Optimizing Control and State Estimation

in a Tubular Polymerization Reactor ⋆

R. Hashemi ∗, S. Engell ∗

∗ Dynamics and Operations Group, Technische Universitat Dortmund,Germany, (e-mail: {Reza.Hashemi,

Sebastian.Engell}@bci.tu-dortmund.de).

Abstract: In this contribution we study the application of non-linear model-based optimizingcontrol to the continuous polymerization of acrylic acid in a tubular reactor. Multiple sideinjections of monomer along the reactor and the reactor temperature which is controlled viacooling/heating jackets provide the means to control the product quantity and the productquality. The homo-polymerization reaction investigated here, can be modeled by a system ofeight pdes which are transformed to an ode system. For this purpose, the spatial domain ofthe pdes is discretized using the weighted essentially non-oscillatory scheme (WENO). Thismethod avoids the need for a very fine discretization grid while reproducing steep fronts well.The controller employs this model and aims at maximizing the product throughput whilesatisfying the product quality constraints. Four temperature measurements along the reactorand a molecular weight measurement, derived from a viscosity measurement, at the outlet ofthe reactor are assumed to be available. A particle filter is implemented that provides the initialcondition of the prediction model. Simulation results show that the controller is robust againstprocess and measurement noise and can meet the product constraints and increases the productthroughput considerably.

Keywords: Optimizing control, NMPC, state estimation, particle filtering, tubular polymerizat-ion reactor.

1. INTRODUCTION

Thanks to the progress in computer hardware and optimiz-ing algorithms, optimization-based controllers can now beapplied to industrial processing units. One of the mainadvantages of these controllers is the ability to imposeconstraints on both manipulated and controlled variables.In this work we investigate the application of such a con-troller to the continuous free radical homo-polymerizationof acrylic acid in a tubular reactor. This process is abenchmark for the transfer of batch polymerizations tocontinuous operation that was investigated in the Euro-pean Project F3 Factory. Due to the plug flow charac-teristic between the inputs and the outputs, the reactorsystem reacts with large time delays to the changes of theinput flow rates. Four side injections of monomer along thereactor can be used to control the product quantity andproduct quality. The uniform jacket temperature is set viaa thermostat and offers an additional control input. Modelpredictive control (MPC) is the suitable choice for control-ling such a multi-input, delayed and constrained system.Standard implementations of MPC aim at tracking somepredefined set points and penalize the violations of thestates or outputs from these set-points. In this work, wefollow the idea of optimizing control and maximize the

⋆ The research leading to these results was funded by the EuropeanCommission in the project F3 Factory (FP7−NMP/2007 − 2013)under the grant agreement n◦ 228867 and by the ERC AdvancedInvestigator Grant MOBOCON (FP7/2012− 2017) under the grantagreement n◦ 291458.

product throughput directly while keeping the productquality constraints (Engell 2007). Four temperature mea-surements at the middle of each segment of reactor and ameasurement of the molecular weight at the outlet of thereactor are assumed to be available. Compared to the num-ber of states, the available measurements are scarce. Theinitial conditions of the prediction model are estimated bya particle filter. In this method a set of weighted samplesrepresents the required posterior density function and theestimation is performed based on this set. Particle filtersemploy the full non-linear model of the process and do notencounter the problems that result from linearization, asthe extended Kalman filter.The rest of this paper is organized as follows: in thesecond section, we discuss the process model, its derivationand numerical methods to solve the pde model of thereactor system. The third section is dedicated to particlefiltering and the evaluation of its performance for oursystem. The simulation results of the optimizing controllerare discussed in section four. Finally conclusions and anoutlook on future work are presented.

2. SIMULATION OF THE SYSTEM

The process which is investigated in this work is thecontinuous production of poly acrylic acid (PAA) in atubular reactor with multiple side injections of monomerand initiator. The reactor consists of eight tubular reactormodules which are connected in series. It has a totallength of four meters. The reactor is equipped with staticmixers to ensure an efficient mixing of the reactants. A

Preprints of the 19th World CongressThe International Federation of Automatic ControlCape Town, South Africa. August 24-29, 2014

Copyright © 2014 IFAC 4873

jacket is used to control the reactor temperature and itstemperature is set uniformly via a thermostat. Figure 1shows the P&ID diagram of this reactor. The reactor isdivided into four segments, each consisting of two modules,and a temperature sensor at the middle of each segment isinstalled. The internal volume of the first two segments is45ml where segments three and four are larger and eachhas a volume of 130ml. A measurement of the viscositymeasurement, is available at the outlet of the reactorwhich can be used to compute the average molecularweight Mw. Based on the energy and component balances

Module1

Module2 TIT1Solvent/

Initiator

Monomer

Module3

Module4 TIT2

Module5

Module6 TIT3

Module7

Module8 TIT4

Product

QIMw

u2

u4

u3

u1

Fig. 1: Flow sheet of the modular continuous polymeriza-tion plant. (u1, u2, u3 and u4): side injections of monomer.The temperature of the reactor is controlled via the cool-ing/heating jacket (T). (T1, T2, T3 and T4): temperaturemeasurements, Mw: molecular weight measurement (de-rived from a viscosity measurement).

and assuming perfect mixing in the radial direction andnegligible axial dispersion, a rigorous model of the reactorwas set up. The free radical polymerization of acrylic acidis modeled by the terminal model approach. The resultingnon-linear partial differential equations (pde) are shown inequations 1 to 8 (Hashemi et al. 2013). The temperaturedependent rate coefficients kd(T ), kp(T ) and ktc(T ) aremodeled by an Arrhenius approach and the method ofmoments is applied to model the polymer chain lengthdistribution (Crowley et al. 1997).

∂cI

∂t= −u

∂cI

∂z+

∂

∂z

(

Dax∂cI

∂z

)

− kdcI (1)

∂cM

∂t= −u

∂cM

∂z+

∂

∂z

(

Dax∂cM

∂z

)

− kpλ0cM (2)

∂λ0

∂t= −u

∂λ0

∂z+

∂

∂z

(

Dax∂λ0

∂z

)

+ 2fkdcI − 2ktcλ02 (3)

∂λ1

∂t= −u

∂λ1

∂z+ 2fkdcI + kpλ0cM − ktcλ0λ1 (4)

∂λ2

∂t= −u

∂λ2

∂z+ 2fkdcI + kpcM (λ0 + 2λ1)− ktcλ0λ2 (5)

∂µ1

∂t= −u

∂µ1

∂z+ ktcλ0λ1 (6)

∂µ2

∂t= −u

∂µ2

∂z+ ktcλ0λ2 + ktcλ1

2 (7)

∂T

∂t= −u

∂T

∂z+

∂

∂z

(

λ∂T

∂z

)

2k

R(Tjac − T )λ0cM (−∆hp) (8)

The weight average molecular weight of the producedpolymer (Mw) results from the moments as:

Mw =µ2 + λ2

µ1 + λ1

. (9)

Numerical methods have to be used to solve the pdemodel of the reactor system. Several methods have beenintroduced for this purpose in the literature. We employthe method of lines here. The main idea of this method isto replace all derivatives in the pde system by algebraicapproximations except of one. Usually all spatial deriva-tives (dimensions) are approximated and do not appearexplicitly in the model any more. Thus the pdes are con-verted to a system of odes. This method assigns an odeto every pde at each discretization point. Well-establishedmethods for solving odes can then be applied to find theapproximate solution of the original system of pdes. Thestandard choice to approximate the spatial derivatives isto use finite differences. Two major numerical issues resultwhen implementing this method for processes with steepfronts, as the one considered here: A first order approx-imation of the spatial derivatives with finite differencesresults in numerical diffusion, i.e. smoothing of the fronts.This problem can be reduced by employing a fine dis-cretization grid or by using higher order finite differences.Higher order approximations, however, result in numericaloscillations and the use of more discretization points doesnot reduce the numerical oscillations (Schiesser et al.2009). Therefor, a first order approximation with a largenumber of discretization points must be used when finitedifferences are employed, leading to large computationtimes. Figure 2 illustrates these numerical problems. Analternative method is to use non-linear approximationsof the spatial derivatives. The class of such non-linearapproximation methods is called high resolution methodsand includes flux limiters and weighted essentially non-oscillatory (WENO) methods (Bouaswaig et al. 2009).The latter one has been applied in this work and it wasobserved that both numerical problems can be avoidedwithout the need for a fine discretization grid. WENO

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5x 106

µ 2 [mol

/m3 ]

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5x 106

Time [s]

µ 2 [mol

/m3 ]

5000 points (simulation time: 204.84 [s])2000 points (simulation time: 48.43 [s])800 points (simulation time: 17.05 [s])400 points (simulation time: 10.08 [s])200 points (simulation time: 4.48 [s])

Fig. 2: Second moment of inactive polymers (µ2) at thereactor outlet. (a) First order finite differences are used toapproximate the spatial domain in different discretizationgirds. This method results in numerical diffusion which canbe avoided partly by fine discretization grids 1 . (b) Highorder finite differences are used to approximate the spatialdomain. This method results in numerical oscillationswhich can not be cured by a finer discretization grid.

schemes implement a dynamic set of stencils and computea low order approximating polynomial in each of them to1 Intel(R) Core(TM) i7-2600 CPU @ 3.40GHz, 24GB RAM

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4874

compute the numerical flux. Each polynomial receives aweight which is determined based on a local smoothnessindicator. The polynomials corresponding to the stencilswhich have a large gradient receive a zero weight giving anon-oscillatory solution at the sharp fronts. The obtainedpolynomials are combined in a non-linear convex fashion,resulting in higher order polynomials at the smooth partsof the solution and in an upwind spatial discretization atthe sharp fronts which avoids interpolation and providesthe necessary dissipation for shock capturing (Borgeset al. 2008). The spatial derivatives then are approxi-mated by a first order finite difference of the computednumerical flux. Different variants of WENO schemes devisedifferent smoothness indicators and determine the weightsof polynomials in different ways. In this work, we haveimplemented the WENO-Z scheme proposed by (Borgeset al. 2008). The details about the determination of thesmoothness indicators and weights can be found in theirwork. The second derivatives of the spatial domain thatappear in the process model can be rewritten by means oftwo first order derivatives using auxiliary variables. Thenthe current WENO scheme can be used to compute thesecond derivatives. (Liu et al. 2011) have proposed anew WENO scheme to compute the second derivativesdirectly. Their scheme has a higher resolution and we haveimplemented their method in our model.The advantage of using the WENO scheme can be seenin figure 3. A Matlab implementation of the model withthe WENO scheme with 200 discretization points needsonly 66.71 seconds and can reach a similar accuracy asthe finite differences with 5000 discretization points whichtakes 204.84 seconds 2 . For the rest of this paper, we willuse the WENO scheme with a uniform discretization gridof 200 points. The obtained ode model, which includes1600 states, is solved using the CVode from the Matlabinterface of SUNDIALS (sundialsTB).

0 1000 2000 3000 4000 5000 60000.5

1

1.5

2

2.5

3

3.5

4

4.5x 106

Time [s]

µ 2 [mol

/m3 ]

Finite Difference with 5000 discretization pointsWENO with 200 discretization points

Fig. 3: Second moment of inactive polymers (µ2) at theoutlet of the reactor.

3. PARTICLE FILTERS

State estimation is an important part of every controlscheme. While the prediction of the behavior of a givensystem requires the knowledge of the initial states, usuallynot all states of the system can be measured or somequantities can not be measured at the requested frequency.

2 The stated simulation times are measured when the Matlabintegrator ode15s has been used. The designed NMPC, uses a mex

implementation of the model and employs CVode from sundialsTB

which increases the simulation speed with a factor of 100.

Also in case of the availability of the suitable sensors,the measurements are always noisy. The extended Kalmanfilter (EKF) is the most widely used state estimation tech-nique for non-linear systems in the process industries. Itemploys a prediction step using the non-linear model anda correction step using the available measurements. TheKalman gain matrix is computed based on a linearizationof the system around the previous estimate (Simon 2006).The extended Kalman filter usually provides a satisfactoryperformance if the non-linearities of the system are nottoo severe otherwise it can show poor performance orbecome unstable. The performance of EKF is stronglydependent on its tuning, a task which can be quite difficultfor large systems. We have implemented the extendedKalman filter for this reactor and observed that it indeedbecame unstable for quite reasonable tuning parameters(results are not shown here). In this work, we apply theparticle filtering (PF). Particle filters are recursive samplebased state estimation algorithms which employ the fullmodel of the system. They start with a given numberof initial guesses of the a priori states (particles). Basedon the probability of the current measurements resultingfrom these particles as a posteriori states, the algorithmassigns a weight to each of them. Finally an a posterioriset of particles is chosen and the mean value of this setis considered as the final estimation of the current state.Particle filters do not need assumptions about the type ofmodel of the system (linear or nonlinear) nor about the dis-tribution of the assumed measurement and process noise.Any suitable system model or noise probability densityfunction can be used. A generic particle filtering algorithmcan be described in the following steps (Arulampalamet al. 2002):

(0) The process and observation equations are given asfollows:

xk = fk(xk−1, uk−1) + ωk−1,

yk = hk(xk) + υk,

x(0) = x0 + ω0.

(10)

where ω and υ are the process and measurement noiseand are white noise signals with known probabilitydensity functions (pdf) and mutually independent.We assumed here that the measurements of the tem-perature and of the molecular weight contain Gaus-sian noise with standard deviations of 0.2 [K] and1 [kg/mol]. The process noise also obeys a Gaussiandistribution and is assumed to have a standard devi-ation of about 0.1% of the range of the states.

(i) From the pdf of the initial states N random particlesare generated. In our case N = 1000. Each particlereceives an importance weight equal to 1

N. The dis-

tribution of the monomer concentrations along thereactor for the initial particles (estimates) and for onesimulation run is shown in figure 4.

(ii) The particles that were generated in the previous stepare propagated via the system model (eq.10).

(iii) The algorithm updates the weights of the particles(for a scalar measurement for simplicity) as follows:

wik = wi∗

k−1 ·p(yk|x

ik) · p(x

ik|x

ik−1

)

q(xik|x

ik−1

, yk)(11)

where q(.) is the importance density function. “k”refers to the time instants and “i” is the counter of

19th IFAC World CongressCape Town, South Africa. August 24-29, 2014

4875

the particles. The superscript “*” denotes the scaledweights. If more than one state is measured, the jointlikelihoods have to be computed.

(iv) The weights are scaled so that the sum of all weightsequals one.

ωi∗k =

ωik

∑N

j=1ωjk

(12)

(v) Particle filters can suffer from the degeneracy problemwhich means that after a few iterations, it is possiblethat all particles except a few have a very smallweight. In such a situation, the set of particles will loseits diversity and it can not represent the likelihooddensity function of the states anymore. A suitablemeasure of the degeneracy of the set of particles isthe effective sample size (Neff ). An estimation of theeffective sample size is:

Neff =1

∑N

j=1(ωi∗

k )2(13)

If Neff is less than a given value, resampling isperformed. A small effective sample size indicatessevere degeneracy.

(vi) A set of a posteriori particles is chosen. This processis called resampling and there are various methods forit. The four most often applied methods are multino-mial, residual, stratified and systematic resampling.In this work, we have tested the multinomial andsystematic methods and observed better results fromthe systematic approach. (Arulampalam et al. 2002)and (Doucet et al. 2008) have also reported a bet-ter performance of the systematic method over theother methods. The systematic resampling methodcan be summarized in the following way: SampleU1 ∼ U [0, 1

N] and define Ui = U1+

i−1

Nfor i = 2, ..., N ,

then set N in =

∣

∣

∣

{

Uj :∑j−1

k=1W k

n ≤ Uj ≤∑i

k=1W k

n

}∣

∣

∣.

The selected set of the a posteriori particles is:

π ={

∑N

i=1

Nin

Nδ(x− xi)

}

(Doucet et al. 2008).

Resampling can drop some of the particle and selectsome of them twice or more.

(vii) Any statistical measure of the a posteriori set can becomputed. Usually the mean value is of interest.

Several variants of particle filters exist which vary in thechoice of the importance sampling density function or inthe resampling step. In this work we have implemented theSequential Importance Resampling (SIR) filter which doesthe resampling at every step and defines the importancesampling density function as follows:

q(xik|x

ik−1, yk) = p(xk|x

ik−1) (14)

Since the resampling is performed at every time instant,the weights of the particles are (Arulampalam et al. 2002):

ωik = p(yk|x

ik) (15)

The SIR algorithm is the most widely used version ofparticle filtering since the importance weights can be easilyevaluated. However it is sensitive to outliers. A simulationresult obtained with this method is presented in figure 5.No model-plant mismatch is considered in this simulation.The distribution of the true initial monomer concentrationalong the reactor and the distribution of the monomerconcentration of all 1000 initial particles are shown infigure 4. The manipulated variables for this simulation are

shown in figure 6. For this simulation all five availablemeasurements have been used. The RMS of the estimationerror of the molecular weight at the outlet (Mw) andmonomer concentration (CM ) are 0.0601 and 1.5976. Thechoice of the initial particles has to be in accordancewith the measurement error. Small measurement noiserequires precise initial particles. The reason for this canbe explained as follows: The algorithm propagates theparticles and computes the predictions of the variableswhich are measured. Then the importance weights areassigned based on the proximity of these values to themeasured ones. If the initial particles are chosen too faraway from the true states, they produce values whichare far from the measured ones and the algorithm willdisqualify all of them.

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

800

1000

1200

1400

length of reactor [m]

CM

[mol

/m3 ]

pariclestrue

Fig. 4: Distribution of the monomer concentrations of theinitial particles and of the true concentration along thereactor.

70

75

80

Mw

[kg/

mol

]

0 0.5 1 1.5 2 2.5 3 3.5 4x 10

4

100

150

200

250

Time [s]

CM

[mol

/m3 ]

trueestimated

truemeasuredestimated

Fig. 5: (a) True, measured and estimated molecular weightat the outlet of the reactor, (b) true and estimatedmonomer concentration at the outlet of the reactor.

0 0.5 1 1.5 2 2.5 3 3.5 4x 10

4

0

50

100

Sid

e in

ject

ions

of m

onom

er (

u 1, u2, u

3, u4)

[%]

Time [s]

0 0.5 1 1.5 2 2.5 3 3.5 4x 10

4

360

380

400

T [K

]

u1

u2

u3

u4 T

Fig. 6: Manipulated variables for the simulation of thestate estimation shown in figure 5.

19th IFAC World CongressCape Town, South Africa. August 24-29, 2014

4876

4. OPTIMIZING CONTROL

4.1 Non-linear Model Predictive Control

The reactor in this work is controlled by a model-basedoptimizing controller i.e. the control moves are optimizedover a finite horizon considering the predicted response ofthe plant and the constraints on the product properties.Standard implementations of NMPC employ tracking costfunctions and penalize the violations of the outputs orstates from a given set point (Findeisen et al. 2004). Inthis work we follow the idea of online optimizing controland aim at maximizing the product throughput under thequality constraints (Engell 2007). For the reactor shownin figure 1, the product throughput is maximized whenthe sum of all flow rates of the monomer side injections aremaximized, but the quality constraints have to be fulfilled.The residual monomer (CM ) is an important productquality indicator and is estimated at the outlet of thereactor. The second product quality constraint is imposedon the molecular weight that is measured indirectly atthe outlet of the reactor (Mw). These constraints can beformulated as hard constraints which is computationallymore demanding but then the quality constraints arestrictly met. In this work we deal with these constraintsas soft constraints and penalize their violations from theconsidered bounds in the cost function. This is based onthe assumption that the produced polymer will be soldin larger quantities hence a considerable degree of mixingoccurs after the production and smoothens short-termconstraint violations. Following this idea, the cost functionis formulated as follows:

minu1k,u2k,u3k,u4k,Tk

Φ(x(tk), u(tk), Nc, Np) (16a)

Φ = −Φ1 + γΦ2 = −Φ1 + γ (Φ21 +Φ22 +Φ23) (16b)

Φ1 =

j=k+Np∑

j=k

(u1 + u2 + u3 + u4) (16c)

Φ21 =

j=k+Np∑

j=k

(max(CMj − CMu, 0))2 (16d)

Φ22 =

j=k+Np∑

j=k

(max(Mwj −Mwu, 0))2 (16e)

Φ23 =

j=k+Np∑

j=k

(min(Mwj −Mwl, 0))2 (16f)

where the subscripts l and u denote the lower and upperbounds of the corresponding variables. Np and Nc are thelength of prediction and control horizons. The residencetime of the reactor for the nominal flow rate is about 2600seconds which implies that the prediction horizon mustbe at least of a similar length. However, the controllermanipulates the total flow rates inside the reactor, causinga shift in the states which postpones or expedites theeffect of a specific control move depending on the previouscontrol move. In order to take this behavior into accountand to ensure a stable behavior, we use a predictionhorizon of 6000 seconds which in this case is a quasi-infinitehorizon. The control horizon is set to one to reduce thenumber of the decision variables. The sampling time is100 seconds. The first part of the cost (Φ1) maximizes the

product throughput while the second part (Φ2) minimizesthe violations of the controlled variables from the givenbounds. γ is a tuning parameter and determines therelative importance of Φ1 and Φ2 in the computed cost.From a numerical point of view, this formulation of thecost function is easier to solve because it does not includeexplicit constraints. The reactor temperature (T) doesnot enter into the optimization problem directly but itis manipulated to fulfill the constraints and to enable anincrease of the throughput. The optimization problem issolved in a sequential approach using the SNOPT solverfrom the TOMLAB package. In each sampling instant, theinitial condition of the model is estimated employing theparticle filter that was introduced in the previous section.

4.2 Simulation Results

In this section we show the application of the opti-mizing controller to the tubular polymerization reactorwhile a particle filter is used to estimate the initialstates of the prediction model. For the base design case(no regulation), the produced polymer has a molecularweight of 77.5 [kg/mol] and a monomer content of about131 [mol/m3] at the reactor outlet. The following choicesof the bounds in (16d− 16f) were made:

CMu = 135, Mwl = 75, Mwu = 80

We assume that the reactor initially is in steady stateand produces the requested polymer with the base-designinputs. At t = 0, the controller is switched on. Consid-ering the normalization of the constructing elements ofthe cost function (Φ1 and Φ2), we have set the tuningparameter γ to 0.02. The computed manipulated variablesby the optimizing controller are shown in figure 7. Com-pared to the base design case, the controller increases thethroughput by 49.0%. As shown in figure 8, the estimatedvariables converge to the true values and the controlledvariables are kept within the specified bounds. The smallviolations from the bounds in the transition phase resultfrom the application of soft constraints. The controllerdrives the reactor to an operation at a higher temperatureas expected and increases the side injections of monomerconsiderably. This system has a very complex behavior anddoes not exhibit a monotonically increasing or decreasingstep response. This is the main reason for the fluctuationof the controlled variables before reaching a steady state.The implemented code of the controller has been partiallyparallelized to apply parallel computation. For the processand measurement noise, the same assumptions as for thesimulation in section 3 were made here. The computationtimes of the optimizing controller and of the particle filterfor this simulation are shown in figure 9.

5. CONCLUSION

We demonstrated the application of model-based optimiz-ing control to a challenging reactor control example. Theinvestigated process is the continuous polymerization ofacrylic acid in a tubular reactor with multiple side injec-tions of monomer. The spatial domain of the pde modelwas discretized and the spatial derivatives were computedusing the WENO scheme. We implemented an economi-cal cost function which aims directly at maximizing theproduct throughput. The product and quality constraints

19th IFAC World CongressCape Town, South Africa. August 24-29, 2014

4877

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

4

380

385

390

395

T [K

]

Time [s]

0

50

100

150

Sid

e in

ject

ions

of m

onom

er (

u 1, u2, u

3, u4)

[%]

u1

u2

u3

u4 Total Flow

Fig. 7: Manipulated variables computed by the optimizingcontroller.

60

80

100

120

140

CM

(@

out

let)

[mol

/m3 ]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

4

72

74

76

78

80

82

84

Time [s]

Mw

(@

out

let)

[kg/

mol

]

truemeasuredestimated

trueestimated

Fig. 8: Controlled variables of the system under optimizingcontroller. A particle filter is employed to provide theinitial conditions of the prediction model.

0

200

400

600

800

Com

puta

tion

time

of o

ptim

izin

g co

ntro

l [s

]

0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

Iteration number

Com

puta

tion

time

of p

artic

le fi

lter

[s]

Fig. 9: (a) Computation times of the optimizing controller.(b) Computation times of the implemented particle filterwith 1000 particles. In this simulation the parallel compu-tation facility of Matlab with 12 workers has been used.

are considered as soft constraints and enter into the costfunction. In order to provide the initial conditions of theprediction model a state estimator must be implemented.State estimation in tubular reactors is a difficult taskbecause compared to the number of the states, the avail-able measurements are scarce. In this work we assumedfour temperature measurements along the reactor and onemolecular weight measurement at the reactor outlet. Thestate estimation was performed using a particle filteringalgorithm. The advantage of particle filtering over theextended Kalman filter is that it utilizes the full non-linearmodel of the process and does not encounter the problemsresulting from linearization. Usually the computation timeof the particle filters is considered as their drawback. How-

ever, as the propagation of each particle is independentfrom the other particles, parallel computation can be ap-plied. With the particle filter estimator, the controller canmeet the specifications of the controlled variables and in-creases the product throughput considerably. A drawbackof the optimizing controller is its long computation times.Reduced or simplified models could be used to reduce thesimulation time of the model. Consideration of parametricinaccuracies besides the measurement and process noiseswill be a further extension of this work.

ACKNOWLEDGEMENTS

We would like to gratefully acknowledge the contributionof Daniel Kohlmann from the Process Dynamics andOperations Group at TU Dortmund to the modeling ofthe process investigated in this work. This work wassupported by the European Union within the project F3Fast Flexible Future Factory and by the ERC AdvancedInvestigator Grant MOBOCON awarded to the secondauthor.

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19th IFAC World CongressCape Town, South Africa. August 24-29, 2014

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