch emulsion polymerization reaction. In the experi-ments conducted in this section, the process variables arethe surfactant and initiator feed rate inputs at distinct


0 5 10 15 20 25 300

1

2

3

4

5

6x 10

−3

free

sur

fact

ant,

mol

L−

1

time, min

free surfactanttargetCMC

(a) Free surfactant profiles.

0 10 20 30 400

0.5

1

1.5

2x 10

−8

part

icle

pop

ulat

ion

dens

ity

particle radius, nm

suboptimaltarget

(b) Particle size distributions.

Figure 3: Synthesis of optimal free surfactant profile using min-max objective. Target and suboptimal distributionsshown are at a batch time of 30 minutes. Initial reactant conditions are as follows: cmc= 0.0039, [I]0 = 0.005 mol/L,[M ]0 = 2.59 mol/L.

time intervals, and the product quality is determined bythe particle size distribution (PSD) at the final time ofthe experiment.

The design of inputs for the emulsion polymerizationprocess is performed on an off-line batch-to-batch ba-sis. The hybrid model, which combines a first-principlesmodel with MPLS, is referred to as the design model.The MPLS model is used to approximate the differencebetween the PSD the first-principles model calculatesand the PSD obtained from executing a given input se-quence in an actual semi-batch reactor, thus capturingthe effect of phenomena for which the first-principlesmodel does not take into account. The optimizationwill be performed by minimizing the sum of the squaredresidual error between the PSD yielded by the designmodel and the target PSD. The design model is a func-tion of the states of the system and the control inputs.A calibration set of data from historical batches having aset of operating conditions similar to that of the desiredPSD will be required to begin the optimization proce-dure. The PSDs in the historical batch data and the tar-get PSD are created from a virtual process model whichsimulates the actual semi-batch reaction in a plant. Thisvirtual process model is structurally different from thefirst-principles model in the design model because it ac-counts for aggregation and has four parameter valuesthat are varied from the design model values by 5%.

The hybrid design model is potentially useful becausethe MPLS model will account for the phenomena suchas particle aggregation which are not included in thefirst-principles model within the design model. Further-

more, the MPLS model can be adjusted to account fornoise that may occur during PSD measurement. In ad-dition, the MPLS model calibration set can be refinedonce a significant amount of batches have been designedby using only a subset of historical batches in the MPLSmodel, and including only those batches most recentlydesigned. A pure MPLS model could also be attempted,but important information can be extracted from thefirst-principles model, so one would not want to aban-don it altogether. The first-principles model in the de-sign model could be useful for optimizing control inputsfor new processes with little batch history, and for opti-mizing control inputs for new grades with no batch his-tory. The efficacy of using a first-principles/MPLS hy-brid model in achieving a target PSD is investigated inthe following work.

While MPCA is used to compress information in theprocess variables to low-dimensional spaces that describethe historical batch operation, MPLS reduces both theprocess variables and product quality variables to low di-mensional spaces, and attempts to find a correlation be-tween these low-dimensional spaces (Neogi and Schlags,1998). PLS attempts to maximize covariance, whichmeans that it focuses on the variance of X that is morepredictive for the product quality Y, rather than focus-ing on the variance of X only (Nomikos and MacGregor,1995). In this investigation, the Matlab PLS Toolbox2.0 by Eigenvector, Inc is used to perform PLS analyses.

Before beginning the optimization algorithm, it is nec-essary to generate historic batch data and a target PSDwith the use of the process model. The target PSD must


10 20 30 40 50 60 700

1

2

3

4

5x 10

−8

particle radius (nm)

Par

ticle

num

ber

dens

ity (

mol

/L×n

m) virtual process

design model

Figure 4: PSD for virtual process and design model.

2 4 6 8 10 12 14 16 18 200

1

2

3

batch number

Sca

led

sum

of s

quar

ed d

evia

tions

from

targ

et

PLS using expanding batch history PLS using only 16 most recent batches

Figure 5: Improvement of PSD tracking error overthe course of 20 successive batches.

be created within the same range of operating conditionsused to generate the batch history, so that it is a differ-ent grade of product, within the same product family.The target PSD and historic PSDs are vectors contain-ing the 48 particle number densities yi corresponding to48 discrete particle radii ri . A control input sequenceof length 20 corresponding to the surfactant and initia-tor feed inputs at 10 evenly spaced time steps for a 100minute batch reaction are used to create the PSDs.

The first step in the optimization algorithm is to ob-tain the scaling data and PLS regression matrix for thecurrent batch history (X and Y blocks) in Matlab.

The second step in the optimization is to design aninput sequence using the following hybrid design model:

ydesign(t = tf ) = yfp(x, u, t) + yresid(x, u)

where

yfp(x, u, tf ) =∫ tf

0

f(x, u)dt

In this equation, tf is the duration time of the semi-batch reaction, x and u are vectors containing the statesof the system and the control inputs of the system, re-spectively yresid is an approximation of the residual PSDbetween the process model PSD and the PSD from thefirst-principles model. yresid(x, u)is calculated using theregression matrix calculated with the current batch his-tory. The first iteration of the optimization will requirean initial value for the control input sequence u. The op-timization is performed by making small perturbationsin u for each iteration. The direction of the perturbationsare determined by the Jacobian matrix calculated fromthe previous iteration of the optimization. The size ofthe perturbation in u is fixed prior to optimization. Thefollowing objective function is minimized by the opti-mization procedure to determine control input sequenceu to be used in the next batch:

ny∑i=1

(ytarget

i − ydesigni

)2

The optimized input sequence is then executed in thevirtual process model, which simulates a batch reactionwithin an actual plant. Representative results of thisstudy are shown below in Figures 4 and 5. The firstfigure shows the mismatch between the simulated processand the fundamental model used for PSD control. Theoverall trend along the batch history is mapped in theform of integral squared error in Figure 5. Clearly, thealgorithm “learns” as the batches proceed.

Case Study II: Control of a Solution Poly-merization Reactor

In this section, we present a comparative study ofmulti-rate control of a jacketed polymerization reactorin which free-radical solution polymerization of styrenetakes place. A multi-rate control system consisting of themulti-rate nonlinear state estimator of (Tatiraju et al.,1999) and a mixed error- and state-feedback controller,is used. The performance of the multi-rate nonlinearcontrol system is shown and compared with those of amulti-rate, PI, parallel cascade, control system and amulti-rate, PI, completely decentralized, control system.

Polymerization Process and the Control Problem

The reactor is a 3 m3, jacketed, continuous, stirred tankreactor in which free-radical solution polymerization ofstyrene takes place. The solvent and initiator are ben-zene and azo-bis-iso-butyro-nitrile, respectively. The re-actor has three feed streams: a pure monomer stream


at a volumetric flow rate of Fm, a pure solvent streamat a volumetric flow rate of Fs, and an initiator stream,which includes solvent and initiator, at a volumetric flowrate of Fi. The volume of the reacting mixture inside thereactor is constant.

We use the same dynamic model described in (Tatirajuet al., 1999) to represent the reactor. The model has theform:

dCi

dt= −

[Ft

V+ ki

]Ci +

FiCii

V

dCs

dt= −FtCs

V+

FiCsi+ FsCss

Vdλ0

dt= −Ftλ0

V+ f3(Ci, Cs, Cm, T )

dλ1

dt= −Ftλ1


dλ2

dt= −Ftλ2


dCm

dt= f6(Ci, Cm, T ) +

FmCmm− FtCm

VdT

dt= f7(Ci, Cm, T, Tj) +

Ft(Tin − T )V

dTj

dt= f8(T, Tj) + αQ

(1)

where Ft = Fi + Fm + Fs, V is the volume of the re-acting mixture inside the reactor, Cm is concentrationof the monomer in the reactor outlet stream, Ci is con-centration of the initiator in the reactor outlet stream,Cs is concentration of the solvent in the reactor outletstream, T is the reactor temperature, Tj is the jackettemperature, Q is the rate of heat input to the reac-tor jacket, and λ0, λ1 and λ2 are the zeroth, first, andsecond moments of the MWD of the polymer product,respectively. The functions f1, · · · , f8 and the parametervalues of the reactor model are given in (Tatiraju et al.,1999); for brevity they are not given here. The first-principles mathematical model of the process describedby (1) is used to represent the actual process.

The number-average and weight-average molecularweights of the polymer product (denoted by Mn and Mw

respectively) are related to the moments according to

Mn =λ1

λ0, Mw =

λ2

λ1

The reacting mixture density, reactor temperature(T ), and jacket temperature (Tj) are assumed to be mea-sured on-line once every 30 seconds and with almost notime delays. The monomer conversion (and thereby themonomer concentration, Cm) can be inferred from thedensity measurement, and thus can be calculated on-line. The zeroth, first, and second moments of the MWDof the polymer product are assumed to be measured atsampling periods of 3 hours and with time delays of 1hour. The rate of heat input to the reactor jacket, Q,

and the flow rate of the initiator feed stream, Fi, canbe set arbitrarily on-line within the following ranges:−20 ≤ Q ≤ 50 kJ.s−1 and 0 ≤ Fi ≤ 3.0×10−5 m3.s−1.

The control problem is to maintain the weight-averagemolecular weight of the polymer, Mw, and the reactortemperature, T , at desired values by manipulating therate of heat input to the reactor jacket, Q, and the flowrate of the initiator feed stream, Fi.

Multi-Rate Nonlinear Control System

State Feedback Synthesis. With Mw and T ascontrolled outputs (y1 = Mw and y2 = T ), and Fi andQ as manipulated inputs (u1 = Fi and u2 = Q), relativeorders (degrees) of the process r1 = 2 and r2 = 1, andthe characteristic (decoupling) matrix of the process isgenerically singular. Because of this generic singularity,we request a state feedback that induces two completelydecoupled, 2nd-order, process output responses of theform:

β12d2Mw

dt2+ β11

dMw

dt+ Mw = Mwsp

(2)

β22d2T

dt2+ β21

dT

dt+ T = Tsp (3)

where β12, β11, β22 and β21 are positive adjustablescalar parameters, Mwsp is the weight-average molecu-lar weight set-point, and Tsp is the reactor temperatureset-point. Substituting for the time derivatives from theprocess model in the preceding two equations, we obtaintwo identities of the forms

φ1(x, u1) =1

β12Mwsp

,

φ2(x, u1, u2, u1) =1

β22Tsp

(4)

where x is the vector of the state variables of the reactor.Let us represent the solution for u = [u1 u2]T of theconstrained minimization problem

minu

{[φ1(x, u1)−

Mwsp

β12

]2

+[φ2(x, u1, u2, 0)− Tsp

β22

]2}

subject to

0 ≤ u1 ≤ 3.0× 10−5 m3.s−1

−20 ≤ u2 ≤ 50 kJ.s−1,

by

u = Ψ(x,Mwsp, Tsp) (5)

Using the identities of (4), we add integral action to thestate feedback of (5) (see Soroush, 1998a, for the details),


leading to the mixed error- and state-feedback controller

η1 = η2

η2 = − 1β12

η1 −β11

β12η2 + φ1(x, u1)

η3 = η4

η4 = − 1β22

η2 −β21

β22η4 + φ2(x, u1, u2, 0)

u = Ψ(x, e1 + η1, e2 + η3)

(6)

where e1 = Mwsp−Mw and e2 = Tsp−T . The controller

of (6) has integral action and inherently includes opti-mal windup and directionality compensators (Soroush,1998a).

Multi-Rate State Observer. Application of themulti-rate nonlinear state estimation method describedin (Tatiraju et al., 1999) to this polymerization reactorleads to the following reduced-order, multi-rate, nonlin-ear, state estimator:

z1

z2

z3

z4

z5

=

−[Ft

V+ ki

]Ci +

FiCii

V

−FtCs

V+

FiCsi + FsCss

V

−Ftλ0


−Ftλ1


−Ftλ2


−K

f6(Ci, Cm, T ) + γ1

f7(Ci, Cm, T, Tj) + γ2

f8(T, Tj) + αQ

+ L

λ∗0 − λ0

λ∗1 − λ1

λ∗2 − λ2

(7)

Ci = z1 + K11Cm + K12T + K13Tj

Cs = z2 + K21Cm + K22T + K23Tj

λ0 = z3 + K31Cm + K32T + K33Tj

λ1 = z4 + K41Cm + K42T + K43Tj

λ2 = z5 + K51Cm + K52T + K53Tj

where L = [Lij ] and K = [Kij ] are the estimator gains,

γ1 =FmCmm

− (Fi + Fm + Fs)Cm

V

γ2 =(Fi + Fm + Fs)(Tin − T )

V

λ∗0(t), λ∗1(t) and λ∗2(t) are the predicted present valuesof the infrequent measurable outputs, each of which isobtained by fitting a least-squared-error line to the mostrecent, three measurements of the moment. These linearregressions are always carried out except when only one

measurement of each slow measurable output is avail-able. In this case, the predicted present value of eachslow measurable output is set equal to the single avail-able measurement. The estimator initial conditions arethe same as those in (Tatiraju et al., 1999). The multi-rate state estimator of (7) can be written in the compactform

z = F (z, y, Y ∗, u)x = Q(z, y)

(8)

where Y ∗ = [λ∗0 λ∗1 λ∗2]T and y = [Cm T Tj ]T .

Multi-Rate Nonlinear Control System. The useof the mixed error- and state-feedback controller of (6),together with the multi-rate state estimator of (7), leadsto a multi-rate nonlinear control system of the form:

z = F (z, y, Y ∗, u)η1 = η2

η2 = − 1β12

η1 −β11

β12η2 + φ1(x, u1)

η3 = η4

η4 = − 1β22

η2 −β21

β22η4 + φ2(x, u1, u2, 0)

u = Ψ(x, e1 + η1, e2 + η3)x = Q(z, y)

(9)

where e1 = Mwsp − λ2/λ1.

Multi-Rate Cascade and Decentralized ControlSystems

We compare the performance of the multi-rate nonlin-ear control system of (9) with a multi-rate, PI, cascade,control system and a multi-rate, PI, completely decen-tralized, control system.

The multi-rate, PI, cascade, control system consistsof two PI controllers. The master PI controller regulatesthe weight-average molecular weight by manipulating thereactor temperature set-point. The master controller isexecuted once every three hours, since the average molec-ular weight measurements are available at that low rate.The slave PI controller regulates the reactor temperatureby manipulating the rate of heat input to the reactorjacket. The slave controller is executed at a much fasterrate (once every 30 seconds). The control system has theform

ξ1(k + 1) =

[1− ∆t

τI1

]ξ1(k) +

∆t

kc1

[Tsp(k)− Tspss ]

ξ2(t) = − 1

τI2

ξ2(t) +1

kc2

[Q(t)−Qss]

Tsp(k) = satT

{Tspss + kc1

[e1(k) +

1

τI1

ξ1(k)

]}Q(t) = satQ

{Qss + kc2

[e2(t) +

1

τI2

ξ2(t)

]}(10)


with ξ1(0) = Mw(0), ξ2(0) = T (0), kc1 = −4.0 × 10−5,kc2 = 1.0×10−4, τI1 = 1.0×106 s, and τI2 = 1.0×106 s,where e1(k) = Mwsp(t) − Mw(k), e2(t) = Tsp(t) − T (t),and ∆t = 1.08× 104 s.

The multi-rate, PI, completely decentralized, controlsystem consists of two completely-decentralized PI con-trollers. One of the PI controllers regulates the weight-average molecular weight by manipulating the flow rateof the initiator feed stream, and the other regulates thereactor temperature by manipulating the rate of heat in-put to the reactor jacket. While the first PI controlleris executed once every three hours (sampling rate of theaverage molecular weight), the second controller is exe-cuted once every 30 seconds (sampling rate of the reactortemperature). The control system has the form

ξ1(k + 1) =

[1− ∆t

τI1

]ξ1(k) +

∆t

kc1

[Fi(k)− Fiss ]

Fi(k) = satFi

{Fiss + kc1

[e1(k) +

1

τI1

ξ1(k)

]}ξ2(t) = − 1

τI2

ξ2(t) +1

kc2

[Q(t)−Qss] ,

Q(t) = satQ

{Qss + kc2

[e2(t) +

1

τI2

ξ2(t)

]}(11)

with ξ1(0) = Mw(0), ξ2(0) = T (0), kc1 = −5.0 × 10−7,kc2 = 2.5×10−4, τI1 = 1.0×107 s, and τI2 = 1.0×106 s.

Each of the preceding PI control systems includestwo PI controllers with windup compensators (Soroush,1998a).

Simulation Results

The performance of the multi-rate nonlinear controlleris evaluated by simulating the following two cases: (a)when there is no measurement noise or model-plant mis-match (nominal case); and (b) when there are mea-surement noise and model-plant mismatch (non-nominalcase). For each case, the performance of the multi-ratenonlinear control (NC) system is compared with those ofthe cascade control (CC) system and the decentralizedcontrol (DC) system. Measurement noise is introducedby adding a white noise signal to each of the momentscalculated by the process model. Each of the noise sig-nals is a 10% deviation from the value of the moment atthat particular time. Model error is simulated by addinga 10% error in the propagation step rate constant. Thefollowing values of tunable parameters are used for thenonlinear controller:

K11 = 1.0, K12 = 0.0, K13 = 1.0

Ki1 = Ki2 = Ki3 = 0.0, i = 2, . . . , 5,

Li1 = Li2 = Li3 = 0.0, i = 1, . . . , 2

Lij = 0.0, i 6= j, i = 3, . . . , 5; j = 1, . . . , 3

310

315

320

325

330

Tem

pera

ture

, K

T (NC)

T (DC)

T (CC)

Tsp

0.0E+00

4.0E+04

8.0E+04

1.2E+05

Mw

, kg.

kmol

-1

Mw (NC)

Mw (NC)Mw (DC)

Mw (CC)Mwsp

(20)

0

20

40

Hea

t Inp

ut, k

J.s-

1

NC

DC

CC

0.00

0.01

0.02

0.03

Inle

t In

itiat

or F

low

Rat

e, l

.s-1

0 10 20 30 40 50

Time, h

NC

DC

CC

Figure 6: Controlled outputs and manipulated inputsunder the multi-rate control systems (nominal case).

The temperature set-point Tsp = 324.0 K, and theweight-average molecular weight set-point Mwsp

=80,000 kg.kmol−1.

Figure 6 shows the profiles of the controlled outputsand the manipulated inputs for the nominal case underthe three multi-rate control systems. The solid line rep-resents the set point. For the Mw graph, the dashedline stands for the continuous estimates of Mw obtainedby using the estimator of the multi-rate NC system,while the bullets stand for the infrequent and delayed“measurements” of the average-molecular weight. Forthis case, L31 = 1.0 × 10−4, L42 = 1.0 × 10−9, andL53 = 1.0 × 10−8. The benefit of using the multi-rate


310

315

320

325

330

Tem

pera

ture

, K

T (NC)

T (DC)

T (CC)

Tsp

0.0E+00

4.0E+04

8.0E+04

1.2E+05

Mw

, kg

.km

ol-1

Mw (NC)

Mw (NC)Mw (DC)

Mw (CC)Mwsp

(20)

0

20

40

Hea

t Inp

ut, k

J.s-

1

NC

DC

CC

0.00

0.01

0.02

0.03

Inle

t In

itiat

or F

low

Rat

e, l

.s-1

0 10 20 30 40 50

Time, h

NC

DC

CC

Figure 7: Controlled outputs and manipulated inputsunder the multi-rate control systems (non-nominalcase).

NC system is obvious in Figure 6. Not only under theNC system are the process output responses faster, butalso they have smaller overshoots in the controlled vari-ables. Under the CC and DC systems it takes morethan 40 hours to take both the temperature and averagemolecular weight to their respective set-points, while un-der the NC system it takes less than 5 hours. The CCsystem regulates Mw only by manipulating Q, but theDC system regulates both Mw and T by manipulatingQ and Fi. From the manipulated input graphs we cansee that the NC system makes the most optimal use ofmanipulated variables. Under the CC and DC systems

the manipulated inputs hardly reach the constraints, butstill the CC and DC systems cannot be tuned to be moreaggressive because the overshoots start increasing. Fig-ure 7 depicts the profiles of controlled outputs and ma-nipulated inputs for the non-nominal case. For this case,L31 = 1.0×10−4, L42 = 2.0×10−7, and L53 = 9.5×10−6.Again, the NC system shows a much superior perfor-mance as compared to the two PI control systems.

Conclusions

In this paper a brief survery of the recent advances thathave led to improvement in polymer product qualitywas presented. Further improvement in polymer prod-uct quality requires solving challenging design, control,and monitoring problems that still exist in polymer pro-cesses.

Lack of sufficient controllability is a barrier to betterproduct quality control in some polymer processes. Inmany polymer processes, better product quality requiresminimizing/maximizing several product quality indicessimultaneously. This multi-objective requirement mayresult in narrow ranges of process trajectories, puttinga premium on the controllability of the process. For in-stance, in coatings, the product’s composition, molec-ular weight, and particle size distributions should bemaintained simultaneously in limited ranges to ensurethe coating has a desired level of film formation, filmstrength, and gloss.

In many batch processes, product quality suffers frombatch-to-batch inconsistency. There is a trend towardsproducts with specific performance, which have highervalue to a formulator or end-user. Furthermore, manyof the current processes result in products with a broadinter-batch variance of molecular and physical charac-teristics, which in turn result in broad variance of per-formance. Blending of these batches usually lowers theaverage performance of the product lots. Segregation of“off-spec” product results in higher costs which may notbe transferable to the customer.

Our understanding of the relationships among the ba-sic plant variables, plant-product quality indices, andend-product quality indices is mostly empirical and qual-itative. Polymer product development in the absence ofqualitative relationships between the recipe, process andthe final performance requires long times. Experimen-tal techniques have been used to develop relationshipsthat hold for the range of the experimental parametersstudied. These products and processes therefore do notreadily lend themselves to optimization, either in termsof productivity or reduction in variance. Having the abil-ity to develop these relationships on a more fundamen-tal basis will allow products to be developed in shortertimes.


Acknowledgments

Financial support from the National Science Foundationthrough the grant CTS-9703278 is gratefully acknowl-edged by M. Soroush. The first author would like to ac-knowledge the financial support of NSF (BES 9896061)as well as the valuable contributions to this work by TimCrowley, Charles Immanuel, and Chris Harrison.

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Control of Product Quality in Polymerization Processes