control of a solution copolymerization reactor using multi-model predictive control

Chemical Engineering Science 58 (2003) 1207–1221www.elsevier.com/locate/ces

Control of a solution copolymerization reactor using multi-modelpredictive control

Leyla 'Ozkana, Mayuresh V. Kotharea ;∗, Christos Georgakisb

aDepartment of Chemical Engineering, Chemical Process Modeling and Control Research Center, 111 Research Drive, Lehigh University,Bethlehem, PA 18015, USA

bDepartment of Chemistry, Chemical, Engineering and Material Science, 728 Rogers Hall, Polytechnic University, Six Metrotech Center,Brooklyn, NY 11201, USA

Received 20 December 2001; received in revised form 4 September 2002; accepted 2 October 2002

Abstract

We study the control of a solution copolymerization reactor using a model predictive control algorithm based on multiple piecewiselinear models. The control algorithm is a receding horizon scheme with a quasi-in7nite horizon objective function which has 7nite andin7nite horizon cost components and uses multiple linear models in its predictions. The 7nite horizon cost consists of free input variablesthat direct the system towards a terminal region which contains the desired operating point. The in7nite horizon cost has an upper boundand takes the system to the 7nal operating point. Simulation results on an industrial scale methyl methacrylate vinyl acetate solutioncopolymerization reactor model demonstrate the ability of the algorithm to rapidly transition the process between di9erent operating points.? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Process control; Polymer; System engineering; Nonlinear dynamics; Model predictive control; Linear matrix inequalities

1. Introduction

In this paper, we study the control of a polymerization re-actor using a multiple model approach. Polymer processingis an important sector of the chemical process industry. Poly-merization reactions are described by complex nonlinear ki-netic mechanisms and polymerization reactors exhibit highlynonlinear behavior. Hence, the control of such processesis complex and challenging. In addition, the demands ofthe global economy and increased competition have forcedpolymer reactors to operate in multiple operating regimesto manufacture several di9erent grades of polymers. A veryimportant control objective is to minimize grade transitiontime, and thereby reduce the amount o9-speci7cation prod-uct produced during transition. However, the nonlinear be-havior of polymer reactors becomes more signi7cant duringthese grade transitions as compared to local operation arounda steady state. As a result, polymerization control providesunique opportunities for employing novel transition controltechniques.

∗ Corresponding author. Tel.: +610-758-6654; fax: +610-758-5057.E-mail address: [email protected] (M. V. Kothare).

The requirement of optimal transition between operatingpoints has stimulated the need to develop Dexible operat-ing strategies. However, the nonlinear nature of chemicalplants makes this task nontrivial. Despite considerable re-search e9orts in developing new methods for modeling andcontrol of nonlinear systems over the last decade, many ofthese advanced techniques have not found applicability inreal industrial problems. This is primarily due to the lack oftheoretical knowledge required to understand these sophis-ticated methods and the need to develop detailed nonlin-ear models to describe complex system behavior. Therefore,there is an increasing awareness of the need for practical ap-proaches to aid engineers to better understand and performcomplex modeling and control tasks. The common strategyin engineering is the decomposition of the problem into sim-pler subproblems. This has lead to development of local andmultiple model/controller approaches to deal with nonlinearand time-varying systems. There are signi7cant bene7ts tobe gained from multiple model/controller approaches:

• the development of local models/controllers is simple;• the model/controller structure is easier to understand andinterpret.

0009-2509/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0009 -2509(02)00559 -6

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1208 L. /Ozkan et al. / Chemical Engineering Science 58 (2003) 1207–1221

Several modeling and control approaches, and applicationexamples based on this philosophy can be found in the recentmonograph by Murray-Smith and Johansen (1997). The ba-sis of these approaches is the decomposition of the system’sfull range of operation into a number of operating regimesin which a simple local model and/or controller is applied.The local models and controllers are then incorporated togive a global model and/or controller.In this paper, we present an application of model pre-

dictive control (MPC) structure using piecewise linearmultiple models on an industrial case study: solution copoly-merization of methyl methacrylate and vinyl acetate. Themain motivation of this paper is twofold: (a) to demonstratethat a multi-model MPC based on linear matrix inequalities(LMIs)/semide7nite programming is applicable to poly-merization control; (b) an LMI-based MPC algorithm isapplicable to a realistic large dimension chemical process-ing application. The control approach implemented hereis based on a receding horizon scheme with the quadraticobjective function split into a 7nite horizon cost and anin7nite horizon (terminal) cost. The 7nite horizon costspeci7es the future control outputs that force the system tomove to the terminal region. The in7nite horizon cost ise9ective starting from the terminal region and the corre-sponding terminal controller takes the system to the desiredsteady-state operating point. We demonstrate the success-ful implementation of the proposed control approach on acopolymerization reactor model. Among other importantissues, we study the e9ect of the number of linearizationpoints and the number of multiple linear models on theperformance of the controller.This paper is organized as follows: Section 2 gives an

overview of relevant work in controlling processes operat-ing at multiple operating points with an emphasis on piece-wise linear systems and the use of multiple models in theMPC formulation. In Section 3, we describe the proposedmulti-model control approach in detail. In Section 4, we de-scribe the polymerization reactor with particular emphasison the use of multiple models in controller design. Simula-tion results indicating controller performance are also dis-cussed. Finally, we present concluding remarks in Section5. Relevant background material on optimization involvingLMIs and the polymerization model is provided in the Ap-pendices.

2. Literature review

In the literature, there are several di9erent approachesdealing with the control of multiple regime processes. Theseapproaches include gain scheduling, multiple-model adap-tive control, and supervisory control. Gain scheduling hasbeen a favorite technique in the control of nonlinear andtime-varying systems since the 1950s because of its sim-plicity and ability to use linear design methods. Researchin gain scheduling has been particularly active over the last

decade. Theoretical properties of this approach can be foundin Rugh and Shamma (2000). Gain scheduling is a power-ful tool if we can relate measurable operating conditions toscheduling variables. It has fast response to changing oper-ating conditions since the controller parameters are changedin an open-loop fashion. On the other hand, it is ad hoc innature, i.e. there is little or no rigorous guideline to achievestability and good performance. The main limitation in thisapproach is the requirement of slow variations in schedulingvariables.Considerable work has been done to overcome these lim-

itations in connection with the progress in nonlinear controltheory. Packard (1994) used linear fractional transforma-tions in gain scheduling. Klatt and Engell (1998) presenteda gain scheduling trajectory control approach that combinesfeedback linearization and gain scheduling. Doyle (III)et al. (1998) proposed a dynamic gain scheduling approachto cope with the limitation of slow variations in schedulingvariables. A similar approach proposed to control multi-ple regime processes is multiple model adaptive control(MMAC). MMAC can be considered as a model based formof gain scheduling. It consists of model/controller pairs anda weighting function that chooses a single model or a com-bination of models that best represents the current plant.The basic di9erence between MMAC and gain scheduling isthat model/controller pairs are 7xed parameters in MMAC.The adaptive structure comes from the fact that as the plantparameters change, di9erent combinations of models areused to represent the plant. The applications of MMAC canbe found in Athans et al. (1977), Yu, Roy, Kaufman, andBequette (1992), Schott and Bequette (1995). One of theadvantages of this approach is its generality in the type ofmodel, control structure or a mixture of model and con-troller types that can be used in the MMAC framework.However, the number of models that can span the wholeoperating region is diLcult to decide. Morse (1996) pro-posed a simple, high-level controller called supervisor thatis capable of selecting the best feedback controller from aset of linear controllers and switching into feedback withan SISO process to cause the output of the process toapproach and track a setpoint. Kosanovich, Charboneau,and Piovoso (1997) demonstrated a hybrid dynamical su-pervisory strategy on an exothermic reactor, based on theswitching strategy proposed by Morse (1996).

Some of the recent work focuses on scheduling of theMPC structure, since MPC has being extensively used inthe processing industry. Within the context of nonlinearMPC, there are several approaches (Lopes, 2000) that ex-plore the transition control problem. Arkun, Banerjee, andLakshmanan (1998) presented a self-scheduling MPCframework for plants described by linear parameter varying(LPV) models. At each operating condition a linear statespace model is obtained and these local models are com-bined into a global model by interpolation that is based onmodel validity functions assigned to each local model. Qin,Martinez, and Foss (1997) proposed an interpolating MPC

L. /Ozkan et al. / Chemical Engineering Science 58 (2003) 1207–1221 1209

approach and applied it to a waste treatment plant. In thisapproach N multiple bounding linear models are obtainedeither by linearization or identi7cation and N linear MPCproblems are solved. The 7nal control sequence is calcu-lated by interpolation of control pro7les.A modeling structure that has attracted researchers re-

cently is piecewise linear systems. This is a class of non-linear systems that has been used to represent a range ofsystem nonlinearities in applications such as saturations, re-lays, overrides and switching logic. A nonlinear system canbe approximated as a family of piecewise linear systems bylinearizations around a set of operating points. The mainmotivation for the use of piecewise linear systems is the sim-plicity of its structure, which is linear in each region. In thisconnection, it is possible to reduce a complex problem to acollection of linear problems that can be solved in a simplermanner. To address stability analysis of piecewise linear sys-tems, Rantzer and Johansson (1998) suggested the construc-tion of piecewise quadratic Lyapunov functions. Hassibi andBoyd (1998) used piecewise quadratic Lyapunov functionsto prove stability of a piecewise linear system with multi-ple equilibrium points. An approach involving a family ofpiecewise linear models has been used in MPC for control-ling steam generator water level in a nuclear plant (Kothare,Mettler, Morari, Bendotti, & Falinower, 2000). The methoduses the power level as a scheduling variable to select theappropriate model from the piecewise linear model fam-ily in a predictive control framework. Chikkula, Lee, andOgunnaike (1998) used the method ofHingingHyperplanesto develop models from input–output data within the sched-uled MPC framework. The Hinging Hyperplane methodallows to divide the system’s dynamic variable space intoseveral linear regimes, based on available data. The result-ing models are piecewise linear approximations of the non-linear system.In the next section, we will develop a multiple model

MPC technique using the theory of LMIs.

3. Multi-model MPC formulation using piecewise linearsystems

Model Predictive Control, also known as moving hori-zon control or receding horizon control has become verysuccessful in process industries, especially in the control ofprocesses that are constrained, multivariable and uncertain.In general, MPC solves online an open-loop optimal con-trol problem subject to system dynamics and constraints ateach time instant and implements only the 7rst element ofthe control pro7le. At each sampling time k, plant measure-ments are obtained and a model of the process is used topredict future outputs of the system. Using these predictions,m control moves u(k + m|k); m= 0; 1; : : : ; n− 1, are com-puted by minimizing a nominal cost Jp(k) over a predictionhorizon p as follows:

minu(k+m|k); m=0; 1;:::; n−1

Jp(k) (1)

subject to constraints on the control input u(k + m|k), m=0; 1; : : : ; n− 1 and on the state x(k + m|k); m= 0; 1; : : : ; p,and the output y(k+m|k); m=1; 2; : : : ; p. It is assumed thatu(k +m|k) = 0, m¿ n, and only the 7rst computed controlmove u(k|k) is implemented. At the next sampling time,the optimization problem (1) is solved again with new mea-surements from the plant. There are several ways to expressthe cost function Jp(k) to achieve speci7c closed-loop char-acteristics. In this work, the following quadratic objectivefunction is considered:

Jp(k) =p∑

m=0

{xT(k + m|k)QIx(k + m|k)

+ uT(k + m|k)Ru(k + m|k)}; (2)

whereQI ¿ 0,R¿ 0 are symmetric weighting matrices andp=∞.Kothare, Balakrishnan, and Morari (1996) presented a

modi7ed form of in7nite horizon MPC algorithm that usesLMI based optimization. The nominal objective function (1)is modi7ed to a minimization of the worst-case in7nite hori-zon objective function taken over a set of uncertain plants.In this work, we extend this formulation using piecewiselinear model.Consider a nonlinear system in the following form with

several operating points:

˙x = f(x; u);

y = g(x; u): (3)

Linearization of the nonlinear system around these op-erating points and discretization gives a bank of piecewiselinear systems of the form

x(k + 1) = Aix(k) + Biu(k) + bi

y(k) = Cix(k) +Diu(k) + di

}x(k)∈Xi; (4)

where i denotes the active linear model and Xi shows the cor-responding operating region in state space in which modeli represents the nonlinear system.The modi7ed MPC law is given by

minu(k+m|k); m=0;1;:::;∞

J∞(k): (5)

As a 7rst step, an upper bound on the objective functionJ∞(k) is derived.Derivation of the upper bound: Consider a quadratic Lya-

punov function for this system at sampling time k of theform V (x(k|k)) = xT(k|k)Px(k|k). Suppose V satis7es thefollowing inequality:

V (x(k + m+ 1|k))− V (x(k + m|k))6− x(k + m|k)TQIx(k + m|k)

− u(k + m|k)TRu(k + m|k) (6)


with the conditions V (x(∞|k)) = 0 and x(∞|k) = 0. Sum-ming (6) from m= 0 to ∞ gives

− V (x(k|k))6− J∞(k): (7)

Then, the upper bound on the objective function J∞(k) isderived as

J∞(k)6V (x(k|k)): (8)

Following Kothare et al. (1996), the upper boundV (x(k|k)) is minimized assuming a constant state-feedbacku(k + m|k) = Kix(k + m|k). As is standard in MPC, onlythe 7rst computed input u(k|k) =Kix(k|k) is implemented.At the next sampling time, the state x(k + 1) is measured,the corresponding linear model and region are selected andthe optimization is repeated to recompute Ki.

The next theorem develops an LMI solution for the mini-mization problem assuming full state feedback u(k+m|k)=Kix(k + m|k) with P = �Q−1 and ellipsoidal local regions

or partitions Xi de7ned as

‖Eix(k + m|k) + ei‖6 1: (9)

Theorem 1. Let x(k|k) be the state of the systemmeasuredat time k in region i. Suppose the operating regions areapproximated as ellipsoids as in Eq. (9). Assume furtherthat there are no input or output constraints. The statefeedback control law u(k + m|k) = Kix(k + m|k) whichminimizes an upper bound on the MPC objective functionis given by the solution of the following linear objectiveminimization problem:

min�;�;Q;Yi

� (10)

s:t:

[1 xT(k|k)

x(k|k) Q

]¿ 0 (11)

and

Q ∗ ∗ ∗ ∗(AiQ+ BiYi) Q+ �bibTi bieTi � 0 0

EiQ �eibTi −�(I − eieTi ) 0 0

Q1=2I Q 0 0 �I 0

R1=2Yi 0 0 0 �I

¿ 0; (12)

where ∗ in the (i; j)th element of the symmetric matrixrepresents the transpose of the (j; i)th element.

Proof. The quadratic function V (x(k|k)) should againsatisfy (6) assuming each region is approximated as an

ellipsoid. So for x(k|k)∈Xi[x(k|k)

1

]T

×[(Ai+BiKi)TP(Ai+BiKi)−P+QI +KT

i RKi ∗bTi P(Ai + BiKi) bTi Pbi

]

×[x(k|k)

1

]6 0 (13)

should be true whenever

[x(k|k)

1

]T [ ETi Ei ET

i ei

eTi Ei −1 + eTi ei

][x(k|k)

1

]6 0: (14)

Using the S-procedure (see Boyd, Ghaoui, Feron, &Balakrishnan, 1994), we get[

(Ai + BiKi)TP(Ai + BiKi) +QI + KTi RKi − P− �ET

i Ei ∗bTi P(Ai + BiKi)− �eTi Ei bTi Pbi − �(−1 + eTi ei)

]6 0: (15)

This can be rewritten as[P+ �ET

i Ei −QI − KTi RKi �ET

i ei

�eTi Ei −�(1− eTi ei)

]

−[(Ai + BiKi)T

bTi

]P[ (Ai + BiKi) bi ]¿ 0: (16)

By Schur complements (see Appendix A), this is equivalenttoP+�ET

i Ei−QI −KiRKi �ETi ei (Ai+BiKi)T

�eTi Ei −�(1− eTi ei) bTi

(Ai + BiKi) bi P−1

¿ 0: (17)

Substituting P= �Q−1 and pre and post multiplying byI 0 0

0 0 I

0 I 0

gives�Q−1+�ET

i Ei−QI −KTi RKi (Ai+BiKi) �ET

i ei

(Ai+BiKi) �−1Q bi

�eTi Ei bTi −�(1−eTi ei)

¿ 0: (18)


Fig. 1. MPC with 7nite and in7nite horizon objective function.

It can be shown, after several steps involving Schur com-plements and algebraic manipulation that this is equi-valent to

Q (AiQ + BiYi)T QETi QQ1=2

I YTi R

1=2

(AiQ + BiYi) Q + �bibTi bieTi � 0 0

EiQ �eibTi −�(I − eieTi ) 0 0

Q1=2I Q 0 0 �I 0

R1=2Yi 0 0 0 �I

¿ 0;(19)

where �= ��−1.

Although derived for a piecewise linear system with el-lipsoidal partitions, the optimization problem in Eq. (10)gives a feasible solution only when bi =0 which means theellipsoidal region i contains the origin (x; u) = (0; 0). Thisis expected since the derivation of the upper bound on theobjective function J∞(k) depends on the assumption thatx(∞|k)=0. For the system outside of the ellipsoidal regionthat contains the origin, the term bi in Eq. (4) is not equalto 0, as a result the assumption x(∞|k) = 0 is not correct.To avoid confusion, the ellipsoidal region that contains theorigin (x; u) = (0; 0) will be denoted as the terminal regionin the rest of the paper and subscript ′t′ will be used in therelated equations.To extend this formulation to a system outside of the ter-

minal region, we consider using a receding horizon controlscheme with the in7nite horizon objective function split intotwo parts as shown below.

J∞(k) =n∑

m=0

[xT(k + m|k)QIx(k + m|k)

+ uT(k + m|k)Ru(k + m|k)]

+∞∑

m=n+1


+ uT(k + m|k)Ru(k + m|k)]: (20)

The 7nite horizon cost speci7es the future control inputsthat will force the states to be in the desired terminal re-gion which contains the origin. The in7nite horizon cost ise9ective starting from the terminal region and bounded bythe terminal cost xT(k + n+ 1|k)Px(k + n+ 1|k). As soonas the system is in the terminal region the local linear statefeedback controller takes it to the origin. Fig. 1 illustratesthis strategy.The optimization problem reduces to

minu(k+m|k);m=0;1;:::; n

n∑m=0


+ uT(k + m|k)Ru(k + m|k)]+ xT(k + n+ 1|k)Px(k + n+ 1|k)

s.t.

V (x(k + m+ 1|k)− V (x(k + m|k)6 {−xT(k + m|k)QIx(k + m|k)

− uT(k + m|k)Ru(k + m|k)}‖Etx(k + m|k) + et‖6 1

m¿ n+ 1: (21)

A similar objective function was implemented by Lu andArkun (2000) for polytopic LPV systems. In their work,only the 7rst move u(k|k) is a free variable and the nonlin-ear system is assumed to vary inside a prescribed polytope.Here, the 7nite horizon objective function is an n stage costand the nonlinear system is represented using piecewise lin-ear models. The problem de7ned in Eq. (21) can be formu-lated as an LMI assuming u(k+m|k); m=0; 1; : : : ; n are free


x1

x 2

OP1

OP2

(x,u)=(0,0)

u),( |kkx )( |kk

Fig. 2. Illustration of multi-model predictive control algorithm.

variables and P= �Q−1.

min � (22)

1 ∗ ∗ ∗ ∗ : : : ∗Q0:5

I x(k|k) �I 0 0 0 0 0

R0:5u(k|k) 0 �I 0 0 0 0

Q0:5I x(k + 1|k) 0 0 �I 0 0 0

R0:5u(k + 1|k) 0 0 0 �I 0 0

.... . .

x(k + n+ 1|k) 0 0 0 0 0 Q

¿ 0; (23)

Q (AtQ + BtYt)T QETt QQ1=2

I YTt R

1=2

(AtQ + BtYt) Q + �btbTt bteTt � 0 0

EtQ �etbTt −�(I − eteTt ) 0 0

Q1=2I Q 0 0 �I 0

R1=2Yt 0 0 0 �I

¿ 0:

(24)

In the implementation of this formulation to multiple re-gions, we assume that we know a priori the sequence ofregions that the states will go through starting from the cur-rent region the system is in, to the terminal region and thatthe piecewise linear models that de7ne the system in eachregion are also known. We also assume that we know thenumber of moves that the system has to take to go fromone region to an adjacent operating region. As a result, wecan express x(k + 1|k); x(k + 2|k); : : : ; x(k + n + 1|k) interms of x(k|k); u(k|k); u(k + 1|k); : : : ; u(k + n|k). Illustra-tion of the control algorithm for multiple regions in the caseof a transition from operating point 1 to operating point2 in two-dimensional state space is shown in Fig. 2. Insummary, the control strategy is equal to the optimization

problem in Eq. (22) subject to constraints (23) and, (24) forx(k|k) �∈ the terminal region. If x(k|k)∈ the terminal regionthen the minimization problem in Eq. (10) is solved subjectto constraints (11) and, (12). Also, additional constraintsregarding the predicted states satisfying region de7nitionsalong the predicted transition trajectory is also included inthe formulation.The multi-model MPC algorithm in Eq. (22) does not in-

corporate input constraints. In order to include input con-straints in the formulation we need to split the input con-straints into two parts. Consider bounds on input at time k:

|uj(k)|6 uj;max k¿ 0 and j = 1; 2; : : : ; nu:

Inputs can be split into sequences: {u(k|k), u(k + 1|k),u(k + 2|k); : : : ; u(k + n|k);Ut} where u(k|k), u(k + 1|k),u(k + 2|k); : : : ; u(k + n|k) are free variables and Ut are fu-ture control moves in the terminal region given by the statefeedback law.

Ut : u(k + m|k) = Ktx(k + m|k); m¿ n+ 1;

Kt = YtQ−1:

Since the u(k|k), u(k + 1|k), u(k + 2|k); : : : ; u(k + n|k) arefree decision variables, constraints on these variables can beimposed directly:

|uj(k + m|k)|6 uj;max;

j = 1; 2; : : : ; nu and m= 0; 1; : : : n: (25)

For the remaining future manipulated variables, the exis-tence of a symmetric matrix X such that[X Yt

YTt Q

]¿ 0 with Xjj6 u2j;max; j = 1; 2; : : : ; nu (26)

guarantees that |uj(k +m|k)|6 uj;max; j=1; 2; : : : ; nu; m=n+ 1; n+ 2; : : : ;∞ (see Kothare et al., 1996).

4. Implementation

4.1. CSTR

We illustrate the application of this algorithm on alow-order continuous stirred tank reactor (CSTR) modelwith an exothermic 7rst-order irreversible reaction ( 'Ozkan,Kothare, & Georgakis, 2000). The example serves to fa-miliarize the reader with the multi-model MPC algorithmdiscussed in Section 3 before demonstrating its applicabil-ity on a high-dimensional complex polymerization reactor.The nonlinear model of a CSTR (Sistu & Bequette, 1991)is given as

dx1d�

=−�x1�(x2) + q(x1f − x1);


Table 1CSTR model parameters

� � � � x1f x2f q

8.0 0.072 0.3 20.0 1.0 0.0 1.0

Fig. 3. State and input variables during the transitions from OP3 to OP2and from OP2 to OP1.

dx2d�

= ��x1�(x2)− (q+ �)x2 + �u+ qx2f;

y = x2;

�(x2) = exp(

x21 + x2=�

);

where x1 is the concentration, x2 is the reactor temperature,u is the cooling jacket temperature. The model parametersare given in Table 1.For the nominal parameters, the reactor exhibits three

output multiplicities that are (xs1; u) = ([0:856; 0:886]; 0),(xs2; u) = ([0:5528; 2:7517]; 0), and (xs3; u) = ([0:2353;4:7050]; 0). Each point is assigned as an operating point.The state space of the system is partitioned into 7ve regionsalong the steady-state curve such that each region containsone of the steady-state points and overlaps with the adja-cent region. The other two steady-state points are selectedas ([0:4140; 3:5];−0:4612) and ([0:6927; 2:0]; 0:4727). Thepartitions are 7rst assumed as slabs and then outer ellipsoidalapproximations are computed. At each steady-state point,local linear models are obtained by Jacobian linearization.As an example, we 7rst consider transitions between

neighboring operating points. Figs. 3 and 4 present theperformance of multi-model MPC. Starting from operatingpoint 3 (OP3) the system 7rst goes through a transitionto operating point 2 (OP2) and then to operating point 1(OP1). In each region, it is 7rst assumed that the systemneeds 15 moves (n = 30 in Eq. (21) since we have tworegions) to reach the terminal region. In the 7rst part of thetransition (OP3 → OP2), in 16 moves the system is moved

Fig. 4. Phase plane characteristics during the transitions from OP3 toOP2 and from OP2 to OP1.

to the terminal region and the local linear state feedbackcontroller takes it to the origin which is OP2 in this case.The second part of transition (OP2 → OP1) takes 26 movesto reach the terminal region. When the system is in theoverlapping region a criterion, which basically comparesthe distance between operating points and the current statevalues, is used to decide which linear model is in e9ectand the appropriate control algorithm to be used. In thisexample, input constraints of

|uj(k)|6 2; k¿ 0 and j = 1; 2; : : : ; nu

are considered.

4.2. MMVA copolymerization reactor

The control of polymerization reactors is a complex taskdue to characteristics that are speci7c to a polymerizationreactor. EliUcabe and Meira (1988) categorized these charac-teristics in the following way:

1. Polymerization reactors have complicated process dy-namics. The dynamic models which are based on massand energy balances are highly nonlinear so linear controltheory cannot be applied directly. In addition, the deter-mination or estimation of model parameters is generallydiLcult.

2. High exothermicities and viscosities are involved. Poly-merization reactions can be highly exothermic and resultin a reactor thermal runway. At high conversions, thereactor mixture may become viscous which introducesmixing problems.

3. The control objectives are diLcult to specify. In poly-merization reactors, product quality is related to molec-ular and/or macroscopic architecture of the product andis determined at the synthesis stage. Thus, it cannot bemodi7ed at the post-polymerization stage. In addition,dynamic models of polymerization reactors do not pro-vide predictions of the end use properties.


4. There are problems with the measurement of the poly-mer structure. Online measurement of polymer structuralproperties such as composition andmolecular weight maynot be easily available.

In the literature, there exist numerous studies related to thecontrol of polymerization reactors. Congalidis and Richards(1998) compiled an extensive list of these studies and clas-si7ed them as follows:

• Temperature control (Chylla & Haase, 1993; Defaye,Reigner, Chabanon, Caralp, & Vidal, 1993; Ni, Debelak,& Hunkeler, 1997).

• Optimization of initiator addition, monomer addition,and/or reactor temperature to achieve desired polymerproperties in minimum time, maximize productivity inbatch or a semibatch reactor, or control the polymermolecular weight distribution (Arzamendi & Asua, 1989;Chen & Jeng, 1978; Chen & Huang, 1981; Louie &Soong, 1985a, b; Maschio, Bello, & Scali, 1992, 1994;Ponnuswamy, Shah, & Kiparissides, 1986; Scali, Carib,Bello, & Maschio, 1995; Secchi, Lima, & Pinto, 1990;Soroush & Kravaris, 1993; Thomas & Kiparissides,1984a, b; Wu, Denton, & Laurence, 1982).

• Online estimators of polymer properties by using calori-metric techniques, a kinetic model in the form of anextended Kalman 7lter, or neural network techniques(Dimitratos, Georgakis, El-Aasser, & Klein, 1991;EliUcabe, 'Ozdeger, & Georgakis, 1995; Ellis, Taylor, &Jensen, 1994; Kim & Choi, 1991; McAuley &MacGregor, 1991; Kozub & MacGregor, 1992; Moritz,1989; Mutha, Cluett, & Penlidis, 1997; Scali et al., 1995;Semino, Manning, & Brambilla, 1995; Zhang, Morris,Martin, & Kiparissides, 1997).

• Evaluation of advanced feedback controllers includingadaptive controllers (Defaye et al., 1993; Houston &Schork, 1987; Kiparissides & Shah, 1983; Wang, Pla, &Corriou, 1995), model predictive controllers (Dittmar,Ogonowski, & Damert, 1991; Gobin, Zullo, & Clavet,1994; Inglis, Cluett, & Penlidis, 1991; Ogunnaike, 1994;Ohshima et al., 1994; Peterson, Hernandez, Arkun, &Schork, 1992), nonlinear controllers (Alvarez, Suarez, &Sanchez, 1990; Gentric, Pla, & Corriou, 1997; Kravaris &Soroush, 1990; McAuley & MacGregor, 1993; Soroush& Kravaris, 1992, 1994).

• Evaluation of statistical techniques based on multiwayprincipal component analysis for polymer reactor moni-toring and control (Nomikos & MacGregor, 1994; Neogi& Schlags, 1997; Yabuki & MacGregor, 1997).

• Investigation of control strategies for reactor startupand product grade transitions (Debling, Han, Kuijpers,BerBurg, & Ray, 1994; Farber & Laurence, 1986;McAuley & MacGregor, 1992).

In this paper, we use the model of the solution copolymer-ization reactor proposed by Congalidis, Richards, and Ray

Monomer (FA)

Monomer (FB)

Initiator (FI)

Transfer Agent (FC)

Solvent (FS)

Inhibitor (FZ)

Coolant

Coolant

PolymerSolventUnreacted feed

Fig. 5. MMVA Copolymerization reactor.

(1989). This model is based on a free radical mechanismwith 27 separate reactions. The nonlinear model has 12 states(Appendix B). The dynamic behavior of the process largelydepends on the monomer feed ratio. Monomers A and B arecontinuously added with initiator, solvent, and chain transferagent (Fig. 5). An inhibitor may also enter with the fresh feedas an impurity. Monomer A is methyl methacrylate (MMA),monomer B is vinyl acetate (VA), the solvent is benzene, theinitiator is azobisisobutyronitrile (AIBN), the chain trans-fer agent is acetaldehyde and inhibitor is m-dinitrobenzene(m-DNB). Congalidis et al. (1989) implemented a combinedfeedforward/feedback control strategy for set point changesand compensation of unmeasured reactor disturbances. Steptests were used to identify the transfer functions and thecontrol structure was selected by ranking various candidatestructures according to the condition number, relative gainarray and minimum singular value. The feedback controlstructure has four SISO PI controllers and an acceptable per-formance is obtained. Maner and Doyle (III) (1997) imple-mented linear MPC on this process using the same transferfunctions as in Congalidis et al. (1989). However, they werenot able to get a signi7cantly better performance than thatobtained from a multiloop PI control strategy. The reason forthis result is explained by the high condition number of thetransfer function matrix. In order to decrease the conditionnumber, Maner and Doyle (III) (1997) proposed to closethe dominant loop with a PI controller. Then, Autoregres-sive plus Volterra models were identi7ed for other loops.A multivariable nonlinear MPC scheme was implementedand performance improvement was observed compared tomultiloop PI control strategy and multivariable linear MPCstructure.Our main motivation in selecting this process is its com-

plexity, hence its similarity to an industrial problem. Asa result, the implementation of MPC using multiple mod-els on this process provides a relevant study in the area ofpolymerization reactor control. In the implementation of theproposed approach, a major question to be addressed wasthe partitioning of the state space of the system so that wecould 7nd local linear models that represented the nonlinearsystem in each partition. Due to the high dimensionality of


Table 2Operating conditions of MMVA solution copolymerization reactor

OP 1 OP 2

InputsMonomer A feed rate (kg/h) 18.0 22.5Monomer B feed rate (kg/h) 90.0 90.0Initiator feed rate (kg/h) 0.18 0.18Solvent feed rate (kg/h) 36.0 36.0Chain transfer agent feed rate (kg/h) 2.7 2.7Inhibitor feed rate (kg/h) 0.0 0.0Reactor jacket temperature (K) 336 336Reactor feed temperature (K) 353 353

OutputsPolymer production rate (kg/h) 23.35 24.9Mole fraction of A in dead polymer 0.56 0.64Weight average molecular weight (kg/kmol)(105) 0.34 0.39Reactor temperature (K) 353.06 353.3

Fig. 6. Open-loop Mw response for a step change in FA=FB; (0:2 → 0:25).

the problem, instead of partitioning the whole state spaceit was decided to get local linear models along a trajectorywhich is obtained by implementing an input pro7le in openloop. As a starting point a step input was introduced to thesystem. As mentioned before, the MMVA solution copoly-merization reactor is highly sensitive to the monomer feedratios. The 7rst operating point given in Congalidis et al.(1989) was obtained for monomer ratio of FA=FB = 0:2. Inthe step input, this ratio was increased to 0.25 keeping otherinput variables and FB constant. The resulting steady-statepoint was selected as the second operating point (Table 2).Along the dynamic trajectory between two steady-state

points, transient points were selected at 7xed time intervals,every 2:5 h in this case (Fig. 6). Jacobian linearization ofthe nonlinear models at these points resulted in a bank ofmodels. For simplicity, only the terminal region was ap-proximated as an ellipsoid. Based on a norm measure whichshows the distance of the current state from the 7rst oper-ating point and transient points, the sequence of the local

353.1

353.3

353.5

T (

K)

3.6

4

Mw

/104 (

kg/k

mol

)

0 10 20 30 40 50

0.56

0.6

0.64

YA

p

t (h)0 10 20 30 40 50

t (h)

23

25

27

Gpi

(kg

/h)

Fig. 7. Closed-loop output response for transition from OP1 to OP2 (twolinear models: dashed lines, open-loop response: solid lines).

models that represent the nonlinear system along the trajec-tory was determined.The manipulated vector was selected as

Wu=

(FA

FB

)+

F+B(

FC

FB

)+

T+j

;

where (+) stands for fractional deviation variable. The inputconstraints were considered to be

| Wu|6

1:0

1:0

1:0

0:1

:

In Fig. 6 one can see that there are 17 points includingthe two steady-state points at both ends of the open-loop dy-namic trajectory. In order to decrease the computation load,we preferred to work with three transient points. Initially,we assumed that two linear models obtained by lineariza-tion at steady-state points were available. Figs. 7 and 8 showthe closed-loop output and input responses of multi-modelMPC during transition from OP1 to OP2, respectively. Thetransition is activated when the origin of the system is madeequal to OP2.Temperature T and production rate Gpi respond faster

to control action compared to molecular weight Mpw andpolymer composition YAp. Multi-model MPC with only twomodels requires almost 15 h for these polymer properties toreach their steady-state values.Figs. 9 and 10 depict the e9ect of number of local models

on the performance of multi-model MPC. It is clear that asthe number of linear models used increases, the response of


FB (

kg/h

)

FA (

kg/h

)F

C (

kg/h

)

Tj (

K)

t (h)0 10 20 30 40 50

t (h)0 10 20 30 40 50

334

335

336

337

338

339

2

2.2

2.4

2.6

2.8

3

70

75

80

85

90

95

15

20

25

30

35

40

Fig. 8. Manipulated variable pro7les for transition from OP1 to OP2 (twolinear models: dashed lines, open-loop response: solid lines).

T (

K)

Mw

/10

4 (k

g/km

ol)

YA

p

t (h)

Gpi

(kg/

h)

0 10 20 300 10 20 30

0 10 20 30t (h)

0 10 20 3022

23

24

25

26

27

0.54

0.56

0.58

0.6

0.62

0.64

3.4

3.6

3.8

4

353

353.1

353.2

353.3

353.4

353.5

Fig. 9. Closed-loop output response for transition from OP1 to OP2 (fourlinear models: dash–dot lines, three linear models: dashed lines, two linearmodels: solid lines).

output variables to control action becomes faster. In partic-ular, in the case of molecular weight Mpw there is a signi7-cant decrease in transition period. This results in a reductionof o9-speci7cation product produced during the transition.In order to examine the economic e9ect of the de-

crease in transition period, we calculated the amount ofo9-speci7cation product produced. Upper and lower boundsthat are equal to ±2:0% of the new set point value foraverage molecular weight Mpw were speci7ed. Then, thepolymer production rate Gpi was integrated over time start-ing from the beginning of the transition until Mpw reachedthe speci7ed range. This calculation gave the amount ofo9-speci7cation product produced during the transition.Table 3 summarizes these results.As a next study, we introduced an unmeasured distur-

bance which is the presence of an inhibitor in the fresh feedduring transition (Congalidis et al., 1989). In open loop, in-hibitor decreases molecular weight production rate and re-

FA

(kg/

h)

FB

(kg/

h)

FC

(kg/

h)

t (h) t (h)

Tj (

K)

0 10 20 300 10 20 30

0 10 20 300 10 20 30

335

336

337

338

2

2.2

2.4

2.6

2.8

3

80

85

90

95

15

20

25

30

35

40

Fig. 10. Manipulated variable pro7les for transition from OP1 to OP2(four linear models: dash–dot lines, three linear models: dashed lines,two linear models: solid lines).

Table 3Amount of o9-speci7cation product (kg)

Open loop 302.4Two models 134.8Three models 110.1Four models 109.8

actor temperature. For an inhibitor disturbance of 4 parts per1000 (mole basis), the controller performance is shown inFigs. 11 and 12. The response of the controller is aggres-sive in rejecting the disturbance e9ect. During the periodwhen the disturbance is active (1.5–3:0 h) we observe largeincrease in monomer Dow rates, even reaching their maxi-mum values. This increase results in overshoots in molecu-lar weight and polymer composition pro7les. In the case ofthe chain transfer agent, its Dow rate gets saturated at somepoint. The transition period does not di9er from simulationresults for the case without disturbance for all output vari-ables except polymer composition. It takes more than 30 hfor polymer composition to reach its desired value. In thesesimulations the number of linear models used is four.The MMVA reactor example is a large problem in terms

of dimensionality. Because of this, some changes had to bemade in the implementation of the control algorithm. Di9er-ent from the CSTR example, linear models were obtainedalong a dynamic trajectory between two operating points.Except in the terminal region which is approximated as anellipsoid, no other regions were de7ned along the trajectory.As a result, the rules that de7ne the sequence of linear mod-els in the control formulation were based on a norm mea-sure between the current state and the nearest linearizationpoints. Due to the high dimensionality of the problem, thetotal number of variables that had to be calculated in a sin-gle iteration varied between 200 and 300. The number ofindependent variables depends primarily on the number offree input variables that will take the system to the terminal


Fig. 11. Closed-loop output response for an unmeasured disturbance ininhibitor during transition from OP1 to OP2.

FA

(kg

/h)

FB

(kg

/h)

FC

(kg

/h)

t (h)

Tj (K

)

0 10 20 30t (h)

0 10 20 30330

332

334

336

338

340

0

1

2

3

4

5

80

100

120

140

160

180

20

30

40

50

60

Fig. 12. Manipulated variable pro7les for an unmeasured disturbance ininhibitor for transition from OP1 to OP2.

region. To the best of our knowledge, we are not aware ofprior demonstration of LMI-based MPC algorithms for suchlarge size problems.

5. Conclusion

We have shown that an industrial solution copolymer-ization reactor can be successfully controlled by using amultiple model predictive control strategy. The e9ect of thenumber of linear models used to represent the nonlinearsystem on the performance of the controller was studied.It was observed that the transition period decreased asthe number of linear models increased which resulted ina reduction of o9-speci7cation product produced duringthe grade transition. The proposed control approach alsosuccessfully rejected disturbances introduced during thetransition.

Notation

A Arrhenius factorC concentration, kmol=m3

E activation energy, kJ/kmolF mass Dow rate, kg/hGpi polymer production rate, kg/hI identity matrixJ cost functionk kinetic rate constant, time instantK state feedback controller gainm summation indexMpw weight average molecular weight, kg/kmolP positive de7nite matrixq dimensionless feed DowrateQ positive de7nite matrixQI weighting matrixR weighting matrixRk reaction rate, kmol=m3

S surface area, m2

t time, hT temperature, Ku input vectorU overall heat transfer coeLcient, kJ=m2 s KU control moves in in7nite horizon cost

functionx1 dimensionless reactor concentrationx2 dimensionless reactor temperaturex state vectorX partitionX symmetric matrixV Lyapunov function, reactor volume, m3

y output vectorY intermediate variable in LMI linearizationYAp polymer composition

Greek letters

� dimensionless heat of reaction� dimensionless activation energy� dimensionless heat transfer coeLcient- initiator eLciency. residence time, h� molar concentration of monomer in polymer, in-

termediate variable in S-procedure� intermediate variable in S-procedure/ density, kg=m3

� dimensionless time� damkohler number (at nominal Dow rate) moment of molecular weight distribution

Superscripts/subscripts

A monomer AB monomer B


C chain transfer agentf feedI initiatori region ij cooling jacket, indexl summation indexn summation indexp prediction horizon, propogationr reactorS solvents steady-state valueT transposet terminal regionZ inhibitor˜ absolute variable

Acknowledgements

Financial support for this research from the industrialsponsors of the Chemical Process Modelling and ControlResearch Center at Lehigh University is gratefully acknowl-edged.

Appendix A. Linear matrix inequalities (LMI)

An LMI is a matrix inequality of the form

F(x) = F0 +l∑

i=1

xiFi ¿ 0; (A.1)

where x1; x2; : : : ; xl are the variables, Fi = FTi ∈Rn×n are

given, and F(x)¿ 0 means that F(x) is positive de7nite.Multiple LMIs Fi(x)¿ 0; : : : ; Fn(x)¿ 0 can be expressedas the single LMI

diag(F1; : : : ; Fn(x))¿ 0: (A.2)

Convex quadratic inequalities are converted to LMI formusing Schur complements. Let Q(x)=Q(x)T; R(x)=R(x)T,and S(x) depend aLnely on x. Then the LMI[Q(x) S(x)

S(x)T R(x)

]¿ 0 (A.3)

is equivalent to the matrix inequalities

R(x)¿ 0; Q(x)− S(x)R(x)−1S(x)T ¿ 0 (A.4)

or, equivalently

Q(x)¿ 0; R(x)− S(x)TQ(x)−1S(x)¿ 0: (A.5)

We often encounter problems in which the variables arematrices, e.g. the Lyapunov inequality ATP+PA¡ 0, whereA∈Rn×n is given and the entries of P=PT are the variables.In such cases, we shall not write out the LMI explicitly inthe form of F(x)¿ 0, but instead make clear which matricesare the variables.Conditional constraints can be combined into a sin-

gle, somewhat conservative matrix inequality using theS-procedure (Boyd et al., 1994). For example, suppose

F0(x)¿ 0 for all Fi(x)¿ 0; i = 1; : : : ; p; (A.6)

where F0; Fi are quadratic functions of x∈Rn. Then, theabove conditions can be combined using the S-procedure tothe existence of �1¿ 0, �2¿ 0; : : : ; �p¿ 0 such that

F0(x)−p∑

i=1

�iFi(x)¿ 0: (A.7)

The LMI-based problem of central importance is that ofminimizing a linear objective function subject to LMI con-straints:

minimize cTx (A.8)

s:t: F(x)¿ 0: (A.9)

Here, F is a symmetric matrix that depends aLnely on theoptimization variable x and c is a real vector of appropriatesize. This is a nonsmooth optimization problem. More infor-mation on this and other LMI-based optimization problemscan be found in Boyd et al. (1994).

Appendix B. MMVA model

In the article by Maner and Doyle (III) (1997), sometypographical errors that had appeared in the original articleby Congalidis et al. (1989) were corrected. The correctedODEs and the values of the parameters (Table 4) are givenin the following:

dCk

dt=

Ckf − Ck

.r− Rk k = a; b; i; s; t; z; (B.1)

dTr

dt=

Trf − Tr

.r

+(−YHpaa)kpaaCaCa· + (−YHpba)kpbaCaCb·

/rcr

+(−YHpab)kpabCbCb· + (−YHpbb)kpbbCbCb·

/rcr

− UrSr(Tr − Tj)Vr/rcr

; (B.2)

�a

dt=

�af − �a

.r+ Ra; (B.3)


Table 4MMVA solution copolymerization reactor model parameters

- 1Ai 4.5 (1/s) Ei 1:255× 105 (kJ/kmol)Acaa 4:209× 1014 (m3/kmol· s) Ecaa 2:69× 104 (”)Acbb 1:61× 1011 (”) Ecbb 4:00× 103 (”)Adaa 0.0 (”) Edaa 0.0 (”)Adbb 0.0 (”) Edbb 0.0 (”)Apaa 3:207× 106 (”) Epaa 2:42× 104 (”)Apab 1:233× 105 (”) Epab 2:42× 104 (”)Apba 2:103× 108 (”) Epba 1:80× 104 (”)Apbb 6:308× 106 (”) Epbb 1:80× 104 (”)Axaa 32.08 (”) Exaa 2:42× 104 (”)Axab 1.234 (”) Exab 2:42× 104 (”)Axas 86.6 (”) Exas 2:42× 104 (”)Axat 2085.0 (”) Exat 2:42× 104 (”)Axba 5:257× 104 (”) Exba 1:80× 104 (”)Axbb 1577 (”) Exbb 1:80× 104 (”)Axbs 1514 (”) Exbs 1:80× 104 (”)Axbt 4:163× 105 (”) Exbt 1:80× 104 (”)Aza 2.2 (”) Eza 0.0 (”)Azb 1:13× 105 (”) Ezb 0.0 (”)

−YHpaa 54:0× 103 (kJ/kmol) −YHpba 54:0× 103 (kJ/kmol)−YHpab 86:0× 103 (”) −YHpbb 86:0× 103 (”)/r 8:79× 102 (kg=m3) cr 2.01 (kJ=kg K)Ur 6:0× 10−2 (kJ=m2 · s K)

Reactor volume, Vr 1 (m3) Reactor heat transfer area, Sr 4.6 (m2)

�b

dt=

�bf − �b

.r+ Rb; (B.4)

d p0

dt=

p0f − p

0

.r+

12kcaa( a·

0 )2 + kcab a·

0 b·0

+12kcbb( b·

0 )2 + L1 a·

0 + L2 b·0 ; (B.5)

d p1

dt=

p1f − p

1

.r+ kcaa a·

0 a·1 + kcab( a·

0 b·1 + b·

0 a·1 )

+ kcbb b·0 b·

1 + L1 a·1 + L2 b·

1 ; (B.6)

d p2

dt=

p2f − p

2

.r+ kcaa{( a·

1 )2 + a·

0 a·2 }

+ kcab(2 a·1 b·

1 + b·2 a·

0 + a·2 b·

0 )

+ kcbb{( b·1 )

2 + b·0 b·

2 }+ L1 a·2 + L2 b·

2 : (B.7)

Outputs are:

Reactor temperature : Tr (B.8)

Weight average molecular weight : Mpw = p2

p1: (B.9)

Mole fraction of monomer A in dead polymer :

YAp =�a

(�a + �b): (B.10)

Production rate : Gpi = (RaMa + RbMa)Vr: (B.11)

The reaction rates Rk are given in Eqs. (4), (5), (6),(7), (8), (9) in Congalidis et al. (1989). Ca· Cb·, a·

0 , a·1 ,

a·2 , b·

0 , b·1 , b·

2 are de7ned in Eqs. (11), (12), (A.5),(A.6), (A.7), (A.8), (A.9), (B.1), respectively, in the samearticle.

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