product property and production rate control of styrene polymerization

Product property and production rate control of styrenepolymerization

Vinay Prasada, Matthias Schleyb, Louis P. Russoc, B. Wayne Bequettea,*aHoward P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

bBASF, GermanycExxonMobil Chemical Company, Baytown, Texas

Received 20 December 2000; received in revised form 14 August 2001; accepted 27 September 2001

Abstract

A multivariable multi-rate nonlinear model predictive control (NMPC) strategy is applied to styrene polymerization. The NMPCalgorithm incorporates a multi-rate Extended Kalman Filter (EKF) to handle state variable and parameter estimation. A fundamentalmodel is developed for the styrene polymerization CSTR, and control of polymer properties such as number average molecular weight

(NAMW) and polydispersity is considered. These properties characterize the final polymer distribution and are strong indicators of thepolymer qualities of interest. Production rate control is also demonstrated. Temperature measurements are available frequently whilelaboratory measurements of concentration and molecular weight distribution are available infrequently with substantial time delays

between sampling and analysis. Observability analysis of the augmented system provides guidelines for the design of the augmenteddisturbance model for use in estimation using the multi-rate EKF. The observability analysis links measurement sets and corre-sponding observable disturbance models, and shows that measurements of moments of the polymer distribution are essential forgood estimation and control. The CSTR is operated at an open-loop unstable steady state. Control simulations are performed

under conditions of plant-model structural mismatch and in the presence of parameter uncertainty and disturbances, and the pro-posed multi-rate NMPC algorithm is shown to provide superior performance compared to linear multi-rate and nonlinear single-rate MPC algorithms. The major contributions of this work are the development of the multi-rate estimator and the measurement

design study based on the observability analysis. # 2002 Published by Elsevier Science Ltd.

Keywords: Multi-rate estimation; Nonlinear model predictive control (NMPC); Styrene polymerization

1. Motivation

Polymerization reactors present challenging controlapplications. Highly exothermic reactions, changingprocess conditions, unknown reaction kinetics and highviscosity often lead to difficult operating conditions.Furthermore, there frequently are strict requirements onproduct quality, underlining the need for tight control.Aiming at high flexibility of these plants, CSTR’s can beoperated at various operating conditions to meet differentproduct requirements.Model-based controllers have been shown to be

effective for chemical processes, especially for processeswith input-output interactions, and with large dead

times and constraints on input and output variables. Aspolymer properties are generally not measurable on-line,model based approaches are required for the control ofthese properties. Furthermore, it is known that changingoperating conditions can have serious impact on controlperformance. Sophisticated multivariable controllers aretherefore needed to ensure the safe operation of thesereaction processes.This paper deals with the control of a styrene poly-

merization continuous stirred tank reactor. We considerthe control of polymer properties such as number averagemolecular weight (NAMW) and polydispersity (PD) orbranching. This requires a model/estimator that can esti-mate additional states, such as molecular weightmoments, even when a limited number of measurements isavailable. If the molecular weight moments are accuratelyestimated, the NAMW and PD can be readily inferredand controlled. Since quality related measurements are

0959-1524/02/$ - see front matter # 2002 Published by Elsevier Science Ltd.

PI I : S0959-1524(01 )00044-0

Journal of Process Control 12 (2002) 353–372

www.elsevier.com/locate/jprocont

* Corresponding author. Tel.: +1-518-276-6683; fax: +1-518-276-

4030.

E-mail address: [email protected] (B. W. Bequette).

usually obtained infrequently and with delays associatedwith a laboratory analysis, a multi-rate estimator basedon using a combination of extended Kalman filters(EKFs) is developed. The multi-rate estimator is coupledwith a nonlinear model predictive control algorithm thatapplies local linearization to predict the effect of futurecontrol moves and solve the constrained nonlinear controlproblem.This paper has the following structure: In Section 2, we

review related work in nonlinear model predictive control,state estimation and styrene polymerization control. Thestyrene reactor modeling and control problem is pre-sented in detail in Section 3. The multi-rate EKF-basedNMPC strategy is explained in Section 4, the observa-bility results for the system are presented in Section 5and simulation results in Section 6. A summary andperspective on future research concludes the paper.

2. Background

2.1. Styrene polymerization reactor control

Synthetic polymer production exceeds one millionmetric tons per year worldwide, with a variety of polymersproduced, ranging from high volume material at roughlyone dollar per kilogram to high-value-added specialtymaterials at several thousand dollars per kilogram[13,25]. The actual desired product property values arerarely measured accurately on-line, so the developmentof model-based estimation and control techniques isconsidered very important; for an example industrialapplication to a terpolymer process [20].Most styrene polymers are produced by batch sus-

pension or continuous mass polymerization processes.Polystyrene mass polymerization is homogeneous(reactions take place in a single phase viscous fluid) andoften a combination of different reactors is used to meetproduct requirements. In general, CSTRs are moreappropriate during the earlier polymerization stageswhere viscosities are relatively low. This corresponds tolow conversions but high reaction rates. A single CSTRoperating at such conversions may not be economicaldue to low space-time yields. A discussion of differentreactors for the polymerization of styrene can be foundin Ref. [10]. Only CSTRs are considered further in thispaper.Hidalgo and Brosilow [11] implement a low-order (4-

state) nonlinear model predictive control strategy toregulate temperature in a styrene polymerization reac-tor. Their procedure is based on using two manipulatedvariables (cooling jacket and monomer flowrates) tocontrol a single output variable (temperature). Doyle etal. [4] apply Volterra model-based MPC to a poly-merization reaction in a CSTR where the controller is alinear controller augmented by an auxiliary loop of

nonlinear corrections. They seek to control the numberaverage molecular weight of the polymer. Russo andBequette [27] perform a complete operability analysis ofthe reactor studied in [11] and show the existence ofinfeasible operating regions where a reactor temperaturecannot be maintained by any single input–single outputcontrol strategy. They also add three moment equationsto understand the effect of process design on importantpolymer properties.Gazi et al. [9] develop a systematic approach to verify

the stability and performance of nonlinear closed-loopsystems under parametric and model structure uncer-tainty, using the styrene polymerization example from[11]. They use a strategy where four manipulated variables(flow rates of initiator, monomer, solvent and coolingwater) are used to regulate the reactor temperature andthe number average molecular weight.

2.2. Nonlinear model predictive control

MPC techniques involving the solution of the non-linear equations for the control move optimization havebeen presented by a number of researchers [6,12,32].Example applications have often focused on systems withoutput multiplicities [32], where processes can move fromopen-loop stable to open-loop unstable operatingpoints, and input multiplicities [34], where the processgain can change sign (and right-half-plane zeros, causing‘‘inverse response’’ behavior, can form) as the operatingconditions change. In [34] Sistu and Bequette show theconnection between right-half-plane zeros and inputmultiplicity, and in [35] they use a numerical Lyapunovapproach to develop regions of attraction; in both papersan isothermal CSTR with series-parallel reactions is usedas an example system.Garcia [7] presented one of the first extensions of linear

model predictive control to include a nonlinear model.At time step k the nonlinear model was integrated fromthe state values at time step k�1, using the known inputapplied over the previous time step. The standard‘‘additive disturbance’’ term was used to compensate forthe difference between the measured states and themodel predictions of those states. The future states werepredicted by integrating the nonlinear model assumingno change in process input, then using the linear stepresponse model (based on state values at the currenttime step) to optimize the set of future control movesusing the standard quadratic dynamic matrix control(QDMC) procedure. This approach is henceforthknown as the NL-QDMC (nonlinear quadratic dynamicmatrix control) technique.In [33] Sistu et al. compared the computation time of

several different methods of solving the nonlinear modeland found that the NL-QDMC approach was significantlyfaster than methods solving the nonlinear model, andhad virtually identical closed-loop performance. Gattu

354 V. Prasad et al. / Journal of Process Control 12 (2002) 353–372

and Zafiriou [8] extended NL-QDMC to include asteady-state Kalman filter, which allows control ofunstable systems. Lee and Ricker [14] noted some lim-itations to this approach and incorporated an extendedKalman filter for state and disturbance estimation.A number of studies have involved measurement

horizon, optimization-based methods for state and para-meter estimation [12,26,31]. In [21], Bequette shows howmulti-rate sampling can be included in the estimationhorizon-based approach. In [2] Ramamurthi et al.develop a two-level strategy for state and parameter esti-mation and show a significant computational savingscompared to a single-level optimization-based approach.A general review of nonlinear process control,

including model predictive approaches, is presented in[1]. More recent nonlinear model predictive controltutorials are presented in [15] and [22].

3. Styrene polymerization model

This work will be focused on a single CSTR with acooling jacket as shown in Fig. 1. Two different modelswill be used to describe the behavior of the reactor. Asimplified model which assumes simpler kinetics, constantphysical properties, constant volume and neglects theso-called ‘‘gel effect’’ will be used for the Kalman filterbased NMPC strategy. A more detailed model whichaccounts for varying physical properties and includes thegel effect will be used to represent the ‘‘plant’’. Thisapproach will demonstrate the ability of the model-basedcontrol strategy to handle model structure uncertainty. Inaddition, control simulations are performed with para-meter mismatch and in the presence of disturbances.

3.1. Plant model

In the free radical solution polymerization of styreneconsidered here, the reaction is initiated by azobisiso-

butyronitrile (AIBN). Isopropyl benzene is used as asolvent to keep both conversion and viscosity low. Theplant model used in the simulations is an extension ofone presented in [11] and [27], modified to include theeffects of varying solvent concentration.The complete plant model includes 9 state variable

differential equations and parameters that vary withconcentration and temperature (density, viscosity, heattransfer coefficient, etc.). The kinetic scheme used is verygeneral, and consists of initiation, propagation, chaintransfer and termination steps. The quasi-steady-stateassumption is applied with respect to the initiator andpolymer radical species since their lifetimes are extremelyshort compared to the other system time constants. Theconcentrations of all growing polymer species (of dif-ferent lengths) are lumped together and calculated as asingle concentration. To account for the gel effect, anempirical relationship taken from Duerksen andHamielec [5] was employed to relate viscosity and thereaction rate constants. The gel effect arises when solu-tions viscosity impairs the mutual termination of growingpolymer. This causes an autoacceleration of the reactionfollowed by a drastic increase in monomer conversion.The overall mass balance for the plant is given by

dm

dt¼ m

: in �m: out ð1Þ

where the mass flow rates represent the sum of the massflow rates of initiator, monomer and solvent. Theinitiator mass balance is given by

dmwI

dt¼ m

:I!If �m

: out!I þ VRI ð2Þ

where the reaction rate is given by

RI ¼ MIrI ¼ MIð�kdCIÞ ð3Þ

Similarly, the monomer and solvent balances aregiven by

dmwM

dt¼ mMwMf

�m: outwm þ VRM ð4Þ

dmws

dt¼ m

:S þm

:Ið1� wIfÞ �m

: outwS þ VRS ð5Þ

where the reaction rates of monomer and solvent aregiven by

RM ¼ MMrM ¼ MMð�2fkdCI � ðkp þ kfÞCMCP�Þ ð6Þ

RS ¼ MSrS ¼ MSð�kfsCP �CSÞ ð7Þ

where CP�, the concentration of polymer radical, isfound from the quasi-steady-state assumption to beFig. 1. Solution polymerization of styrene in a CSTR.

V. Prasad et al. / Journal of Process Control 12 (2002) 353–372 355

CP�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fkdCI

ktc þ ktd

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fkdCI

kt

s ð8Þ

The probability of propagation, �, can be defined as

� ¼kpCM

kpCM þ kfsCS þ kfCM þ ktCP�ð9Þ

and will be used to characterize the molecular weightdistribution.All reaction rate constants are given by the Arrhenius

relation:

k ¼ k0e� E

RT ð10Þ

The CSTR energy balance assumes well-mixedness,negligible shaft work from the agitator and negligibleheats of initiation and termination compared to the heatof polymerization. Under highly viscous conditions, theassumptions of negligible shaft work and well-mixed-ness will not be valid. The energy balance is given by

dðmcpTÞ

dt¼ m

: incpTf �m: outcpTþ ð��HrÞVkpCMCP�

�UAðT� TcÞ ð11Þ

The specific heat, cp, and the overall heat transfercoefficient, UA, are assumed to be functions of thereactor temperature alone. The functional relations forcp and U are of the form

cp ¼ a1 þ a2Tþ a3T2 ð12Þ

U ¼ U0ð1� �wPÞ ð13Þ

The jacket energy balance is given by

dTc

dt¼

Q:cðTcf � TcÞ

Vcþ

UA

�ccpcVcðT� TcÞ ð14Þ

where the density and heat capacity of the jacket fluidare assumed to be constant.The method of moments [23] is used to characterize the

polymer property distribution. The kth moment of thenumber chain length distribution (NCLD) is given by

lk ¼X1n¼1

nkPn

while the kth moment of the weight chain length dis-tribution (WCLD) is

lk ¼ MM

X1n¼1

nkþ1Pn

¼ MMlkþ1

where l0 is a measure of the total concentration, and l1represents the total weight of polymer present. Thenumber average molecular weight (NAMW) is given by

NAMW ¼ MMl1l0

ð15Þ

The breadth of the molecular weight distribution canbe characterized by the polydispersity (PD), which isgiven by

PD ¼l2l0l21

ð16Þ

The differential equations describing the first threemoments of the polymer distribution are obtained bysuitable modification of expressions derived by Ray [23].They are of the form

dl0dt

¼ ðkfskfCSCM þ ktdCP�Þ�CP �þ

1

2ktcCP �

2 �Q:out

Vl0 ð17Þ

dl1dt

¼CP

1� �ðkfsCS þ kfCM þ ktdCP�Þð2�� 2Þ þ ktcCP��

�Q:out

Vl1

ð18Þ

dl2dt

¼CP�

ð1� �Þ2ðkfsCS þ kfCM þ ktdCP�Þð4�� 3�2 þ �3Þ�

þ ktcCP �ð2þ �Þ��Q:out

Vl2 ð19Þ

The values of the parameters used in the above equa-tions are given in Table 1.

3.2. Simplified model

The plant model described in the previous subsectionis used in the control simulations to act as the ‘‘true’’plant. A simplified model is used in the estimation andnonlinear MPC algorithms. In addition to the quasi-steady-state approximation, the long-chain hypothesis ismade and the gel effect is neglected. This means that themonomer reaction rate can be approximated by

rM ¼ �kpCMCP� ð20Þ


The chain transfer reactions are assumed negligiblewith respect to propagation and termination reactions,and the probability of propagation is simplified to

� ¼kpCM

kpCM þ ktCP�ð21Þ

All chain transfer reactions are neglected in the sol-vent balance equation, and density, specific heat andheat transfer coefficient changes with temperature areneglected. A constant volume assumption thus results.With the assumptions made above the initiator balancecan be written as

dCI

dt¼

Q:ICIf �Q

:outCI

V� kdCI ð22Þ

In the same manner the monomer material balancecan be simplified to

dCM

dt¼

Q:MCMf �Q

:outCM

V� kpCMCP ð23Þ

As all chain transfer reactions are neglected the sol-vent balance equation simplifies to

dCS

dt¼

Q:SCSf þQ

:ICIf �Q

:outCS

Vð24Þ

The energy balance can be simplified to

dT

dt¼

Q:outðTf � TÞ

Vþð��HrÞ

�~c~pkpCMCP

�U�A

�~c~pVðT� TcÞ

ð25Þ

where U~ ; c~p and �~ are the constant values for the heattransfer coefficient, heat capacity and density assumedby the simplified model. The jacket balance is

dTc

dt¼

Q:cðTcf � TcÞ

Vcþ

U~A

�~cc~pcVcðT� TcÞ ð26Þ

The moment equations are also simplified even thoughchain transfer can not be completely neglected. With � �

1 the following moment equations are obtained:

dl0dt

¼ ðkfsCSkfCMÞ�CP �þ1

2ktcCP �

2 �Q:out

Vl0 ð27Þ

dl1dt

¼C2

P�

1� ��Q:out

Vl1 ð28Þ

dl2dt

¼3C2

P�

ð1� �Þ2�Q:out

Vl2 ð29Þ

The eight state variables in the model are initiatorconcentration, monomer concentration, solvent con-centration, reactor temperature, cooling jacket tem-perature, and the first three moments of the molecularweight distribution:

xT ¼ CICMCSTTcl0l1l2½

Note that the model obtained as described above isrewritten in terms of dimensionless quantities to avoidnumerical problems in the simulations. The possiblemanipulated inputs are the coolant, initiator, monomerand solvent flowrates:

uT ¼ QcQIQMQS½ :

3.3. Operating conditions

For the operating conditions studied in this paper, thereactor has three steady-states; the selected operatingpoint is the middle, unstable steady-state. Thermal initia-tion of secondary reactions is an important factor at thehigh temperature steady-state, while the conversion is toolow at the low temperature steady-state. The middlesteady-state is chosen because of the high conversion itprovides. Note that the use of a single manipulatedvariable, cooling jacket flowrate, will not be successfulsince the operating point is near an infeasible tempera-ture region [27]. This makes it imperative that a multi-variable strategy be used for control. The variables thatcan be controlled are the reactor temperature, the pro-duction rate, the number average molecular weight andthe polydispersity of the molecular weight distribution.

4. EKF-based multi-rate NMPC

This section presents an algorithm for the use of amulti-rate extended Kalman filter with nonlinear modelpredictive control. The multi-rate EKF algorithm isexplained in Section 4.2.

Table 1

Plant parameters

wIf=0.1 �=0.15

Tf=330 K �Hr=�69919.56 J/mol

Tcf=295 K f �¼0=0.6

Ed=123853.658 J/mol �ccpc=4045 J/(l K)

Ep=29572.898 J/mol UA0=293.076 W/K

Et=7017.27 J/mol Vc=3100 m3

Ef=53020.29 J/mol MI=164 g/mol

Efs=91457.3 J/mol MM=104.15 g/mol

kd0=5.95*10131/s MS=106.17 g/mol

kp0=1.06*107 l/(mol s) R=8.314 J/(mol K)

k�t0=0=1.25*109 l/(mol s) �1=1.938 J/(gK)

kf0=53020.29 l(mol s) �2=�3.77*10�3 J/(gK2)

kfs0=91457.3 l(mol s) �3=1.05*10�5 J/(gK3)


4.1. Extended Kalman filter

Consider a process that can be described by the fol-lowing equations

x:¼ fðx; u; dÞ ð30Þ

y ¼ gðx; dÞ ð31Þ

which can be discretized to obtain the discrete model

xk ¼ Ftsðxk�1; uk�1; dk�1Þ ð32Þ

yk ¼ gðxk; dkÞ ð33Þ

The following stochastic difference equation is aug-mented to the original system states and is used toexpress the unmeasured disturbance signal d:

xwk ¼ Awxwk�1 þ Bwwk�1 ð34Þ

dk ¼ Cwxwk ð35Þ

The augmented model may therefore be expressed as

xkþ1

xwkþ1

� �¼

Ftsðxk; uk;Cwxwk Þ

Awxwk

� �þ

0Bw

� �wk ð36Þ

yk ¼ gðxk;Cwxwk Þ þ vk ð37Þ

where wk and vk are discrete-time white noise signalswith covariances Rw and Rv respectively.For this model formulation, the following extended

Kalman filter provides optimal estimates for single-ratesystems [14]:Model prediction:

xkjk�1

xwkjk�1

� �¼

Ftsðxk�1jk�1;uk�1;Cwxwk�1jk�1Þ

Awxwk�1jk�1

� �ð38Þ

Xkjk�1 ¼ �k�1

Xk�1jk�1�

Tk�1 þ wRwð wÞ

Tð39Þ

Measurement correction:

xkjkxwkjk

� �¼

xkjk�1

xwkjk�1

� �þ Lkðyk � ykÞ ð40Þ

Here, Lk is the Kalman gain,P

k is the error covar-iance, and w, � and ! are given by

w ¼0Bw

� �ð41Þ

�k�1 ¼Ak�1B

dk�1C

w

0 Aw

� �ð42Þ

!k ¼ CkCdkC

w� �

ð43Þ

Ak�1 ¼ expðAk�1tsÞ;A~ k�1

¼@fðx; u; dÞ

@xjðx¼xk�1jk�1;u¼uk�1;d¼Cwxw

k�1jk�1Þ ð44Þ

Bdk�1 ¼

ðts0

expðA~ k�1Þd �B~dk�1;B

~ dk�1

¼@fðx; u; dÞ

@djðx¼xk�1jk�1;u¼uk�1;d¼Cwxw

k�1jk�1Þ ð45Þ

Ck ¼@gðx; dÞ

@xjðx¼xkjk�1;d¼Cwxw

kjk�1Þ ð46Þ

Cdk ¼

@gðx; dÞ

@djðx¼xkjk�1;d¼Cwxw

kjk�1Þ ð47Þ

4.2. Multi-rate extended Kalman filter

We now consider the case of a system with multi-ratesampling. The original extended Kalman filter is aug-mented with an additional updating Kalman filterwhich only operates when laboratory data is available,thus providing a relatively easily implemented suboptimalsolution to the multi-rate estimation problem. Theimportant feature of this implementation is the conserva-tion of the estimated data from the fast EKF tier. Thestructure of this multi-rate EKF is shown in Fig. 2. Atthe fast sample rate, few measurements yk are availableand only EKF1 operates. The equations for the single-rate EKF described above apply to EKF1 withoutmodification. The Kalman gain matrix for EKF1 is ofsmaller dimension since only the fast measurements areavailable, and only part of the state vector is observable.The unobservable subsystem has states updated only inan open-loop fashion.The delay of the laboratory measurements is denoted

by dlab. As soon as laboratory measurements becomeavailable the following state estimation step is performedby EKF*1 and EKF*2. EKF*2 uses the complete statemeasurement vector y�k�dlab

to estimate the plant statex�k�dlab jk�dlab

for the time tk�dlab when the laboratorymeasurement was taken. EKF*2 has a larger size Kalmangain vector since the measurement set is larger.The new estimated state vector x�k�dlab jk�dlab

is thenforwarded to EKF*1 where the current, ‘‘corrected’’state vector xcorrkjk is evaluated in a recursive manner byEKF*1. Stored past measurements and past controlmoves used by EKF1 are now used by EKF*1. EKF*1has the same structure as EKF1, and just uses theimproved ‘‘initial’’ state estimate at time (k-dlab) pro-vided by EKF�

2ðx�k�dlab jk�dlab

Þ to recalculate state esti-mates for the period between tk�dlab and tk. Fig. 3 shows


the functioning of the updating algorithm. The resultsfrom the updating algorithm are plotted as dotted lines.Note that the corrected estimates xcorrkjk do not lie on thetrue states. This is due to the model based nature of theKalman filter, which will, depending on the tuning,more or less ‘trust’ the inferred a priori estimates. Alsonote that the original system state vector is augmentedwith ‘disturbance states’ to create an augmented statevector. These disturbance states are estimated alongwith the original system states, and they provide a way

to account for parameter or structural uncertainty in themodel and the effect of disturbances on the plant.The NMPC algorithm used in this paper uses the

nonlinear system model and the current state and dis-turbance estimates to calculate the effect of past controlmoves on the future outputs. A local linearization of thenonlinear model is used to approximate the effect offuture (undecided) control moves on the future outputs.It is similar to the approach of Lee and Ricker [14].However, the current state estimates are obtained fromthe multi-rate extended Kalman filter, and the predic-tion algorithm uses these estimates in future state andcontrolled output predictions.

5. Observability results

A detailed observability analysis was performed onthe original system and on the extended system formedby augmenting the state vector with disturbances. Lin-ear observability analysis was performed on the linear-ized system matrices at the unstable steady stateoperating point. The results obtained are summarized inthis section. These results provide some insight into thechoice of measurements for the control of the styrenepolymerization CSTR at the chosen operating conditions.This choice may be based not only on the observability ofthe original system states, but also on whether the aug-mented disturbance states are observable or not. If weexpect disturbances or uncertainty to affect certain stateequations, we might choose measurements that ensure

Fig. 2. Multi-rate EKF.

Fig. 3. Updating Kalman filter estimates.


observability of that state and the disturbance variables.This is important because observability of the aug-mented disturbance states is essential to ensure goodstate estimate updates. This is because observability ofthe original system states in itself only ensures that theestimator updates all the states, and does not guaranteethat the expressions for the state equations are correct,i.e., the disturbance estimates affect the state estimatessignificantly.There are many options in modeling the effect of dis-

turbances on the original state equations. Since we aredealing with an open-loop unstable system, we rejectany possibilities that involve output disturbance assump-tions. This is to ensure stability of the estimation andcontrol scheme [17]. The disturbances are thus modeled asinput or state disturbances. What this means is that eachaugmented disturbance affects a state equation in thefollowing form. Let the original model state equation berepresented by

x:i ¼ fiðx; uÞ ð48Þ

The disturbance dj is simply assumed to change thestate equation to

x:i ¼ fiðx; uÞ þ dj ð49Þ

In the absence of true knowledge of the actual dis-turbances that affect the plant, this is the only feasibleconstruction of the disturbance variables for our particularplant. The disturbance states are then augmented to theoriginal state vector so that the states of the augmentedsystem are given by

X ¼ x1 � � � xNd1 � � � dM½

This augmented state vector is then estimated usingthe EKF. Note that our assumption of integrated whitenoise disturbances implies that the augmented dis-turbance model has Aw=I, Bw=I, Cw=I. In themeasurement correction step of the EKF, the dis-turbance estimates get updated along with the states ofthe original system model.Another possibility is to use system parameters rather

than disturbances as augmented states to be estimated.The choice of parameters to be estimated is a difficulttask. Also, a particular parameter chosen as an aug-mented state to be estimated may affect more than onesystem state in a correlated fashion, which is not alwaysdesirable. Choosing an appropriate set of parameters toaugment as appended states requires great physicalinsight into the system, which is not always feasible.The approach we take towards studying observability

of the system augmented with the disturbance states isas follows. We consider a particular available measure-ment vector, and consider all possible combinations of

disturbances that may be appended to the state equa-tions. The number of disturbances that may be appen-ded and estimated must be less than or equal to thenumber of measurements [16]. This comes from a rankcondition on the observability matrix for the augmentedsystem. For our analysis, we select the number ofappended disturbances to be equal to the number ofmeasurements. For each choice of possible disturbances,we construct the null space of the observability matrixof the augmented system. The null space of the aug-mented observability matrix provides the set of basisvectors for the unobservable modes of the system forthe particular choice of measurements and appendeddisturbances. Note that the observability matrix for asystem with state transition matrix A and output matrixC is given by

G ¼ CTATCTðATÞ2CT . . . ðATÞ

n�1CT� �

where n is the number of states, i.e. the dimension of thesquare matrix A. We perform a linear observabilityanalysis, and our results are only valid for the specificsystem and operating conditions we choose. If a certaindisturbance is a ‘‘likely’’ disturbance for the true plant,which might affect the state estimates and the controlperformance, then we need to select a set of measure-ments (and appended disturbances) that will ensureobservability of the augmented system (or at least mini-mize the unobservable subspace), so that good stateestimation is ensured.We restrict ourselves to the study of four possible com-

binations of available measurements. Other combinationsof measurements are rejected on practical grounds, forexample, if one moment of the polymer distribution isavailable, then the other two moments will also be avail-able, since the moments are typically calculated by astatistical analysis of gel permeation chromatography(GPC) measurements.

5.1. Temperature measurements

The first possible measurement set is TTc½ , i.e., onlytemperature measurements. In this case, only the sub-system CICMTTc½ is observable. The solvent con-centration and the three moments of the polymerdistribution are not observable, and the possibilities fordisturbance estimation are very limited. Note that in thesimulation results presented in Section 6, it will beassumed in every simulation that temperature measure-ments are available at the fast sampling instants.

5.2. Concentration and temperature measurements

The next measurement set that we consider isCICMCSTTc½ . In this case, temperatures are availableat a fast sampling rate and concentrations are available


at a slow sampling rate from laboratory measurements(with time delay). The three moments of the polymerdistribution are not observable even with the slow mea-surements, and the unobservable modes of the aug-mented system include the moments themselves and anydisturbances that affect the state equations for themoments.

5.3. Moment and temperature measurements

The next measurement set considered is TTcl0l1l2½ ,i.e. temperature and moment measurements. The origi-nal system is completely observable in this case, leadingto the conclusion that moment (or molecular weight)measurements are essential for observability of themoments. This finding is in agreement with the litera-ture on observability in polymerization reactor systems[24,36,28]. Ray [24] makes the point that observability isa much more desirable property than detectability, sincethe convergence of estimates of states that are obser-vable can be controlled by the tuning of the estimator,whereas estimates of states that are only detectable tendto show sluggish dynamic convergence. The observa-bility analysis of the augmented system shows us that adisturbance in the solvent concentration (CS) equationis the most likely disturbance to be unobservable. Forsome combinations of estimated disturbances, a dis-turbance in the monomer concentration (CM) equationis also unobservable.

5.4. Moment, temperature and CS measurements

In the next case we consider the solvent concentration(CS) to be measured along with the temperatures andmoments. The original states are all observable, as expec-ted. In addition, the number of disturbance combinationsthat are not fully observable is reduced drastically. Dis-turbances in the solvent concentration equation arealways observable. The monomer concentration is themost likely disturbance to be unobservable. By using theadditional measurement of the solvent concentration,we obtain better state estimates on two counts, onebeing the larger set of measurements for the EKF to useto update state estimates, and the other being betterdisturbance estimation. If all states are measured, thenboth the original and the augmented systems are com-pletely observable.

5.5. Analysis

The above results allow us to decide on an appropriateset of measurements for our particular application. For agiven measurement set, the number of different combina-tions of possible disturbances that have the augmentedobservability matrix less than full rank can be obtained,and the null space of the augmented observability

matrix provides us with the unobservable disturbancestates. This is illustrated for the case with temperatureand moment measurements ðT;Tc; l0; l1; l2Þ in Table 2and for the case with six measurementsðCS;T;Tc; l0; l1; l2Þ in Table 3.The obvious difference between the two cases is that it

is possible to use six augmented disturbances in one caseas opposed to five in the other. The additional aug-mented disturbance provides an additional degree offreedom in the state estimation. However, not all com-binations of five disturbances (for the case with tem-perature and moment measurements) lead to observableaugmented systems. Table 2 shows the combinations ofaugmented disturbances for which the augmentedobservability matrix is not of full rank (13 in this case).Note that in most case, the disturbance affecting thesolvent concentration equation lies in the unobservablesubspace. A large number of disturbance combinationsresult in partially unobservable augmented systems.Now consider the case with six measurements. In this

case, at most six augmented disturbances can be esti-mated, and the combinations of disturbances that result

Table 2

Augmented system unobservability ðT;Tc; l0; l1; l2 measured)

Disturbances Obs.

matrix rank

Unobservable

subspace

dCI; dCM

; dCS; dT; dl0 12 dCS

dCI; dCM


dCI; dCM


dCI; dCM

; dCS; dTc

; dl0 12 dCS

dCI; dCM

; dCS; dl0 ; dl1 12 dCS

dCI; dCM


dCI; dCM


dCI; dCM

; dT; dl0 ; dl1 12 dCM

dCI; dCM


dCI; dCM


dCI; dCM

; dl0 ; dl1 ; dl2 12 dCM

dCI; dCM

; dT; dTc; dl0 12 dCS

dCI; dCM

; dT; dl0 ; dl1 12 dCS

dCI; dCM


dCI; dCM


dCI; dCM

; dTc; dl0 ; dl1 12 dCS

dCI; dCM


dCI; dCM

; dl0 ; dl1 ; dl2 12 dCS

dCI; dT; dl0 ; dl1 ; dl2 12 0:22CM � 0:73T

þ 0:60dCI� 0:26dT

dCM; dCS

; dT; dTc; dl0 12 dCS

dCM; dCS


dCM; dCS


dCM; dCS

; dT; dl1 ; dl2 12 0:18CM � 0:29CS

þ 0:55dCM� 0:76dCS

dCM; dCS


dCM; dCS


dCM; dCS


dCM; dT; dl0 ; dl1 ; dl2 12 dCM

dCM; dCS

; dl0 ; dl1 ; dl2 12 dCS

dCM; dT; dl0 ; dl1 ; dl2 12 dCS

dCS; dT; dTc

; dl0 ; dl1 12 dCS

dCS; dTc

; dl0 ; dl1 ; dl2 12 dCS


in a not-fully observable augmented system are shownin Table 3. The number of unobservable disturbancecombinations is greatly reduced. As can be seen, themajority of the cases involve disturbances in the mono-mer concentration equation being unobservable. Thismeans that if monomer concentration disturbances areconsidered to be likely to affect the plant, the estimatorwith six measurements will not be able to estimate dis-turbances in the monomer concentration if the aug-mented disturbance model matches any one of the casesin Table 3. There are thus two options available: includeadditional measurements at additional cost, or modelthe augmented disturbance set differently from thesecombinations. A similar analysis may be performed forother possible measurement sets. The observabilityanalysis thus provides a basis for measurement selectionand choice of augmented disturbances.In this context, another approach that may be used to

select a combination of measurements for a given sys-tem is described by Morari and Stephanopoulos [16].For the case of slowly varying persistent disturbances,they assume the process to exhibit quasi-steady-statebehavior with respect to the disturbances. They notethat in most cases, problems of structural observabilityof the augmented system exist if all possible dis-turbances are considered and suggest an optimal way toresolve them. This essentially involves minimizing theinfluence of the unobservable subspace of the vector ofobservations (measurements) on the vector of estimatedoutputs. Our approach treats this problem slightly dif-ferently, since we select a set of available measurementsand then use an observability analysis to investigate thedisturbances that can not be estimated with themeasurement set. Physical knowledge of the plant andthe possible disturbances that may affect it then lets us

choose the measurement set that we require to providesatisfactory state estimation.

6. Control results

The results shown in this section are based upon thesimultaneous control of four outputs: reactor tempera-ture, l0, NAMW and PD. The choice of these as con-trolled outputs reflects the desire to control polymerproduct quality. The number averaged molecular weightand polydispersity directly reflect the composition of thepolymer distribution and influence it’s end-use proper-ties. l0, the zeroth moment of the polymer distribution,is a measure of the total polymer concentration. Thereactor temperature is also included as a controlledoutput to explicitly account for the open-loop instabilityat the chosen operating point.

6.1. Preliminaries

In addition to the NMPC algorithm presented in thispaper, a linear MPC approach has also been consideredfor this particular control problem. However, the poly-merization CSTR shows highly nonlinear behavior,which results in the linear MPC controller not beingable to provide good performance over a wide range ofoperating conditions. The linear MPC scheme does notprovide good control for setpoint changes similar tothose shown in this section, though it provides satisfac-tory control for disturbance rejection (regulatory con-trol) so long as the disturbances are of low magnitude.We do not show linear MPC results for the sake ofbrevity. A comparison of NMPC and a decentralizedcontrol strategy for concentration and temperature con-trol has also been performed and can be found in Schleyet al. [29]. In our study, as in Schley et al. [29], the per-formance criterion for evaluation of the control strate-gies is the integral squared error between the referenceand plant outputs. We focus on inferential controlresults in this section, i.e. the control of NAMW andPD, which are not directly measured outputs. Also, wepresent results on reference trajectory tracking (in thepresence of modeling uncertainty and disturbances) andnot on pure regulatory control, i.e. the rejection of dis-turbances. In most cases, the reference trajectories aremodeled as second order transfer functions. The readershould be aware that the choice of a feasible set-pointcombination for this system is a non-trivial exercise.Russo et al. [27] provide an analysis of the operabilitycharacteristics of the styrene polymerization CSTR, andtheir analysis may be used to decide on feasible, achiev-able set-points for all the controlled variables.The simulations are performed under the following

conditions: temperatures are sampled frequently (every5 min), whereas the slow measurements (concentrations

Table 3

Augmented system unobservability (Cs;T;Tc; l0; l1; l2 measured)

Disturbances Obs.

matrix rank

Unobservable

subspace

dCI; dCM

; dCS; dT; dl0 ; dl1 13 dCM

dCI; dCM


dCI; dCM


dCI; dCM

; dCS; dTc

; dl0 ; dl2 13 dCM

dCI; dCM

; dCS; dl0 ; dl1 ; dl2 13 dCM

dCI; dCM

; dT; dTc; dl0 ; dl1 13 dCM

dCI; dCM


dCI; dCM


dCI; dCM

; dT; dl0 ; dl1 ; dl2 12 0:22CI � 0:67CM þ 0:62dCI

þ 0:23dCM� 0:27dT; dCM

dCI; dCM

; dTc; dl0 ; dl1 ; dl2 13 dCM

dCI; dCS

; dT; dl0 ; dl1 ; dl2 13 � 0:22CI þ 0:73CM � 0:60dCI

þ 0:26dTdCI

; dT; dTc; dl0 ; dl1 ; dl2 13 � 0:22CI þ 0:73CM � 0:60dCI

þ 0:26dTdCM

; dCS; dT; dl0 ; dl1 ; dl2 13 dCM

dCM; dT; dTc

; dl0 ; dl1 ; dl2 13 dCM


and/or the leading moments of the distribution) areobtained via laboratory analysis every 30 min with a delayof 30 min. It is not necessary that the measurements beavailable at periodic intervals, but the simulations areperformed under such conditions for simplicity. Theexecution time per sample of the estimation/controlscheme is of the order of 1 s on a 550 MHz Pentium II,so the computational burden is low. It is assumed thattemperature measurements are disturbed by normallydistributed noise of standard deviation 0.1 K. Measure-ments of concentrations and leading moments areassumed to be corrupted with Gaussian noise of standarddeviation 2% of the nominal value of the variable. Theprediction horizon is chosen to be 20 whereas an inputhorizon of 2 is implemented for all simulations. Thecontrol results do not differ significantly in most casesfor input horizons of 1 and 3. Finite horizon MPC is

used in this paper, and though this means that nominalstability is not guaranteed, there were no problems withclosed-loop stability in any of the simulations with thechoice of prediction and control horizons describedabove. All simulations have an initial condition mismatchbetween the plant and the model, since it is very unlikelythat accurate estimates of the plant states will be availableat the outset.Proper tuning is a very important factor that affects

the performance of estimators, especially that of theextended Kalman filter. The process and measurementnoise covariances, along with the initial error covarianceof the states are the tuning parameters in the EKF. Forthe simulations in this study, the EKFs were tunedessentially by trial and error, and the performance verifiedby simulation. Valappil and Georgakis [37] present twomethods for tuning the process noise covariance matrix,

Fig. 4. Plot of state estimates for multi-rate estimation and control. Polymer PD is being controlled.


one based on a Taylor series expansion of the nonlinearmodel equations around nominal parameter values, andthe other based on Monte Carlo simulations for one timestep using an assumed parameter covariance distribution.These tuning methods, however, do not provide estimatesof the process noise covariance of the augmented dis-turbance states. Only the elements of the process noisecovariance matrix that are associated with the originalsystem states are updated. Tuning of the process noisecovariance matrix for the styrene polymerization CSTRusing these methods did not provide any improvementover the case where tuning was done in a more ad hocmanner. This can be attributed to the fact that the cov-ariance associated with the augmented states is notaffected by these tuning methods. Another factor maybe that the process noise covariance matrix is relativelyrobust to parameter uncertainty or variability. The tuningparameters for the EKF for the six-measurement caseare given in Appendix A.

6.2. Multi-rate versus single-rate estimation

Figs. 4–7 show simulation results for control of poly-dispersity (PD) using multi-rate estimation and control.In this case, the available measurement vector (at theslow sampling instants) is assumed to be CSTTcl0l1l2½ .Only temperature measurements are available at fastsampling instants. Disturbances affect the jacket inlettemperature, and the monomer and initiator concentra-tions. The estimator model augments disturbances inthe balances for the initiator, reactor and jacket tem-peratures, and the three moments of the polymer dis-tribution. Fig. 4 shows the state estimates and Fig. 5 themanipulated input profiles for this case. The polymerproperties, NAMW and PD are shown in Fig. 6. Notethat the polydispersity is controlled while keeping thetemperature and NAMW constant at their originalsteady-state values. While l0 is also a potential controlledvariable, it is left uncontrolled because of the difficulty

Fig. 5. Plot of manipulated variables for multi-rate estimation and control. Polymer PD is being controlled.


in specifying a feasible and meaningful setpoint for it inconjunction with the setpoints on the other controlledoutputs. Accordingly, no weighting is given to errors in l0in the objective function for the controller QP optimiza-tion. Approximations of the initial and final polymer dis-tributions are presented in Fig. 7. Since the polydispersityis reduced, the final distribution is not as broad as theinitial distribution. Also, since the NAMW is kept con-stant with a narrower distribution, the peak of the finaldistribution is higher.A comparison is nowmade with a single-rate estimation

and control scheme. Here, estimation updates are onlymade when the slow measurements are available. Inaddition, control action is only taken when estimationupdates are made, and this adds to the deterioration incontrol performance. Only the NAMW and PD controlresults are shown for brevity. Fig. 8 shows the control ofNAMW and PD. Very poor control is achieved with thesingle rate strategy, indicating the need for multi-rateestimation and control. The delay in convergence ofstate estimates in the single rate strategy contributes inlarge part to the deterioration in control performance.

6.3. Comparison of measurement sets

In the next set of simulations, we provide results forthe simultaneous control of PD and NAMW whilekeeping the reactor temperature constant. We seek toraise the number average molecular weight and reducethe polydispersity at the same time. In these simulations,

Fig. 6. Plot of polymer properties for multi-rate estimation and control. Polymer PD is being controlled.

Fig. 7. Plot of polymer distribution for multi-rate estimation and

control. Polymer PD is being controlled.


disturbances affect each of the plant state equations. Wecompare the results with four possible measurement sets,set I being temperature and concentration measurements,set II being temperature and moment measurements, setIII having temperature and moment measurementsalong with solvent concentration measurements and set

IV being the case with all states being measured. Figs. 9,10, 12 and 13 show the results for NAMW and PDcontrol for each of these cases. We see improvements inthe control performance as we proceed from temperatureand concentration measurements to temperature andmoment measurements. Further (slight) improvement in

Fig. 8. Plot of polymer properties for single rate estimation and control. Polymer PD is being controlled.

Fig. 9. Plot of polymer properties for set I measurements. Both NAMW and PD are being controlled.


the control of NAMW and PD is made as we include thesolvent concentration as a measurement. The improve-ment made by having all states measured is not very sig-nificant. State estimates for the case with six measurementsare shown in Fig. 11. These may be taken to be repre-sentative of the other cases. Temperature is again keptconstant at its original setpoint, and l0 is left uncontrolled.We seek to explain these control results by means of

an observability analysis. The estimator with set I mea-surements assumes the augmented disturbances to affectthe monomer, reactor temperature and three polymermoments. The moments of the polymer distribution areunobservable, as are the augmented disturbances affectingthe moment equations. This means that the polymerproperty control is essentially open-loop in this case.The estimator with set II measurements assumes the

same augmented disturbance model as the previous case.In this case, the polymer moments are observable, andthe unobservable subspace of the augmented system liesalong a combination of the monomer concentration, themonomer disturbance and the temperature disturbancedirections. The polymer properties control is, therefore,better in this case than in the previous case.The estimator with set III measurements (solvent

concentration, reactor and jacket temperatures andpolymer moments) augments disturbances to the initiator,reactor and jacket temperatures and the polymer momentequations. The unobservable subspace of the augmentedsystem lies along the same direction as the previous case,but the extra measurement (solvent concentration) leadsto better estimation and control of polymer NAMWand PD.

The estimator with full state measurement (set IV) hasaugmented disturbances entering all the state equations,and the augmented system is fully observable. Thepolymer property control is improved very slightly inthis case.As mentioned earlier, the results obtained above may

also be explained by studying the influence of theunobservable subspace of the vector of observations(measurements) on the vector of estimated (and con-trolled) outputs, as proposed by Morari and Stephano-poulos [16]. They assume slowly-varying disturbanceswith respect to the process. In the simulations describedabove, disturbances of constant magnitude affect theplant, and these fit the description of slowly-varying dis-turbances. For each of the four sets of measurements, weconstruct a disturbance vector, compute the unobservablesubspace, and compute the effect of the unobservablesubspace on the controlled outputs. Appendix B outlinesthe calculation of the numerical measure of unobserva-bility. The optimal set of measurements has the lowesteffect of the unobservable subspace on the controlledoutputs. For the styrene polymerization CSTR, thisnumerical value of the unobservability is practically thesame for all the cases that have temperatures andmoments being measured. The unobservability indexdoes not vary significantly if additional (concentration)measurements are available. This indicates that thesedisturbances do not greatly affect the estimates of theoutput variables that are to be controlled. When onlyconcentrations and temperatures are measured, thevalue is orders of magnitude higher, thus explaining thepoorer control performance in that case.

Fig. 10. Plot of polymer properties for set II measurements. Both NAMW and PD are being controlled.


6.4. Production rate control

Production rate control is an important considerationin the chemical process industry. In the next set of simu-lations, we show the control of the production rate in thereactor. A measure of the production rate is the productof the flow rate and the total weight of polymer (l0) inthe reactor. Since the flow rates of monomer, initiatorand solvent are manipulated variables, the controlledoutput equation is modified to include the direct effectof inputs on the controlled outputs. The prediction equa-tions for the NMPC controller are also modified appro-priately for this case. For brevity, only the production rateincrease is shown (Fig. 14). This corresponds to a dou-bling of the original production rate. A higher reactiontemperature is specified, but NAMW and PD are held

at their original values, and the polymer distribution isunchanged. The available measurement vector in thiscase is CSTTcl0l1l2½ , and the modeled disturbancesaffect initiator concentration, temperature, coolant Tand all three moments. The actual disturbances affectthe coolant feed temperature and the initiator andmonomer feed concentration. Though not shown, allthe state estimates converge to their true values, and thefinal steady states for all manipulated variable flow ratesare higher than their initial values.

7. Summary

A model-based strategy for the control of a styrenepolymerization reactor is presented in this paper. The

Fig. 11. State estimates for set III measurements. Both NAMW and PD are being controlled.


reactor is operated at an open-loop unstable steady-statethat is close to an infeasible temperature region (if only asingle manipulated variable is considered). The manipu-lated variables considered in this paper are the coolant,initiator, monomer and solvent flow rates. The mostimportant quantities desired to be controlled are relatedto product quality (NAMW, PD) and are usually not

directly measured. The system was assumed to have twosets of measurements—fast on-line measurements avail-able instantaneously, and slow laboratory measure-ments with significant analysis delay. This involves thedesign of a multi-rate estimator based on extendedKalman filtering, which is coupled with a nonlinear MPCalgorithm to provide inferential control for the system.

Fig. 12. Plot of polymer properties for set III measurements. Both NAMW and PD are being controlled.

Fig. 13. Plot of polymer properties for set IV measurements. Both NAMW and PD are being controlled.


The multi-rate estimation and control strategy providesgood control under conditions of structural and para-metric uncertainty and in the presence of disturbances. Acomparison with a single (slow) rate estimation schemeshows the improved control performance obtained withthe multi-rate estimator.The estimator appends disturbances as states aug-

mented to the original system, which provides amechanism for disturbance estimation. We perform anobservability analysis on the original and the aug-mented system, which provides us with insight about thedisturbances and states that may be estimated with aparticular set of measurements. This is used to addressthe (design) problem of measurement selection for theparticular system.

Acknowledgements

We gratefully acknowledge support from MerckResearch Laboratories, Rahway, NJ, USA.

Appendix A: Extended Kalman filter parameters

The initial disturbance vector was chosen as follows

xw0 ¼ �0:0038 0:3568� 0:34410:0385� 0:6924½

� 0:0240� 0:0072 0:0847T

The disturbances affect the plant equations pertainingto the initiator, monomer and solvent balances, thereactor and jacket temperatures and the three momentequations. The values refer to the scaled variables in theplant equations. The magnitude of each disturbancereflects a percentage of the steady state value of the statevariable it affects.For the six measurement case, the estimator appends

disturbances in the state equations for the initiator,reactor temperature, jacket temperature, and the threemoments of the polymer distribution (in that order).The covariance matrix of the estimated disturbances(process noise covariance) is a matrix with all non-diagonal elements being zero, and is given by

Rw ¼ diag 0:001 0:1 0:01 0:001 0:001 0:006½

The covariance matrices for the measurements at thefast sampling intervals Rv

1 and the slow sampling inter-vals Rv

2 are

Rv1 ¼ diag 0:0033 0:0033½

Rv2 ¼ diag 0:0033 0:0033 0:05 0:0016 0:0016 0:006½

where the measurement vectors are

y1 ¼ TTc½ T; y2 ¼ CSTTcl0l1l2½ T

The initial covariance matrix for the model states waschosen asX

0 ¼ diag 0:005 0:4 0:4 0:001 0:001 0:0001 0:1 1:0½

The model state vector x is

x ¼ CICMCSTTcl0l1l2½ T

Appendix B: Unobservability index

This section describes the numerical measure ofunobservability of Morari and Stephanopoulos [16].Consider a linear time-invariant system given by:

x:¼ Axþ Buþ Gd

y ¼ Cx

z ¼ Fx

where d is the vector of unknown inputs (disturbances),y is the vector of observations and z the vector wewould like to estimate (or control). The objective is toobtain estimates of z and d. d is assumed to consist ofslowly varying disturbances with respect to the plant.An augmented system can be created in the form:

Fig. 14. Controlled production rate.


x:

d:

� �¼

A F

0 0

� �x

u

� �þ

B

0

� �u

y ¼ C 0� � x

u

� �

z ¼ F 0� � x

u

� �

Slow disturbances are assumed and a static estimatoris constructed. The following quantities are defined:

ST ¼ CA�1G

TT ¼ FA�1G

R ¼ TTT� TTSðSTSÞ�1STT

R is the expectation of estimation error of the estimated(controlled) outputs, and TTT is the covariance matrix ofthe estimates of z. The observations (measurements) arethen selected to solve the following problem:

mintraceðRÞ

traceðTTTÞ

The above quantity provides a numerical measure ofthe unobservability of the augmented system for a par-ticular set of measurements. Further details may beobtained from Morari and Stephanopoulos [16].

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